Subgroups — #67765

Order 384.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#4458	C₂	1	1	yes	maximal, characteristic
4	#1332	—	3	1	no	maximal, Sylow
5	C₂×C₃×Q₁₆	C₄	1	1	yes	characteristic
6	C₃×C₄×Q₈	—	2	1	no	—
7	C₃×#107	C₄	1	1	yes	characteristic
8	C₃×#103	—	2	1	no	—
9	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₃⋊C₁₆	—	2	1	no	—
11	C₄⋊(C₃⋊C₈)	—	2	1	no	—
12	C₄×Q₁₆	D₃	1	1	yes	characteristic
13	C₄×C₁₆	—	3	1	no	—
14	#335	—	3	1	no	—
15	C₂×C₃×Q₈	—	2	1	no	—
16	C₃×Q₁₆	—	2	1	no	—
17	C₃×Q₁₆	C₈	1	2	yes	complemented
18	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₃×C₈	D₄	1	1	yes	characteristic
20	C₃×C₄⋊C₄	—	2	1	no	—
21	C₃×C₄⋊C₄	—	4	1	no	—
22	C₃×C₄²	—	4	1	no	—
23	C₃×C₄²	D₄	1	1	yes	characteristic
24	C₂×C₃⋊C₈	—	4	1	no	—
25	C₃⋊C₁₆	—	2	2	no	—
26	C₂×Q₁₆	Dic₃	1	1	yes	characteristic
27	C₄×Q₈	—	2	1	no	—
28	C₂×C₁₆	—	6	1	no	—
29	#107	Dic₃	1	1	yes	characteristic
30	C₄⋊C₈	—	6	1	no	—
31	#103	—	2	1	no	—
32	C₄×C₈	C₂×D₃	1	1	yes	characteristic
33	C₃×Q₈	—	2	2	no	—
34	C₃×Q₈	—	4	1	no	—
35	C₂×C₃×C₄	—	4	1	no	—
36	C₂×C₃×C₄	—	4	1	no	—
37	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₃×C₄	D₈	1	1	yes	characteristic
39	C₂×C₃×C₄	SD₁₆	1	1	yes	characteristic
40	C₃×C₈	—	2	1	no	—
41	C₃×C₈	C₂×C₈	1	1	yes	characteristic
42	C₃×C₈	M₁₆	1	1	yes	characteristic
43	C₃⋊C₈	—	8	1	no	—
44	C₂×Q₈	—	2	1	no	—
45	Q₁₆	—	2	1	no	—
46	Q₁₆	C₃⋊C₈	1	2	yes	complemented
47	C₂×C₈	—	12	1	no	—
48	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
49	C₂×C₈	#67	1	1	yes	characteristic
50	C₄⋊C₄	—	2	1	no	—
51	C₄⋊C₄	—	4	1	no	—
52	C₄²	—	4	1	no	—
53	C₄²	#67	1	1	yes	characteristic
54	C₁₆	—	6	2	no	—
55	C₂²×C₃	#102	1	1	yes	characteristic
56	C₃×C₄	—	2	1	no	—
57	C₃×C₄	—	4	2	no	—
58	C₃×C₄	—	8	1	no	—
59	C₃×C₄	C₂²⋊C₈	1	1	yes	characteristic
60	C₃×C₄	#104	1	1	yes	characteristic
61	C₃×C₄	SD₃₂	1	1	yes	characteristic
62	C₃×C₄	Q₃₂	1	1	yes	characteristic
63	Q₈	—	2	2	no	—
64	Q₈	—	4	1	no	—
65	C₂×C₄	—	4	1	no	—
66	C₂×C₄	—	4	1	no	—
67	C₂×C₄	#213	1	1	yes	characteristic
68	C₂×C₄	#214	1	1	yes	characteristic
69	C₂×C₄	#217	1	1	yes	characteristic
70	C₈	—	2	1	no	—
71	C₈	—	24	1	no	—
72	C₈	C₂×C₃⋊C₈	1	1	yes	characteristic
73	C₈	#208	1	1	yes	characteristic
74	C₂×C₃	D₄⋊C₈	1	1	yes	characteristic
75	C₂×C₃	#358	1	1	yes	characteristic
76	C₂×C₃	#359	1	1	yes	characteristic
77	C₂²	#832	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	4	2	no	—
80	C₄	—	8	1	no	—
81	C₄	#828	1	1	yes	characteristic
82	C₄	#829	1	1	yes	characteristic
83	C₄	#830	1	1	yes	characteristic
84	C₄	#837	1	1	yes	characteristic
85	C₃	#1332	1	1	yes	characteristic, complemented, Sylow
86	C₂	D₄⋊(C₃⋊C₈)	1	1	yes	characteristic
87	C₂	#4564	1	1	yes	characteristic
88	C₂	#4567	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#213
commutator	C₃×C₈	C₂×C₈
Frattini	C₄×C₈	C₂×D₃
Fitting	C₃×C₄×Q₁₆	C₂
socle, prime radical	C₂²×C₃	#102
cosocle	C₃×C₄×C₈	C₂²
hypercenter	C₄×Q₁₆	D₃
hypocenter	C₃	#1332
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
17. C₃×Q₁₆	C₈	semidirect
46. Q₁₆	C₃⋊C₈	semidirect
85. C₃	#1332	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×Q₁₆	2
C₄×C₃⋊C₁₆	2
#4458	2
#1332	3