Subgroups — #74356

Order 384.

648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#888	C₂	1	6	yes	maximal
2	C₃×#384	C₂	1	1	yes	maximal, characteristic
3	#4975	C₂	1	6	yes	maximal
4	#4935	C₂	1	1	yes	maximal, characteristic
5	C₄²×Dic₃	C₂	1	1	yes	maximal, characteristic
6	#2868	—	3	1	no	maximal, Sylow
7	C₂²×Dic₆	C₂²	1	4	yes	—
8	C₂×C₃×C₄⋊C₄	C₂²	1	6	yes	—
9	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₃	C₂²	1	6	yes	—
11	C₂×Dic₃⋊C₄	C₂²	1	12	yes	—
12	C₂×C₄×Dic₃	C₂²	1	3	yes	—
13	C₂×C₄×Dic₃	C₂²	1	3	yes	—
14	#888	—	2	24	no	—
15	C₂×C₄⋊Q₈	—	3	6	no	—
16	#384	—	3	6	no	—
17	#384	—	3	1	no	—
18	#384	D₃	1	1	yes	characteristic
19	C₄³	—	3	1	no	—
20	C₂²×C₃×C₄	C₂³	1	4	yes	—
21	C₂²×C₃×C₄	C₂³	1	3	yes	—
22	C₂²×Dic₃	C₂³	1	4	yes	—
23	C₂²×Dic₃	C₂³	1	1	yes	characteristic
24	C₂²×Dic₃	C₂³	1	3	yes	—
25	C₂×Dic₆	—	2	24	no	—
26	C₃×C₄⋊C₄	—	2	12	no	—
27	C₃×C₄²	Q₈	1	4	yes	complemented
28	C₄⋊Dic₃	—	2	12	no	—
29	Dic₃⋊C₄	—	2	24	no	—
30	C₄×Dic₃	D₄	1	12	yes	—
31	C₄×Dic₃	Q₈	1	12	yes	complemented
32	C₂²×Q₈	—	3	4	no	—
33	C₄⋊Q₈	—	6	24	no	—
34	C₂×C₄⋊C₄	—	3	12	no	—
35	C₂×C₄⋊C₄	—	3	6	no	—
36	C₂×C₄⋊C₄	C₂×D₃	1	6	yes	—
37	C₂×C₄²	—	3	3	no	—
38	C₂×C₄²	—	3	3	no	—
39	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
40	C₂³×C₃	C₂⁴	1	1	yes	characteristic
41	C₂×C₃×C₄	—	2	12	no	—
42	C₂×C₃×C₄	C₂×D₄	1	6	yes	—
43	C₂×C₃×C₄	C₂×Q₈	1	6	yes	—
44	C₂×C₃×C₄	C₂×Q₈	1	6	yes	—
45	C₂×Dic₃	—	2	12	no	—
46	C₂×Dic₃	C₂×D₄	1	6	yes	—
47	C₂×Dic₃	C₂×D₄	1	6	yes	—
48	C₂×Dic₃	C₂×Q₈	1	12	yes	—
49	Dic₆	—	4	16	no	—
50	C₂×Q₈	—	6	24	no	—
51	C₂²×C₄	—	3	3	no	—
52	C₂²×C₄	—	3	1	no	—
53	C₂²×C₄	—	3	4	no	—
54	C₂²×C₄	C₂²×D₃	1	4	yes	—
55	C₂²×C₄	C₂²×D₃	1	3	yes	—
56	C₄⋊C₄	—	2	12	no	—
57	C₄⋊C₄	—	6	24	no	—
58	C₄⋊C₄	—	6	12	no	—
59	C₄²	—	3	12	no	—
60	C₄²	—	3	12	no	—
61	C₄²	Dic₆	1	4	yes	complemented
62	C₂²×C₃	C₂²×D₄	1	3	yes	—
63	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
64	C₂²×C₃	C₂²×Q₈	1	3	yes	—
65	C₃×C₄	—	4	4	no	—
66	C₃×C₄	C₄⋊Q₈	1	12	yes	complemented
67	Dic₃	—	4	4	no	—
68	Dic₃	#127	1	4	yes	—
69	Dic₃	C₄⋊Q₈	1	12	yes	complemented
70	C₂³	C₂³×D₃	1	1	yes	characteristic
71	Q₈	—	12	16	no	—
72	C₂×C₄	—	2	12	no	—
73	C₂×C₄	—	3	6	no	—
74	C₂×C₄	—	3	6	no	—
75	C₂×C₄	—	3	12	no	—
76	C₂×C₄	—	6	12	no	—
77	C₂×C₄	C₂×Dic₆	1	6	yes	—
78	C₂×C₄	D₃×D₄	1	6	yes	—
79	C₂×C₄	D₃×Q₈	1	6	yes	—
80	C₂×C₃	C₂×#127	1	1	yes	characteristic
81	C₂×C₃	C₂×C₄⋊Q₈	1	3	yes	—
82	C₂×C₃	C₂×C₄⋊Q₈	1	3	yes	—
83	C₂²	C₂²×Dic₆	1	1	yes	characteristic
84	C₂²	C₂×D₃×D₄	1	3	yes	—
85	C₂²	C₂×D₃×Q₈	1	3	yes	—
86	C₄	—	3	4	no	—
87	C₄	—	3	12	no	—
88	C₄	—	4	4	no	—
89	C₄	—	12	4	no	—
90	C₄	#888	1	12	yes	complemented
91	C₃	#2868	1	1	yes	characteristic, complemented, Sylow
92	C₂	C₂×#888	1	3	yes	—
93	C₂	D₃×#127	1	1	yes	characteristic
94	C₂	D₃×C₄⋊Q₈	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator, socle, prime radical, cosocle	C₂³×C₃	C₂⁴
Fitting	C₃×#384	C₂
hypercenter	#384	D₃
hypocenter	C₃	#2868
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ #384

Complements

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

N	H ≅ G/N	type
27. C₃×C₄²	Q₈	semidirect
31. C₄×Dic₃	Q₈	semidirect
61. C₄²	Dic₆	semidirect
66. C₃×C₄	C₄⋊Q₈	semidirect
69. Dic₃	C₄⋊Q₈	semidirect
90. C₄	#888	semidirect
91. C₃	#2868	semidirect

Maximal subgroups

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

subgroup	index
C₂×#888	2
C₃×#384	2
#4975	2
#4935	2
C₄²×Dic₃	2
#2868	3