Finite Group Databasebeta

← #24611

Subgroups — #24611

Order 256.

394 conjugacy classes of subgroups.

The subgroup lattice has 311 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#485C211yesmaximal, characteristic
2C2×#432C211yesmaximal, characteristic
3#2855C211yesmaximal, characteristic
4#2296C211yesmaximal, characteristic
5#2088C211yesmaximal, characteristic, complemented
6#2087C211yesmaximal, characteristic, complemented
7#2068C211yesmaximal, characteristic, complemented
8#2066C211yesmaximal, characteristic, complemented
9#1944C211yesmaximal, characteristic, complemented
10#1943C211yesmaximal, characteristic, complemented
11#1935C211yesmaximal, characteristic, complemented
12#1935C211yesmaximal, characteristic, complemented
13#1873C211yesmaximal, characteristic
14#1854C211yesmaximal, characteristic
15#1825C211yesmaximal, characteristic
16C2×C22⋊Q8C2211yescharacteristic
17C2×#117C2211yescharacteristic
18C2×#117C2211yescharacteristic
19C22×C42C2211yescharacteristic
20#485—24no—
21#432—24no—
22C2×#107C2211yescharacteristic
23C2×#107C2211yescharacteristic
24C2×#106C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×#97C2211yescharacteristic
35C2×#97C2211yescharacteristic
36#391C2211yescharacteristic
37#391C2211yescharacteristic
38#389C2211yescharacteristic
39#389C2211yescharacteristic
40#389C2211yescharacteristic
41#389C2211yescharacteristic
42#389C2211yescharacteristic
43#389C2211yescharacteristic
44#387—21no—
45#387—21no—
46#385—21no—
47#385—21no—
48#385—21no—
49#381—21no—
50#380C2211yescharacteristic
51C4×C4⋊C4—21no—
52C4×C4⋊C4—21no—
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C23×C4D411yescharacteristic
63C23×C4D411yescharacteristic
64C23×C4C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22⋊Q8—22no—
70C22⋊Q8—22no—
71#117—22no—
72#117—22no—
73#117—22no—
74#117—22no—
75C2×C4⋊C4—21no—
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⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C42—21no—
94C2×C42—22no—
95C2×C42—22no—
96C2×C42—21no—
97C2×C42D411yescharacteristic
98C2×C42D411yescharacteristic
99C2×C42C2311yescharacteristic
100C2×C42C2311yescharacteristic
101#107—22no—
102#107—22no—
103#106—22no—
104#106—22no—
105C4⋊C8—22no—
106C4⋊C8—22no—
107#103—22no—
108#103—22no—
109#103—22no—
110#103—22no—
111C22⋊C8—22no—
112C22⋊C8D414yes—
113#97—22no—
114#97—22no—
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#95—21no—
125#95—21no—
126C24C2×D411yescharacteristic
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D412yes—
156C22×C4C2×D411yescharacteristic
157C22×C4#4112yes—
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×C8C2×D412yes—
170C2×C8C2×D412yes—
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—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4#4112yes—
191C4⋊C4#4112yes—
192C22⋊C4—24no—
193C22⋊C4—42no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C42—24no—
197C42—22no—
198C42—22no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—21no—
203C42—42no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23#12112yes—
211C23C22×D411yescharacteristic
212Q8—41no—
213Q8—41no—
214Q8—41no—
215Q8—41no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—44no—
236C2×C4—44no—
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—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#12112yes—
260C2×C4#13712yes—
261C2×C4C22×D411yescharacteristic
262C2×C4C2×#4111yescharacteristic
263C2×C4C2×#4111yescharacteristic
264C2×C4C2×#4111yescharacteristic
265C2×C42−1+411yescharacteristic
266C2×C42−1+411yescharacteristic
267C8—22no—
268C8—22no—
269C8—41no—
270C8—41no—
271C22—22no—
272C22—22no—
273C22—22no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22C2×#12111yescharacteristic
279C22#52511yescharacteristic
280C22#53611yescharacteristic
281C22C2×#13711yescharacteristic
282C22#57511yescharacteristic
283C22#57511yescharacteristic
284C22#57511yescharacteristic
285C4—22no—
286C4—22no—
287C4—42no—
288C4—42no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—81no—
293C4—81no—
294C4—81no—
295C4—81no—
296C4—81no—
297C4—81no—
298C4#54711yescharacteristic
299C4#54711yescharacteristic
300C4#55211yescharacteristic
301C4#55211yescharacteristic
302C2—42no—
303C2#241311yescharacteristic
304C2#305411yescharacteristic
305C2#305611yescharacteristic
306C2#311111yescharacteristic
307C2#311611yescharacteristic
308C2#323711yescharacteristic
309C2#324011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #2088C2semidirect
6. #2087C2semidirect
7. #2068C2semidirect
8. #2066C2semidirect
9. #1944C2semidirect
10. #1943C2semidirect
11. #1935C2semidirect
12. #1935C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4852
C2×#4322
#28552
#22962
#20882
#20872
#20682
#20662
#19442
#19432
#19352
#18732
#18542
#18252