Finite Group Databasebeta

← #27706

Subgroups — #27706

Order 256.

371 conjugacy classes of subgroups.

The subgroup lattice has 307 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2499C211yesmaximal, characteristic
2#2440C211yesmaximal, characteristic
3#2101C211yesmaximal, characteristic, complemented
4#2099C211yesmaximal, characteristic, complemented
5#2072C211yesmaximal, characteristic, complemented
6#2024C211yesmaximal, characteristic, complemented
7#1993C211yesmaximal, characteristic, complemented
8#1935C211yesmaximal, characteristic, complemented
9#1933C211yesmaximal, characteristic, complemented
10#1894C211yesmaximal, characteristic
11#1872C211yesmaximal, characteristic
12#1834C211yesmaximal, characteristic, complemented
13#1825C211yesmaximal, characteristic
14#1753C211yesmaximal, characteristic
15#1736C211yesmaximal, characteristic
16C22×M16C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×C22⋊Q8—21no—
19C2×C22⋊Q8C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22C2×#103C2211yescharacteristic
23C2×#103C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×C22⋊C8C2211yescharacteristic
27C2×#97C2211yescharacteristic
28C2×C4×C8C2211yescharacteristic
29#398—21no—
30#397—21no—
31#397—21no—
32#395—21no—
33#391C2211yescharacteristic
34#391C2211yescharacteristic
35#389C2211yescharacteristic
36#389C2211yescharacteristic
37#387—21no—
38#387—21no—
39#385—21no—
40#384C2211yescharacteristic
41#384C2211yescharacteristic
42#382C2211yescharacteristic
43#380C2211yescharacteristic
44C4×C22⋊C4C2211yescharacteristic
45#340C2211yescharacteristic
46#340C2211yescharacteristic
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×Q8—21no—
60C22×Q8C2311yescharacteristic
61C23×C4—21no—
62C23×C4C2311yescharacteristic
63C2×M16—22no—
64C2×M16—22no—
65C2×M16—41no—
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C4⋊Q8—22no—
71C4⋊Q8—41no—
72C22⋊Q8—22no—
73C22⋊Q8—22no—
74C22⋊Q8—44no—
75#117—22no—
76#117—22no—
77C2×C4⋊C4—22no—
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⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—41no—
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⋊C4D411yescharacteristic
99C2×C22⋊C4D411yescharacteristic
100C2×C42—21no—
101C2×C42C2311yescharacteristic
102C2×C42C2311yescharacteristic
103#103—22no—
104#103—22no—
105#103—41no—
106#103—41no—
107C22⋊C8—22no—
108#97—41no—
109C4×C8—22no—
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#95—21no—
119#95D411yescharacteristic
120#95D411yescharacteristic
121C24C2×D411yescharacteristic
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—42no—
129C2×Q8—42no—
130C2×Q8—81no—
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×C4—21no—
142C22×C4—21no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—81no—
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4#4112yes—
152C22×C4#4112yes—
153C22×C4C2411yescharacteristic
154M16—44no—
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—
166C2×C8#4112yes—
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—21no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—81no—
189C4⋊C4#4112yes—
190C22⋊C4—22no—
191C22⋊C4—22no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—42no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—42no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23#12412yes—
207C23C22×D411yescharacteristic
208Q8—41no—
209Q8—41no—
210Q8—41no—
211Q8—41no—
212Q8—84no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
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×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—84no—
254C2×C4#12412yes—
255C2×C4AGL(1,ℤ/8)12yes—
256C2×C4#13712yes—
257C2×C4C2×#4111yescharacteristic
258C2×C4C2×#4111yescharacteristic
259C2×C4C2×#4111yescharacteristic
260C2×C4C2×#4111yescharacteristic
261C2×C4H(5,𝔽2)11yescharacteristic
262C2×C42−1+411yescharacteristic
263C8—22no—
264C8—41no—
265C8—41no—
266C8—41no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22C2×#12411yescharacteristic
275C22#54611yescharacteristic
276C22#54711yescharacteristic
277C22C2×AGL(1,ℤ/8)11yescharacteristic
278C22C2×#13711yescharacteristic
279C22#57711yescharacteristic
280C22#57811yescharacteristic
281C4—22no—
282C4—22no—
283C4—22no—
284C4—41no—
285C4—41no—
286C4—42no—
287C4—81no—
288C4—81no—
289C4—81no—
290C4—81no—
291C4—81no—
292C4—81no—
293C4—81no—
294C4#52911yescharacteristic
295C4#53911yescharacteristic
296C4#55111yescharacteristic
297C4#55211yescharacteristic
298C2—42no—
299C2#248811yescharacteristic
300C2#313711yescharacteristic
301C2#314111yescharacteristic
302C2#332411yescharacteristic
303C2#332811yescharacteristic
304C2#334311yescharacteristic
305C2#334411yescharacteristic
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. #2101C2semidirect
4. #2099C2semidirect
5. #2072C2semidirect
6. #2024C2semidirect
7. #1993C2semidirect
8. #1935C2semidirect
9. #1933C2semidirect
12. #1834C2semidirect

Maximal subgroups

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

subgroupindex
#24992
#24402
#21012
#20992
#20722
#20242
#19932
#19352
#19332
#18942
#18722
#18342
#18252
#17532
#17362