Finite Group Databasebeta

← #32307

Subgroups — #32307

Order 256.

313 conjugacy classes of subgroups.

The subgroup lattice has 295 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2499C211yesmaximal, characteristic
2#2495C211yesmaximal, characteristic
3#2100C211yesmaximal, characteristic, complemented
4#2097C211yesmaximal, characteristic, complemented
5#2085C211yesmaximal, characteristic, complemented
6#2066C211yesmaximal, characteristic, complemented
7#2036C211yesmaximal, characteristic
8#2035C211yesmaximal, characteristic
9#1949C211yesmaximal, characteristic, complemented
10#1935C211yesmaximal, characteristic, complemented
11#1928C211yesmaximal, characteristic, complemented
12#1924C211yesmaximal, characteristic
13#1854C211yesmaximal, characteristic
14#1854C211yesmaximal, characteristic
15#1847C211yesmaximal, characteristic, complemented
16C2×C22⋊Q8C2211yescharacteristic
17C2×#117C2211yescharacteristic
18C2×#107C2211yescharacteristic
19C2×#107C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×C22⋊C8C2211yescharacteristic
30C2×C22⋊C8C2211yescharacteristic
31C2×#97C2211yescharacteristic
32#400—21no—
33#397—21no—
34#397—21no—
35#397—21no—
36#391C2211yescharacteristic
37#391C2211yescharacteristic
38#389C2211yescharacteristic
39#389C2211yescharacteristic
40#388—21no—
41#387—21no—
42#387—21no—
43#384C2211yescharacteristic
44#383—21no—
45#382C2211yescharacteristic
46#382C2211yescharacteristic
47#380C2211yescharacteristic
48#380C2211yescharacteristic
49C4×C4⋊C4C2211yescharacteristic
50C4×C22⋊C4C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C23×C4C2311yescharacteristic
61C22×C8C2311yescharacteristic
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22⋊Q8—41no—
66C22⋊Q8—41no—
67#117—41no—
68#117—41no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2311yescharacteristic
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C22⋊C4—21no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4D411yescharacteristic
86C2×C22⋊C4D411yescharacteristic
87C2×C42—21no—
88C2×C42—21no—
89C2×C42C2311yescharacteristic
90C2×C42C2311yescharacteristic
91#107—41no—
92#107—41no—
93#107—41no—
94#106—41no—
95#106—41no—
96#106—41no—
97C4⋊C8—22no—
98#103—22no—
99#103—22no—
100#103—41no—
101#103—41no—
102C22⋊C8—22no—
103C22⋊C8—22no—
104#97—22no—
105#95—21no—
106#95—21no—
107#95—21no—
108#95—21no—
109#95—21no—
110#95—21no—
111#95—21no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95D411yescharacteristic
119#95D411yescharacteristic
120C24C2×D411yescharacteristic
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×Q8—21no—
126C2×Q8—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
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×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2411yescharacteristic
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8#4112yes—
158C2×C8#4112yes—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—81no—
177C4⋊C4—81no—
178C22⋊C4—41no—
179C22⋊C4—41no—
180C22⋊C4—42no—
181C22⋊C4—42no—
182C22⋊C4—42no—
183C22⋊C4—81no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42—21no—
188C42—42no—
189C42—42no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23—21no—
196C23—21no—
197C23C22×D411yescharacteristic
198Q8—41no—
199Q8—41no—
200Q8—41no—
201Q8—41no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4C2×#4111yescharacteristic
246C2×C4C2×#4111yescharacteristic
247C2×C4H(5,𝔽2)11yescharacteristic
248C2×C4H(5,𝔽2)11yescharacteristic
249C2×C42−1+411yescharacteristic
250C2×C42−1+411yescharacteristic
251C8—22no—
252C8—22no—
253C8—41no—
254C8—41no—
255C22—41no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22#53511yescharacteristic
263C22#54711yescharacteristic
264C22#54711yescharacteristic
265C22#57511yescharacteristic
266C22#57511yescharacteristic
267C22#57711yescharacteristic
268C22#57811yescharacteristic
269C4—41no—
270C4—41no—
271C4—41no—
272C4—41no—
273C4—41no—
274C4—81no—
275C4—81no—
276C4—81no—
277C4—81no—
278C4—81no—
279C4—81no—
280C4—81no—
281C4—81no—
282C4#53911yescharacteristic
283C4#54211yescharacteristic
284C4#55311yescharacteristic
285C4#56311yescharacteristic
286C2—81no—
287C2#271811yescharacteristic
288C2#321911yescharacteristic
289C2#322511yescharacteristic
290C2#334811yescharacteristic
291C2#334911yescharacteristic
292C2#335011yescharacteristic
293C2#335211yescharacteristic
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
3. #2100C2semidirect
4. #2097C2semidirect
5. #2085C2semidirect
6. #2066C2semidirect
9. #1949C2semidirect
10. #1935C2semidirect
11. #1928C2semidirect
15. #1847C2semidirect

Maximal subgroups

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

subgroupindex
#24992
#24952
#21002
#20972
#20852
#20662
#20362
#20352
#19492
#19352
#19282
#19242
#18542
#18472