Finite Group Databasebeta

← #25915

Subgroups — #25915

Order 256.

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

normal subgroups only

This group has 555 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, complemented
2C2×#490C211yesmaximal, characteristic, complemented
3C2×#486C211yesmaximal, characteristic, complemented
4#2619C211yesmaximal, characteristic
5#2025C212yesmaximal, complemented
6#2023C212yesmaximal, complemented
7#1993C211yesmaximal, characteristic, complemented
8#1991C211yesmaximal, characteristic, complemented
9#1875C212yesmaximal, complemented
10#1874C212yesmaximal, complemented
11#1750C211yesmaximal, characteristic, complemented
12#544—24no—
13#544—24no—
14C2×C4⋊Q8—21no—
15C2×C4⋊Q8C2211yescharacteristic
16C2×C4⋊Q8C2211yescharacteristic
17C2×#127C2211yescharacteristic
18C2×#125—21no—
19C2×#125C2211yescharacteristic
20C2×#125C2211yescharacteristic
21C2×C22⋊Q8—21no—
22C2×C22⋊Q8—21no—
23C2×#121C2212yes—
24C2×#117C2211yescharacteristic
25C2×#117C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27#490—24no—
28#486—24no—
29C2×#102C2212yes—
30C2×#102C2212yes—
31C2×#102C2212yes—
32C2×#102C2212yes—
33C2×C22⋊C8C2212yes—
34C2×#97C2211yescharacteristic
35C2×C4×C8C2211yescharacteristic
36#394—21no—
37#394—21no—
38#392—21no—
39#392—21no—
40#390C2212yes—
41#382C2211yescharacteristic
42#340C2212yes—
43#340C2212yes—
44#340C2212yes—
45#340C2212yes—
46#336C2212yes—
47C22×Q8—21no—
48C22×Q8—21no—
49C22×D4—22no—
50C22×D4—21no—
51C22×D4—21no—
52C22×D4C2312yes—
53C23×C4—21no—
54C23×C4C2311yescharacteristic
55C22×C8C2312yes—
56C22×C8C2312yes—
57C4⋊Q8—28no—
58C4⋊Q8—22no—
59C4⋊Q8—22no—
60C4⋊Q8—22no—
61C4⋊Q8D414yescomplemented
62#127—22no—
63#127—22no—
64#125—28no—
65#125—22no—
66#125—22no—
67#125—22no—
68#125D414yescomplemented
69C22⋊Q8—28no—
70C22⋊Q8—28no—
71#121—28no—
72#117—22no—
73#117—22no—
74#117—22no—
75#117—22no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
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⋊C4—22no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4D411yescharacteristic
97C2×C22⋊C4D411yescharacteristic
98C2×C42—21no—
99C2×C42C2311yescharacteristic
100C2×C42C2311yescharacteristic
101#102—24no—
102#102—24no—
103#102—24no—
104#102—24no—
105C22⋊C8—24no—
106#97—22no—
107C4×C8—22no—
108#95—21no—
109#95—21no—
110C24—22no—
111C24C2×D411yescharacteristic
112C2×Q8—24no—
113C2×Q8—24no—
114C2×Q8—82no—
115C2×Q8—82no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—24no—
119C2×D4—24no—
120C2×D4—44no—
121C2×D4—42no—
122C2×D4—42no—
123C2×D4—82no—
124C2×D4—82no—
125C2×D4—82no—
126C2×D4—82no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—24no—
135C22×C4—22no—
136C22×C4—82no—
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142C22×C4#4114yes—
143C22×C4C2411yescharacteristic
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C4⋊C4—24no—
151C4⋊C4—24no—
152C4⋊C4—24no—
153C4⋊C4—24no—
154C4⋊C4—24no—
155C4⋊C4—24no—
156C4⋊C4—24no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—24no—
164C4⋊C4C2×D412yes—
165C4⋊C4C2×D412yes—
166C4⋊C4C2×D412yes—
167C4⋊C4C2×D412yes—
168C22⋊C4—24no—
169C22⋊C4—22no—
170C22⋊C4—22no—
171C22⋊C4—44no—
172C22⋊C4—82no—
173C22⋊C4—82no—
174C42—24no—
175C42—21no—
176C42—21no—
177C42C2×D412yes—
178C42C2×D412yes—
179C23—21no—
180C23—21no—
181C23—22no—
182C23—22no—
183C23—42no—
184C23—42no—
185C23—42no—
186C23—42no—
187C23—42no—
188C23—42no—
189C23—42no—
190C23#12412yes—
191C23C22×D411yescharacteristic
192Q8—84no—
193Q8—84no—
194D4—44no—
195D4—44no—
196D4—84no—
197D4—84no—
198D4—88no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—24no—
204C2×C4—24no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—84no—
221C2×C4#12012yes—
222C2×C4#12012yes—
223C2×C4#12412yes—
224C2×C4C22×D411yescharacteristic
225C2×C4C22×D411yescharacteristic
226C2×C4C2×#4112yes—
227C2×C42−1+412yes—
228C8—42no—
229C8—42no—
230C22—24no—
231C22—22no—
232C22—42no—
233C22—41no—
234C22—41no—
235C22—82no—
236C22—82no—
237C22—82no—
238C22—82no—
239C22—82no—
240C22—82no—
241C22—82no—
242C22C2×#12011yescharacteristic
243C22C2×#12411yescharacteristic
244C22#53611yescharacteristic
245C22#57612yes—
246C22#57712yes—
247C4—22no—
248C4—22no—
249C4—22no—
250C4—41no—
251C4—42no—
252C4—42no—
253C4—42no—
254C4—41no—
255C4—42no—
256C4—82no—
257C4#54812yes—
258C4#54812yes—
259C2—42no—
260C2—162no—
261C2#243611yescharacteristic
262C2#300811yescharacteristic
263C2#301011yescharacteristic
264C2#313612yes—
265C2#324112yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2619C2
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
1. C2×#544C2semidirect
2. C2×#490C2semidirect
3. C2×#486C2semidirect
5. #2025C2semidirect
6. #2023C2semidirect
7. #1993C2semidirect
8. #1991C2semidirect
9. #1875C2semidirect
10. #1874C2semidirect
11. #1750C2semidirect
61. C4⋊Q8D4semidirect
68. #125D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5442
C2×#4902
C2×#4862
#26192
#20252
#20232
#19932
#19912
#18752
#18742
#17502