Subgroups — #69394

Order 384.

300 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄³	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₃⋊C₈)	C₂	1	6	yes	maximal
3	#1842	—	3	1	no	maximal, Sylow
4	C₂×C₃×C₄²	C₄	1	3	yes	—
5	C₂×C₃×C₄²	C₄	1	1	yes	characteristic
6	C₂×C₃×C₄²	C₂²	1	3	yes	—
7	C₂²×C₃⋊C₈	C₂²	1	4	yes	—
8	C₄⋊(C₃⋊C₈)	—	2	12	no	—
9	C₂×C₄⋊C₈	—	3	6	no	—
10	C₄³	D₃	1	1	yes	characteristic
11	C₂²×C₃×C₄	C₂×C₄	1	3	yes	—
12	C₂²×C₃×C₄	C₂×C₄	1	3	yes	—
13	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
14	C₃×C₄²	—	2	6	no	—
15	C₃×C₄²	C₈	1	4	yes	complemented
16	C₃×C₄²	D₄	1	6	yes	—
17	C₃×C₄²	Q₈	1	6	yes	—
18	C₂×C₃⋊C₈	—	2	12	no	—
19	C₂²×C₈	—	3	4	no	—
20	C₂×C₄²	Dic₃	1	3	yes	—
21	C₂×C₄²	Dic₃	1	1	yes	characteristic
22	C₂×C₄²	C₂×D₃	1	3	yes	—
23	C₄⋊C₈	—	6	12	no	—
24	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
25	C₂×C₃×C₄	—	2	6	no	—
26	C₂×C₃×C₄	C₄⋊C₄	1	6	yes	—
27	C₂×C₃×C₄	C₄⋊C₄	1	6	yes	—
28	C₂×C₃×C₄	C₂×C₈	1	6	yes	—
29	C₂×C₃×C₄	M₁₆	1	6	yes	—
30	C₂×C₃×C₄	C₂×D₄	1	3	yes	—
31	C₂×C₃×C₄	C₂×Q₈	1	3	yes	—
32	C₃⋊C₈	—	4	4	no	—
33	C₂²×C₄	C₂×Dic₃	1	3	yes	—
34	C₂²×C₄	C₂×Dic₃	1	3	yes	—
35	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
36	C₂×C₈	—	6	12	no	—
37	C₄²	—	2	6	no	—
38	C₄²	C₃⋊C₈	1	4	yes	complemented
39	C₄²	Dic₆	1	6	yes	—
40	C₄²	D₁₂	1	6	yes	—
41	C₂²×C₃	C₂×C₄⋊C₄	1	3	yes	—
42	C₂²×C₃	C₂²×C₈	1	1	yes	characteristic
43	C₂²×C₃	C₂×M₁₆	1	3	yes	—
44	C₃×C₄	—	2	6	no	—
45	C₃×C₄	C₄⋊C₈	1	12	yes	complemented
46	C₃×C₄	#127	1	1	yes	characteristic
47	C₃×C₄	C₄⋊Q₈	1	3	yes	—
48	C₂³	C₂²×Dic₃	1	1	yes	characteristic
49	C₂×C₄	—	2	6	no	—
50	C₂×C₄	C₂×C₃⋊C₈	1	6	yes	—
51	C₂×C₄	#208	1	6	yes	—
52	C₂×C₄	C₄⋊Dic₃	1	6	yes	—
53	C₂×C₄	C₄⋊Dic₃	1	6	yes	—
54	C₂×C₄	C₂×Dic₆	1	3	yes	—
55	C₂×C₄	C₂×D₁₂	1	3	yes	—
56	C₈	—	12	4	no	—
57	C₂×C₃	#384	1	1	yes	characteristic
58	C₂×C₃	C₂×C₄⋊C₈	1	3	yes	—
59	C₂×C₃	#423	1	3	yes	—
60	C₂²	C₂²×C₃⋊C₈	1	1	yes	characteristic
61	C₂²	C₂×#208	1	3	yes	—
62	C₂²	C₂×C₄⋊Dic₃	1	3	yes	—
63	C₄	—	2	6	no	—
64	C₄	C₄⋊(C₃⋊C₈)	1	12	yes	complemented
65	C₄	#869	1	3	yes	—
66	C₄	#874	1	1	yes	characteristic
67	C₃	#1842	1	1	yes	characteristic, complemented, Sylow
68	C₂	C₂×C₄⋊(C₃⋊C₈)	1	3	yes	—
69	C₂	#4924	1	3	yes	—
70	C₂	#4935	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₈
Fitting	C₃×C₄³	C₂
socle, prime radical	C₂³×C₃	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₄³	D₃
hypocenter	C₃	#1842
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
15. C₃×C₄²	C₈	semidirect
38. C₄²	C₃⋊C₈	semidirect
45. C₃×C₄	C₄⋊C₈	semidirect
64. C₄	C₄⋊(C₃⋊C₈)	semidirect
67. C₃	#1842	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄³	2
C₂×C₄⋊(C₃⋊C₈)	2
#1842	3