Finite Group Databasebeta

← #31956

Subgroups — #31956

Order 256.

354 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2502C211yesmaximal, characteristic
2#2421C211yesmaximal, characteristic
3#2096C211yesmaximal, characteristic, complemented
4#2084C211yesmaximal, characteristic, complemented
5#2074C211yesmaximal, characteristic, complemented
6#2066C211yesmaximal, characteristic, complemented
7#2035C211yesmaximal, characteristic
8#2001C211yesmaximal, characteristic
9#1971C211yesmaximal, characteristic, complemented
10#1924C211yesmaximal, characteristic
11#1878C211yesmaximal, characteristic, complemented
12#1872C211yesmaximal, characteristic
13#1853C211yesmaximal, characteristic
14#1836C211yesmaximal, characteristic, complemented
15#1803C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C2×C22⋊Q8C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×#97C2211yescharacteristic
34#400—21no—
35#398—21no—
36#398—21no—
37#397—21no—
38#391C2211yescharacteristic
39#391C2211yescharacteristic
40#391C2211yescharacteristic
41#389C2211yescharacteristic
42#388—21no—
43#387—21no—
44#385—21no—
45#385—21no—
46#384C2211yescharacteristic
47#384C2211yescharacteristic
48#382C2211yescharacteristic
49#380C2211yescharacteristic
50C4×C22⋊C4C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×Q8C2311yescharacteristic
61C23×C4—21no—
62C23×C4C2311yescharacteristic
63C2×Q16—22no—
64C2×Q16—22no—
65C2×Q16—41no—
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22⋊Q8—22no—
71C22⋊Q8—41no—
72C22⋊Q8—41no—
73C22⋊Q8—41no—
74C4×Q8—22no—
75C4×Q8—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⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4D411yescharacteristic
97C2×C22⋊C4D411yescharacteristic
98C2×C42—21no—
99C2×C42C2311yescharacteristic
100C2×C42C2311yescharacteristic
101#106—41no—
102#106—41no—
103#106—41no—
104C4⋊C8—41no—
105#103—41no—
106#103—41no—
107#103—41no—
108#103—41no—
109#103—41no—
110#103—41no—
111C22⋊C8—22no—
112C22⋊C8—41no—
113#97—22no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95D411yescharacteristic
125#95D411yescharacteristic
126C24C2×D411yescharacteristic
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8#4112yes—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
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—42no—
154C22×C4—81no—
155C22×C4—81no—
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2411yescharacteristic
162Q16—44no—
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—21no—
174C2×C8#4112yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
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—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—81no—
193C4⋊C4—81no—
194C4⋊C4—81no—
195C4⋊C4—81no—
196C22⋊C4—22no—
197C22⋊C4—22no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—42no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23#13712yes—
213C23C22×D411yescharacteristic
214Q8—22no—
215Q8—22no—
216Q8—41no—
217Q8—41no—
218Q8—41no—
219Q8—41no—
220Q8—41no—
221Q8—41no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
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—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—84no—
265C2×C4#13712yes—
266C2×C4C2×#4111yescharacteristic
267C2×C4C2×#4111yescharacteristic
268C2×C4H(5,𝔽2)11yescharacteristic
269C2×C4H(5,𝔽2)11yescharacteristic
270C2×C4H(5,𝔽2)11yescharacteristic
271C2×C42−1+411yescharacteristic
272C8—22no—
273C8—41no—
274C8—41no—
275C8—41no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22#53511yescharacteristic
284C22#54611yescharacteristic
285C22#54711yescharacteristic
286C22C2×#13711yescharacteristic
287C22C2×#13711yescharacteristic
288C22#57711yescharacteristic
289C22#57711yescharacteristic
290C4—22no—
291C4—41no—
292C4—41no—
293C4—41no—
294C4—42no—
295C4—41no—
296C4—81no—
297C4—81no—
298C4—81no—
299C4—81no—
300C4—81no—
301C4—81no—
302C4—81no—
303C4#53911yescharacteristic
304C4#54011yescharacteristic
305C4#55111yescharacteristic
306C4#56211yescharacteristic
307C2—42no—
308C2#270611yescharacteristic
309C2#321811yescharacteristic
310C2#322211yescharacteristic
311C2#331611yescharacteristic
312C2#331611yescharacteristic
313C2#334411yescharacteristic
314C2#335611yescharacteristic
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. #2096C2semidirect
4. #2084C2semidirect
5. #2074C2semidirect
6. #2066C2semidirect
9. #1971C2semidirect
11. #1878C2semidirect
14. #1836C2semidirect
15. #1803C2semidirect

Maximal subgroups

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

subgroupindex
#25022
#24212
#20962
#20842
#20742
#20662
#20352
#20012
#19712
#19242
#18782
#18722
#18532
#18362
#18032