Subgroups — C₂×C₃²×#470 #11770102

Order 1152.

1476 conjugacy classes of subgroups.

The subgroup lattice has 252 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	C₂²×C₃²×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃²×C₂²⋊Q₈	C₂	1	2	yes	maximal
4	C₃²×#470	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃²×#106	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃²×#103	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×#470	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	C₂×C₃²×M₁₆	—	2	4	no	—
13	C₂×C₃²×M₁₆	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×#106	C₂²	1	4	yes	complemented
22	C₃²×#103	C₂²	1	8	yes	complemented
23	C₂²×C₃×Q₁₆	C₂×C₃	1	4	yes	complemented
24	C₂²×C₃×M₁₆	C₂×C₃	1	4	yes	—
25	C₂×C₃×C₂²⋊Q₈	C₂×C₃	1	8	yes	—
26	C₃×#470	C₂×C₃	1	32	yes	direct product factor
27	C₂×C₃×#106	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	C₃²×M₁₆	—	2	8	no	—
44	C₂×C₃²×C₈	—	2	2	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×#470	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	C₂×C₃×M₁₆	—	2	16	no	—
58	C₂×C₃×M₁₆	C₂²×C₃	1	16	yes	—
59	C₂²×C₃×C₈	C₂²×C₃	1	4	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×#106	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	4	no	—
70	C₂³×C₃²	C₂×D₄	1	2	yes	—
71	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
72	C₃²×Q₈	—	2	4	no	—
73	C₃²×Q₈	—	2	4	no	—
74	C₃²×Q₈	—	4	4	no	—
75	C₂×C₃²×C₄	—	2	2	no	—
76	C₂×C₃²×C₄	—	2	4	no	—
77	C₂×C₃²×C₄	—	2	2	no	—
78	C₂×C₃²×C₄	—	2	2	no	—
79	C₂×C₃²×C₄	—	2	4	no	—
80	C₂×C₃²×C₄	—	2	4	no	—
81	C₂×C₃²×C₄	—	4	4	no	—
82	C₂×C₃²×C₄	—	4	4	no	—
83	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
84	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
86	C₂×C₃²×C₄	#41	1	2	yes	—
87	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
88	C₃²×C₈	—	2	2	no	—
89	C₃²×C₈	C₂×D₄	1	4	yes	—
90	C₂²×Q₁₆	C₂×C₃²	1	1	yes	characteristic, complemented
91	C₂²×M₁₆	C₂×C₃²	1	1	yes	characteristic
92	C₂×C₂²⋊Q₈	C₂×C₃²	1	2	yes	—
93	#470	C₂×C₃²	1	8	yes	direct product factor
94	C₂×#106	C₂×C₃²	1	1	yes	characteristic, complemented
95	C₂×#103	C₂×C₃²	1	2	yes	complemented
96	C₂⁴×C₃	C₃×D₄	1	4	yes	—
97	C₂×C₃×Q₈	—	2	8	no	—
98	C₂×C₃×Q₈	—	2	16	no	—
99	C₂×C₃×Q₈	—	2	16	no	—
100	C₂×C₃×Q₈	C₂³×C₃	1	16	yes	—
101	C₂²×C₃×C₄	—	2	8	no	—
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	C₃×D₄	1	4	yes	—
106	C₂²×C₃×C₄	C₃×D₄	1	8	yes	—
107	C₂²×C₃×C₄	C₂³×C₃	1	8	yes	—
108	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
109	C₃×Q₁₆	—	2	32	no	—
110	C₃×M₁₆	—	2	32	no	—
111	C₂×C₃×C₈	—	2	8	no	—
112	C₂×C₃×C₈	C₃×D₄	1	16	yes	—
113	C₂×C₃×C₈	C₂³×C₃	1	8	yes	—
114	C₂×C₃×C₈	C₂³×C₃	1	8	yes	—
115	C₃×C₄⋊C₄	—	2	32	no	—
116	C₃×C₄⋊C₄	—	2	8	no	—
117	C₃×C₄⋊C₄	C₂³×C₃	1	16	yes	—
118	C₃×C₂²⋊C₄	—	2	32	no	—
119	C₂²×C₃²	—	2	2	no	—
120	C₂²×C₃²	—	2	4	no	—
121	C₂²×C₃²	—	4	4	no	—
122	C₂²×C₃²	#121	1	2	yes	—
123	C₂²×C₃²	#137	1	4	yes	—
124	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
125	C₃²×C₄	—	2	2	no	—
126	C₃²×C₄	—	4	4	no	—
127	C₃²×C₄	—	4	4	no	—
128	C₃²×C₄	#121	1	2	yes	—
129	C₃²×C₄	C₂²×D₄	1	1	yes	characteristic
130	C₃²×C₄	C₂×#41	1	1	yes	characteristic
131	C₂²×Q₈	C₂²×C₃²	1	2	yes	—
132	C₂³×C₄	C₂²×C₃²	1	1	yes	characteristic
133	C₂×Q₁₆	—	2	4	no	—
134	C₂×Q₁₆	C₂²×C₃²	1	4	yes	complemented
135	C₂×M₁₆	—	2	4	no	—
136	C₂×M₁₆	C₂²×C₃²	1	4	yes	—
