Subgroups — #71496

Order 384.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	Dic₃×Q₁₆	C₂	1	2	yes	maximal
3	#4564	C₂	1	2	yes	maximal
4	#4505	C₂	1	1	yes	maximal, characteristic
5	#4502	C₂	1	1	yes	maximal, characteristic
6	#2176	—	3	1	no	maximal, Sylow
7	C₂×C₃×Q₁₆	C₂²	1	2	yes	—
8	Q₈×Dic₃	—	2	2	no	—
9	C₃×Q₃₂	C₄	1	4	yes	complemented
10	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
11	#835	—	2	2	no	—
12	#818	C₂²	1	2	yes	—
13	C₈×Dic₃	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂×Q₃₂	D₃	1	1	yes	characteristic
16	C₄×Q₁₆	—	3	2	no	—
17	#367	—	3	1	no	—
18	#358	—	3	2	no	—
19	#346	—	3	1	no	—
20	C₂×C₃×Q₈	—	2	2	no	—
21	C₃×Q₁₆	—	2	2	no	—
22	C₃×Q₁₆	C₂×C₄	1	4	yes	—
23	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
24	C₄⋊Dic₃	—	2	2	no	—
25	C₄⋊Dic₃	—	4	2	no	—
26	C₄×Dic₃	—	4	2	no	—
27	C₄×Dic₃	D₄	1	1	yes	characteristic
28	C₂×C₃⋊C₈	D₄	1	1	yes	characteristic
29	C₃×C₁₆	C₂×C₄	1	2	yes	—
30	C₃⋊C₁₆	—	2	1	no	—
31	C₂×Q₁₆	C₂×D₃	1	2	yes	—
32	C₄×Q₈	—	6	2	no	—
33	Q₃₂	Dic₃	1	4	yes	complemented
34	C₂×C₁₆	—	3	1	no	—
35	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
36	#107	—	3	2	no	—
37	#103	—	6	2	no	—
38	C₄×C₈	—	3	1	no	—
39	C₃×Q₈	—	2	4	no	—
40	C₃×Q₈	—	4	2	no	—
41	C₂×C₃×C₄	—	4	2	no	—
42	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
43	C₂×Dic₃	—	4	2	no	—
44	C₂×Dic₃	D₈	1	2	yes	—
45	C₃×C₈	C₂²×C₄	1	1	yes	characteristic
46	C₃×C₈	#41	1	1	yes	characteristic
47	C₃⋊C₈	—	2	1	no	—
48	C₂×Q₈	—	2	2	no	—
49	Q₁₆	—	2	2	no	—
50	Q₁₆	C₂×Dic₃	1	4	yes	—
51	C₂×C₈	—	3	1	no	—
52	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
53	C₄⋊C₄	—	6	2	no	—
54	C₄⋊C₄	—	12	2	no	—
55	C₄²	—	3	1	no	—
56	C₄²	—	12	2	no	—
57	C₁₆	—	6	1	no	—
58	C₁₆	C₂×Dic₃	1	2	yes	—
59	C₂²×C₃	C₂×D₈	1	1	yes	characteristic
60	C₃×C₄	—	4	4	no	—
61	C₃×C₄	C₄×D₄	1	1	yes	characteristic
62	C₃×C₄	#135	1	1	yes	characteristic
63	Dic₃	—	2	2	no	—
64	Dic₃	—	8	2	no	—
65	Q₈	—	2	4	no	—
66	Q₈	—	4	2	no	—
67	C₂×C₄	—	3	2	no	—
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	—	12	2	no	—
70	C₂×C₄	D₃×D₄	1	1	yes	characteristic
71	C₈	—	6	1	no	—
72	C₈	#237	1	1	yes	characteristic
73	C₈	C₂²×Dic₃	1	1	yes	characteristic
74	C₂×C₃	C₄×D₈	1	1	yes	characteristic
75	C₂×C₃	#510	1	2	yes	—
76	C₂²	D₃×D₈	1	1	yes	characteristic
77	C₄	—	4	4	no	—
78	C₄	—	6	2	no	—
79	C₄	—	24	2	no	—
80	C₄	#912	1	1	yes	characteristic
81	C₄	D₄×Dic₃	1	1	yes	characteristic
82	C₃	#2176	1	1	yes	characteristic, complemented, Sylow
83	C₂	#4918	1	2	yes	—
84	C₂	Dic₃×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×D₈
commutator	C₃×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₂×C₃×Q₃₂	C₂
socle, prime radical	C₂²×C₃	C₂×D₈
cosocle	C₂×C₃×C₈	C₂³
hypercenter	C₂×Q₃₂	D₃
hypocenter	C₃	#2176
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
9. C₃×Q₃₂	C₄	semidirect
33. Q₃₂	Dic₃	semidirect
82. C₃	#2176	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×Q₃₂	2
Dic₃×Q₁₆	2
#4564	2
#4505	2
#4502	2
#2176	3