Subgroups — C₂×C₃²×#466 #11770097

Order 1152.

1848 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×D₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₃²×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃²×#121	C₂	1	2	yes	maximal, complemented
4	C₃²×#466	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃²×#107	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃²×#102	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×#466	C₃	1	4	yes	maximal, direct product factor
8	C₂²×C₃²×D₄	—	2	2	no	—
9	C₂²×C₃²×D₄	C₂²	1	2	yes	—
10	C₂³×C₃²×C₄	C₂²	1	1	yes	characteristic
11	C₂×C₃²×D₈	—	2	4	no	—
12	C₂×C₃²×D₈	C₂²	1	4	yes	complemented
13	C₂²×C₃²×C₈	—	2	4	no	—
14	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
15	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
16	C₂²×C₃²×C₈	C₂²	1	4	yes	complemented
17	C₃²×#121	—	2	4	no	—
18	C₃²×#121	C₂²	1	8	yes	complemented
19	C₂×C₃²×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₃²×C₂²⋊C₄	—	2	2	no	—
21	C₃²×#107	C₂²	1	4	yes	complemented
22	C₃²×#102	C₂²	1	8	yes	complemented
23	C₂²×C₃×D₈	C₂×C₃	1	4	yes	complemented
24	C₂³×C₃×C₈	C₂×C₃	1	4	yes	complemented
25	C₂×C₃×#121	C₂×C₃	1	8	yes	complemented
26	C₃×#466	C₂×C₃	1	32	yes	direct product factor
27	C₂×C₃×#107	C₂×C₃	1	4	yes	complemented
28	C₂×C₃×#102	C₂×C₃	1	8	yes	complemented
29	C₂⁴×C₃²	—	2	2	no	—
30	C₂⁴×C₃²	D₄	1	1	yes	characteristic
31	C₂×C₃²×D₄	—	2	8	no	—
32	C₂×C₃²×D₄	—	2	4	no	—
33	C₂×C₃²×D₄	—	2	2	no	—
34	C₂×C₃²×D₄	—	2	4	no	—
35	C₂×C₃²×D₄	—	4	8	no	—
36	C₂×C₃²×D₄	C₂³	1	4	yes	—
37	C₂²×C₃²×C₄	—	2	2	no	—
38	C₂²×C₃²×C₄	—	2	2	no	—
39	C₂²×C₃²×C₄	—	2	2	no	—
40	C₂²×C₃²×C₄	D₄	1	2	yes	—
41	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
42	C₂²×C₃²×C₄	C₂³	1	2	yes	—
43	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
44	C₃²×D₈	—	2	8	no	—
45	C₂×C₃²×C₈	—	2	2	no	—
46	C₂×C₃²×C₈	—	2	8	no	—
47	C₂×C₃²×C₈	D₄	1	4	yes	complemented
48	C₂×C₃²×C₈	C₂³	1	2	yes	—
49	C₂×C₃²×C₈	C₂³	1	2	yes	—
50	C₃²×C₄⋊C₄	—	2	2	no	—
51	C₃²×C₄⋊C₄	C₂³	1	4	yes	—
52	C₃²×C₂²⋊C₄	—	2	8	no	—
53	C₂×#466	C₃²	1	1	yes	characteristic, direct product factor, Sylow
54	C₂²×C₃×D₄	—	2	8	no	—
55	C₂²×C₃×D₄	C₂²×C₃	1	8	yes	—
56	C₂³×C₃×C₄	C₂²×C₃	1	4	yes	—
57	C₂×C₃×D₈	—	2	16	no	—
58	C₂×C₃×D₈	C₂²×C₃	1	16	yes	complemented
59	C₂²×C₃×C₈	—	2	16	no	—
60	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
61	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
62	C₂²×C₃×C₈	C₂²×C₃	1	16	yes	complemented
63	C₃×#121	—	2	16	no	—
64	C₃×#121	C₂²×C₃	1	32	yes	complemented
65	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	8	yes	—
66	C₂×C₃×C₂²⋊C₄	—	2	8	no	—
67	C₃×#107	C₂²×C₃	1	16	yes	complemented
68	C₃×#102	C₂²×C₃	1	32	yes	complemented
69	C₂³×C₃²	—	2	2	no	—
70	C₂³×C₃²	—	2	2	no	—
71	C₂³×C₃²	—	2	4	no	—
72	C₂³×C₃²	—	4	4	no	—
73	C₂³×C₃²	—	4	4	no	—
