Subgroups — #10593788

Order 576.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#64289	C₂	1	2	yes	maximal, complemented
2	#64128	C₂	1	1	yes	maximal, characteristic
3	#63785	C₂	1	2	yes	maximal, complemented
4	#63770	C₂	1	1	yes	maximal, characteristic, complemented
5	#63767	C₂	1	1	yes	maximal, characteristic, complemented
6	#3832	C₂²	1	2	yes	—
7	C₄×#157	C₂²	1	1	yes	characteristic
8	#3788	—	2	2	no	—
9	#3788	C₂²	1	2	yes	—
10	#3787	—	2	2	no	—
11	#3787	C₂²	1	1	yes	characteristic
12	C₃²×Q₁₆	C₄	1	1	yes	characteristic, complemented
13	#3755	C₄	1	2	yes	complemented
14	#3745	C₄	1	1	yes	characteristic, complemented
15	C₈×C₃⋊D₃	C₂²	1	1	yes	characteristic
16	#919	—	6	2	no	—
17	C₂×#157	—	2	1	no	—
18	C₃²×Q₈	C₂×C₄	1	2	yes	—
19	C₄⋊(C₃⋊D₃)	—	2	2	no	—
20	C₄⋊(C₃⋊D₃)	C₂×C₄	1	2	yes	—
21	C₄×C₃⋊D₃	—	2	2	no	—
22	C₄×C₃⋊D₃	C₂³	1	1	yes	characteristic
23	#622	—	2	2	no	—
24	#622	D₄	1	2	yes	—
25	C₃²×C₈	C₂×C₄	1	1	yes	characteristic
26	C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
27	#444	—	9	1	no	maximal, Sylow
28	#239	—	6	4	no	—
29	C₃×Q₁₆	—	2	2	no	—
30	#215	—	6	4	no	—
31	D₂₄	—	6	2	no	—
32	D₃×C₈	—	6	2	no	—
33	C₂×C₃⋊D₃	—	2	2	no	—
34	C₂×C₃⋊D₃	#41	1	1	yes	characteristic
35	#157	—	4	1	no	—
36	C₃²×C₄	—	2	2	no	—
37	C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
38	C₃⋊Dic₃	C₂×D₄	1	1	yes	characteristic
39	#135	—	9	1	no	—
40	#131	—	9	2	no	—
41	#108	—	9	1	no	—
42	#104	—	9	2	no	—
43	#97	—	9	1	no	—
44	C₃×Q₈	—	2	4	no	—
45	D₁₂	—	6	4	no	—
46	D₁₂	—	12	4	no	—
47	C₄×D₃	—	6	2	no	—
48	C₄×D₃	—	12	4	no	—
49	C₃×C₈	—	2	2	no	—
50	C₃⋊C₈	—	6	2	no	—
51	C₂×C₃²	C₄×D₄	1	1	yes	characteristic
52	C₃⋊D₃	—	2	1	no	—
53	C₃⋊D₃	—	4	2	no	—
54	#41	—	9	2	no	—
55	Q₁₆	#157	1	1	yes	characteristic, complemented
56	SD₁₆	—	9	2	no	—
57	D₈	—	9	1	no	—
58	M₁₆	—	9	2	no	—
59	M₁₆	—	18	2	no	—
60	C₂×C₈	—	9	1	no	—
61	C₂×C₈	—	9	1	no	—
62	C₂×C₈	—	18	2	no	—
63	C₄²	—	9	1	no	—
64	C₂×D₃	—	6	2	no	—
65	C₂×D₃	—	12	4	no	—
66	C₃×C₄	—	2	2	no	—
67	C₃×C₄	—	4	4	no	—
68	Dic₃	—	6	2	no	—
69	C₃²	#444	1	1	yes	characteristic, complemented, Sylow
70	Q₈	C₂×#157	1	2	yes	—
71	D₄	—	9	2	no	—
72	D₄	—	18	2	no	—
73	C₂×C₄	—	9	1	no	—
74	C₂×C₄	—	18	2	no	—
75	C₂×C₄	—	18	1	no	—
76	C₈	—	9	2	no	—
77	C₈	—	9	1	no	—
78	C₈	—	18	2	no	—
79	C₈	C₂×#157	1	1	yes	characteristic
80	C₂×C₃	—	2	2	no	—
81	D₃	—	12	2	no	—
82	D₃	—	24	4	no	—
83	C₂²	—	9	1	no	—
84	C₂²	—	18	2	no	—
85	C₄	—	2	2	no	—
86	C₄	—	9	1	no	—
87	C₄	—	36	1	no	—
88	C₄	C₂²×#157	1	1	yes	characteristic
89	C₃	—	2	2	no	—
90	C₂	—	18	1	no	—
91	C₂	—	36	2	no	—
92	C₂	D₄×#157	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#157
commutator	C₃²×C₄	C₂²×C₄
Frattini	C₄	C₂²×#157
Fitting	C₃²×Q₁₆	C₄
socle	C₂×C₃²	C₄×D₄
prime radical	#64128	C₂
cosocle	C₄×C₃⋊D₃	C₂³
hypercenter	Q₁₆	#157
hypocenter	C₃²	#444
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
1. #64289	C₂	semidirect
3. #63785	C₂	semidirect
4. #63770	C₂	semidirect
5. #63767	C₂	semidirect
12. C₃²×Q₁₆	C₄	semidirect
13. #3755	C₄	semidirect
14. #3745	C₄	semidirect
55. Q₁₆	#157	semidirect
69. C₃²	#444	semidirect

Maximal subgroups

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

subgroup	index
#64289	2
#64128	2
#63785	2
#63770	2
#63767	2
#444	9