Subgroups — C₃²×#3506 #11770555

Order 1152.

3420 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#559	C₂	1	4	yes	maximal, complemented
2	C₃²×#556	C₂	1	2	yes	maximal, complemented
3	C₃²×#555	C₂	1	2	yes	maximal, complemented
4	C₃²×#553	C₂	1	4	yes	maximal, complemented
5	C₃²×#550	C₂	1	2	yes	maximal, complemented
6	C₃²×#547	C₂	1	2	yes	maximal, complemented
7	C₃²×#538	C₂	1	4	yes	maximal, complemented
8	C₃²×#535	C₂	1	2	yes	maximal, complemented
9	C₃²×#533	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃²×#532	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃²×#525	C₂	1	4	yes	maximal, complemented
12	C₃²×#520	C₂	1	2	yes	maximal, complemented
13	C₂×C₃²×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃×#3506	C₃	1	4	yes	maximal, direct product factor
15	C₂×C₃²×#41	C₂²	1	2	yes	—
16	C₂×C₃²×#41	C₂²	1	4	yes	complemented
17	C₂²×C₃²×D₄	C₂²	1	1	yes	characteristic
18	C₂³×C₃²×C₄	C₂²	1	2	yes	—
19	C₃²×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
20	C₃²×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
21	C₃²×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
22	C₃²×#127	C₂²	1	4	yes	—
23	C₃²×#127	C₂²	1	1	yes	characteristic, complemented
24	C₃²×#126	C₂²	1	8	yes	complemented
25	C₃²×#125	C₂²	1	2	yes	complemented
26	C₃²×#124	C₂²	1	4	yes	—
27	C₃²×#124	C₂²	1	8	yes	complemented
28	C₃²×#124	C₂²	1	2	yes	complemented
29	C₃²×#123	C₂²	1	8	yes	complemented
30	C₃²×#123	C₂²	1	8	yes	complemented
31	C₃²×C₂²⋊Q₈	C₂²	1	2	yes	—
32	C₃²×C₂²⋊Q₈	C₂²	1	2	yes	—
33	C₃²×C₂²⋊Q₈	C₂²	1	4	yes	complemented
34	C₃²×#121	C₂²	1	4	yes	—
35	C₃²×#121	C₂²	1	4	yes	—
36	C₃²×#121	C₂²	1	2	yes	—
37	C₃²×#121	C₂²	1	2	yes	—
38	C₃²×#121	C₂²	1	8	yes	complemented
39	C₃²×#121	C₂²	1	4	yes	complemented
40	C₃²×#120	C₂²	1	8	yes	—
41	C₃²×C₄×Q₈	C₂²	1	4	yes	complemented
42	C₃²×C₄×Q₈	C₂²	1	2	yes	complemented
43	C₃²×C₄×Q₈	C₂²	1	2	yes	complemented
44	C₃²×C₄×D₄	C₂²	1	2	yes	—
45	C₃²×C₄×D₄	C₂²	1	2	yes	—
46	C₃²×C₄×D₄	C₂²	1	2	yes	—
47	C₃²×C₄×D₄	C₂²	1	2	yes	—
48	C₃²×C₄×D₄	C₂²	1	8	yes	—
49	C₃²×C₄×D₄	C₂²	1	2	yes	complemented
50	C₃²×C₄×D₄	C₂²	1	8	yes	complemented
51	C₃²×C₄×D₄	C₂²	1	2	yes	complemented
52	C₃²×C₄×D₄	C₂²	1	4	yes	complemented
53	C₃²×#117	C₂²	1	2	yes	—
54	C₃²×#117	C₂²	1	4	yes	complemented
55	C₃²×#117	C₂²	1	4	yes	complemented
56	C₂×C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
57	C₂×C₃²×C₄⋊C₄	C₂²	1	4	yes	complemented
58	C₂×C₃²×C₂²⋊C₄	C₂²	1	2	yes	—
59	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
60	C₃×#559	C₂×C₃	1	16	yes	complemented
61	C₃×#556	C₂×C₃	1	8	yes	complemented
62	C₃×#555	C₂×C₃	1	8	yes	complemented
