Subgroups — #71298

Order 384.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#494	C₂	1	1	yes	maximal, characteristic
2	#5078	C₂	1	4	yes	maximal
3	#4699	C₂	1	1	yes	maximal, characteristic
4	C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#1713	—	3	1	no	maximal, Sylow
6	C₂×C₃×Q₁₆	—	2	2	no	—
7	C₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×Dic₁₂	—	2	2	no	—
9	#869	C₂²	1	2	yes	—
10	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
11	#807	—	2	8	no	—
12	C₄×C₃⋊C₈	C₂²	1	2	yes	—
13	#500	—	3	4	no	—
14	#494	—	3	1	no	—
15	#494	D₃	1	1	yes	characteristic
16	C₈²	—	3	1	no	—
17	C₂×C₃×Q₈	—	2	2	no	—
18	C₂×Dic₆	—	2	2	no	—
19	C₃×Q₁₆	—	4	4	no	—
20	C₂×C₃×C₈	D₄	1	2	yes	—
21	C₃×C₄⋊C₄	—	2	4	no	—
22	C₃×C₄²	C₂³	1	1	yes	characteristic
23	C₄⋊Dic₃	—	2	4	no	—
24	C₂×C₃⋊C₈	D₄	1	4	yes	—
25	Dic₁₂	—	4	4	no	—
26	C₂×Q₁₆	—	2	2	no	—
27	C₂×Q₁₆	—	6	2	no	—
28	C₄⋊Q₈	—	3	2	no	—
29	C₄⋊Q₈	C₂×D₃	1	2	yes	—
30	#107	—	6	8	no	—
31	C₄×C₈	—	3	2	no	—
32	C₄×C₈	C₂×D₃	1	1	yes	characteristic
33	C₃×Q₈	—	4	4	no	—
34	C₂×C₃×C₄	—	4	2	no	—
35	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
36	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
37	C₂×Dic₃	—	4	2	no	—
38	Dic₆	—	4	4	no	—
39	C₃×C₈	D₈	1	4	yes	—
40	C₃⋊C₈	Q₁₆	1	8	yes	complemented
41	C₂×Q₈	—	2	2	no	—
42	C₂×Q₈	—	6	2	no	—
43	Q₁₆	—	4	4	no	—
44	Q₁₆	—	12	4	no	—
45	C₂×C₈	—	3	4	no	—
46	C₂×C₈	#67	1	2	yes	—
47	C₄⋊C₄	—	2	4	no	—
48	C₄⋊C₄	—	6	4	no	—
49	C₄²	C₂²×D₃	1	1	yes	characteristic
50	C₂²×C₃	#127	1	1	yes	characteristic
51	C₃×C₄	—	8	2	no	—
52	C₃×C₄	C₂×D₈	1	2	yes	—
53	C₃×C₄	C₂×Q₁₆	1	4	yes	—
54	Dic₃	—	8	2	no	—
55	Q₈	—	4	4	no	—
56	Q₈	—	12	4	no	—
57	C₂×C₄	—	4	2	no	—
58	C₂×C₄	—	12	2	no	—
59	C₂×C₄	D₃×D₄	1	2	yes	—
60	C₂×C₄	C₂×#67	1	1	yes	characteristic
61	C₈	—	3	8	no	—
62	C₈	#213	1	4	yes	—
63	C₂×C₃	#493	1	1	yes	characteristic
64	C₂×C₃	#494	1	2	yes	—
65	C₂²	#940	1	1	yes	characteristic
66	C₄	—	8	2	no	—
67	C₄	—	24	2	no	—
68	C₄	D₃×Q₁₆	1	4	yes	—
69	C₄	C₂×#213	1	2	yes	—
70	C₃	#1713	1	1	yes	characteristic, complemented, Sylow
71	C₂	#5073	1	1	yes	characteristic
72	C₂	#5183	1	2	yes	—

	subgroup	G/H
center	C₂²	#940
commutator, cosocle	C₃×C₄²	C₂³
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#494	C₂
socle, prime radical	C₂²×C₃	#127
hypercenter	#494	D₃
hypocenter	C₃	#1713
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
40. C₃⋊C₈	Q₁₆	semidirect
70. C₃	#1713	semidirect

Maximal subgroups

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

subgroup	index
C₃×#494	2
#5078	2
#4699	2
C₈×C₃⋊C₈	2
#1713	3