Finite Group Databasebeta

← #24840

Subgroups — #24840

Order 256.

430 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
1C2×#479C211yesmaximal, characteristic, complemented
2C22×C4⋊C8C211yesmaximal, characteristic
3#2851C211yesmaximal, characteristic
4C4×C22⋊Q8C211yesmaximal, characteristic
5#2088C211yesmaximal, characteristic, complemented
6#2086C211yesmaximal, characteristic, complemented
7#2068C211yesmaximal, characteristic, complemented
8#2066C211yesmaximal, characteristic, complemented
9#1989C211yesmaximal, characteristic, complemented
10#1949C211yesmaximal, characteristic, complemented
11#1892C211yesmaximal, characteristic
12#1892C211yesmaximal, characteristic
13#1808C211yesmaximal, characteristic
14#1807C211yesmaximal, characteristic
15#1805C211yesmaximal, characteristic, complemented
16C23×C8C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×#125C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C22×C42C2211yescharacteristic
22#479—24no—
23C2×#107C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×C4⋊C8—24no—
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36#391C2211yescharacteristic
37#389C2211yescharacteristic
38#389C2211yescharacteristic
39#389C2211yescharacteristic
40#387—21no—
41#387—21no—
42#387—21no—
43#385—21no—
44#385—21no—
45#385—21no—
46#382C2211yescharacteristic
47#380C2211yescharacteristic
48#380C2211yescharacteristic
49#380C2211yescharacteristic
50C4×C4⋊C4—21no—
51C4×C4⋊C4C2211yescharacteristic
52C4×C22⋊C4—21no—
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C23×C4D411yescharacteristic
63C23×C4D411yescharacteristic
64C23×C4C2311yescharacteristic
65C22×C8—22no—
66C22×C8—22no—
67C22×C8—22no—
68C22×C8—22no—
69C22×C8—41no—
70C22×C8—41no—
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75#125—22no—
76#125—22no—
77C22⋊Q8—22no—
78C22⋊Q8Q814yes—
79C4×Q8—22no—
80C4×Q8—22no—
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×C4⋊C4C2311yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C42—22no—
100C2×C42—22no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42D411yescharacteristic
104C2×C42D411yescharacteristic
105C2×C42C2311yescharacteristic
106C2×C42C2311yescharacteristic
107#107—22no—
108#107—41no—
109#106—22no—
110#106—41no—
111C4⋊C8—22no—
112C4⋊C8—22no—
113C4⋊C8—22no—
114C4⋊C8D414yes—
115#103—22no—
116#103—22no—
117#103—41no—
118#103—41no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130C24C2×D411yescharacteristic
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8C2×Q812yes—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—41no—
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×Q812yes—
161C22×C4#4112yes—
162C22×C4C2411yescharacteristic
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—44no—
175C2×C8C2×D412yes—
176C2×C8C2×D412yes—
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—21no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4C2×Q812yes—
198C22⋊C4—24no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C42—22no—
203C42—22no—
204C42—21no—
205C42—21no—
206C42—42no—
207C42—42no—
208C42C2×D412yes—
209C42#4112yes—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23C22⋊Q812yes—
215C23#13512yes—
216C23#13712yes—
217C23C22×D411yescharacteristic
218Q8—22no—
219Q8—22no—
220Q8—41no—
221Q8—41no—
222C2×C4—28no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—44no—
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#12112yes—
264C2×C4#12112yes—
265C2×C4C22⋊Q812yes—
266C2×C4#13512yes—
267C2×C4C22×D411yescharacteristic
268C2×C4C22×Q811yescharacteristic
269C2×C4C2×#4111yescharacteristic
270C2×C4C2×#4111yescharacteristic
271C2×C4H(5,𝔽2)11yescharacteristic
272C2×C42−1+411yescharacteristic
273C8—22no—
274C8—22no—
275C8—41no—
276C8—41no—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—22no—
281C22—22no—
282C22—22no—
283C22#47914yescomplemented
284C22C2×#12111yescharacteristic
285C22C2×C22⋊Q811yescharacteristic
286C22#53711yescharacteristic
287C22C2×#13511yescharacteristic
288C22C2×#13511yescharacteristic
289C22C2×#13711yescharacteristic
290C22#57511yescharacteristic
291C4—22no—
292C4—22no—
293C4—22no—
294C4—22no—
295C4—42no—
296C4—42no—
297C4—41no—
298C4—81no—
299C4—81no—
300C4—81no—
301C4—81no—
302C4—81no—
303C4—81no—
304C4#54611yescharacteristic
305C4D4×Q811yescharacteristic
306C4#55211yescharacteristic
307C4#55411yescharacteristic
308C2—24no—
309C2#241711yescharacteristic
310C2#305011yescharacteristic
311C2#305611yescharacteristic
312C2C2×#47911yescharacteristic
313C2#307611yescharacteristic
314C2#324811yescharacteristic
315C2#325011yescharacteristic
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
1. C2×#479C2semidirect
5. #2088C2semidirect
6. #2086C2semidirect
7. #2068C2semidirect
8. #2066C2semidirect
9. #1989C2semidirect
10. #1949C2semidirect
15. #1805C2semidirect
283. C22#479semidirect

Maximal subgroups

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

subgroupindex
C2×#4792
C22×C4⋊C82
#28512
C4×C22⋊Q82
#20882
#20862
#20682
#20662
#19892
#19492
#18922
#18082
#18072
#18052