Finite Group Databasebeta

← #22778

Subgroups — #22778

Order 256.

447 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
1C2×#440C211yesmaximal, characteristic
2#2356C211yesmaximal, characteristic, complemented
3D4×C4⋊C4C211yesmaximal, 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#1804C211yesmaximal, characteristic
12#1803C211yesmaximal, characteristic, complemented
13#1771C211yesmaximal, characteristic, complemented
14#1762C211yesmaximal, characteristic
15C4×#102C211yesmaximal, characteristic
16C22×SD16C2211yescharacteristic
17C2×C4×Q8C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C22×C4⋊C4—21no—
22#440C418yescomplemented
23C2×#107C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#97C2211yescharacteristic
32C2×#97C2211yescharacteristic
33C2×C4×C8C2211yescharacteristic
34#391C2211yescharacteristic
35#391C2211yescharacteristic
36#389C2211yescharacteristic
37#389C2211yescharacteristic
38#387—21no—
39#387—21no—
40#385—21no—
41#385—21no—
42#384C2211yescharacteristic
43#383—21no—
44#382C2211yescharacteristic
45#380—21no—
46C4×C4⋊C4C2211yescharacteristic
47C4×C4⋊C4C2211yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49#376—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⋊C4—44no—
86C2×C4⋊C4D411yescharacteristic
87C2×C4⋊C4D411yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42C2311yescharacteristic
101C2×C42C2311yescharacteristic
102C2×C42C2311yescharacteristic
103#107C2×C414yescomplemented
104C4⋊C8—22no—
105#103—41no—
106#103—41no—
107#103C2×C414yescomplemented
108#102—22no—
109#102—22no—
110#102C2×C414yes—
111#97—41no—
112#97C2×C414yescomplemented
113C4×C8—41no—
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
126C24—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8C22×C412yes—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4C22×C412yes—
137C2×D4#4112yes—
138C2×D4#4112yes—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—24no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—81no—
154C22×C4—81no—
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2411yescharacteristic
162SD16—28no—
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×C8C22×C412yes—
174C2×C8C22×C412yes—
175C4⋊C4—24no—
176C4⋊C4—22no—
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—41no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—81no—
192C4⋊C4C22×C412yes—
193C4⋊C4C22×C412yes—
194C22⋊C4—24no—
195C22⋊C4—42no—
196C22⋊C4—42no—
197C42—24no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—42no—
203C42—42no—
204C42C22×C412yes—
205C42#4112yes—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23C22×D411yescharacteristic
214Q8—22no—
215Q8—22no—
216Q8—41no—
217Q8—41no—
218D4—22no—
219D4—22no—
220D4—22no—
221D4#11714yes—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
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—21no—
240C2×C4—48no—
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—84no—
260C2×C4—84no—
261C2×C4C4×D412yes—
262C2×C4C4×D412yes—
263C2×C4AGL(1,ℤ/8)12yes—
264C2×C4C23×C411yescharacteristic
265C2×C4C2×#4111yescharacteristic
266C2×C4C2×#4111yescharacteristic
267C2×C4C2×#4111yescharacteristic
268C2×C4H(5,𝔽2)11yescharacteristic
269C2×C42−1+411yescharacteristic
270C8—22no—
271C8—22no—
272C8—41no—
273C8—41no—
274C22—24no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22C2×C4×D411yescharacteristic
282C22#54611yescharacteristic
283C22#54811yescharacteristic
284C22C2×AGL(1,ℤ/8)11yescharacteristic
285C22#57511yescharacteristic
286C22#57711yescharacteristic
287C22#57811yescharacteristic
288C4—22no—
289C4—22no—
290C4—22no—
291C4—42no—
292C4—42no—
293C4—42no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—81no—
298C4—81no—
299C4—81no—
300C4—81no—
301C4C2×#11711yescharacteristic
302C4#52011yescharacteristic
303C4#55211yescharacteristic
304C4#55311yescharacteristic
305C2—44no—
306C2#235211yescharacteristic
307C2C4×AGL(1,ℤ/8)11yescharacteristic
308C2#294711yescharacteristic
309C2#331011yescharacteristic
310C2#331511yescharacteristic
311C2#336711yescharacteristic
312C2#337611yescharacteristic
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. #2356C2semidirect
4. #1935C2semidirect
5. #1933C2semidirect
6. #1921C2semidirect
7. #1918C2semidirect
10. #1836C2semidirect
12. #1803C2semidirect
13. #1771C2semidirect
22. #440C4semidirect
73. C4×Q8C2×C4semidirect
103. #107C2×C4semidirect
107. #103C2×C4semidirect
112. #97C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4402
#23562
D4×C4⋊C42
#19352
#19332
#19212
#19182
#18772
#18362
#18042
#18032
#17712
#17622
C4×#1022