Subgroups — #12482

Order 256.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2241	C₂	1	1	yes	maximal, characteristic
2	#2183	C₂	1	1	yes	maximal, characteristic
3	#1674	C₂	1	1	yes	maximal, characteristic
4	#1670	C₂	1	1	yes	maximal, characteristic
5	#1371	C₂	1	1	yes	maximal, characteristic
6	#1364	C₂	1	1	yes	maximal, characteristic
7	#1362	C₂	1	1	yes	maximal, characteristic
8	#500	C₂²	1	1	yes	characteristic
9	#494	C₂²	1	1	yes	characteristic
10	#475	—	2	1	no	—
11	#462	—	2	1	no	—
12	C₈×Q₈	C₂²	1	1	yes	characteristic
13	C₄⋊C₁₆	—	2	1	no	—
14	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	#358	—	2	1	no	—
16	#358	—	2	1	no	—
17	#346	—	2	1	no	—
18	C₄×C₁₆	C₂²	1	1	yes	characteristic
19	#335	C₂²	1	1	yes	characteristic
20	#335	C₂²	1	1	yes	characteristic
21	#332	—	2	1	no	—
22	#332	—	2	1	no	—
23	C₂×Q₁₆	—	2	1	no	—
24	C₂×Q₁₆	—	2	1	no	—
25	C₂×Q₁₆	—	4	1	no	—
26	C₄⋊Q₈	—	2	1	no	—
27	C₄⋊Q₈	—	2	1	no	—
28	C₄×Q₈	D₄	1	1	yes	characteristic
29	C₂×C₁₆	—	2	1	no	—
30	C₂×C₁₆	D₄	1	1	yes	characteristic
31	C₂×C₁₆	D₄	1	1	yes	characteristic
32	#107	—	2	1	no	—
33	#107	—	2	1	no	—
34	#106	—	4	1	no	—
35	C₄⋊C₈	—	2	1	no	—
36	C₄⋊C₈	—	2	1	no	—
37	C₄⋊C₈	—	2	1	no	—
38	C₄⋊C₈	D₄	1	1	yes	characteristic
39	#103	—	4	1	no	—
40	#103	—	4	1	no	—
41	C₄×C₈	—	2	1	no	—
42	C₄×C₈	C₂³	1	1	yes	characteristic
43	C₂×Q₈	—	4	1	no	—
44	C₂×Q₈	—	4	1	no	—
45	C₂×Q₈	D₈	1	1	yes	characteristic
46	Q₁₆	—	4	1	no	—
47	Q₁₆	—	4	1	no	—
48	Q₁₆	—	8	1	no	—
49	Q₁₆	—	8	2	no	—
50	C₂×C₈	—	2	1	no	—
51	C₂×C₈	—	4	1	no	—
52	C₂×C₈	—	4	1	no	—
53	C₂×C₈	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈	#41	1	1	yes	characteristic
55	C₄⋊C₄	—	2	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	D₈	1	1	yes	characteristic
61	C₄²	—	2	1	no	—
62	C₄²	C₂×D₄	1	1	yes	characteristic
63	C₁₆	—	2	1	no	—
64	C₁₆	—	2	1	no	—
65	C₁₆	—	4	1	no	—
66	Q₈	—	2	1	no	—
67	Q₈	—	2	1	no	—
68	Q₈	—	8	1	no	—
69	Q₈	—	8	1	no	—
70	Q₈	—	8	1	no	—
71	Q₈	—	8	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	8	1	no	—
75	C₂×C₄	—	8	1	no	—
76	C₂×C₄	#121	1	1	yes	characteristic
77	C₂×C₄	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄	#135	1	1	yes	characteristic
79	C₈	—	2	1	no	—
80	C₈	—	4	1	no	—
81	C₈	—	8	1	no	—
82	C₈	—	8	1	no	—
83	C₈	#135	1	1	yes	characteristic
84	C₈	#137	1	1	yes	characteristic
85	C₂²	#466	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	4	1	no	—
88	C₄	—	4	1	no	—
89	C₄	—	16	1	no	—
90	C₄	—	16	1	no	—
91	C₄	#464	1	1	yes	characteristic
92	C₄	#471	1	1	yes	characteristic
93	C₄	#509	1	1	yes	characteristic
94	C₄	#510	1	1	yes	characteristic
95	C₂	#1682	1	1	yes	characteristic
96	C₂	#2220	1	1	yes	characteristic
97	C₂	#2222	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	#466
commutator, Frattini, cosocle	C₄×C₈	C₂³
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ 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
#2241	2
#2183	2
#1674	2
#1670	2
#1371	2
#1364	2
#1362	2