Finite Group Databasebeta

← #27401

Subgroups — #27401

Order 256.

363 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2429C211yesmaximal, characteristic
2#2347C211yesmaximal, characteristic
3#2086C211yesmaximal, characteristic, complemented
4#2043C211yesmaximal, characteristic, complemented
5#2036C211yesmaximal, characteristic
6#2035C211yesmaximal, characteristic
7#1981C211yesmaximal, characteristic, complemented
8#1933C211yesmaximal, characteristic, complemented
9#1928C211yesmaximal, characteristic, complemented
10#1923C211yesmaximal, characteristic
11#1919C211yesmaximal, characteristic, complemented
12#1873C211yesmaximal, characteristic
13#1873C211yesmaximal, characteristic
14#1865C211yesmaximal, characteristic, complemented
15C4×#103C211yesmaximal, characteristic, complemented
16C2×C4⋊Q8C2211yescharacteristic
17C2×C22⋊Q8C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×C4×C8C2211yescharacteristic
35#398—21no—
36#396—21no—
37#391C2211yescharacteristic
38#389—21no—
39#389C2211yescharacteristic
40#389C2211yescharacteristic
41#389C2211yescharacteristic
42#388—21no—
43#388—21no—
44#388—21no—
45#383—21no—
46#380C2211yescharacteristic
47C4×C4⋊C4—21no—
48C4×C4⋊C4C2211yescharacteristic
49C4×C4⋊C4C2211yescharacteristic
50C4×C22⋊C4C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×Q8C2311yescharacteristic
61C23×C4C2311yescharacteristic
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C4⋊Q8—22no—
67C4⋊Q8—41no—
68C22⋊Q8—22no—
69C22⋊Q8—22no—
70C22⋊Q8—41no—
71C22⋊Q8—41no—
72C4×Q8—22no—
73C4×Q8—41no—
74#117—22no—
75#117—22no—
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⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4D411yescharacteristic
95C2×C22⋊C4D411yescharacteristic
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42C2311yescharacteristic
101C2×C42C2311yescharacteristic
102#107—41no—
103#106—22no—
104C4⋊C8—41no—
105#103—22no—
106#103—22no—
107#103—22no—
108#103—22no—
109#103—22no—
110#103—41no—
111#103—41no—
112#103—41no—
113C22⋊C8—41no—
114C22⋊C8—41no—
115C4×C8—41no—
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×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2411yescharacteristic
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—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#4112yes—
189C4⋊C4#4112yes—
190C4⋊C4#4112yes—
191C22⋊C4—24no—
192C22⋊C4—22no—
193C22⋊C4—22no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C22⋊C4—81no—
197C42—24no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—42no—
203C42—81no—
204C42—81no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23C22×D411yescharacteristic
213Q8—22no—
214Q8—22no—
215Q8—41no—
216Q8—41no—
217Q8—41no—
218Q8—41no—
219Q8—41no—
220Q8—41no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
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#13512yes—
263C2×C4#13712yes—
264C2×C4C2×#4111yescharacteristic
265C2×C4C2×#4111yescharacteristic
266C2×C4C2×#4111yescharacteristic
267C2×C4C2×#4111yescharacteristic
268C2×C4H(5,𝔽2)11yescharacteristic
269C2×C4H(5,𝔽2)11yescharacteristic
270C8—41no—
271C8—41no—
272C8—41no—
273C8—41no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22#52511yescharacteristic
282C22#54611yescharacteristic
283C22#54611yescharacteristic
284C22C2×#13511yescharacteristic
285C22C2×#13711yescharacteristic
286C22#57711yescharacteristic
287C22#57811yescharacteristic
288C4—22no—
289C4—22no—
290C4—41no—
291C4—42no—
292C4—42no—
293C4—41no—
294C4—42no—
295C4—41no—
296C4—42no—
297C4—81no—
298C4—81no—
299C4—81no—
300C4—81no—
301C4#52911yescharacteristic
302C4#54011yescharacteristic
303C4#55311yescharacteristic
304C4#55511yescharacteristic
305C2—81no—
306C2#247611yescharacteristic
307C2#310711yescharacteristic
308C2#312011yescharacteristic
309C2#330111yescharacteristic
310C2#330211yescharacteristic
311C2#332011yescharacteristic
312C2#332811yescharacteristic
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. #2086C2semidirect
4. #2043C2semidirect
7. #1981C2semidirect
8. #1933C2semidirect
9. #1928C2semidirect
11. #1919C2semidirect
14. #1865C2semidirect
15. C4×#103C2semidirect

Maximal subgroups

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

subgroupindex
#24292
#23472
#20862
#20432
#20362
#20352
#19812
#19332
#19282
#19232
#19192
#18732
#18652
C4×#1032