Subgroups — #12502

Order 256.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2227	C₂	1	1	yes	maximal, characteristic
2	#2225	C₂	1	1	yes	maximal, characteristic
3	#1710	C₂	1	1	yes	maximal, characteristic
4	#1591	C₂	1	1	yes	maximal, characteristic
5	#1367	C₂	1	1	yes	maximal, characteristic
6	#1365	C₂	1	1	yes	maximal, characteristic
7	#1364	C₂	1	1	yes	maximal, characteristic
8	#500	C₂²	1	1	yes	characteristic
9	#500	C₂²	1	1	yes	characteristic
10	#487	—	2	1	no	—
11	#487	—	2	1	no	—
12	#446	—	2	1	no	—
13	C₄×Q₁₆	C₂²	1	1	yes	characteristic
14	#367	—	2	1	no	—
15	#366	—	2	1	no	—
16	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
17	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
18	#358	—	2	1	no	—
19	#358	—	2	1	no	—
20	#335	C₂²	1	1	yes	characteristic
21	Q₈⋊C₈	—	2	1	no	—
22	C₈²	C₂²	1	1	yes	characteristic
23	C₂×Q₁₆	D₄	1	1	yes	characteristic
24	C₄⋊Q₈	—	2	1	no	—
25	C₄⋊Q₈	—	2	1	no	—
26	C₄×Q₈	—	2	1	no	—
27	C₂×C₁₆	—	2	1	no	—
28	C₂×C₁₆	—	2	1	no	—
29	#107	—	2	1	no	—
30	#107	—	2	1	no	—
31	#107	—	2	1	no	—
32	#107	—	2	1	no	—
33	#107	D₄	1	1	yes	characteristic
34	C₄⋊C₈	—	2	1	no	—
35	C₄⋊C₈	—	4	1	no	—
36	#103	—	2	1	no	—
37	#103	—	4	2	no	—
38	#103	—	4	1	no	—
39	#103	—	4	1	no	—
40	#97	—	4	1	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	—	2	1	no	—
45	C₂×Q₈	—	4	1	no	—
46	C₂×Q₈	—	4	1	no	—
47	Q₁₆	—	2	1	no	—
48	Q₁₆	—	2	1	no	—
49	C₂×C₈	—	2	1	no	—
50	C₂×C₈	—	4	1	no	—
51	C₂×C₈	D₈	1	1	yes	characteristic
52	C₂×C₈	D₈	1	1	yes	characteristic
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	—	4	1	no	—
61	C₄²	—	4	1	no	—
62	C₄²	C₂×D₄	1	1	yes	characteristic
63	C₁₆	—	4	1	no	—
64	C₁₆	—	4	1	no	—
65	Q₈	—	4	1	no	—
66	Q₈	—	4	1	no	—
67	Q₈	—	8	2	no	—
68	Q₈	—	8	1	no	—
69	Q₈	—	8	1	no	—
70	C₂×C₄	—	4	1	no	—
71	C₂×C₄	—	4	1	no	—
72	C₂×C₄	—	8	1	no	—
73	C₂×C₄	—	8	1	no	—
74	C₂×C₄	#121	1	1	yes	characteristic
75	C₂×C₄	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	2	1	no	—
79	C₈	—	2	1	no	—
80	C₈	—	4	1	no	—
81	C₈	—	8	1	no	—
82	C₈	#137	1	1	yes	characteristic
83	C₈	#137	1	1	yes	characteristic
84	C₂²	#459	1	1	yes	characteristic
85	C₄	—	2	1	no	—
86	C₄	—	8	1	no	—
87	C₄	—	8	1	no	—
88	C₄	—	16	1	no	—
89	C₄	—	16	1	no	—
90	C₄	#470	1	1	yes	characteristic
91	C₄	#473	1	1	yes	characteristic
92	C₄	#508	1	1	yes	characteristic
93	C₄	#510	1	1	yes	characteristic
94	C₂	#1689	1	1	yes	characteristic
95	C₂	#2211	1	1	yes	characteristic
96	C₂	#2212	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	#459
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
#2227	2
#2225	2
#1710	2
#1591	2
#1367	2
#1365	2
#1364	2