Subgroups — C₂×C₃²×#462 #11770082

Order 1152.

1440 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	2	yes	maximal
2	C₂×C₃²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃²×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
4	C₃²×#462	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃²×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃²×#103	C₂	1	2	yes	maximal
7	C₂×C₃×#462	C₃	1	4	yes	maximal, direct product factor
8	C₂²×C₃²×Q₈	C₂²	1	2	yes	—
9	C₂²×C₃²×Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₃²×Q₁₆	—	2	8	no	—
11	C₂×C₃²×Q₁₆	C₂²	1	8	yes	—
12	C₂²×C₃²×C₈	C₂²	1	2	yes	—
13	C₃²×C₄⋊Q₈	—	2	2	no	—
14	C₃²×C₄⋊Q₈	C₂²	1	4	yes	—
15	C₃²×C₄×Q₈	—	2	2	no	—
16	C₃²×C₄×Q₈	C₂²	1	4	yes	—
17	C₂×C₃²×C₄⋊C₄	—	2	1	no	—
18	C₂×C₃²×C₄⋊C₄	—	2	1	no	—
19	C₂×C₃²×C₄⋊C₄	—	2	1	no	—
20	C₂×C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₃²×C₄²	—	2	1	no	—
22	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
23	C₃²×C₄⋊C₈	C₂²	1	4	yes	—
24	C₃²×#103	C₂²	1	8	yes	—
25	C₂²×C₃×Q₁₆	C₂×C₃	1	8	yes	—
26	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	4	yes	—
27	C₂×C₃×C₄×Q₈	C₂×C₃	1	4	yes	—
28	C₃×#462	C₂×C₃	1	32	yes	direct product factor
29	C₂×C₃×C₄⋊C₈	C₂×C₃	1	4	yes	—
30	C₂×C₃×#103	C₂×C₃	1	8	yes	—
31	C₂×C₃²×Q₈	—	2	1	no	—
32	C₂×C₃²×Q₈	—	2	2	no	—
33	C₂×C₃²×Q₈	—	2	2	no	—
34	C₂×C₃²×Q₈	—	2	4	no	—
35	C₂×C₃²×Q₈	—	2	4	no	—
36	C₂×C₃²×Q₈	D₄	1	4	yes	—
37	C₂×C₃²×Q₈	C₂³	1	4	yes	—
38	C₂×C₃²×Q₈	C₂³	1	2	yes	—
39	C₂²×C₃²×C₄	—	2	1	no	—
40	C₂²×C₃²×C₄	—	2	2	no	—
41	C₂²×C₃²×C₄	—	2	1	no	—
42	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
43	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
44	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
45	C₃²×Q₁₆	—	2	16	no	—
46	C₂×C₃²×C₈	—	2	4	no	—
47	C₂×C₃²×C₈	C₂³	1	4	yes	—
48	C₃²×C₄⋊C₄	—	2	4	no	—
49	C₃²×C₄⋊C₄	—	2	4	no	—
50	C₃²×C₄⋊C₄	—	2	4	no	—
51	C₃²×C₄⋊C₄	—	2	1	no	—
52	C₃²×C₄⋊C₄	C₂³	1	2	yes	—
53	C₃²×C₄²	—	2	4	no	—
54	C₃²×C₄²	D₄	1	2	yes	—
55	C₃²×C₄²	C₂³	1	2	yes	—
56	C₂×#462	C₃²	1	1	yes	characteristic, direct product factor, Sylow
57	C₂²×C₃×Q₈	C₂²×C₃	1	8	yes	—
58	C₂²×C₃×Q₈	C₂²×C₃	1	4	yes	—
59	C₂×C₃×Q₁₆	—	2	32	no	—
60	C₂×C₃×Q₁₆	C₂²×C₃	1	32	yes	—
61	C₂²×C₃×C₈	C₂²×C₃	1	8	yes	—
62	C₃×C₄⋊Q₈	—	2	8	no	—
63	C₃×C₄⋊Q₈	C₂²×C₃	1	16	yes	—
64	C₃×C₄×Q₈	—	2	8	no	—
65	C₃×C₄×Q₈	C₂²×C₃	1	16	yes	—
66	C₂×C₃×C₄⋊C₄	—	2	4	no	—
