Finite Group Databasebeta

← #25206

Subgroups — #25206

Order 256.

429 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#484C211yesmaximal, characteristic
2C2×#467C211yesmaximal, characteristic
3#2502C211yesmaximal, characteristic
4#2360C211yesmaximal, characteristic
5#2083C211yesmaximal, characteristic, complemented
6#2072C211yesmaximal, characteristic, complemented
7#2068C211yesmaximal, characteristic, complemented
8#2056C211yesmaximal, characteristic, complemented
9#1947C211yesmaximal, characteristic, complemented
10#1928C211yesmaximal, characteristic, complemented
11#1878C211yesmaximal, characteristic, complemented
12#1872C211yesmaximal, characteristic
13#1850C211yesmaximal, characteristic
14#1803C211yesmaximal, characteristic, complemented
15C2×#340C211yesmaximal, characteristic
16C22×Q16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23#484—24no—
24#467—24no—
25C2×#107C2211yescharacteristic
26C2×#107C2211yescharacteristic
27C2×#107C2211yescharacteristic
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#389—21no—
43#389C2211yescharacteristic
44#387—21no—
45#385—21no—
46#385—21no—
47#385—21no—
48#384C2211yescharacteristic
49#384C2211yescharacteristic
50C4×C4⋊C4C2211yescharacteristic
51C4×C22⋊C4—21no—
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×C4—21no—
65C23×C4C2311yescharacteristic
66C2×Q16—22no—
67C2×Q16—22no—
68C2×Q16—41no—
69C22×C8—22no—
70C22×C8—22no—
71C22×C8—41no—
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22⋊Q8—22no—
77C22⋊Q8—22no—
78C22⋊Q8—22no—
79C22⋊Q8D414yes—
80C4×Q8—22no—
81C4×Q8—41no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—22no—
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⋊C4—21no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4D411yescharacteristic
105C2×C22⋊C4D411yescharacteristic
106C2×C42—21no—
107C2×C42—21no—
108C2×C42C2311yescharacteristic
109C2×C42C2311yescharacteristic
110#107—22no—
111#107—22no—
112#107—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—42no—
159C22×C4—42no—
160C22×C4—81no—
161C22×C4—81no—
162C22×C4—81no—
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D412yes—
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4#4112yes—
170C22×C4C2411yescharacteristic
171Q16—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—21no—
182C2×C8—21no—
183C2×C8—44no—
184C2×C8#4112yes—
185C4⋊C4—24no—
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—42no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—81no—
201C4⋊C4—81no—
202C4⋊C4—81no—
203C4⋊C4—81no—
204C4⋊C4—81no—
205C4⋊C4C2×D412yes—
206C4⋊C4#4112yes—
207C22⋊C4—24no—
208C22⋊C4—22no—
209C22⋊C4—42no—
210C22⋊C4Q1614yescomplemented
211C42—21no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—42no—
216C42—81no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23#12112yes—
222C23C2×Q1612yes—
223C23#13712yes—
224C23C22×D411yescharacteristic
225Q8—22no—
226Q8—22no—
227Q8—41no—
228Q8—41no—
229Q8—41no—
230Q8—41no—
231Q8—41no—
232Q8—41no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
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—84no—
274C2×C4—84no—
275C2×C4#12112yes—
276C2×C4C2×Q1612yes—
277C2×C4C2×Q1612yes—
278C2×C4C22×D411yescharacteristic
279C2×C4C2×#4111yescharacteristic
280C2×C4C2×#4111yescharacteristic
281C2×C4C2×#4111yescharacteristic
282C2×C4H(5,𝔽2)11yescharacteristic
283C2×C42−1+411yescharacteristic
284C8—22no—
285C8—41no—
286C8—41no—
287C8—41no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22#46214yes—
295C22C2×#12111yescharacteristic
296C22#54611yescharacteristic
297C22#54711yescharacteristic
298C22C22×Q1611yescharacteristic
299C22C2×#13711yescharacteristic
300C22#57611yescharacteristic
301C22#57711yescharacteristic
302C4—22no—
303C4—22no—
304C4—22no—
305C4—41no—
306C4—42no—
307C4—42no—
308C4—41no—
309C4—42no—
310C4—81no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—81no—
315C4#52911yescharacteristic
316C4#53911yescharacteristic
317C4#54611yescharacteristic
318C4#54711yescharacteristic
319C2—24no—
320C2#242511yescharacteristic
321C2C2×#46211yescharacteristic
322C2#304511yescharacteristic
323C2#329911yescharacteristic
324C2#331411yescharacteristic
325C2#333811yescharacteristic
326C2#334411yescharacteristic
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. #2083C2semidirect
6. #2072C2semidirect
7. #2068C2semidirect
8. #2056C2semidirect
9. #1947C2semidirect
10. #1928C2semidirect
11. #1878C2semidirect
14. #1803C2semidirect
210. C22⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#4842
C2×#4672
#25022
#23602
#20832
#20722
#20682
#20562
#19472
#19282
#18782
#18722
#18502
#18032
C2×#3402