Finite Group Databasebeta

← #24586

Subgroups — #24586

Order 256.

381 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#485C211yesmaximal, characteristic
2C2×#427C211yesmaximal, characteristic
3#2499C211yesmaximal, characteristic
4#2346C211yesmaximal, characteristic
5#2085C211yesmaximal, characteristic, complemented
6#2084C211yesmaximal, characteristic, complemented
7#2068C211yesmaximal, characteristic, complemented
8#2066C211yesmaximal, characteristic, complemented
9#1949C211yesmaximal, characteristic, complemented
10#1947C211yesmaximal, characteristic, complemented
11#1928C211yesmaximal, characteristic, complemented
12#1927C211yesmaximal, characteristic, complemented
13#1892C211yesmaximal, characteristic
14#1850C211yesmaximal, characteristic
15#1825C211yesmaximal, characteristic
16C23×C8C2211yescharacteristic
17C2×C22⋊Q8C2211yescharacteristic
18C2×#117C2211yescharacteristic
19C2×#117C2211yescharacteristic
20#485—24no—
21#427—24no—
22C2×#107C2211yescharacteristic
23C2×#107C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35#397—21no—
36#397—21no—
37#391C2211yescharacteristic
38#391C2211yescharacteristic
39#389C2211yescharacteristic
40#389C2211yescharacteristic
41#388—21no—
42#388—21no—
43#387—21no—
44#387—21no—
45#384C2211yescharacteristic
46#382—21no—
47#382C2211yescharacteristic
48#380C2211yescharacteristic
49C4×C4⋊C4C2211yescharacteristic
50C4×C4⋊C4C2211yescharacteristic
51C4×C22⋊C4C2211yescharacteristic
52#376—21no—
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C23×C4C2311yescharacteristic
63C22×C8—22no—
64C22×C8—22no—
65C22×C8—22no—
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22⋊Q8—22no—
71C22⋊Q8—22no—
72#117—22no—
73#117—22no—
74#117—22no—
75#117D414yes—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4D411yescharacteristic
93C2×C22⋊C4D411yescharacteristic
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42C2311yescharacteristic
99C2×C42C2311yescharacteristic
100#107—22no—
101#107—22no—
102#107—22no—
103#106—22no—
104#106—22no—
105#106—22no—
106C4⋊C8—22no—
107C4⋊C8—22no—
108#103—22no—
109#103—22no—
110#103—22no—
111#103—22no—
112C22⋊C8—22no—
113#95—21no—
114#95—21no—
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#95D411yescharacteristic
125#95D411yescharacteristic
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—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D412yes—
150C22×C4C2×D411yescharacteristic
151C22×C4#4112yes—
152C22×C4C2411yescharacteristic
153C2×C8—28no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8#4112yes—
165C2×C8#4112yes—
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—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—81no—
182C4⋊C4—81no—
183C4⋊C4C2×D412yes—
184C4⋊C4C2×D412yes—
185C22⋊C4—24no—
186C22⋊C4—22no—
187C22⋊C4—41no—
188C22⋊C4—42no—
189C22⋊C4—42no—
190C22⋊C4—81no—
191C42—24no—
192C42—21no—
193C42—21no—
194C42—21no—
195C42—21no—
196C42—42no—
197C42—42no—
198C42—42no—
199C23—21no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23#12112yes—
206C23C22×D411yescharacteristic
207Q8—41no—
208Q8—41no—
209Q8—41no—
210Q8—41no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
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—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4#12112yes—
253C2×C4#13512yes—
254C2×C4C22×D411yescharacteristic
255C2×C4C2×#4111yescharacteristic
256C2×C4C2×#4111yescharacteristic
257C2×C4C2×#4111yescharacteristic
258C2×C42−1+411yescharacteristic
259C2×C42−1+411yescharacteristic
260C8—22no—
261C8—22no—
262C8—41no—
263C8—41no—
264C22—22no—
265C22—22no—
266C22—22no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22C2×#12111yescharacteristic
272C22#54711yescharacteristic
273C22#54711yescharacteristic
274C22C2×#13511yescharacteristic
275C22#57511yescharacteristic
276C22#57711yescharacteristic
277C22#57811yescharacteristic
278C4—22no—
279C4—22no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—42no—
284C4—42no—
285C4—81no—
286C4—81no—
287C4—81no—
288C4—81no—
289C4—81no—
290C4—81no—
291C4#52511yescharacteristic
292C4#53611yescharacteristic
293C4#55211yescharacteristic
294C4#55211yescharacteristic
295C2—42no—
296C2#241311yescharacteristic
297C2#303511yescharacteristic
298C2#304611yescharacteristic
299C2#333611yescharacteristic
300C2#334011yescharacteristic
301C2#334811yescharacteristic
302C2#334911yescharacteristic
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. #2085C2semidirect
6. #2084C2semidirect
7. #2068C2semidirect
8. #2066C2semidirect
9. #1949C2semidirect
10. #1947C2semidirect
11. #1928C2semidirect
12. #1927C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4852
C2×#4272
#24992
#23462
#20852
#20842
#20682
#20662
#19492
#19472
#19282
#19272
#18922
#18502
#18252