74	C₂³×C₃²	—	4	4	no	—
75	C₂³×C₃²	D₈	1	4	yes	complemented
76	C₂³×C₃²	C₂×D₄	1	2	yes	—
77	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
78	C₃²×D₄	—	2	4	no	—
79	C₃²×D₄	—	2	4	no	—
80	C₃²×D₄	—	4	16	no	—
81	C₃²×D₄	—	4	4	no	—
82	C₂×C₃²×C₄	—	2	2	no	—
83	C₂×C₃²×C₄	—	2	4	no	—
84	C₂×C₃²×C₄	—	2	2	no	—
85	C₂×C₃²×C₄	—	2	2	no	—
86	C₂×C₃²×C₄	—	2	4	no	—
87	C₂×C₃²×C₄	—	4	4	no	—
88	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
89	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
90	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
91	C₂×C₃²×C₄	#41	1	2	yes	—
92	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
93	C₃²×C₈	—	2	2	no	—
94	C₃²×C₈	C₂×D₄	1	4	yes	complemented
95	C₂²×D₈	C₂×C₃²	1	1	yes	characteristic, complemented
96	C₂³×C₈	C₂×C₃²	1	1	yes	characteristic, complemented
97	C₂×#121	C₂×C₃²	1	2	yes	complemented
98	#466	C₂×C₃²	1	8	yes	direct product factor
99	C₂×#107	C₂×C₃²	1	1	yes	characteristic, complemented
100	C₂×#102	C₂×C₃²	1	2	yes	complemented
101	C₂⁴×C₃	—	2	8	no	—
102	C₂⁴×C₃	C₃×D₄	1	4	yes	—
103	C₂×C₃×D₄	—	2	32	no	—
104	C₂×C₃×D₄	—	2	16	no	—
105	C₂×C₃×D₄	—	2	8	no	—
106	C₂×C₃×D₄	—	2	16	no	—
107	C₂×C₃×D₄	—	4	32	no	—
108	C₂×C₃×D₄	C₂³×C₃	1	16	yes	—
109	C₂²×C₃×C₄	—	2	8	no	—
110	C₂²×C₃×C₄	—	2	8	no	—
111	C₂²×C₃×C₄	—	2	8	no	—
112	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
113	C₂²×C₃×C₄	C₃×D₄	1	8	yes	—
114	C₂²×C₃×C₄	C₂³×C₃	1	8	yes	—
115	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
116	C₃×D₈	—	2	32	no	—
117	C₂×C₃×C₈	—	2	8	no	—
118	C₂×C₃×C₈	—	2	32	no	—
119	C₂×C₃×C₈	C₃×D₄	1	16	yes	complemented
120	C₂×C₃×C₈	C₂³×C₃	1	8	yes	—
121	C₂×C₃×C₈	C₂³×C₃	1	8	yes	—
122	C₃×C₄⋊C₄	—	2	8	no	—
123	C₃×C₄⋊C₄	C₂³×C₃	1	16	yes	—
124	C₃×C₂²⋊C₄	—	2	32	no	—
125	C₂²×C₃²	—	2	2	no	—
126	C₂²×C₃²	—	2	2	no	—
127	C₂²×C₃²	—	2	8	no	—
128	C₂²×C₃²	—	4	4	no	—
129	C₂²×C₃²	—	4	4	no	—
130	C₂²×C₃²	—	4	4	no	—
131	C₂²×C₃²	—	8	8	no	—
132	C₂²×C₃²	#121	1	2	yes	—
133	C₂²×C₃²	C₂×D₈	1	2	yes	—
134	C₂²×C₃²	C₂×D₈	1	4	yes	complemented
135	C₂²×C₃²	#135	1	2	yes	—
136	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
137	C₃²×C₄	—	2	2	no	—
138	C₃²×C₄	—	4	4	no	—
139	C₃²×C₄	#121	1	2	yes	—
140	C₃²×C₄	C₂²×D₄	1	1	yes	characteristic
141	C₃²×C₄	C₂×#41	1	1	yes	characteristic
142	C₂²×D₄	—	2	2	no	—
143	C₂²×D₄	C₂²×C₃²	1	2	yes	—
144	C₂³×C₄	C₂²×C₃²	1	1	yes	characteristic
145	C₂×D₈	—	2	4	no	—
146	C₂×D₈	C₂²×C₃²	1	4	yes	complemented
147	C₂²×C₈	—	2	4	no	—
148	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
149	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
150	C₂²×C₈	C₂²×C₃²	1	4	yes	complemented
151	#121	—	2	4	no	—
152	#121	C₂²×C₃²	1	8	yes	complemented
153	C₂×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
154	C₂×C₂²⋊C₄	—	2	2	no	—
155	#107	C₂²×C₃²	1	4	yes	complemented
156	#102	C₂²×C₃²	1	8	yes	complemented
157	C₂³×C₃	—	2	8	no	—
