Finite Group Databasebeta

← #25219

Subgroups — #25219

Order 256.

412 conjugacy classes of subgroups.

The subgroup lattice has 325 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#483C211yesmaximal, characteristic
2C2×#470C211yesmaximal, characteristic
3#2502C211yesmaximal, characteristic
4#2362C211yesmaximal, characteristic
5#2084C211yesmaximal, characteristic, complemented
6#2072C211yesmaximal, characteristic, complemented
7#2066C211yesmaximal, characteristic, complemented
8#2056C211yesmaximal, characteristic, complemented
9#1948C211yesmaximal, characteristic, complemented
10#1927C211yesmaximal, characteristic, complemented
11#1878C211yesmaximal, characteristic, complemented
12#1873C211yesmaximal, characteristic
13#1851C211yesmaximal, characteristic
14#1803C211yesmaximal, characteristic, complemented
15#1736C211yesmaximal, characteristic
16C22×Q16C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C2×#117C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23#483—24no—
24#470—24no—
25C2×#106C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37#398—21no—
38#398—21no—
39#391C2211yescharacteristic
40#391C2211yescharacteristic
41#391C2211yescharacteristic
42#389C2211yescharacteristic
43#388—21no—
44#388—21no—
45#387—21no—
46#385—21no—
47#385—21no—
48#384C2211yescharacteristic
49#384C2211yescharacteristic
50C4×C4⋊C4—21no—
51C4×C4⋊C4C2211yescharacteristic
52C4×C22⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×Q8C2311yescharacteristic
63C23×C4—21no—
64C23×C4C2311yescharacteristic
65C2×Q16—22no—
66C2×Q16—22no—
67C2×Q16—41no—
68C2×M16—22no—
69C2×M16—22no—
70C2×M16—41no—
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22⋊Q8—22no—
76C22⋊Q8—22no—
77C22⋊Q8—22no—
78C22⋊Q8D414yes—
79C4×Q8—22no—
80C4×Q8—41no—
81#117—22no—
82#117—41no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4D411yescharacteristic
104C2×C22⋊C4D411yescharacteristic
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42C2311yescharacteristic
109C2×C42C2311yescharacteristic
110#106—22no—
111#106—22no—
112#106—22no—
113C4⋊C8—22no—
114C4⋊C8—41no—
115#103—22no—
116#103—22no—
117#103—22no—
118#103—22no—
119#103—22no—
120#103—41no—
121C22⋊C8—22no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95D411yescharacteristic
131#95D411yescharacteristic
132C24C2×D411yescharacteristic
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8C2×D412yes—
145C22×C4—21no—
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—42no—
158C22×C4—81no—
159C22×C4—81no—
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D412yes—
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4#4112yes—
167C22×C4C2411yescharacteristic
168Q16—44no—
169M16—44no—
170C2×C8—21no—
171C2×C8—21no—
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#4112yes—
182C4⋊C4—24no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—81no—
198C4⋊C4—81no—
199C4⋊C4—81no—
200C4⋊C4—81no—
201C4⋊C4C2×D412yes—
202C4⋊C4#4112yes—
203C22⋊C4—24no—
204C22⋊C4—22no—
205C22⋊C4—22no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—42no—
213C42—42no—
214C42—42no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23#12112yes—
222C23C22×D411yescharacteristic
223Q8—22no—
224Q8—22no—
225Q8—41no—
226Q8—41no—
227Q8—41no—
228Q8—41no—
229Q8—41no—
230Q8—41no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—21no—
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—84no—
272C2×C4#12112yes—
273C2×C4#13712yes—
274C2×C4#13712yes—
275C2×C4C22×D411yescharacteristic
276C2×C4C2×#4111yescharacteristic
277C2×C4C2×#4111yescharacteristic
278C2×C4C2×#4111yescharacteristic
279C2×C4H(5,𝔽2)11yescharacteristic
280C2×C42−1+411yescharacteristic
281C8—22no—
282C8—41no—
283C8—41no—
284C8—41no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22C2×#12111yescharacteristic
293C22#54611yescharacteristic
294C22#54711yescharacteristic
295C22C2×#13711yescharacteristic
296C22C2×#13711yescharacteristic
297C22#57711yescharacteristic
298C22#57711yescharacteristic
299C4—22no—
300C4—22no—
301C4—22no—
302C4—41no—
303C4—42no—
304C4—41no—
305C4—42no—
306C4—42no—
307C4—81no—
308C4—81no—
309C4—81no—
310C4—81no—
311C4—81no—
312C4#52911yescharacteristic
313C4#53911yescharacteristic
314C4#54611yescharacteristic
315C4#54711yescharacteristic
316C2—42no—
317C2#242511yescharacteristic
318C2#303911yescharacteristic
319C2#304311yescharacteristic
320C2#331611yescharacteristic
321C2#333111yescharacteristic
322C2#334411yescharacteristic
323C2#335611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #2084C2semidirect
6. #2072C2semidirect
7. #2066C2semidirect
8. #2056C2semidirect
9. #1948C2semidirect
10. #1927C2semidirect
11. #1878C2semidirect
14. #1803C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4832
C2×#4702
#25022
#23622
#20842
#20722
#20662
#20562
#19482
#19272
#18782
#18732
#18512
#18032
#17362