Subgroups — #71362

Order 384.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#500	C₂	1	1	yes	maximal, characteristic
2	#5092	C₂	1	2	yes	maximal
3	#5078	C₂	1	2	yes	maximal
4	#4682	C₂	1	1	yes	maximal, characteristic
5	C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#1713	—	3	1	no	maximal, Sylow
7	C₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×#216	—	2	4	no	—
9	#869	C₂²	1	2	yes	—
10	C₃×#107	—	2	2	no	—
11	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	#818	—	2	2	no	—
13	#807	—	2	4	no	—
14	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	#500	—	3	2	no	—
17	#500	—	3	1	no	—
18	#500	D₃	1	1	yes	characteristic
19	#494	—	3	2	no	—
20	C₈²	—	3	1	no	—
21	C₂×C₃×Q₈	—	2	2	no	—
22	C₂×Dic₆	—	2	2	no	—
23	C₂×C₃×C₈	D₄	1	2	yes	—
24	C₃×C₄⋊C₄	—	2	2	no	—
25	C₃×C₄⋊C₄	—	2	2	no	—
26	C₃×C₄²	C₂³	1	1	yes	characteristic
27	#216	—	4	8	no	—
28	C₄⋊Dic₃	—	2	2	no	—
29	C₄⋊Dic₃	—	2	2	no	—
30	C₂×C₃⋊C₈	D₄	1	2	yes	—
31	C₂×C₃⋊C₈	D₄	1	2	yes	—
32	C₂×Q₁₆	—	6	4	no	—
33	C₄⋊Q₈	—	3	2	no	—
34	C₄⋊Q₈	C₂×D₃	1	2	yes	—
35	#107	—	2	2	no	—
36	#107	—	6	4	no	—
37	#107	—	6	2	no	—
38	C₄×C₈	—	3	1	no	—
39	C₄×C₈	—	3	1	no	—
40	C₄×C₈	C₂×D₃	1	1	yes	characteristic
41	C₃×Q₈	—	4	4	no	—
42	C₂×C₃×C₄	—	4	2	no	—
43	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
44	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
45	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
46	C₂×Dic₃	—	4	2	no	—
47	Dic₆	—	4	4	no	—
48	C₃×C₈	Q₁₆	1	4	yes	complemented
49	C₃⋊C₈	D₈	1	4	yes	—
50	C₃⋊C₈	Q₁₆	1	4	yes	complemented
51	C₂×Q₈	—	2	2	no	—
52	C₂×Q₈	—	6	2	no	—
53	Q₁₆	—	12	8	no	—
54	C₂×C₈	—	3	2	no	—
55	C₂×C₈	—	3	2	no	—
56	C₂×C₈	#67	1	2	yes	—
57	C₄⋊C₄	—	2	2	no	—
58	C₄⋊C₄	—	2	2	no	—
59	C₄⋊C₄	—	6	2	no	—
60	C₄⋊C₄	—	6	2	no	—
61	C₄²	C₂²×D₃	1	1	yes	characteristic
62	C₂²×C₃	#127	1	1	yes	characteristic
63	C₃×C₄	—	8	2	no	—
64	C₃×C₄	C₂×D₈	1	2	yes	—
65	C₃×C₄	C₂×Q₁₆	1	2	yes	—
66	C₃×C₄	C₂×Q₁₆	1	2	yes	—
67	Dic₃	—	8	2	no	—
68	Q₈	—	4	4	no	—
69	Q₈	—	12	4	no	—
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	—	12	2	no	—
72	C₂×C₄	D₃×D₄	1	1	yes	characteristic
73	C₂×C₄	D₃×D₄	1	1	yes	characteristic
74	C₂×C₄	C₂×#67	1	1	yes	characteristic
75	C₈	—	3	4	no	—
76	C₈	—	3	4	no	—
77	C₈	#216	1	4	yes	complemented
78	C₂×C₃	#493	1	1	yes	characteristic
79	C₂×C₃	#494	1	1	yes	characteristic
80	C₂×C₃	#494	1	1	yes	characteristic
81	C₂²	#940	1	1	yes	characteristic
82	C₄	—	8	2	no	—
83	C₄	—	24	2	no	—
84	C₄	D₃×D₈	1	2	yes	—
85	C₄	D₃×Q₁₆	1	2	yes	—
86	C₄	C₂×#216	1	2	yes	—
87	C₃	#1713	1	1	yes	characteristic, complemented, Sylow
88	C₂	#5092	1	1	yes	characteristic
89	C₂	#5151	1	1	yes	characteristic
90	C₂	#5183	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#940
commutator, cosocle	C₃×C₄²	C₂³
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#500	C₂
socle, prime radical	C₂²×C₃	#127
hypercenter	#500	D₃
hypocenter	C₃	#1713
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

Complements

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

N	H ≅ G/N	type
48. C₃×C₈	Q₁₆	semidirect
50. C₃⋊C₈	Q₁₆	semidirect
77. C₈	#216	semidirect
87. 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×#500	2
#5092	2
#5078	2
#4682	2
C₈×C₃⋊C₈	2
#1713	3