Subgroups — #72134

Order 384.

736 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃⋊#95	C₂	1	6	yes	maximal
3	#1446	—	3	1	no	maximal, Sylow
4	C₂²×C₃×Q₈	—	2	12	no	—
5	C₂²×C₃×Q₈	C₄	1	4	yes	complemented
6	C₂³×C₃×C₄	C₂²	1	3	yes	—
7	C₂³×Dic₃	C₂²	1	4	yes	—
8	C₃⋊#95	—	2	12	no	—
9	C₂³×Q₈	D₃	1	1	yes	characteristic
10	C₂×#95	—	3	6	no	—
11	C₂⁴×C₃	C₂³	1	1	yes	characteristic
12	C₂×C₃×Q₈	—	2	32	no	—
13	C₂×C₃×Q₈	—	2	24	no	—
14	C₂²×C₃×C₄	—	2	12	no	—
15	C₂²×C₃×C₄	C₂×C₄	1	6	yes	—
16	C₂²×C₃×C₄	D₄	1	12	yes	—
17	C₂²×Dic₃	—	2	12	no	—
18	C₂²×Dic₃	—	2	12	no	—
19	C₂²×Dic₃	Q₈	1	8	yes	complemented
20	C₂²×Q₈	—	2	12	no	—
21	C₂²×Q₈	Dic₃	1	4	yes	complemented
22	C₂³×C₄	—	3	4	no	—
23	C₂³×C₄	C₂×D₃	1	3	yes	—
24	#95	—	6	12	no	—
25	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
26	C₂³×C₃	C₂×D₄	1	6	yes	—
27	C₂³×C₃	C₂×Q₈	1	4	yes	—
28	C₂³×C₃	#41	1	4	yes	—
29	C₃×Q₈	—	2	32	no	—
30	C₂×C₃×C₄	—	2	12	no	—
31	C₂×C₃×C₄	—	2	24	no	—
32	C₂×C₃×C₄	C₂²⋊C₄	1	12	yes	—
33	C₂×Dic₃	—	2	24	no	—
34	C₂×Dic₃	—	4	4	no	—
35	C₂×Dic₃	—	4	12	no	—
36	C₂⁴	C₂²×D₃	1	1	yes	characteristic
37	C₂×Q₈	—	2	32	no	—
38	C₂×Q₈	—	2	24	no	—
39	C₂²×C₄	—	2	12	no	—
40	C₂²×C₄	—	3	8	no	—
41	C₂²×C₄	—	6	12	no	—
42	C₂²×C₄	—	6	12	no	—
43	C₂²×C₄	C₂×Dic₃	1	6	yes	—
44	C₂²×C₄	#67	1	12	yes	—
45	C₂²×C₃	C₂×C₂²⋊C₄	1	3	yes	—
46	C₂²×C₃	C₄×Q₈	1	4	yes	—
47	C₂²×C₃	#120	1	4	yes	—
48	C₂²×C₃	C₂²⋊Q₈	1	12	yes	—
49	C₂²×C₃	#124	1	6	yes	—
50	C₂²×C₃	C₄⋊Q₈	1	6	yes	—
51	C₃×C₄	—	2	12	no	—
52	Dic₃	—	4	8	no	—
53	C₂³	D₃×Q₈	1	4	yes	—
54	C₂³	#239	1	4	yes	—
55	C₂³	C₂²×Dic₃	1	1	yes	characteristic
56	C₂³	C₂×#67	1	6	yes	—
57	Q₈	—	2	32	no	—
58	C₂×C₄	—	2	12	no	—
59	C₂×C₄	—	2	24	no	—
60	C₂×C₄	—	6	24	no	—
61	C₂×C₄	—	12	12	no	—
62	C₂×C₄	—	12	4	no	—
63	C₂×C₄	#217	1	12	yes	—
64	C₂×C₃	#379	1	1	yes	characteristic
65	C₂×C₃	#391	1	6	yes	—
66	C₂×C₃	#393	1	4	yes	—
67	C₂×C₃	#395	1	4	yes	—
68	C₂²	#944	1	6	yes	—
69	C₂²	Q₈×Dic₃	1	4	yes	—
70	C₂²	#946	1	12	yes	—
71	C₂²	#947	1	6	yes	—
72	C₂²	C₂×#217	1	3	yes	—
73	C₂²	#953	1	4	yes	—
74	C₄	—	2	12	no	—
75	C₄	—	12	8	no	—
76	C₃	#1446	1	1	yes	characteristic, complemented, Sylow
77	C₂	#5229	1	6	yes	—
78	C₂	#5230	1	4	yes	—
79	C₂	#5231	1	4	yes	—
80	C₂	#5247	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂⁴	C₂²×D₃
commutator	C₂³×C₃	C₂²×C₄
Fitting	C₂³×C₃×Q₈	C₂
socle, prime radical, cosocle	C₂⁴×C₃	C₂³
hypercenter	C₂³×Q₈	D₃
hypocenter	C₃	#1446
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
5. C₂²×C₃×Q₈	C₄	semidirect
19. C₂²×Dic₃	Q₈	semidirect
21. C₂²×Q₈	Dic₃	semidirect
76. C₃	#1446	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₃×Q₈	2
C₂×C₃⋊#95	2
#1446	3