Subgroups — #71942

Order 384.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×Q₈²	C₂	1	1	yes	maximal, characteristic, complemented
2	#5095	C₂	1	3	yes	maximal
3	#4548	C₂	1	3	yes	maximal, complemented
4	#2205	—	3	1	no	maximal, Sylow
5	#1007	C₂²	1	1	yes	characteristic
6	C₃×C₄⋊Q₈	—	2	3	no	—
7	C₃×C₄⋊Q₈	C₂²	1	3	yes	—
8	C₃×C₄×Q₈	—	2	3	no	—
9	#942	—	2	3	no	—
10	#836	C₂²	1	3	yes	—
11	#805	—	2	3	no	—
12	Q₈²	D₃	1	1	yes	characteristic, complemented
13	#456	—	3	3	no	—
14	#356	—	3	3	no	—
15	C₂×C₃×Q₈	—	2	1	no	—
16	C₂×C₃×Q₈	—	2	1	no	—
17	C₂×C₃×Q₈	D₄	1	3	yes	—
18	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
19	#239	—	4	1	no	—
20	D₃×Q₈	—	4	1	no	—
21	#235	—	2	3	no	—
22	C₃×C₄⋊C₄	—	2	3	no	—
23	C₃×C₄⋊C₄	—	2	6	no	—
24	C₃×C₄²	—	2	3	no	—
25	C₃×C₄²	D₄	1	3	yes	—
26	#216	—	4	3	no	—
27	#215	—	4	3	no	—
28	#208	—	2	3	no	—
29	2₋¹⁺⁴	—	3	1	no	—
30	#137	—	6	3	no	—
31	C₄⋊Q₈	—	2	3	no	—
32	C₄⋊Q₈	C₂×D₃	1	3	yes	—
33	C₄×Q₈	—	2	3	no	—
34	#104	—	6	3	no	—
35	#101	—	3	3	no	—
36	C₃×Q₈	—	2	4	no	—
37	C₃×Q₈	—	4	1	no	—
38	C₃×Q₈	—	4	3	no	—
39	C₃×Q₈	—	8	1	no	—
40	C₂×C₃×C₄	—	2	3	no	—
41	C₂×C₃×C₄	—	2	3	no	—
42	C₂×C₃×C₄	C₂×D₄	1	3	yes	—
43	#67	—	4	1	no	—
44	D₁₂	—	2	3	no	—
45	C₄×D₃	—	4	3	no	—
46	Dic₆	—	2	3	no	—
47	C₃⋊C₈	—	4	3	no	—
48	#41	—	6	3	no	—
49	#41	—	12	1	no	—
50	C₂×Q₈	—	2	1	no	—
51	C₂×Q₈	—	2	1	no	—
52	C₂×Q₈	—	12	1	no	—
53	C₂×Q₈	#67	1	3	yes	—
54	C₂×Q₈	C₂²×D₃	1	1	yes	characteristic
55	Q₁₆	—	12	3	no	—
56	SD₁₆	—	12	3	no	—
57	M₁₆	—	6	3	no	—
58	C₄⋊C₄	—	2	3	no	—
59	C₄⋊C₄	—	2	6	no	—
60	C₄²	—	2	3	no	—
61	C₄²	#67	1	3	yes	—
62	C₂²×C₃	#120	1	1	yes	characteristic
63	C₂×D₃	—	4	1	no	—
64	C₃×C₄	—	2	6	no	—
65	C₃×C₄	—	2	3	no	—
66	C₃×C₄	—	4	3	no	—
67	Dic₃	—	4	1	no	—
68	Q₈	—	2	4	no	—
69	Q₈	—	4	1	no	—
70	Q₈	—	4	3	no	—
71	Q₈	—	6	3	no	—
72	Q₈	—	8	1	no	—
73	D₄	—	6	3	no	—
74	D₄	—	12	1	no	—
75	C₂×C₄	—	2	3	no	—
76	C₂×C₄	—	2	3	no	—
77	C₂×C₄	—	12	3	no	—
78	C₂×C₄	C₂×#67	1	3	yes	—
79	C₈	—	12	3	no	—
80	C₂×C₃	—	2	1	no	—
81	C₂×C₃	B(4,𝔽₂)	1	1	yes	characteristic
82	D₃	—	8	1	no	—
83	C₂²	—	12	1	no	—
84	C₂²	#953	1	1	yes	characteristic
85	C₄	—	2	6	no	—
86	C₄	—	2	3	no	—
87	C₄	—	4	3	no	—
88	C₄	—	12	1	no	—
89	C₃	#2205	1	1	yes	characteristic, complemented, Sylow
90	C₂	—	2	1	no	—
91	C₂	—	24	1	no	—
92	C₂	#5244	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#5244
commutator, cosocle	C₂×C₃×Q₈	C₂³
Frattini	C₂×Q₈	C₂²×D₃
Fitting	C₃×Q₈²	C₂
socle	C₂×C₃	B(4,𝔽₂)
prime radical	#1007	C₂²
hypercenter	Q₈²	D₃
hypocenter	C₃	#2205
LCS	G ▷ C₂×C₃×Q₈ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ Q₈²

Complements

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

N	H ≅ G/N	type
1. C₃×Q₈²	C₂	semidirect
3. #4548	C₂	semidirect
12. Q₈²	D₃	semidirect
89. C₃	#2205	semidirect

Maximal subgroups

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

subgroup	index
C₃×Q₈²	2
#5095	2
#4548	2
#2205	3