Finite Group Databasebeta

← #21843

Subgroups — #21843

Order 256.

485 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#544C211yesmaximal, characteristic
2C2×#433C211yesmaximal, characteristic
3#2314C211yesmaximal, characteristic
4#1993C211yesmaximal, characteristic, complemented
5#1992C211yesmaximal, characteristic, complemented
6#1991C211yesmaximal, characteristic, complemented
7#1990C211yesmaximal, characteristic, complemented
8#1989C211yesmaximal, characteristic, complemented
9#1989C211yesmaximal, characteristic, complemented
10#1988C211yesmaximal, characteristic, complemented
11#1987C211yesmaximal, characteristic, complemented
12#1825C211yesmaximal, characteristic
13#1825C211yesmaximal, characteristic
14#1824C211yesmaximal, characteristic
15#1823C211yesmaximal, characteristic
16#544—24no—
17#544C418yescomplemented
18C2×C4⋊Q8—21no—
19C2×C4⋊Q8C2211yescharacteristic
20C2×C4⋊Q8C2211yescharacteristic
21C2×#125—21no—
22C2×#125C2211yescharacteristic
23C2×#125C2211yescharacteristic
24C2×C22⋊Q8—21no—
25C2×C22⋊Q8—21no—
26C2×#117C2211yescharacteristic
27C2×#117C2211yescharacteristic
28C22×C4⋊C4C2211yescharacteristic
29#433—24no—
30C2×C4⋊C8C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×#97C2211yescharacteristic
35C2×C4×C8C2211yescharacteristic
36#388—21no—
37#388—21no—
38#387—21no—
39#387—21no—
40#384C2211yescharacteristic
41#382C2211yescharacteristic
42#380C2211yescharacteristic
43#380C2211yescharacteristic
44C4×C4⋊C4C2211yescharacteristic
45C4×C4⋊C4C2211yescharacteristic
46#340C2211yescharacteristic
47#340C2211yescharacteristic
48#340C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62C22×Q8—21no—
63C22×Q8—21no—
64C23×C4—21no—
65C23×C4C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C4⋊Q8—28no—
71C4⋊Q8—22no—
72C4⋊Q8—22no—
73C4⋊Q8C2×C414yes—
74C4⋊Q8C2×C414yes—
75#125—28no—
76#125—22no—
77#125—22no—
78#125C2×C414yes—
79#125C2×C414yes—
80C22⋊Q8—28no—
81C22⋊Q8—28no—
82#117—22no—
83#117C2×C414yes—
84#117C2×C414yes—
85#117D414yes—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4C2×C414yes—
97C2×C4⋊C4D411yescharacteristic
98C2×C4⋊C4D411yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4D411yescharacteristic
111C2×C22⋊C4D411yescharacteristic
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42C2311yescharacteristic
116C2×C42C2311yescharacteristic
117C4⋊C8—22no—
118C4⋊C8—22no—
119C22⋊C8—22no—
120C22⋊C8—22no—
121#97—22no—
122C4×C8—22no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129C24C2×D411yescharacteristic
130C2×Q8—24no—
131C2×Q8—24no—
132C2×Q8—81no—
133C2×Q8—81no—
134C2×Q8—81no—
135C2×Q8—81no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—24no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—81no—
150C22×C4—81no—
151C22×C4C22×C412yes—
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D412yes—
158C22×C4C2411yescharacteristic
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C4⋊C4—24no—
172C4⋊C4—24no—
173C4⋊C4—24no—
174C4⋊C4—24no—
175C4⋊C4—24no—
176C4⋊C4—24no—
177C4⋊C4—24no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—81no—
191C4⋊C4—81no—
192C4⋊C4C22⋊C414yes—
193C4⋊C4C22⋊C414yes—
194C4⋊C4C22×C412yes—
195C4⋊C4C22×C412yes—
196C4⋊C4C22×C412yes—
197C4⋊C4C22×C412yes—
198C22⋊C4—24no—
199C22⋊C4—81no—
200C22⋊C4—81no—
201C22⋊C4C22⋊C414yes—
202C22⋊C4C22⋊C414yes—
203C42—24no—
204C42—81no—
205C42—81no—
206C42C22×C412yes—
207C42C22×C412yes—
208C42C2×D412yes—
209C42C2×D412yes—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23C2×C22⋊C412yes—
217C23C22×D411yescharacteristic
218Q8—84no—
219Q8—84no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—84no—
259C2×C4C2×C22⋊C412yes—
260C2×C4C2×C22⋊C412yes—
261C2×C4C2×C22⋊C412yes—
262C2×C4C2×C22⋊C412yes—
263C2×C4C2×C22⋊C412yes—
264C2×C4C23×C411yescharacteristic
265C2×C4C22×D411yescharacteristic
266C2×C42−1+411yescharacteristic
267C2×C42−1+411yescharacteristic
268C2×C42−1+411yescharacteristic
269C2×C42−1+411yescharacteristic
270C8—41no—
271C8—41no—
272C8—41no—
273C8—41no—
274C22—22no—
275C22—22no—
276C22—22no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22C22×C22⋊C411yescharacteristic
282C22#53611yescharacteristic
283C22#53611yescharacteristic
284C22#57611yescharacteristic
285C22#57711yescharacteristic
286C22#57711yescharacteristic
287C22#57811yescharacteristic
288C4—22no—
289C4—22no—
290C4—22no—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—42no—
295C4—42no—
296C4—42no—
297C4—81no—
298C4—81no—
299C4—81no—
300C4—81no—
301C4#51911yescharacteristic
302C4#51911yescharacteristic
303C4#53611yescharacteristic
304C4#53611yescharacteristic
305C2—42no—
306C2#231011yescharacteristic
307C2#289711yescharacteristic
308C2#289811yescharacteristic
309C2#324111yescharacteristic
310C2#324211yescharacteristic
311C2#324311yescharacteristic
312C2#324411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C2×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
4. #1993C2semidirect
5. #1992C2semidirect
6. #1991C2semidirect
7. #1990C2semidirect
8. #1989C2semidirect
9. #1989C2semidirect
10. #1988C2semidirect
11. #1987C2semidirect
17. #544C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5442
C2×#4332
#23142
#19932
#19922
#19912
#19902
#19892
#19882
#19872
#18252
#18242
#18232