Subgroups — #71385

Order 384.

128 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	#4682	C₂	1	1	yes	maximal, characteristic
3	#4490	C₂	1	4	yes	maximal
4	C₄×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	#2254	—	3	1	no	maximal, Sylow
6	C₃×C₄⋊Q₈	—	2	1	no	—
7	#869	—	2	1	no	—
8	C₃×#107	—	2	1	no	—
9	C₃×#107	C₂²	1	2	yes	—
10	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
11	#818	—	2	1	no	—
12	#818	C₂²	1	2	yes	—
13	C₂×C₃⋊C₁₆	C₂²	1	2	yes	—
14	#500	—	3	1	no	—
15	#500	D₃	1	1	yes	characteristic
16	#367	—	3	4	no	—
17	C₄×C₁₆	—	3	1	no	—
18	C₂×C₃×Q₈	—	4	1	no	—
19	C₂×Dic₆	—	4	1	no	—
20	C₂×C₃×C₈	D₄	1	1	yes	characteristic
21	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
22	C₃×C₄⋊C₄	—	2	2	no	—
23	C₃×C₄⋊C₄	—	4	1	no	—
24	C₃×C₄²	D₄	1	1	yes	characteristic
25	C₄⋊Dic₃	—	2	2	no	—
26	C₄⋊Dic₃	—	4	1	no	—
27	C₃⋊C₁₆	Q₈	1	4	yes	complemented
28	C₄⋊Q₈	—	2	1	no	—
29	C₄⋊Q₈	—	6	1	no	—
30	C₂×C₁₆	—	3	2	no	—
31	#107	—	2	1	no	—
32	#107	—	3	2	no	—
33	#107	—	6	1	no	—
34	#107	C₂×D₃	1	2	yes	—
35	C₄×C₈	C₂×D₃	1	1	yes	characteristic
36	C₃×Q₈	—	8	2	no	—
37	C₂×C₃×C₄	—	4	2	no	—
38	C₂×C₃×C₄	D₈	1	1	yes	characteristic
39	C₂×C₃×C₄	D₈	1	1	yes	characteristic
40	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
41	C₂×Dic₃	—	4	2	no	—
42	Dic₆	—	8	2	no	—
43	C₃×C₈	Q₁₆	1	2	yes	—
44	C₃×C₈	C₂×Q₈	1	2	yes	—
45	C₂×Q₈	—	4	1	no	—
46	C₂×Q₈	—	12	1	no	—
47	C₂×C₈	#67	1	1	yes	characteristic
48	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
49	C₄⋊C₄	—	2	2	no	—
50	C₄⋊C₄	—	4	1	no	—
51	C₄⋊C₄	—	6	2	no	—
52	C₄⋊C₄	—	12	1	no	—
53	C₄²	#67	1	1	yes	characteristic
54	C₁₆	—	3	4	no	—
55	C₂²×C₃	C₂×D₈	1	1	yes	characteristic
56	C₃×C₄	—	8	2	no	—
57	C₃×C₄	SD₃₂	1	2	yes	—
58	C₃×C₄	SD₃₂	1	2	yes	—
59	C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
60	C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
61	Dic₃	—	8	2	no	—
62	Q₈	—	8	2	no	—
63	Q₈	—	24	2	no	—
64	C₂×C₄	—	4	2	no	—
65	C₂×C₄	—	12	2	no	—
66	C₂×C₄	#213	1	1	yes	characteristic
67	C₂×C₄	#213	1	1	yes	characteristic
68	C₂×C₄	C₂×#67	1	1	yes	characteristic
69	C₈	#216	1	2	yes	—
70	C₈	D₃×Q₈	1	2	yes	—
71	C₂×C₃	#500	1	1	yes	characteristic
72	C₂×C₃	C₂×SD₃₂	1	1	yes	characteristic
73	C₂×C₃	C₂×SD₃₂	1	1	yes	characteristic
74	C₂²	C₂×#213	1	1	yes	characteristic
75	C₄	—	8	2	no	—
76	C₄	—	24	2	no	—
77	C₄	#827	1	2	yes	—
78	C₄	#828	1	2	yes	—
79	C₄	C₂×#216	1	1	yes	characteristic
80	C₄	#944	1	1	yes	characteristic
81	C₃	#2254	1	1	yes	characteristic, complemented, Sylow
82	C₂	#5078	1	1	yes	characteristic
83	C₂	C₂×#827	1	1	yes	characteristic
84	C₂	C₂×#828	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#213
commutator, cosocle	C₂×C₃×C₈	C₂³
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₃×#500	C₂
socle, prime radical	C₂²×C₃	C₂×D₈
hypercenter	#500	D₃
hypocenter	C₃	#2254
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

Complements

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

N	H ≅ G/N	type
27. C₃⋊C₁₆	Q₈	semidirect
81. C₃	#2254	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
#4682	2
#4490	2
C₄×C₃⋊C₁₆	2
#2254	3