Finite Group Databasebeta

← #19864

Subgroups — #19864

Order 256.

289 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3360C211yesmaximal, characteristic, complemented
2#3357C211yesmaximal, characteristic, complemented
3#3341C211yesmaximal, characteristic
4#3340C211yesmaximal, characteristic, complemented
5#1723C211yesmaximal, characteristic, complemented
6#1691C211yesmaximal, characteristic, complemented
7#1683C211yesmaximal, characteristic, complemented
8#1676C211yesmaximal, characteristic, complemented
9#1675C211yesmaximal, characteristic, complemented
10#1646C211yesmaximal, characteristic, complemented
11#1638C211yesmaximal, characteristic, complemented
12#1634C211yesmaximal, characteristic, complemented
13#1632C211yesmaximal, characteristic, complemented
14#1581C211yesmaximal, characteristic, complemented
15SD16⋊C8C211yesmaximal, characteristic, complemented
16#556C2211yescharacteristic
17#555C2211yescharacteristic
18#555C2211yescharacteristic
19#554C2211yescharacteristic
20#500C2211yescharacteristic
21#500C2211yescharacteristic
22#498C2211yescharacteristic
23#494C2211yescharacteristic
24#488—21no—
25#487C2211yescharacteristic
26#487C2211yescharacteristic
27#485—21no—
28#485—21no—
29#482—21no—
30#482—21no—
31#482—21no—
32#480—21no—
33#479—21no—
34#477C2211yescharacteristic
35#477C2211yescharacteristic
36#476C2211yescharacteristic
37#476C2211yescharacteristic
38#476C2211yescharacteristic
39#476C2211yescharacteristic
40#474C2211yescharacteristic
41#468—21no—
42#466—21no—
43#465—21no—
44#463—21no—
45#461C2211yescharacteristic
46C8×Q8C2211yescharacteristic
47C4×SD16—21no—
48C4×SD16C2211yescharacteristic
49C4×D8C2211yescharacteristic
50#436C2211yescharacteristic
51C8×D4C2211yescharacteristic
52C8×D4C2211yescharacteristic
53#428—21no—
54#427—21no—
55#427—21no—
56#335C2211yescharacteristic
57#335C2211yescharacteristic
58#333C2211yescharacteristic
59#333C2211yescharacteristic
60#333C2211yescharacteristic
61#332C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×Q16—41no—
68C2×SD16—21no—
69C2×SD16D411yescharacteristic
70C2×D8D411yescharacteristic
71C2×M16—21no—
72C22×C8—21no—
73C22×C8—21no—
74C4⋊Q8C2311yescharacteristic
75C4⋊Q8C2311yescharacteristic
76C4⋊Q8C2311yescharacteristic
77C4⋊Q8C2311yescharacteristic
78#126—41no—
79#125—21no—
80#124—21no—
81#124—41no—
82#124—41no—
83C22⋊Q8—21no—
84C22⋊Q8—21no—
85C22⋊Q8—41no—
86#121—21no—
87#121—21no—
88#121—21no—
89#121—21no—
90C4×Q8—21no—
91C4×Q8C2311yescharacteristic
92C4×D4—21no—
93C4×D4C2311yescharacteristic
94C4×D4C2311yescharacteristic
95C4×D4C2311yescharacteristic
96#117—21no—
97#117—21no—
98#117—21no—
99#117—21no—
100#117—21no—
101C2×C4⋊C4—21no—
102#107—21no—
103#107—21no—
104#107—21no—
105#107—21no—
106#107—21no—
107#107—21no—
108#107—21no—
109#107D411yescharacteristic
110#106—21no—
111#106—21no—
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—41no—
128#103D411yescharacteristic
129#102—21no—
130#102—21no—
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
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8C2×D411yescharacteristic
149C2×D4—41no—
150C2×D4—41no—
151C2×D4C2×D411yescharacteristic
152C2×D4C2×D411yescharacteristic
153C2×D4C2×D411yescharacteristic
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—41no—
158Q16—82no—
159SD16—22no—
160SD16—42no—
161D8—41no—
162M16—42no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—42no—
169C2×C8—42no—
170C2×C8C2×D411yescharacteristic
171C2×C8C2×D411yescharacteristic
172C2×C8C2×D411yescharacteristic
173C2×C8C2×D411yescharacteristic
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4C2×D411yescharacteristic
187C4⋊C4C2×D411yescharacteristic
188C4⋊C4C2×D411yescharacteristic
189C4⋊C4C2×D411yescharacteristic
190C22⋊C4—21no—
191C22⋊C4—21no—
192C22⋊C4—21no—
193C22⋊C4—41no—
194C22⋊C4—41no—
195C22⋊C4—41no—
196C22⋊C4—41no—
197C22⋊C4—41no—
198C22⋊C4—41no—
199C22⋊C4—41no—
200C42—21no—
201C42—41no—
202C42—41no—
203C42—41no—
204C42C2411yescharacteristic
205C23—21no—
206C23—21no—
207C23—21no—
208Q8—21no—
209Q8—21no—
210Q8—41no—
211Q8—41no—
212Q8—81no—
213Q8—81no—
214Q8—81no—
215D4—21no—
216D4—21no—
217D4—21no—
218D4—21no—
219D4—21no—
220D4—21no—
221D4—161no—
222D4—161no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—161no—
239C2×C4C22×D411yescharacteristic
240C2×C4C22×D411yescharacteristic
241C2×C4H(5,𝔽2)11yescharacteristic
242C8—21no—
243C8—21no—
244C8—21no—
245C8—41no—
246C8—41no—
247C8—41no—
248C8—41no—
249C8—41no—
250C8#13512yes—
251C22—41no—
252C22—41no—
253C22—41no—
254C22—41no—
255C22—41no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22D4211yescharacteristic
261C4—21no—
262C4—41no—
263C4—41no—
264C4—41no—
265C4—41no—
266C4—41no—
267C4—81no—
268C4—81no—
269C4—81no—
270C4—81no—
271C4C2×#13511yescharacteristic
272C4#57511yescharacteristic
273C4#57811yescharacteristic
274C4#57811yescharacteristic
275C2—81no—
276C2—81no—
277C2—81no—
278C2#326911yescharacteristic
279C2#328311yescharacteristic
280C2#328811yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3341C2
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. #3360C2semidirect
2. #3357C2semidirect
4. #3340C2semidirect
5. #1723C2semidirect
6. #1691C2semidirect
7. #1683C2semidirect
8. #1676C2semidirect
9. #1675C2semidirect
10. #1646C2semidirect
11. #1638C2semidirect
12. #1634C2semidirect
13. #1632C2semidirect
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
#33602
#33572
#33412
#33402
#17232
#16912
#16832
#16762
#16752
#16462
#16382
#16342
#16322
#15812
SD16⋊C82