Subgroups — C₃²×D₃×#104 #13162256

Order 1728.

1196 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×D₃×#41	C₂	1	1	yes	maximal, characteristic
2	C₃²×D₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×C₄²×D₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃³×#104	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×#837	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#825	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃²×#805	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃²×#104	—	3	1	no	maximal
9	C₃×D₃×#104	C₃	1	4	yes	maximal, direct product factor
10	C₃³×#41	C₂²	1	1	yes	characteristic
11	C₃²×#239	C₄	1	1	yes	characteristic, complemented
12	C₃²×D₃×Q₈	C₄	1	1	yes	characteristic, complemented
13	C₃²×#237	—	2	1	no	—
14	C₃²×#237	C₄	1	1	yes	characteristic, complemented
15	C₃²×D₃×D₄	—	2	1	no	—
16	C₃²×D₃×D₄	C₄	1	1	yes	characteristic, complemented
17	C₃²×#235	—	2	1	no	—
18	C₃²×#235	C₂²	1	1	yes	characteristic
19	C₂×C₃²×C₄×D₃	—	2	1	no	—
20	C₂×C₃²×C₄×D₃	—	2	1	no	—
21	C₂×C₃²×C₄×D₃	C₂²	1	1	yes	characteristic
22	C₃³×M₁₆	C₂²	1	1	yes	characteristic, complemented
23	C₃³×C₄²	C₂²	1	1	yes	characteristic, complemented
24	C₃²×C₄×Dic₃	—	2	1	no	—
25	C₃²×C₄×Dic₃	C₂²	1	1	yes	characteristic, complemented
26	C₃²×#208	C₂²	1	1	yes	characteristic, complemented
27	C₃²×ΓL(1,𝔽₂₅)	—	2	1	no	—
28	C₃²×D₃×C₈	—	2	1	no	—
29	C₂×C₃²×#41	—	3	1	no	—
30	C₃×D₃×#41	C₂×C₃	1	4	yes	—
31	C₂×C₃²×M₁₆	—	3	1	no	—
32	C₂×C₃²×C₄²	—	3	1	no	—
33	C₃×D₃×M₁₆	C₂×C₃	1	4	yes	complemented
34	C₃×C₄²×D₃	C₂×C₃	1	4	yes	complemented
35	C₃²×#104	—	2	4	no	—
36	C₃²×#104	—	3	1	no	—
37	C₃²×#104	—	3	1	no	—
38	C₃²×#104	—	3	1	no	—
39	C₃²×#104	D₃	1	1	yes	characteristic, direct product factor
40	C₃²×#104	C₂×C₃	1	4	yes	complemented
41	C₃×#837	C₂×C₃	1	4	yes	complemented
42	C₃×#825	C₂×C₃	1	4	yes	complemented
43	C₃×#805	C₂×C₃	1	4	yes	complemented
44	C₂²×C₃²×D₃	—	2	1	no	—
45	C₂²×C₃²×D₃	D₄	1	1	yes	characteristic
46	C₃³×Q₈	C₂×C₄	1	1	yes	characteristic, complemented
47	C₃³×D₄	—	2	1	no	—
48	C₃³×D₄	C₂×C₄	1	1	yes	characteristic, complemented
49	C₂×C₃³×C₄	—	2	1	no	—
50	C₂×C₃³×C₄	—	2	1	no	—
51	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
52	C₃²×#67	—	2	1	no	—
53	C₃²×#67	—	2	1	no	—
54	C₃²×#67	—	2	1	no	—
55	C₂×C₃²×Dic₃	—	2	1	no	—
56	C₂×C₃²×Dic₃	—	2	1	no	—
57	C₂×C₃²×Dic₃	D₄	1	1	yes	characteristic
