Subgroups — C₃²×#3054 #11770103

Order 1152.

1500 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#135	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃²×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#525	C₂	1	2	yes	maximal, complemented
4	C₃²×#470	C₂	1	2	yes	maximal, complemented
5	C₃²×#469	C₂	1	2	yes	maximal, complemented
6	C₃²×#468	C₂	1	4	yes	maximal, complemented
7	C₃²×#427	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃²×#416	C₂	1	2	yes	maximal, complemented
9	C₃×#3054	C₃	1	4	yes	maximal, direct product factor
10	C₂×C₃²×#41	C₂²	1	2	yes	—
11	C₂³×C₃²×C₄	C₂²	1	1	yes	characteristic
12	C₃²×#135	—	2	4	no	—
13	C₂×C₃²×Q₁₆	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₃²×SD₁₆	C₂²	1	2	yes	complemented
15	C₂×C₃²×D₈	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₃²×M₁₆	—	2	4	no	—
17	C₂×C₃²×M₁₆	C₂²	1	2	yes	—
18	C₂×C₃²×M₁₆	C₂²	1	2	yes	—
19	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
20	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
21	C₃²×#123	—	2	2	no	—
22	C₃²×C₂²⋊Q₈	C₂²	1	4	yes	—
23	C₃²×#121	C₂²	1	4	yes	—
24	C₃²×#120	—	2	2	no	—
25	C₃²×C₄×D₄	—	2	4	no	—
26	C₃²×#117	C₂²	1	2	yes	—
27	C₃²×#107	C₂²	1	2	yes	complemented
28	C₃²×#106	C₂²	1	2	yes	complemented
29	C₃²×#103	C₂²	1	4	yes	complemented
30	C₃²×#102	C₂²	1	4	yes	complemented
31	C₂×C₃×#135	C₂×C₃	1	4	yes	complemented
32	C₂²×C₃×M₁₆	C₂×C₃	1	4	yes	complemented
33	C₃×#525	C₂×C₃	1	8	yes	complemented
34	C₃×#470	C₂×C₃	1	8	yes	complemented
35	C₃×#469	C₂×C₃	1	8	yes	complemented
36	C₃×#468	C₂×C₃	1	16	yes	complemented
37	C₃×#427	C₂×C₃	1	4	yes	complemented
38	C₃×#416	C₂×C₃	1	8	yes	complemented
39	C₂⁴×C₃²	D₄	1	1	yes	characteristic
40	C₃²×#41	—	2	4	no	—
41	C₃²×#41	—	2	4	no	—
42	C₂×C₃²×Q₈	C₂³	1	2	yes	—
43	C₂×C₃²×D₄	—	2	2	no	—
44	C₂×C₃²×D₄	—	2	4	no	—
45	C₂×C₃²×D₄	C₂³	1	2	yes	—
46	C₂²×C₃²×C₄	—	2	2	no	—
47	C₂²×C₃²×C₄	—	2	2	no	—
48	C₂²×C₃²×C₄	—	2	2	no	—
49	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
50	C₂²×C₃²×C₄	D₄	1	2	yes	—
51	C₂²×C₃²×C₄	C₂³	1	2	yes	—
52	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
53	C₃²×Q₁₆	—	2	2	no	—
54	C₃²×SD₁₆	—	2	4	no	—
55	C₃²×D₈	—	2	2	no	—
56	C₃²×M₁₆	—	2	4	no	—
57	C₃²×M₁₆	—	2	4	no	—
58	C₂×C₃²×C₈	—	2	2	no	—
59	C₂×C₃²×C₈	D₄	1	4	yes	complemented
60	C₂×C₃²×C₈	C₂³	1	2	yes	—
61	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
62	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
63	C₃²×C₄⋊C₄	—	2	4	no	—
