Finite Group Databasebeta

← #26355

Subgroups — #26355

Order 256.

424 conjugacy classes of subgroups.

The subgroup lattice has 319 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#477C211yesmaximal, characteristic, complemented
2C2×#428C211yesmaximal, characteristic
3#2440C211yesmaximal, characteristic
4#2362C211yesmaximal, characteristic
5#2096C211yesmaximal, characteristic, complemented
6#2088C211yesmaximal, characteristic, complemented
7#2066C211yesmaximal, characteristic, complemented
8#2058C211yesmaximal, characteristic, complemented
9#1947C211yesmaximal, characteristic, complemented
10#1918C211yesmaximal, characteristic, complemented
11#1894C211yesmaximal, characteristic
12#1872C211yesmaximal, characteristic
13#1863C211yesmaximal, characteristic, complemented
14#1854C211yesmaximal, characteristic
15#1851C211yesmaximal, characteristic
16C22×M16C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×C22⋊Q8—21no—
19C2×C22⋊Q8C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C2×#117C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23#477—24no—
24#428—24no—
25C2×#107C2211yescharacteristic
26C2×#107C2211yescharacteristic
27C2×#107C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×#106C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×C4⋊C8C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38#398—21no—
39#395—21no—
40#391C2211yescharacteristic
41#389C2211yescharacteristic
42#389C2211yescharacteristic
43#389C2211yescharacteristic
44#388—21no—
45#388—21no—
46#387—21no—
47#385—21no—
48#384C2211yescharacteristic
49#380C2211yescharacteristic
50C4×C4⋊C4—21no—
51C4×C4⋊C4C2211yescharacteristic
52C4×C22⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×Q8C2311yescharacteristic
63C23×C4—21no—
64C23×C4C2311yescharacteristic
65C2×M16—22no—
66C2×M16—22no—
67C2×M16—22no—
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C4⋊Q8—22no—
73C4⋊Q8—41no—
74C22⋊Q8—22no—
75C22⋊Q8—44no—
76C22⋊Q8Q814yescomplemented
77C4×Q8—22no—
78C4×Q8—41no—
79#117—22no—
80#117—41no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—22no—
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×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4D411yescharacteristic
103C2×C22⋊C4D411yescharacteristic
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42C2311yescharacteristic
108C2×C42C2311yescharacteristic
109#107—22no—
110#107—22no—
111#107D414yes—
112#106—22no—
113#106—22no—
114#106—41no—
115C4⋊C8—22no—
116C4⋊C8—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#95D411yescharacteristic
130#95D411yescharacteristic
131C24C2×D411yescharacteristic
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—81no—
140C2×Q8C2×Q812yes—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
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—42no—
154C22×C4—42no—
155C22×C4—81no—
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2×Q812yes—
162C22×C4#4112yes—
163C22×C4C2411yescharacteristic
164M16—28no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8C2×D412yes—
176C2×C8#4112yes—
177C4⋊C4—24no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—81no—
197C4⋊C4C2×D412yes—
198C4⋊C4C2×D412yes—
199C4⋊C4C2×Q812yes—
200C22⋊C4—24no—
201C22⋊C4—22no—
202C22⋊C4—22no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—42no—
210C42—42no—
211C42—42no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23C22⋊Q812yes—
219C23C22×D411yescharacteristic
220Q8—22no—
221Q8—22no—
222Q8—41no—
223Q8—41no—
224Q8—84no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
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—41no—
264C2×C4—41no—
265C2×C4—84no—
266C2×C4C22⋊Q812yes—
267C2×C4AGL(1,ℤ/8)12yes—
268C2×C4#13712yes—
269C2×C4C22×D411yescharacteristic
270C2×C4C22×Q811yescharacteristic
271C2×C4C2×#4111yescharacteristic
272C2×C4C2×#4111yescharacteristic
273C2×C4H(5,𝔽2)11yescharacteristic
274C2×C42−1+411yescharacteristic
275C8—22no—
276C8—22no—
277C8—41no—
278C8—41no—
279C22—22no—
280C22—22no—
281C22—22no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22C2×C22⋊Q811yescharacteristic
287C22D4211yescharacteristic
288C22#54711yescharacteristic
289C22C2×AGL(1,ℤ/8)11yescharacteristic
290C22C2×#13711yescharacteristic
291C22#57711yescharacteristic
292C22#57811yescharacteristic
293C4—22no—
294C4—22no—
295C4—22no—
296C4—42no—
297C4—41no—
298C4—42no—
299C4—41no—
300C4—42no—
301C4—42no—
302C4—81no—
303C4—81no—
304C4—81no—
305C4—81no—
306C4#52511yescharacteristic
307C4#53911yescharacteristic
308C4D4×Q811yescharacteristic
309C4#55411yescharacteristic
310C2—42no—
311C2#244911yescharacteristic
312C2#307911yescharacteristic
313C2#308211yescharacteristic
314C2#327011yescharacteristic
315C2#327211yescharacteristic
316C2#334311yescharacteristic
317C2#335611yescharacteristic
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×#477C2semidirect
5. #2096C2semidirect
6. #2088C2semidirect
7. #2066C2semidirect
8. #2058C2semidirect
9. #1947C2semidirect
10. #1918C2semidirect
13. #1863C2semidirect
76. C22⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#4772
C2×#4282
#24402
#23622
#20962
#20882
#20662
#20582
#19472
#19182
#18942
#18722
#18632
#18542
#18512