58	C₃²×D₁₂	—	2	1	no	—
59	C₃²×D₁₂	C₂×C₄	1	1	yes	characteristic, complemented
60	C₃²×C₄×D₃	—	2	4	no	—
61	C₃²×C₄×D₃	—	2	1	no	—
62	C₃²×C₄×D₃	—	2	1	no	—
63	C₃²×C₄×D₃	—	2	1	no	—
64	C₃²×C₄×D₃	C₂×C₄	1	1	yes	characteristic
65	C₃²×C₄×D₃	C₂×C₄	1	1	yes	characteristic
66	C₃²×C₄×D₃	D₄	1	1	yes	characteristic
67	C₃²×C₄×D₃	D₄	1	1	yes	characteristic
68	C₃²×Dic₆	—	2	1	no	—
69	C₃²×Dic₆	C₂×C₄	1	1	yes	characteristic, complemented
70	C₃³×C₈	—	2	1	no	—
71	C₃²×C₃⋊C₈	—	2	1	no	—
72	C₂×C₃×#104	—	3	4	no	—
73	D₃×#104	C₃²	1	1	yes	characteristic, direct product factor
74	C₃²×#41	—	2	4	no	—
75	C₃²×#41	—	3	1	no	—
76	C₃²×#41	—	3	1	no	—
77	C₃²×#41	—	3	1	no	—
78	C₃²×#41	—	6	1	no	—
79	C₃²×#41	—	6	1	no	—
80	C₃²×#41	C₂×D₃	1	1	yes	characteristic
81	C₃²×#41	C₂²×C₃	1	4	yes	—
82	C₂×C₃²×Q₈	—	3	1	no	—
83	C₂×C₃²×D₄	—	3	1	no	—
84	C₂×C₃²×D₄	—	6	1	no	—
85	C₂²×C₃²×C₄	—	3	1	no	—
86	C₂²×C₃²×C₄	—	6	1	no	—
87	C₂²×C₃²×C₄	—	6	1	no	—
88	C₃×#239	C₃×C₄	1	4	yes	complemented
89	C₃×D₃×Q₈	C₃×C₄	1	4	yes	complemented
90	C₃×#237	—	2	4	no	—
91	C₃×#237	C₃×C₄	1	4	yes	complemented
92	C₃×D₃×D₄	—	2	4	no	—
93	C₃×D₃×D₄	C₃×C₄	1	4	yes	complemented
94	C₃×#235	—	2	4	no	—
95	C₃×#235	C₂²×C₃	1	4	yes	—
96	C₂×C₃×C₄×D₃	—	2	4	no	—
97	C₂×C₃×C₄×D₃	—	2	4	no	—
98	C₂×C₃×C₄×D₃	C₂²×C₃	1	4	yes	—
99	C₃²×M₁₆	—	2	4	no	—
100	C₃²×M₁₆	—	3	1	no	—
101	C₃²×M₁₆	—	6	1	no	—
102	C₃²×M₁₆	C₂×D₃	1	1	yes	characteristic, complemented
103	C₃²×M₁₆	C₂²×C₃	1	4	yes	complemented
104	C₂×C₃²×C₈	—	6	1	no	—
105	C₃²×C₄²	—	2	4	no	—
106	C₃²×C₄²	—	3	1	no	—
107	C₃²×C₄²	—	6	1	no	—
108	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic, complemented
109	C₃²×C₄²	C₂²×C₃	1	4	yes	complemented
110	C₃×C₄×Dic₃	—	2	4	no	—
111	C₃×C₄×Dic₃	C₂²×C₃	1	4	yes	complemented
112	C₃×#208	C₂²×C₃	1	4	yes	complemented
113	C₃×ΓL(1,𝔽₂₅)	—	2	4	no	—
114	C₃×D₃×C₈	—	2	4	no	—
115	C₂²×C₃³	—	2	1	no	—
116	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
117	C₂×C₃²×D₃	—	2	2	no	—
118	C₂×C₃²×D₃	—	2	1	no	—
119	C₂×C₃²×D₃	—	4	2	no	—
120	C₂×C₃²×D₃	C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₃²×D₃	C₂²⋊C₄	1	1	yes	characteristic
122	C₃³×C₄	—	2	2	no	—
123	C₃³×C₄	—	2	1	no	—
124	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
125	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
126	C₃²×Dic₃	—	2	2	no	—
127	C₃²×Dic₃	—	2	1	no	—
128	C₃²×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
