Finite Group Databasebeta

← #25914

Subgroups — #25914

Order 256.

505 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 505 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.

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

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#544C211yesmaximal, characteristic
2C2×#491C211yesmaximal, characteristic, complemented
3C2×#487C211yesmaximal, characteristic, complemented
4#2621C211yesmaximal, characteristic
5#2026C212yesmaximal, complemented
6#2024C212yesmaximal, complemented
7#1993C211yesmaximal, characteristic, complemented
8#1991C211yesmaximal, characteristic, complemented
9#1873C212yesmaximal
10#1872C212yesmaximal
11#1750C211yesmaximal, characteristic
12#544—24no—
13#544—24no—
14C2×C4⋊Q8—21no—
15C2×C4⋊Q8C2211yescharacteristic
16C2×C4⋊Q8C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×#125—21no—
19C2×#125C2211yescharacteristic
20C2×#125C2211yescharacteristic
21C2×C22⋊Q8—21no—
22C2×C22⋊Q8—21no—
23C2×C22⋊Q8C2212yes—
24C2×#117C2211yescharacteristic
25C2×#117C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27#491—24no—
28#487—24no—
29C2×#103C2212yes—
30C2×#103C2212yes—
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×C22⋊C8C2212yes—
34C2×#97C2211yescharacteristic
35C2×C4×C8C2211yescharacteristic
36#398—21no—
37#398—21no—
38#397—21no—
39#397—21no—
40#391C2212yes—
41#382C2211yescharacteristic
42#340C2212yes—
43#340C2212yes—
44#340C2212yes—
45#340C2212yes—
46#336C2212yes—
47C22×Q8—21no—
48C22×Q8—21no—
49C22×Q8C2312yes—
50C23×C4—21no—
51C23×C4C2311yescharacteristic
52C22×C8C2312yes—
53C22×C8C2312yes—
54C4⋊Q8—28no—
55C4⋊Q8—22no—
56C4⋊Q8—22no—
57C4⋊Q8—22no—
58C4⋊Q8—22no—
59C4⋊Q8—22no—
60C4⋊Q8D414yes—
61#125—28no—
62#125—22no—
63#125—22no—
64#125—22no—
65#125D414yes—
66C22⋊Q8—28no—
67C22⋊Q8—28no—
68C22⋊Q8—28no—
69#117—22no—
70#117—22no—
71#117—22no—
72#117—22no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4D411yescharacteristic
86C2×C4⋊C4D411yescharacteristic
87C2×C4⋊C4C2312yes—
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C22⋊C4—21no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4D411yescharacteristic
95C2×C22⋊C4D411yescharacteristic
96C2×C42—21no—
97C2×C42C2311yescharacteristic
98C2×C42C2311yescharacteristic
99#103—24no—
100#103—24no—
101#103—24no—
102#103—24no—
103C22⋊C8—24no—
104#97—22no—
105C4×C8—22no—
106#95—21no—
107#95—21no—
108#95—22no—
109#95—22no—
110C24C2×D411yescharacteristic
111C2×Q8—24no—
112C2×Q8—24no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—24no—
116C2×Q8—24no—
117C2×Q8—82no—
118C2×Q8—82no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—24no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—82no—
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4#4114yes—
136C22×C4C2411yescharacteristic
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C4⋊C4—24no—
144C4⋊C4—24no—
145C4⋊C4—24no—
146C4⋊C4—24no—
147C4⋊C4—24no—
148C4⋊C4—24no—
149C4⋊C4—24no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—24no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—44no—
160C4⋊C4C2×D412yes—
161C4⋊C4C2×D412yes—
162C4⋊C4C2×D412yes—
163C4⋊C4C2×D412yes—
164C22⋊C4—24no—
165C22⋊C4—22no—
166C22⋊C4—22no—
167C22⋊C4—44no—
168C42—24no—
169C42—21no—
170C42—21no—
171C42C2×D412yes—
172C42C2×D412yes—
173C23—21no—
174C23—21no—
175C23—22no—
176C23—22no—
177C23#12412yes—
178C23C22×D411yescharacteristic
179Q8—44no—
180Q8—44no—
181Q8—84no—
182Q8—84no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—84no—
208C2×C4#12012yes—
209C2×C4#12012yes—
210C2×C4#12412yes—
211C2×C4C22×D411yescharacteristic
212C2×C4C22×D411yescharacteristic
213C2×C4C2×#4112yes—
214C2×C42−1+412yes—
215C8—42no—
216C8—42no—
217C22—24no—
218C22—22no—
219C22—42no—
220C22—41no—
221C22—41no—
222C22C2×#12011yescharacteristic
223C22C2×#12411yescharacteristic
224C22#53611yescharacteristic
225C22#57712yes—
226C22#57812yes—
227C4—22no—
228C4—22no—
229C4—22no—
230C4—41no—
231C4—42no—
232C4—42no—
233C4—42no—
234C4—41no—
235C4—42no—
236C4—82no—
237C4—82no—
238C4#54812yes—
239C4#54812yes—
240C2—42no—
241C2#243611yescharacteristic
242C2#300911yescharacteristic
243C2#301111yescharacteristic
244C2#313712yes—
245C2#324212yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#491C2semidirect
3. C2×#487C2semidirect
5. #2026C2semidirect
6. #2024C2semidirect
7. #1993C2semidirect
8. #1991C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5442
C2×#4912
C2×#4872
#26212
#20262
#20242
#19932
#19912
#18732
#18722
#17502