Finite Group Databasebeta

← #28977

Subgroups — #28977

Order 256.

375 conjugacy classes of subgroups.

The subgroup lattice has 317 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2492C211yesmaximal, characteristic
2#2423C211yesmaximal, characteristic
3#2076C211yesmaximal, characteristic
4#2072C211yesmaximal, characteristic, complemented
5#2024C211yesmaximal, characteristic, complemented
6#2001C211yesmaximal, characteristic
7#1990C211yesmaximal, characteristic, complemented
8#1969C211yesmaximal, characteristic, complemented
9#1933C211yesmaximal, characteristic, complemented
10#1921C211yesmaximal, characteristic, complemented
11#1892C211yesmaximal, characteristic
12#1892C211yesmaximal, characteristic
13#1878C211yesmaximal, characteristic, complemented
14C4×#103C211yesmaximal, characteristic, complemented
15C8×C22⋊C4C211yesmaximal, characteristic
16C22×Q16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×C4×Q8C2211yescharacteristic
22C2×#107C2211yescharacteristic
23C2×#103C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×C22⋊C8C2211yescharacteristic
30C2×C4×C8C2211yescharacteristic
31C2×C4×C8C2211yescharacteristic
32#400—21no—
33#399—21no—
34#398—21no—
35#393—21no—
36#391—21no—
37#391C2211yescharacteristic
38#391C2211yescharacteristic
39#391C2211yescharacteristic
40#389C2211yescharacteristic
41#388—21no—
42#388—21no—
43#387—21no—
44#384C2211yescharacteristic
45#380C2211yescharacteristic
46#380C2211yescharacteristic
47C4×C4⋊C4C2211yescharacteristic
48C4×C22⋊C4C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8—21no—
60C22×Q8C2311yescharacteristic
61C22×Q8C2311yescharacteristic
62C23×C4C2311yescharacteristic
63C2×Q16—22no—
64C2×Q16—22no—
65C2×Q16—41no—
66C22×C8—22no—
67C22×C8—41no—
68C22×C8—41no—
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C4⋊Q8—22no—
74C4⋊Q8—41no—
75C22⋊Q8—22no—
76C22⋊Q8—22no—
77C22⋊Q8—41no—
78C22⋊Q8—41no—
79C4×Q8—22no—
80C4×Q8—41no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4D411yescharacteristic
99C2×C22⋊C4D411yescharacteristic
100C2×C42—21no—
101C2×C42—21no—
102C2×C42C2311yescharacteristic
103C2×C42C2311yescharacteristic
104#107—22no—
105#103—22no—
106#103—22no—
107#103—22no—
108#103—41no—
109#103—41no—
110#103—41no—
111C22⋊C8—22no—
112C4×C8—41no—
113C4×C8—41no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95D411yescharacteristic
125#95D411yescharacteristic
126C24C2×D411yescharacteristic
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—42no—
139C2×Q8—42no—
140C2×Q8—81no—
141C2×Q8#4112yes—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2411yescharacteristic
162Q16—44no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—44no—
174C2×C8#4112yes—
175C2×C8#4112yes—
176C4⋊C4—21no—
177C4⋊C4—21no—
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—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—81no—
193C4⋊C4—81no—
194C4⋊C4#4112yes—
195C22⋊C4—22no—
196C22⋊C4—42no—
197C22⋊C4—42no—
198C22⋊C4—81no—
199C22⋊C4—81no—
200C22⋊C4Q1614yescomplemented
201C42—21no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—42no—
206C42—81no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23C2×Q1612yes—
214C23C22×D411yescharacteristic
215Q8—22no—
216Q8—22no—
217Q8—41no—
218Q8—41no—
219Q8—41no—
220Q8—41no—
221Q8—41no—
222Q8—41no—
223Q8—84no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
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—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4C2×Q1612yes—
265C2×C4C2×Q1612yes—
266C2×C4#13512yes—
267C2×C4C2×#4111yescharacteristic
268C2×C4C2×#4111yescharacteristic
269C2×C4C2×#4111yescharacteristic
270C2×C4C2×#4111yescharacteristic
271C2×C4H(5,𝔽2)11yescharacteristic
272C2×C4H(5,𝔽2)11yescharacteristic
273C8—22no—
274C8—22no—
275C8—41no—
276C8—41no—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22#53211yescharacteristic
285C22#54611yescharacteristic
286C22#54611yescharacteristic
287C22C22×Q1611yescharacteristic
288C22C2×#13511yescharacteristic
289C22#57711yescharacteristic
290C22#57711yescharacteristic
291C4—22no—
292C4—22no—
293C4—22no—
294C4—42no—
295C4—41no—
296C4—41no—
297C4—42no—
298C4—81no—
299C4—81no—
300C4—81no—
301C4—81no—
302C4—81no—
303C4—81no—
304C4#52911yescharacteristic
305C4#53811yescharacteristic
306C4#55111yescharacteristic
307C4#55911yescharacteristic
308C2—42no—
309C2#255711yescharacteristic
310C2#317011yescharacteristic
311C2#317311yescharacteristic
312C2#329611yescharacteristic
313C2#329911yescharacteristic
314C2#329911yescharacteristic
315C2#330411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #2072C2semidirect
5. #2024C2semidirect
7. #1990C2semidirect
8. #1969C2semidirect
9. #1933C2semidirect
10. #1921C2semidirect
13. #1878C2semidirect
14. C4×#103C2semidirect
200. C22⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
#24922
#24232
#20762
#20722
#20242
#20012
#19902
#19692
#19332
#19212
#18922
#18782
C4×#1032
C8×C22⋊C42