Finite Group Databasebeta

← #30798

Subgroups — #30798

Order 256.

355 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#501C211yesmaximal, characteristic, complemented
2C2×#485C211yesmaximal, characteristic
3#2489C211yesmaximal, characteristic
4#2439C211yesmaximal, characteristic
5#2097C211yesmaximal, characteristic, complemented
6#2076C211yesmaximal, characteristic
7#2068C211yesmaximal, characteristic, complemented
8#2059C211yesmaximal, characteristic, complemented
9#2035C211yesmaximal, characteristic
10#2024C211yesmaximal, characteristic, complemented
11#1949C211yesmaximal, characteristic, complemented
12#1935C211yesmaximal, characteristic, complemented
13#1924C211yesmaximal, characteristic
14#1919C211yesmaximal, characteristic, complemented
15#1854C211yesmaximal, characteristic
16C2×C4⋊Q8C2211yescharacteristic
17C2×#125C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×#117C2211yescharacteristic
20#501—24no—
21#485—24no—
22C2×#107C2211yescharacteristic
23C2×#107C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35C2×#97C2211yescharacteristic
36#400—21no—
37#397—21no—
38#396—21no—
39#393—21no—
40#391—21no—
41#391C2211yescharacteristic
42#391C2211yescharacteristic
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#388—21no—
46#387—21no—
47#382—21no—
48#382C2211yescharacteristic
49#380C2211yescharacteristic
50#380C2211yescharacteristic
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×C4C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C4⋊Q8—22no—
69C4⋊Q8—41no—
70#125—22no—
71#125—41no—
72C22⋊Q8—22no—
73C22⋊Q8—41no—
74#117—22no—
75#117—41no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4D411yescharacteristic
94C2×C22⋊C4D411yescharacteristic
95C2×C42—21no—
96C2×C42—21no—
97C2×C42C2311yescharacteristic
98C2×C42C2311yescharacteristic
99#107—22no—
100#107—22no—
101#107—22no—
102#106—22no—
103#106—41no—
104#106D414yes—
105C4⋊C8—22no—
106#103—22no—
107#103—22no—
108#103—22no—
109#103—41no—
110C22⋊C8—22no—
111C22⋊C8—22no—
112#97—22no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95D411yescharacteristic
125#95D411yescharacteristic
126C24C2×D411yescharacteristic
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—42no—
134C2×Q8—42no—
135C2×Q8—81no—
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—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2411yescharacteristic
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8C2×D412yes—
167C2×C8#4112yes—
168C4⋊C4—21no—
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—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—81no—
186C4⋊C4C2×D412yes—
187C4⋊C4C2×D412yes—
188C22⋊C4—41no—
189C22⋊C4—41no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C22⋊C4—42no—
193C22⋊C4—81no—
194C42—21no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—42no—
199C42—42no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23C22×D411yescharacteristic
208Q8—41no—
209Q8—41no—
210Q8—41no—
211Q8—41no—
212Q8—84no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
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×C4C22×D411yescharacteristic
257C2×C4C2×#4111yescharacteristic
258C2×C4H(5,𝔽2)11yescharacteristic
259C2×C42−1+411yescharacteristic
260C2×C42−1+411yescharacteristic
261C2×C42−1+411yescharacteristic
262C8—22no—
263C8—22no—
264C8—41no—
265C8—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22#53611yescharacteristic
274C22D4211yescharacteristic
275C22#54711yescharacteristic
276C22#57511yescharacteristic
277C22#57511yescharacteristic
278C22#57711yescharacteristic
279C22#57811yescharacteristic
280C4—41no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—42no—
285C4—41no—
286C4—42no—
287C4—81no—
288C4—81no—
289C4—81no—
290C4—81no—
291C4—81no—
292C4—81no—
293C4#53611yescharacteristic
294C4#54211yescharacteristic
295C4#54811yescharacteristic
296C4#56311yescharacteristic
297C2—81no—
298C2#267411yescharacteristic
299C2#324011yescharacteristic
300C2#324211yescharacteristic
301C2#326911yescharacteristic
302C2#327511yescharacteristic
303C2#334811yescharacteristic
304C2#334911yescharacteristic
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. #2097C2semidirect
7. #2068C2semidirect
8. #2059C2semidirect
10. #2024C2semidirect
11. #1949C2semidirect
12. #1935C2semidirect
14. #1919C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5012
C2×#4852
#24892
#24392
#20972
#20762
#20682
#20592
#20352
#20242
#19492
#19352
#19242
#19192
#18542