64	C₃²×C₄⋊C₄	C₂³	1	4	yes	—
65	C₃²×C₂²⋊C₄	—	2	2	no	—
66	C₃²×C₂²⋊C₄	—	2	4	no	—
67	C₃²×C₂²⋊C₄	—	2	4	no	—
68	C₃²×C₄²	—	2	2	no	—
69	#3054	C₃²	1	1	yes	characteristic, direct product factor, Sylow
70	C₂×C₃×#41	C₂²×C₃	1	8	yes	—
71	C₂³×C₃×C₄	C₂²×C₃	1	4	yes	—
72	C₃×#135	—	2	16	no	—
73	C₂×C₃×Q₁₆	C₂²×C₃	1	4	yes	complemented
74	C₂×C₃×SD₁₆	C₂²×C₃	1	8	yes	complemented
75	C₂×C₃×D₈	C₂²×C₃	1	4	yes	complemented
76	C₂×C₃×M₁₆	—	2	16	no	—
77	C₂×C₃×M₁₆	C₂²×C₃	1	8	yes	—
78	C₂×C₃×M₁₆	C₂²×C₃	1	8	yes	—
79	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
80	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
81	C₃×#123	—	2	8	no	—
82	C₃×C₂²⋊Q₈	C₂²×C₃	1	16	yes	—
83	C₃×#121	C₂²×C₃	1	16	yes	—
84	C₃×#120	—	2	8	no	—
85	C₃×C₄×D₄	—	2	16	no	—
86	C₃×#117	C₂²×C₃	1	8	yes	—
87	C₃×#107	C₂²×C₃	1	8	yes	complemented
88	C₃×#106	C₂²×C₃	1	8	yes	complemented
89	C₃×#103	C₂²×C₃	1	16	yes	complemented
90	C₃×#102	C₂²×C₃	1	16	yes	complemented
91	C₂³×C₃²	—	2	2	no	—
92	C₂³×C₃²	—	2	2	no	—
93	C₂³×C₃²	—	4	2	no	—
94	C₂³×C₃²	C₂×D₄	1	2	yes	—
95	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
96	C₃²×Q₈	—	2	2	no	—
97	C₃²×Q₈	—	2	2	no	—
98	C₃²×D₄	—	2	2	no	—
99	C₃²×D₄	—	2	2	no	—
100	C₃²×D₄	—	4	8	no	—
101	C₃²×D₄	—	4	4	no	—
102	C₂×C₃²×C₄	—	2	2	no	—
103	C₂×C₃²×C₄	—	2	4	no	—
104	C₂×C₃²×C₄	—	2	2	no	—
105	C₂×C₃²×C₄	—	2	2	no	—
106	C₂×C₃²×C₄	—	2	4	no	—
107	C₂×C₃²×C₄	—	2	2	no	—
108	C₂×C₃²×C₄	—	4	4	no	—
109	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
110	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
111	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
112	C₂×C₃²×C₄	#41	1	2	yes	—
113	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
114	C₃²×C₈	—	2	2	no	—
115	C₃²×C₈	C₂×D₄	1	2	yes	—
116	C₃²×C₈	C₂×D₄	1	2	yes	—
117	C₂×#135	C₂×C₃²	1	1	yes	characteristic, complemented
118	C₂²×M₁₆	C₂×C₃²	1	1	yes	characteristic, complemented
119	#525	C₂×C₃²	1	2	yes	complemented
120	#470	C₂×C₃²	1	2	yes	complemented
121	#469	C₂×C₃²	1	2	yes	complemented
122	#468	C₂×C₃²	1	4	yes	complemented
123	#427	C₂×C₃²	1	1	yes	characteristic, complemented
124	#416	C₂×C₃²	1	2	yes	complemented
125	C₂⁴×C₃	C₃×D₄	1	4	yes	—
126	C₃×#41	—	2	16	no	—
127	C₃×#41	—	2	16	no	—
128	C₂×C₃×Q₈	C₂³×C₃	1	8	yes	—
129	C₂×C₃×D₄	—	2	16	no	—
130	C₂×C₃×D₄	—	2	8	no	—
131	C₂×C₃×D₄	C₂³×C₃	1	8	yes	—
132	C₂²×C₃×C₄	—	2	8	no	—
133	C₂²×C₃×C₄	—	2	8	no	—
134	C₂²×C₃×C₄	—	2	8	no	—
135	C₂²×C₃×C₄	C₃×D₄	1	8	yes	—
