Finite Group Databasebeta

← #25201

Subgroups — #25201

Order 256.

403 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#485C211yesmaximal, characteristic
2C2×#467C211yesmaximal, characteristic
3#2499C211yesmaximal, characteristic
4#2362C211yesmaximal, characteristic
5#2083C211yesmaximal, characteristic, complemented
6#2074C211yesmaximal, characteristic, complemented
7#2066C211yesmaximal, characteristic, complemented
8#2056C211yesmaximal, characteristic, complemented
9#1949C211yesmaximal, characteristic, complemented
10#1928C211yesmaximal, characteristic, complemented
11#1878C211yesmaximal, characteristic, complemented
12#1873C211yesmaximal, characteristic
13#1850C211yesmaximal, characteristic
14#1803C211yesmaximal, characteristic, complemented
15#1735C211yesmaximal, characteristic
16C22×Q16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C2×#117C2211yescharacteristic
22C2×#117C2211yescharacteristic
23#485—24no—
24#467—24no—
25C2×#107C2211yescharacteristic
26C2×#107C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37#397—21no—
38#397—21no—
39#391C2211yescharacteristic
40#391C2211yescharacteristic
41#391C2211yescharacteristic
42#389C2211yescharacteristic
43#388—21no—
44#388—21no—
45#387—21no—
46#387—21no—
47#387—21no—
48#384C2211yescharacteristic
49#382C2211yescharacteristic
50C4×C4⋊C4—21no—
51C4×C4⋊C4C2211yescharacteristic
52C4×C22⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×Q8C2311yescharacteristic
63C23×C4C2311yescharacteristic
64C2×Q16—22no—
65C2×Q16—22no—
66C2×Q16—41no—
67C22×C8—22no—
68C22×C8—22no—
69C22×C8—41no—
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22⋊Q8—22no—
75C22⋊Q8—22no—
76C22⋊Q8—22no—
77C22⋊Q8D414yes—
78C4×Q8—22no—
79C4×Q8—41no—
80#117—22no—
81#117—22no—
82#117—22no—
83#117—41no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4D411yescharacteristic
101C2×C22⋊C4D411yescharacteristic
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42C2311yescharacteristic
107C2×C42C2311yescharacteristic
108#107—22no—
109#107—22no—
110#106—22no—
111C4⋊C8—22no—
112C4⋊C8—41no—
113#103—22no—
114#103—22no—
115#103—22no—
116#103—22no—
117#103—22no—
118#103—41no—
119C22⋊C8—22no—
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—
130#95D411yescharacteristic
131#95D411yescharacteristic
132C24C2×D411yescharacteristic
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8C2×D412yes—
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×C4—21no—
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D412yes—
162C22×C4C2×D411yescharacteristic
163C22×C4#4112yes—
164C22×C4C2411yescharacteristic
165Q16—44no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—44no—
178C2×C8#4112yes—
179C4⋊C4—24no—
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—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4C2×D412yes—
196C4⋊C4#4112yes—
197C22⋊C4—24no—
198C22⋊C4—22no—
199C22⋊C4—41no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—42no—
208C42—42no—
209C42—42no—
210C42—42no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23#12112yes—
218C23C22×D411yescharacteristic
219Q8—22no—
220Q8—22no—
221Q8—41no—
222Q8—41no—
223Q8—41no—
224Q8—41no—
225Q8—41no—
226Q8—41no—
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—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
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×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4#12112yes—
269C2×C4#13512yes—
270C2×C4C22×D411yescharacteristic
271C2×C4C2×#4111yescharacteristic
272C2×C4C2×#4111yescharacteristic
273C2×C4C2×#4111yescharacteristic
274C2×C4H(5,𝔽2)11yescharacteristic
275C2×C42−1+411yescharacteristic
276C8—22no—
277C8—41no—
278C8—41no—
279C8—41no—
280C22—22no—
281C22—22no—
282C22—22no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—41no—
287C22C2×#12111yescharacteristic
288C22#54611yescharacteristic
289C22#54711yescharacteristic
290C22C2×#13511yescharacteristic
291C22#57511yescharacteristic
292C22#57811yescharacteristic
293C22#57811yescharacteristic
294C4—22no—
295C4—22no—
296C4—41no—
297C4—41no—
298C4—42no—
299C4—42no—
300C4—41no—
301C4—42no—
302C4—81no—
303C4—81no—
304C4—81no—
305C4—81no—
306C4—81no—
307C4#52911yescharacteristic
308C4#53911yescharacteristic
309C4#54611yescharacteristic
310C4#54711yescharacteristic
311C2—42no—
312C2#242511yescharacteristic
313C2#303511yescharacteristic
314C2#304411yescharacteristic
315C2#330011yescharacteristic
316C2#331511yescharacteristic
317C2#334211yescharacteristic
318C2#334911yescharacteristic
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
5. #2083C2semidirect
6. #2074C2semidirect
7. #2066C2semidirect
8. #2056C2semidirect
9. #1949C2semidirect
10. #1928C2semidirect
11. #1878C2semidirect
14. #1803C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4852
C2×#4672
#24992
#23622
#20832
#20742
#20662
#20562
#19492
#19282
#18782
#18732
#18502
#18032
#17352