Finite Group Databasebeta

← #19863

Subgroups — #19863

Order 256.

297 conjugacy classes of subgroups.

The subgroup lattice has 289 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3393C211yesmaximal, characteristic, complemented
2#3353C211yesmaximal, characteristic, complemented
3#3340C211yesmaximal, characteristic, complemented
4#3303C211yesmaximal, characteristic
5#1723C211yesmaximal, characteristic, complemented
6#1691C211yesmaximal, characteristic, complemented
7#1684C211yesmaximal, characteristic, complemented
8#1673C211yesmaximal, characteristic, complemented
9#1669C211yesmaximal, characteristic, complemented
10#1650C211yesmaximal, characteristic, complemented
11#1647C211yesmaximal, characteristic, complemented
12#1646C211yesmaximal, characteristic, complemented
13#1638C211yesmaximal, characteristic, complemented
14#1581C211yesmaximal, characteristic, complemented
15SD16⋊C8C211yesmaximal, characteristic, complemented
16#556C2211yescharacteristic
17#556C2211yescharacteristic
18#555C2211yescharacteristic
19#547C2211yescharacteristic
20#500C2211yescharacteristic
21#500C2211yescharacteristic
22#498C2211yescharacteristic
23#495—21no—
24#494C2211yescharacteristic
25#487C2211yescharacteristic
26#487C2211yescharacteristic
27#485—21no—
28#485—21no—
29#484—21no—
30#482—21no—
31#482—21no—
32#477C2211yescharacteristic
33#476C2211yescharacteristic
34#475C2211yescharacteristic
35#474C2211yescharacteristic
36#474C2211yescharacteristic
37#468—21no—
38#465—21no—
39#465—21no—
40#464—21no—
41#464—21no—
42#461C2211yescharacteristic
43#461C2211yescharacteristic
44#461C2211yescharacteristic
45#449—21no—
46C8×Q8C2211yescharacteristic
47C4×Q16—21no—
48C4×SD16C2211yescharacteristic
49C4×D8C2211yescharacteristic
50#436C2211yescharacteristic
51C8×D4C2211yescharacteristic
52C8×D4C2211yescharacteristic
53#428—21no—
54#427—21no—
55#416—21no—
56#335C2211yescharacteristic
57#335C2211yescharacteristic
58#333C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#332C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×Q16—21no—
69C2×Q16—21no—
70C2×SD16—21no—
71C2×SD16—21no—
72C2×SD16—21no—
73C2×SD16D411yescharacteristic
74C2×D8D411yescharacteristic
75C2×M16—21no—
76C22×C8—21no—
77C22×C8—21no—
78C4⋊Q8C2311yescharacteristic
79C4⋊Q8C2311yescharacteristic
80C4⋊Q8C2311yescharacteristic
81C4⋊Q8C2311yescharacteristic
82#126—41no—
83#126—41no—
84#124—21no—
85#124—21no—
86#124—41no—
87#123—41no—
88C22⋊Q8—21no—
89C22⋊Q8—21no—
90C22⋊Q8—21no—
91#121—21no—
92#121—21no—
93#121—21no—
94C4×Q8—21no—
95C4×Q8—21no—
96C4×Q8C2311yescharacteristic
97C4×D4C2311yescharacteristic
98C4×D4C2311yescharacteristic
99C4×D4C2311yescharacteristic
100#117—21no—
101#117—21no—
102#117—21no—
103#117—21no—
104C2×C4⋊C4—21no—
105#107—21no—
106#107—21no—
107#107—21no—
108#107—21no—
109#107—21no—
110#107—41no—
111#107D411yescharacteristic
112#106—21no—
113#106—21no—
114#106—21no—
115#106—21no—
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—41no—
130#103D411yescharacteristic
131#102—21no—
132#102—21no—
133#102—21no—
134#102D411yescharacteristic
135#102D411yescharacteristic
136#102D411yescharacteristic
137C22⋊C8—21no—
138C22⋊C8—21no—
139C22⋊C8—21no—
140C4×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×Q8C2×D411yescharacteristic
151C2×D4—41no—
152C2×D4—41no—
153C2×D4C2×D411yescharacteristic
154C2×D4C2×D411yescharacteristic
155C2×D4C2×D411yescharacteristic
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—41no—
160Q16—42no—
161Q16—81no—
162SD16—22no—
163SD16—81no—
164SD16—81no—
165SD16—81no—
166D8—41no—
167M16—42no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—42no—
174C2×C8—81no—
175C2×C8C2×D411yescharacteristic
176C2×C8C2×D411yescharacteristic
177C2×C8C2×D411yescharacteristic
178C2×C8C2×D411yescharacteristic
179C4⋊C4—21no—
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⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4C2×D411yescharacteristic
192C4⋊C4C2×D411yescharacteristic
193C4⋊C4C2×D411yescharacteristic
194C4⋊C4C2×D411yescharacteristic
195C22⋊C4—21no—
196C22⋊C4—21no—
197C22⋊C4—21no—
198C22⋊C4—41no—
199C22⋊C4—41no—
200C22⋊C4—41no—
201C22⋊C4—41no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C42—21no—
206C42—41no—
207C42—41no—
208C42—41no—
209C42C2411yescharacteristic
210C23—21no—
211C23—21no—
212C23—21no—
213Q8—21no—
214Q8—21no—
215Q8—41no—
216Q8—41no—
217Q8—41no—
218Q8—41no—
219Q8—41no—
220Q8—41no—
221Q8—81no—
222D4—21no—
223D4—21no—
224D4—21no—
225D4—21no—
226D4—21no—
227D4—21no—
228D4—82no—
229D4—161no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
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—
267C22D4211yescharacteristic
268C4—21no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—41no—
273C4—41no—
274C4—81no—
275C4—81no—
276C4—81no—
277C4—81no—
278C4C2×#13511yescharacteristic
279C4#57511yescharacteristic
280C4#57611yescharacteristic
281C4#57811yescharacteristic
282C2—81no—
283C2—81no—
284C2—81no—
285C2#326811yescharacteristic
286C2#328311yescharacteristic
287C2#328911yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3303C2
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. #3393C2semidirect
2. #3353C2semidirect
3. #3340C2semidirect
5. #1723C2semidirect
6. #1691C2semidirect
7. #1684C2semidirect
8. #1673C2semidirect
9. #1669C2semidirect
10. #1650C2semidirect
11. #1647C2semidirect
12. #1646C2semidirect
13. #1638C2semidirect
14. #1581C2semidirect
15. SD16⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
#33932
#33532
#33402
#33032
#17232
#16912
#16842
#16732
#16692
#16502
#16472
#16462
#16382
#15812
SD16⋊C82