Subgroups — #10688

Order 256.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1940	C₂	1	1	yes	maximal, characteristic
2	#1849	C₂	1	1	yes	maximal, characteristic
3	#1303	C₂	1	2	yes	maximal
4	#1301	C₂	1	2	yes	maximal
5	#1275	C₂	1	1	yes	maximal, characteristic
6	#431	C₂²	1	2	yes	—
7	C₂×#108	—	2	1	no	—
8	C₂×#108	—	2	1	no	—
9	#423	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈	—	2	1	no	—
11	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×#97	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	#336	—	2	1	no	—
15	C₂×M₁₆	—	2	2	no	—
16	C₂²×C₈	—	2	1	no	—
17	C₂²×C₈	D₄	1	1	yes	characteristic
18	C₂²×C₈	D₄	1	1	yes	characteristic
19	C₂×C₄⋊C₄	—	2	1	no	—
20	C₂×C₄²	—	2	1	no	—
21	C₂×C₄²	C₂³	1	1	yes	characteristic
22	#108	—	4	2	no	—
23	#108	—	4	2	no	—
24	C₄⋊C₈	—	2	2	no	—
25	C₄⋊C₈	—	2	4	no	—
26	C₄⋊C₈	—	2	2	no	—
27	C₄⋊C₈	—	4	2	no	—
28	C₂²⋊C₈	—	2	2	no	—
29	#97	—	2	2	no	—
30	C₄×C₈	—	2	2	no	—
31	#95	Q₈	1	1	yes	characteristic
32	#95	Q₈	1	1	yes	characteristic
33	C₂²×C₄	—	2	1	no	—
34	C₂²×C₄	—	2	1	no	—
35	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
37	C₂²×C₄	#41	1	1	yes	characteristic
38	M₁₆	—	4	2	no	—
39	M₁₆	—	4	2	no	—
40	C₂×C₈	—	2	1	no	—
41	C₂×C₈	—	2	1	no	—
42	C₂×C₈	—	2	1	no	—
43	C₂×C₈	—	2	1	no	—
44	C₂×C₈	—	4	1	no	—
45	C₂×C₈	—	4	2	no	—
46	C₂×C₈	—	4	2	no	—
47	C₂×C₈	—	4	1	no	—
48	C₂×C₈	—	4	1	no	—
49	C₂×C₈	SD₁₆	1	2	yes	—
50	C₂×C₈	Q₁₆	1	2	yes	—
51	C₄⋊C₄	—	4	2	no	—
52	C₄²	—	4	1	no	—
53	C₄²	—	4	1	no	—
54	C₄²	#41	1	2	yes	—
55	C₄²	#41	1	2	yes	—
56	C₂³	C₂²⋊Q₈	1	1	yes	characteristic
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	4	1	no	—
59	C₂×C₄	—	4	1	no	—
60	C₂×C₄	—	4	1	no	—
61	C₂×C₄	—	4	1	no	—
62	C₂×C₄	—	4	1	no	—
63	C₂×C₄	—	4	1	no	—
64	C₂×C₄	—	4	1	no	—
65	C₂×C₄	#123	1	1	yes	characteristic
66	C₂×C₄	#124	1	1	yes	characteristic
67	C₂×C₄	#125	1	2	yes	—
68	C₂×C₄	#126	1	2	yes	—
69	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
70	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₈	—	2	2	no	—
74	C₈	—	2	2	no	—
75	C₈	—	8	2	no	—
76	C₈	—	8	1	no	—
77	C₈	—	8	2	no	—
78	C₂²	—	2	1	no	—
79	C₂²	—	2	1	no	—
80	C₂²	#400	1	1	yes	characteristic
81	C₂²	#476	1	1	yes	characteristic
82	C₂²	#477	1	1	yes	characteristic
83	C₄	—	2	2	no	—
84	C₄	—	2	2	no	—
85	C₄	—	4	2	no	—
86	C₄	—	8	1	no	—
87	C₄	#481	1	1	yes	characteristic
88	C₄	#484	1	1	yes	characteristic
89	C₄	#487	1	1	yes	characteristic
90	C₄	#490	1	1	yes	characteristic
91	C₂	—	2	1	no	—
92	C₂	—	2	1	no	—
93	C₂	#2100	1	1	yes	characteristic
94	C₂	#2102	1	1	yes	characteristic
95	C₂	#2103	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#124
commutator, Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#400
prime radical	C₂³	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Maximal subgroups

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

subgroup	index
#1940	2
#1849	2
#1303	2
#1301	2
#1275	2