Subgroups — #73010

Order 384.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#226	C₂	1	1	yes	maximal, characteristic
2	C₈×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic
3	SL(2,𝔽₃)⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#1584	—	3	1	no	maximal, Sylow
5	C₂×#226	C₄	1	1	yes	characteristic
6	C₄×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
7	#859	C₄	1	1	yes	characteristic
8	#815	—	4	1	no	maximal
9	#446	—	3	1	no	—
10	C₈×Q₈	D₃	1	1	yes	characteristic
11	C₄×Q₁₆	—	3	1	no	—
12	#335	—	3	1	no	—
13	Q₈⋊C₈	—	3	1	no	—
14	Q₈⋊C₈	—	3	1	no	—
15	#322	—	3	1	no	—
16	C₂×SL(2,𝔽₃)	C₂×C₄	1	1	yes	characteristic
17	#226	—	2	1	no	—
18	C₂×C₃×C₈	—	4	1	no	—
19	C₄×Dic₃	—	4	1	no	—
20	C₂×C₃⋊C₈	—	4	1	no	—
21	C₂×Q₁₆	—	3	1	no	—
22	C₄×Q₈	—	3	1	no	—
23	C₄×Q₈	C₂×D₃	1	1	yes	characteristic
24	#107	—	3	1	no	—
25	C₄⋊C₈	—	3	1	no	—
26	C₄⋊C₈	—	3	1	no	—
27	C₄⋊C₈	—	6	1	no	—
28	#103	—	3	1	no	—
29	#103	—	3	1	no	—
30	#97	—	6	1	no	—
31	C₄×C₈	—	3	1	no	—
32	C₄×C₈	—	3	1	no	—
33	C₄×C₈	—	3	1	no	—
34	C₂×C₃×C₄	—	4	1	no	—
35	C₂×Dic₃	—	4	1	no	—
36	C₂×Dic₃	—	4	1	no	—
37	SL(2,𝔽₃)	M₁₆	1	1	yes	characteristic
38	C₃×C₈	—	4	2	no	—
39	C₃⋊C₈	—	4	2	no	—
40	C₂×Q₈	—	3	1	no	—
41	C₂×Q₈	C₄×D₃	1	1	yes	characteristic
42	Q₁₆	—	6	1	no	—
43	Q₁₆	—	6	1	no	—
44	C₂×C₈	—	3	1	no	—
45	C₂×C₈	—	3	1	no	—
46	C₂×C₈	—	3	1	no	—
47	C₂×C₈	—	6	1	no	—
48	C₂×C₈	—	6	1	no	—
49	C₂×C₈	S₄	1	1	yes	characteristic
50	C₄⋊C₄	—	3	1	no	—
51	C₄⋊C₄	—	3	1	no	—
52	C₄⋊C₄	—	6	1	no	—
53	C₄²	—	3	1	no	—
54	C₄²	—	6	1	no	—
55	C₂²×C₃	—	4	1	no	—
56	C₃×C₄	—	4	1	no	—
57	C₃×C₄	—	4	1	no	—
58	Dic₃	—	8	1	no	—
59	Dic₃	—	8	1	no	—
60	Q₈	—	3	1	no	—
61	Q₈	—	6	1	no	—
62	Q₈	—	6	1	no	—
63	Q₈	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
64	C₂×C₄	—	3	1	no	—
65	C₂×C₄	—	3	1	no	—
66	C₂×C₄	—	6	1	no	—
67	C₂×C₄	—	6	1	no	—
68	C₂×C₄	C₂×S₄	1	1	yes	characteristic
69	C₈	—	6	1	no	—
70	C₈	—	6	1	no	—
71	C₈	—	6	1	no	—
72	C₈	—	12	1	no	—
73	C₈	—	12	1	no	—
74	C₈	#226	1	2	yes	complemented
75	C₂×C₃	—	4	1	no	—
76	C₂×C₃	—	4	1	no	—
77	C₂×C₃	—	4	1	no	—
78	C₂²	C₄×S₄	1	1	yes	characteristic
79	C₄	—	3	1	no	—
80	C₄	—	3	1	no	—
81	C₄	—	6	1	no	—
82	C₄	—	12	1	no	—
83	C₄	—	12	1	no	—
84	C₄	C₂×#226	1	1	yes	characteristic
85	C₄	#985	1	1	yes	characteristic
86	C₃	—	4	1	no	Sylow
87	C₂	C₄×#226	1	1	yes	characteristic
88	C₂	#5400	1	1	yes	characteristic
89	C₂	#5405	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×S₄
commutator, prime radical	C₂×SL(2,𝔽₃)	C₂×C₄
Fitting	C₈×Q₈	D₃
socle	C₂²	C₄×S₄
cosocle	C₄×SL(2,𝔽₃)	C₂²
hypercenter	C₂×C₈	S₄
hypocenter	SL(2,𝔽₃)	M₁₆
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
74. 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
SL(2,𝔽₃)⋊C₈	2
#1584	3
#815	4