Subgroups — #67560

Order 384.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4695	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#4458	C₂	1	1	yes	maximal, characteristic, complemented
4	#1361	—	3	1	no	maximal, Sylow
5	C₂×D₂₄	—	2	1	no	—
6	C₂×D₂₄	C₄	1	2	yes	—
7	#874	—	2	1	no	—
8	C₂×C₃×C₁₆	—	2	1	no	—
9	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₄⋊(C₃⋊C₈)	—	2	1	no	—
11	#493	—	3	1	no	—
12	C₄⋊C₁₆	D₃	1	1	yes	characteristic, complemented
13	#335	—	3	1	no	—
14	C₂×D₁₂	—	2	2	no	—
15	C₂×D₁₂	—	4	1	no	—
16	C₂×D₁₂	—	4	1	no	—
17	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃×C₈	D₄	1	1	yes	characteristic
19	C₃×C₄²	D₄	1	1	yes	characteristic
20	C₂×C₃⋊C₈	—	4	1	no	—
21	D₂₄	—	4	2	no	—
22	D₂₄	—	4	1	no	—
23	D₂₄	—	4	1	no	—
24	C₃×C₁₆	—	4	1	no	—
25	C₂×D₈	—	3	2	no	—
26	C₂×D₈	—	6	1	no	—
27	#127	—	6	1	no	—
28	C₂×C₁₆	—	2	1	no	—
29	C₄⋊C₈	—	6	1	no	—
30	C₄×C₈	C₂×D₃	1	1	yes	characteristic
31	C₂²×D₃	—	4	2	no	—
32	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₃×C₄	D₈	1	1	yes	characteristic
34	C₂×C₃×C₄	SD₁₆	1	1	yes	characteristic
35	D₁₂	—	4	2	no	—
36	D₁₂	—	4	2	no	—
37	D₁₂	—	8	2	no	—
38	D₁₂	—	8	1	no	—
39	D₁₂	—	8	1	no	—
40	C₃×C₈	—	2	1	no	—
41	C₃×C₈	D₈	1	1	yes	characteristic
42	C₃×C₈	SD₁₆	1	1	yes	characteristic
43	C₃⋊C₈	—	8	1	no	—
44	C₂×D₄	—	6	2	no	—
45	C₂×D₄	—	12	1	no	—
46	C₂×D₄	—	12	1	no	—
47	D₈	—	12	2	no	—
48	D₈	—	12	1	no	—
49	D₈	—	12	1	no	—
50	C₂×C₈	—	12	1	no	—
51	C₂×C₈	C₄×D₃	1	1	yes	characteristic
52	C₂×C₈	#67	1	1	yes	characteristic
53	C₄²	D₁₂	1	1	yes	characteristic
54	C₁₆	—	4	1	no	—
55	C₂²×C₃	#102	1	1	yes	characteristic
56	C₂×D₃	—	8	2	no	—
57	C₂×D₃	—	8	2	no	—
58	C₂×D₃	—	8	2	no	—
59	C₃×C₄	—	2	1	no	—
60	C₃×C₄	#100	1	1	yes	characteristic
61	C₃×C₄	#102	1	1	yes	characteristic
62	C₃×C₄	D₁₆	1	1	yes	characteristic
63	C₃×C₄	SD₃₂	1	1	yes	characteristic
64	C₂³	—	12	2	no	—
65	D₄	—	12	2	no	—
66	D₄	—	12	2	no	—
67	D₄	—	24	2	no	—
68	D₄	—	24	1	no	—
69	D₄	—	24	1	no	—
70	C₂×C₄	#204	1	1	yes	characteristic
71	C₂×C₄	D₂₄	1	1	yes	characteristic
72	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
73	C₈	—	2	1	no	—
74	C₈	—	24	1	no	—
75	C₈	#213	1	1	yes	characteristic
76	C₈	#215	1	1	yes	characteristic
77	C₂×C₃	#331	1	1	yes	characteristic
78	C₂×C₃	#357	1	1	yes	characteristic
79	C₂×C₃	#361	1	1	yes	characteristic
80	D₃	—	16	2	no	—
81	C₂²	—	24	2	no	—
82	C₂²	—	24	2	no	—
83	C₂²	—	24	2	no	—
84	C₂²	#821	1	1	yes	characteristic
85	C₄	—	2	1	no	—
86	C₄	D₄₈	1	1	yes	characteristic
87	C₄	#800	1	1	yes	characteristic
88	C₄	#809	1	1	yes	characteristic
89	C₄	#823	1	1	yes	characteristic
90	C₃	#1361	1	1	yes	characteristic, complemented, Sylow
91	C₂	—	48	2	no	—
92	C₂	#4485	1	1	yes	characteristic
93	C₂	#4509	1	1	yes	characteristic
94	C₂	#4516	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#821
commutator	C₂×C₃×C₈	C₂×C₄
Frattini	C₄×C₈	C₂×D₃
Fitting	C₃×C₄⋊C₁₆	C₂
socle	C₂²×C₃	#102
prime radical	#4695	C₂
cosocle	C₃×C₄×C₈	C₂²
hypercenter	C₄⋊C₁₆	D₃
hypocenter	C₃	#1361
LCS	G ▷ C₂×C₃×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₃×C₄⋊C₁₆	C₂	semidirect
3. #4458	C₂	semidirect
12. C₄⋊C₁₆	D₃	semidirect
90. C₃	#1361	semidirect

Maximal subgroups

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

subgroup	index
#4695	2
C₃×C₄⋊C₁₆	2
#4458	2
#1361	3