Finite Group Databasebeta

← #25677

Subgroups — #25677

Order 256.

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

normal subgroups only

This group has 519 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×#533C211yesmaximal, characteristic
2C2×#501C211yesmaximal, characteristic, complemented
3C2×#491C211yesmaximal, characteristic, complemented
4#2851C211yesmaximal, characteristic
5#2036C212yesmaximal
6#2035C212yesmaximal
7#1993C211yesmaximal, characteristic, complemented
8#1960C211yesmaximal, characteristic, complemented
9#1865C212yesmaximal, complemented
10#1863C212yesmaximal, complemented
11#1749C211yesmaximal, characteristic
12#533—24no—
13#533—24no—
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4⋊Q8C2211yescharacteristic
16C2×#125—21no—
17C2×#125C2211yescharacteristic
18C2×C22⋊Q8—21no—
19C2×C22⋊Q8C2212yes—
20C2×C4×Q8—21no—
21C2×C4×Q8C2212yes—
22C2×#117—21no—
23C22×C42C2211yescharacteristic
24#501—24no—
25#491—24no—
26C2×#107C2212yes—
27C2×#106C2212yes—
28C2×#103C2212yes—
29C2×#103C2212yes—
30C2×#103C2212yes—
31C2×#103C2212yes—
32C2×C22⋊C8C2212yes—
33C2×#97C2211yescharacteristic
34C2×#97C2211yescharacteristic
35#389C2211yescharacteristic
36#389C2211yescharacteristic
37#387—21no—
38#387—21no—
39#385—21no—
40#385—21no—
41#382C2211yescharacteristic
42#380C2211yescharacteristic
43#380C2211yescharacteristic
44#340C2212yes—
45#340C2212yes—
46#336C2212yes—
47C22×Q8—21no—
48C22×Q8C2312yes—
49C23×C4D411yescharacteristic
50C23×C4D411yescharacteristic
51C23×C4C2311yescharacteristic
52C22×C8C2312yes—
53C22×C8C2312yes—
54C4⋊Q8—22no—
55C4⋊Q8—22no—
56C4⋊Q8D414yes—
57C4⋊Q8D414yes—
58#125—28no—
59#125—22no—
60#125—22no—
61C22⋊Q8—28no—
62C22⋊Q8—24no—
63C22⋊Q8—24no—
64C4×Q8—28no—
65C4×Q8—24no—
66C4×Q8—24no—
67#117—28no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2312yes—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C42—22no—
86C2×C42—22no—
87C2×C42—22no—
88C2×C42D411yescharacteristic
89C2×C42D411yescharacteristic
90C2×C42C2311yescharacteristic
91C2×C42C2311yescharacteristic
92#107—24no—
93#106—24no—
94#103—24no—
95#103—24no—
96#103—24no—
97#103—24no—
98C22⋊C8—24no—
99#97—22no—
100#97—22no—
101#95—21no—
102#95—21no—
103#95—21no—
104#95—21no—
105#95—21no—
106#95—21no—
107C24C2×D411yescharacteristic
108C2×Q8—24no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8—22no—
114C2×Q8—42no—
115C2×Q8—42no—
116C2×Q8C2×D414yes—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4C2411yescharacteristic
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8#4114yes—
145C4⋊C4—28no—
146C4⋊C4—28no—
147C4⋊C4—24no—
148C4⋊C4—24no—
149C4⋊C4—24no—
150C4⋊C4—22no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4C2×D414yes—
162C22⋊C4—28no—
163C22⋊C4—42no—
164C22⋊C4—42no—
165C42—28no—
166C42—22no—
167C42—22no—
168C42—21no—
169C42—21no—
170C42C2×D412yes—
171C42C2×D412yes—
172C23—21no—
173C23—21no—
174C23—21no—
175C23—21no—
176C23—21no—
177C23—21no—
178C23#13712yes—
179C23C22×D411yescharacteristic
180Q8—24no—
181Q8—24no—
182Q8—48no—
183Q8—42no—
184Q8—42no—
185C2×C4—28no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—44no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4#12012yes—
220C2×C4#12012yes—
221C2×C4#13712yes—
222C2×C4C22×D411yescharacteristic
223C2×C4C22×D411yescharacteristic
224C2×C4C2×#4112yes—
225C2×C42−1+412yes—
226C8—24no—
227C8—42no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22C2×#12011yescharacteristic
236C22#53211yescharacteristic
237C22#53611yescharacteristic
238C22C2×#13711yescharacteristic
239C22C2×#13711yescharacteristic
240C22#57511yescharacteristic
241C22#57511yescharacteristic
242C4—22no—
243C4—22no—
244C4—22no—
245C4—41no—
246C4—44no—
247C4—41no—
248C4—44no—
249C4—81no—
250C4—81no—
251C4—81no—
252C4—81no—
253C4#54712yes—
254C4#54812yes—
255C2—42no—
256C2#243211yescharacteristic
257C2#300411yescharacteristic
258C2#300511yescharacteristic
259C2#317211yescharacteristic
260C2#318111yescharacteristic
261C2#323611yescharacteristic
262C2#323711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#501C2semidirect
3. C2×#491C2semidirect
7. #1993C2semidirect
8. #1960C2semidirect
9. #1865C2semidirect
10. #1863C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5332
C2×#5012
C2×#4912
#28512
#20362
#20352
#19932
#19602
#18652
#18632
#17492