137	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
138	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
139	C₂²⋊Q₈	—	2	4	no	—
140	C₂²⋊Q₈	C₂²×C₃²	1	8	yes	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
143	C₂×C₂²⋊C₄	—	2	2	no	—
144	#106	C₂²×C₃²	1	4	yes	complemented
145	#103	C₂²×C₃²	1	8	yes	complemented
146	C₂³×C₃	—	2	8	no	—
147	C₂³×C₃	—	2	16	no	—
148	C₂³×C₃	C₂×C₃×D₄	1	8	yes	—
149	C₂³×C₃	C₂×C₃×D₄	1	4	yes	—
150	C₃×Q₈	—	2	16	no	—
151	C₃×Q₈	—	2	16	no	—
152	C₃×Q₈	—	4	16	no	—
153	C₂×C₃×C₄	—	2	8	no	—
154	C₂×C₃×C₄	—	2	16	no	—
155	C₂×C₃×C₄	—	2	8	no	—
156	C₂×C₃×C₄	—	2	8	no	—
157	C₂×C₃×C₄	—	2	16	no	—
158	C₂×C₃×C₄	—	2	16	no	—
159	C₂×C₃×C₄	—	4	16	no	—
160	C₂×C₃×C₄	—	4	16	no	—
161	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
162	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
163	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
164	C₂×C₃×C₄	C₃×#41	1	8	yes	—
165	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
166	C₃×C₈	—	2	8	no	—
167	C₃×C₈	C₂×C₃×D₄	1	16	yes	—
168	C₂×C₃²	—	4	2	no	—
169	C₂×C₃²	#470	1	4	yes	direct product factor
170	C₂×C₃²	C₂×#121	1	1	yes	characteristic
171	C₂×C₃²	C₂×#137	1	2	yes	—
172	C₂⁴	C₃²×D₄	1	1	yes	characteristic
173	C₂×Q₈	—	2	2	no	—
174	C₂×Q₈	—	2	4	no	—
175	C₂×Q₈	—	2	4	no	—
176	C₂×Q₈	C₂³×C₃²	1	4	yes	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	C₃²×D₄	1	1	yes	characteristic
182	C₂²×C₄	C₃²×D₄	1	2	yes	—
183	C₂²×C₄	C₂³×C₃²	1	2	yes	—
184	C₂²×C₄	C₂³×C₃²	1	1	yes	characteristic
185	Q₁₆	—	2	8	no	—
186	M₁₆	—	2	8	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	C₃²×D₄	1	4	yes	—
189	C₂×C₈	C₂³×C₃²	1	2	yes	—
190	C₂×C₈	C₂³×C₃²	1	2	yes	—
191	C₄⋊C₄	—	2	8	no	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	C₂³×C₃²	1	4	yes	—
194	C₂²⋊C₄	—	2	8	no	—
195	C₂²×C₃	—	2	16	no	—
196	C₂²×C₃	—	2	8	no	—
197	C₂²×C₃	—	4	16	no	—
198	C₂²×C₃	C₃×#121	1	8	yes	—
199	C₂²×C₃	C₃×#137	1	16	yes	—
200	C₂²×C₃	C₂²×C₃×D₄	1	4	yes	—
201	C₃×C₄	—	2	8	no	—
202	C₃×C₄	—	4	16	no	—
203	C₃×C₄	—	4	16	no	—
204	C₃×C₄	C₃×#121	1	8	yes	—
205	C₃×C₄	C₂²×C₃×D₄	1	4	yes	—
206	C₃×C₄	C₂×C₃×#41	1	4	yes	—
207	C₃²	C₂×#470	1	1	yes	characteristic, direct product factor, Sylow
208	C₂³	—	2	2	no	—
209	C₂³	—	2	4	no	—
210	C₂³	C₂×C₃²×D₄	1	2	yes	—
211	C₂³	C₂×C₃²×D₄	1	1	yes	characteristic
212	Q₈	—	2	4	no	—
213	Q₈	—	2	4	no	—
214	Q₈	—	4	4	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
224	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
225	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
226	C₂×C₄	C₃²×#41	1	2	yes	—
227	C₂×C₄	C₂⁴×C₃²	1	1	yes	characteristic
228	C₈	—	2	2	no	—
229	C₈	C₂×C₃²×D₄	1	4	yes	—
230	C₂×C₃	—	4	8	no	—
231	C₂×C₃	C₃×#470	1	16	yes	direct product factor
232	C₂×C₃	C₂×C₃×#121	1	4	yes	—
233	C₂×C₃	C₂×C₃×#137	1	8	yes	—
234	C₂²	—	2	4	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	4	4	no	—
237	C₂²	C₃²×#121	1	2	yes	—
238	C₂²	C₃²×#137	1	4	yes	—
239	C₂²	C₂²×C₃²×D₄	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	4	4	no	—
242	C₄	—	4	4	no	—
243	C₄	C₃²×#121	1	2	yes	—
244	C₄	C₂²×C₃²×D₄	1	1	yes	characteristic
245	C₄	C₂×C₃²×#41	1	1	yes	characteristic
246	C₃	C₂×C₃×#470	1	4	yes	direct product factor
247	C₂	—	4	2	no	—
248	C₂	C₃²×#470	1	4	yes	direct product factor
249	C₂	C₂×C₃²×#121	1	1	yes	characteristic
250	C₂	C₂×C₃²×#137	1	2	yes	—

