Finite Group Databasebeta

← #19901

Subgroups — #19901

Order 256.

324 conjugacy classes of subgroups.

The subgroup lattice has 305 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3360C211yesmaximal, characteristic, complemented
2#3331C211yesmaximal, characteristic, complemented
3#3324C211yesmaximal, characteristic, complemented
4#3294C211yesmaximal, characteristic
5#1708C211yesmaximal, characteristic, complemented
6#1703C211yesmaximal, characteristic, complemented
7#1701C211yesmaximal, characteristic, complemented
8#1683C211yesmaximal, characteristic, complemented
9#1682C211yesmaximal, characteristic, complemented
10#1654C211yesmaximal, characteristic, complemented
11#1652C211yesmaximal, characteristic, complemented
12#1629C211yesmaximal, characteristic, complemented
13#1622C211yesmaximal, characteristic, complemented
14#1589C211yesmaximal, characteristic, complemented
15C8×SD16C211yesmaximal, characteristic, complemented
16#556C2211yescharacteristic
17#554C2211yescharacteristic, complemented
18#547C2211yescharacteristic
19D42C2211yescharacteristic
20#498C2211yescharacteristic, complemented
21#492C2211yescharacteristic
22#488—21no—
23#487C2211yescharacteristic, complemented
24#487C2211yescharacteristic, complemented
25#486C2211yescharacteristic
26#486C2211yescharacteristic
27#483—21no—
28#482—21no—
29#480—21no—
30#479—21no—
31#476C2211yescharacteristic
32#476C2211yescharacteristic
33#474C2211yescharacteristic
34#474C2211yescharacteristic
35#470—21no—
36#468—21no—
37#466—21no—
38#463—21no—
39#462C2211yescharacteristic, complemented
40#461C2211yescharacteristic, complemented
41#460C2211yescharacteristic
42#459C2211yescharacteristic
43#452—21no—
44#450—21no—
45#447—21no—
46C8×Q8C2211yescharacteristic, complemented
47C4×SD16—21no—
48C4×SD16C2211yescharacteristic
49C4×D8C2211yescharacteristic
50#436C2211yescharacteristic
51#436C2211yescharacteristic, complemented
52C8×D4C2211yescharacteristic
53#418—21no—
54#417—21no—
55C2×#102—21no—
56#335C2211yescharacteristic, complemented
57#334C2211yescharacteristic, complemented
58#333C2211yescharacteristic, complemented
59#332C2211yescharacteristic, complemented
60#332C2211yescharacteristic, complemented
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic, complemented
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic, complemented
66C82C2211yescharacteristic, complemented
67C2×#41—21no—
68C22×D4—21no—
69C22×D4—21no—
70C2×Q16—21no—
71C2×SD16—21no—
72C2×SD16—21no—
73C2×SD16—41no—
74C2×SD16D411yescharacteristic
75C2×D8—21no—
76C2×D8D411yescharacteristic
77C2×M16—21no—
78C2×M16—21no—
79C22×C8—21no—
80C4⋊Q8C2311yescharacteristic
81C4⋊Q8C2311yescharacteristic
82C4⋊Q8C2311yescharacteristic
83#127C2311yescharacteristic
84#126—41no—
85#125—21no—
86#124—21no—
87#123—41no—
88C22⋊Q8—21no—
89C22⋊Q8—21no—
90C22⋊Q8—21no—
91C22⋊Q8—41no—
92#121—21no—
93#121—21no—
94#121—21no—
95#120—41no—
96C4×Q8—21no—
97C4×Q8C2311yescharacteristic
98C4×D4—21no—
99C4×D4C2311yescharacteristic
100C4×D4C2311yescharacteristic
101C4×D4C2311yescharacteristic
102#117—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105#107—21no—
106#107—21no—
107#107D411yescharacteristic
108#106—21no—
109#106—21no—
110#106—41no—
111#106D411yescharacteristic
112C4⋊C8—21no—
113C4⋊C8C2311yescharacteristic
114C4⋊C8C2311yescharacteristic
115C4⋊C8C2311yescharacteristic
116C4⋊C8C2311yescharacteristic
117#103—21no—
118#103—21no—
119#103—21no—
120#103—21no—
121#103—21no—
122#103—21no—
123#103D411yescharacteristic