136	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
137	C₂²×C₃×C₄	C₂³×C₃	1	8	yes	—
138	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
139	C₃×Q₁₆	—	2	8	no	—
140	C₃×SD₁₆	—	2	16	no	—
141	C₃×D₈	—	2	8	no	—
142	C₃×M₁₆	—	2	16	no	—
143	C₃×M₁₆	—	2	16	no	—
144	C₂×C₃×C₈	—	2	8	no	—
145	C₂×C₃×C₈	C₃×D₄	1	16	yes	complemented
146	C₂×C₃×C₈	C₂³×C₃	1	8	yes	—
147	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
148	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
149	C₃×C₄⋊C₄	—	2	16	no	—
150	C₃×C₄⋊C₄	C₂³×C₃	1	16	yes	—
151	C₃×C₂²⋊C₄	—	2	8	no	—
152	C₃×C₂²⋊C₄	—	2	16	no	—
153	C₃×C₂²⋊C₄	—	2	16	no	—
154	C₃×C₄²	—	2	8	no	—
155	C₂²×C₃²	—	2	2	no	—
156	C₂²×C₃²	—	2	2	no	—
157	C₂²×C₃²	—	2	4	no	—
158	C₂²×C₃²	—	4	4	no	—
159	C₂²×C₃²	—	4	2	no	—
160	C₂²×C₃²	—	4	4	no	—
161	C₂²×C₃²	#121	1	2	yes	—
162	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
163	C₃²×C₄	—	2	2	no	—
164	C₃²×C₄	—	4	2	no	—
165	C₃²×C₄	—	4	4	no	—
166	C₃²×C₄	#121	1	2	yes	—
167	C₃²×C₄	C₂²×D₄	1	1	yes	characteristic
168	C₃²×C₄	C₂×#41	1	1	yes	characteristic
169	C₂×#41	C₂²×C₃²	1	2	yes	—
170	C₂³×C₄	C₂²×C₃²	1	1	yes	characteristic
171	#135	—	2	4	no	—
172	C₂×Q₁₆	C₂²×C₃²	1	1	yes	characteristic, complemented
173	C₂×SD₁₆	C₂²×C₃²	1	2	yes	complemented
174	C₂×D₈	C₂²×C₃²	1	1	yes	characteristic, complemented
175	C₂×M₁₆	—	2	4	no	—
176	C₂×M₁₆	C₂²×C₃²	1	2	yes	—
177	C₂×M₁₆	C₂²×C₃²	1	2	yes	—
178	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
179	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
180	#123	—	2	2	no	—
181	C₂²⋊Q₈	C₂²×C₃²	1	4	yes	—
182	#121	C₂²×C₃²	1	4	yes	—
183	#120	—	2	2	no	—
184	C₄×D₄	—	2	4	no	—
185	#117	C₂²×C₃²	1	2	yes	—
186	#107	C₂²×C₃²	1	2	yes	complemented
187	#106	C₂²×C₃²	1	2	yes	complemented
188	#103	C₂²×C₃²	1	4	yes	complemented
189	#102	C₂²×C₃²	1	4	yes	complemented
190	C₂³×C₃	—	2	8	no	—
191	C₂³×C₃	—	2	8	no	—
192	C₂³×C₃	—	4	8	no	—
193	C₂³×C₃	C₂×C₃×D₄	1	8	yes	—
194	C₂³×C₃	C₂×C₃×D₄	1	4	yes	—
195	C₃×Q₈	—	2	8	no	—
196	C₃×Q₈	—	2	8	no	—
197	C₃×D₄	—	2	8	no	—
198	C₃×D₄	—	2	8	no	—
199	C₃×D₄	—	4	32	no	—
200	C₃×D₄	—	4	16	no	—
201	C₂×C₃×C₄	—	2	8	no	—
202	C₂×C₃×C₄	—	2	16	no	—
203	C₂×C₃×C₄	—	2	8	no	—
204	C₂×C₃×C₄	—	2	8	no	—
205	C₂×C₃×C₄	—	2	16	no	—
206	C₂×C₃×C₄	—	2	8	no	—
207	C₂×C₃×C₄	—	4	16	no	—
208	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
209	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
210	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
211	C₂×C₃×C₄	C₃×#41	1	8	yes	—