	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×#470	C₂	direct (N × H)
5. C₂×C₃²×#106	C₂	semidirect
6. C₂×C₃²×#103	C₂	semidirect
7. C₂×C₃×#470	C₃	direct (N × H)
11. C₂×C₃²×Q₁₆	C₂²	semidirect
21. C₃²×#106	C₂²	semidirect
22. C₃²×#103	C₂²	semidirect
23. C₂²×C₃×Q₁₆	C₂×C₃	semidirect
26. C₃×#470	C₂×C₃	direct (N × H)
27. C₂×C₃×#106	C₂×C₃	semidirect
28. C₂×C₃×#103	C₂×C₃	semidirect
52. C₂×#470	C₃²	direct (N × H)
56. C₂×C₃×Q₁₆	C₂²×C₃	semidirect
66. C₃×#106	C₂²×C₃	semidirect
67. C₃×#103	C₂²×C₃	semidirect
90. C₂²×Q₁₆	C₂×C₃²	semidirect
93. #470	C₂×C₃²	direct (N × H)
94. C₂×#106	C₂×C₃²	semidirect
95. C₂×#103	C₂×C₃²	semidirect
134. C₂×Q₁₆	C₂²×C₃²	semidirect
144. #106	C₂²×C₃²	semidirect
145. #103	C₂²×C₃²	semidirect
169. C₂×C₃²	#470	direct (N × H)
207. C₃²	C₂×#470	direct (N × H)
231. C₂×C₃	C₃×#470	direct (N × H)
246. C₃	C₂×C₃×#470	direct (N × H)
248. C₂	C₃²×#470	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
C₂²×C₃²×M₁₆	2
C₂×C₃²×C₂²⋊Q₈	2
C₃²×#470	2
C₂×C₃²×#106	2
C₂×C₃²×#103	2
C₂×C₃×#470	3

Subgroups — C2×C32×#470 #11770102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×#470 #11770102