Subgroups — #13283

Order 256.

195 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×#346	C₂	1	1	yes	maximal, characteristic
3	#1846	C₂	1	1	yes	maximal, characteristic
4	#1380	C₂	1	4	yes	maximal
5	C₂×C₄⋊Q₈	—	2	1	no	—
6	C₂²×C₁₆	C₂²	1	2	yes	—
7	#500	—	2	2	no	—
8	#500	C₄	1	4	yes	complemented
9	C₂×#107	—	2	1	no	—
10	C₂×#107	C₂²	1	2	yes	—
11	C₂×#107	C₂²	1	2	yes	—
12	C₂×#106	—	2	1	no	—
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	#382	—	2	1	no	—
15	#346	—	2	2	no	—
16	C₂²×Q₈	—	4	1	no	—
17	C₂²×C₈	D₄	1	1	yes	characteristic
18	C₂²×C₈	C₂³	1	1	yes	characteristic
19	C₄⋊Q₈	—	2	4	no	—
20	C₄⋊Q₈	—	4	2	no	—
21	C₂×C₄⋊C₄	—	2	2	no	—
22	C₂×C₄⋊C₄	—	2	2	no	—
23	C₂×C₄⋊C₄	—	4	1	no	—
24	C₂×C₄²	D₄	1	1	yes	characteristic
25	C₂×C₁₆	—	2	2	no	—
26	C₂×C₁₆	—	2	2	no	—
27	C₂×C₁₆	Q₈	1	4	yes	complemented
28	#107	—	2	4	no	—
29	#107	—	2	2	no	—
30	#107	—	2	4	no	—
31	#107	C₂×C₄	1	4	yes	—
32	#106	—	4	2	no	—
33	C₄×C₈	C₂×C₄	1	2	yes	—
34	C₄×C₈	D₄	1	2	yes	—
35	#95	—	4	1	no	—
36	#95	—	4	1	no	—
37	C₂×Q₈	—	4	4	no	—
38	C₂×Q₈	—	8	4	no	—
39	C₂²×C₄	—	4	2	no	—
40	C₂²×C₄	—	4	2	no	—
41	C₂²×C₄	D₈	1	1	yes	characteristic
42	C₂²×C₄	D₈	1	1	yes	characteristic
43	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈	C₂²⋊C₄	1	2	yes	—
45	C₂×C₈	SD₁₆	1	2	yes	—
46	C₂×C₈	Q₁₆	1	2	yes	—
47	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈	C₂×Q₈	1	2	yes	—
50	C₂×C₈	#41	1	2	yes	—
51	C₄⋊C₄	—	2	4	no	—
52	C₄⋊C₄	—	4	4	no	—
53	C₄⋊C₄	—	4	2	no	—
54	C₄⋊C₄	—	4	2	no	—
55	C₄⋊C₄	—	4	2	no	—
56	C₄²	C₂²⋊C₄	1	2	yes	—
57	C₄²	SD₁₆	1	2	yes	—
58	C₁₆	—	2	4	no	—
59	C₂³	C₂×D₈	1	1	yes	characteristic
60	Q₈	—	8	8	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	4	4	no	—
64	C₂×C₄	—	8	4	no	—
65	C₂×C₄	—	8	2	no	—
66	C₂×C₄	—	8	2	no	—
67	C₂×C₄	—	8	2	no	—
68	C₂×C₄	#102	1	2	yes	—
69	C₂×C₄	#102	1	2	yes	—
70	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄	#124	1	1	yes	characteristic
72	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
76	C₈	#103	1	4	yes	—
77	C₈	C₄×Q₈	1	2	yes	—
78	C₈	C₂²⋊Q₈	1	2	yes	—
79	C₂²	C₂×#102	1	1	yes	characteristic
80	C₂²	#486	1	1	yes	characteristic
81	C₂²	#500	1	1	yes	characteristic
82	C₂²	#509	1	2	yes	—
83	C₂²	#510	1	2	yes	—
84	C₄	—	2	2	no	—
85	C₄	—	2	2	no	—
86	C₄	—	8	4	no	—
87	C₄	—	16	2	no	—
88	C₄	#391	1	1	yes	characteristic
89	C₄	C₂×#103	1	1	yes	characteristic
90	C₄	#487	1	1	yes	characteristic
91	C₄	#498	1	1	yes	characteristic
92	C₂	#1990	1	1	yes	characteristic
93	C₂	#2140	1	1	yes	characteristic
94	C₂	#2141	1	1	yes	characteristic
95	C₂	#2243	1	2	yes	—
96	C₂	#2255	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₈
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
8. #500	C₄	semidirect
27. C₂×C₁₆	Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#500	2
C₂×#346	2
#1846	2
#1380	2