Subgroups — C₂×C₃²×#467 #11770098

Order 1152.

1536 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₃²×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃²×C₂²⋊Q₈	C₂	1	2	yes	maximal
4	C₃²×#467	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃²×#107	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃²×#103	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×#467	C₃	1	4	yes	maximal, direct product factor
8	C₂²×C₃²×Q₈	C₂²	1	2	yes	—
9	C₂³×C₃²×C₄	C₂²	1	1	yes	characteristic
10	C₂×C₃²×Q₁₆	—	2	4	no	—
11	C₂×C₃²×Q₁₆	C₂²	1	4	yes	complemented
12	C₂²×C₃²×C₈	—	2	4	no	—
13	C₂²×C₃²×C₈	C₂²	1	4	yes	—
14	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
15	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
16	C₃²×C₂²⋊Q₈	—	2	4	no	—
17	C₃²×C₂²⋊Q₈	C₂²	1	8	yes	—
18	C₂×C₃²×C₄⋊C₄	—	2	2	no	—
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×#103	C₂²	1	8	yes	complemented
23	C₂²×C₃×Q₁₆	C₂×C₃	1	4	yes	complemented
24	C₂³×C₃×C₈	C₂×C₃	1	4	yes	—
25	C₂×C₃×C₂²⋊Q₈	C₂×C₃	1	8	yes	—
26	C₃×#467	C₂×C₃	1	32	yes	direct product factor
27	C₂×C₃×#107	C₂×C₃	1	4	yes	complemented
28	C₂×C₃×#103	C₂×C₃	1	8	yes	complemented
29	C₂⁴×C₃²	D₄	1	1	yes	characteristic
30	C₂×C₃²×Q₈	—	2	2	no	—
31	C₂×C₃²×Q₈	—	2	4	no	—
32	C₂×C₃²×Q₈	—	2	4	no	—
33	C₂×C₃²×Q₈	C₂³	1	4	yes	—
34	C₂²×C₃²×C₄	—	2	2	no	—
35	C₂²×C₃²×C₄	—	2	2	no	—
36	C₂²×C₃²×C₄	—	2	2	no	—
37	C₂²×C₃²×C₄	—	2	2	no	—
38	C₂²×C₃²×C₄	D₄	1	2	yes	—
39	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
40	C₂²×C₃²×C₄	C₂³	1	2	yes	—
41	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
42	C₃²×Q₁₆	—	2	8	no	—
43	C₂×C₃²×C₈	—	2	2	no	—
44	C₂×C₃²×C₈	—	2	8	no	—
45	C₂×C₃²×C₈	D₄	1	4	yes	—
46	C₂×C₃²×C₈	C₂³	1	2	yes	—
47	C₂×C₃²×C₈	C₂³	1	2	yes	—
48	C₃²×C₄⋊C₄	—	2	8	no	—
49	C₃²×C₄⋊C₄	—	2	2	no	—
50	C₃²×C₄⋊C₄	C₂³	1	4	yes	—
51	C₃²×C₂²⋊C₄	—	2	8	no	—
52	C₂×#467	C₃²	1	1	yes	characteristic, direct product factor, Sylow
53	C₂²×C₃×Q₈	C₂²×C₃	1	8	yes	—
54	C₂³×C₃×C₄	C₂²×C₃	1	4	yes	—
55	C₂×C₃×Q₁₆	—	2	16	no	—
56	C₂×C₃×Q₁₆	C₂²×C₃	1	16	yes	complemented
57	C₂²×C₃×C₈	—	2	16	no	—
58	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
59	C₂²×C₃×C₈	C₂²×C₃	1	16	yes	—
60	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
61	C₃×C₂²⋊Q₈	—	2	16	no	—
62	C₃×C₂²⋊Q₈	C₂²×C₃	1	32	yes	—
63	C₂×C₃×C₄⋊C₄	—	2	8	no	—
64	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	8	yes	—
65	C₂×C₃×C₂²⋊C₄	—	2	8	no	—
66	C₃×#107	C₂²×C₃	1	16	yes	complemented
67	C₃×#103	C₂²×C₃	1	32	yes	complemented
68	C₂³×C₃²	—	2	2	no	—
