Finite Group Databasebeta

← #26558

Subgroups — #26558

Order 256.

408 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×#483C211yesmaximal, characteristic
2C2×C4×Q16C211yesmaximal, characteristic, complemented
3#2421C211yesmaximal, characteristic
4#2360C211yesmaximal, characteristic
5#2097C211yesmaximal, characteristic, complemented
6#2056C211yesmaximal, characteristic, complemented
7#2035C211yesmaximal, characteristic
8#2001C211yesmaximal, characteristic
9#1949C211yesmaximal, characteristic, complemented
10#1941C211yesmaximal, characteristic, complemented
11#1927C211yesmaximal, characteristic, complemented
12#1923C211yesmaximal, characteristic
13#1865C211yesmaximal, characteristic, complemented
14#1863C211yesmaximal, characteristic, complemented
15#1852C211yesmaximal, characteristic
16C22×Q16C2211yescharacteristic
17C2×C22⋊Q8C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22#483—24no—
23C4×Q16—24no—
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×C4×C8C2211yescharacteristic
39#400—21no—
40#397—21no—
41#391C2211yescharacteristic
42#391C2211yescharacteristic
43#389—21no—
44#389C2211yescharacteristic
45#389C2211yescharacteristic
46#388—21no—
47#387—21no—
48#387—21no—
49#385—21no—
50#382C2211yescharacteristic
51#380C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53C4×C22⋊C4—21no—
54C4×C22⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×Q8C2311yescharacteristic
63C23×C4—21no—
64C23×C4C2311yescharacteristic
65C2×Q16—22no—
66C2×Q16—22no—
67C2×Q16—22no—
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22⋊Q8—22no—
73C22⋊Q8—22no—
74C22⋊Q8—22no—
75C22⋊Q8—41no—
76C4×Q8—22no—
77C4×Q8—22no—
78C4×Q8—41no—
79C4×Q8—41no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—22no—
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—41no—
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4D411yescharacteristic
99C2×C22⋊C4D411yescharacteristic
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42C2311yescharacteristic
104C2×C42C2311yescharacteristic
105#107—22no—
106#107—41no—
107#106—22no—
108#106—22no—
109#106—22no—
110C4⋊C8—22no—
111#103—22no—
112#103—22no—
113#103—22no—
114#103—22no—
115#103—41no—
116#103D414yes—
117C22⋊C8—22no—
118C22⋊C8—22no—
119C4×C8—22no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95D411yescharacteristic
131#95D411yescharacteristic
132C24C2×D411yescharacteristic
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8C2×D412yes—
144C2×Q8#4112yes—
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—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—81no—
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2411yescharacteristic
168Q16—28no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8C2×D412yes—
180C2×C8#4112yes—
181C4⋊C4—24no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—81no—
198C4⋊C4C2×D412yes—
199C4⋊C4#4112yes—
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—81no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23#13512yes—
219C23C22×D411yescharacteristic
220Q8—22no—
221Q8—22no—
222Q8—22no—
223Q8—22no—
224Q8—41no—
225Q8—41no—
226Q8—41no—
227Q8—41no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
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—21no—
245C2×C4—21no—
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—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—84no—
271C2×C4#13712yes—
272C2×C4C22×D411yescharacteristic
273C2×C4C2×#4111yescharacteristic
274C2×C4C2×#4111yescharacteristic
275C2×C4C2×#4111yescharacteristic
276C2×C4H(5,𝔽2)11yescharacteristic
277C2×C42−1+411yescharacteristic
278C8—22no—
279C8—22no—
280C8—41no—
281C8—41no—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22#52511yescharacteristic
290C22D4211yescharacteristic
291C22#54711yescharacteristic
292C22C2×#13511yescharacteristic
293C22C2×#13711yescharacteristic
294C22#57711yescharacteristic
295C22#57811yescharacteristic
296C4—22no—
297C4—41no—
298C4—41no—
299C4—42no—
300C4—41no—
301C4—42no—
302C4—42no—
303C4—41no—
304C4—42no—
305C4—81no—
306C4—81no—
307C4—81no—
308C4—81no—
309C4#52511yescharacteristic
310C4#53911yescharacteristic
311C4#54711yescharacteristic
312C4#55911yescharacteristic
313C2—42no—
314C2#245511yescharacteristic
315C2#310811yescharacteristic
316C2#312011yescharacteristic
317C2#327011yescharacteristic
318C2#328211yescharacteristic
319C2#333611yescharacteristic
320C2#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
2. C2×C4×Q16C2semidirect
5. #2097C2semidirect
6. #2056C2semidirect
9. #1949C2semidirect
10. #1941C2semidirect
11. #1927C2semidirect
13. #1865C2semidirect
14. #1863C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4832
C2×C4×Q162
#24212
#23602
#20972
#20562
#20352
#20012
#19492
#19412
#19272
#19232
#18652
#18632
#18522