Finite Group Databasebeta

← #30779

Subgroups — #30779

Order 256.

391 conjugacy classes of subgroups.

The subgroup lattice has 315 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#500C211yesmaximal, characteristic, complemented
2C2×#484C211yesmaximal, characteristic
3#2492C211yesmaximal, characteristic
4#2440C211yesmaximal, characteristic
5#2095C211yesmaximal, characteristic, complemented
6#2076C211yesmaximal, characteristic
7#2068C211yesmaximal, characteristic, complemented
8#2057C211yesmaximal, characteristic, complemented
9#2036C211yesmaximal, characteristic
10#2024C211yesmaximal, characteristic, complemented
11#1947C211yesmaximal, characteristic, complemented
12#1933C211yesmaximal, characteristic, complemented
13#1923C211yesmaximal, characteristic
14#1921C211yesmaximal, characteristic, complemented
15#1852C211yesmaximal, characteristic
16C2×C4⋊Q8C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×C22⋊Q8—21no—
19C2×C22⋊Q8C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21#500—24no—
22#484—24no—
23C2×#107C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#107C2211yescharacteristic
27C2×#107C2211yescharacteristic
28C2×#107C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×C4×C8C2211yescharacteristic
37#400—21no—
38#398—21no—
39#395—21no—
40#393—21no—
41#391—21no—
42#391C2211yescharacteristic
43#391C2211yescharacteristic
44#389C2211yescharacteristic
45#389C2211yescharacteristic
46#388—21no—
47#385—21no—
48#384C2211yescharacteristic
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×Q8—21no—
63C22×Q8C2311yescharacteristic
64C23×C4—21no—
65C23×C4C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C4⋊Q8—22no—
71C4⋊Q8—22no—
72C4⋊Q8—41no—
73C4⋊Q8—41no—
74C22⋊Q8—22no—
75C22⋊Q8—44no—
76C22⋊Q8—41no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4D411yescharacteristic
98C2×C22⋊C4D411yescharacteristic
99C2×C42—21no—
100C2×C42C2311yescharacteristic
101C2×C42C2311yescharacteristic
102#107—22no—
103#107—22no—
104#107—22no—
105#107—22no—
106#107—41no—
107#107D414yes—
108C4⋊C8—22no—
109#103—22no—
110#103—22no—
111#103—22no—
112#103—41no—
113C22⋊C8—22no—
114C22⋊C8—22no—
115C4×C8—22no—
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#95—21no—
125#95D411yescharacteristic
126#95D411yescharacteristic
127C24C2×D411yescharacteristic
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8—81no—
139C2×Q8—81no—
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×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×C4C2411yescharacteristic
163C2×C8—21no—
164C2×C8—21no—
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×C8#4112yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
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—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—81no—
195C4⋊C4C2×D412yes—
196C4⋊C4C2×D412yes—
197C22⋊C4—22no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—81no—
201C22⋊C4Q1614yes—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—81no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23C2×Q1612yes—
214C23C22×D411yescharacteristic
215Q8—41no—
216Q8—41no—
217Q8—41no—
218Q8—41no—
219Q8—84no—
220Q8—84no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
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×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—84no—
262C2×C4C2×Q1612yes—
263C2×C4C2×Q1612yes—
264C2×C4AGL(1,ℤ/8)12yes—
265C2×C4C22×D411yescharacteristic
266C2×C4C2×#4111yescharacteristic
267C2×C4H(5,𝔽2)11yescharacteristic
268C2×C42−1+411yescharacteristic
269C2×C42−1+411yescharacteristic
270C2×C42−1+411yescharacteristic
271C8—22no—
272C8—22no—
273C8—41no—
274C8—41no—
275C22—22no—
276C22—22no—
277C22—22no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22#53611yescharacteristic
283C22D4211yescharacteristic
284C22#54711yescharacteristic
285C22C22×Q1611yescharacteristic
286C22C2×AGL(1,ℤ/8)11yescharacteristic
287C22#57611yescharacteristic
288C22#57811yescharacteristic
289C4—22no—
290C4—22no—
291C4—22no—
292C4—41no—
293C4—41no—
294C4—42no—
295C4—42no—
296C4—81no—
297C4—81no—
298C4—81no—
299C4—81no—
300C4—81no—
301C4—81no—
302C4#53611yescharacteristic
303C4#54211yescharacteristic
304C4#54811yescharacteristic
305C4#56311yescharacteristic
306C2—42no—
307C2#267411yescharacteristic
308C2#323011yescharacteristic
309C2#324311yescharacteristic
310C2#326411yescharacteristic
311C2D4×Q1611yescharacteristic
312C2#333811yescharacteristic
313C2#335311yescharacteristic
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×#500C2semidirect
5. #2095C2semidirect
7. #2068C2semidirect
8. #2057C2semidirect
10. #2024C2semidirect
11. #1947C2semidirect
12. #1933C2semidirect
14. #1921C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5002
C2×#4842
#24922
#24402
#20952
#20762
#20682
#20572
#20362
#20242
#19472
#19332
#19232
#19212
#18522