Finite Group Databasebeta

← #25356

Subgroups — #25356

Order 256.

420 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#475C211yesmaximal, characteristic, complemented
2C2×#435C211yesmaximal, characteristic
3#2563C211yesmaximal, characteristic
4#2536C211yesmaximal, characteristic
5#2077C211yesmaximal, characteristic, complemented
6#2077C211yesmaximal, characteristic, complemented
7#2036C211yesmaximal, characteristic, complemented
8#2035C211yesmaximal, characteristic, complemented
9#1993C211yesmaximal, characteristic, complemented
10#1949C211yesmaximal, characteristic, complemented
11#1894C211yesmaximal, characteristic, complemented
12#1892C211yesmaximal, characteristic, complemented
13#1809C211yesmaximal, characteristic, complemented
14#1807C211yesmaximal, characteristic, complemented
15#1763C211yesmaximal, characteristic, complemented
16C22×M16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×C4×Q8C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23#475—24no—
24#435—24no—
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×#97C2211yescharacteristic
37#398—21no—
38#394—21no—
39#393—21no—
40#393—21no—
41#390—21no—
42#388—21no—
43#388—21no—
44#382—21no—
45#382C2211yescharacteristic
46#380C2211yescharacteristic
47#380C2211yescharacteristic
48#380C2211yescharacteristic
49#380C2211yescharacteristic
50#380C2211yescharacteristic
51#380C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×Q8—21no—
62C22×Q8C2311yescharacteristic
63C22×D4D411yescharacteristic
64C23×C4C2311yescharacteristic
65C23×C4C2311yescharacteristic
66C2×M16—22no—
67C2×M16—22no—
68C2×M16—41no—
69C22×C8—22no—
70C22×C8—22no—
71C22×C8—41no—
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C4⋊Q8—22no—
77C4⋊Q8—22no—
78C22⋊Q8—22no—
79C22⋊Q8—22no—
80C22⋊Q8—41no—
81C22⋊Q8—41no—
82C4×Q8—22no—
83C4×Q8—22no—
84C4×D4—22no—
85C4×D4—22no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4D411yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4D411yescharacteristic
107C2×C22⋊C4D411yescharacteristic
108C2×C42—21no—
109C2×C42—21no—
110C2×C42C2311yescharacteristic
111#107—41no—
112#106—22no—
113#106—22no—
114#106—41no—
115C4⋊C8D414yes—
116#103—22no—
117#103—22no—
118#103—22no—
119#103—22no—
120C22⋊C8—22no—
121C22⋊C8—22no—
122#97—22no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132C24C2×D411yescharacteristic
133C24C2×D411yescharacteristic
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—42no—
140C2×Q8—81no—
141C2×Q8—81no—
142C2×Q8#4112yes—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—41no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2411yescharacteristic
171M16—44no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—44no—
182C2×C8C2×D412yes—
183C2×C8C2×D412yes—
184C2×C8#4112yes—
185C4⋊C4—22no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C22⋊C4—22no—
205C22⋊C4—22no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—81no—
211C22⋊C4—81no—
212C22⋊C4—81no—
213C22⋊C4—81no—
214C42—42no—
215C42—42no—
216C42C2×D412yes—
217C42#4112yes—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23#13712yes—
232C23C22×D411yescharacteristic
233Q8—22no—
234Q8—22no—
235Q8—41no—
236Q8—41no—
237Q8—84no—
238D4—44no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
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—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4#12112yes—
282C2×C4#12112yes—
283C2×C4C22×D411yescharacteristic
284C2×C4C2×#4111yescharacteristic
285C2×C4C2×#4111yescharacteristic
286C2×C4C2×#4111yescharacteristic
287C2×C4H(5,𝔽2)11yescharacteristic
288C2×C42−1+411yescharacteristic
289C8—22no—
290C8—22no—
291C8—22no—
292C8—41no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22C2×#12111yescharacteristic
308C22#54611yescharacteristic
309C22#54711yescharacteristic
310C22C2×#13711yescharacteristic
311C22#57511yescharacteristic
312C22#57711yescharacteristic
313C22#57711yescharacteristic
314C4—22no—
315C4—22no—
316C4—41no—
317C4—41no—
318C4—42no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4—81no—
323C4—81no—
324C4—81no—
325C4—81no—
326C4#52911yescharacteristic
327C4#54211yescharacteristic
328C4#54611yescharacteristic
329C4#54811yescharacteristic
330C2—42no—
331C2—81no—
332C2#242711yescharacteristic
333C2#305611yescharacteristic
334C2#305711yescharacteristic
335C2#331611yescharacteristic
336C2#332111yescharacteristic
337C2#334411yescharacteristic
338C2#334811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#475C2semidirect
5. #2077C2semidirect
6. #2077C2semidirect
7. #2036C2semidirect
8. #2035C2semidirect
9. #1993C2semidirect
10. #1949C2semidirect
11. #1894C2semidirect
12. #1892C2semidirect
13. #1809C2semidirect
14. #1807C2semidirect
15. #1763C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4752
C2×#4352
#25632
#25362
#20772
#20362
#20352
#19932
#19492
#18942
#18922
#18092
#18072
#17632