Finite Group Databasebeta

← #26486

Subgroups — #26486

Order 256.

411 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×#501C211yesmaximal, characteristic, complemented
2C2×C4×SD16C211yesmaximal, characteristic
3#2453C211yesmaximal, characteristic, complemented
4#2452C211yesmaximal, characteristic
5#2068C211yesmaximal, characteristic, complemented
6#2065C211yesmaximal, characteristic
7#2025C211yesmaximal, characteristic
8#2024C211yesmaximal, characteristic, complemented
9#1973C211yesmaximal, characteristic
10#1973C211yesmaximal, characteristic
11#1944C211yesmaximal, characteristic, complemented
12#1942C211yesmaximal, characteristic, complemented
13#1865C211yesmaximal, characteristic, complemented
14#1862C211yesmaximal, characteristic
15#1771C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×#125C2211yescharacteristic
19C2×C22⋊Q8—21no—
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22#501—24no—
23C4×SD16—24no—
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#107C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#97C2211yescharacteristic
37C2×#97C2211yescharacteristic
38C2×C4×C8C2211yescharacteristic
39#400—21no—
40#398—21no—
41#396—21no—
42#393—21no—
43#391C2211yescharacteristic
44#391C2211yescharacteristic
45#390C2211yescharacteristic
46#390C2211yescharacteristic
47#389—21no—
48#389C2211yescharacteristic
49#389C2211yescharacteristic
50#389C2211yescharacteristic
51#389C2211yescharacteristic
52#387—21no—
53#385—21no—
54C4×C4⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×Q8C2311yescharacteristic
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C2×SD16—22no—
66C2×SD16—22no—
67C2×SD16Q814yescomplemented
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C4⋊Q8—22no—
73C4⋊Q8—41no—
74#125—22no—
75#125—41no—
76C22⋊Q8—44no—
77C4×Q8—22no—
78C4×Q8—41no—
79C4×D4—22no—
80C4×D4—41no—
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⋊C4D411yescharacteristic
90C2×C4⋊C4D411yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C42—21no—
103C2×C42C2311yescharacteristic
104C2×C42C2311yescharacteristic
105C2×C42C2311yescharacteristic
106#107—22no—
107#107—22no—
108#107—22no—
109#106—22no—
110#106D414yes—
111C4⋊C8—22no—
112#103—22no—
113#103—22no—
114#103—41no—
115#102—22no—
116#102—22no—
117#102—41no—
118#97—22no—
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#95—21no—
128#95—21no—
129#95D411yescharacteristic
130#95D411yescharacteristic
131C24—21no—
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—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4C2×Q812yes—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—81no—
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2411yescharacteristic
164SD16—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×C8C2×D412yes—
174C2×C8C2×Q812yes—
175C2×C8#4112yes—
176C2×C8#4112yes—
177C4⋊C4—22no—
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—41no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4C2×D412yes—
198C4⋊C4C2×D412yes—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—81no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—42no—
211C23—22no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23C22×D411yescharacteristic
219Q8—22no—
220Q8—22no—
221Q8—41no—
222Q8—41no—
223Q8—84no—
224D4—22no—
225D4—22no—
226D4—41no—
227D4—41no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
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—21no—
247C2×C4—21no—
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—84no—
267C2×C4AGL(1,ℤ/8)12yes—
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—
277C8C22⋊Q814yescomplemented
278C22—42no—
279C22—42no—
280C22—42no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22#53211yescharacteristic
286C22D4211yescharacteristic
287C22D4×Q811yescharacteristic
288C22C2×AGL(1,ℤ/8)11yescharacteristic
289C22#57511yescharacteristic
290C22#57711yescharacteristic
291C22#57811yescharacteristic
292C4—22no—
293C4—41no—
294C4—41no—
295C4—42no—
296C4—41no—
297C4—42no—
298C4—41no—
299C4—41no—
300C4—42no—
301C4—81no—
302C4—81no—
303C4—81no—
304C4—81no—
305C4C2×C22⋊Q811yescharacteristic
306C4#54211yescharacteristic
307C4#54811yescharacteristic
308C4#55411yescharacteristic
309C2—82no—
310C2#245211yescharacteristic
311C2#317111yescharacteristic
312C2#317811yescharacteristic
313C2#326511yescharacteristic
314C2#326911yescharacteristic
315C2#338611yescharacteristic
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
1. C2×#501C2semidirect
3. #2453C2semidirect
5. #2068C2semidirect
8. #2024C2semidirect
11. #1944C2semidirect
12. #1942C2semidirect
13. #1865C2semidirect
15. #1771C2semidirect
67. C2×SD16Q8semidirect
277. 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×C4×SD162
#24532
#24522
#20682
#20652
#20252
#20242
#19732
#19442
#19422
#18652
#18622
#17712