129	C₃²×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₃×#41	—	3	4	no	—
131	D₃×#41	C₂×C₃²	1	1	yes	characteristic
132	C₂×C₃×M₁₆	—	3	4	no	—
133	C₂×C₃×C₄²	—	3	4	no	—
134	D₃×M₁₆	C₂×C₃²	1	1	yes	characteristic, complemented
135	C₄²×D₃	C₂×C₃²	1	1	yes	characteristic, complemented
136	C₃×#104	—	2	4	no	—
137	C₃×#104	—	3	4	no	—
138	C₃×#104	—	3	4	no	—
139	C₃×#104	—	3	4	no	—
140	C₃×#104	C₃×D₃	1	4	yes	direct product factor
141	C₃×#104	C₂×C₃²	1	1	yes	characteristic, complemented
142	#837	C₂×C₃²	1	1	yes	characteristic, complemented
143	#825	C₂×C₃²	1	1	yes	characteristic, complemented
144	#805	C₂×C₃²	1	1	yes	characteristic, complemented
145	C₂³×C₃²	—	3	1	no	—
146	C₂³×C₃²	—	6	1	no	—
147	C₂²×C₃×D₃	—	2	4	no	—
148	C₂²×C₃×D₃	C₃×D₄	1	4	yes	—
149	C₃²×Q₈	—	2	4	no	—
150	C₃²×Q₈	—	3	1	no	—
151	C₃²×Q₈	—	6	1	no	—
152	C₃²×Q₈	C₄×D₃	1	1	yes	characteristic, complemented
153	C₃²×Q₈	C₂×C₃×C₄	1	4	yes	complemented
154	C₃²×D₄	—	2	1	no	—
155	C₃²×D₄	—	2	4	no	—
156	C₃²×D₄	—	2	4	no	—
157	C₃²×D₄	—	3	1	no	—
158	C₃²×D₄	—	4	4	no	—
159	C₃²×D₄	—	6	1	no	—
160	C₃²×D₄	—	6	1	no	—
161	C₃²×D₄	—	6	1	no	—
162	C₃²×D₄	—	6	1	no	—
163	C₃²×D₄	C₄×D₃	1	1	yes	characteristic, complemented
164	C₃²×D₄	C₂×C₃×C₄	1	4	yes	complemented
165	C₂×C₃²×C₄	—	2	1	no	—
166	C₂×C₃²×C₄	—	2	1	no	—
167	C₂×C₃²×C₄	—	2	4	no	—
168	C₂×C₃²×C₄	—	2	4	no	—
169	C₂×C₃²×C₄	—	2	4	no	—
170	C₂×C₃²×C₄	—	3	1	no	—
171	C₂×C₃²×C₄	—	3	1	no	—
172	C₂×C₃²×C₄	—	3	1	no	—
173	C₂×C₃²×C₄	—	3	1	no	—
174	C₂×C₃²×C₄	—	3	1	no	—
175	C₂×C₃²×C₄	—	4	4	no	—
176	C₂×C₃²×C₄	—	4	4	no	—
177	C₂×C₃²×C₄	—	6	1	no	—
178	C₂×C₃²×C₄	—	6	1	no	—
179	C₂×C₃²×C₄	—	6	4	no	—
180	C₂×C₃²×C₄	—	6	1	no	—
181	C₂×C₃²×C₄	—	6	1	no	—
182	C₂×C₃²×C₄	—	6	1	no	—
183	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
184	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
185	C₃×#67	—	2	4	no	—
186	C₃×#67	—	2	4	no	—
187	C₃×#67	—	2	4	no	—
188	C₂×C₃×Dic₃	—	2	4	no	—
189	C₂×C₃×Dic₃	—	2	4	no	—
190	C₂×C₃×Dic₃	C₃×D₄	1	4	yes	—
191	C₃×D₁₂	—	2	4	no	—
192	C₃×D₁₂	C₂×C₃×C₄	1	4	yes	complemented
193	C₃×C₄×D₃	—	2	16	no	—
194	C₃×C₄×D₃	—	2	4	no	—
195	C₃×C₄×D₃	—	2	4	no	—
196	C₃×C₄×D₃	—	2	4	no	—
197	C₃×C₄×D₃	C₂×C₃×C₄	1	4	yes	—
198	C₃×C₄×D₃	C₂×C₃×C₄	1	4	yes	—
199	C₃×C₄×D₃	C₃×D₄	1	4	yes	—
200	C₃×C₄×D₃	C₃×D₄	1	4	yes	—
201	C₃×Dic₆	—	2	4	no	—
202	C₃×Dic₆	C₂×C₃×C₄	1	4	yes	complemented
203	C₃²×C₈	—	2	1	no	—
