Subgroups — #69318

Order 384.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₄	C₂	1	1	yes	maximal, characteristic
2	#4873	C₂	1	2	yes	maximal
3	C₃×#366	C₂	1	1	yes	maximal, characteristic
4	Dic₃×C₁₆	C₂	1	1	yes	maximal, characteristic
5	#4492	C₂	1	2	yes	maximal
6	C₄×Q₃₂	—	3	1	no	maximal, Sylow
7	C₂×Dic₁₂	C₂²	1	2	yes	—
8	#887	—	2	2	no	—
9	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
10	C₃×#107	C₂²	1	2	yes	—
11	C₈×Dic₃	C₂²	1	1	yes	characteristic
12	C₂×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
13	#810	—	2	2	no	—
14	Dic₂₄	C₄	1	4	yes	complemented
15	C₂×Q₃₂	—	3	1	no	—
16	C₄×Q₁₆	—	3	2	no	—
17	#366	D₃	1	1	yes	characteristic
18	#358	—	3	2	no	—
19	C₄×C₁₆	—	3	1	no	—
20	C₂×Dic₆	—	2	2	no	—
21	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
22	C₃×C₄⋊C₄	—	2	2	no	—
23	Dic₃⋊C₄	—	4	2	no	—
24	C₄×Dic₃	—	4	2	no	—
25	C₄×Dic₃	D₄	1	1	yes	characteristic
26	C₂×C₃⋊C₈	D₄	1	1	yes	characteristic
27	Dic₁₂	—	2	2	no	—
28	Dic₁₂	C₂×C₄	1	4	yes	—
29	C₃×C₁₆	C₂×C₄	1	2	yes	—
30	C₃⋊C₁₆	—	2	1	no	—
31	C₂×Q₁₆	—	3	2	no	—
32	C₄×Q₈	—	6	2	no	—
33	Q₃₂	—	3	4	no	—
34	C₂×C₁₆	—	3	1	no	—
35	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
36	#107	C₂×D₃	1	2	yes	—
37	#103	—	6	2	no	—
38	C₄×C₈	—	3	1	no	—
39	C₂×C₃×C₄	—	4	2	no	—
40	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
41	C₂×Dic₃	—	4	2	no	—
42	C₂×Dic₃	D₈	1	1	yes	characteristic
43	C₂×Dic₃	D₈	1	1	yes	characteristic
44	Dic₆	—	2	4	no	—
45	Dic₆	—	4	2	no	—
46	C₃×C₈	C₂²×C₄	1	1	yes	characteristic
47	C₃×C₈	#41	1	1	yes	characteristic
48	C₃⋊C₈	—	2	1	no	—
49	C₂×Q₈	—	6	2	no	—
50	Q₁₆	—	3	4	no	—
51	Q₁₆	—	6	2	no	—
52	C₂×C₈	—	3	1	no	—
53	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
54	C₄⋊C₄	—	2	2	no	—
55	C₄⋊C₄	—	12	2	no	—
56	C₄²	—	3	1	no	—
57	C₄²	—	12	2	no	—
58	C₁₆	—	6	1	no	—
59	C₁₆	C₄×D₃	1	2	yes	—
60	C₂²×C₃	C₂×D₈	1	1	yes	characteristic
61	C₃×C₄	—	8	2	no	—
62	C₃×C₄	C₄×D₄	1	1	yes	characteristic
63	C₃×C₄	#135	1	1	yes	characteristic
64	Dic₃	—	2	1	no	—
65	Dic₃	—	4	4	no	—
66	Dic₃	Q₃₂	1	2	yes	complemented
67	Q₈	—	6	4	no	—
68	Q₈	—	12	2	no	—
69	C₂×C₄	—	3	1	no	—
70	C₂×C₄	—	3	1	no	—
71	C₂×C₄	—	4	2	no	—
72	C₂×C₄	—	12	2	no	—
73	C₂×C₄	D₃×D₄	1	1	yes	characteristic
74	C₈	—	6	1	no	—
75	C₈	C₂×C₄×D₃	1	1	yes	characteristic
76	C₈	#239	1	1	yes	characteristic
77	C₂×C₃	C₄×D₈	1	1	yes	characteristic
78	C₂×C₃	C₂×Q₃₂	1	1	yes	characteristic
79	C₂×C₃	#508	1	1	yes	characteristic
80	C₂²	D₃×D₈	1	1	yes	characteristic
81	C₄	—	3	2	no	—
82	C₄	—	6	1	no	—
83	C₄	—	8	2	no	—
84	C₄	—	12	4	no	—
85	C₄	#893	1	1	yes	characteristic
86	C₄	#919	1	1	yes	characteristic
87	C₃	C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
88	C₂	#4872	1	1	yes	characteristic
89	C₂	#4912	1	1	yes	characteristic
90	C₂	D₃×Q₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×D₈
commutator	C₃×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₃×#366	C₂
socle, prime radical	C₂²×C₃	C₂×D₈
cosocle	C₂×C₃×C₈	C₂³
hypercenter	#366	D₃
hypocenter	C₃	C₄×Q₃₂
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

Complements

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

N	H ≅ G/N	type
14. Dic₂₄	C₄	semidirect
66. Dic₃	Q₃₂	semidirect
87. C₃	C₄×Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₄	2
#4873	2
C₃×#366	2
Dic₃×C₁₆	2
#4492	2
C₄×Q₃₂	3