Finite Group Databasebeta

← #24822

Subgroups — #24822

Order 256.

425 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#478C211yesmaximal, characteristic
2C2×C4×SD16C211yesmaximal, characteristic
3#2565C211yesmaximal, characteristic, complemented
4#2349C211yesmaximal, characteristic
5#2085C211yesmaximal, characteristic, complemented
6#2084C211yesmaximal, characteristic, complemented
7#2068C211yesmaximal, characteristic, complemented
8#2066C211yesmaximal, characteristic, complemented
9#1991C211yesmaximal, characteristic, complemented
10#1939C211yesmaximal, characteristic, complemented
11#1877C211yesmaximal, characteristic
12#1877C211yesmaximal, characteristic
13#1805C211yesmaximal, characteristic, complemented
14#1804C211yesmaximal, characteristic
15#1802C211yesmaximal, characteristic
16C22×SD16C2211yescharacteristic
17C2×#125C2211yescharacteristic
18C2×C4×Q8C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22#478—24no—
23C4×SD16—24no—
24C2×#107C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×C4×C8C2211yescharacteristic
37#398—21no—
38#398—21no—
39#391C2211yescharacteristic
40#391C2211yescharacteristic
41#389C2211yescharacteristic
42#389C2211yescharacteristic
43#387—21no—
44#387—21no—
45#385—21no—
46#385—21no—
47#384C2211yescharacteristic
48#382C2211yescharacteristic
49#382C2211yescharacteristic
50#380—21no—
51C4×C4⋊C4C2211yescharacteristic
52C2×#95—21no—
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4C2311yescharacteristic
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C2×SD16—22no—
67C2×SD16—22no—
68C2×SD16Q814yes—
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73#125—22no—
74#125—22no—
75C4×Q8—22no—
76C4×Q8—22no—
77C4×D4—22no—
78C4×D4—22no—
79C4×D4—22no—
80C4×D4—41no—
81C4×D4—41no—
82C4×D4D414yes—
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⋊C4D411yescharacteristic
91C2×C4⋊C4D411yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C42—21no—
102C2×C42C2311yescharacteristic
103C2×C42C2311yescharacteristic
104C2×C42C2311yescharacteristic
105#107—22no—
106#106—22no—
107#106—22no—
108C4⋊C8—22no—
109C4⋊C8—41no—
110C4⋊C8—41no—
111#103—22no—
112#103—41no—
113#103—41no—
114#102—22no—
115#102—22no—
116#102—22no—
117C4×C8—22no—
118#95—21no—
119#95—21no—
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#95D411yescharacteristic
129#95D411yescharacteristic
130C24—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8C2×Q812yes—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4C2×D412yes—
141C2×D4C2×Q812yes—
142C2×D4#4112yes—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—24no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—81no—
156C22×C4—81no—
157C22×C4—81no—
158C22×C4—81no—
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2411yescharacteristic
166SD16—28no—
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×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8C2×Q812yes—
179C4⋊C4—22no—
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—21no—
190C4⋊C4—21no—
191C4⋊C4—41no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—81no—
198C4⋊C4—81no—
199C4⋊C4C2×D412yes—
200C22⋊C4—24no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—42no—
208C42C2×D412yes—
209C42#4112yes—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23C22×D411yescharacteristic
218Q8—22no—
219Q8—22no—
220Q8—41no—
221Q8—41no—
222D4—22no—
223D4—22no—
224D4—22no—
225D4C22⋊Q814yes—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
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—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—84no—
263C2×C4—84no—
264C2×C4—84no—
265C2×C4#12112yes—
266C2×C4#12112yes—
267C2×C4#13512yes—
268C2×C4C22×D411yescharacteristic
269C2×C4C22×Q811yescharacteristic
270C2×C4C2×#4111yescharacteristic
271C2×C4C2×#4111yescharacteristic
272C2×C4H(5,𝔽2)11yescharacteristic
273C2×C42−1+411yescharacteristic
274C8—22no—
275C8—41no—
276C8—41no—
277C8—41no—
278C22—24no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22C2×#12111yescharacteristic
286C22#54611yescharacteristic
287C22D4×Q811yescharacteristic
288C22C2×#13511yescharacteristic
289C22#57511yescharacteristic
290C22#57711yescharacteristic
291C22#57711yescharacteristic
292C4—22no—
293C4—22no—
294C4—22no—
295C4—42no—
296C4—41no—
297C4—42no—
298C4—41no—
299C4—41no—
300C4—81no—
301C4—81no—
302C4—81no—
303C4—81no—
304C4—81no—
305C4C2×C22⋊Q811yescharacteristic
306C4#53711yescharacteristic
307C4#55211yescharacteristic
308C4#55411yescharacteristic
309C2—44no—
310C2#241711yescharacteristic
311C2#303511yescharacteristic
312C2#304211yescharacteristic
313C2#329611yescharacteristic
314C2#332311yescharacteristic
315C2#338111yescharacteristic
316C2#338811yescharacteristic
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
5. #2085C2semidirect
6. #2084C2semidirect
7. #2068C2semidirect
8. #2066C2semidirect
9. #1991C2semidirect
10. #1939C2semidirect
13. #1805C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4782
C2×C4×SD162
#25652
#23492
#20852
#20842
#20682
#20662
#19912
#19392
#18772
#18052
#18042
#18022