Finite Group Databasebeta

← #24585

Subgroups — #24585

Order 256.

402 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#482C211yesmaximal, characteristic
2C2×#427C211yesmaximal, characteristic, complemented
3#2500C211yesmaximal, characteristic
4#2347C211yesmaximal, characteristic, complemented
5#2085C211yesmaximal, characteristic, complemented
6#2084C211yesmaximal, characteristic, complemented
7#2067C211yesmaximal, characteristic, complemented
8#2065C211yesmaximal, characteristic, complemented
9#1949C211yesmaximal, characteristic, complemented
10#1947C211yesmaximal, characteristic, complemented
11#1926C211yesmaximal, characteristic, complemented
12#1925C211yesmaximal, characteristic, complemented
13#1893C211yesmaximal, characteristic
14#1850C211yesmaximal, characteristic, complemented
15#1825C211yesmaximal, characteristic, complemented
16C23×C8C2211yescharacteristic
17C2×#121C2211yescharacteristic
18C2×#117C2211yescharacteristic
19C2×#117C2211yescharacteristic
20#482—24no—
21#427—24no—
22C2×#107C2211yescharacteristic
23C2×#107C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35#394—21no—
36#394—21no—
37#390C2211yescharacteristic
38#390C2211yescharacteristic
39#389C2211yescharacteristic
40#389C2211yescharacteristic
41#388—21no—
42#388—21no—
43#388—21no—
44#388—21no—
45#384C2211yescharacteristic
46#383—21no—
47#382C2211yescharacteristic
48#380C2211yescharacteristic
49C4×C4⋊C4—21no—
50C4×C4⋊C4C2211yescharacteristic
51C4×C4⋊C4C2211yescharacteristic
52C4×C22⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×D4—21no—
62C22×D4C2311yescharacteristic
63C23×C4C2311yescharacteristic
64C22×C8—22no—
65C22×C8—22no—
66C22×C8—22no—
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71#121—22no—
72#121—22no—
73#117—22no—
74#117—22no—
75#117—22no—
76#117D414yes—
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⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4D411yescharacteristic
97C2×C22⋊C4D411yescharacteristic
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42C2311yescharacteristic
103C2×C42C2311yescharacteristic
104#107—22no—
105#107—22no—
106#107—22no—
107#106—22no—
108#106—22no—
109#106—22no—
110C4⋊C8—22no—
111C4⋊C8—22no—
112#102—22no—
113#102—22no—
114#102—22no—
115#102—22no—
116C22⋊C8—22no—
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
126C24—21no—
127C24C2×D411yescharacteristic
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—42no—
135C2×D4—81no—
136C2×D4—81no—
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×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D412yes—
153C22×C4C2×D411yescharacteristic
154C22×C4#4112yes—
155C22×C4C2411yescharacteristic
156C2×C8—28no—
157C2×C8—21no—
158C2×C8—21no—
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#4112yes—
168C2×C8#4112yes—
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—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—81no—
184C4⋊C4—81no—
185C4⋊C4C2×D412yes—
186C4⋊C4C2×D412yes—
187C22⋊C4—24no—
188C22⋊C4—22no—
189C22⋊C4—41no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—81no—
195C22⋊C4—81no—
196C22⋊C4—81no—
197C42—24no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—42no—
203C42—81no—
204C42—81no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23#12112yes—
219C23C22×D411yescharacteristic
220D4—41no—
221D4—41no—
222D4—41no—
223D4—41no—
224D4—84no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
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—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4#12112yes—
264C2×C4#13512yes—
265C2×C4C22×D411yescharacteristic
266C2×C4C2×#4111yescharacteristic
267C2×C4C2×#4111yescharacteristic
268C2×C4C2×#4111yescharacteristic
269C2×C42−1+411yescharacteristic
270C2×C42−1+411yescharacteristic
271C8—22no—
272C8—22no—
273C8—41no—
274C8—41no—
275C22—22no—
276C22—22no—
277C22—22no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22—81no—
289C22C2×#12111yescharacteristic
290C22#54711yescharacteristic
291C22#54711yescharacteristic
292C22C2×#13511yescharacteristic
293C22#57511yescharacteristic
294C22#57611yescharacteristic
295C22#57711yescharacteristic
296C4—22no—
297C4—22no—
298C4—41no—
299C4—41no—
300C4—41no—
301C4—42no—
302C4—42no—
303C4—81no—
304C4—81no—
305C4—81no—
306C4—81no—
307C4—81no—
308C4#52511yescharacteristic
309C4#53611yescharacteristic
310C4#55211yescharacteristic
311C4#55211yescharacteristic
312C2—42no—
313C2—161no—
314C2#241311yescharacteristic
315C2#303511yescharacteristic
316C2#304511yescharacteristic
317C2#333911yescharacteristic
318C2#334011yescharacteristic
319C2#334711yescharacteristic
320C2#335011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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
2. C2×#427C2semidirect
4. #2347C2semidirect
5. #2085C2semidirect
6. #2084C2semidirect
7. #2067C2semidirect
8. #2065C2semidirect
9. #1949C2semidirect
10. #1947C2semidirect
11. #1926C2semidirect
12. #1925C2semidirect
14. #1850C2semidirect
15. #1825C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4822
C2×#4272
#25002
#23472
#20852
#20842
#20672
#20652
#19492
#19472
#19262
#19252
#18932
#18502
#18252