204	C₃²×C₈	—	2	4	no	—
205	C₃²×C₈	—	4	4	no	—
206	C₃²×C₈	—	6	1	no	—
207	C₃×C₃⋊C₈	—	2	4	no	—
208	C₂×#104	—	3	1	no	Sylow
209	C₂×C₃³	—	2	1	no	—
210	C₂×C₃³	—	4	1	no	—
211	C₂×C₃³	C₂×C₂²⋊C₄	1	1	yes	characteristic
212	C₃²×D₃	—	2	1	no	—
213	C₃²×D₃	—	4	1	no	—
214	C₃²×D₃	#104	1	2	yes	direct product factor
215	C₃×#41	—	2	4	no	—
216	C₃×#41	—	3	4	no	—
217	C₃×#41	—	3	4	no	—
218	C₃×#41	—	3	4	no	—
219	C₃×#41	—	6	4	no	—
220	C₃×#41	—	6	4	no	—
221	C₃×#41	C₂×C₃×D₃	1	4	yes	—
222	C₃×#41	C₂²×C₃²	1	1	yes	characteristic
223	C₂×C₃×Q₈	—	3	4	no	—
224	C₂×C₃×D₄	—	3	4	no	—
225	C₂×C₃×D₄	—	6	4	no	—
226	C₂²×C₃×C₄	—	3	4	no	—
227	C₂²×C₃×C₄	—	6	4	no	—
228	C₂²×C₃×C₄	—	6	4	no	—
229	#239	C₃²×C₄	1	1	yes	characteristic, complemented
230	D₃×Q₈	C₃²×C₄	1	1	yes	characteristic, complemented
231	#237	—	2	1	no	—
232	#237	C₃²×C₄	1	1	yes	characteristic, complemented
233	D₃×D₄	—	2	1	no	—
234	D₃×D₄	C₃²×C₄	1	1	yes	characteristic, complemented
235	#235	—	2	1	no	—
236	#235	C₂²×C₃²	1	1	yes	characteristic
237	C₂×C₄×D₃	—	2	1	no	—
238	C₂×C₄×D₃	—	2	1	no	—
239	C₂×C₄×D₃	C₂²×C₃²	1	1	yes	characteristic
240	C₃×M₁₆	—	2	4	no	—
241	C₃×M₁₆	—	3	4	no	—
242	C₃×M₁₆	—	6	4	no	—
243	C₃×M₁₆	C₂×C₃×D₃	1	4	yes	complemented
244	C₃×M₁₆	C₂²×C₃²	1	1	yes	characteristic, complemented
245	C₂×C₃×C₈	—	6	4	no	—
246	C₃×C₄²	—	2	4	no	—
247	C₃×C₄²	—	3	4	no	—
248	C₃×C₄²	—	6	4	no	—
249	C₃×C₄²	C₂×C₃×D₃	1	4	yes	complemented
250	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic, complemented
251	C₄×Dic₃	—	2	1	no	—
252	C₄×Dic₃	C₂²×C₃²	1	1	yes	characteristic, complemented
253	#208	C₂²×C₃²	1	1	yes	characteristic, complemented
254	ΓL(1,𝔽₂₅)	—	2	1	no	—
255	D₃×C₈	—	2	1	no	—
256	C₂²×C₃²	—	2	1	no	—
257	C₂²×C₃²	—	2	4	no	—
258	C₂²×C₃²	—	2	4	no	—
259	C₂²×C₃²	—	3	1	no	—
260	C₂²×C₃²	—	3	1	no	—
261	C₂²×C₃²	—	4	4	no	—
262	C₂²×C₃²	—	6	2	no	—
263	C₂²×C₃²	—	6	1	no	—
264	C₂²×C₃²	—	12	2	no	—
265	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
266	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
267	C₂×C₃×D₃	—	2	8	no	—
268	C₂×C₃×D₃	—	2	4	no	—
269	C₂×C₃×D₃	—	4	8	no	—
270	C₂×C₃×D₃	C₃×C₂²⋊C₄	1	4	yes	—
271	C₂×C₃×D₃	C₃×C₂²⋊C₄	1	4	yes	—
272	C₃²×C₄	—	2	2	no	—
273	C₃²×C₄	—	2	1	no	—
274	C₃²×C₄	—	2	4	no	—
275	C₃²×C₄	—	2	4	no	—
276	C₃²×C₄	—	2	8	no	—
277	C₃²×C₄	—	2	4	no	—
278	C₃²×C₄	—	3	1	no	—
279	C₃²×C₄	—	3	1	no	—
280	C₃²×C₄	—	4	4	no	—
281	C₃²×C₄	—	4	8	no	—
