Finite Group Databasebeta

← #20077

Subgroups — #20077

Order 256.

315 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3369C211yesmaximal, characteristic, complemented
2#3362C211yesmaximal, characteristic, complemented
3#3353C211yesmaximal, characteristic, complemented
4#3284C211yesmaximal, characteristic
5#1719C211yesmaximal, characteristic, complemented
6#1697C211yesmaximal, characteristic, complemented
7#1686C211yesmaximal, characteristic, complemented
8#1683C211yesmaximal, characteristic, complemented
9#1674C211yesmaximal, characteristic, complemented
10#1645C211yesmaximal, characteristic, complemented
11#1641C211yesmaximal, characteristic, complemented
12#1635C211yesmaximal, characteristic, complemented
13#1628C211yesmaximal, characteristic, complemented
14#1588C211yesmaximal, characteristic, complemented
15#1585C211yesmaximal, characteristic, complemented
16C2×#135—21no—
17Q82C2211yescharacteristic
18#556C2211yescharacteristic
19#550C2211yescharacteristic
20#547C2211yescharacteristic
21#500C2211yescharacteristic
22#492C2211yescharacteristic
23#489—21no—
24#487C2211yescharacteristic
25#487C2211yescharacteristic
26#486—21no—
27#486C2211yescharacteristic
28#486C2211yescharacteristic
29#484—21no—
30#482—21no—
31#476—21no—
32#476C2211yescharacteristic
33#476C2211yescharacteristic
34#475—21no—
35#475C2211yescharacteristic
36#474C2211yescharacteristic
37#468—21no—
38#465—21no—
39#464—21no—
40#462C2211yescharacteristic
41#462C2211yescharacteristic
42#460—21no—
43#460C2211yescharacteristic
44#459C2211yescharacteristic
45#452—21no—
46#449—21no—
47#446C2211yescharacteristic
48C8×Q8C2211yescharacteristic
49#441—21no—
50C4×SD16—21no—
51C4×SD16C2211yescharacteristic
52C4×SD16C2211yescharacteristic
53#436C2211yescharacteristic
54C8×D4C2211yescharacteristic
55#428—21no—
56#335C2211yescharacteristic
57#335C2211yescharacteristic
58#333C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×#41—21no—
69#135—44no—
70C2×Q16—21no—
71C2×Q16—21no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16D411yescharacteristic
76C2×SD16D411yescharacteristic
77C2×D8—21no—
78C2×M16—21no—
79C22×C8—21no—
80C4⋊Q8—21no—
81C4⋊Q8—21no—
82C4⋊Q8—41no—
83C4⋊Q8C2311yescharacteristic
84C4⋊Q8C2311yescharacteristic
85C4⋊Q8C2311yescharacteristic
86#127—21no—
87#127C2311yescharacteristic
88#126—41no—
89#124—21no—
90#123—41no—
91C22⋊Q8—21no—
92C22⋊Q8—21no—
93#121—21no—
94#121—21no—
95#121—41no—
96C4×Q8—21no—
97C4×Q8—21no—
98C4×Q8—21no—
99C4×Q8C2311yescharacteristic
100C4×Q8C2311yescharacteristic
101C4×D4—21no—
102C4×D4C2311yescharacteristic
103C4×D4C2311yescharacteristic
104#117—21no—
105C2×C4⋊C4—21no—
106#107—21no—
107#107—21no—
108#107—21no—
109#106—21no—
110#106—21no—
111#106—21no—
112#106D411yescharacteristic
113#106D411yescharacteristic
114C4⋊C8—21no—
115C4⋊C8—21no—
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120#103—21no—
121#103—21no—
122#103—21no—
123#103—21no—
124#103—21no—
125#103—41no—
126#103D411yescharacteristic
127#103D411yescharacteristic
128#102—21no—
129#102—21no—
130#102—21no—
131#102—21no—
132#102—21no—
133#102—21no—
134#102—21no—
135#102—41no—
136#102D411yescharacteristic
137#102D411yescharacteristic
138C22⋊C8—21no—
139C22⋊C8—21no—
140#97—21no—
141C4×C8—21no—
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144C4×C8C2311yescharacteristic
145#41—42no—
146#41—42no—
147#41—81no—
148#41—81no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—41no—
153C2×Q8C2×D411yescharacteristic
154C2×Q8C2×D411yescharacteristic
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4C2×D411yescharacteristic
162C2×D4C2×D411yescharacteristic
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—41no—
166C22×C4—41no—
167Q16—42no—
168Q16—42no—
169SD16—22no—
170SD16—41no—
171SD16—42no—
172SD16—42no—
173SD16—81no—
174D8—42no—
175M16—42no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—42no—
182C2×C8C2×D411yescharacteristic
183C2×C8C2×D411yescharacteristic
184C2×C8C2×D411yescharacteristic
185C2×C8C2×D411yescharacteristic
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4C2×D411yescharacteristic
199C4⋊C4C2×D411yescharacteristic
200C4⋊C4C2×D411yescharacteristic
201C4⋊C4C2×D411yescharacteristic
202C22⋊C4—21no—
203C22⋊C4—21no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C42—21no—
209C42—21no—
210C42—41no—
211C42—41no—
212C42C2411yescharacteristic
213C23—21no—
214C23—21no—
215C23—41no—
216Q8—21no—
217Q8—21no—
218Q8—21no—
219Q8—41no—
220Q8—41no—
221Q8—41no—
222Q8—41no—
223Q8—81no—
224Q8—161no—
225Q8AGL(1,ℤ/8)12yes—
226D4—21no—
227D4—21no—
228D4—21no—
229D4—21no—
230D4—41no—
231D4—41no—
232D4—41no—
233D4—41no—
234D4—82no—
235D4—81no—
236D4—82no—
237D4—161no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4C22×D411yescharacteristic
254C2×C4C22×D411yescharacteristic
255C2×C4H(5,𝔽2)11yescharacteristic
256C8—21no—
257C8—21no—
258C8—21no—
259C8—21no—
260C8—41no—
261C8—41no—
262C8—41no—
263C8—41no—
264C8—41no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—81no—
272C22—81no—
273C22—81no—
274C22D4211yescharacteristic
275C4—22no—
276C4—21no—
277C4—41no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—81no—
283C4—81no—
284C4—81no—
285C4C2×AGL(1,ℤ/8)11yescharacteristic
286C4#57511yescharacteristic
287C4#57711yescharacteristic
288C4#57811yescharacteristic
289C2—81no—
290C2—81no—
291C2—161no—
292C2#326511yescharacteristic
293C2#326911yescharacteristic
294C2#329011yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3284C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3369C2semidirect
2. #3362C2semidirect
3. #3353C2semidirect
5. #1719C2semidirect
6. #1697C2semidirect
7. #1686C2semidirect
8. #1683C2semidirect
9. #1674C2semidirect
10. #1645C2semidirect
11. #1641C2semidirect
12. #1635C2semidirect
13. #1628C2semidirect
14. #1588C2semidirect
15. #1585C2semidirect

Maximal subgroups

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

subgroupindex
#33692
#33622
#33532
#32842
#17192
#16972
#16862
#16832
#16742
#16452
#16412
#16352
#16282
#15882
#15852