69	C₂³×C₃²	—	2	2	no	—
70	C₂³×C₃²	Q₁₆	1	4	yes	complemented
71	C₂³×C₃²	C₂×D₄	1	2	yes	—
72	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
73	C₃²×Q₈	—	2	4	no	—
74	C₃²×Q₈	—	2	4	no	—
75	C₃²×Q₈	—	4	4	no	—
76	C₂×C₃²×C₄	—	2	2	no	—
77	C₂×C₃²×C₄	—	2	4	no	—
78	C₂×C₃²×C₄	—	2	2	no	—
79	C₂×C₃²×C₄	—	2	2	no	—
80	C₂×C₃²×C₄	—	2	4	no	—
81	C₂×C₃²×C₄	—	2	4	no	—
82	C₂×C₃²×C₄	—	4	4	no	—
83	C₂×C₃²×C₄	—	4	4	no	—
84	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
85	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
87	C₂×C₃²×C₄	#41	1	2	yes	—
88	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
89	C₃²×C₈	—	2	2	no	—
90	C₃²×C₈	C₂×D₄	1	4	yes	—
91	C₂²×Q₁₆	C₂×C₃²	1	1	yes	characteristic, complemented
92	C₂³×C₈	C₂×C₃²	1	1	yes	characteristic
93	C₂×C₂²⋊Q₈	C₂×C₃²	1	2	yes	—
94	#467	C₂×C₃²	1	8	yes	direct product factor
95	C₂×#107	C₂×C₃²	1	1	yes	characteristic, complemented
96	C₂×#103	C₂×C₃²	1	2	yes	complemented
97	C₂⁴×C₃	C₃×D₄	1	4	yes	—
98	C₂×C₃×Q₈	—	2	8	no	—
99	C₂×C₃×Q₈	—	2	16	no	—
100	C₂×C₃×Q₈	—	2	16	no	—
101	C₂×C₃×Q₈	C₂³×C₃	1	16	yes	—
102	C₂²×C₃×C₄	—	2	8	no	—
103	C₂²×C₃×C₄	—	2	8	no	—
104	C₂²×C₃×C₄	—	2	8	no	—
105	C₂²×C₃×C₄	—	2	8	no	—
106	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
107	C₂²×C₃×C₄	C₃×D₄	1	8	yes	—
108	C₂²×C₃×C₄	C₂³×C₃	1	8	yes	—
109	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
110	C₃×Q₁₆	—	2	32	no	—
111	C₂×C₃×C₈	—	2	8	no	—
112	C₂×C₃×C₈	—	2	32	no	—
113	C₂×C₃×C₈	C₃×D₄	1	16	yes	—
114	C₂×C₃×C₈	C₂³×C₃	1	8	yes	—
115	C₂×C₃×C₈	C₂³×C₃	1	8	yes	—
116	C₃×C₄⋊C₄	—	2	32	no	—
117	C₃×C₄⋊C₄	—	2	8	no	—
118	C₃×C₄⋊C₄	C₂³×C₃	1	16	yes	—
119	C₃×C₂²⋊C₄	—	2	32	no	—
120	C₂²×C₃²	—	2	2	no	—
121	C₂²×C₃²	—	2	2	no	—
122	C₂²×C₃²	—	2	8	no	—
123	C₂²×C₃²	#121	1	2	yes	—
124	C₂²×C₃²	C₂×Q₁₆	1	2	yes	—
125	C₂²×C₃²	C₂×Q₁₆	1	4	yes	complemented
126	C₂²×C₃²	#135	1	2	yes	—
127	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
128	C₃²×C₄	—	2	2	no	—
129	C₃²×C₄	—	4	4	no	—
130	C₃²×C₄	—	4	4	no	—
131	C₃²×C₄	#121	1	2	yes	—
132	C₃²×C₄	C₂²×D₄	1	1	yes	characteristic
133	C₃²×C₄	C₂×#41	1	1	yes	characteristic
134	C₂²×Q₈	C₂²×C₃²	1	2	yes	—
135	C₂³×C₄	C₂²×C₃²	1	1	yes	characteristic
136	C₂×Q₁₆	—	2	4	no	—
137	C₂×Q₁₆	C₂²×C₃²	1	4	yes	complemented
138	C₂²×C₈	—	2	4	no	—
139	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
140	C₂²×C₈	C₂²×C₃²	1	4	yes	—
141	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
142	C₂²⋊Q₈	—	2	4	no	—
143	C₂²⋊Q₈	C₂²×C₃²	1	8	yes	—
144	C₂×C₄⋊C₄	—	2	2	no	—
