Subgroups — #73012

Order 384.

91 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5391	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#226	C₂	1	1	yes	maximal, characteristic
3	C₈×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic
4	#1672	—	3	1	no	maximal, Sylow
5	C₂×#226	—	2	1	no	—
6	C₄×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
7	#859	—	2	1	no	—
8	#817	—	4	1	no	maximal
9	#498	—	3	1	no	—
10	#477	—	3	1	no	—
11	#462	—	3	1	no	—
12	C₈×Q₈	D₃	1	1	yes	characteristic
13	#334	—	3	1	no	—
14	#333	—	3	1	no	—
15	#332	—	3	1	no	—
16	C₂×SL(2,𝔽₃)	D₄	1	1	yes	characteristic
17	#226	—	4	1	no	—
18	C₂×C₃×C₈	—	4	1	no	—
19	C₄⋊Dic₃	—	4	1	no	—
20	C₄⋊Dic₃	—	4	1	no	—
21	C₂×Q₁₆	—	6	1	no	—
22	C₄⋊Q₈	—	3	1	no	—
23	C₄⋊Q₈	—	3	1	no	—
24	C₄×Q₈	C₂×D₃	1	1	yes	characteristic
25	#107	—	6	1	no	—
26	#106	—	6	1	no	—
27	#106	—	6	1	no	—
28	C₄⋊C₈	—	3	1	no	—
29	C₄⋊C₈	—	3	1	no	—
30	C₄⋊C₈	—	3	1	no	—
31	#103	—	6	1	no	—
32	#103	—	6	1	no	—
33	C₄×C₈	—	3	1	no	—
34	C₂×C₃×C₄	—	4	1	no	—
35	C₂×Dic₃	—	8	1	no	—
36	C₂×Dic₃	—	8	1	no	—
37	SL(2,𝔽₃)	SD₁₆	1	1	yes	characteristic
38	C₃×C₈	—	4	2	no	—
39	C₂×Q₈	—	6	1	no	—
40	C₂×Q₈	—	6	1	no	—
41	C₂×Q₈	D₁₂	1	1	yes	characteristic
42	Q₁₆	—	12	1	no	—
43	Q₁₆	—	12	1	no	—
44	C₂×C₈	—	3	1	no	—
45	C₂×C₈	—	6	1	no	—
46	C₂×C₈	—	6	1	no	—
47	C₂×C₈	S₄	1	1	yes	characteristic
48	C₄⋊C₄	—	3	1	no	—
49	C₄⋊C₄	—	6	1	no	—
50	C₄⋊C₄	—	6	1	no	—
51	C₄⋊C₄	—	6	1	no	—
52	C₄⋊C₄	—	6	1	no	—
53	C₄²	—	3	1	no	—
54	C₂²×C₃	—	4	1	no	—
55	C₃×C₄	—	4	1	no	—
56	C₃×C₄	—	4	1	no	—
57	Dic₃	—	16	1	no	—
58	Dic₃	—	16	1	no	—
59	Q₈	—	3	1	no	—
60	Q₈	—	12	1	no	—
61	Q₈	—	12	1	no	—
62	Q₈	—	24	1	no	—
63	Q₈	#204	1	1	yes	characteristic
64	C₂×C₄	—	3	1	no	—
65	C₂×C₄	—	3	1	no	—
66	C₂×C₄	—	12	1	no	—
67	C₂×C₄	—	12	1	no	—
68	C₂×C₄	C₂×S₄	1	1	yes	characteristic
69	C₈	—	6	1	no	—
70	C₈	—	12	1	no	—
71	C₈	—	12	1	no	—
72	C₈	#226	1	2	yes	complemented
73	C₂×C₃	—	4	1	no	—
74	C₂×C₃	—	4	1	no	—
75	C₂×C₃	—	4	1	no	—
76	C₂²	C₄⋊S₄	1	1	yes	characteristic
77	C₄	—	3	1	no	—
78	C₄	—	3	1	no	—
79	C₄	—	6	1	no	—
80	C₄	—	24	1	no	—
81	C₄	—	24	1	no	—
82	C₄	C₂×#226	1	1	yes	characteristic
83	C₄	#984	1	1	yes	characteristic
84	C₃	—	4	1	no	Sylow
85	C₂	C₄⋊#226	1	1	yes	characteristic
86	C₂	#5401	1	1	yes	characteristic
87	C₂	#5406	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)	SD₁₆
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
72. C₈	#226	semidirect

Maximal subgroups

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

subgroup	index
#5391	2
C₄⋊#226	2
C₈×SL(2,𝔽₃)	2
#1672	3
#817	4