Subgroups — #69434

Order 384.

328 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#376	C₂	1	1	yes	maximal, characteristic
2	C₂×#803	C₂	1	3	yes	maximal
3	C₂×C₄×C₃⋊C₈	C₂	1	3	yes	maximal
4	#1726	—	3	1	no	maximal, Sylow
5	C₂×C₃×C₄²	C₂²	1	3	yes	—
6	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₃⋊C₈	C₂²	1	3	yes	—
8	C₃×#95	C₄	1	4	yes	—
9	#803	C₄	1	12	yes	complemented
10	C₄×C₃⋊C₈	C₄	1	12	yes	complemented
11	C₂×#97	—	3	3	no	—
12	C₂×C₄×C₈	—	3	3	no	—
13	#376	D₃	1	1	yes	characteristic
14	C₂²×C₃×C₄	C₂×C₄	1	6	yes	—
15	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
16	C₃×C₄²	C₂×C₄	1	6	yes	—
17	C₃×C₄²	C₂×C₄	1	6	yes	—
18	C₂×C₃⋊C₈	C₂×C₄	1	6	yes	—
19	C₂×C₃⋊C₈	C₂×C₄	1	6	yes	—
20	C₂×C₃⋊C₈	C₂×C₄	1	12	yes	—
21	C₂²×C₈	—	3	1	no	—
22	C₂²×C₈	—	3	3	no	—
23	C₂×C₄²	C₂×D₃	1	3	yes	—
24	#97	—	3	12	no	—
25	C₄×C₈	—	3	12	no	—
26	#95	Dic₃	1	4	yes	—
27	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₃×C₄	—	2	12	no	—
29	C₂×C₃×C₄	C₄²	1	12	yes	—
30	C₂×C₃×C₄	C₂²×C₄	1	3	yes	—
31	C₂×C₃×C₄	C₂²×C₄	1	3	yes	—
32	C₃⋊C₈	C₄²	1	4	yes	—
33	C₃⋊C₈	C₄²	1	12	yes	—
34	C₂²×C₄	C₂×Dic₃	1	6	yes	—
35	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
36	C₂×C₈	—	3	6	no	—
37	C₂×C₈	—	3	6	no	—
38	C₂×C₈	—	3	12	no	—
39	C₄²	C₄×D₃	1	6	yes	—
40	C₄²	C₄×D₃	1	6	yes	—
41	C₂²×C₃	C₂×C₄²	1	3	yes	—
42	C₂²×C₃	#131	1	4	yes	—
43	C₃×C₄	—	2	12	no	—
44	C₃×C₄	C₂×C₄²	1	1	yes	characteristic
45	C₃×C₄	C₂×C₄²	1	3	yes	—
46	C₂³	C₂²×Dic₃	1	1	yes	characteristic
47	C₂×C₄	—	2	12	no	—
48	C₂×C₄	C₄×Dic₃	1	12	yes	—
49	C₂×C₄	C₂×C₄×D₃	1	3	yes	—
50	C₂×C₄	C₂×C₄×D₃	1	3	yes	—
51	C₈	—	3	4	no	—
52	C₈	—	3	12	no	—
53	C₂×C₃	C₄³	1	1	yes	characteristic
54	C₂×C₃	#405	1	6	yes	—
55	C₂²	C₂×C₄×Dic₃	1	3	yes	—
56	C₂²	#948	1	4	yes	—
57	C₄	—	2	12	no	—
58	C₄	C₄²×D₃	1	1	yes	characteristic
59	C₄	C₄²×D₃	1	3	yes	—
60	C₃	#1726	1	1	yes	characteristic, complemented, Sylow
61	C₂	C₄²×Dic₃	1	1	yes	characteristic
62	C₂	#4997	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃
commutator	C₂×C₃	C₄³
Fitting	C₃×#376	C₂
socle, prime radical	C₂³×C₃	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#376	D₃
hypocenter	C₃	#1726
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
9. #803	C₄	semidirect
10. C₄×C₃⋊C₈	C₄	semidirect
60. C₃	#1726	semidirect

Maximal subgroups

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

subgroup	index
C₃×#376	2
C₂×#803	2
C₂×C₄×C₃⋊C₈	2
#1726	3