Subgroups — #67658

Order 384.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#812	C₂	1	2	yes	maximal
3	#1376	—	3	1	no	maximal, Sylow
4	C₂²×C₃×C₈	C₄	1	1	yes	characteristic
5	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
6	C₂×C₃×C₄²	C₄	1	1	yes	characteristic
7	C₃×C₄×C₈	—	2	2	no	—
8	#812	—	2	2	no	—
9	#812	C₄	1	4	yes	complemented
10	C₂×C₃⋊C₁₆	—	2	2	no	—
11	C₂×M₃₂	—	3	2	no	—
12	C₂×C₄×C₈	D₃	1	1	yes	characteristic
13	C₂²×C₃×C₄	—	2	1	no	—
14	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
15	C₂×C₃×C₈	—	2	1	no	—
16	C₂×C₃×C₈	—	2	1	no	—
17	C₂×C₃×C₈	C₈	1	2	yes	—
18	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
19	C₂×C₃×C₈	D₄	1	1	yes	characteristic
20	C₂×C₃×C₈	D₄	1	2	yes	—
21	C₂×C₃×C₈	Q₈	1	1	yes	characteristic
22	C₃×C₄²	—	2	1	no	—
23	C₃×C₄²	C₈	1	2	yes	—
24	C₃⋊C₁₆	—	2	4	no	—
25	C₂²×C₈	Dic₃	1	1	yes	characteristic
26	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
27	C₂×C₄²	Dic₃	1	1	yes	characteristic
28	M₃₂	—	3	4	no	—
29	M₃₂	—	6	2	no	—
30	C₂×C₁₆	—	6	2	no	—
31	C₄×C₈	—	2	2	no	—
32	C₂³×C₃	M₁₆	1	1	yes	characteristic
33	C₂×C₃×C₄	—	2	4	no	—
34	C₂×C₃×C₄	—	2	2	no	—
35	C₂×C₃×C₄	C₄²	1	1	yes	characteristic
36	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
39	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
40	C₂×C₃×C₄	M₁₆	1	1	yes	characteristic
41	C₃×C₈	—	2	2	no	—
42	C₃×C₈	C₂²⋊C₄	1	2	yes	—
43	C₃×C₈	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄	—	2	1	no	—
45	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
46	C₂×C₈	—	2	1	no	—
47	C₂×C₈	—	2	1	no	—
48	C₂×C₈	C₃⋊C₈	1	2	yes	—
49	C₂×C₈	Dic₆	1	1	yes	characteristic
50	C₂×C₈	C₄×D₃	1	2	yes	—
51	C₂×C₈	D₁₂	1	1	yes	characteristic
52	C₂×C₈	#67	1	2	yes	—
53	C₄²	—	2	1	no	—
54	C₄²	C₃⋊C₈	1	2	yes	—
55	C₁₆	—	6	4	no	—
56	C₂²×C₃	—	2	2	no	—
57	C₂²×C₃	#97	1	1	yes	characteristic
58	C₂²×C₃	C₂²⋊C₈	1	1	yes	characteristic
59	C₂²×C₃	C₄⋊C₈	1	1	yes	characteristic
60	C₃×C₄	—	2	4	no	—
61	C₃×C₄	#95	1	1	yes	characteristic
62	C₃×C₄	C₄×C₈	1	1	yes	characteristic
63	C₃×C₄	C₂²⋊C₈	1	1	yes	characteristic
64	C₃×C₄	C₄⋊C₈	1	1	yes	characteristic
65	C₂³	#208	1	1	yes	characteristic
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
69	C₂×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
70	C₂×C₄	#208	1	1	yes	characteristic
71	C₂×C₄	C₄×Dic₃	1	1	yes	characteristic
72	C₂×C₄	C₄⋊Dic₃	1	1	yes	characteristic
73	C₂×C₄	#217	1	1	yes	characteristic
74	C₈	—	2	2	no	—
75	C₈	Dic₃⋊C₄	1	2	yes	—
76	C₈	(C₂×D₃)⋊C₄	1	2	yes	—
77	C₂×C₃	—	2	1	no	—
78	C₂×C₃	—	2	1	no	—
79	C₂×C₃	#336	1	1	yes	characteristic
80	C₂×C₃	#364	1	2	yes	—
81	C₂²	—	2	2	no	—
82	C₂²	#803	1	1	yes	characteristic
83	C₂²	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
84	C₂²	#830	1	1	yes	characteristic
85	C₄	—	2	4	no	—
86	C₄	C₄×C₃⋊C₈	1	1	yes	characteristic
87	C₄	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
88	C₄	#830	1	1	yes	characteristic
89	C₄	C₃⋊#95	1	1	yes	characteristic
90	C₃	#1376	1	1	yes	characteristic, complemented, Sylow
91	C₂	—	2	1	no	—
92	C₂	—	2	1	no	—
93	C₂	#4463	1	2	yes	—
94	C₂	#4524	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₁₂
commutator	C₃×C₄	C₄×C₈
Frattini	C₂²×C₈	C₂×D₃
Fitting	C₂×C₃×C₄×C₈	C₂
socle	C₂²×C₃	C₄⋊C₈
prime radical	C₂³×C₃	M₁₆
cosocle	C₂²×C₃×C₈	C₂²
hypercenter	C₂×C₄×C₈	D₃
hypocenter	C₃	#1376
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
9. #812	C₄	semidirect
90. C₃	#1376	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₈	2
C₂×#812	2
#1376	3