124#102—21no—
125#102—21no—
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#102D411yescharacteristic
135#102D411yescharacteristic
136#102D411yescharacteristic
137C22⋊C8—21no—
138C22⋊C8—21no—
139C22⋊C8—21no—
140C4×C8—21no—
141C4×C8C2311yescharacteristic
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144C24—41no—
145#41—42no—
146#41—81no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8C2×D411yescharacteristic
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—41no—
155C2×D4—41no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—41no—
159C2×D4—81no—
160C2×D4—81no—
161C2×D4C2×D411yescharacteristic
162C2×D4C2×D411yescharacteristic
163C2×D4C2×D411yescharacteristic
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—41no—
168C22×C4—41no—
169Q16—81no—
170SD16—22no—
171SD16—42no—
172SD16—82no—
173SD16—81no—
174D8—41no—
175D8—81no—
176M16—81no—
177M16—81no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—81no—
184C2×C8C2×D411yescharacteristic
185C2×C8C2×D411yescharacteristic
186C2×C8C2×D411yescharacteristic
187C2×C8C2×D411yescharacteristic
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4C2×D411yescharacteristic
200C4⋊C4C2×D411yescharacteristic
201C4⋊C4C2×D411yescharacteristic
202C4⋊C4C2×D411yescharacteristic
203C22⋊C4—21no—
204C22⋊C4—21no—
205C22⋊C4—21no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C42—21no—
212C42—41no—
213C42C2411yescharacteristic
214C23—21no—
215C23—21no—
216C23—21no—
217C23—41no—
218C23—82no—
219C23—82no—
220C23—161no—
221Q8—21no—
222Q8—21no—
223Q8—41no—
224Q8—41no—
225Q8—41no—
226Q8—41no—
227Q8—81no—
228D4—21no—
229D4—21no—
230D4—21no—
231D4—21no—
232D4—21no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—41no—
237D4—81no—
238D4—82no—
239D4—82no—
240D4—82no—
241D4#13512yes—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—82no—
257C2×C4—161no—
258C2×C4C22×D411yescharacteristic
259C2×C4C22×D411yescharacteristic
260C2×C4H(5,𝔽2)11yescharacteristic
261C8—22no—
262C8—21no—
263C8—21no—
264C8—21no—
265C8—21no—
266C8—41no—
267C8—41no—
268C8—41no—
269C8—41no—
270C22—22no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—81no—
280C22—81no—
281C22—81no—
282C22—164no—
283C22D4211yescharacteristic
284C4—21no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—81no—
291C4—81no—
292C4—81no—
293C4C2×#13511yescharacteristic
294C4#57511yescharacteristic
295C4#57611yescharacteristic
296C4#57711yescharacteristic
297C2—42no—
298C2—81no—
299C2—81no—
300C2—161no—
301C2#327511yescharacteristic
302C2#328011yescharacteristic
303C2#329111yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3294C2
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. #3331C2semidirect
3. #3324C2semidirect
5. #1708C2semidirect
6. #1703C2semidirect
7. #1701C2semidirect
8. #1683C2semidirect
9. #1682C2semidirect
10. #1654C2semidirect
11. #1652C2semidirect
12. #1629C2semidirect
13. #1622C2semidirect
14. #1589C2semidirect
15. C8×SD16C2semidirect
17. #554C22semidirect
20. #498C22semidirect
23. #487C22semidirect
24. #487C22semidirect
39. #462C22semidirect
40. #461C22semidirect
46. C8×Q8C22semidirect
51. #436C22semidirect
56. #335C22semidirect
57. #334C22semidirect
58. #333C22semidirect
59. #332C22semidirect
60. #332C22semidirect
62. Q8⋊C8C22semidirect
65. D4⋊C8C22semidirect
66. C82C22semidirect

Maximal subgroups

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

subgroupindex
#33602
#33312
#33242
#32942
#17082
#17032
#17012
#16832
#16822
#16542
#16522
#16292
#16222
#15892
C8×SD162