Subgroups — #67572

Order 384.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4699	C₂	1	1	yes	maximal, characteristic
2	C₃×#364	C₂	1	1	yes	maximal, characteristic
3	#4463	C₂	1	1	yes	maximal, characteristic
4	#1358	—	3	1	no	maximal, Sylow
5	C₂×Dic₁₂	—	2	1	no	—
6	C₂×Dic₁₂	—	2	1	no	—
7	#869	C₄	1	2	yes	—
8	C₃×M₃₂	—	2	1	no	—
9	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	#812	—	2	1	no	—
11	#494	—	3	1	no	—
12	#364	—	3	1	no	—
13	#364	D₃	1	1	yes	characteristic
14	C₂×Dic₆	—	2	2	no	—
15	C₂×C₃×C₈	D₄	1	1	yes	characteristic
16	C₂×C₃×C₈	D₄	1	1	yes	characteristic
17	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
18	C₄⋊Dic₃	—	4	2	no	—
19	Dic₁₂	—	4	1	no	—
20	Dic₁₂	—	4	1	no	—
21	Dic₁₂	—	4	1	no	—
22	Dic₁₂	—	4	1	no	—
23	C₃×C₁₆	—	4	1	no	—
24	C₃⋊C₁₆	—	4	1	no	—
25	C₂×Q₁₆	—	6	1	no	—
26	C₂×Q₁₆	—	6	1	no	—
27	C₄⋊Q₈	—	3	2	no	—
28	M₃₂	—	2	1	no	—
29	M₃₂	—	6	1	no	—
30	C₄×C₈	C₂×D₃	1	1	yes	characteristic
31	C₂×C₃×C₄	—	2	1	no	—
32	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×Dic₃	—	4	2	no	—
34	Dic₆	—	4	2	no	—
35	Dic₆	—	4	2	no	—
36	C₃×C₈	D₈	1	1	yes	characteristic
37	C₃×C₈	D₈	1	1	yes	characteristic
38	C₃×C₈	SD₁₆	1	1	yes	characteristic
39	C₃×C₈	SD₁₆	1	1	yes	characteristic
40	C₂×Q₈	—	6	2	no	—
41	Q₁₆	—	12	1	no	—
42	Q₁₆	—	12	1	no	—
43	Q₁₆	—	12	1	no	—
44	Q₁₆	—	12	1	no	—
45	C₂×C₈	D₁₂	1	1	yes	characteristic
46	C₂×C₈	#67	1	1	yes	characteristic
47	C₄⋊C₄	—	12	2	no	—
48	C₄²	C₄×D₃	1	1	yes	characteristic
49	C₁₆	—	4	1	no	—
50	C₁₆	—	12	1	no	—
51	C₂²×C₃	#100	1	1	yes	characteristic
52	C₃×C₄	—	2	2	no	—
53	C₃×C₄	#102	1	1	yes	characteristic
54	C₃×C₄	#102	1	1	yes	characteristic
55	Dic₃	—	8	2	no	—
56	Q₈	—	12	2	no	—
57	Q₈	—	12	2	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	12	2	no	—
60	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
61	C₈	#204	1	1	yes	characteristic
62	C₈	D₂₄	1	1	yes	characteristic
63	C₈	#213	1	1	yes	characteristic
64	C₈	#215	1	1	yes	characteristic
65	C₂×C₃	—	2	1	no	—
66	C₂×C₃	#331	1	1	yes	characteristic
67	C₂²	#823	1	1	yes	characteristic
68	C₄	—	2	2	no	—
69	C₄	—	24	2	no	—
70	C₄	#809	1	1	yes	characteristic
71	C₄	#821	1	1	yes	characteristic
72	C₃	#1358	1	1	yes	characteristic, complemented, Sylow
73	C₂	—	2	1	no	—
74	C₂	#4485	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#4485
commutator	C₃×C₄²	C₂×C₄
Frattini	C₄×C₈	C₂×D₃
Fitting	C₃×#364	C₂
socle	C₂×C₃	#331
prime radical	C₂²×C₃	#100
cosocle	C₃×C₄×C₈	C₂²
hypercenter	#364	D₃
hypocenter	C₃	#1358
LCS	G ▷ C₃×C₄² ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
72. C₃	#1358	semidirect

Maximal subgroups

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

subgroup	index
#4699	2
C₃×#364	2
#4463	2
#1358	3