212	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
213	C₃×C₈	—	2	8	no	—
214	C₃×C₈	C₂×C₃×D₄	1	8	yes	—
215	C₃×C₈	C₂×C₃×D₄	1	8	yes	—
216	C₂×C₃²	—	2	2	no	—
217	C₂×C₃²	—	4	2	no	—
218	C₂×C₃²	—	8	2	no	—
219	C₂×C₃²	C₂×#121	1	1	yes	characteristic
220	C₂×C₃²	#575	1	2	yes	—
221	C₂⁴	C₃²×D₄	1	1	yes	characteristic
222	#41	—	2	4	no	—
223	#41	—	2	4	no	—
224	C₂×Q₈	C₂³×C₃²	1	2	yes	—
225	C₂×D₄	—	2	4	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	C₂³×C₃²	1	2	yes	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	C₃²×D₄	1	2	yes	—
232	C₂²×C₄	C₃²×D₄	1	1	yes	characteristic
233	C₂²×C₄	C₂³×C₃²	1	2	yes	—
234	C₂²×C₄	C₂³×C₃²	1	1	yes	characteristic
235	Q₁₆	—	2	2	no	—
236	SD₁₆	—	2	4	no	—
237	D₈	—	2	2	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	2	4	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	C₃²×D₄	1	4	yes	complemented
242	C₂×C₈	C₂³×C₃²	1	2	yes	—
243	C₂×C₈	C₂³×C₃²	1	1	yes	characteristic
244	C₂×C₈	C₂³×C₃²	1	1	yes	characteristic
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	C₂³×C₃²	1	4	yes	—
247	C₂²⋊C₄	—	2	2	no	—
248	C₂²⋊C₄	—	2	4	no	—
249	C₂²⋊C₄	—	2	4	no	—
250	C₄²	—	2	2	no	—
251	C₂²×C₃	—	2	8	no	—
252	C₂²×C₃	—	2	16	no	—
253	C₂²×C₃	—	2	8	no	—
254	C₂²×C₃	—	4	16	no	—
255	C₂²×C₃	—	4	16	no	—
256	C₂²×C₃	—	4	8	no	—
257	C₂²×C₃	C₃×#121	1	8	yes	—
258	C₂²×C₃	C₂²×C₃×D₄	1	4	yes	—
259	C₃×C₄	—	2	8	no	—
260	C₃×C₄	—	4	16	no	—
261	C₃×C₄	—	4	8	no	—
262	C₃×C₄	C₃×#121	1	8	yes	—
263	C₃×C₄	C₂²×C₃×D₄	1	4	yes	—
264	C₃×C₄	C₂×C₃×#41	1	4	yes	—
265	C₃²	#3054	1	1	yes	characteristic, direct product factor, Sylow
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	4	2	no	—
269	C₂³	C₂×C₃²×D₄	1	2	yes	—
270	C₂³	C₂×C₃²×D₄	1	1	yes	characteristic
271	Q₈	—	2	2	no	—
272	Q₈	—	2	2	no	—
273	D₄	—	2	2	no	—
274	D₄	—	2	2	no	—
275	D₄	—	4	8	no	—
276	D₄	—	4	4	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
285	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
286	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
287	C₂×C₄	C₃²×#41	1	2	yes	—
288	C₂×C₄	C₂⁴×C₃²	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	C₂×C₃²×D₄	1	2	yes	—
291	C₈	C₂×C₃²×D₄	1	2	yes	—
292	C₂×C₃	—	2	8	no	—
293	C₂×C₃	—	4	8	no	—
294	C₂×C₃	—	8	8	no	—
295	C₂×C₃	C₂×C₃×#121	1	4	yes	—
296	C₂×C₃	C₃×#575	1	8	yes	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	4	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	4	4	no	—
301	C₂²	—	4	4	no	—
302	C₂²	—	4	2	no	—
