Subgroups — #13284

Order 256.

187 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#499	C₂	1	1	yes	maximal, characteristic
2	C₂×#346	C₂	1	1	yes	maximal, characteristic
3	#1845	C₂	1	1	yes	maximal, characteristic
4	#1380	C₂	1	4	yes	maximal
5	C₂×#125	—	2	1	no	—
6	C₂²×C₁₆	C₂²	1	2	yes	—
7	#499	—	2	2	no	—
8	#499	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	#384	—	2	1	no	—
15	#346	—	2	2	no	—
16	C₂²×C₈	D₄	1	1	yes	characteristic
17	C₂²×C₈	C₂³	1	1	yes	characteristic
18	#125	—	2	4	no	—
19	#125	—	4	2	no	—
20	C₂×C₄⋊C₄	—	2	2	no	—
21	C₂×C₄⋊C₄	—	2	2	no	—
22	C₂×C₄⋊C₄	—	4	2	no	—
23	C₂×C₄⋊C₄	—	4	2	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	—
28	#107	—	2	2	no	—
29	#107	—	2	4	no	—
30	#107	—	4	2	no	—
31	#107	C₂×C₄	1	4	yes	—
32	#106	—	2	4	no	—
33	C₄×C₈	C₂×C₄	1	2	yes	—
34	C₄×C₈	D₄	1	2	yes	—
35	C₂²×C₄	—	4	2	no	—
36	C₂²×C₄	—	4	2	no	—
37	C₂²×C₄	D₈	1	2	yes	—
38	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈	C₂²⋊C₄	1	2	yes	—
40	C₂×C₈	SD₁₆	1	2	yes	—
41	C₂×C₈	Q₁₆	1	2	yes	—
42	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈	C₂×Q₈	1	2	yes	—
45	C₂×C₈	#41	1	2	yes	—
46	C₄⋊C₄	—	2	4	no	—
47	C₄⋊C₄	—	4	8	no	—
48	C₄⋊C₄	—	4	2	no	—
49	C₄⋊C₄	—	4	2	no	—
50	C₄⋊C₄	—	4	2	no	—
51	C₄⋊C₄	—	8	4	no	—
52	C₄²	C₂²⋊C₄	1	2	yes	—
53	C₄²	SD₁₆	1	2	yes	—
54	C₁₆	—	2	4	no	—
55	C₂³	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	4	4	no	—
59	C₂×C₄	—	8	2	no	—
60	C₂×C₄	—	8	2	no	—
61	C₂×C₄	—	8	2	no	—
62	C₂×C₄	—	8	4	no	—
63	C₂×C₄	#102	1	4	yes	—
64	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄	#124	1	1	yes	characteristic
66	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
68	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
70	C₈	#103	1	4	yes	—
71	C₈	C₄×Q₈	1	2	yes	—
72	C₈	C₂²⋊Q₈	1	2	yes	—
73	C₂²	C₂×#102	1	1	yes	characteristic
74	C₂²	#486	1	1	yes	characteristic
75	C₂²	#500	1	1	yes	characteristic
76	C₂²	#509	1	2	yes	—
77	C₂²	#510	1	2	yes	—
78	C₄	—	2	4	no	—
79	C₄	—	8	4	no	—
80	C₄	—	16	2	no	—
81	C₄	#391	1	1	yes	characteristic
82	C₄	C₂×#103	1	1	yes	characteristic
83	C₄	#487	1	1	yes	characteristic
84	C₄	#498	1	1	yes	characteristic
85	C₂	#1990	1	1	yes	characteristic
86	C₂	#2139	1	2	yes	—
87	C₂	#2244	1	1	yes	characteristic
88	C₂	#2245	1	1	yes	characteristic
89	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. #499	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#499	2
C₂×#346	2
#1845	2
#1380	2