Subgroups — #67718

Order 384.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#424	C₂	1	1	yes	maximal, characteristic
2	#4549	C₂	1	2	yes	maximal
3	#1323	—	3	1	no	maximal, Sylow
4	C₂×C₃×M₁₆	C₄	1	1	yes	characteristic
5	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
6	C₃×#117	C₄	1	1	yes	characteristic
7	C₃×C₄⋊C₈	—	2	2	no	—
8	C₂×C₃⋊C₁₆	—	2	2	no	—
9	#424	D₃	1	1	yes	characteristic
10	C₂²⋊C₁₆	—	3	2	no	—
11	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
12	C₃×M₁₆	—	4	1	no	—
13	C₂×C₃×C₈	—	2	1	no	—
14	C₂×C₃×C₈	—	4	1	no	—
15	C₂×C₃×C₈	D₄	1	2	yes	—
16	C₃×C₄⋊C₄	C₈	1	2	yes	—
17	C₃×C₂²⋊C₄	—	2	1	no	—
18	C₃×C₄²	—	2	1	no	—
19	C₃⋊C₁₆	—	4	2	no	—
20	C₂×M₁₆	Dic₃	1	1	yes	characteristic
21	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
22	#117	Dic₃	1	1	yes	characteristic
23	C₂×C₁₆	—	6	2	no	—
24	C₄⋊C₈	—	2	2	no	—
25	C₂³×C₃	M₁₆	1	1	yes	characteristic
26	C₂×C₃×C₄	—	2	1	no	—
27	C₂×C₃×C₄	—	2	1	no	—
28	C₂×C₃×C₄	—	2	2	no	—
29	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
31	C₃×C₈	—	2	2	no	—
32	C₃×C₈	—	4	1	no	—
33	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
34	M₁₆	—	4	1	no	—
35	C₂×C₈	—	2	1	no	—
36	C₂×C₈	—	4	1	no	—
37	C₂×C₈	#67	1	2	yes	—
38	C₄⋊C₄	C₃⋊C₈	1	2	yes	—
39	C₂²⋊C₄	—	2	1	no	—
40	C₄²	—	2	1	no	—
41	C₁₆	—	12	2	no	—
42	C₂²×C₃	—	2	1	no	—
43	C₂²×C₃	—	2	2	no	—
44	C₂²×C₃	C₂²⋊C₈	1	1	yes	characteristic
45	C₃×C₄	—	2	1	no	—
46	C₃×C₄	—	4	2	no	—
47	C₃×C₄	#99	1	1	yes	characteristic
48	C₃×C₄	#101	1	1	yes	characteristic
49	C₂³	#208	1	1	yes	characteristic
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
54	C₂×C₄	#217	1	1	yes	characteristic
55	C₈	—	2	2	no	—
56	C₈	—	4	1	no	—
57	C₂×C₃	—	4	1	no	—
58	C₂×C₃	#324	1	1	yes	characteristic
59	C₂×C₃	#350	1	2	yes	—
60	C₂²	—	2	1	no	—
61	C₂²	—	2	2	no	—
62	C₂²	#830	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	—	4	2	no	—
65	C₄	#834	1	1	yes	characteristic
66	C₄	#836	1	1	yes	characteristic
67	C₃	#1323	1	1	yes	characteristic, complemented, Sylow
68	C₂	—	4	1	no	—
69	C₂	#4528	1	1	yes	characteristic
70	C₂	#4561	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#217
commutator	C₂×C₃×C₄	C₂×C₈
Frattini	C₂²×C₈	C₂×D₃
Fitting	C₃×#424	C₂
socle	C₂²×C₃	C₂²⋊C₈
prime radical	C₂³×C₃	M₁₆
cosocle	C₂²×C₃×C₈	C₂²
hypercenter	#424	D₃
hypocenter	C₃	#1323
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #424

Complements

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

N	H ≅ G/N	type
67. C₃	#1323	semidirect

Maximal subgroups

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

subgroup	index
C₃×#424	2
#4549	2
#1323	3