303	C₂²	C₃²×#121	1	2	yes	—
304	C₂²	C₂²×C₃²×D₄	1	1	yes	characteristic
305	C₄	—	2	2	no	—
306	C₄	—	4	4	no	—
307	C₄	—	4	2	no	—
308	C₄	C₃²×#121	1	2	yes	—
309	C₄	C₂²×C₃²×D₄	1	1	yes	characteristic
310	C₄	C₂×C₃²×#41	1	1	yes	characteristic
311	C₃	C₃×#3054	1	4	yes	direct product factor
312	C₂	—	2	2	no	—
313	C₂	—	4	2	no	—
314	C₂	—	8	2	no	—
315	C₂	C₂×C₃²×#121	1	1	yes	characteristic
316	C₂	C₃²×#575	1	2	yes	—

	subgroup	G/H
center	C₂×C₃²×C₄	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃²
socle	C₂²×C₃²	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₄ ◁ C₂³×C₃²×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×#135	C₂	semidirect
2. C₂²×C₃²×M₁₆	C₂	semidirect
3. C₃²×#525	C₂	semidirect
4. C₃²×#470	C₂	semidirect
5. C₃²×#469	C₂	semidirect
6. C₃²×#468	C₂	semidirect
7. C₃²×#427	C₂	semidirect
8. C₃²×#416	C₂	semidirect
9. C₃×#3054	C₃	direct (N × H)
13. C₂×C₃²×Q₁₆	C₂²	semidirect
14. C₂×C₃²×SD₁₆	C₂²	semidirect
15. C₂×C₃²×D₈	C₂²	semidirect
27. C₃²×#107	C₂²	semidirect
28. C₃²×#106	C₂²	semidirect
29. C₃²×#103	C₂²	semidirect
30. C₃²×#102	C₂²	semidirect
31. C₂×C₃×#135	C₂×C₃	semidirect
32. C₂²×C₃×M₁₆	C₂×C₃	semidirect
33. C₃×#525	C₂×C₃	semidirect
34. C₃×#470	C₂×C₃	semidirect
35. C₃×#469	C₂×C₃	semidirect
36. C₃×#468	C₂×C₃	semidirect
37. C₃×#427	C₂×C₃	semidirect
38. C₃×#416	C₂×C₃	semidirect
59. C₂×C₃²×C₈	D₄	semidirect
69. #3054	C₃²	direct (N × H)
73. C₂×C₃×Q₁₆	C₂²×C₃	semidirect
74. C₂×C₃×SD₁₆	C₂²×C₃	semidirect
75. C₂×C₃×D₈	C₂²×C₃	semidirect
87. C₃×#107	C₂²×C₃	semidirect
88. C₃×#106	C₂²×C₃	semidirect
89. C₃×#103	C₂²×C₃	semidirect
90. C₃×#102	C₂²×C₃	semidirect
117. C₂×#135	C₂×C₃²	semidirect
118. C₂²×M₁₆	C₂×C₃²	semidirect
119. #525	C₂×C₃²	semidirect
120. #470	C₂×C₃²	semidirect
121. #469	C₂×C₃²	semidirect
122. #468	C₂×C₃²	semidirect
123. #427	C₂×C₃²	semidirect
124. #416	C₂×C₃²	semidirect
145. C₂×C₃×C₈	C₃×D₄	semidirect
172. C₂×Q₁₆	C₂²×C₃²	semidirect
173. C₂×SD₁₆	C₂²×C₃²	semidirect
174. C₂×D₈	C₂²×C₃²	semidirect
186. #107	C₂²×C₃²	semidirect
187. #106	C₂²×C₃²	semidirect
188. #103	C₂²×C₃²	semidirect
189. #102	C₂²×C₃²	semidirect
241. C₂×C₈	C₃²×D₄	semidirect
265. C₃²	#3054	direct (N × H)
311. C₃	C₃×#3054	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×#135	2
C₂²×C₃²×M₁₆	2
C₃²×#525	2
C₃²×#470	2
C₃²×#469	2
C₃²×#468	2
C₃²×#427	2
C₃²×#416	2
C₃×#3054	3

Subgroups — C32×#3054 #11770103

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#3054 #11770103