67	C₂×C₃×C₄⋊C₄	—	2	4	no	—
68	C₂×C₃×C₄⋊C₄	—	2	4	no	—
69	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
70	C₂×C₃×C₄²	—	2	4	no	—
71	C₂×C₃×C₄²	C₂²×C₃	1	4	yes	—
72	C₃×C₄⋊C₈	C₂²×C₃	1	16	yes	—
73	C₃×#103	C₂²×C₃	1	32	yes	—
74	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
75	C₃²×Q₈	—	2	4	no	—
76	C₃²×Q₈	—	2	2	no	—
77	C₃²×Q₈	—	2	4	no	—
78	C₃²×Q₈	—	2	4	no	—
79	C₃²×Q₈	—	4	4	no	—
80	C₃²×Q₈	C₂×D₄	1	4	yes	—
81	C₂×C₃²×C₄	—	2	4	no	—
82	C₂×C₃²×C₄	—	2	2	no	—
83	C₂×C₃²×C₄	—	2	2	no	—
84	C₂×C₃²×C₄	—	2	2	no	—
85	C₂×C₃²×C₄	—	2	4	no	—
86	C₂×C₃²×C₄	—	4	4	no	—
87	C₂×C₃²×C₄	—	4	2	no	—
88	C₂×C₃²×C₄	Q₁₆	1	4	yes	complemented
89	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
90	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
91	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
92	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
93	C₂×C₃²×C₄	#41	1	2	yes	—
94	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
95	C₃²×C₈	—	2	4	no	—
96	C₂²×Q₁₆	C₂×C₃²	1	2	yes	—
97	C₂×C₄⋊Q₈	C₂×C₃²	1	1	yes	characteristic
98	C₂×C₄×Q₈	C₂×C₃²	1	1	yes	characteristic
99	#462	C₂×C₃²	1	8	yes	direct product factor
100	C₂×C₄⋊C₈	C₂×C₃²	1	1	yes	characteristic
101	C₂×#103	C₂×C₃²	1	2	yes	—
102	C₂×C₃×Q₈	—	2	4	no	—
103	C₂×C₃×Q₈	—	2	8	no	—
104	C₂×C₃×Q₈	—	2	8	no	—
105	C₂×C₃×Q₈	—	2	16	no	—
106	C₂×C₃×Q₈	—	2	16	no	—
107	C₂×C₃×Q₈	C₃×D₄	1	16	yes	—
108	C₂×C₃×Q₈	C₂³×C₃	1	16	yes	—
109	C₂×C₃×Q₈	C₂³×C₃	1	8	yes	—
110	C₂²×C₃×C₄	—	2	4	no	—
111	C₂²×C₃×C₄	—	2	4	no	—
112	C₂²×C₃×C₄	—	2	8	no	—
113	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
114	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
115	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
116	C₃×Q₁₆	—	2	64	no	—
117	C₂×C₃×C₈	—	2	16	no	—
118	C₂×C₃×C₈	C₂³×C₃	1	16	yes	—
119	C₃×C₄⋊C₄	—	2	16	no	—
120	C₃×C₄⋊C₄	—	2	16	no	—
121	C₃×C₄⋊C₄	—	2	16	no	—
122	C₃×C₄⋊C₄	—	2	4	no	—
123	C₃×C₄⋊C₄	C₂³×C₃	1	8	yes	—
124	C₃×C₄²	—	2	16	no	—
125	C₃×C₄²	C₃×D₄	1	8	yes	—
126	C₃×C₄²	C₂³×C₃	1	8	yes	—
127	C₂²×C₃²	#121	1	2	yes	—
128	C₂²×C₃²	C₂×Q₁₆	1	2	yes	—
129	C₂²×C₃²	#137	1	2	yes	—
130	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
131	C₃²×C₄	—	2	4	no	—
132	C₃²×C₄	—	2	2	no	—
133	C₃²×C₄	—	4	4	no	—
134	C₃²×C₄	—	4	2	no	—
135	C₃²×C₄	#121	1	2	yes	—
136	C₃²×C₄	C₂×Q₁₆	1	4	yes	complemented
137	C₃²×C₄	C₂²×D₄	1	1	yes	characteristic
138	C₃²×C₄	C₂×#41	1	1	yes	characteristic
