Subgroups — #13541

Order 256.

127 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1944	C₂	1	1	yes	maximal, characteristic
2	#1849	C₂	1	1	yes	maximal, characteristic
3	#1387	C₂	1	2	yes	maximal
4	#1383	C₂	1	2	yes	maximal
5	#1377	C₂	1	1	yes	maximal, characteristic
6	C₂×M₃₂	C₂²	1	2	yes	—
7	C₂×#108	—	2	1	no	—
8	C₂×#108	C₂²	1	2	yes	—
9	C₂×#107	C₂²	1	1	yes	characteristic
10	C₂×#106	C₂²	1	1	yes	characteristic
11	#423	—	2	1	no	—
12	C₂×#97	C₂²	1	1	yes	characteristic
13	#389	—	2	1	no	—
14	#340	—	2	1	no	—
15	C₂×M₁₆	—	2	2	no	—
16	C₂²×C₈	D₄	1	1	yes	characteristic
17	C₂²×C₈	C₂³	1	1	yes	characteristic
18	C₂×C₄⋊C₄	—	2	1	no	—
19	C₂×C₄⋊C₄	—	2	1	no	—
20	C₂×C₄⋊C₄	—	4	1	no	—
21	C₂×C₄²	D₄	1	1	yes	characteristic
22	M₃₂	—	2	2	no	—
23	M₃₂	—	2	2	no	—
24	C₂×C₁₆	—	2	2	no	—
25	#108	—	2	4	no	—
26	#108	—	4	2	no	—
27	#107	—	2	2	no	—
28	#106	—	2	1	no	—
29	#106	Q₈	1	2	yes	—
30	C₄⋊C₈	—	4	2	no	—
31	#97	—	2	2	no	—
32	#95	—	4	1	no	—
33	C₂²×C₄	—	4	1	no	—
34	C₂²×C₄	—	4	1	no	—
35	C₂²×C₄	SD₁₆	1	1	yes	characteristic
36	C₂²×C₄	SD₁₆	1	1	yes	characteristic
37	C₂²×C₄	C₂×D₄	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	—	4	2	no	—
43	C₂×C₈	Q₁₆	1	2	yes	—
44	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈	#41	1	2	yes	—
46	C₂×C₈	#41	1	2	yes	—
47	C₂×C₈	#41	1	1	yes	characteristic
48	C₄⋊C₄	—	2	2	no	—
49	C₄⋊C₄	—	4	1	no	—
50	C₄⋊C₄	—	4	1	no	—
51	C₄⋊C₄	—	4	1	no	—
52	C₄⋊C₄	—	8	2	no	—
53	C₄²	—	2	1	no	—
54	C₄²	—	2	1	no	—
55	C₁₆	—	4	2	no	—
56	C₂³	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	4	1	no	—
60	C₂×C₄	—	4	1	no	—
61	C₂×C₄	—	4	2	no	—
62	C₂×C₄	—	8	1	no	—
63	C₂×C₄	—	8	1	no	—
64	C₂×C₄	—	8	1	no	—
65	C₂×C₄	—	8	1	no	—
66	C₂×C₄	—	8	1	no	—
67	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂×C₄	#123	1	1	yes	characteristic
69	C₂×C₄	#124	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	—	8	2	no	—
75	C₈	#125	1	2	yes	—
76	C₈	#126	1	2	yes	—
77	C₂²	—	2	1	no	—
78	C₂²	—	2	1	no	—
79	C₂²	#476	1	1	yes	characteristic
80	C₂²	#481	1	1	yes	characteristic
81	C₂²	#487	1	1	yes	characteristic
82	C₄	—	4	1	no	—
83	C₄	—	4	1	no	—
84	C₄	—	8	2	no	—
85	C₄	—	16	1	no	—
86	C₄	#400	1	1	yes	characteristic
87	C₄	#477	1	1	yes	characteristic
88	C₄	#484	1	1	yes	characteristic
89	C₄	#490	1	1	yes	characteristic
90	C₂	—	2	1	no	—
91	C₂	—	2	1	no	—
92	C₂	#2100	1	1	yes	characteristic
93	C₂	#2238	1	1	yes	characteristic
94	C₂	#2239	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#481
commutator, Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂³	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#1944	2
#1849	2
#1387	2
#1383	2
#1377	2