Finite Group Databasebeta

← #24690

Subgroups — #24690

Order 256.

404 conjugacy classes of subgroups.

The subgroup lattice has 311 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#478C211yesmaximal, characteristic
2C2×#440C211yesmaximal, characteristic
3#2565C211yesmaximal, characteristic, complemented
4#2354C211yesmaximal, characteristic
5#2073C211yesmaximal, characteristic
6#2071C211yesmaximal, characteristic
7#2055C211yesmaximal, characteristic
8#2055C211yesmaximal, characteristic
9#1992C211yesmaximal, characteristic, complemented
10#1949C211yesmaximal, characteristic, complemented
11#1947C211yesmaximal, characteristic, complemented
12#1940C211yesmaximal, characteristic, complemented
13#1928C211yesmaximal, characteristic, complemented
14#1927C211yesmaximal, characteristic, complemented
15#1803C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C2×#125C2211yescharacteristic
18C2×C4×Q8C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×#117—21no—
21#478—24no—
22#440—24no—
23C2×#107C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#97C2211yescharacteristic
36#398—21no—
37#398—21no—
38#391C2211yescharacteristic
39#391C2211yescharacteristic
40#390C2211yescharacteristic
41#390C2211yescharacteristic
42#388—21no—
43#388—21no—
44#387—21no—
45#387—21no—
46#384C2211yescharacteristic
47#384C2211yescharacteristic
48#382C2211yescharacteristic
49#380—21no—
50C4×C4⋊C4C2211yescharacteristic
51C4×C4⋊C4C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4C2311yescharacteristic
63C23×C4—21no—
64C2×SD16—22no—
65C2×SD16—22no—
66C2×SD16—22no—
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71#125—22no—
72#125—22no—
73C4×Q8—22no—
74C4×Q8—22no—
75C4×D4—22no—
76C4×D4D414yes—
77#117—44no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4D411yescharacteristic
86C2×C4⋊C4D411yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42C2311yescharacteristic
102C2×C42C2311yescharacteristic
103C2×C42C2311yescharacteristic
104#107—22no—
105#107—22no—
106#106—22no—
107C4⋊C8—22no—
108C4⋊C8—22no—
109C4⋊C8—22no—
110#103—22no—
111#103—22no—
112#103—22no—
113#102—22no—
114#102—22no—
115#102—22no—
116#97—22no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95D411yescharacteristic
126#95D411yescharacteristic
127C24—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8#4112yes—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4C2×D412yes—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—24no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2411yescharacteristic
157SD16—28no—
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—21no—
168C2×C8—21no—
169C2×C8#4112yes—
170C4⋊C4—22no—
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—21no—
182C4⋊C4—41no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—81no—
190C4⋊C4C2×D412yes—
191C22⋊C4—24no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—81no—
195C22⋊C4—81no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—42no—
201C42—42no—
202C42—42no—
203C42C2×D412yes—
204C42#4112yes—
205C23—22no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23C22×D411yescharacteristic
213Q8—22no—
214Q8—22no—
215Q8—41no—
216Q8—41no—
217D4—22no—
218D4—22no—
219D4—41no—
220D4—41no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—48no—
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—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4#12112yes—
259C2×C4#12112yes—
260C2×C4AGL(1,ℤ/8)12yes—
261C2×C4C22×D411yescharacteristic
262C2×C4C2×#4111yescharacteristic
263C2×C4C2×#4111yescharacteristic
264C2×C4C2×#4111yescharacteristic
265C2×C42−1+411yescharacteristic
266C2×C42−1+411yescharacteristic
267C8—22no—
268C8—41no—
269C8—41no—
270C8—41no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—81no—
275C22—81no—
276C22—81no—
277C22—81no—
278C22C2×#12111yescharacteristic
279C22#54711yescharacteristic
280C22#54811yescharacteristic
281C22C2×AGL(1,ℤ/8)11yescharacteristic
282C22#57511yescharacteristic
283C22#57711yescharacteristic
284C22#57811yescharacteristic
285C4—22no—
286C4—22no—
287C4—22no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—42no—
292C4—42no—
293C4—81no—
294C4—81no—
295C4—81no—
296C4—81no—
297C4—81no—
298C4#52511yescharacteristic
299C4#53611yescharacteristic
300C4#55211yescharacteristic
301C4#55611yescharacteristic
302C2—82no—
303C2#241511yescharacteristic
304C2#303611yescharacteristic
305C2#304611yescharacteristic
306C2#334311yescharacteristic
307C2#334811yescharacteristic
308C2#336811yescharacteristic
309C2#337711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2565C2semidirect
9. #1992C2semidirect
10. #1949C2semidirect
11. #1947C2semidirect
12. #1940C2semidirect
13. #1928C2semidirect
14. #1927C2semidirect
15. #1803C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4782
C2×#4402
#25652
#23542
#20732
#20712
#20552
#19922
#19492
#19472
#19402
#19282
#19272
#18032