63	C₃×#553	C₂×C₃	1	16	yes	complemented
64	C₃×#550	C₂×C₃	1	8	yes	complemented
65	C₃×#547	C₂×C₃	1	8	yes	complemented
66	C₃×#538	C₂×C₃	1	16	yes	complemented
67	C₃×#535	C₂×C₃	1	8	yes	complemented
68	C₃×#533	C₂×C₃	1	4	yes	complemented
69	C₃×#532	C₂×C₃	1	4	yes	complemented
70	C₃×#525	C₂×C₃	1	16	yes	complemented
71	C₃×#520	C₂×C₃	1	8	yes	complemented
72	C₂×C₃×C₄×D₄	C₂×C₃	1	4	yes	complemented
73	C₂⁴×C₃²	C₂³	1	2	yes	—
74	C₃²×#41	—	2	8	no	—
75	C₃²×#41	—	4	8	no	—
76	C₂×C₃²×Q₈	C₂³	1	2	yes	—
77	C₂×C₃²×Q₈	C₂³	1	2	yes	—
78	C₂×C₃²×Q₈	C₂³	1	2	yes	—
79	C₂×C₃²×D₄	—	2	4	no	—
80	C₂×C₃²×D₄	C₂³	1	2	yes	—
81	C₂×C₃²×D₄	C₂³	1	2	yes	—
82	C₂×C₃²×D₄	C₂³	1	4	yes	—
83	C₂×C₃²×D₄	C₂³	1	2	yes	—
84	C₂×C₃²×D₄	C₂³	1	2	yes	—
85	C₂×C₃²×D₄	C₂³	1	8	yes	—
86	C₂×C₃²×D₄	C₂³	1	8	yes	—
87	C₂×C₃²×D₄	C₂³	1	2	yes	—
88	C₂²×C₃²×C₄	—	4	4	no	—
89	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
90	C₂²×C₃²×C₄	C₂³	1	4	yes	—
91	C₂²×C₃²×C₄	C₂³	1	4	yes	—
92	C₂²×C₃²×C₄	C₂³	1	2	yes	—
93	C₂²×C₃²×C₄	C₂³	1	4	yes	—
94	C₂²×C₃²×C₄	C₂³	1	8	yes	—
95	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
96	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
97	C₃²×C₄⋊C₄	C₂³	1	2	yes	—
98	C₃²×C₄⋊C₄	C₂³	1	2	yes	—
99	C₃²×C₄⋊C₄	C₂³	1	2	yes	—
100	C₃²×C₄⋊C₄	C₂³	1	8	yes	—
101	C₃²×C₄⋊C₄	C₂³	1	2	yes	—
102	C₃²×C₄⋊C₄	C₂³	1	2	yes	—
103	C₃²×C₄⋊C₄	C₂³	1	2	yes	—
104	C₃²×C₄⋊C₄	C₂³	1	4	yes	—
105	C₃²×C₄⋊C₄	C₂³	1	2	yes	—
106	C₃²×C₄⋊C₄	C₂³	1	4	yes	—
107	C₃²×C₄⋊C₄	C₂³	1	2	yes	—
108	C₃²×C₂²⋊C₄	C₂³	1	4	yes	—
109	C₃²×C₂²⋊C₄	C₂³	1	4	yes	—
110	C₃²×C₂²⋊C₄	C₂³	1	4	yes	—
111	C₃²×C₂²⋊C₄	C₂³	1	8	yes	—
112	C₃²×C₂²⋊C₄	C₂³	1	8	yes	—
113	C₃²×C₂²⋊C₄	C₂³	1	4	yes	—
114	C₃²×C₂²⋊C₄	C₂³	1	8	yes	—
115	C₃²×C₂²⋊C₄	C₂³	1	4	yes	—
116	C₃²×C₄²	C₂³	1	1	yes	characteristic
117	C₃²×C₄²	C₂³	1	1	yes	characteristic
118	C₃²×C₄²	C₂³	1	1	yes	characteristic
119	C₃²×C₄²	C₂³	1	1	yes	characteristic
120	C₃²×C₄²	C₂³	1	4	yes	—
121	C₃²×C₄²	C₂³	1	8	yes	—
122	#3506	C₃²	1	1	yes	characteristic, direct product factor, Sylow
123	C₂×C₃×#41	C₂²×C₃	1	8	yes	—
124	C₂×C₃×#41	C₂²×C₃	1	16	yes	complemented
125	C₂²×C₃×D₄	C₂²×C₃	1	4	yes	—
126	C₂³×C₃×C₄	C₂²×C₃	1	8	yes	—
127	C₃×C₄⋊Q₈	C₂²×C₃	1	4	yes	complemented
128	C₃×C₄⋊Q₈	C₂²×C₃	1	4	yes	complemented
129	C₃×C₄⋊Q₈	C₂²×C₃	1	4	yes	complemented
130	C₃×#127	C₂²×C₃	1	16	yes	—
131	C₃×#127	C₂²×C₃	1	4	yes	complemented
132	C₃×#126	C₂²×C₃	1	32	yes	complemented
133	C₃×#125	C₂²×C₃	1	8	yes	complemented
134	C₃×#124	C₂²×C₃	1	16	yes	—
135	C₃×#124	C₂²×C₃	1	8	yes	complemented
136	C₃×#124	C₂²×C₃	1	32	yes	complemented
137	C₃×#123	C₂²×C₃	1	32	yes	complemented
138	C₃×#123	C₂²×C₃	1	32	yes	complemented
139	C₃×C₂²⋊Q₈	C₂²×C₃	1	8	yes	—
