Finite Group Databasebeta

← #20291

Subgroups — #20291

Order 256.

320 conjugacy classes of subgroups.

The subgroup lattice has 299 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3367C211yesmaximal, characteristic, complemented
2#3353C211yesmaximal, characteristic, complemented
3#3330C211yesmaximal, characteristic
4#3276C211yesmaximal, characteristic, complemented
5#1707C211yesmaximal, characteristic, complemented
6#1700C211yesmaximal, characteristic, complemented
7#1699C211yesmaximal, characteristic, complemented
8#1693C211yesmaximal, characteristic, complemented
9#1689C211yesmaximal, characteristic, complemented
10#1654C211yesmaximal, characteristic, complemented
11#1654C211yesmaximal, characteristic, complemented
12#1652C211yesmaximal, characteristic, complemented
13#1651C211yesmaximal, characteristic, complemented
14#1590C211yesmaximal, characteristic, complemented
15#1589C211yesmaximal, characteristic, complemented
16C2×#137—21no—
17#556C2211yescharacteristic
18D4×Q8C2211yescharacteristic
19#547C2211yescharacteristic
20#547C2211yescharacteristic
21#500C2211yescharacteristic
22#498C2211yescharacteristic
23#498C2211yescharacteristic
24#494C2211yescharacteristic
25#489—21no—
26#487C2211yescharacteristic
27#487C2211yescharacteristic
28#484—21no—
29#484—21no—
30#482—21no—
31#477—21no—
32#475C2211yescharacteristic
33#474C2211yescharacteristic
34#474C2211yescharacteristic
35#474C2211yescharacteristic
36#470—21no—
37#468—21no—
38#468—21no—
39#463—21no—
40#462C2211yescharacteristic
41#461C2211yescharacteristic
42#461C2211yescharacteristic
43#461C2211yescharacteristic
44#452—21no—
45#451—21no—
46#449—21no—
47#446C2211yescharacteristic
48#440—21no—
49C4×SD16C2211yescharacteristic
50C4×D8C2211yescharacteristic
51#436C2211yescharacteristic
52#436C2211yescharacteristic
53#436C2211yescharacteristic
54#428—21no—
55#417—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#332C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#332C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66C82C2211yescharacteristic
67C2×#41—21no—
68C2×#41—21no—
69C22×Q8—21no—
70#137—44no—
71C2×Q16—21no—
72C2×Q16—21no—
73C2×Q16—21no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×SD16—21no—
77C2×SD16D411yescharacteristic
78C2×D8D411yescharacteristic
79C2×M16—21no—
80C2×M16—21no—
81C2×M16—21no—
82C4⋊Q8—21no—
83C4⋊Q8C2311yescharacteristic
84C4⋊Q8C2311yescharacteristic
85C4⋊Q8C2311yescharacteristic
86C4⋊Q8C2311yescharacteristic
87#126—41no—
88#124—21no—
89#123—41no—
90#123—41no—
91C22⋊Q8—21no—
92C22⋊Q8—21no—
93C22⋊Q8—21no—
94C22⋊Q8—21no—
95C22⋊Q8—41no—
96#121—21no—
97#121—21no—
98C4×Q8—21no—
99C4×Q8C2311yescharacteristic
100C4×D4—21no—
101C4×D4C2311yescharacteristic
102C4×D4C2311yescharacteristic
103C4×D4C2311yescharacteristic
104#117—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107#107—21no—
108#107—21no—
109#107—21no—
110#107—21no—
111#107—21no—
112#107D411yescharacteristic
113#106—21no—
114#106—41no—
115#106—41no—
116#106D411yescharacteristic
117C4⋊C8—21no—
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121C4⋊C8C2311yescharacteristic
122#103—21no—
123#103—21no—
124#103—21no—
125#103—21no—
126#103—21no—
127#103—21no—
128#103—21no—
129#103—21no—
130#103—21no—
131#103D411yescharacteristic
132#102—21no—
133#102—21no—
134#102—21no—
135#102D411yescharacteristic
136#102D411yescharacteristic
137#102D411yescharacteristic
138C22⋊C8—21no—
139C22⋊C8—21no—
140C22⋊C8—21no—
141#97—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—21no—
153C2×Q8—41no—
154C2×Q8—42no—
155C2×Q8—81no—
156C2×Q8C2×D411yescharacteristic
157C2×D4—41no—
158C2×D4—41no—
159C2×D4C2×D411yescharacteristic
160C2×D4C2×D411yescharacteristic
161C2×D4C2×D411yescharacteristic
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168Q16—42no—
169Q16—42no—
170Q16—81no—
171SD16—22no—
172SD16—42no—
173SD16—42no—
174SD16—81no—
175D8—41no—
176M16—42no—
177M16—42no—
178M16—81no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8C2×D411yescharacteristic
185C2×C8C2×D411yescharacteristic
186C2×C8C2×D411yescharacteristic
187C2×C8C2×D411yescharacteristic
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4C2×D411yescharacteristic
202C4⋊C4C2×D411yescharacteristic
203C4⋊C4C2×D411yescharacteristic
204C4⋊C4C2×D411yescharacteristic
205C22⋊C4—21no—
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C42—21no—
214C42—41no—
215C42C2411yescharacteristic
216C23—21no—
217C23—21no—
218C23—21no—
219Q8—21no—
220Q8—21no—
221Q8—41no—
222Q8—41no—
223Q8—41no—
224Q8—41no—
225Q8—41no—
226Q8—41no—
227Q8—41no—
228Q8—41no—
229Q8—82no—
230D4—21no—
231D4—21no—
232D4—21no—
233D4—21no—
234D4—21no—
235D4—21no—
236D4—82no—
237D4—82no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
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—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4C22×D411yescharacteristic
257C2×C4C22×D411yescharacteristic
258C2×C4H(5,𝔽2)11yescharacteristic
259C8—22no—
260C8—21no—
261C8—21no—
262C8—21no—
263C8—41no—
264C8—41no—
265C8—41no—
266C8—41no—
267C8AGL(1,ℤ/8)12yes—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22D4211yescharacteristic
278C4—21no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—81no—
285C4—81no—
286C4—81no—
287C4—81no—
288C4C2×AGL(1,ℤ/8)11yescharacteristic
289C4#57511yescharacteristic
290C4#57711yescharacteristic
291C4#57711yescharacteristic
292C2—81no—
293C2—81no—
294C2—81no—
295C2#327211yescharacteristic
296C2#327511yescharacteristic
297C2#329311yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3330C2
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. #3367C2semidirect
2. #3353C2semidirect
4. #3276C2semidirect
5. #1707C2semidirect
6. #1700C2semidirect
7. #1699C2semidirect
8. #1693C2semidirect
9. #1689C2semidirect
10. #1654C2semidirect
11. #1654C2semidirect
12. #1652C2semidirect
13. #1651C2semidirect
14. #1590C2semidirect
15. #1589C2semidirect

Maximal subgroups

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

subgroupindex
#33672
#33532
#33302
#32762
#17072
#17002
#16992
#16932
#16892
#16542
#16522
#16512
#15902
#15892