Subgroups — #10593511

Order 576.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#622	C₂	1	1	yes	maximal, characteristic
2	#63957	C₂	1	2	yes	maximal
3	#63743	C₂	1	2	yes	maximal, complemented
4	#63742	C₂	1	2	yes	maximal, complemented
5	C₂²×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
6	C₂×#622	C₄	1	4	yes	complemented
7	C₂×#622	C₂²	1	1	yes	characteristic
8	C₂×#622	C₂²	1	1	yes	characteristic
9	C₃×#217	—	2	2	no	—
10	#3724	C₂²	1	4	yes	—
11	Dic₃²	—	2	2	no	—
12	#877	—	6	2	no	—
13	C₂³×C₃²	D₄	1	1	yes	characteristic
14	C₂×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
15	C₂×C₃⋊Dic₃	D₄	1	1	yes	characteristic
16	C₂×C₃⋊Dic₃	Q₈	1	2	yes	—
17	C₂×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
18	C₂×C₃×Dic₃	—	2	4	no	—
19	#622	C₂×C₄	1	2	yes	—
20	#622	C₂×C₄	1	2	yes	—
21	#427	—	9	1	no	maximal, Sylow
22	C₂²×Dic₃	—	6	2	no	—
23	C₃×C₂²⋊C₄	—	6	2	no	—
24	#217	—	2	2	no	—
25	Dic₃⋊C₄	—	6	4	no	—
26	C₄×Dic₃	—	6	4	no	—
27	C₂²×C₃²	—	2	2	no	—
28	C₂²×C₃²	C₄⋊C₄	1	2	yes	—
29	C₂²×C₃²	C₂×D₄	1	1	yes	characteristic
30	C₃⋊Dic₃	C₄⋊C₄	1	2	yes	—
31	C₃⋊Dic₃	C₂²×C₄	1	1	yes	characteristic
32	C₃⋊Dic₃	C₂×Q₈	1	1	yes	characteristic
33	C₃×Dic₃	—	4	4	no	—
34	C₂²×C₈	—	9	1	no	—
35	#117	—	9	2	no	—
36	#107	—	9	2	no	—
37	#106	—	9	2	no	—
38	C₂³×C₃	—	2	2	no	—
39	C₂×C₃×C₄	—	6	4	no	—
40	C₂×Dic₃	—	2	4	no	—
41	C₂×Dic₃	—	6	4	no	—
42	C₂×Dic₃	—	6	2	no	—
43	C₂×Dic₃	—	6	4	no	—
44	C₂×Dic₃	—	6	2	no	—
45	C₂×C₃²	—	2	2	no	—
46	C₂×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₃²	#135	1	2	yes	—
48	C₂²×C₄	—	9	1	no	—
49	C₂×C₈	—	9	1	no	—
50	C₂×C₈	—	9	1	no	—
51	C₂×C₈	—	9	4	no	—
52	C₄⋊C₄	—	9	4	no	—
53	C₂²⋊C₄	—	6	2	no	—
54	C₄²	—	18	2	no	—
55	C₂²×C₃	—	2	2	no	—
56	C₂²×C₃	—	2	4	no	—
57	C₂²×C₃	—	4	4	no	—
58	C₃×C₄	—	12	4	no	—
59	Dic₃	—	4	4	no	—
60	Dic₃	—	6	4	no	—
61	Dic₃	—	6	2	no	—
62	Dic₃	—	6	2	no	—
63	C₃²	#427	1	1	yes	characteristic, complemented, Sylow
64	C₂³	GO⁺(4,𝔽₂)	1	1	yes	characteristic
65	C₂×C₄	—	6	4	no	—
66	C₂×C₄	—	9	1	no	—
67	C₂×C₄	—	9	1	no	—
68	C₂×C₄	—	9	2	no	—
69	C₂×C₄	—	9	2	no	—
70	C₈	—	9	2	no	—
71	C₈	—	9	2	no	—
72	C₂×C₃	—	2	2	no	—
73	C₂×C₃	—	2	4	no	—
74	C₂×C₃	—	4	4	no	—
75	C₂²	—	2	2	no	—
76	C₂²	#3773	1	2	yes	—
77	C₂²	C₂×GO⁺(4,𝔽₂)	1	1	yes	characteristic
78	C₄	—	9	1	no	—
79	C₄	—	9	2	no	—
80	C₄	—	9	1	no	—
81	C₄	—	12	4	no	—
82	C₃	—	2	2	no	—
83	C₂	—	2	2	no	—
84	C₂	C₂×#3773	1	1	yes	characteristic
85	C₂	#64237	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂×GO⁺(4,𝔽₂)
commutator	C₃⋊Dic₃	C₂²×C₄
Fitting, prime radical	C₂³×C₃²	D₄
socle	C₂²×C₃²	C₂×D₄
cosocle	C₂×C₃⋊Dic₃	C₂³
hypercenter	C₂³	GO⁺(4,𝔽₂)
hypocenter	C₃²	#427
LCS	G ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂³

Complements

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

N	H ≅ G/N	type
3. #63743	C₂	semidirect
4. #63742	C₂	semidirect
6. C₂×#622	C₄	semidirect
63. C₃²	#427	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#622	2
#63957	2
#63743	2
#63742	2
#427	9