Finite Group Databasebeta

← #26078

Subgroups — #26078

Order 256.

409 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#501C211yesmaximal, characteristic, complemented
2C2×#428C211yesmaximal, characteristic
3#2441C211yesmaximal, characteristic
4#2440C211yesmaximal, characteristic
5#2084C211yesmaximal, characteristic, complemented
6#2083C211yesmaximal, characteristic, complemented
7#2059C211yesmaximal, characteristic, complemented
8#2058C211yesmaximal, characteristic, complemented
9#1943C211yesmaximal, characteristic, complemented
10#1942C211yesmaximal, characteristic, complemented
11#1854C211yesmaximal, characteristic
12#1853C211yesmaximal, characteristic
13#1844C211yesmaximal, characteristic, complemented
14#1809C211yesmaximal, characteristic
15#1753C211yesmaximal, characteristic
16C22×M16C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×#125C2211yescharacteristic
19C2×C22⋊Q8—21no—
20C2×#117C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22#501—24no—
23#428—24no—
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#107C2211yescharacteristic
27C2×#107C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×#106C2211yescharacteristic
31C2×#106C2211yescharacteristic
32C2×#106C2211yescharacteristic
33C2×#106C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35C2×#97C2211yescharacteristic
36C2×C4×C8C2211yescharacteristic
37#399—21no—
38#398—21no—
39#396—21no—
40#395—21no—
41#389—21no—
42#389C2211yescharacteristic
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#389C2211yescharacteristic
46#387—21no—
47#385—21no—
48#384C2211yescharacteristic
49#384C2211yescharacteristic
50#380C2211yescharacteristic
51#380C2211yescharacteristic
52C4×C22⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C23×C4—21no—
63C23×C4C2311yescharacteristic
64C2×M16—22no—
65C2×M16—22no—
66C2×M16Q814yescomplemented
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C4⋊Q8—22no—
72C4⋊Q8—41no—
73#125—22no—
74#125—41no—
75C22⋊Q8—44no—
76#117—22no—
77#117—41no—
78C2×C4⋊C4—22no—
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⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
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×C42C2311yescharacteristic
107C2×C42C2311yescharacteristic
108#107—22no—
109#107—22no—
110#107—22no—
111#107D414yes—
112#106—22no—
113#106—22no—
114#106—22no—
115#106—22no—
116#106—41no—
117#106—41no—
118C22⋊C8Q814yescomplemented
119#97—22no—
120C4×C8—22no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95D411yescharacteristic
128#95D411yescharacteristic
129C24C2×D411yescharacteristic
130C2×Q8—42no—
131C2×Q8—42no—
132C2×Q8—81no—
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×C4—42no—
146C22×C4—42no—
147C22×C4—81no—
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×Q812yes—
154C22×C4C2×Q812yes—
155C22×C4C2411yescharacteristic
156M16—28no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8C2×D412yes—
164C2×C8C2×Q812yes—
165C2×C8C2×Q812yes—
166C2×C8C2×Q812yes—
167C2×C8C2×Q812yes—
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—21no—
182C4⋊C4—21no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—81no—
193C4⋊C4—81no—
194C4⋊C4—81no—
195C4⋊C4C2×D412yes—
196C4⋊C4C2×D412yes—
197C22⋊C4—22no—
198C22⋊C4—22no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—81no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—42no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23C4⋊Q812yes—
214C23C22×D411yescharacteristic
215Q8—84no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
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—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—84no—
257C2×C4C4⋊Q812yes—
258C2×C4AGL(1,ℤ/8)12yes—
259C2×C4#13712yes—
260C2×C4C22×D411yescharacteristic
261C2×C4C22×Q811yescharacteristic
262C2×C4C22×Q811yescharacteristic
263C2×C4C2×#4111yescharacteristic
264C2×C4H(5,𝔽2)11yescharacteristic
265C2×C42−1+411yescharacteristic
266C8—22no—
267C8—22no—
268C8—22no—
269C8C22⋊Q814yescomplemented
270C22—22no—
271C22—22no—
272C22—22no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22C2×C4⋊Q811yescharacteristic
278C22D4211yescharacteristic
279C22#54711yescharacteristic
280C22C2×AGL(1,ℤ/8)11yescharacteristic
281C22C2×#13711yescharacteristic
282C22#57711yescharacteristic
283C22#57711yescharacteristic
284C4—22no—
285C4—22no—
286C4—22no—
287C4—41no—
288C4—42no—
289C4—42no—
290C4—41no—
291C4—81no—
292C4—81no—
293C4—81no—
294C4—81no—
295C4—81no—
296C4—81no—
297C4C2×C22⋊Q811yescharacteristic
298C4#54411yescharacteristic
299C4D4×Q811yescharacteristic
300C4#55411yescharacteristic
301C2—42no—
302C2#244011yescharacteristic
303C2#316311yescharacteristic
304C2#316511yescharacteristic
305C2#327211yescharacteristic
306C2#327811yescharacteristic
307C2#334411yescharacteristic
308C2#334511yescharacteristic
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
1. C2×#501C2semidirect
5. #2084C2semidirect
6. #2083C2semidirect
7. #2059C2semidirect
8. #2058C2semidirect
9. #1943C2semidirect
10. #1942C2semidirect
13. #1844C2semidirect
66. C2×M16Q8semidirect
118. C22⋊C8Q8semidirect
269. C8C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5012
C2×#4282
#24412
#24402
#20842
#20832
#20592
#20582
#19432
#19422
#18542
#18532
#18442
#18092
#17532