Finite Group Databasebeta

← #22776

Subgroups — #22776

Order 256.

433 conjugacy classes of subgroups.

The subgroup lattice has 313 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#440C211yesmaximal, characteristic
2#2355C211yesmaximal, characteristic, complemented
3#2349C211yesmaximal, characteristic
4#1935C211yesmaximal, characteristic, complemented
5#1933C211yesmaximal, characteristic, complemented
6#1921C211yesmaximal, characteristic, complemented
7#1918C211yesmaximal, characteristic, complemented
8#1877C211yesmaximal, characteristic
9#1877C211yesmaximal, characteristic
10#1836C211yesmaximal, characteristic, complemented
11#1805C211yesmaximal, characteristic, complemented
12#1802C211yesmaximal, characteristic
13#1771C211yesmaximal, characteristic, complemented
14#1762C211yesmaximal, characteristic
15C4×#102C211yesmaximal, characteristic
16C22×SD16C2211yescharacteristic
17C2×C4×Q8C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21#440C418yescomplemented
22C2×#107C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#97C2211yescharacteristic
31C2×#97C2211yescharacteristic
32C2×C4×C8C2211yescharacteristic
33#391C2211yescharacteristic
34#391C2211yescharacteristic
35#389C2211yescharacteristic
36#389C2211yescharacteristic
37#387—21no—
38#387—21no—
39#385—21no—
40#385—21no—
41#384C2211yescharacteristic
42#382—21no—
43#382C2211yescharacteristic
44#380—21no—
45C4×C4⋊C4—21no—
46C4×C4⋊C4C2211yescharacteristic
47C4×C4⋊C4C2211yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49C2×#95—21no—
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60C22×Q8C2311yescharacteristic
61C22×D4C2311yescharacteristic
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C2×SD16—22no—
66C2×SD16—22no—
67C2×SD16C2×C414yes—
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C4×Q8—22no—
73C4×Q8C2×C414yescomplemented
74C4×D4—22no—
75C4×D4—22no—
76C4×D4—22no—
77C4×D4—41no—
78C4×D4—41no—
79C4×D4C2×C414yes—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4D411yescharacteristic
86C2×C4⋊C4D411yescharacteristic
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×C42—21no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42C2311yescharacteristic
100C2×C42C2311yescharacteristic
101C2×C42C2311yescharacteristic
102#107C2×C414yescomplemented
103C4⋊C8—22no—
104#103—41no—
105#103—41no—
106#103C2×C414yescomplemented
107#102—22no—
108#102—22no—
109#102C2×C414yes—
110#97—41no—
111#97C2×C414yescomplemented
112C4×C8—41no—
113#95—21no—
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#95D411yescharacteristic
124#95D411yescharacteristic
125C24—21no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8C22×C412yes—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4C22×C412yes—
136C2×D4#4112yes—
137C2×D4#4112yes—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—24no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—81no—
151C22×C4—81no—
152C22×C4—81no—
153C22×C4—81no—
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2411yescharacteristic
161SD16—28no—
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—
171C2×C8—21no—
172C2×C8C22×C412yes—
173C2×C8C22×C412yes—
174C4⋊C4—24no—
175C4⋊C4—22no—
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—41no—
187C4⋊C4—42no—
188C4⋊C4—81no—
189C4⋊C4—81no—
190C4⋊C4—81no—
191C4⋊C4C22×C412yes—
192C4⋊C4C22×C412yes—
193C22⋊C4—24no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C42—24no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—81no—
202C42—81no—
203C42C22×C412yes—
204C42#4112yes—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23C22×D411yescharacteristic
213Q8—22no—
214Q8—22no—
215Q8—41no—
216Q8—41no—
217D4—22no—
218D4—22no—
219D4—22no—
220D4#11714yes—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
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—84no—
258C2×C4—84no—
259C2×C4—84no—
260C2×C4C4×D412yes—
261C2×C4C4×D412yes—
262C2×C4#13712yes—
263C2×C4C23×C411yescharacteristic
264C2×C4C2×#4111yescharacteristic
265C2×C4C2×#4111yescharacteristic
266C2×C4C2×#4111yescharacteristic
267C2×C4H(5,𝔽2)11yescharacteristic
268C2×C42−1+411yescharacteristic
269C8—22no—
270C8—22no—
271C8—41no—
272C8—41no—
273C22—24no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22C2×C4×D411yescharacteristic
281C22#54611yescharacteristic
282C22#54811yescharacteristic
283C22C2×#13711yescharacteristic
284C22#57511yescharacteristic
285C22#57611yescharacteristic
286C22#57711yescharacteristic
287C4—22no—
288C4—22no—
289C4—22no—
290C4—42no—
291C4—42no—
292C4—42no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—81no—
297C4—81no—
298C4—81no—
299C4—81no—
300C4C2×#11711yescharacteristic
301C4#52011yescharacteristic
302C4#55211yescharacteristic
303C4#55311yescharacteristic
304C2—44no—
305C2#235211yescharacteristic
306C2C4×#13711yescharacteristic
307C2#294611yescharacteristic
308C2#331111yescharacteristic
309C2#331411yescharacteristic
310C2#337011yescharacteristic
311C2#337211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #2355C2semidirect
4. #1935C2semidirect
5. #1933C2semidirect
6. #1921C2semidirect
7. #1918C2semidirect
10. #1836C2semidirect
11. #1805C2semidirect
13. #1771C2semidirect
21. #440C4semidirect
73. C4×Q8C2×C4semidirect
102. #107C2×C4semidirect
106. #103C2×C4semidirect
111. #97C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4402
#23552
#23492
#19352
#19332
#19212
#19182
#18772
#18362
#18052
#18022
#17712
#17622
C4×#1022