158	C₂³×C₃	—	2	8	no	—
159	C₂³×C₃	—	2	16	no	—
160	C₂³×C₃	—	4	16	no	—
161	C₂³×C₃	—	4	16	no	—
162	C₂³×C₃	—	4	16	no	—
163	C₂³×C₃	C₃×D₈	1	16	yes	complemented
164	C₂³×C₃	C₂×C₃×D₄	1	8	yes	—
165	C₂³×C₃	C₂×C₃×D₄	1	4	yes	—
166	C₃×D₄	—	2	16	no	—
167	C₃×D₄	—	2	16	no	—
168	C₃×D₄	—	4	64	no	—
169	C₃×D₄	—	4	16	no	—
170	C₂×C₃×C₄	—	2	8	no	—
171	C₂×C₃×C₄	—	2	16	no	—
172	C₂×C₃×C₄	—	2	8	no	—
173	C₂×C₃×C₄	—	2	8	no	—
174	C₂×C₃×C₄	—	2	16	no	—
175	C₂×C₃×C₄	—	4	16	no	—
176	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
177	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
178	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
179	C₂×C₃×C₄	C₃×#41	1	8	yes	—
180	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
181	C₃×C₈	—	2	8	no	—
182	C₃×C₈	C₂×C₃×D₄	1	16	yes	complemented
183	C₂×C₃²	—	2	4	no	—
184	C₂×C₃²	—	8	4	no	—
185	C₂×C₃²	#466	1	4	yes	direct product factor
186	C₂×C₃²	C₂×#121	1	1	yes	characteristic
187	C₂×C₃²	C₂²×D₈	1	1	yes	characteristic
188	C₂×C₃²	C₂×#135	1	1	yes	characteristic
189	C₂⁴	—	2	2	no	—
190	C₂⁴	C₃²×D₄	1	1	yes	characteristic
191	C₂×D₄	—	2	8	no	—
192	C₂×D₄	—	2	4	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	2	4	no	—
195	C₂×D₄	—	4	8	no	—
196	C₂×D₄	C₂³×C₃²	1	4	yes	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	C₃²×D₄	1	1	yes	characteristic
201	C₂²×C₄	C₃²×D₄	1	2	yes	—
202	C₂²×C₄	C₂³×C₃²	1	2	yes	—
203	C₂²×C₄	C₂³×C₃²	1	1	yes	characteristic
204	D₈	—	2	8	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	8	no	—
207	C₂×C₈	C₃²×D₄	1	4	yes	complemented
208	C₂×C₈	C₂³×C₃²	1	2	yes	—
209	C₂×C₈	C₂³×C₃²	1	2	yes	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	C₂³×C₃²	1	4	yes	—
212	C₂²⋊C₄	—	2	8	no	—
213	C₂²×C₃	—	2	8	no	—
214	C₂²×C₃	—	2	32	no	—
215	C₂²×C₃	—	2	8	no	—
216	C₂²×C₃	—	4	16	no	—
217	C₂²×C₃	—	4	16	no	—
218	C₂²×C₃	—	4	16	no	—
219	C₂²×C₃	—	8	32	no	—
220	C₂²×C₃	C₃×#121	1	8	yes	—
221	C₂²×C₃	C₂×C₃×D₈	1	8	yes	—
222	C₂²×C₃	C₂×C₃×D₈	1	16	yes	complemented
223	C₂²×C₃	C₃×#135	1	8	yes	—
224	C₂²×C₃	C₂²×C₃×D₄	1	4	yes	—
225	C₃×C₄	—	2	8	no	—
226	C₃×C₄	—	4	16	no	—
227	C₃×C₄	C₃×#121	1	8	yes	—
228	C₃×C₄	C₂²×C₃×D₄	1	4	yes	—
229	C₃×C₄	C₂×C₃×#41	1	4	yes	—
230	C₃²	C₂×#466	1	1	yes	characteristic, direct product factor, Sylow
231	C₂³	—	2	2	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	4	no	—
234	C₂³	—	4	4	no	—
235	C₂³	—	4	4	no	—
236	C₂³	—	4	4	no	—
237	C₂³	C₃²×D₈	1	4	yes	complemented
238	C₂³	C₂×C₃²×D₄	1	2	yes	—
239	C₂³	C₂×C₃²×D₄	1	1	yes	characteristic
240	D₄	—	2	4	no	—
241	D₄	—	2	4	no	—
242	D₄	—	4	16	no	—
243	D₄	—	4	4	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
251	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
252	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
253	C₂×C₄	C₃²×#41	1	2	yes	—
254	C₂×C₄	C₂⁴×C₃²	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	C₂×C₃²×D₄	1	4	yes	complemented