145	C₂×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
146	C₂×C₂²⋊C₄	—	2	2	no	—
147	#107	C₂²×C₃²	1	4	yes	complemented
148	#103	C₂²×C₃²	1	8	yes	complemented
149	C₂³×C₃	—	2	8	no	—
150	C₂³×C₃	—	2	8	no	—
151	C₂³×C₃	C₃×Q₁₆	1	16	yes	complemented
152	C₂³×C₃	C₂×C₃×D₄	1	8	yes	—
153	C₂³×C₃	C₂×C₃×D₄	1	4	yes	—
154	C₃×Q₈	—	2	16	no	—
155	C₃×Q₈	—	2	16	no	—
156	C₃×Q₈	—	4	16	no	—
157	C₂×C₃×C₄	—	2	8	no	—
158	C₂×C₃×C₄	—	2	16	no	—
159	C₂×C₃×C₄	—	2	8	no	—
160	C₂×C₃×C₄	—	2	8	no	—
161	C₂×C₃×C₄	—	2	16	no	—
162	C₂×C₃×C₄	—	2	16	no	—
163	C₂×C₃×C₄	—	4	16	no	—
164	C₂×C₃×C₄	—	4	16	no	—
165	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
166	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
167	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
168	C₂×C₃×C₄	C₃×#41	1	8	yes	—
169	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
170	C₃×C₈	—	2	8	no	—
171	C₃×C₈	C₂×C₃×D₄	1	16	yes	—
172	C₂×C₃²	—	2	4	no	—
173	C₂×C₃²	#467	1	4	yes	direct product factor
174	C₂×C₃²	C₂×#121	1	1	yes	characteristic
175	C₂×C₃²	C₂²×Q₁₆	1	1	yes	characteristic
176	C₂×C₃²	C₂×#135	1	1	yes	characteristic
177	C₂⁴	C₃²×D₄	1	1	yes	characteristic
178	C₂×Q₈	—	2	2	no	—
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	—	2	4	no	—
181	C₂×Q₈	C₂³×C₃²	1	4	yes	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	C₃²×D₄	1	1	yes	characteristic
187	C₂²×C₄	C₃²×D₄	1	2	yes	—
188	C₂²×C₄	C₂³×C₃²	1	2	yes	—
189	C₂²×C₄	C₂³×C₃²	1	1	yes	characteristic
190	Q₁₆	—	2	8	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	8	no	—
193	C₂×C₈	C₃²×D₄	1	4	yes	—
194	C₂×C₈	C₂³×C₃²	1	2	yes	—
195	C₂×C₈	C₂³×C₃²	1	2	yes	—
196	C₄⋊C₄	—	2	8	no	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	C₂³×C₃²	1	4	yes	—
199	C₂²⋊C₄	—	2	8	no	—
200	C₂²×C₃	—	2	8	no	—
201	C₂²×C₃	—	2	8	no	—
202	C₂²×C₃	—	2	32	no	—
203	C₂²×C₃	C₃×#121	1	8	yes	—
204	C₂²×C₃	C₂×C₃×Q₁₆	1	8	yes	—
205	C₂²×C₃	C₂×C₃×Q₁₆	1	16	yes	complemented
206	C₂²×C₃	C₃×#135	1	8	yes	—
207	C₂²×C₃	C₂²×C₃×D₄	1	4	yes	—
208	C₃×C₄	—	2	8	no	—
209	C₃×C₄	—	4	16	no	—
210	C₃×C₄	—	4	16	no	—
211	C₃×C₄	C₃×#121	1	8	yes	—
212	C₃×C₄	C₂²×C₃×D₄	1	4	yes	—
213	C₃×C₄	C₂×C₃×#41	1	4	yes	—
214	C₃²	C₂×#467	1	1	yes	characteristic, direct product factor, Sylow
215	C₂³	—	2	2	no	—
216	C₂³	—	2	2	no	—
217	C₂³	C₃²×Q₁₆	1	4	yes	complemented
218	C₂³	C₂×C₃²×D₄	1	2	yes	—
219	C₂³	C₂×C₃²×D₄	1	1	yes	characteristic
220	Q₈	—	2	4	no	—
221	Q₈	—	2	4	no	—
222	Q₈	—	4	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
232	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