282	C₃²×C₄	—	6	2	no	—
283	C₃²×C₄	—	6	1	no	—
284	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
285	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
286	C₃²×C₄	C₂²×C₃×C₄	1	4	yes	—
287	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
288	C₃×Dic₃	—	2	8	no	—
289	C₃×Dic₃	—	2	4	no	—
290	C₃×Dic₃	C₃×C₂²⋊C₄	1	4	yes	—
291	C₃×Dic₃	C₃×C₂²⋊C₄	1	4	yes	—
292	C₂×#41	—	3	1	no	—
293	C₂×M₁₆	—	3	1	no	—
294	C₂×C₄²	—	3	1	no	—
295	#104	—	3	1	no	—
296	#104	—	3	1	no	—
297	#104	—	3	1	no	—
298	#104	C₃²×D₃	1	1	yes	characteristic, direct product factor
299	C₃³	C₂×#104	1	1	yes	characteristic, complemented, Sylow
300	C₂³×C₃	—	3	4	no	—
301	C₂³×C₃	—	6	4	no	—
302	C₂²×D₃	—	2	1	no	—
303	C₂²×D₃	C₃²×D₄	1	1	yes	characteristic
304	C₃×Q₈	—	2	4	no	—
305	C₃×Q₈	—	3	4	no	—
306	C₃×Q₈	—	6	4	no	—
307	C₃×Q₈	C₃×C₄×D₃	1	4	yes	complemented
308	C₃×Q₈	C₂×C₃²×C₄	1	1	yes	characteristic, complemented
309	C₃×D₄	—	2	4	no	—
310	C₃×D₄	—	2	1	no	—
311	C₃×D₄	—	2	4	no	—
312	C₃×D₄	—	3	4	no	—
313	C₃×D₄	—	4	4	no	—
314	C₃×D₄	—	6	4	no	—
315	C₃×D₄	—	6	4	no	—
316	C₃×D₄	—	6	4	no	—
317	C₃×D₄	—	6	4	no	—
318	C₃×D₄	C₃×C₄×D₃	1	4	yes	complemented
319	C₃×D₄	C₂×C₃²×C₄	1	1	yes	characteristic, complemented
320	C₂×C₃×C₄	—	2	4	no	—
321	C₂×C₃×C₄	—	2	1	no	—
322	C₂×C₃×C₄	—	2	4	no	—
323	C₂×C₃×C₄	—	2	1	no	—
324	C₂×C₃×C₄	—	2	4	no	—
325	C₂×C₃×C₄	—	3	4	no	—
326	C₂×C₃×C₄	—	3	4	no	—
327	C₂×C₃×C₄	—	3	4	no	—
328	C₂×C₃×C₄	—	3	4	no	—
329	C₂×C₃×C₄	—	3	4	no	—
330	C₂×C₃×C₄	—	4	4	no	—
331	C₂×C₃×C₄	—	4	4	no	—
332	C₂×C₃×C₄	—	6	4	no	—
333	C₂×C₃×C₄	—	6	4	no	—
334	C₂×C₃×C₄	—	6	16	no	—
335	C₂×C₃×C₄	—	6	4	no	—
336	C₂×C₃×C₄	—	6	4	no	—
337	C₂×C₃×C₄	—	6	4	no	—
338	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
339	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
340	#67	—	2	1	no	—
341	#67	—	2	1	no	—
342	#67	—	2	1	no	—
343	C₂×Dic₃	—	2	1	no	—
344	C₂×Dic₃	—	2	1	no	—
345	C₂×Dic₃	C₃²×D₄	1	1	yes	characteristic
346	D₁₂	—	2	1	no	—
347	D₁₂	C₂×C₃²×C₄	1	1	yes	characteristic, complemented
348	C₄×D₃	—	2	4	no	—
349	C₄×D₃	—	2	1	no	—
350	C₄×D₃	—	2	1	no	—
351	C₄×D₃	—	2	1	no	—
352	C₄×D₃	C₂×C₃²×C₄	1	1	yes	characteristic
353	C₄×D₃	C₂×C₃²×C₄	1	1	yes	characteristic
354	C₄×D₃	C₃²×D₄	1	1	yes	characteristic
355	C₄×D₃	C₃²×D₄	1	1	yes	characteristic
356	Dic₆	—	2	1	no	—
357	Dic₆	C₂×C₃²×C₄	1	1	yes	characteristic, complemented
358	C₃×C₈	—	2	4	no	—