140	C₃×C₂²⋊Q₈	C₂²×C₃	1	8	yes	—
141	C₃×C₂²⋊Q₈	C₂²×C₃	1	16	yes	complemented
142	C₃×#121	C₂²×C₃	1	8	yes	—
143	C₃×#121	C₂²×C₃	1	8	yes	—
144	C₃×#121	C₂²×C₃	1	16	yes	—
145	C₃×#121	C₂²×C₃	1	16	yes	—
146	C₃×#121	C₂²×C₃	1	16	yes	complemented
147	C₃×#121	C₂²×C₃	1	32	yes	complemented
148	C₃×#120	C₂²×C₃	1	32	yes	—
149	C₃×C₄×Q₈	C₂²×C₃	1	8	yes	complemented
150	C₃×C₄×Q₈	C₂²×C₃	1	16	yes	complemented
151	C₃×C₄×Q₈	C₂²×C₃	1	8	yes	complemented
152	C₃×C₄×D₄	C₂²×C₃	1	8	yes	—
153	C₃×C₄×D₄	C₂²×C₃	1	8	yes	—
154	C₃×C₄×D₄	C₂²×C₃	1	8	yes	—
155	C₃×C₄×D₄	C₂²×C₃	1	8	yes	—
156	C₃×C₄×D₄	C₂²×C₃	1	32	yes	—
157	C₃×C₄×D₄	C₂²×C₃	1	16	yes	complemented
158	C₃×C₄×D₄	C₂²×C₃	1	32	yes	complemented
159	C₃×C₄×D₄	C₂²×C₃	1	8	yes	complemented
160	C₃×C₄×D₄	C₂²×C₃	1	8	yes	complemented
161	C₃×#117	C₂²×C₃	1	8	yes	—
162	C₃×#117	C₂²×C₃	1	16	yes	complemented
163	C₃×#117	C₂²×C₃	1	16	yes	complemented
164	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
165	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	16	yes	complemented
166	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
167	C₂×C₃×C₄²	C₂²×C₃	1	4	yes	—
168	C₂³×C₃²	—	2	4	no	—
169	C₂³×C₃²	—	4	2	no	—
170	C₂³×C₃²	—	4	2	no	—
171	C₂³×C₃²	C₂⁴	1	4	yes	—
172	C₂³×C₃²	C₂⁴	1	4	yes	—
173	C₂³×C₃²	C₂⁴	1	1	yes	characteristic
174	C₃²×Q₈	—	2	4	no	—
175	C₃²×Q₈	—	2	4	no	—
176	C₃²×Q₈	—	4	2	no	—
177	C₃²×D₄	—	2	8	no	—
178	C₃²×D₄	—	2	4	no	—
179	C₃²×D₄	—	2	4	no	—
180	C₃²×D₄	—	2	4	no	—
181	C₃²×D₄	—	2	4	no	—
182	C₃²×D₄	—	4	2	no	—
183	C₃²×D₄	—	4	8	no	—
184	C₃²×D₄	—	4	8	no	—
185	C₂×C₃²×C₄	—	2	8	no	—
186	C₂×C₃²×C₄	—	2	2	no	—
187	C₂×C₃²×C₄	—	2	8	no	—
188	C₂×C₃²×C₄	—	4	2	no	—
189	C₂×C₃²×C₄	—	4	4	no	—
190	C₂×C₃²×C₄	—	4	1	no	—
191	C₂×C₃²×C₄	—	4	8	no	—
192	C₂×C₃²×C₄	#41	1	4	yes	complemented
193	C₂×C₃²×C₄	C₂⁴	1	4	yes	—
194	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
195	C₂×C₃²×C₄	C₂⁴	1	4	yes	—
196	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
197	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
198	C₂×C₃²×C₄	C₂⁴	1	4	yes	—
199	C₂×C₃²×C₄	C₂⁴	1	4	yes	—
200	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
201	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
202	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
203	#559	C₂×C₃²	1	4	yes	complemented
204	#556	C₂×C₃²	1	2	yes	complemented
205	#555	C₂×C₃²	1	2	yes	complemented
206	#553	C₂×C₃²	1	4	yes	complemented
207	#550	C₂×C₃²	1	2	yes	complemented
208	#547	C₂×C₃²	1	2	yes	complemented
209	#538	C₂×C₃²	1	4	yes	complemented
210	#535	C₂×C₃²	1	2	yes	complemented
211	#533	C₂×C₃²	1	1	yes	characteristic, complemented
212	#532	C₂×C₃²	1	1	yes	characteristic, complemented