257	C₂×C₃	—	2	16	no	—
258	C₂×C₃	—	8	16	no	—
259	C₂×C₃	C₃×#466	1	16	yes	direct product factor
260	C₂×C₃	C₂×C₃×#121	1	4	yes	—
261	C₂×C₃	C₂²×C₃×D₈	1	4	yes	—
262	C₂×C₃	C₂×C₃×#135	1	4	yes	—
263	C₂²	—	2	8	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	4	4	no	—
267	C₂²	—	4	4	no	—
268	C₂²	—	4	4	no	—
269	C₂²	—	8	8	no	—
270	C₂²	C₃²×#121	1	2	yes	—
271	C₂²	C₂×C₃²×D₈	1	2	yes	—
272	C₂²	C₂×C₃²×D₈	1	4	yes	complemented
273	C₂²	C₃²×#135	1	2	yes	—
274	C₂²	C₂²×C₃²×D₄	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	4	4	no	—
277	C₄	C₃²×#121	1	2	yes	—
278	C₄	C₂²×C₃²×D₄	1	1	yes	characteristic
279	C₄	C₂×C₃²×#41	1	1	yes	characteristic
280	C₃	C₂×C₃×#466	1	4	yes	direct product factor
281	C₂	—	2	4	no	—
282	C₂	—	8	4	no	—
283	C₂	C₃²×#466	1	4	yes	direct product factor
284	C₂	C₂×C₃²×#121	1	1	yes	characteristic
285	C₂	C₂²×C₃²×D₈	1	1	yes	characteristic
286	C₂	C₂×C₃²×#135	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃²×D₈	C₂	semidirect
2. C₂³×C₃²×C₈	C₂	semidirect
3. C₂×C₃²×#121	C₂	semidirect
4. C₃²×#466	C₂	direct (N × H)
5. C₂×C₃²×#107	C₂	semidirect
6. C₂×C₃²×#102	C₂	semidirect
7. C₂×C₃×#466	C₃	direct (N × H)
12. C₂×C₃²×D₈	C₂²	semidirect
16. C₂²×C₃²×C₈	C₂²	semidirect
18. C₃²×#121	C₂²	semidirect
21. C₃²×#107	C₂²	semidirect
22. C₃²×#102	C₂²	semidirect
23. C₂²×C₃×D₈	C₂×C₃	semidirect
24. C₂³×C₃×C₈	C₂×C₃	semidirect
25. C₂×C₃×#121	C₂×C₃	semidirect
26. C₃×#466	C₂×C₃	direct (N × H)
27. C₂×C₃×#107	C₂×C₃	semidirect
28. C₂×C₃×#102	C₂×C₃	semidirect
47. C₂×C₃²×C₈	D₄	semidirect
53. C₂×#466	C₃²	direct (N × H)
58. C₂×C₃×D₈	C₂²×C₃	semidirect
62. C₂²×C₃×C₈	C₂²×C₃	semidirect
64. C₃×#121	C₂²×C₃	semidirect
67. C₃×#107	C₂²×C₃	semidirect
68. C₃×#102	C₂²×C₃	semidirect
75. C₂³×C₃²	D₈	semidirect
94. C₃²×C₈	C₂×D₄	semidirect
95. C₂²×D₈	C₂×C₃²	semidirect
96. C₂³×C₈	C₂×C₃²	semidirect
97. C₂×#121	C₂×C₃²	semidirect
98. #466	C₂×C₃²	direct (N × H)
99. C₂×#107	C₂×C₃²	semidirect
100. C₂×#102	C₂×C₃²	semidirect
119. C₂×C₃×C₈	C₃×D₄	semidirect
134. C₂²×C₃²	C₂×D₈	semidirect
146. C₂×D₈	C₂²×C₃²	semidirect
150. C₂²×C₈	C₂²×C₃²	semidirect
152. #121	C₂²×C₃²	semidirect
155. #107	C₂²×C₃²	semidirect
156. #102	C₂²×C₃²	semidirect
163. C₂³×C₃	C₃×D₈	semidirect
182. C₃×C₈	C₂×C₃×D₄	semidirect
185. C₂×C₃²	#466	direct (N × H)
207. C₂×C₈	C₃²×D₄	semidirect
222. C₂²×C₃	C₂×C₃×D₈	semidirect
230. C₃²	C₂×#466	direct (N × H)
237. C₂³	C₃²×D₈	semidirect
256. C₈	C₂×C₃²×D₄	semidirect
259. C₂×C₃	C₃×#466	direct (N × H)
272. C₂²	C₂×C₃²×D₈	semidirect
280. C₃	C₂×C₃×#466	direct (N × H)
283. C₂	C₃²×#466	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃²×D₈	2
C₂³×C₃²×C₈	2
C₂×C₃²×#121	2
C₃²×#466	2
C₂×C₃²×#107	2
C₂×C₃²×#102	2
C₂×C₃×#466	3

Subgroups — C2×C32×#466 #11770097

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×#466 #11770097