Finite Group Databasebeta

← #19862

Subgroups — #19862

Order 256.

301 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3354C211yesmaximal, characteristic, complemented
2#3341C211yesmaximal, characteristic, complemented
3#3341C211yesmaximal, characteristic, complemented
4#3303C211yesmaximal, characteristic, complemented
5#1723C211yesmaximal, characteristic, complemented
6#1695C211yesmaximal, characteristic, complemented
7#1683C211yesmaximal, characteristic, complemented
8#1675C211yesmaximal, characteristic, complemented
9#1671C211yesmaximal, characteristic, complemented
10#1646C211yesmaximal, characteristic, complemented
11#1635C211yesmaximal, characteristic, complemented
12#1634C211yesmaximal, characteristic, complemented
13#1634C211yesmaximal, characteristic, complemented
14#1581C211yesmaximal, characteristic, complemented
15#1580C211yesmaximal, characteristic, complemented
16#555C2211yescharacteristic
17#555C2211yescharacteristic
18#555C2211yescharacteristic
19#547C2211yescharacteristic
20#500C2211yescharacteristic
21#500C2211yescharacteristic
22#498C2211yescharacteristic
23#494C2211yescharacteristic
24#487C2211yescharacteristic
25#487C2211yescharacteristic
26#485—21no—
27#482—21no—
28#482—21no—
29#482—21no—
30#482—21no—
31#482—21no—
32#481—21no—
33#476C2211yescharacteristic
34#476C2211yescharacteristic
35#476C2211yescharacteristic
36#476C2211yescharacteristic
37#476C2211yescharacteristic
38#476C2211yescharacteristic
39#474C2211yescharacteristic
40#469—21no—
41#466—21no—
42#466—21no—
43#465—21no—
44#461C2211yescharacteristic
45#449—21no—
46C4×D8C2211yescharacteristic
47C4×D8C2211yescharacteristic
48#436C2211yescharacteristic
49C8×D4C2211yescharacteristic
50C8×D4C2211yescharacteristic
51C8×D4C2211yescharacteristic
52#428—21no—
53#427—21no—
54#427—21no—
55#416—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#333C2211yescharacteristic
59#333C2211yescharacteristic
60#333C2211yescharacteristic
61#332C2211yescharacteristic
62D4⋊C8C2211yescharacteristic
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×Q16—41no—
69C2×SD16—21no—
70C2×D8D411yescharacteristic
71C2×D8D411yescharacteristic
72C2×M16—21no—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—21no—
76C4⋊Q8C2311yescharacteristic
77C4⋊Q8C2311yescharacteristic
78C4⋊Q8C2311yescharacteristic
79C4⋊Q8C2311yescharacteristic
80#124—41no—
81#124—41no—
82#124—41no—
83#123—41no—
84C22⋊Q8—21no—
85#121—21no—
86#121—21no—
87#121—21no—
88#121—21no—
89#121—21no—
90#121—21no—
91#121—21no—
92C4×D4C2311yescharacteristic
93C4×D4C2311yescharacteristic
94C4×D4C2311yescharacteristic
95C4×D4C2311yescharacteristic
96#117—21no—
97#117—21no—
98#117—21no—
99#117—21no—
100#117—21no—
101#117—21no—
102C2×C4⋊C4—21no—
103#107—21no—
104#107—21no—
105#107—21no—
106#107—21no—
107#107—21no—
108#107D411yescharacteristic
109#107D411yescharacteristic
110#106—21no—
111#106—21no—
112#106—21no—
113#106—21no—
114#106—21no—
115#106—21no—
116#106—21no—
117#106—21no—
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121C4⋊C8C2311yescharacteristic
122#103—21no—
123#103—21no—
124#103—21no—
125#103—41no—
126#102—21no—
127#102—21no—
128#102—21no—
129#102—21no—
130#102—21no—
131#102—21no—
132#102—21no—
133#102D411yescharacteristic
134#102D411yescharacteristic
135#102D411yescharacteristic
136#102D411yescharacteristic
137C22⋊C8—21no—
138C22⋊C8—21no—
139C22⋊C8—21no—
140C22⋊C8—21no—
141C4×C8C2311yescharacteristic
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144#41—42no—
145#41—81no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×D4—41no—
151C2×D4—41no—
152C2×D4—41no—
153C2×D4—41no—
154C2×D4C2×D411yescharacteristic
155C2×D4C2×D411yescharacteristic
156C2×D4C2×D411yescharacteristic
157C2×D4C2×D411yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—41no—
163Q16—82no—
164SD16—81no—
165D8—22no—
166D8—41no—
167M16—42no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—81no—
176C2×C8C2×D411yescharacteristic
177C2×C8C2×D411yescharacteristic
178C2×C8C2×D411yescharacteristic
179C2×C8C2×D411yescharacteristic
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—41no—
189C4⋊C4C2×D411yescharacteristic
190C4⋊C4C2×D411yescharacteristic
191C4⋊C4C2×D411yescharacteristic
192C4⋊C4C2×D411yescharacteristic
193C22⋊C4—21no—
194C22⋊C4—21no—
195C22⋊C4—21no—
196C22⋊C4—21no—
197C22⋊C4—41no—
198C22⋊C4—41no—
199C22⋊C4—41no—
200C22⋊C4—41no—
201C22⋊C4—41no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C42—41no—
205C42—41no—
206C42—41no—
207C42C2411yescharacteristic
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212Q8—41no—
213Q8—41no—
214Q8—81no—
215Q8—81no—
216Q8—81no—
217D4—21no—
218D4—21no—
219D4—21no—
220D4—21no—
221D4—21no—
222D4—21no—
223D4—21no—
224D4—21no—
225D4—82no—
226D4—161no—
227D4—161no—
228D4—161no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—41no—
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—82no—
246C2×C4C22×D411yescharacteristic
247C2×C4C22×D411yescharacteristic
248C2×C4H(5,𝔽2)11yescharacteristic
249C8—21no—
250C8—21no—
251C8—21no—
252C8—41no—
253C8—41no—
254C8—41no—
255C8—41no—
256C8—41no—
257C8#13512yes—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22D4211yescharacteristic
271C4—21no—
272C4—41no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—81no—
277C4—81no—
278C4—81no—
279C4—81no—
280C4C2×#13511yescharacteristic
281C4#57511yescharacteristic
282C4#57711yescharacteristic
283C4#57811yescharacteristic
284C2—81no—
285C2—81no—
286C2—81no—
287C2—81no—
288C2#326711yescharacteristic
289C2#328311yescharacteristic
290C2#328811yescharacteristic
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. #3354C2semidirect
2. #3341C2semidirect
3. #3341C2semidirect
4. #3303C2semidirect
5. #1723C2semidirect
6. #1695C2semidirect
7. #1683C2semidirect
8. #1675C2semidirect
9. #1671C2semidirect
10. #1646C2semidirect
11. #1635C2semidirect
12. #1634C2semidirect
13. #1634C2semidirect
14. #1581C2semidirect
15. #1580C2semidirect

Maximal subgroups

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

subgroupindex
#33542
#33412
#33032
#17232
#16952
#16832
#16752
#16712
#16462
#16352
#16342
#15812
#15802