Subgroups — #13699

Order 256.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×SD₆₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#2216	C₂	1	1	yes	maximal, characteristic, complemented
4	#2215	C₂	1	1	yes	maximal, characteristic
5	#1423	C₂	1	1	yes	maximal, characteristic, complemented
6	#1416	C₂	1	1	yes	maximal, characteristic, complemented
7	#1415	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₃₂	C₂²	1	1	yes	characteristic
9	C₂×D₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆	C₂²	1	1	yes	characteristic
11	#467	—	2	1	no	—
12	#466	—	2	1	no	—
13	SD₆₄	—	2	2	no	—
14	M₆₄	—	2	2	no	—
15	C₂×C₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₃₂	C₂²	1	1	yes	characteristic
17	#366	C₂²	1	1	yes	characteristic
18	#366	C₂²	1	1	yes	characteristic
19	#358	—	2	1	no	—
20	#357	—	2	1	no	—
21	C₂×Q₁₆	—	2	1	no	—
22	C₂×D₈	—	2	1	no	—
23	C₂²×C₈	D₄	1	1	yes	characteristic
24	C₂²⋊Q₈	—	4	1	no	—
25	#121	—	4	1	no	—
26	Q₃₂	—	2	1	no	—
27	Q₃₂	—	2	1	no	—
28	D₁₆	—	2	1	no	—
29	D₁₆	—	2	1	no	—
30	C₂×C₁₆	—	2	1	no	—
31	C₂×C₁₆	—	2	1	no	—
32	C₂×C₁₆	D₄	1	1	yes	characteristic
33	C₂×C₁₆	C₂³	1	1	yes	characteristic
34	#107	—	2	1	no	—
35	#107	—	2	1	no	—
36	#103	—	4	1	no	—
37	#102	—	4	1	no	—
38	C₃₂	—	2	1	no	—
39	C₃₂	D₄	1	2	yes	complemented
40	C₂×Q₈	—	4	1	no	—
41	C₂×D₄	—	4	1	no	—
42	C₂×D₄	—	8	1	no	—
43	C₂²×C₄	D₈	1	1	yes	characteristic
44	Q₁₆	—	4	1	no	—
45	Q₁₆	—	4	1	no	—
46	D₈	—	4	1	no	—
47	D₈	—	4	1	no	—
48	C₂×C₈	—	2	1	no	—
49	C₂×C₈	—	2	1	no	—
50	C₂×C₈	D₈	1	1	yes	characteristic
51	C₂×C₈	C₂×D₄	1	1	yes	characteristic
52	C₄⋊C₄	—	4	1	no	—
53	C₄⋊C₄	—	4	1	no	—
54	C₄⋊C₄	—	8	1	no	—
55	C₂²⋊C₄	—	8	1	no	—
56	C₂²⋊C₄	—	8	1	no	—
57	C₁₆	—	2	1	no	—
58	C₁₆	C₂×D₄	1	1	yes	characteristic
59	C₁₆	#41	1	1	yes	characteristic
60	C₂³	—	8	1	no	—
61	C₂³	D₁₆	1	1	yes	characteristic
62	Q₈	—	8	1	no	—
63	Q₈	—	8	1	no	—
64	D₄	—	8	1	no	—
65	D₄	—	8	1	no	—
66	D₄	—	16	2	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	8	1	no	—
70	C₂×C₄	—	8	1	no	—
71	C₂×C₄	—	8	1	no	—
72	C₂×C₄	D₁₆	1	1	yes	characteristic
73	C₂×C₄	C₂×D₈	1	1	yes	characteristic
74	C₈	—	2	1	no	—
75	C₈	#121	1	1	yes	characteristic
76	C₈	#135	1	1	yes	characteristic
77	C₂²	—	2	1	no	—
78	C₂²	—	2	1	no	—
79	C₂²	—	2	1	no	—
80	C₂²	—	16	1	no	—
81	C₂²	—	16	1	no	—
82	C₂²	—	16	1	no	—
83	C₂²	C₂×D₁₆	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	16	1	no	—
86	C₄	—	16	1	no	—
87	C₄	—	16	1	no	—
88	C₄	#466	1	1	yes	characteristic
89	C₄	#508	1	1	yes	characteristic
90	C₂	—	4	1	no	—
91	C₂	—	32	1	no	—
92	C₂	#2215	1	1	yes	characteristic
93	C₂	#2263	1	1	yes	characteristic
94	C₂	#2264	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³
prime radical	#2215	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×SD₆₄	C₂	semidirect
2. C₂×M₆₄	C₂	semidirect
3. #2216	C₂	semidirect
5. #1423	C₂	semidirect
6. #1416	C₂	semidirect
7. #1415	C₂	semidirect
39. C₃₂	D₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×SD₆₄	2
C₂×M₆₄	2
#2216	2
#2215	2
#1423	2
#1416	2
#1415	2