359	C₃×C₈	—	2	1	no	—
360	C₃×C₈	—	4	4	no	—
361	C₃×C₈	—	6	4	no	—
362	C₃⋊C₈	—	2	1	no	—
363	C₂×C₃²	—	2	1	no	—
364	C₂×C₃²	—	2	4	no	—
365	C₂×C₃²	—	2	4	no	—
366	C₂×C₃²	—	3	2	no	—
367	C₂×C₃²	—	4	4	no	—
368	C₂×C₃²	—	4	1	no	—
369	C₂×C₃²	—	4	4	no	—
370	C₂×C₃²	—	6	1	no	—
371	C₂×C₃²	—	8	4	no	—
372	C₂×C₃²	—	12	1	no	—
373	C₂×C₃²	D₃×C₂²⋊C₄	1	1	yes	characteristic
374	C₂×C₃²	C₂×C₃×C₂²⋊C₄	1	4	yes	—
375	C₃×D₃	—	2	4	no	—
376	C₃×D₃	—	4	4	no	—
377	C₃×D₃	C₃×#104	1	8	yes	direct product factor
378	#41	—	3	1	no	—
379	#41	—	3	1	no	—
380	#41	—	3	1	no	—
381	#41	—	6	1	no	—
382	#41	—	6	1	no	—
383	#41	C₂×C₃²×D₃	1	1	yes	characteristic
384	C₂×Q₈	—	3	1	no	—
385	C₂×D₄	—	3	1	no	—
386	C₂×D₄	—	6	1	no	—
387	C₂²×C₄	—	3	1	no	—
388	C₂²×C₄	—	6	1	no	—
389	C₂²×C₄	—	6	1	no	—
390	M₁₆	—	3	1	no	—
391	M₁₆	—	6	1	no	—
392	M₁₆	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
393	C₂×C₈	—	6	1	no	—
394	C₄²	—	3	1	no	—
395	C₄²	—	6	1	no	—
396	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
397	C₂²×C₃	—	2	4	no	—
398	C₂²×C₃	—	2	1	no	—
399	C₂²×C₃	—	2	4	no	—
400	C₂²×C₃	—	3	4	no	—
401	C₂²×C₃	—	3	4	no	—
402	C₂²×C₃	—	4	4	no	—
403	C₂²×C₃	—	6	8	no	—
404	C₂²×C₃	—	6	4	no	—
405	C₂²×C₃	—	12	8	no	—
406	C₂²×C₃	C₃×D₃×D₄	1	4	yes	—
407	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
408	C₂×D₃	—	2	2	no	—
409	C₂×D₃	—	2	1	no	—
410	C₂×D₃	—	4	2	no	—
411	C₂×D₃	C₃²×C₂²⋊C₄	1	1	yes	characteristic
412	C₂×D₃	C₃²×C₂²⋊C₄	1	1	yes	characteristic
413	C₃×C₄	—	2	2	no	—
414	C₃×C₄	—	2	8	no	—
415	C₃×C₄	—	2	4	no	—
416	C₃×C₄	—	2	1	no	—
417	C₃×C₄	—	2	4	no	—
418	C₃×C₄	—	2	4	no	—
419	C₃×C₄	—	3	4	no	—
420	C₃×C₄	—	3	4	no	—
421	C₃×C₄	—	4	4	no	—
422	C₃×C₄	—	4	8	no	—
423	C₃×C₄	—	6	8	no	—
424	C₃×C₄	—	6	4	no	—
425	C₃×C₄	C₂×C₃×C₄×D₃	1	4	yes	—
426	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
427	C₃×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
428	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
429	Dic₃	—	2	2	no	—
430	Dic₃	—	2	1	no	—
431	Dic₃	C₃²×C₂²⋊C₄	1	1	yes	characteristic
432	Dic₃	C₃²×C₂²⋊C₄	1	1	yes	characteristic
433	C₃²	—	2	4	no	—
434	C₃²	D₃×#104	1	1	yes	characteristic, direct product factor
435	C₃²	C₂×C₃×#104	1	4	yes	complemented
436	C₂³	—	3	1	no	—
437	C₂³	—	6	1	no	—
438	Q₈	—	3	1	no	—
439	Q₈	—	6	1	no	—
440	Q₈	C₃²×C₄×D₃	1	1	yes	characteristic, complemented
441	D₄	—	2	1	no	—
442	D₄	—	3	1	no	—
