Finite Group Databasebeta

← #19949

Subgroups — #19949

Order 256.

311 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3399C211yesmaximal, characteristic, complemented
2#3361C211yesmaximal, characteristic, complemented
3#3357C211yesmaximal, characteristic, complemented
4#3305C211yesmaximal, characteristic, complemented
5#1706C211yesmaximal, characteristic, complemented
6#1696C211yesmaximal, characteristic, complemented
7#1687C211yesmaximal, characteristic, complemented
8#1678C211yesmaximal, characteristic, complemented
9#1677C211yesmaximal, characteristic, complemented
10#1631C211yesmaximal, characteristic, complemented
11#1630C211yesmaximal, characteristic, complemented
12#1624C211yesmaximal, characteristic, complemented
13#1622C211yesmaximal, characteristic, complemented
14#1590C211yesmaximal, characteristic, complemented
15C8×SD16C211yesmaximal, characteristic, complemented
16#559C2211yescharacteristic
17#555C2211yescharacteristic
18#554C2211yescharacteristic
19#550C2211yescharacteristic
20#500C2211yescharacteristic
21#496—21no—
22#493C2211yescharacteristic
23#492C2211yescharacteristic
24#492C2211yescharacteristic
25#488—21no—
26#486C2211yescharacteristic
27#486C2211yescharacteristic
28#485—21no—
29#482—21no—
30#480—21no—
31#478—21no—
32#477C2211yescharacteristic
33#477C2211yescharacteristic
34#477C2211yescharacteristic
35#476C2211yescharacteristic
36#468—21no—
37#466—21no—
38#463—21no—
39#463—21no—
40#460C2211yescharacteristic
41#459—21no—
42#459C2211yescharacteristic
43#459C2211yescharacteristic
44#459C2211yescharacteristic
45#449—21no—
46#446C2211yescharacteristic
47C8×Q8C2211yescharacteristic
48#442—21no—
49C4×SD16C2211yescharacteristic
50C4×SD16C2211yescharacteristic
51C4×D8—21no—
52#436C2211yescharacteristic
53C8×D4C2211yescharacteristic
54#428—21no—
55#416—21no—
56#334C2211yescharacteristic
57#334C2211yescharacteristic
58#333C2211yescharacteristic
59#333C2211yescharacteristic
60#331C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66C82C2211yescharacteristic
67C2×#41—21no—
68C2×SD16—21no—
69C2×SD16—41no—
70C2×SD16—41no—
71C2×SD16D411yescharacteristic
72C2×SD16D411yescharacteristic
73C2×D8—21no—
74C2×D8—21no—
75C2×D8—21no—
76C2×D8—21no—
77C2×D8—21no—
78C2×M16—21no—
79C22×C8—21no—
80C4⋊Q8C2311yescharacteristic
81C4⋊Q8C2311yescharacteristic
82#127—21no—
83#127C2311yescharacteristic
84#127C2311yescharacteristic
85#125—21no—
86#124—21no—
87#124—21no—
88#124—41no—
89#123—41no—
90C22⋊Q8—21no—
91C22⋊Q8—41no—
92#121—21no—
93#121—21no—
94#121—21no—
95#121—41no—
96C4×Q8C2311yescharacteristic
97C4×Q8C2311yescharacteristic
98C4×D4—21no—
99C4×D4—21no—
100C4×D4—21no—
101C4×D4—21no—
102C4×D4C2311yescharacteristic
103C4×D4C2311yescharacteristic
104#117—21no—
105#117—21no—
106C2×C4⋊C4—21no—
107#107—21no—
108#107—21no—
109#107—21no—
110#107—21no—
111#107—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#103D411yescharacteristic
125#103D411yescharacteristic
126#102—21no—
127#102—21no—
128#102—21no—
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—
139#97—21no—
140C4×C8—21no—
141C4×C8C2311yescharacteristic
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144#41—42no—
145#41—81no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8C2×D411yescharacteristic
149C2×Q8C2×D411yescharacteristic
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
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—
166SD16—22no—
167SD16—41no—
168SD16—82no—
169SD16—82no—
170SD16—81no—
171D8—42no—
172D8—42no—
173D8—81no—
174D8—81no—
175D8—81no—
176M16—42no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—81no—
183C2×C8C2×D411yescharacteristic
184C2×C8C2×D411yescharacteristic
185C2×C8C2×D411yescharacteristic
186C2×C8C2×D411yescharacteristic
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4C2×D411yescharacteristic
196C4⋊C4C2×D411yescharacteristic
197C4⋊C4C2×D411yescharacteristic
198C4⋊C4C2×D411yescharacteristic
199C22⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—41no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C42—21no—
209C42—21no—
210C42—41no—
211C42C2411yescharacteristic
212C23—21no—
213C23—21no—
214C23—41no—
215C23—41no—
216Q8—21no—
217Q8—21no—
218Q8—21no—
219Q8—21no—
220Q8—81no—
221Q8—81no—
222D4—21no—
223D4—21no—
224D4—21no—
225D4—21no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—41no—
230D4—41no—
231D4—41no—
232D4—41no—
233D4—41no—
234D4—81no—
235D4—81no—
236D4—82no—
237D4—161no—
238D4—161no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—82no—
252C2×C4—161no—
253C2×C4—161no—
254C2×C4C22×D411yescharacteristic
255C2×C4C22×D411yescharacteristic
256C2×C4H(5,𝔽2)11yescharacteristic
257C8—22no—
258C8—21no—
259C8—21no—
260C8—21no—
261C8—41no—
262C8—41no—
263C8—41no—
264C8—41no—
265C8#13512yes—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—81no—
273C22—81no—
274C22—81no—
275C22—81no—
276C22—81no—
277C22—81no—
278C22D4211yescharacteristic
279C4—21no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—41no—
285C4—41no—
286C4—81no—
287C4—81no—
288C4C2×#13511yescharacteristic
289C4#57511yescharacteristic
290C4#57711yescharacteristic
291C4#57711yescharacteristic
292C2—81no—
293C2—81no—
294C2—161no—
295C2—161no—
296C2#327511yescharacteristic
297C2#328411yescharacteristic
298C2#329211yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
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. #3399C2semidirect
2. #3361C2semidirect
3. #3357C2semidirect
4. #3305C2semidirect
5. #1706C2semidirect
6. #1696C2semidirect
7. #1687C2semidirect
8. #1678C2semidirect
9. #1677C2semidirect
10. #1631C2semidirect
11. #1630C2semidirect
12. #1624C2semidirect
13. #1622C2semidirect
14. #1590C2semidirect
15. C8×SD16C2semidirect

Maximal subgroups

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

subgroupindex
#33992
#33612
#33572
#33052
#17062
#16962
#16872
#16782
#16772
#16312
#16302
#16242
#16222
#15902
C8×SD162