Finite Group Databasebeta

Subgroups — #28005

Order 256.

317 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 317 conjugacy classes of subgroups — too many to draw. Only its 127 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	#2506	C₂	1	2	yes	maximal
2	#2099	C₂	1	2	yes	maximal
3	#2087	C₂	1	2	yes	maximal
4	#1944	C₂	1	1	yes	maximal, characteristic
5	#1944	C₂	1	1	yes	maximal, characteristic
6	#1943	C₂	1	1	yes	maximal, characteristic
7	#1941	C₂	1	1	yes	maximal, characteristic
8	#1921	C₂	1	2	yes	maximal
9	#1845	C₂	1	1	yes	maximal, characteristic
10	#1776	C₂	1	1	yes	maximal, characteristic
11	#1771	C₂	1	1	yes	maximal, characteristic
12	C₂×#125	—	2	2	no	—
13	C₂×#107	C₂²	1	2	yes	—
14	C₂×#107	C₂²	1	1	yes	characteristic
15	C₂×#107	C₂²	1	1	yes	characteristic
16	C₂×#106	C₂²	1	1	yes	characteristic
17	C₂×#106	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×#97	C₂²	1	1	yes	characteristic
20	C₂×#97	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
22	#400	—	2	2	no	—
23	#398	—	2	2	no	—
24	#389	C₂²	1	2	yes	—
25	#389	C₂²	1	2	yes	—
26	#389	C₂²	1	2	yes	—
27	#389	C₂²	1	2	yes	—
28	#387	—	2	2	no	—
29	#384	C₂²	1	2	yes	—
30	C₄×C₄⋊C₄	C₂²	1	2	yes	—
31	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
32	#340	C₂²	1	2	yes	—
33	#340	C₂²	1	2	yes	—
34	#340	C₂²	1	2	yes	—
35	#340	C₂²	1	2	yes	—
36	#340	C₂²	1	2	yes	—
37	#336	C₂²	1	1	yes	characteristic
38	#336	C₂²	1	1	yes	characteristic
39	C₂²×C₈	C₂³	1	1	yes	characteristic
40	C₂²×C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₈	C₂³	1	1	yes	characteristic
43	#125	—	4	8	no	—
44	C₂×C₄⋊C₄	—	2	2	no	—
45	C₂×C₄⋊C₄	—	2	2	no	—
46	C₂×C₄⋊C₄	—	2	2	no	—
47	C₂×C₄⋊C₄	—	2	2	no	—
48	C₂×C₄⋊C₄	—	2	2	no	—
49	C₂×C₄⋊C₄	—	2	2	no	—
50	C₂×C₄⋊C₄	D₄	1	2	yes	—
51	C₂×C₄⋊C₄	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄	C₂³	1	2	yes	—
55	C₂×C₄²	—	2	2	no	—
56	C₂×C₄²	C₂³	1	1	yes	characteristic
57	C₂×C₄²	C₂³	1	1	yes	characteristic
58	C₂×C₄²	C₂³	1	1	yes	characteristic
59	#107	—	2	4	no	—
60	#107	—	2	2	no	—
61	#107	—	2	2	no	—
62	#106	—	2	2	no	—
63	#106	Q₈	1	4	yes	—
64	C₄⋊C₈	—	2	2	no	—
65	#97	—	2	2	no	—
66	#97	—	2	2	no	—
67	C₄×C₈	—	2	2	no	—
68	#95	—	2	2	no	—
69	#95	—	2	2	no	—
70	#95	—	2	2	no	—
71	#95	—	2	2	no	—
72	#95	—	2	2	no	—
73	#95	D₄	1	2	yes	—
74	C₂²×C₄	—	2	2	no	—
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	—	2	2	no	—
77	C₂²×C₄	—	2	2	no	—
78	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄	C₂×D₄	1	2	yes	—
80	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄	C₂×D₄	1	2	yes	—
82	C₂²×C₄	C₂⁴	1	1	yes	characteristic
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	C₂×Q₈	1	2	yes	—
92	C₂×C₈	#41	1	2	yes	—
93	C₂×C₈	#41	1	2	yes	—
94	C₂×C₈	#41	1	2	yes	—
95	C₄⋊C₄	—	2	4	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	2	2	no	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	4	4	no	—
104	C₄⋊C₄	—	4	4	no	—
105	C₄⋊C₄	—	4	4	no	—
106	C₄⋊C₄	—	4	4	no	—
107	C₄⋊C₄	—	4	4	no	—
108	C₄⋊C₄	—	4	4	no	—
109	C₄⋊C₄	C₂×Q₈	1	4	yes	—
110	C₄²	—	2	1	no	—
111	C₄²	—	2	1	no	—
112	C₄²	—	2	1	no	—
113	C₄²	—	2	1	no	—
114	C₄²	—	2	1	no	—
115	C₄²	—	2	1	no	—
116	C₄²	—	4	4	no	—
117	C₂³	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	AGL(1,ℤ/8)	1	4	yes	—
140	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
141	C₂×C₄	C₂×#41	1	1	yes	characteristic
142	C₂×C₄	C₂×#41	1	1	yes	characteristic
143	C₂×C₄	C₂×#41	1	1	yes	characteristic
144	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
145	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
146	C₈	—	2	2	no	—
147	C₈	—	4	1	no	—
148	C₈	—	4	1	no	—
149	C₈	#125	1	4	yes	complemented
150	C₂²	#532	1	1	yes	characteristic
151	C₂²	#547	1	1	yes	characteristic
152	C₂²	D₄×Q₈	1	1	yes	characteristic
153	C₂²	C₂×AGL(1,ℤ/8)	1	2	yes	—
154	C₂²	#578	1	2	yes	—
155	C₄	—	2	4	no	—
156	C₄	—	4	2	no	—
157	C₄	—	4	4	no	—
158	C₄	—	4	1	no	—
159	C₄	—	4	1	no	—
160	C₄	—	8	2	no	—
161	C₄	—	8	2	no	—
162	C₄	—	8	2	no	—
163	C₄	C₂×#125	1	1	yes	characteristic
164	C₄	#541	1	1	yes	characteristic
165	C₄	#552	1	1	yes	characteristic
166	C₄	#557	1	1	yes	characteristic
167	C₂	#2506	1	1	yes	characteristic
168	C₂	#3177	1	1	yes	characteristic
169	C₂	#3184	1	1	yes	characteristic
170	C₂	#3343	1	2	yes	—
171	C₂	#3386	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
149. C₈	#125	semidirect

Maximal subgroups

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

subgroup	index
#2506	2
#2099	2
#2087	2
#1944	2
#1943	2
#1941	2
#1921	2
#1845	2
#1776	2
#1771	2