443	D₄	—	6	1	no	—
444	D₄	—	6	1	no	—
445	D₄	—	6	1	no	—
446	D₄	—	6	1	no	—
447	D₄	C₃²×C₄×D₃	1	1	yes	characteristic, complemented
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	3	1	no	—
451	C₂×C₄	—	3	1	no	—
452	C₂×C₄	—	3	1	no	—
453	C₂×C₄	—	3	1	no	—
454	C₂×C₄	—	3	1	no	—
455	C₂×C₄	—	6	1	no	—
456	C₂×C₄	—	6	1	no	—
457	C₂×C₄	—	6	4	no	—
458	C₂×C₄	—	6	1	no	—
459	C₂×C₄	—	6	1	no	—
460	C₂×C₄	—	6	1	no	—
461	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
462	C₈	—	2	1	no	—
463	C₈	—	6	1	no	—
464	C₂×C₃	—	2	4	no	—
465	C₂×C₃	—	2	1	no	—
466	C₂×C₃	—	2	4	no	—
467	C₂×C₃	—	3	8	no	—
468	C₂×C₃	—	4	4	no	—
469	C₂×C₃	—	4	4	no	—
470	C₂×C₃	—	4	1	no	—
471	C₂×C₃	—	6	4	no	—
472	C₂×C₃	—	8	4	no	—
473	C₂×C₃	—	12	4	no	—
474	C₂×C₃	C₃×D₃×C₂²⋊C₄	1	4	yes	—
475	C₂×C₃	C₂×C₃²×C₂²⋊C₄	1	1	yes	characteristic
476	D₃	—	2	1	no	—
477	D₃	—	4	1	no	—
478	D₃	C₃²×#104	1	2	yes	direct product factor
479	C₂²	—	2	1	no	—
480	C₂²	—	3	1	no	—
481	C₂²	—	3	1	no	—
482	C₂²	—	6	2	no	—
483	C₂²	—	6	1	no	—
484	C₂²	—	12	2	no	—
485	C₂²	C₃²×D₃×D₄	1	1	yes	characteristic
486	C₄	—	2	2	no	—
487	C₄	—	2	1	no	—
488	C₄	—	3	1	no	—
489	C₄	—	3	1	no	—
490	C₄	—	6	2	no	—
491	C₄	—	6	1	no	—
492	C₄	C₂×C₃²×C₄×D₃	1	1	yes	characteristic
493	C₄	C₃²×D₃×D₄	1	1	yes	characteristic
494	C₃	—	2	4	no	—
495	C₃	C₃×D₃×#104	1	4	yes	direct product factor
496	C₃	C₂×C₃²×#104	1	1	yes	characteristic, complemented
497	C₂	—	2	1	no	—
498	C₂	—	3	2	no	—
499	C₂	—	4	1	no	—
500	C₂	—	6	1	no	—
501	C₂	—	12	1	no	—
502	C₂	C₃²×D₃×C₂²⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₃²×C₄	D₃×D₄
commutator	C₃×C₄	C₂²×C₃²×C₄
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₃³×#104	C₂
socle	C₂×C₃³	C₂×C₂²⋊C₄
prime radical	C₃²×D₃×#41	C₂
cosocle	C₂×C₃×C₄	C₂³×C₃²
hypercenter	C₃²×#104	D₃
hypocenter	C₃	C₂×C₃²×#104
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃²×C₄ ◁ C₂×C₃²×C₄ ◁ C₃²×#104

Complements

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

N	H ≅ G/N	type
2. C₃²×D₃×M₁₆	C₂	semidirect
3. C₃²×C₄²×D₃	C₂	semidirect
4. C₃³×#104	C₂	semidirect
5. C₃²×#837	C₂	semidirect
6. C₃²×#825	C₂	semidirect
7. C₃²×#805	C₂	semidirect
9. C₃×D₃×#104	C₃	direct (N × H)
11. C₃²×#239	C₄	semidirect
12. C₃²×D₃×Q₈	C₄	semidirect
14. C₃²×#237	C₄	semidirect
16. C₃²×D₃×D₄	C₄	semidirect
22. C₃³×M₁₆	C₂²	semidirect
23. C₃³×C₄²	C₂²	semidirect
25. C₃²×C₄×Dic₃	C₂²	semidirect
