Subgroups — #73041

Order 384.

85 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5407	C₂	1	1	yes	maximal, characteristic
2	#5403	C₂	1	1	yes	maximal, characteristic, complemented
3	#504⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#2223	—	3	1	no	maximal, Sylow
5	#986	—	2	1	no	—
6	#984	—	2	1	no	—
7	#131⋊C₃	C₂²	1	1	yes	characteristic
8	#800	—	4	1	no	maximal
9	C₂×SD₃₂	—	3	1	no	—
10	#504	D₃	1	1	yes	characteristic, complemented
11	#473	—	3	1	no	—
12	#472	—	3	1	no	—
13	#368	—	3	1	no	—
14	#362	—	3	1	no	—
15	#361	—	3	1	no	—
16	#41⋊C₃	D₄	1	1	yes	characteristic
17	GL(2,𝔽₃)	—	4	1	no	—
18	#226	—	4	1	no	—
19	Dic₁₂	—	4	1	no	—
20	D₂₄	—	4	1	no	—
21	C₃×C₁₆	—	4	1	no	—
22	#137	—	6	1	no	—
23	AGL(1,ℤ/8)	—	6	1	no	—
24	C₂×Q₁₆	—	3	1	no	—
25	C₂×D₈	—	3	1	no	—
26	#131	C₂×D₃	1	1	yes	characteristic
27	SD₃₂	—	6	1	no	—
28	SD₃₂	—	6	1	no	—
29	M₃₂	—	3	1	no	—
30	C₂×C₁₆	—	3	1	no	—
31	#108	—	3	1	no	—
32	#108	—	3	1	no	—
33	#101	—	6	1	no	—
34	#100	—	6	1	no	—
35	D₁₂	—	8	1	no	—
36	Dic₆	—	8	1	no	—
37	SL(2,𝔽₃)	D₈	1	1	yes	characteristic
38	C₃×C₈	—	4	1	no	—
39	#41	D₁₂	1	1	yes	characteristic
40	C₂×Q₈	—	6	1	no	—
41	C₂×D₄	—	6	1	no	—
42	Q₁₆	—	6	1	no	—
43	Q₁₆	—	6	1	no	—
44	Q₁₆	—	12	1	no	—
45	SD₁₆	—	12	1	no	—
46	SD₁₆	—	12	1	no	—
47	D₈	—	6	1	no	—
48	D₈	—	6	1	no	—
49	D₈	—	12	1	no	—
50	M₁₆	—	3	1	no	—
51	M₁₆	—	6	1	no	—
52	M₁₆	—	6	1	no	—
53	C₂×C₈	—	3	1	no	—
54	C₁₆	—	3	1	no	—
55	C₁₆	S₄	1	1	yes	characteristic
56	C₂×D₃	—	16	1	no	—
57	C₃×C₄	—	4	1	no	—
58	Dic₃	—	16	1	no	—
59	C₂³	—	12	1	no	—
60	Q₈	—	12	1	no	—
61	Q₈	—	12	1	no	—
62	Q₈	D₂₄	1	1	yes	characteristic
63	D₄	—	3	1	no	—
64	D₄	—	12	1	no	—
65	D₄	—	12	1	no	—
66	C₂×C₄	—	3	1	no	—
67	C₂×C₄	—	12	1	no	—
68	C₈	—	3	1	no	—
69	C₈	—	12	1	no	—
70	C₈	—	12	1	no	—
71	C₈	C₂×S₄	1	1	yes	characteristic
72	C₂×C₃	—	4	1	no	—
73	D₃	—	32	1	no	—
74	C₂²	—	3	1	no	—
75	C₂²	—	24	1	no	—
76	C₂²	—	48	1	no	—
77	C₄	—	3	1	no	—
78	C₄	—	24	1	no	—
79	C₄	C₄⋊S₄	1	1	yes	characteristic
80	C₃	—	4	1	no	Sylow
81	C₂	—	6	1	no	—
82	C₂	—	48	1	no	—
83	C₂	#5402	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#5402
commutator, cosocle	#131⋊C₃	C₂²
Frattini	C₈	C₂×S₄
Fitting	#504	D₃
prime radical	#5407	C₂
hypercenter	C₁₆	S₄
hypocenter	SL(2,𝔽₃)	D₈
LCS	G ▷ #131⋊C₃ ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

Complements

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

N	H ≅ G/N	type
2. #5403	C₂	semidirect
3. #504⋊C₃	C₂	semidirect
10. #504	D₃	semidirect

Maximal subgroups

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

subgroup	index
#5407	2
#5403	2
#504⋊C₃	2
#2223	3
#800	4