233	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
234	C₂×C₄	C₃²×#41	1	2	yes	—
235	C₂×C₄	C₂⁴×C₃²	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	C₂×C₃²×D₄	1	4	yes	—
238	C₂×C₃	—	2	16	no	—
239	C₂×C₃	C₃×#467	1	16	yes	direct product factor
240	C₂×C₃	C₂×C₃×#121	1	4	yes	—
241	C₂×C₃	C₂²×C₃×Q₁₆	1	4	yes	—
242	C₂×C₃	C₂×C₃×#135	1	4	yes	—
243	C₂²	—	2	8	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	C₃²×#121	1	2	yes	—
247	C₂²	C₂×C₃²×Q₁₆	1	2	yes	—
248	C₂²	C₂×C₃²×Q₁₆	1	4	yes	complemented
249	C₂²	C₃²×#135	1	2	yes	—
250	C₂²	C₂²×C₃²×D₄	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	4	4	no	—
253	C₄	—	4	4	no	—
254	C₄	C₃²×#121	1	2	yes	—
255	C₄	C₂²×C₃²×D₄	1	1	yes	characteristic
256	C₄	C₂×C₃²×#41	1	1	yes	characteristic
257	C₃	C₂×C₃×#467	1	4	yes	direct product factor
258	C₂	—	2	4	no	—
259	C₂	C₃²×#467	1	4	yes	direct product factor
260	C₂	C₂×C₃²×#121	1	1	yes	characteristic
261	C₂	C₂²×C₃²×Q₁₆	1	1	yes	characteristic
262	C₂	C₂×C₃²×#135	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃²	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃²
prime radical	C₂⁴×C₃²	D₄
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×Q₁₆	C₂	semidirect
4. C₃²×#467	C₂	direct (N × H)
5. C₂×C₃²×#107	C₂	semidirect
6. C₂×C₃²×#103	C₂	semidirect
7. C₂×C₃×#467	C₃	direct (N × H)
11. C₂×C₃²×Q₁₆	C₂²	semidirect
21. C₃²×#107	C₂²	semidirect
22. C₃²×#103	C₂²	semidirect
23. C₂²×C₃×Q₁₆	C₂×C₃	semidirect
26. C₃×#467	C₂×C₃	direct (N × H)
27. C₂×C₃×#107	C₂×C₃	semidirect
28. C₂×C₃×#103	C₂×C₃	semidirect
52. C₂×#467	C₃²	direct (N × H)
56. C₂×C₃×Q₁₆	C₂²×C₃	semidirect
66. C₃×#107	C₂²×C₃	semidirect
67. C₃×#103	C₂²×C₃	semidirect
70. C₂³×C₃²	Q₁₆	semidirect
91. C₂²×Q₁₆	C₂×C₃²	semidirect
94. #467	C₂×C₃²	direct (N × H)
95. C₂×#107	C₂×C₃²	semidirect
96. C₂×#103	C₂×C₃²	semidirect
125. C₂²×C₃²	C₂×Q₁₆	semidirect
137. C₂×Q₁₆	C₂²×C₃²	semidirect
147. #107	C₂²×C₃²	semidirect
148. #103	C₂²×C₃²	semidirect
151. C₂³×C₃	C₃×Q₁₆	semidirect
173. C₂×C₃²	#467	direct (N × H)
205. C₂²×C₃	C₂×C₃×Q₁₆	semidirect
214. C₃²	C₂×#467	direct (N × H)
217. C₂³	C₃²×Q₁₆	semidirect
239. C₂×C₃	C₃×#467	direct (N × H)
248. C₂²	C₂×C₃²×Q₁₆	semidirect
257. C₃	C₂×C₃×#467	direct (N × H)
259. C₂	C₃²×#467	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×Q₁₆	2
C₂³×C₃²×C₈	2
C₂×C₃²×C₂²⋊Q₈	2
C₃²×#467	2
C₂×C₃²×#107	2
C₂×C₃²×#103	2
C₂×C₃×#467	3

Subgroups — C2×C32×#467 #11770098

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×#467 #11770098