Subgroups — #73015

Order 384.

105 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5386	C₂	1	2	yes	maximal
2	C₈×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic
3	#1670	—	3	1	no	maximal, Sylow
4	C₂×Dic₁₂	—	4	1	no	maximal
5	C₄×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
6	#859	—	2	2	no	—
7	#494	—	3	1	no	—
8	#475	—	3	2	no	—
9	C₈×Q₈	D₃	1	1	yes	characteristic
10	#335	—	3	1	no	—
11	#332	—	3	2	no	—
12	C₂×Dic₆	—	4	2	no	—
13	C₂×SL(2,𝔽₃)	D₄	1	1	yes	characteristic
14	C₂×C₃×C₈	—	4	1	no	—
15	Dic₁₂	—	4	4	no	—
16	C₂×Q₁₆	—	6	1	no	—
17	C₂×Q₁₆	—	6	1	no	—
18	C₄⋊Q₈	—	3	2	no	—
19	C₄×Q₈	C₂×D₃	1	1	yes	characteristic
20	#106	—	6	2	no	—
21	C₄⋊C₈	—	3	1	no	—
22	C₄⋊C₈	—	3	2	no	—
23	#103	—	6	2	no	—
24	C₄×C₈	—	3	1	no	—
25	C₂×C₃×C₄	—	4	1	no	—
26	C₂×Dic₃	—	8	2	no	—
27	Dic₆	—	4	4	no	—
28	Dic₆	—	8	2	no	—
29	SL(2,𝔽₃)	Q₁₆	1	1	yes	characteristic, complemented
30	C₃×C₈	—	4	2	no	—
31	C₂×Q₈	—	6	2	no	—
32	C₂×Q₈	D₁₂	1	1	yes	characteristic
33	Q₁₆	—	12	2	no	—
34	Q₁₆	—	12	2	no	—
35	C₂×C₈	—	3	1	no	—
36	C₂×C₈	—	6	2	no	—
37	C₂×C₈	S₄	1	1	yes	characteristic
38	C₄⋊C₄	—	3	1	no	—
39	C₄⋊C₄	—	6	2	no	—
40	C₄⋊C₄	—	6	2	no	—
41	C₄²	—	3	1	no	—
42	C₂²×C₃	—	4	1	no	—
43	C₃×C₄	—	4	1	no	—
44	C₃×C₄	—	4	1	no	—
45	Dic₃	—	8	4	no	—
46	Q₈	—	3	1	no	—
47	Q₈	—	12	2	no	—
48	Q₈	—	12	2	no	—
49	Q₈	Dic₁₂	1	1	yes	characteristic, complemented
50	C₂×C₄	—	3	1	no	—
51	C₂×C₄	—	3	1	no	—
52	C₂×C₄	—	12	2	no	—
53	C₂×C₄	C₂×S₄	1	1	yes	characteristic
54	C₈	—	6	1	no	—
55	C₈	—	12	2	no	—
56	C₈	GL(2,𝔽₃)	1	2	yes	—
57	C₂×C₃	—	4	1	no	—
58	C₂×C₃	—	4	1	no	—
59	C₂×C₃	—	4	1	no	—
60	C₂²	C₄⋊S₄	1	1	yes	characteristic
61	C₄	—	3	1	no	—
62	C₄	—	3	1	no	—
63	C₄	—	6	1	no	—
64	C₄	—	24	2	no	—
65	C₄	C₂×GL(2,𝔽₃)	1	1	yes	characteristic
66	C₄	#986	1	1	yes	characteristic
67	C₃	—	4	1	no	Sylow
68	C₂	C₄⋊GL(2,𝔽₃)	1	1	yes	characteristic
69	C₂	#5398	1	1	yes	characteristic
70	C₂	#5403	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₄⋊S₄
commutator, cosocle	C₄×SL(2,𝔽₃)	C₂²
Frattini	C₂×C₄	C₂×S₄
Fitting	C₈×Q₈	D₃
prime radical	C₂×SL(2,𝔽₃)	D₄
hypercenter	C₂×C₈	S₄
hypocenter	SL(2,𝔽₃)	Q₁₆
LCS	G ▷ C₄×SL(2,𝔽₃) ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
29. SL(2,𝔽₃)	Q₁₆	semidirect
49. Q₈	Dic₁₂	semidirect

Maximal subgroups

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

subgroup	index
#5386	2
C₈×SL(2,𝔽₃)	2
#1670	3
C₂×Dic₁₂	4