Subgroups — #12079

Order 256.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2176	C₂	1	1	yes	maximal, characteristic
2	#1593	C₂	1	1	yes	maximal, characteristic
3	#1579	C₂	1	1	yes	maximal, characteristic
4	#1368	C₂	1	1	yes	maximal, characteristic
5	#1334	C₂	1	1	yes	maximal, characteristic
6	#1334	C₂	1	1	yes	maximal, characteristic
7	#1313	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₃₂	C₄	1	1	yes	characteristic
9	#446	—	2	1	no	—
10	C₈×Q₈	—	2	1	no	—
11	C₄×Q₁₆	C₂²	1	1	yes	characteristic
12	C₄×Q₁₆	C₂²	1	1	yes	characteristic
13	#367	C₄	1	1	yes	characteristic
14	#358	C₄	1	1	yes	characteristic
15	#358	C₄	1	1	yes	characteristic
16	#346	C₂²	1	1	yes	characteristic
17	#346	C₂²	1	1	yes	characteristic
18	#334	C₂²	1	1	yes	characteristic
19	#334	C₂²	1	1	yes	characteristic
20	Q₈⋊C₈	—	2	1	no	—
21	Q₈⋊C₈	—	2	1	no	—
22	#322	C₂²	1	1	yes	characteristic
23	C₂×Q₁₆	C₂×C₄	1	1	yes	characteristic
24	C₂×Q₁₆	C₂×C₄	1	1	yes	characteristic
25	C₄×Q₈	—	2	1	no	—
26	C₄×Q₈	—	2	1	no	—
27	Q₃₂	—	2	2	no	—
28	C₂×C₁₆	—	2	1	no	—
29	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
31	#107	C₂×C₄	1	1	yes	characteristic
32	#107	C₂×C₄	1	1	yes	characteristic
33	C₄⋊C₈	—	2	1	no	—
34	C₄⋊C₈	—	2	1	no	—
35	C₄⋊C₈	—	4	1	no	—
36	C₄⋊C₈	—	4	1	no	—
37	#103	—	2	1	no	—
38	#103	—	2	1	no	—
39	#97	—	4	1	no	—
40	C₄×C₈	—	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	—	2	1	no	—
46	Q₁₆	—	2	1	no	—
47	Q₁₆	—	2	1	no	—
48	Q₁₆	—	2	1	no	—
49	Q₁₆	M₁₆	1	2	yes	—
50	C₂×C₈	—	2	1	no	—
51	C₂×C₈	—	2	1	no	—
52	C₂×C₈	—	4	1	no	—
53	C₂×C₈	—	4	1	no	—
54	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈	#41	1	1	yes	characteristic
56	C₄⋊C₄	—	2	1	no	—
57	C₄⋊C₄	—	2	1	no	—
58	C₄⋊C₄	—	4	1	no	—
59	C₄⋊C₄	—	4	1	no	—
60	C₄²	—	4	1	no	—
61	C₄²	—	4	1	no	—
62	C₄²	C₂×D₄	1	1	yes	characteristic
63	C₁₆	—	2	2	no	—
64	C₁₆	—	2	1	no	—
65	C₁₆	—	2	1	no	—
66	Q₈	—	2	2	no	—
67	Q₈	—	4	1	no	—
68	Q₈	—	4	1	no	—
69	Q₈	—	4	1	no	—
70	C₂×C₄	—	4	1	no	—
71	C₂×C₄	—	4	1	no	—
72	C₂×C₄	—	4	1	no	—
73	C₂×C₄	—	4	1	no	—
74	C₂×C₄	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
76	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	4	1	no	—
79	C₈	—	4	1	no	—
80	C₈	—	8	1	no	—
81	C₈	—	8	1	no	—
82	C₈	C₂×M₁₆	1	1	yes	characteristic
83	C₈	#131	1	1	yes	characteristic
84	C₂²	#442	1	1	yes	characteristic
85	C₄	—	2	1	no	—
86	C₄	—	2	1	no	—
87	C₄	—	4	2	no	—
88	C₄	—	8	1	no	—
89	C₄	—	8	1	no	—
90	C₄	—	8	1	no	—
91	C₄	#435	1	1	yes	characteristic
92	C₄	#443	1	1	yes	characteristic
93	C₂	#1586	1	1	yes	characteristic
94	C₂	#2180	1	1	yes	characteristic
95	C₂	#2180	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	#442
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₄×C₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322 ◁ 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
#2176	2
#1593	2
#1579	2
#1368	2
#1334	2
#1313	2