Subgroups — #73013

Order 384.

95 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#226	C₂	1	2	yes	maximal
2	C₈×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic
3	#1674	—	3	1	no	maximal, Sylow
4	C₂×#226	—	2	2	no	—
5	C₄×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
6	#818	—	4	1	no	maximal
7	#500	—	3	1	no	—
8	#462	—	3	2	no	—
9	C₈×Q₈	D₃	1	1	yes	characteristic
10	#335	—	3	1	no	—
11	#332	—	3	2	no	—
12	C₂×SL(2,𝔽₃)	D₄	1	1	yes	characteristic
13	#226	—	4	2	no	—
14	C₂×C₃×C₈	—	4	1	no	—
15	C₄⋊Dic₃	—	4	2	no	—
16	C₂×Q₁₆	—	6	2	no	—
17	C₄⋊Q₈	—	3	2	no	—
18	C₄×Q₈	C₂×D₃	1	1	yes	characteristic
19	#107	—	6	1	no	—
20	#107	—	6	1	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	SL(2,𝔽₃)	D₈	1	1	yes	characteristic
28	C₃×C₈	—	4	2	no	—
29	C₂×Q₈	—	6	2	no	—
30	C₂×Q₈	D₁₂	1	1	yes	characteristic
31	Q₁₆	—	12	2	no	—
32	Q₁₆	—	12	2	no	—
33	C₂×C₈	—	3	1	no	—
34	C₂×C₈	—	6	2	no	—
35	C₂×C₈	S₄	1	1	yes	characteristic
36	C₄⋊C₄	—	3	1	no	—
37	C₄⋊C₄	—	6	2	no	—
38	C₄⋊C₄	—	6	2	no	—
39	C₄²	—	3	1	no	—
40	C₂²×C₃	—	4	1	no	—
41	C₃×C₄	—	4	1	no	—
42	C₃×C₄	—	4	1	no	—
43	Dic₃	—	16	2	no	—
44	Q₈	—	3	1	no	—
45	Q₈	—	12	2	no	—
46	Q₈	—	12	2	no	—
47	Q₈	D₂₄	1	1	yes	characteristic
48	C₂×C₄	—	3	1	no	—
49	C₂×C₄	—	3	1	no	—
50	C₂×C₄	—	12	2	no	—
51	C₂×C₄	C₂×S₄	1	1	yes	characteristic
52	C₈	—	6	1	no	—
53	C₈	—	12	2	no	—
54	C₈	#226	1	2	yes	complemented
55	C₂×C₃	—	4	1	no	—
56	C₂×C₃	—	4	1	no	—
57	C₂×C₃	—	4	1	no	—
58	C₂²	C₄⋊S₄	1	1	yes	characteristic
59	C₄	—	3	1	no	—
60	C₄	—	3	1	no	—
61	C₄	—	6	1	no	—
62	C₄	—	24	2	no	—
63	C₄	C₂×#226	1	1	yes	characteristic
64	C₄	#984	1	1	yes	characteristic
65	C₃	—	4	1	no	Sylow
66	C₂	C₄⋊#226	1	1	yes	characteristic
67	C₂	#5402	1	1	yes	characteristic
68	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)	D₈
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
54. C₈	#226	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#226	2
C₈×SL(2,𝔽₃)	2
#1674	3
#818	4