213	#525	C₂×C₃²	1	4	yes	complemented
214	#520	C₂×C₃²	1	2	yes	complemented
215	C₂×C₄×D₄	C₂×C₃²	1	1	yes	characteristic, complemented
216	C₂⁴×C₃	C₂³×C₃	1	8	yes	—
217	C₃×#41	—	2	32	no	—
218	C₃×#41	—	4	32	no	—
219	C₂×C₃×Q₈	C₂³×C₃	1	8	yes	—
220	C₂×C₃×Q₈	C₂³×C₃	1	8	yes	—
221	C₂×C₃×Q₈	C₂³×C₃	1	8	yes	—
222	C₂×C₃×D₄	—	2	16	no	—
223	C₂×C₃×D₄	C₂³×C₃	1	16	yes	—
224	C₂×C₃×D₄	C₂³×C₃	1	8	yes	—
225	C₂×C₃×D₄	C₂³×C₃	1	8	yes	—
226	C₂×C₃×D₄	C₂³×C₃	1	8	yes	—
227	C₂×C₃×D₄	C₂³×C₃	1	8	yes	—
228	C₂×C₃×D₄	C₂³×C₃	1	32	yes	—
229	C₂×C₃×D₄	C₂³×C₃	1	8	yes	—
230	C₂×C₃×D₄	C₂³×C₃	1	32	yes	—
231	C₂²×C₃×C₄	—	4	16	no	—
232	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
233	C₂²×C₃×C₄	C₂³×C₃	1	16	yes	—
234	C₂²×C₃×C₄	C₂³×C₃	1	16	yes	—
235	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
236	C₂²×C₃×C₄	C₂³×C₃	1	8	yes	—
237	C₂²×C₃×C₄	C₂³×C₃	1	32	yes	—
238	C₂²×C₃×C₄	C₂³×C₃	1	16	yes	—
239	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
240	C₃×C₄⋊C₄	C₂³×C₃	1	8	yes	—
241	C₃×C₄⋊C₄	C₂³×C₃	1	8	yes	—
242	C₃×C₄⋊C₄	C₂³×C₃	1	8	yes	—
243	C₃×C₄⋊C₄	C₂³×C₃	1	32	yes	—
244	C₃×C₄⋊C₄	C₂³×C₃	1	8	yes	—
245	C₃×C₄⋊C₄	C₂³×C₃	1	8	yes	—
246	C₃×C₄⋊C₄	C₂³×C₃	1	8	yes	—
247	C₃×C₄⋊C₄	C₂³×C₃	1	16	yes	—
248	C₃×C₄⋊C₄	C₂³×C₃	1	8	yes	—
249	C₃×C₄⋊C₄	C₂³×C₃	1	16	yes	—
250	C₃×C₄⋊C₄	C₂³×C₃	1	8	yes	—
251	C₃×C₂²⋊C₄	C₂³×C₃	1	16	yes	—
252	C₃×C₂²⋊C₄	C₂³×C₃	1	16	yes	—
253	C₃×C₂²⋊C₄	C₂³×C₃	1	16	yes	—
254	C₃×C₂²⋊C₄	C₂³×C₃	1	32	yes	—
255	C₃×C₂²⋊C₄	C₂³×C₃	1	32	yes	—
256	C₃×C₂²⋊C₄	C₂³×C₃	1	16	yes	—
257	C₃×C₂²⋊C₄	C₂³×C₃	1	32	yes	—
258	C₃×C₂²⋊C₄	C₂³×C₃	1	16	yes	—
259	C₃×C₄²	C₂³×C₃	1	4	yes	—
260	C₃×C₄²	C₂³×C₃	1	4	yes	—
261	C₃×C₄²	C₂³×C₃	1	4	yes	—
262	C₃×C₄²	C₂³×C₃	1	4	yes	—
263	C₃×C₄²	C₂³×C₃	1	16	yes	—
264	C₃×C₄²	C₂³×C₃	1	32	yes	—
265	C₂²×C₃²	—	2	1	no	—
266	C₂²×C₃²	—	2	1	no	—
267	C₂²×C₃²	—	2	4	no	—
268	C₂²×C₃²	—	2	4	no	—
269	C₂²×C₃²	—	2	4	no	—
270	C₂²×C₃²	—	2	4	no	—
271	C₂²×C₃²	—	2	4	no	—
272	C₂²×C₃²	—	2	4	no	—
273	C₂²×C₃²	—	4	8	no	—
274	C₂²×C₃²	C₂×#41	1	2	yes	—
275	C₂²×C₃²	C₂⁵	1	1	yes	characteristic
276	C₃²×C₄	—	2	4	no	—
277	C₃²×C₄	—	2	1	no	—
278	C₃²×C₄	—	2	4	no	—
279	C₃²×C₄	—	2	4	no	—
280	C₃²×C₄	—	2	1	no	—
281	C₃²×C₄	—	2	1	no	—
282	C₃²×C₄	—	2	4	no	—
283	C₃²×C₄	C₂×#41	1	2	yes	—
284	C₃²×C₄	C₂×#41	1	2	yes	—
285	C₃²×C₄	H(5,𝔽₂)	1	2	yes	—
286	C₂×#41	C₂²×C₃²	1	2	yes	—
287	C₂×#41	C₂²×C₃²	1	4	yes	complemented
288	C₂²×D₄	C₂²×C₃²	1	1	yes	characteristic
289	C₂³×C₄	C₂²×C₃²	1	2	yes	—
290	C₄⋊Q₈	C₂²×C₃²	1	1	yes	characteristic, complemented
291	C₄⋊Q₈	C₂²×C₃²	1	1	yes	characteristic, complemented
292	C₄⋊Q₈	C₂²×C₃²	1	1	yes	characteristic, complemented
293	#127	C₂²×C₃²	1	4	yes	—
