Subgroups — #67788

Order 384.

110 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	#4545	C₂	1	2	yes	maximal
3	#1409	—	3	1	no	maximal, Sylow
4	C₂×C₃×SD₁₆	—	2	2	no	—
5	C₃×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
6	C₃×#127	C₂²	1	1	yes	characteristic
7	C₃×C₄×C₈	C₄	1	1	yes	characteristic, complemented
8	#834	—	2	2	no	—
9	#492	D₃	1	1	yes	characteristic
10	#353	—	3	2	no	—
11	C₂×C₃×Q₈	—	2	1	no	—
12	C₂×C₃×D₄	—	4	1	no	—
13	C₂×C₃×D₄	D₄	1	2	yes	—
14	C₃×SD₁₆	—	4	4	no	—
15	C₂×C₃×C₈	—	2	1	no	—
16	C₃×C₄⋊C₄	—	4	1	no	—
17	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
18	#217	—	4	2	no	—
19	C₂×SD₁₆	—	2	2	no	—
20	C₄⋊Q₈	Dic₃	1	1	yes	characteristic, complemented
21	#127	C₂×D₃	1	1	yes	characteristic
22	#99	—	6	2	no	—
23	C₄×C₈	Dic₃	1	1	yes	characteristic, complemented
24	C₂³×C₃	—	2	2	no	—
25	C₃×Q₈	—	4	2	no	—
26	C₃×D₄	—	4	2	no	—
27	C₃×D₄	—	8	2	no	—
28	C₂×C₃×C₄	—	2	1	no	—
29	C₂×C₃×C₄	—	4	1	no	—
30	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×Dic₃	—	8	2	no	—
32	C₃×C₈	—	2	2	no	—
33	C₂×Q₈	—	2	1	no	—
34	C₂×D₄	—	4	1	no	—
35	C₂×D₄	#67	1	2	yes	—
36	SD₁₆	—	4	4	no	—
37	C₂×C₈	—	2	1	no	—
38	C₄⋊C₄	—	4	1	no	—
39	C₂²⋊C₄	—	12	2	no	—
40	C₄²	C₂×Dic₃	1	1	yes	characteristic
41	C₂²×C₃	—	4	2	no	—
42	C₂²×C₃	—	8	2	no	—
43	C₂²×C₃	#99	1	1	yes	characteristic
44	C₃×C₄	—	2	2	no	—
45	C₃×C₄	—	2	1	no	—
46	C₃×C₄	—	8	1	no	—
47	Dic₃	—	16	2	no	—
48	C₂³	—	2	2	no	—
49	Q₈	—	4	2	no	—
50	D₄	—	4	2	no	—
51	D₄	—	8	2	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	4	1	no	—
54	C₂×C₄	—	24	2	no	—
55	C₂×C₄	#217	1	1	yes	characteristic
56	C₈	—	2	2	no	—
57	C₂×C₃	—	2	1	no	—
58	C₂×C₃	—	8	2	no	—
59	C₂×C₃	#353	1	1	yes	characteristic
60	C₂²	—	4	2	no	—
61	C₂²	—	8	2	no	—
62	C₂²	#834	1	1	yes	characteristic
63	C₄	—	2	2	no	—
64	C₄	—	2	1	no	—
65	C₄	—	8	1	no	—
66	C₄	—	48	2	no	—
67	C₃	#1409	1	1	yes	characteristic, complemented, Sylow
68	C₂	—	2	1	no	—
69	C₂	—	8	2	no	—
70	C₂	#4545	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#4545
commutator	C₃×C₄²	C₂×C₄
Frattini	#127	C₂×D₃
Fitting	C₃×#492	C₂
socle	C₂×C₃	#353
prime radical, cosocle	C₃×#127	C₂²
hypercenter	#492	D₃
hypocenter	C₃	#1409
LCS	G ▷ C₃×C₄² ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #127 ◁ #492

Complements

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

N	H ≅ G/N	type
5. C₃×C₄⋊Q₈	C₄	semidirect
7. C₃×C₄×C₈	C₄	semidirect
20. C₄⋊Q₈	Dic₃	semidirect
23. C₄×C₈	Dic₃	semidirect
67. C₃	#1409	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
#4545	2
#1409	3