Subgroups — #10596646

Order 576.

422 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
2	Q₈×#157	C₂	1	8	yes	maximal, complemented
3	#64286	C₂	1	6	yes	maximal
4	C₂×C₃²×Q₈	C₄	1	1	yes	characteristic, complemented
5	Q₈×C₃⋊D₃	C₄	1	4	yes	complemented
6	Q₈×C₃⋊D₃	C₂²	1	4	yes	—
7	C₂×C₄×C₃⋊D₃	C₂²	1	3	yes	—
8	C₂×#634	C₄	1	3	yes	complemented
9	C₂²⋊#157	C₂²	1	4	yes	—
10	C₄⋊#157	C₂²	1	12	yes	—
11	C₄×#157	C₂²	1	12	yes	—
12	C₂×D₃×Q₈	—	6	2	no	—
13	C₂²×C₃⋊D₃	C₂³	1	1	yes	characteristic
14	C₂×#157	C₂³	1	8	yes	—
15	C₃²×Q₈	C₂×C₄	1	4	yes	—
16	C₂×C₃²×C₄	C₂×C₄	1	3	yes	—
17	C₂×C₃⋊Dic₃	C₂×C₄	1	3	yes	—
18	C₄×C₃⋊D₃	C₂×C₄	1	6	yes	—
19	C₄×C₃⋊D₃	C₂³	1	6	yes	—
20	#634	C₂×C₄	1	12	yes	—
21	#519	—	9	1	no	maximal, Sylow
22	C₂×C₃×Q₈	—	2	2	no	—
23	D₃×Q₈	—	6	8	no	—
24	D₃×Q₈	—	6	8	no	—
25	C₂×C₄×D₃	—	6	6	no	—
26	C₂×Dic₆	—	6	6	no	—
27	C₂²×C₃²	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₃⋊D₃	—	2	2	no	—
29	C₂×C₃⋊D₃	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₃⋊D₃	C₂⁴	1	1	yes	characteristic
31	#157	—	2	8	no	—
32	C₃²×C₄	C₂²×C₄	1	6	yes	—
33	C₃⋊Dic₃	C₂²×C₄	1	6	yes	—
34	C₂²×Q₈	—	9	1	no	—
35	C₄×Q₈	—	9	8	no	—
36	#117	—	9	6	no	—
37	C₂²×D₃	—	6	2	no	—
38	C₃×Q₈	—	2	8	no	—
39	C₂×C₃×C₄	—	2	6	no	—
40	C₂×Dic₃	—	6	6	no	—
41	C₄×D₃	—	6	12	no	—
42	C₄×D₃	—	6	12	no	—
43	Dic₆	—	6	24	no	—
44	C₂×C₃²	—	2	1	no	—
45	C₂×C₃²	C₂³×C₄	1	1	yes	characteristic
46	C₃⋊D₃	—	2	1	no	—
47	C₃⋊D₃	2₋¹⁺⁴	1	2	yes	—
48	C₂×Q₈	—	9	4	no	—
49	C₂×Q₈	—	9	4	no	—
50	C₂×Q₈	—	9	3	no	—
51	C₂×Q₈	#157	1	1	yes	characteristic, complemented
52	C₂²×C₄	—	9	3	no	—
53	C₄⋊C₄	—	9	12	no	—
54	C₂²⋊C₄	—	9	4	no	—
55	C₄²	—	9	12	no	—
56	C₂²×C₃	—	2	2	no	—
57	C₂×D₃	—	6	2	no	—
58	C₂×D₃	—	6	2	no	—
59	C₂×D₃	—	6	8	no	—
60	C₃×C₄	—	2	12	no	—
61	Dic₃	—	6	12	no	—
62	C₃²	#519	1	1	yes	characteristic, complemented, Sylow
63	C₂³	—	9	1	no	—
64	Q₈	—	9	12	no	—
65	Q₈	C₂×#157	1	4	yes	—
66	C₂×C₄	—	9	3	no	—
67	C₂×C₄	—	9	6	no	—
68	C₂×C₄	—	9	6	no	—
69	C₂×C₄	—	9	8	no	—
70	C₂×C₄	C₂×#157	1	3	yes	—
71	C₂×C₃	—	2	4	no	—
72	C₂×C₃	—	2	2	no	—
73	D₃	—	6	4	no	—
74	D₃	—	6	4	no	—
75	C₂²	—	9	1	no	—
76	C₂²	—	9	1	no	—
77	C₂²	—	18	2	no	—
78	C₂²	C₂²×#157	1	1	yes	characteristic
79	C₄	—	9	6	no	—
80	C₄	—	18	8	no	—
81	C₄	C₂²×#157	1	6	yes	—
82	C₃	—	2	2	no	—
83	C₂	—	2	1	no	—
84	C₂	—	9	2	no	—
85	C₂	—	18	1	no	—
86	C₂	C₂³×#157	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×#157
commutator, socle	C₂×C₃²	C₂³×C₄
Fitting	C₂×C₃²×Q₈	C₄
prime radical	C₂²×C₃⋊D₃	C₂³
cosocle	C₂×C₃⋊D₃	C₂⁴
hypercenter	C₂×Q₈	#157
hypocenter	C₃²	#519
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×#157	C₂	semidirect
4. C₂×C₃²×Q₈	C₄	semidirect
5. Q₈×C₃⋊D₃	C₄	semidirect
8. C₂×#634	C₄	semidirect
51. C₂×Q₈	#157	semidirect
62. C₃²	#519	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₃⋊D₃	2
Q₈×#157	2
#64286	2
#519	9