294	#127	C₂²×C₃²	1	1	yes	characteristic, complemented
295	#126	C₂²×C₃²	1	8	yes	complemented
296	#125	C₂²×C₃²	1	2	yes	complemented
297	#124	C₂²×C₃²	1	4	yes	—
298	#124	C₂²×C₃²	1	2	yes	complemented
299	#124	C₂²×C₃²	1	8	yes	complemented
300	#123	C₂²×C₃²	1	8	yes	complemented
301	#123	C₂²×C₃²	1	8	yes	complemented
302	C₂²⋊Q₈	C₂²×C₃²	1	2	yes	—
303	C₂²⋊Q₈	C₂²×C₃²	1	2	yes	—
304	C₂²⋊Q₈	C₂²×C₃²	1	4	yes	complemented
305	#121	C₂²×C₃²	1	2	yes	—
306	#121	C₂²×C₃²	1	2	yes	—
307	#121	C₂²×C₃²	1	4	yes	—
308	#121	C₂²×C₃²	1	4	yes	—
309	#121	C₂²×C₃²	1	4	yes	complemented
310	#121	C₂²×C₃²	1	8	yes	complemented
311	#120	C₂²×C₃²	1	8	yes	—
312	C₄×Q₈	C₂²×C₃²	1	4	yes	complemented
313	C₄×Q₈	C₂²×C₃²	1	2	yes	complemented
314	C₄×Q₈	C₂²×C₃²	1	2	yes	complemented
315	C₄×D₄	C₂²×C₃²	1	2	yes	—
316	C₄×D₄	C₂²×C₃²	1	2	yes	—
317	C₄×D₄	C₂²×C₃²	1	2	yes	—
318	C₄×D₄	C₂²×C₃²	1	2	yes	—
319	C₄×D₄	C₂²×C₃²	1	8	yes	—
320	C₄×D₄	C₂²×C₃²	1	4	yes	complemented
321	C₄×D₄	C₂²×C₃²	1	8	yes	complemented
322	C₄×D₄	C₂²×C₃²	1	2	yes	complemented
323	C₄×D₄	C₂²×C₃²	1	2	yes	complemented
324	#117	C₂²×C₃²	1	2	yes	—
325	#117	C₂²×C₃²	1	4	yes	complemented
326	#117	C₂²×C₃²	1	4	yes	complemented
327	C₂×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
328	C₂×C₄⋊C₄	C₂²×C₃²	1	4	yes	complemented
329	C₂×C₂²⋊C₄	C₂²×C₃²	1	2	yes	—
330	C₂×C₄²	C₂²×C₃²	1	1	yes	characteristic
331	C₂³×C₃	—	2	16	no	—
332	C₂³×C₃	—	4	8	no	—
333	C₂³×C₃	—	4	8	no	—
334	C₂³×C₃	C₂⁴×C₃	1	16	yes	—
335	C₂³×C₃	C₂⁴×C₃	1	16	yes	—
336	C₂³×C₃	C₂⁴×C₃	1	4	yes	—
337	C₃×Q₈	—	2	16	no	—
338	C₃×Q₈	—	2	16	no	—
339	C₃×Q₈	—	4	8	no	—
340	C₃×D₄	—	2	32	no	—
341	C₃×D₄	—	2	16	no	—
342	C₃×D₄	—	2	16	no	—
343	C₃×D₄	—	2	16	no	—
344	C₃×D₄	—	2	16	no	—
345	C₃×D₄	—	4	8	no	—
346	C₃×D₄	—	4	32	no	—
347	C₃×D₄	—	4	32	no	—
348	C₂×C₃×C₄	—	2	32	no	—
349	C₂×C₃×C₄	—	2	8	no	—
350	C₂×C₃×C₄	—	2	32	no	—
351	C₂×C₃×C₄	—	4	8	no	—
352	C₂×C₃×C₄	—	4	16	no	—
353	C₂×C₃×C₄	—	4	4	no	—
354	C₂×C₃×C₄	—	4	32	no	—
355	C₂×C₃×C₄	C₃×#41	1	16	yes	complemented
356	C₂×C₃×C₄	C₂⁴×C₃	1	16	yes	—
357	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
358	C₂×C₃×C₄	C₂⁴×C₃	1	16	yes	—
359	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
360	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
361	C₂×C₃×C₄	C₂⁴×C₃	1	16	yes	—
362	C₂×C₃×C₄	C₂⁴×C₃	1	16	yes	—
363	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
364	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
365	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
366	C₂×C₃²	—	2	2	no	—
367	C₂×C₃²	—	4	4	no	—
368	C₂×C₃²	—	4	4	no	—
369	C₂×C₃²	C₂²×#41	1	1	yes	characteristic
370	C₂×C₃²	C₂×H(5,𝔽₂)	1	1	yes	characteristic