26. C₃²×#208	C₂²	semidirect
33. C₃×D₃×M₁₆	C₂×C₃	semidirect
34. C₃×C₄²×D₃	C₂×C₃	semidirect
39. C₃²×#104	D₃	direct (N × H)
40. C₃²×#104	C₂×C₃	semidirect
41. C₃×#837	C₂×C₃	semidirect
42. C₃×#825	C₂×C₃	semidirect
43. C₃×#805	C₂×C₃	semidirect
46. C₃³×Q₈	C₂×C₄	semidirect
48. C₃³×D₄	C₂×C₄	semidirect
59. C₃²×D₁₂	C₂×C₄	semidirect
69. C₃²×Dic₆	C₂×C₄	semidirect
73. D₃×#104	C₃²	direct (N × H)
88. C₃×#239	C₃×C₄	semidirect
89. C₃×D₃×Q₈	C₃×C₄	semidirect
91. C₃×#237	C₃×C₄	semidirect
93. C₃×D₃×D₄	C₃×C₄	semidirect
102. C₃²×M₁₆	C₂×D₃	semidirect
103. C₃²×M₁₆	C₂²×C₃	semidirect
108. C₃²×C₄²	C₂×D₃	semidirect
109. C₃²×C₄²	C₂²×C₃	semidirect
111. C₃×C₄×Dic₃	C₂²×C₃	semidirect
112. C₃×#208	C₂²×C₃	semidirect
134. D₃×M₁₆	C₂×C₃²	semidirect
135. C₄²×D₃	C₂×C₃²	semidirect
140. C₃×#104	C₃×D₃	direct (N × H)
141. C₃×#104	C₂×C₃²	semidirect
142. #837	C₂×C₃²	semidirect
143. #825	C₂×C₃²	semidirect
144. #805	C₂×C₃²	semidirect
152. C₃²×Q₈	C₄×D₃	semidirect
153. C₃²×Q₈	C₂×C₃×C₄	semidirect
163. C₃²×D₄	C₄×D₃	semidirect
164. C₃²×D₄	C₂×C₃×C₄	semidirect
192. C₃×D₁₂	C₂×C₃×C₄	semidirect
202. C₃×Dic₆	C₂×C₃×C₄	semidirect
214. C₃²×D₃	#104	direct (N × H)
229. #239	C₃²×C₄	semidirect
230. D₃×Q₈	C₃²×C₄	semidirect
232. #237	C₃²×C₄	semidirect
234. D₃×D₄	C₃²×C₄	semidirect
243. C₃×M₁₆	C₂×C₃×D₃	semidirect
244. C₃×M₁₆	C₂²×C₃²	semidirect
249. C₃×C₄²	C₂×C₃×D₃	semidirect
250. C₃×C₄²	C₂²×C₃²	semidirect
252. C₄×Dic₃	C₂²×C₃²	semidirect
253. #208	C₂²×C₃²	semidirect
298. #104	C₃²×D₃	direct (N × H)
299. C₃³	C₂×#104	semidirect
307. C₃×Q₈	C₃×C₄×D₃	semidirect
308. C₃×Q₈	C₂×C₃²×C₄	semidirect
318. C₃×D₄	C₃×C₄×D₃	semidirect
319. C₃×D₄	C₂×C₃²×C₄	semidirect
347. D₁₂	C₂×C₃²×C₄	semidirect
357. Dic₆	C₂×C₃²×C₄	semidirect
377. C₃×D₃	C₃×#104	direct (N × H)
392. M₁₆	C₂×C₃²×D₃	semidirect
396. C₄²	C₂×C₃²×D₃	semidirect
434. C₃²	D₃×#104	direct (N × H)
435. C₃²	C₂×C₃×#104	semidirect
440. Q₈	C₃²×C₄×D₃	semidirect
447. D₄	C₃²×C₄×D₃	semidirect
478. D₃	C₃²×#104	direct (N × H)
495. C₃	C₃×D₃×#104	direct (N × H)
496. C₃	C₂×C₃²×#104	semidirect

Maximal subgroups

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

subgroup	index
C₃²×D₃×#41	2
C₃²×D₃×M₁₆	2
C₃²×C₄²×D₃	2
C₃³×#104	2
C₃²×#837	2
C₃²×#825	2
C₃²×#805	2
C₂×C₃²×#104	3
C₃×D₃×#104	3

Subgroups — C32×D3×#104 #13162256

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×D₃×#104 #13162256