Subgroups — #64226

Order 288.

99 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3798	C₂	1	1	yes	maximal, characteristic
2	#3797	C₂	1	1	yes	maximal, characteristic, complemented
3	#3788	C₂	1	1	yes	maximal, characteristic, complemented
4	#3776	C₂	1	2	yes	maximal, complemented
5	#3775	C₂	1	2	yes	maximal, complemented
6	C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
7	C₃×C₄×D₃	—	2	1	no	—
8	C₃×Dic₆	—	2	1	no	—
9	#627	C₂²	1	2	yes	—
10	#627	C₂²	1	1	yes	characteristic
11	#626	—	2	1	no	—
12	#625	C₂²	1	1	yes	characteristic
13	#624	—	2	1	no	—
14	#622	C₂²	1	2	yes	—
15	D₃×Q₈	—	6	1	no	—
16	#235	—	6	1	no	—
17	C₂×C₃⋊D₃	D₄	1	1	yes	characteristic
18	C₂×C₃×D₃	—	2	1	no	—
19	C₃²×C₄	D₄	1	1	yes	characteristic
20	C₃⋊Dic₃	C₂³	1	1	yes	characteristic
21	C₃×Dic₃	—	2	2	no	—
22	C₃×Dic₃	—	2	1	no	—
23	#137	—	9	1	no	maximal, Sylow
24	C₃×Q₈	—	6	1	no	—
25	C₂×C₃×C₄	—	6	1	no	—
26	#67	—	6	1	no	—
27	#67	—	6	1	no	—
28	D₁₂	—	6	1	no	—
29	C₄×D₃	—	2	1	no	—
30	C₄×D₃	—	6	1	no	—
31	C₄×D₃	—	6	1	no	—
32	C₄×D₃	—	6	2	no	—
33	Dic₆	—	2	1	no	—
34	Dic₆	—	6	1	no	—
35	Dic₆	—	6	2	no	—
36	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
37	C₃⋊D₃	—	2	1	no	—
38	C₃×D₃	—	4	1	no	—
39	#41	—	9	1	no	—
40	C₂×Q₈	—	9	1	no	—
41	Q₁₆	—	9	2	no	—
42	SD₁₆	—	9	2	no	—
43	M₁₆	—	9	1	no	—
44	C₂²×C₃	—	6	1	no	—
45	C₂×D₃	—	2	1	no	—
46	C₂×D₃	—	6	1	no	—
47	C₂×D₃	—	6	1	no	—
48	C₃×C₄	—	2	1	no	—
49	C₃×C₄	—	2	1	no	—
50	C₃×C₄	—	6	2	no	—
51	C₃×C₄	—	6	1	no	—
52	Dic₃	—	2	2	no	—
53	Dic₃	—	2	1	no	—
54	Dic₃	—	6	1	no	—
55	Dic₃	—	6	1	no	—
56	C₃²	#137	1	1	yes	characteristic, complemented, Sylow
57	Q₈	—	6	1	no	—
58	Q₈	—	9	2	no	—
59	Q₈	—	9	1	no	—
60	D₄	—	9	1	no	—
61	D₄	—	18	1	no	—
62	C₂×C₄	—	6	1	no	—
63	C₂×C₄	—	9	1	no	—
64	C₂×C₄	—	18	1	no	—
65	C₈	—	9	2	no	—
66	C₂×C₃	—	2	1	no	—
67	C₂×C₃	—	2	1	no	—
68	C₂×C₃	—	12	1	no	—
69	D₃	—	4	1	no	—
70	D₃	—	6	2	no	—
71	D₃	—	12	1	no	—
72	C₂²	—	6	1	no	—
73	C₂²	—	9	1	no	—
74	C₄	—	6	1	no	—
75	C₄	—	6	2	no	—
76	C₄	—	9	1	no	—
77	C₄	GO⁺(4,𝔽₂)	1	1	yes	characteristic
78	C₃	—	2	1	no	—
79	C₃	—	2	1	no	—
80	C₂	—	12	1	no	—
81	C₂	—	18	1	no	—
82	C₂	C₂×GO⁺(4,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×GO⁺(4,𝔽₂)
commutator, cosocle	C₃⋊Dic₃	C₂³
Fitting	C₃²×C₄	D₄
socle	C₂×C₃²	C₂×D₄
prime radical	#3798	C₂
hypercenter	C₄	GO⁺(4,𝔽₂)
hypocenter	C₃²	#137
LCS	G ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. #3797	C₂	semidirect
3. #3788	C₂	semidirect
4. #3776	C₂	semidirect
5. #3775	C₂	semidirect
56. C₃²	#137	semidirect

Maximal subgroups

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

subgroup	index
#3798	2
#3797	2
#3788	2
#3776	2
#3775	2
#137	9