Finite Group Databasebeta

← #28715

Subgroups — #28715

Order 256.

398 conjugacy classes of subgroups.

The subgroup lattice has 325 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#501C211yesmaximal, characteristic, complemented
2C2×#435C211yesmaximal, characteristic
3#2563C211yesmaximal, characteristic
4#2562C211yesmaximal, characteristic
5#2077C211yesmaximal, characteristic, complemented
6#2077C211yesmaximal, characteristic, complemented
7#2076C211yesmaximal, characteristic, complemented
8#2075C211yesmaximal, characteristic, complemented
9#1989C211yesmaximal, characteristic, complemented
10#1988C211yesmaximal, characteristic, complemented
11#1809C211yesmaximal, characteristic, complemented
12#1809C211yesmaximal, characteristic, complemented
13#1808C211yesmaximal, characteristic, complemented
14#1807C211yesmaximal, characteristic, complemented
15#1776C211yesmaximal, characteristic, complemented
16C22×M16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×#125C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21#501—24no—
22#435—24no—
23C2×#107C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×C22⋊C8C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×#97C2211yescharacteristic
33#400—21no—
34#394—21no—
35#393—21no—
36#393—21no—
37#390—21no—
38#388—21no—
39#388—21no—
40#382—21no—
41#380C2211yescharacteristic
42#380C2211yescharacteristic
43#380C2211yescharacteristic
44#380C2211yescharacteristic
45#380C2211yescharacteristic
46#380C2211yescharacteristic
47#380C2211yescharacteristic
48#380C2211yescharacteristic
49C4×C4⋊C4C2211yescharacteristic
50C4×C4⋊C4C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×Q8—21no—
62C22×D4D411yescharacteristic
63C23×C4C2311yescharacteristic
64C23×C4C2311yescharacteristic
65C2×M16—22no—
66C2×M16—22no—
67C2×M16—22no—
68C22×C8—22no—
69C22×C8—22no—
70C22×C8—22no—
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C4⋊Q8—22no—
76C4⋊Q8—22no—
77#125—22no—
78#125—22no—
79C4×D4—22no—
80C4×D4—22no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4D411yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4D411yescharacteristic
102C2×C22⋊C4D411yescharacteristic
103C2×C42—21no—
104C2×C42—21no—
105C2×C42C2311yescharacteristic
106#107—22no—
107#107—22no—
108#107—22no—
109#106—22no—
110#106—22no—
111#106—22no—
112C4⋊C8—22no—
113C22⋊C8—22no—
114C22⋊C8—22no—
115#97D414yes—
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#95—21no—
126C24C2×D411yescharacteristic
127C24C2×D411yescharacteristic
128C2×Q8—42no—
129C2×Q8—81no—
130C2×Q8—81no—
131C2×D4—22no—
132C2×D4—22no—
133C2×D4—41no—
134C22×C4—21no—
135C22×C4—21no—
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×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2411yescharacteristic
159M16—44no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—44no—
169C2×C8C2×D412yes—
170C2×C8C2×D412yes—
171C2×C8#4112yes—
172C2×C8#4112yes—
173C4⋊C4—22no—
174C4⋊C4—21no—
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—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—42no—
196C22⋊C4—42no—
197C22⋊C4—81no—
198C22⋊C4—81no—
199C22⋊C4—81no—
200C22⋊C4—81no—
201C22⋊C4—81no—
202C22⋊C4—81no—
203C42—42no—
204C42—81no—
205C42C2×D412yes—
206C42#4112yes—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23#13712yes—
221C23C22×D411yescharacteristic
222Q8—84no—
223D4—44no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
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×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#12112yes—
267C2×C4#12112yes—
268C2×C4C22×D411yescharacteristic
269C2×C4C2×#4111yescharacteristic
270C2×C4C2×#4111yescharacteristic
271C2×C4C2×#4111yescharacteristic
272C2×C42−1+411yescharacteristic
273C2×C42−1+411yescharacteristic
274C8—22no—
275C8—41no—
276C8—41no—
277C8#12414yes—
278C22—22no—
279C22—22no—
280C22—22no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22C2×#12111yescharacteristic
293C22#54711yescharacteristic
294C22#54711yescharacteristic
295C22C2×#13711yescharacteristic
296C22#57511yescharacteristic
297C22#57611yescharacteristic
298C22#57711yescharacteristic
299C4—22no—
300C4—22no—
301C4—41no—
302C4—41no—
303C4—81no—
304C4—81no—
305C4—81no—
306C4—81no—
307C4—81no—
308C4—81no—
309C4—81no—
310C4—81no—
311C4C2×#12411yescharacteristic
312C4#53611yescharacteristic
313C4#55611yescharacteristic
314C4#55611yescharacteristic
315C2—42no—
316C2—81no—
317C2#254711yescharacteristic
318C2#305611yescharacteristic
319C2#306211yescharacteristic
320C2#334411yescharacteristic
321C2#334611yescharacteristic
322C2#334711yescharacteristic
323C2#334811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#501C2semidirect
5. #2077C2semidirect
6. #2077C2semidirect
7. #2076C2semidirect
8. #2075C2semidirect
9. #1989C2semidirect
10. #1988C2semidirect
11. #1809C2semidirect
12. #1809C2semidirect
13. #1808C2semidirect
14. #1807C2semidirect
15. #1776C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5012
C2×#4352
#25632
#25622
#20772
#20762
#20752
#19892
#19882
#18092
#18082
#18072
#17762