Subgroups — #71355

Order 384.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#499	C₂	1	1	yes	maximal, characteristic
2	#4682	C₂	1	1	yes	maximal, characteristic
3	#4490	C₂	1	2	yes	maximal
4	#4489	C₂	1	2	yes	maximal
5	#4461	C₂	1	1	yes	maximal, characteristic
6	#2255	—	3	1	no	maximal, Sylow
7	C₃×#125	—	2	1	no	—
8	#869	—	2	1	no	—
9	C₃×#107	C₂²	1	1	yes	characteristic
10	C₃×#107	C₂²	1	1	yes	characteristic
11	C₃×#106	—	2	1	no	—
12	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	#818	—	2	1	no	—
14	#818	C₂²	1	2	yes	—
15	C₂×C₃⋊C₁₆	C₂²	1	2	yes	—
16	#500	—	3	1	no	—
17	#499	D₃	1	1	yes	characteristic
18	#367	—	3	2	no	—
19	#366	—	3	2	no	—
20	#346	—	3	1	no	—
21	C₂×Dic₆	—	4	1	no	—
22	C₂×C₃×C₈	D₄	1	1	yes	characteristic
23	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
24	C₃×C₄⋊C₄	—	2	1	no	—
25	C₃×C₄⋊C₄	—	2	1	no	—
26	C₃×C₄⋊C₄	—	4	1	no	—
27	C₃×C₄⋊C₄	—	4	1	no	—
28	C₃×C₄²	D₄	1	1	yes	characteristic
29	C₄⋊Dic₃	—	2	2	no	—
30	C₄⋊Dic₃	—	4	1	no	—
31	C₃⋊C₁₆	Q₈	1	4	yes	complemented
32	C₄⋊Q₈	—	6	1	no	—
33	#125	—	2	1	no	—
34	C₂×C₁₆	—	3	2	no	—
35	#107	—	3	2	no	—
36	#107	—	6	1	no	—
37	#107	C₂×D₃	1	1	yes	characteristic
38	#107	C₂×D₃	1	1	yes	characteristic
39	#106	—	2	1	no	—
40	C₄×C₈	C₂×D₃	1	1	yes	characteristic
41	C₂×C₃×C₄	—	4	1	no	—
42	C₂×C₃×C₄	—	4	1	no	—
43	C₂×C₃×C₄	D₈	1	1	yes	characteristic
44	C₂×C₃×C₄	D₈	1	1	yes	characteristic
45	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
46	C₂×Dic₃	—	4	2	no	—
47	Dic₆	—	8	2	no	—
48	C₃×C₈	Q₁₆	1	2	yes	—
49	C₃×C₈	C₂×Q₈	1	2	yes	—
50	C₂×Q₈	—	12	1	no	—
51	C₂×C₈	#67	1	1	yes	characteristic
52	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
53	C₄⋊C₄	—	2	1	no	—
54	C₄⋊C₄	—	2	1	no	—
55	C₄⋊C₄	—	4	1	no	—
56	C₄⋊C₄	—	4	1	no	—
57	C₄⋊C₄	—	6	2	no	—
58	C₄⋊C₄	—	12	1	no	—
59	C₄²	#67	1	1	yes	characteristic
60	C₁₆	—	3	4	no	—
61	C₂²×C₃	C₂×D₈	1	1	yes	characteristic
62	C₃×C₄	—	2	1	no	—
63	C₃×C₄	—	2	1	no	—
64	C₃×C₄	—	8	1	no	—
65	C₃×C₄	—	8	1	no	—
66	C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
67	C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
68	Dic₃	—	8	2	no	—
69	Q₈	—	24	2	no	—
70	C₂×C₄	—	4	1	no	—
71	C₂×C₄	—	4	1	no	—
72	C₂×C₄	—	12	2	no	—
73	C₂×C₄	#213	1	1	yes	characteristic
74	C₂×C₄	#213	1	1	yes	characteristic
75	C₂×C₄	C₂×#67	1	1	yes	characteristic
76	C₈	#216	1	2	yes	—
77	C₈	D₃×Q₈	1	2	yes	—
78	C₂×C₃	#500	1	1	yes	characteristic
79	C₂×C₃	#509	1	1	yes	characteristic
80	C₂×C₃	#510	1	1	yes	characteristic
81	C₂²	C₂×#213	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	2	1	no	—
84	C₄	—	8	1	no	—
85	C₄	—	8	1	no	—
86	C₄	—	24	2	no	—
87	C₄	C₂×#216	1	1	yes	characteristic
88	C₄	#944	1	1	yes	characteristic
89	C₃	#2255	1	1	yes	characteristic, complemented, Sylow
90	C₂	#5078	1	1	yes	characteristic
91	C₂	#5193	1	1	yes	characteristic
92	C₂	#5195	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#213
commutator, cosocle	C₂×C₃×C₈	C₂³
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₃×#499	C₂
socle, prime radical	C₂²×C₃	C₂×D₈
hypercenter	#499	D₃
hypocenter	C₃	#2255
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #499

Complements

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

N	H ≅ G/N	type
31. C₃⋊C₁₆	Q₈	semidirect
89. C₃	#2255	semidirect

Maximal subgroups

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

subgroup	index
C₃×#499	2
#4682	2
#4490	2
#4489	2
#4461	2
#2255	3