Finite Group Databasebeta

← #31369

Subgroups — #31369

Order 256.

367 conjugacy classes of subgroups.

The subgroup lattice has 321 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2441C211yesmaximal, characteristic, complemented
2#2422C211yesmaximal, characteristic
3#2099C211yesmaximal, characteristic, complemented
4#2088C211yesmaximal, characteristic, complemented
5#2084C211yesmaximal, characteristic, complemented
6#2077C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic
8#2025C211yesmaximal, characteristic, complemented
9#1992C211yesmaximal, characteristic, complemented
10#1936C211yesmaximal, characteristic, complemented
11#1924C211yesmaximal, characteristic, complemented
12#1875C211yesmaximal, characteristic
13#1864C211yesmaximal, characteristic, complemented
14#1825C211yesmaximal, characteristic, complemented
15#1802C211yesmaximal, characteristic, complemented
16C2×#125C2211yescharacteristic
17C2×#123—21no—
18C2×#121C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#102C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×C22⋊C8C2211yescharacteristic
29C2×C22⋊C8C2211yescharacteristic
30C2×#97C2211yescharacteristic
31#399—21no—
32#397—21no—
33#396—21no—
34#394—21no—
35#390C2211yescharacteristic
36#389—21no—
37#389C2211yescharacteristic
38#389C2211yescharacteristic
39#389C2211yescharacteristic
40#387—21no—
41#385—21no—
42#384C2211yescharacteristic
43#384C2211yescharacteristic
44#380C2211yescharacteristic
45#380C2211yescharacteristic
46C4×C22⋊C4C2211yescharacteristic
47#340C2211yescharacteristic
48#340C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×D4—21no—
60C22×D4C2311yescharacteristic
61C23×C4—21no—
62C23×C4C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67#125—22no—
68#125—41no—
69#123—44no—
70#121—22no—
71#121—41no—
72C4×D4—22no—
73C4×D4—41no—
74#117—22no—
75#117—41no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4D411yescharacteristic
100C2×C22⋊C4D411yescharacteristic
101C2×C42—21no—
102C2×C42C2311yescharacteristic
103C2×C42C2311yescharacteristic
104#106—41no—
105#106—41no—
106C4⋊C8—41no—
107#102—22no—
108#102—22no—
109#102—41no—
110#102—41no—
111C22⋊C8—41no—
112C22⋊C8—41no—
113#97—41no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95D411yescharacteristic
120#95D411yescharacteristic
121C24—21no—
122C24C2×D411yescharacteristic
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—42no—
129C2×D4—42no—
130C2×D4—81no—
131C2×D4#4112yes—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
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—21no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—81no—
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2411yescharacteristic
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—81no—
186C4⋊C4—81no—
187C4⋊C4—81no—
188C4⋊C4#4112yes—
189C22⋊C4—22no—
190C22⋊C4—41no—
191C22⋊C4—42no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C22⋊C4—42no—
197C22⋊C4—81no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—42no—
203C23—21no—
204C23—21no—
205C23—22no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23C22×D411yescharacteristic
218D4—22no—
219D4—22no—
220D4—41no—
221D4—41no—
222D4—84no—
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—21no—
232C2×C4—21no—
233C2×C4—21no—
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—84no—
263C2×C4AGL(1,ℤ/8)12yes—
264C2×C4C2×#4111yescharacteristic
265C2×C4C2×#4111yescharacteristic
266C2×C4H(5,𝔽2)11yescharacteristic
267C2×C4H(5,𝔽2)11yescharacteristic
268C2×C42−1+411yescharacteristic
269C2×C42−1+411yescharacteristic
270C8—41no—
271C8—41no—
272C8—41no—
273C8—41no—
274C22—41no—
275C22—41no—
276C22—42no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—42no—
282C22—42no—
283C22—41no—
284C22—81no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22#53611yescharacteristic
289C22#54611yescharacteristic
290C22#54611yescharacteristic
291C22C2×AGL(1,ℤ/8)11yescharacteristic
292C22#57511yescharacteristic
293C22#57711yescharacteristic
294C22#57711yescharacteristic
295C4—22no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—42no—
301C4—81no—
302C4—81no—
303C4—81no—
304C4—81no—
305C4—81no—
306C4—81no—
307C4#53911yescharacteristic
308C4#54211yescharacteristic
309C4#55211yescharacteristic
310C4#56211yescharacteristic
311C2—81no—
312C2—82no—
313C2#268711yescharacteristic
314C2#323511yescharacteristic
315C2#324411yescharacteristic
316C2#330711yescharacteristic
317C2#330711yescharacteristic
318C2#331011yescharacteristic
319C2#333211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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
1. #2441C2semidirect
3. #2099C2semidirect
4. #2088C2semidirect
5. #2084C2semidirect
6. #2077C2semidirect
8. #2025C2semidirect
9. #1992C2semidirect
10. #1936C2semidirect
11. #1924C2semidirect
13. #1864C2semidirect
14. #1825C2semidirect
15. #1802C2semidirect

Maximal subgroups

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

subgroupindex
#24412
#24222
#20992
#20882
#20842
#20772
#20342
#20252
#19922
#19362
#19242
#18752
#18642
#18252
#18022