Finite Group Databasebeta

← #20109

Subgroups — #20109

Order 256.

308 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#3392C211yesmaximal, characteristic, complemented
2#3381C211yesmaximal, characteristic, complemented
3#3332C211yesmaximal, characteristic
4#3331C211yesmaximal, characteristic, complemented
5#1714C211yesmaximal, characteristic, complemented
6#1691C211yesmaximal, characteristic, complemented
7#1685C211yesmaximal, characteristic, complemented
8#1684C211yesmaximal, characteristic, complemented
9#1682C211yesmaximal, characteristic, complemented
10#1648C211yesmaximal, characteristic, complemented
11#1647C211yesmaximal, characteristic, complemented
12#1631C211yesmaximal, characteristic, complemented
13#1622C211yesmaximal, characteristic, complemented
14#1588C211yesmaximal, characteristic, complemented
15#1588C211yesmaximal, characteristic, complemented
16#557C2211yescharacteristic
17#554C2211yescharacteristic
18#550C2211yescharacteristic
19#547C2211yescharacteristic
20#500C2211yescharacteristic
21#499—21no—
22#498C2211yescharacteristic
23#492—21no—
24#492C2211yescharacteristic
25#492C2211yescharacteristic
26#487C2211yescharacteristic
27#486C2211yescharacteristic
28#483—21no—
29#481—21no—
30#479—21no—
31#478—21no—
32#477C2211yescharacteristic
33#476C2211yescharacteristic
34#475C2211yescharacteristic
35#474C2211yescharacteristic
36#470—21no—
37#469—21no—
38#462C2211yescharacteristic
39#461—21no—
40#461C2211yescharacteristic
41#460—21no—
42#460C2211yescharacteristic
43#459C2211yescharacteristic
44#452—21no—
45#449—21no—
46C8×Q8C2211yescharacteristic
47C8×Q8C2211yescharacteristic
48C4×SD16—21no—
49C4×SD16—21no—
50C4×SD16C2211yescharacteristic
51C4×SD16C2211yescharacteristic
52#436C2211yescharacteristic
53#436C2211yescharacteristic
54#417—21no—
55#417—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—
69C2×Q16—21no—
70C2×SD16—21no—
71C2×SD16—21no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—41no—
75C2×SD16D411yescharacteristic
76C2×SD16D411yescharacteristic
77C2×D8—21no—
78C2×M16—21no—
79C2×M16—21no—
80C4⋊Q8—21no—
81C4⋊Q8C2311yescharacteristic
82C4⋊Q8C2311yescharacteristic
83C4⋊Q8C2311yescharacteristic
84#127—21no—
85#127C2311yescharacteristic
86#125—21no—
87#125—21no—
88#125—41no—
89#123—41no—
90C22⋊Q8—21no—
91C22⋊Q8—21no—
92C22⋊Q8—41no—
93#121—21no—
94#121—21no—
95#121—41no—
96C4×Q8—21no—
97C4×Q8—21no—
98C4×Q8C2311yescharacteristic
99C4×Q8C2311yescharacteristic
100C4×D4—21no—
101C4×D4—21no—
102C4×D4C2311yescharacteristic
103C4×D4C2311yescharacteristic
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106#107—21no—
107#107—21no—
108#107—41no—
109#106—21no—
110#106—21no—
111#106—21no—
112#106—21no—
113#106D411yescharacteristic
114#106D411yescharacteristic
115C4⋊C8—21no—
116C4⋊C8—21no—
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121#103—21no—
122#103—21no—
123#103—21no—
124#103—21no—
125#103—21no—
126#103—21no—
127#103D411yescharacteristic
128#103D411yescharacteristic
129#102—21no—
130#102—21no—
131#102—21no—
132#102—21no—
133#102—21no—
134#102—21no—
135#102D411yescharacteristic
136#102D411yescharacteristic
137C22⋊C8—21no—
138C22⋊C8—21no—
139C4×C8—21no—
140C4×C8—21no—
141C4×C8C2311yescharacteristic
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144#41—42no—
145#41—42no—
146#41—81no—
147#41—81no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8C2×D411yescharacteristic
152C2×Q8C2×D411yescharacteristic
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4C2×D411yescharacteristic
160C2×D4C2×D411yescharacteristic
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166Q16—81no—
167SD16—22no—
168SD16—41no—
169SD16—42no—
170SD16—81no—
171SD16—82no—
172SD16—81no—
173SD16—81no—
174D8—81no—
175M16—81no—
176M16—81no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
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⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4C2×D411yescharacteristic
201C4⋊C4C2×D411yescharacteristic
202C4⋊C4C2×D411yescharacteristic
203C4⋊C4C2×D411yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—21no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C42—21no—
210C42—21no—
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#13712yes—
225D4—21no—
226D4—21no—
227D4—21no—
228D4—21no—
229D4—41no—
230D4—41no—
231D4—41no—
232D4—41no—
233D4—82no—
234D4—81no—
235D4—82no—
236D4—161no—
237C2×C4—21no—
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—82no—
251C2×C4—82no—
252C2×C4—161no—
253C2×C4C22×D411yescharacteristic
254C2×C4C22×D411yescharacteristic
255C2×C4H(5,𝔽2)11yescharacteristic
256C8—22no—
257C8—21no—
258C8—21no—
259C8—21no—
260C8—21no—
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×#13711yescharacteristic
286C4#57511yescharacteristic
287C4#57611yescharacteristic
288C4#57711yescharacteristic
289C2—81no—
290C2—81no—
291C2—161no—
292C2#326611yescharacteristic
293C2#326811yescharacteristic
294C2#329111yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3332C2
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. #3392C2semidirect
2. #3381C2semidirect
4. #3331C2semidirect
5. #1714C2semidirect
6. #1691C2semidirect
7. #1685C2semidirect
8. #1684C2semidirect
9. #1682C2semidirect
10. #1648C2semidirect
11. #1647C2semidirect
12. #1631C2semidirect
13. #1622C2semidirect
14. #1588C2semidirect
15. #1588C2semidirect

Maximal subgroups

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

subgroupindex
#33922
#33812
#33322
#33312
#17142
#16912
#16852
#16842
#16822
#16482
#16472
#16312
#16222
#15882