139	C₂²×Q₈	C₂²×C₃²	1	2	yes	—
140	C₂²×Q₈	C₂²×C₃²	1	1	yes	characteristic
141	C₂×Q₁₆	—	2	8	no	—
142	C₂×Q₁₆	C₂²×C₃²	1	8	yes	—
143	C₂²×C₈	C₂²×C₃²	1	2	yes	—
144	C₄⋊Q₈	—	2	2	no	—
145	C₄⋊Q₈	C₂²×C₃²	1	4	yes	—
146	C₄×Q₈	—	2	2	no	—
147	C₄×Q₈	C₂²×C₃²	1	4	yes	—
148	C₂×C₄⋊C₄	—	2	1	no	—
149	C₂×C₄⋊C₄	—	2	1	no	—
150	C₂×C₄⋊C₄	—	2	1	no	—
151	C₂×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
152	C₂×C₄²	—	2	1	no	—
153	C₂×C₄²	C₂²×C₃²	1	1	yes	characteristic
154	C₄⋊C₈	C₂²×C₃²	1	4	yes	—
155	#103	C₂²×C₃²	1	8	yes	—
156	C₂³×C₃	C₂×C₃×D₄	1	4	yes	—
157	C₃×Q₈	—	2	16	no	—
158	C₃×Q₈	—	2	8	no	—
159	C₃×Q₈	—	2	16	no	—
160	C₃×Q₈	—	2	16	no	—
161	C₃×Q₈	—	4	16	no	—
162	C₃×Q₈	C₂×C₃×D₄	1	16	yes	—
163	C₂×C₃×C₄	—	2	16	no	—
164	C₂×C₃×C₄	—	2	8	no	—
165	C₂×C₃×C₄	—	2	8	no	—
166	C₂×C₃×C₄	—	2	8	no	—
167	C₂×C₃×C₄	—	2	16	no	—
168	C₂×C₃×C₄	—	4	16	no	—
169	C₂×C₃×C₄	—	4	8	no	—
170	C₂×C₃×C₄	C₃×Q₁₆	1	16	yes	complemented
171	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
172	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
173	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
174	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
175	C₂×C₃×C₄	C₃×#41	1	8	yes	—
176	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
177	C₃×C₈	—	2	16	no	—
178	C₂×C₃²	#462	1	4	yes	direct product factor
179	C₂×C₃²	C₂×#121	1	1	yes	characteristic
180	C₂×C₃²	C₂²×Q₁₆	1	1	yes	characteristic
181	C₂×C₃²	C₂×#137	1	1	yes	characteristic
182	C₂×Q₈	—	2	1	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×Q₈	—	2	2	no	—
185	C₂×Q₈	—	2	4	no	—
186	C₂×Q₈	—	2	4	no	—
187	C₂×Q₈	C₃²×D₄	1	4	yes	—
188	C₂×Q₈	C₂³×C₃²	1	4	yes	—
189	C₂×Q₈	C₂³×C₃²	1	2	yes	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	C₃²×D₄	1	1	yes	characteristic
194	C₂²×C₄	C₃²×D₄	1	1	yes	characteristic
195	C₂²×C₄	C₂³×C₃²	1	1	yes	characteristic
196	Q₁₆	—	2	16	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	C₂³×C₃²	1	4	yes	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	2	4	no	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	—	2	1	no	—
203	C₄⋊C₄	C₂³×C₃²	1	2	yes	—
204	C₄²	—	2	4	no	—
205	C₄²	C₃²×D₄	1	2	yes	—
206	C₄²	C₂³×C₃²	1	2	yes	—
207	C₂²×C₃	C₃×#121	1	8	yes	—
208	C₂²×C₃	C₂×C₃×Q₁₆	1	8	yes	—
209	C₂²×C₃	C₃×#137	1	8	yes	—
210	C₂²×C₃	C₂²×C₃×D₄	1	4	yes	—
211	C₃×C₄	—	2	8	no	—
212	C₃×C₄	—	2	16	no	—
213	C₃×C₄	—	4	8	no	—
