Finite Group Databasebeta

← #27825

Subgroups — #27825

Order 256.

371 conjugacy classes of subgroups.

The subgroup lattice has 303 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2566C211yesmaximal, characteristic
2#2551C211yesmaximal, characteristic, complemented
3#2101C211yesmaximal, characteristic, complemented
4#2100C211yesmaximal, characteristic, complemented
5#2023C211yesmaximal, characteristic
6#2023C211yesmaximal, characteristic
7#1993C211yesmaximal, characteristic, complemented
8#1992C211yesmaximal, characteristic, complemented
9#1990C211yesmaximal, characteristic, complemented
10#1989C211yesmaximal, characteristic, complemented
11#1936C211yesmaximal, characteristic
12#1934C211yesmaximal, characteristic
13#1802C211yesmaximal, characteristic
14#1771C211yesmaximal, characteristic, complemented
15#1762C211yesmaximal, characteristic
16C2×C4⋊Q8—21no—
17C2×C4⋊Q8C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×#125C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×#102C2211yescharacteristic
23C2×#102C2211yescharacteristic
24C2×#102C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#97C2211yescharacteristic
27C2×#97C2211yescharacteristic
28C2×C4×C8C2211yescharacteristic
29#396—21no—
30#396—21no—
31#395—21no—
32#395—21no—
33#390C2211yescharacteristic
34#390C2211yescharacteristic
35#389—21no—
36#389C2211yescharacteristic
37#389C2211yescharacteristic
38#385—21no—
39#385—21no—
40#384C2211yescharacteristic
41#384C2211yescharacteristic
42#382C2211yescharacteristic
43C4×C4⋊C4C2211yescharacteristic
44C4×C4⋊C4C2211yescharacteristic
45#340C2211yescharacteristic
46#340C2211yescharacteristic
47#340C2211yescharacteristic
48#340C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8—21no—
60C22×Q8—21no—
61C22×D4C2311yescharacteristic
62C23×C4—21no—
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C4⋊Q8—22no—
68C4⋊Q8—22no—
69C4⋊Q8—41no—
70C4⋊Q8—41no—
71C4⋊Q8—44no—
72#125—22no—
73#125—22no—
74C4×D4—22no—
75C4×D4—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⋊C4—21no—
85C2×C4⋊C4—21no—
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×C4⋊C4C2311yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C42—21no—
101C2×C42C2311yescharacteristic
102C2×C42C2311yescharacteristic
103C2×C42C2311yescharacteristic
104C4⋊C8—22no—
105#102—22no—
106#102—41no—
107#102—41no—
108#102Q814yescomplemented
109#97—22no—
110#97—41no—
111C4×C8—41no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95D411yescharacteristic
118#95D411yescharacteristic
119C24—21no—
120C2×Q8—42no—
121C2×Q8—42no—
122C2×Q8—42no—
123C2×Q8—42no—
124C2×Q8—81no—
125C2×Q8—81no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4C2×Q812yes—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—42no—
140C22×C4—81no—
141C22×C4—81no—
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2411yescharacteristic
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8C2×Q812yes—
161C4⋊C4—22no—
162C4⋊C4—22no—
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—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—81no—
183C4⋊C4—81no—
184C4⋊C4—81no—
185C4⋊C4C2×Q812yes—
186C4⋊C4#4112yes—
187C22⋊C4—42no—
188C22⋊C4—42no—
189C22⋊C4—42no—
190C22⋊C4—81no—
191C22⋊C4—81no—
192C42—21no—
193C42—21no—
194C42—21no—
195C42—21no—
196C42—42no—
197C42#4112yes—
198C42#4112yes—
199C23—22no—
200C23—41no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23C22×D411yescharacteristic
207Q8—84no—
208Q8—84no—
209D4—22no—
210D4—22no—
211D4—41no—
212D4—41no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
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—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—84no—
249C2×C4#12412yes—
250C2×C4#12412yes—
251C2×C4AGL(1,ℤ/8)12yes—
252C2×C4#13712yes—
253C2×C4C22×Q811yescharacteristic
254C2×C4C2×#4111yescharacteristic
255C2×C4C2×#4111yescharacteristic
256C2×C4C2×#4111yescharacteristic
257C2×C4H(5,𝔽2)11yescharacteristic
258C2×C42−1+411yescharacteristic
259C8—22no—
260C8—41no—
261C8—41no—
262C8—41no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—81no—
267C22—81no—
268C22—81no—
269C22—81no—
270C22C2×#12411yescharacteristic
271C22#54611yescharacteristic
272C22D4×Q811yescharacteristic
273C22C2×AGL(1,ℤ/8)11yescharacteristic
274C22C2×#13711yescharacteristic
275C22#57611yescharacteristic
276C22#57711yescharacteristic
277C4—22no—
278C4—22no—
279C4—22no—
280C4—22no—
281C4—42no—
282C4—41no—
283C4—42no—
284C4—41no—
285C4—81no—
286C4—81no—
287C4—81no—
288C4—81no—
289C4—81no—
290C4#53311yescharacteristic
291C4#54211yescharacteristic
292C4#55411yescharacteristic
293C4#55611yescharacteristic
294C2—82no—
295C2#249211yescharacteristic
296C2#313611yescharacteristic
297C2#314111yescharacteristic
298C2#331811yescharacteristic
299C2#332011yescharacteristic
300C2#338411yescharacteristic
301C2#338511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×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. #2551C2semidirect
3. #2101C2semidirect
4. #2100C2semidirect
7. #1993C2semidirect
8. #1992C2semidirect
9. #1990C2semidirect
10. #1989C2semidirect
14. #1771C2semidirect
108. #102Q8semidirect

Maximal subgroups

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

subgroupindex
#25662
#25512
#21012
#21002
#20232
#19932
#19922
#19902
#19892
#19362
#19342
#18022
#17712
#17622