Finite Group Databasebeta

Subgroups — #19629

Order 256.

525 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 525 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3205	C₂	1	2	yes	maximal, complemented
2	#3169	C₂	1	1	yes	maximal, characteristic, complemented
3	#3151	C₂	1	2	yes	maximal, complemented
4	C₂×#493	C₂	1	1	yes	maximal, characteristic, complemented
5	#1721	C₂	1	4	yes	maximal, complemented
6	#1712	C₂	1	2	yes	maximal, complemented
7	#1706	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×AGL(1,ℤ/8)	—	2	4	no	—
10	C₂²×D₈	—	2	2	no	—
11	#532	C₂²	1	2	yes	—
12	C₂×#127	C₂²	1	2	yes	—
13	#500	C₂²	1	1	yes	characteristic, complemented
14	#496	—	2	4	no	—
15	#493	—	2	2	no	—
16	#493	C₂²	1	2	yes	—
17	#493	C₂²	1	2	yes	—
18	#493	C₂²	1	4	yes	complemented
19	#493	C₂²	1	1	yes	characteristic, complemented
20	#493	C₂²	1	4	yes	complemented
21	#492	C₂²	1	4	yes	complemented
22	#486	C₂²	1	2	yes	complemented
23	#486	C₂²	1	4	yes	complemented
24	#466	—	2	2	no	—
25	#459	—	2	4	no	—
26	#447	—	2	4	no	—
27	C₄×D₈	—	2	2	no	—
28	#431	C₂²	1	1	yes	characteristic
29	C₄×M₁₆	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
31	#322	C₂²	1	2	yes	complemented
32	C₈²	C₂²	1	2	yes	complemented
33	C₂×#41	—	2	2	no	—
34	C₂²×D₄	—	2	2	no	—
35	C₂²×D₄	—	2	2	no	—
36	C₂²×D₄	—	2	2	no	—
37	AGL(1,ℤ/8)	—	4	16	no	—
38	C₂×SD₁₆	—	2	8	no	—
39	C₂×D₈	—	2	4	no	—
40	C₂×D₈	—	2	4	no	—
41	C₂×D₈	—	2	8	no	—
42	C₂×D₈	—	2	2	no	—
43	C₂×D₈	—	2	8	no	—
44	C₂×D₈	—	4	8	no	—
45	C₂×M₁₆	D₄	1	2	yes	—
46	C₂²×C₈	D₄	1	2	yes	—
47	C₄⋊Q₈	C₂³	1	2	yes	—
48	#127	—	2	2	no	—
49	#127	—	2	2	no	—
50	#127	—	2	2	no	—
51	#127	C₂³	1	4	yes	—
52	#127	C₂³	1	2	yes	—
53	#124	—	2	2	no	—
54	#121	—	2	2	no	—
55	#121	—	2	2	no	—
56	C₄×D₄	—	2	2	no	—
57	C₄×D₄	—	2	2	no	—
58	C₂×C₄²	C₂³	1	1	yes	characteristic
59	#107	—	2	2	no	—
60	C₄⋊C₈	D₄	1	2	yes	—
61	#102	—	2	4	no	—
62	#102	—	2	8	no	—
63	C₂²⋊C₈	D₄	1	2	yes	—
64	#97	D₄	1	2	yes	—
65	C₄×C₈	D₄	1	2	yes	—
66	C₄×C₈	C₂³	1	1	yes	characteristic
67	C₄×C₈	C₂³	1	2	yes	—
68	C₄×C₈	C₂³	1	2	yes	—
69	C₄×C₈	C₂³	1	1	yes	characteristic
70	C₂⁴	—	4	2	no	—
71	#41	—	4	8	no	—
72	C₂×Q₈	—	2	2	no	—
73	C₂×D₄	—	2	2	no	—
74	C₂×D₄	—	2	4	no	—
75	C₂×D₄	—	2	2	no	—
76	C₂×D₄	—	2	2	no	—
77	C₂×D₄	—	2	4	no	—
78	C₂×D₄	—	2	4	no	—
79	C₂×D₄	—	4	8	no	—
80	C₂×D₄	—	4	2	no	—
81	C₂×D₄	—	4	2	no	—
82	C₂×D₄	—	4	2	no	—
83	C₂×D₄	—	4	2	no	—
84	C₂×D₄	—	4	8	no	—
85	C₂×D₄	—	8	4	no	—
86	C₂²×C₄	—	4	2	no	—
87	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