214	C₃×C₄	—	4	16	no	—
215	C₃×C₄	C₃×#121	1	8	yes	—
216	C₃×C₄	C₂×C₃×Q₁₆	1	16	yes	complemented
217	C₃×C₄	C₂²×C₃×D₄	1	4	yes	—
218	C₃×C₄	C₂×C₃×#41	1	4	yes	—
219	C₃²	C₂×#462	1	1	yes	characteristic, direct product factor, Sylow
220	C₂³	C₂×C₃²×D₄	1	1	yes	characteristic
221	Q₈	—	2	2	no	—
222	Q₈	—	2	4	no	—
223	Q₈	—	2	4	no	—
224	Q₈	—	2	4	no	—
225	Q₈	—	4	4	no	—
226	Q₈	C₂×C₃²×D₄	1	4	yes	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	C₃²×Q₁₆	1	4	yes	complemented
235	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
236	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
237	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
238	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
239	C₂×C₄	C₃²×#41	1	2	yes	—
240	C₂×C₄	C₂⁴×C₃²	1	1	yes	characteristic
241	C₈	—	2	4	no	—
242	C₂×C₃	C₃×#462	1	16	yes	direct product factor
243	C₂×C₃	C₂×C₃×#121	1	4	yes	—
244	C₂×C₃	C₂²×C₃×Q₁₆	1	4	yes	—
245	C₂×C₃	C₂×C₃×#137	1	4	yes	—
246	C₂²	C₃²×#121	1	2	yes	—
247	C₂²	C₂×C₃²×Q₁₆	1	2	yes	—
248	C₂²	C₃²×#137	1	2	yes	—
249	C₂²	C₂²×C₃²×D₄	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	4	no	—
252	C₄	—	4	2	no	—
253	C₄	—	4	4	no	—
254	C₄	C₃²×#121	1	2	yes	—
255	C₄	C₂×C₃²×Q₁₆	1	4	yes	complemented
256	C₄	C₂²×C₃²×D₄	1	1	yes	characteristic
257	C₄	C₂×C₃²×#41	1	1	yes	characteristic
258	C₃	C₂×C₃×#462	1	4	yes	direct product factor
259	C₂	C₃²×#462	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×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃²	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃² ◁ C₂×C₃²×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃²×#462	C₂	direct (N × H)
7. C₂×C₃×#462	C₃	direct (N × H)
28. C₃×#462	C₂×C₃	direct (N × H)
56. C₂×#462	C₃²	direct (N × H)
88. C₂×C₃²×C₄	Q₁₆	semidirect
99. #462	C₂×C₃²	direct (N × H)
136. C₃²×C₄	C₂×Q₁₆	semidirect
170. C₂×C₃×C₄	C₃×Q₁₆	semidirect
178. C₂×C₃²	#462	direct (N × H)
216. C₃×C₄	C₂×C₃×Q₁₆	semidirect
219. C₃²	C₂×#462	direct (N × H)
234. C₂×C₄	C₃²×Q₁₆	semidirect
242. C₂×C₃	C₃×#462	direct (N × H)
255. C₄	C₂×C₃²×Q₁₆	semidirect
258. C₃	C₂×C₃×#462	direct (N × H)
259. C₂	C₃²×#462	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
C₂×C₃²×C₄⋊Q₈	2
C₂×C₃²×C₄×Q₈	2
C₃²×#462	2
C₂×C₃²×C₄⋊C₈	2
C₂×C₃²×#103	2
C₂×C₃×#462	3

Subgroups — C2×C32×#462 #11770082

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×#462 #11770082