371	C₂×C₃²	#585	1	1	yes	characteristic
372	C₂⁴	C₂³×C₃²	1	2	yes	—
373	#41	—	2	8	no	—
374	#41	—	4	8	no	—
375	C₂×Q₈	C₂³×C₃²	1	2	yes	—
376	C₂×Q₈	C₂³×C₃²	1	2	yes	—
377	C₂×Q₈	C₂³×C₃²	1	2	yes	—
378	C₂×D₄	—	2	4	no	—
379	C₂×D₄	C₂³×C₃²	1	4	yes	—
380	C₂×D₄	C₂³×C₃²	1	2	yes	—
381	C₂×D₄	C₂³×C₃²	1	2	yes	—
382	C₂×D₄	C₂³×C₃²	1	2	yes	—
383	C₂×D₄	C₂³×C₃²	1	2	yes	—
384	C₂×D₄	C₂³×C₃²	1	8	yes	—
385	C₂×D₄	C₂³×C₃²	1	2	yes	—
386	C₂×D₄	C₂³×C₃²	1	8	yes	—
387	C₂²×C₄	—	4	4	no	—
388	C₂²×C₄	C₂³×C₃²	1	1	yes	characteristic
389	C₂²×C₄	C₂³×C₃²	1	4	yes	—
390	C₂²×C₄	C₂³×C₃²	1	4	yes	—
391	C₂²×C₄	C₂³×C₃²	1	1	yes	characteristic
392	C₂²×C₄	C₂³×C₃²	1	2	yes	—
393	C₂²×C₄	C₂³×C₃²	1	8	yes	—
394	C₂²×C₄	C₂³×C₃²	1	4	yes	—
395	C₂²×C₄	C₂³×C₃²	1	1	yes	characteristic
396	C₄⋊C₄	C₂³×C₃²	1	2	yes	—
397	C₄⋊C₄	C₂³×C₃²	1	2	yes	—
398	C₄⋊C₄	C₂³×C₃²	1	2	yes	—
399	C₄⋊C₄	C₂³×C₃²	1	8	yes	—
400	C₄⋊C₄	C₂³×C₃²	1	2	yes	—
401	C₄⋊C₄	C₂³×C₃²	1	2	yes	—
402	C₄⋊C₄	C₂³×C₃²	1	2	yes	—
403	C₄⋊C₄	C₂³×C₃²	1	4	yes	—
404	C₄⋊C₄	C₂³×C₃²	1	2	yes	—
405	C₄⋊C₄	C₂³×C₃²	1	4	yes	—
406	C₄⋊C₄	C₂³×C₃²	1	2	yes	—
407	C₂²⋊C₄	C₂³×C₃²	1	4	yes	—
408	C₂²⋊C₄	C₂³×C₃²	1	4	yes	—
409	C₂²⋊C₄	C₂³×C₃²	1	4	yes	—
410	C₂²⋊C₄	C₂³×C₃²	1	8	yes	—
411	C₂²⋊C₄	C₂³×C₃²	1	8	yes	—
412	C₂²⋊C₄	C₂³×C₃²	1	4	yes	—
413	C₂²⋊C₄	C₂³×C₃²	1	8	yes	—
414	C₂²⋊C₄	C₂³×C₃²	1	4	yes	—
415	C₄²	C₂³×C₃²	1	1	yes	characteristic
416	C₄²	C₂³×C₃²	1	1	yes	characteristic
417	C₄²	C₂³×C₃²	1	1	yes	characteristic
418	C₄²	C₂³×C₃²	1	1	yes	characteristic
419	C₄²	C₂³×C₃²	1	4	yes	—
420	C₄²	C₂³×C₃²	1	8	yes	—
421	C₂²×C₃	—	2	4	no	—
422	C₂²×C₃	—	2	4	no	—
423	C₂²×C₃	—	2	16	no	—
424	C₂²×C₃	—	2	16	no	—
425	C₂²×C₃	—	2	16	no	—
426	C₂²×C₃	—	2	16	no	—
427	C₂²×C₃	—	2	16	no	—
428	C₂²×C₃	—	2	16	no	—
429	C₂²×C₃	—	4	32	no	—
430	C₂²×C₃	C₂×C₃×#41	1	8	yes	—
431	C₂²×C₃	C₂⁵×C₃	1	4	yes	—
432	C₃×C₄	—	2	16	no	—
433	C₃×C₄	—	2	4	no	—
434	C₃×C₄	—	2	16	no	—
435	C₃×C₄	—	2	16	no	—
436	C₃×C₄	—	2	4	no	—
437	C₃×C₄	—	2	4	no	—
438	C₃×C₄	—	2	16	no	—
439	C₃×C₄	C₂×C₃×#41	1	8	yes	—
440	C₃×C₄	C₂×C₃×#41	1	8	yes	—
441	C₃×C₄	C₃×H(5,𝔽₂)	1	8	yes	—
442	C₃²	#3506	1	1	yes	characteristic, direct product factor, Sylow
443	C₂³	—	2	4	no	—
444	C₂³	—	4	2	no	—
445	C₂³	—	4	2	no	—
446	C₂³	C₂⁴×C₃²	1	4	yes	—
447	C₂³	C₂⁴×C₃²	1	4	yes	—
448	C₂³	C₂⁴×C₃²	1	1	yes	characteristic
449	Q₈	—	2	4	no	—
450	Q₈	—	2	4	no	—
451	Q₈	—	4	2	no	—
452	D₄	—	2	8	no	—
453	D₄	—	2	4	no	—
454	D₄	—	2	4	no	—
455	D₄	—	2	4	no	—
456	D₄	—	2	4	no	—
457	D₄	—	4	8	no	—
458	D₄	—	4	2	no	—
459	D₄	—	4	8	no	—
460	C₂×C₄	—	2	8	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	8	no	—
463	C₂×C₄	—	4	2	no	—
464	C₂×C₄	—	4	4	no	—
465	C₂×C₄	—	4	8	no	—
466	C₂×C₄	—	4	1	no	—
