Subgroups — #12533

Order 256.

119 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2254	C₂	1	1	yes	maximal, characteristic
2	#2252	C₂	1	1	yes	maximal, characteristic
3	#2241	C₂	1	1	yes	maximal, characteristic
4	#2241	C₂	1	1	yes	maximal, characteristic
5	#1710	C₂	1	1	yes	maximal, characteristic
6	#1707	C₂	1	1	yes	maximal, characteristic
7	#1312	C₂	1	1	yes	maximal, characteristic
8	#500	C₂²	1	1	yes	characteristic
9	#500	C₂²	1	1	yes	characteristic
10	#500	C₂²	1	1	yes	characteristic
11	#498	—	2	1	no	—
12	#494	C₂²	1	1	yes	characteristic
13	#487	—	2	1	no	—
14	#487	—	2	1	no	—
15	#487	—	2	1	no	—
16	#367	—	2	2	no	—
17	#366	—	2	2	no	—
18	#358	—	2	2	no	—
19	#358	—	2	2	no	—
20	C₄×C₁₆	C₂²	1	1	yes	characteristic
21	C₄×C₁₆	C₂²	1	1	yes	characteristic
22	C₈²	C₂²	1	1	yes	characteristic
23	C₂×Q₁₆	—	2	1	no	—
24	C₂×Q₁₆	—	2	1	no	—
25	C₄⋊Q₈	—	2	1	no	—
26	C₄⋊Q₈	—	2	1	no	—
27	C₄⋊Q₈	—	2	1	no	—
28	C₄⋊Q₈	—	2	1	no	—
29	C₂×C₁₆	D₄	1	2	yes	—
30	C₂×C₁₆	D₄	1	2	yes	—
31	#107	—	2	1	no	—
32	#107	—	2	1	no	—
33	#107	—	2	1	no	—
34	#107	—	2	1	no	—
35	#107	—	2	1	no	—
36	#107	—	2	1	no	—
37	#106	—	4	2	no	—
38	#103	—	4	2	no	—
39	#103	—	4	2	no	—
40	#103	—	4	2	no	—
41	C₄×C₈	D₄	1	1	yes	characteristic
42	C₄×C₈	D₄	1	1	yes	characteristic
43	C₄×C₈	C₂³	1	1	yes	characteristic
44	C₂×Q₈	—	4	1	no	—
45	C₂×Q₈	—	4	1	no	—
46	C₂×Q₈	—	4	1	no	—
47	C₂×Q₈	—	4	1	no	—
48	Q₁₆	—	4	2	no	—
49	Q₁₆	—	4	2	no	—
50	C₂×C₈	D₈	1	2	yes	—
51	C₂×C₈	D₈	1	2	yes	—
52	C₂×C₈	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈	C₂×D₄	1	1	yes	characteristic
54	C₄⋊C₄	—	4	1	no	—
55	C₄⋊C₄	—	4	1	no	—
56	C₄⋊C₄	—	4	1	no	—
57	C₄⋊C₄	—	4	1	no	—
58	C₄⋊C₄	—	4	1	no	—
59	C₄⋊C₄	—	4	1	no	—
60	C₄⋊C₄	—	4	1	no	—
61	C₄⋊C₄	—	4	1	no	—
62	C₄²	C₂×D₄	1	1	yes	characteristic
63	C₁₆	—	2	2	no	—
64	C₁₆	—	2	2	no	—
65	Q₈	—	8	1	no	—
66	Q₈	—	8	1	no	—
67	Q₈	—	8	2	no	—
68	Q₈	—	8	2	no	—
69	Q₈	—	8	2	no	—
70	C₂×C₄	—	8	1	no	—
71	C₂×C₄	—	8	1	no	—
72	C₂×C₄	—	8	1	no	—
73	C₂×C₄	—	8	1	no	—
74	C₂×C₄	#127	1	1	yes	characteristic
75	C₂×C₄	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄	C₂×D₈	1	1	yes	characteristic
77	C₈	—	2	2	no	—
78	C₈	—	2	2	no	—
79	C₈	#137	1	2	yes	—
80	C₈	#137	1	2	yes	—
81	C₂²	#493	1	1	yes	characteristic
82	C₄	—	16	1	no	—
83	C₄	—	16	1	no	—
84	C₄	—	16	1	no	—
85	C₄	—	16	1	no	—
86	C₄	#497	1	1	yes	characteristic
87	C₄	#497	1	1	yes	characteristic
88	C₄	#508	1	2	yes	—
89	C₄	#510	1	2	yes	—
90	C₂	#1722	1	1	yes	characteristic
91	C₂	#2249	1	1	yes	characteristic
92	C₂	#2251	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#2254	2
#2252	2
#2241	2
#1710	2
#1707	2
#1312	2