90	SD₁₆	—	4	16	no	—
91	D₈	—	4	8	no	—
92	D₈	—	4	16	no	—
93	D₈	—	4	4	no	—
94	D₈	—	4	8	no	—
95	D₈	—	8	8	no	—
96	M₁₆	D₈	1	8	yes	complemented
97	C₂×C₈	—	2	4	no	—
98	C₂×C₈	C₂×D₄	1	2	yes	—
99	C₂×C₈	C₂×D₄	1	4	yes	—
100	C₂×C₈	C₂×D₄	1	4	yes	—
101	C₂×C₈	C₂×D₄	1	2	yes	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₂²⋊C₄	—	4	2	no	—
105	C₂²⋊C₄	—	4	2	no	—
106	C₄²	C₂×D₄	1	1	yes	characteristic
107	C₄²	C₂×D₄	1	1	yes	characteristic
108	C₄²	C₂×D₄	1	1	yes	characteristic
109	C₄²	C₂⁴	1	1	yes	characteristic
110	C₂³	—	4	2	no	—
111	C₂³	—	4	4	no	—
112	C₂³	—	8	4	no	—
113	C₂³	—	8	4	no	—
114	C₂³	—	8	4	no	—
115	C₂³	#127	1	1	yes	characteristic
116	Q₈	—	4	4	no	—
117	D₄	—	4	4	no	—
118	D₄	—	4	8	no	—
119	D₄	—	4	8	no	—
120	D₄	—	4	4	no	—
121	D₄	—	4	4	no	—
122	D₄	—	4	8	no	—
123	D₄	—	8	4	no	—
124	D₄	—	8	4	no	—
125	D₄	—	8	4	no	—
126	D₄	—	8	4	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	8	4	no	—
131	C₂×C₄	#127	1	1	yes	characteristic
132	C₂×C₄	#127	1	1	yes	characteristic
133	C₂×C₄	#127	1	1	yes	characteristic
134	C₂×C₄	C₂×D₈	1	4	yes	—
135	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
138	C₈	—	2	2	no	—
139	C₈	—	2	4	no	—
140	C₈	C₂×D₈	1	8	yes	complemented
141	C₈	AGL(1,ℤ/8)	1	4	yes	complemented
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	8	2	no	—
145	C₂²	—	8	4	no	—
146	C₂²	—	8	4	no	—
147	C₂²	—	8	2	no	—
148	C₂²	—	8	4	no	—
149	C₂²	—	8	2	no	—
150	C₂²	—	16	8	no	—
151	C₂²	#493	1	2	yes	—
152	C₂²	C₂×#127	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	8	2	no	—
156	C₄	#493	1	2	yes	—
157	C₄	C₂²×D₈	1	2	yes	—
158	C₄	C₂×AGL(1,ℤ/8)	1	2	yes	—
159	C₄	#576	1	2	yes	—
160	C₂	—	2	2	no	—
161	C₂	—	16	2	no	—
162	C₂	—	16	4	no	—
163	C₂	C₂×#493	1	1	yes	characteristic
164	C₂	#3151	1	1	yes	characteristic
165	C₂	#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #3205	C₂	semidirect
2. #3169	C₂	semidirect
3. #3151	C₂	semidirect
4. C₂×#493	C₂	semidirect
5. #1721	C₂	semidirect
6. #1712	C₂	semidirect
7. #1706	C₂	semidirect
8. C₈×M₁₆	C₂	semidirect
13. #500	C₂²	semidirect
18. #493	C₂²	semidirect
19. #493	C₂²	semidirect
20. #493	C₂²	semidirect
21. #492	C₂²	semidirect
22. #486	C₂²	semidirect
23. #486	C₂²	semidirect
31. #322	C₂²	semidirect
32. C₈²	C₂²	semidirect
96. M₁₆	D₈	semidirect
140. C₈	C₂×D₈	semidirect
141. C₈	AGL(1,ℤ/8)	semidirect

Maximal subgroups

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

subgroup	index
#3205	2
#3169	2
#3151	2
C₂×#493	2
#1721	2
#1712	2
#1706	2
C₈×M₁₆	2