Finite Group Databasebeta

← #23075

Subgroups — #23075

Order 256.

403 conjugacy classes of subgroups.

The subgroup lattice has 297 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#476C211yesmaximal, characteristic
2#2365C211yesmaximal, characteristic, complemented
3#2357C211yesmaximal, characteristic
4#1993C211yesmaximal, characteristic, complemented
5#1949C211yesmaximal, characteristic, complemented
6#1936C211yesmaximal, characteristic
7#1934C211yesmaximal, characteristic
8#1926C211yesmaximal, characteristic
9#1925C211yesmaximal, characteristic
10#1920C211yesmaximal, characteristic, complemented
11#1919C211yesmaximal, characteristic, complemented
12#1776C211yesmaximal, characteristic, complemented
13C4×#106C211yesmaximal, characteristic, complemented
14#1771C211yesmaximal, characteristic, complemented
15#1762C211yesmaximal, characteristic
16C2×C4⋊Q8C2211yescharacteristic
17C2×C4×Q8—21no—
18C2×C4×D4C2211yescharacteristic
19#476C418yescomplemented
20C2×#107C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#97C2211yescharacteristic
30C2×#97C2211yescharacteristic
31C2×C4×C8C2211yescharacteristic
32#391—21no—
33#390C2211yescharacteristic
34#390C2211yescharacteristic
35#389C2211yescharacteristic
36#389C2211yescharacteristic
37#388—21no—
38#388—21no—
39#387—21no—
40#387—21no—
41#382C2211yescharacteristic
42#381—21no—
43C4×C4⋊C4C2211yescharacteristic
44C4×C4⋊C4C2211yescharacteristic
45C4×C4⋊C4C2211yescharacteristic
46C4×C4⋊C4C2211yescharacteristic
47C4×C4⋊C4C2211yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49C4×C22⋊C4—21no—
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60C22×Q8—21no—
61C22×D4C2311yescharacteristic
62C23×C4—21no—
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C4⋊Q8—22no—
68C4⋊Q8C2×C414yes—
69C4×Q8—44no—
70C4×D4—22no—
71C4×D4C2×C414yes—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4D411yescharacteristic
76C2×C4⋊C4D411yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42C2311yescharacteristic
95C2×C42C2311yescharacteristic
96C2×C42C2311yescharacteristic
97#107—22no—
98#106—22no—
99#106C2×C414yes—
100#106C2×C414yes—
101C4⋊C8C2×C414yes—
102#102—22no—
103#102—22no—
104#102C2×C414yes—
105#102C2×C414yes—
106#97—22no—
107#97—22no—
108C4×C8—22no—
109#95—21no—
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#95D411yescharacteristic
119#95D411yescharacteristic
120C24—21no—
121C2×Q8—24no—
122C2×Q8—42no—
123C2×Q8—42no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4C22×C412yes—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—24no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—81no—
139C22×C4—81no—
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2411yescharacteristic
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8C22×C412yes—
157C2×C8C22×C412yes—
158C2×C8#4112yes—
159C4⋊C4—24no—
160C4⋊C4—22no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—41no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4C22×C412yes—
175C4⋊C4C22×C412yes—
176C4⋊C4C22×C412yes—
177C4⋊C4#4112yes—
178C22⋊C4—24no—
179C22⋊C4—42no—
180C22⋊C4—42no—
181C22⋊C4—81no—
182C22⋊C4—81no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42—42no—
188C42—42no—
189C42—42no—
190C42—81no—
191C42—81no—
192C42C22×C412yes—
193C42#4112yes—
194C23—22no—
195C23—41no—
196C23—41no—
197C23—41no—
198C23—41no—
199C23—41no—
200C23—41no—
201C23C22×D411yescharacteristic
202Q8—48no—
203D4—22no—
204D4—22no—
205D4—41no—
206D4—41no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
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—84no—
244C2×C4C4×D412yes—
245C2×C4C4×D412yes—
246C2×C4AGL(1,ℤ/8)12yes—
247C2×C4C23×C411yescharacteristic
248C2×C4C2×#4111yescharacteristic
249C2×C4C2×#4111yescharacteristic
250C2×C4C2×#4111yescharacteristic
251C2×C42−1+411yescharacteristic
252C2×C42−1+411yescharacteristic
253C8—22no—
254C8—22no—
255C8—22no—
256C8—41no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—81no—
261C22—81no—
262C22—81no—
263C22—81no—
264C22C2×C4×D411yescharacteristic
265C22#54711yescharacteristic
266C22#54811yescharacteristic
267C22C2×AGL(1,ℤ/8)11yescharacteristic
268C22#57511yescharacteristic
269C22#57711yescharacteristic
270C22#57711yescharacteristic
271C4—22no—
272C4—22no—
273C4—22no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—42no—
278C4—42no—
279C4—42no—
280C4—42no—
281C4—81no—
282C4—81no—
283C4—81no—
284C4C4×#4111yescharacteristic
285C4#51911yescharacteristic
286C4#55211yescharacteristic
287C4#55611yescharacteristic
288C2—82no—
289C2#236211yescharacteristic
290C2C4×AGL(1,ℤ/8)11yescharacteristic
291C2#294911yescharacteristic
292C2#334511yescharacteristic
293C2#335011yescharacteristic
294C2#337111yescharacteristic
295C2#337311yescharacteristic
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. #2365C2semidirect
4. #1993C2semidirect
5. #1949C2semidirect
10. #1920C2semidirect
11. #1919C2semidirect
12. #1776C2semidirect
13. C4×#106C2semidirect
14. #1771C2semidirect
19. #476C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4762
#23652
#23572
#19932
#19492
#19362
#19342
#19262
#19252
#19202
#19192
#17762
C4×#1062
#17712
#17622