467	C₂×C₄	C₃²×#41	1	4	yes	complemented
468	C₂×C₄	C₂⁴×C₃²	1	4	yes	—
469	C₂×C₄	C₂⁴×C₃²	1	1	yes	characteristic
470	C₂×C₄	C₂⁴×C₃²	1	4	yes	—
471	C₂×C₄	C₂⁴×C₃²	1	1	yes	characteristic
472	C₂×C₄	C₂⁴×C₃²	1	1	yes	characteristic
473	C₂×C₄	C₂⁴×C₃²	1	4	yes	—
474	C₂×C₄	C₂⁴×C₃²	1	4	yes	—
475	C₂×C₄	C₂⁴×C₃²	1	1	yes	characteristic
476	C₂×C₄	C₂⁴×C₃²	1	1	yes	characteristic
477	C₂×C₄	C₂⁴×C₃²	1	1	yes	characteristic
478	C₂×C₃	—	2	8	no	—
479	C₂×C₃	—	4	16	no	—
480	C₂×C₃	—	4	16	no	—
481	C₂×C₃	C₂²×C₃×#41	1	4	yes	—
482	C₂×C₃	C₂×C₃×H(5,𝔽₂)	1	4	yes	—
483	C₂×C₃	C₃×#585	1	4	yes	—
484	C₂²	—	2	1	no	—
485	C₂²	—	2	1	no	—
486	C₂²	—	2	4	no	—
487	C₂²	—	2	4	no	—
488	C₂²	—	2	4	no	—
489	C₂²	—	2	4	no	—
490	C₂²	—	2	4	no	—
491	C₂²	—	2	4	no	—
492	C₂²	—	4	8	no	—
493	C₂²	C₂×C₃²×#41	1	2	yes	—
494	C₂²	C₂⁵×C₃²	1	1	yes	characteristic
495	C₄	—	2	4	no	—
496	C₄	—	2	1	no	—
497	C₄	—	2	4	no	—
498	C₄	—	2	4	no	—
499	C₄	—	2	1	no	—
500	C₄	—	2	1	no	—
501	C₄	—	2	4	no	—
502	C₄	C₂×C₃²×#41	1	2	yes	—
503	C₄	C₂×C₃²×#41	1	2	yes	—
504	C₄	C₃²×H(5,𝔽₂)	1	2	yes	—
505	C₃	C₃×#3506	1	4	yes	direct product factor
506	C₂	—	2	2	no	—
507	C₂	—	4	4	no	—
508	C₂	—	4	4	no	—
509	C₂	C₂²×C₃²×#41	1	1	yes	characteristic
510	C₂	C₂×C₃²×H(5,𝔽₂)	1	1	yes	characteristic
511	C₂	C₃²×#585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃²	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃²×#559	C₂	semidirect
2. C₃²×#556	C₂	semidirect
3. C₃²×#555	C₂	semidirect
4. C₃²×#553	C₂	semidirect
5. C₃²×#550	C₂	semidirect
6. C₃²×#547	C₂	semidirect
7. C₃²×#538	C₂	semidirect
8. C₃²×#535	C₂	semidirect
9. C₃²×#533	C₂	semidirect
10. C₃²×#532	C₂	semidirect
11. C₃²×#525	C₂	semidirect
12. C₃²×#520	C₂	semidirect
13. C₂×C₃²×C₄×D₄	C₂	semidirect
14. C₃×#3506	C₃	direct (N × H)
16. C₂×C₃²×#41	C₂²	semidirect
19. C₃²×C₄⋊Q₈	C₂²	semidirect
20. C₃²×C₄⋊Q₈	C₂²	semidirect
21. C₃²×C₄⋊Q₈	C₂²	semidirect
23. C₃²×#127	C₂²	semidirect
24. C₃²×#126	C₂²	semidirect
25. C₃²×#125	C₂²	semidirect
27. C₃²×#124	C₂²	semidirect
28. C₃²×#124	C₂²	semidirect
29. C₃²×#123	C₂²	semidirect
30. C₃²×#123	C₂²	semidirect
33. C₃²×C₂²⋊Q₈	C₂²	semidirect
38. C₃²×#121	C₂²	semidirect
39. C₃²×#121	C₂²	semidirect
41. C₃²×C₄×Q₈	C₂²	semidirect
42. C₃²×C₄×Q₈	C₂²	semidirect
43. C₃²×C₄×Q₈	C₂²	semidirect
49. C₃²×C₄×D₄	C₂²	semidirect
50. C₃²×C₄×D₄	C₂²	semidirect
51. C₃²×C₄×D₄	C₂²	semidirect
52. C₃²×C₄×D₄	C₂²	semidirect
54. C₃²×#117	C₂²	semidirect
55. C₃²×#117	C₂²	semidirect
57. C₂×C₃²×C₄⋊C₄	C₂²	semidirect
60. C₃×#559	C₂×C₃	semidirect
61. C₃×#556	C₂×C₃	semidirect
62. C₃×#555	C₂×C₃	semidirect
63. C₃×#553	C₂×C₃	semidirect
64. C₃×#550	C₂×C₃	semidirect
65. C₃×#547	C₂×C₃	semidirect
