Subgroups — #7180

Order 256.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#364	C₂	1	1	yes	maximal, characteristic
2	#1772	C₂	1	1	yes	maximal, characteristic
3	#1450	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂	—	2	1	no	—
5	#431	—	2	1	no	—
6	C₂×#108	C₄	1	1	yes	characteristic
7	#427	C₄	1	1	yes	characteristic
8	#405	—	2	1	no	—
9	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
10	#364	—	2	1	no	—
11	#364	C₄	1	2	yes	complemented
12	#339	—	2	1	no	—
13	#322	—	2	1	no	—
14	C₈²	C₄	1	2	yes	complemented
15	C₂×M₁₆	—	2	1	no	—
16	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈	D₄	1	1	yes	characteristic
18	#117	—	2	1	no	—
19	C₂×C₄²	D₄	1	1	yes	characteristic
20	M₃₂	—	2	2	no	—
21	M₃₂	—	4	1	no	—
22	C₂×C₁₆	—	4	1	no	—
23	#108	—	2	1	no	—
24	#108	C₈	1	2	yes	complemented
25	#107	—	2	1	no	—
26	#106	C₈	1	2	yes	complemented
27	C₄⋊C₈	—	2	2	no	—
28	C₂²⋊C₈	—	2	2	no	—
29	#97	—	4	1	no	—
30	C₄×C₈	—	2	2	no	—
31	C₄×C₈	—	4	1	no	—
32	C₄×C₈	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈	D₄	1	1	yes	characteristic
35	C₄×C₈	Q₈	1	1	yes	characteristic
36	C₂²×C₄	—	2	1	no	—
37	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
38	M₁₆	—	2	2	no	—
39	M₁₆	—	4	1	no	—
40	C₂×C₈	—	2	4	no	—
41	C₂×C₈	—	4	1	no	—
42	C₂×C₈	C₄²	1	1	yes	characteristic
43	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈	C₂×C₈	1	1	yes	characteristic
47	C₂×C₈	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈	M₁₆	1	1	yes	characteristic
49	C₂×C₈	M₁₆	1	1	yes	characteristic
50	C₂×C₈	D₈	1	1	yes	characteristic
51	C₂×C₈	SD₁₆	1	1	yes	characteristic
52	C₂×C₈	SD₁₆	1	1	yes	characteristic
53	C₂×C₈	Q₁₆	1	1	yes	characteristic
54	C₄⋊C₄	—	2	2	no	—
55	C₂²⋊C₄	—	4	1	no	—
56	C₄²	—	2	1	no	—
57	C₄²	—	4	1	no	—
58	C₄²	C₂²⋊C₄	1	1	yes	characteristic
59	C₄²	C₄⋊C₄	1	1	yes	characteristic
60	C₁₆	—	4	2	no	—
61	C₂³	#104	1	1	yes	characteristic
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	4	2	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	#95	1	1	yes	characteristic
67	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
68	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
69	C₂×C₄	#102	1	1	yes	characteristic
70	C₂×C₄	#103	1	1	yes	characteristic
71	C₂×C₄	#104	1	1	yes	characteristic
72	C₈	—	4	4	no	—
73	C₈	—	4	2	no	—
74	C₈	C₄×C₈	1	1	yes	characteristic
75	C₈	#97	1	1	yes	characteristic
76	C₈	#102	1	1	yes	characteristic
77	C₈	#103	1	1	yes	characteristic
78	C₈	C₄⋊C₈	1	1	yes	characteristic
79	C₈	C₄⋊C₈	1	1	yes	characteristic
80	C₈	#106	1	1	yes	characteristic
81	C₈	#107	1	1	yes	characteristic
82	C₂²	—	4	1	no	—
83	C₂²	D₄⋊C₈	1	1	yes	characteristic
84	C₂²	Q₈⋊C₈	1	1	yes	characteristic
85	C₂²	#339	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	4	1	no	—
88	C₄	—	8	2	no	—
89	C₄	D₄⋊C₈	1	1	yes	characteristic
90	C₄	Q₈⋊C₈	1	1	yes	characteristic
91	C₄	#336	1	1	yes	characteristic
92	C₄	#340	1	1	yes	characteristic
93	C₂	—	2	1	no	—
94	C₂	—	2	1	no	—
95	C₂	—	2	1	no	—
96	C₂	#1279	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#106
commutator	C₈	C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂²
socle	C₂	#1279
prime radical	C₂³	#104
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
11. #364	C₄	semidirect
14. C₈²	C₄	semidirect
24. #108	C₈	semidirect
26. #106	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#364	2
#1772	2
#1450	2