Subgroups — #67787

Order 384.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#492	C₂	1	1	yes	maximal, characteristic
2	#4463	C₂	1	2	yes	maximal, complemented
3	#1359	—	3	1	no	maximal, Sylow
4	C₂×C₃×SD₁₆	—	2	2	no	—
5	C₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₃×#127	C₄	1	1	yes	characteristic
7	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
8	#812	—	2	2	no	—
9	#492	D₃	1	1	yes	characteristic
10	#364	—	3	2	no	—
11	C₂×C₃×Q₈	—	2	1	no	—
12	C₂×C₃×D₄	—	2	1	no	—
13	C₂×C₃×D₄	—	4	1	no	—
14	C₃×SD₁₆	—	4	2	no	—
15	C₃×SD₁₆	—	4	2	no	—
16	C₂×C₃×C₈	D₄	1	2	yes	—
17	C₃×C₄⋊C₄	—	4	1	no	—
18	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
19	C₃⋊C₁₆	—	4	2	no	—
20	C₂×SD₁₆	—	2	2	no	—
21	C₄⋊Q₈	Dic₃	1	1	yes	characteristic
22	#127	Dic₃	1	1	yes	characteristic
23	M₃₂	—	6	2	no	—
24	C₄×C₈	C₂×D₃	1	1	yes	characteristic
25	C₂³×C₃	—	4	1	no	—
26	C₃×Q₈	—	4	2	no	—
27	C₃×D₄	—	4	2	no	—
28	C₃×D₄	—	8	2	no	—
29	C₂×C₃×C₄	—	2	1	no	—
30	C₂×C₃×C₄	—	4	1	no	—
31	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
32	C₃×C₈	D₈	1	2	yes	—
33	C₃×C₈	SD₁₆	1	2	yes	—
34	C₂×Q₈	—	2	1	no	—
35	C₂×D₄	—	2	1	no	—
36	C₂×D₄	—	4	1	no	—
37	SD₁₆	—	4	2	no	—
38	SD₁₆	—	4	2	no	—
39	C₂×C₈	#67	1	2	yes	—
40	C₄⋊C₄	—	4	1	no	—
41	C₄²	C₂×Dic₃	1	1	yes	characteristic
42	C₁₆	—	12	2	no	—
43	C₂²×C₃	—	8	2	no	—
44	C₂²×C₃	—	8	1	no	—
45	C₂²×C₃	#100	1	1	yes	characteristic
46	C₃×C₄	—	2	2	no	—
47	C₃×C₄	—	8	1	no	—
48	C₃×C₄	#102	1	2	yes	—
49	C₂³	—	4	1	no	—
50	Q₈	—	4	2	no	—
51	D₄	—	4	2	no	—
52	D₄	—	8	2	no	—
53	C₂×C₄	—	2	1	no	—
54	C₂×C₄	—	4	1	no	—
55	C₂×C₄	#217	1	1	yes	characteristic
56	C₈	#213	1	2	yes	—
57	C₈	#214	1	2	yes	—
58	C₂×C₃	—	2	1	no	—
59	C₂×C₃	—	16	1	no	—
60	C₂×C₃	#331	1	1	yes	characteristic
61	C₂²	—	8	2	no	—
62	C₂²	—	8	1	no	—
63	C₂²	#833	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	8	1	no	—
66	C₄	#832	1	2	yes	—
67	C₃	#1359	1	1	yes	characteristic, complemented, Sylow
68	C₂	—	2	1	no	—
69	C₂	—	16	1	no	—
70	C₂	#4544	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#4544
commutator	C₃×C₄²	C₂×C₄
Frattini	C₄×C₈	C₂×D₃
Fitting	C₃×#492	C₂
socle	C₂×C₃	#331
prime radical	C₃×#127	C₄
cosocle	C₃×C₄×C₈	C₂²
hypercenter	#492	D₃
hypocenter	C₃	#1359
LCS	G ▷ C₃×C₄² ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #492

Complements

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

N	H ≅ G/N	type
2. #4463	C₂	semidirect
67. C₃	#1359	semidirect

Maximal subgroups

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

subgroup	index
C₃×#492	2
#4463	2
#1359	3