66. C₃×#538	C₂×C₃	semidirect
67. C₃×#535	C₂×C₃	semidirect
68. C₃×#533	C₂×C₃	semidirect
69. C₃×#532	C₂×C₃	semidirect
70. C₃×#525	C₂×C₃	semidirect
71. C₃×#520	C₂×C₃	semidirect
72. C₂×C₃×C₄×D₄	C₂×C₃	semidirect
122. #3506	C₃²	direct (N × H)
124. C₂×C₃×#41	C₂²×C₃	semidirect
127. C₃×C₄⋊Q₈	C₂²×C₃	semidirect
128. C₃×C₄⋊Q₈	C₂²×C₃	semidirect
129. C₃×C₄⋊Q₈	C₂²×C₃	semidirect
131. C₃×#127	C₂²×C₃	semidirect
132. C₃×#126	C₂²×C₃	semidirect
133. C₃×#125	C₂²×C₃	semidirect
135. C₃×#124	C₂²×C₃	semidirect
136. C₃×#124	C₂²×C₃	semidirect
137. C₃×#123	C₂²×C₃	semidirect
138. C₃×#123	C₂²×C₃	semidirect
141. C₃×C₂²⋊Q₈	C₂²×C₃	semidirect
146. C₃×#121	C₂²×C₃	semidirect
147. C₃×#121	C₂²×C₃	semidirect
149. C₃×C₄×Q₈	C₂²×C₃	semidirect
150. C₃×C₄×Q₈	C₂²×C₃	semidirect
151. C₃×C₄×Q₈	C₂²×C₃	semidirect
157. C₃×C₄×D₄	C₂²×C₃	semidirect
158. C₃×C₄×D₄	C₂²×C₃	semidirect
159. C₃×C₄×D₄	C₂²×C₃	semidirect
160. C₃×C₄×D₄	C₂²×C₃	semidirect
162. C₃×#117	C₂²×C₃	semidirect
163. C₃×#117	C₂²×C₃	semidirect
165. C₂×C₃×C₄⋊C₄	C₂²×C₃	semidirect
192. C₂×C₃²×C₄	#41	semidirect
203. #559	C₂×C₃²	semidirect
204. #556	C₂×C₃²	semidirect
205. #555	C₂×C₃²	semidirect
206. #553	C₂×C₃²	semidirect
207. #550	C₂×C₃²	semidirect
208. #547	C₂×C₃²	semidirect
209. #538	C₂×C₃²	semidirect
210. #535	C₂×C₃²	semidirect
211. #533	C₂×C₃²	semidirect
212. #532	C₂×C₃²	semidirect
213. #525	C₂×C₃²	semidirect
214. #520	C₂×C₃²	semidirect
215. C₂×C₄×D₄	C₂×C₃²	semidirect
287. C₂×#41	C₂²×C₃²	semidirect
290. C₄⋊Q₈	C₂²×C₃²	semidirect
291. C₄⋊Q₈	C₂²×C₃²	semidirect
292. C₄⋊Q₈	C₂²×C₃²	semidirect
294. #127	C₂²×C₃²	semidirect
295. #126	C₂²×C₃²	semidirect
296. #125	C₂²×C₃²	semidirect
298. #124	C₂²×C₃²	semidirect
299. #124	C₂²×C₃²	semidirect
300. #123	C₂²×C₃²	semidirect
301. #123	C₂²×C₃²	semidirect
304. C₂²⋊Q₈	C₂²×C₃²	semidirect
309. #121	C₂²×C₃²	semidirect
310. #121	C₂²×C₃²	semidirect
312. C₄×Q₈	C₂²×C₃²	semidirect
313. C₄×Q₈	C₂²×C₃²	semidirect
314. C₄×Q₈	C₂²×C₃²	semidirect
320. C₄×D₄	C₂²×C₃²	semidirect
321. C₄×D₄	C₂²×C₃²	semidirect
322. C₄×D₄	C₂²×C₃²	semidirect
323. C₄×D₄	C₂²×C₃²	semidirect
325. #117	C₂²×C₃²	semidirect
326. #117	C₂²×C₃²	semidirect
328. C₂×C₄⋊C₄	C₂²×C₃²	semidirect
355. C₂×C₃×C₄	C₃×#41	semidirect
442. C₃²	#3506	direct (N × H)
467. C₂×C₄	C₃²×#41	semidirect
505. C₃	C₃×#3506	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃²×#559	2
C₃²×#556	2
C₃²×#555	2
C₃²×#553	2
C₃²×#550	2
C₃²×#547	2
C₃²×#538	2
C₃²×#535	2
C₃²×#533	2
C₃²×#532	2
C₃²×#525	2
C₃²×#520	2
C₂×C₃²×C₄×D₄	2
C₃×#3506	3

Subgroups — C32×#3506 #11770555

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#3506 #11770555