Finite Group Databasebeta

← #19774

Subgroups — #19774

Order 256.

325 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3380C211yesmaximal, characteristic, complemented
2#3331C211yesmaximal, characteristic, complemented
3#3295C211yesmaximal, characteristic, complemented
4#3294C211yesmaximal, characteristic
5#1719C211yesmaximal, characteristic, complemented
6#1697C211yesmaximal, characteristic, complemented
7#1683C211yesmaximal, characteristic, complemented
8#1682C211yesmaximal, characteristic, complemented
9#1675C211yesmaximal, characteristic, complemented
10#1644C211yesmaximal, characteristic, complemented
11#1635C211yesmaximal, characteristic, complemented
12#1629C211yesmaximal, characteristic, complemented
13#1622C211yesmaximal, characteristic, complemented
14#1581C211yesmaximal, characteristic, complemented
15SD16⋊C8C211yesmaximal, characteristic, complemented
16#554C2211yescharacteristic
17#554C2211yescharacteristic, complemented
18#547C2211yescharacteristic
19D42C2211yescharacteristic
20#500C2211yescharacteristic, complemented
21#499—21no—
22#492C2211yescharacteristic
23#487C2211yescharacteristic, complemented
24#487C2211yescharacteristic, complemented
25#486C2211yescharacteristic
26#486C2211yescharacteristic
27#483—21no—
28#481—21no—
29#480—21no—
30#478—21no—
31#478—21no—
32#476C2211yescharacteristic
33#476C2211yescharacteristic
34#476C2211yescharacteristic
35#476C2211yescharacteristic, complemented
36#474C2211yescharacteristic
37#470—21no—
38#466—21no—
39#465—21no—
40#462C2211yescharacteristic, complemented
41#460C2211yescharacteristic
42#459C2211yescharacteristic
43#452—21no—
44#450—21no—
45#447—21no—
46C8×Q8C2211yescharacteristic, complemented
47C4×SD16C2211yescharacteristic
48C4×D8—21no—
49C4×D8C2211yescharacteristic
50#436C2211yescharacteristic, complemented
51C8×D4C2211yescharacteristic
52C8×D4C2211yescharacteristic
53#417—21no—
54C2×#102—21no—
55C2×#102—21no—
56#335C2211yescharacteristic, complemented
57#335C2211yescharacteristic, complemented
58#333C2211yescharacteristic, complemented
59#333C2211yescharacteristic, complemented
60#332C2211yescharacteristic, complemented
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic, complemented
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic, complemented
66#322C2211yescharacteristic, complemented
67C2×#41—21no—
68C22×D4—21no—
69C22×D4—21no—
70C2×Q16—21no—
71C2×SD16—21no—
72C2×SD16—41no—
73C2×SD16D411yescharacteristic
74C2×D8—21no—
75C2×D8D411yescharacteristic
76C2×M16—21no—
77C22×C8—21no—
78C22×C8—21no—
79C4⋊Q8C2311yescharacteristic
80C4⋊Q8C2311yescharacteristic
81C4⋊Q8C2311yescharacteristic
82#127C2311yescharacteristic
83#125—21no—
84#125—21no—
85#123—41no—
86C22⋊Q8—21no—
87C22⋊Q8—21no—
88C22⋊Q8—21no—
89C22⋊Q8—41no—
90C22⋊Q8—41no—
91#121—21no—
92#121—21no—
93#121—21no—
94#120—41no—
95C4×Q8C2311yescharacteristic
96C4×D4—21no—
97C4×D4—21no—
98C4×D4C2311yescharacteristic
99C4×D4C2311yescharacteristic
100C4×D4C2311yescharacteristic
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104#107—21no—
105#107—21no—
106#107—21no—
107#107D411yescharacteristic
108#106—21no—
109#106—21no—
110#106—21no—
111#106—21no—
112#106D411yescharacteristic
113C4⋊C8—21no—
114C4⋊C8C2311yescharacteristic
115C4⋊C8C2311yescharacteristic
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118#103—21no—
119#103—21no—
120#103—21no—
121#103—41no—
122#103D411yescharacteristic
123#102—21no—
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—
169C22×C4—41no—
170Q16—81no—
171SD16—22no—
172SD16—81no—
173SD16—82no—
174D8—22no—
175D8—81no—
176M16—81no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—81no—
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⋊C4—41no—
200C4⋊C4C2×D411yescharacteristic
201C4⋊C4C2×D411yescharacteristic
202C4⋊C4C2×D411yescharacteristic
203C4⋊C4C2×D411yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—21no—
206C22⋊C4—21no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C42—21no—
212C42C2411yescharacteristic
213C23—21no—
214C23—21no—
215C23—21no—
216C23—41no—
217C23—82no—
218C23—82no—
219C23—161no—
220Q8—21no—
221Q8—21no—
222Q8—41no—
223Q8—41no—
224Q8—81no—
225Q8—81no—
226D4—21no—
227D4—21no—
228D4—21no—
229D4—21no—
230D4—41no—
231D4—41no—
232D4—41no—
233D4—41no—
234D4—81no—
235D4—82no—
236D4—82no—
237D4—82no—
238D4#13512yes—
239D4#13712yes—
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—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—82no—
255C2×C4—161no—
256C2×C4—161no—
257C2×C4C22×D411yescharacteristic
258C2×C4C22×D411yescharacteristic
259C2×C4H(5,𝔽2)11yescharacteristic
260C8—21no—
261C8—21no—
262C8—21no—
263C8—21no—
264C8—41no—
265C8—41no—
266C8—41no—
267C8—41no—
268C8—41no—
269C22—22no—
270C22—22no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22—164no—
282C22D4211yescharacteristic
283C4—21no—
284C4—41no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—81no—
290C4—81no—
291C4—81no—
292C4C2×#13511yescharacteristic
293C4C2×#13711yescharacteristic
294C4#57611yescharacteristic
295C4#57711yescharacteristic
296C2—42no—
297C2—42no—
298C2—81no—
299C2—161no—
300C2#326611yescharacteristic
301C2#328011yescharacteristic
302C2#328811yescharacteristic
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. #3380C2semidirect
2. #3331C2semidirect
3. #3295C2semidirect
5. #1719C2semidirect
6. #1697C2semidirect
7. #1683C2semidirect
8. #1682C2semidirect
9. #1675C2semidirect
10. #1644C2semidirect
11. #1635C2semidirect
12. #1629C2semidirect
13. #1622C2semidirect
14. #1581C2semidirect
15. SD16⋊C8C2semidirect
17. #554C22semidirect
20. #500C22semidirect
23. #487C22semidirect
24. #487C22semidirect
35. #476C22semidirect
40. #462C22semidirect
46. C8×Q8C22semidirect
50. #436C22semidirect
56. #335C22semidirect
57. #335C22semidirect
58. #333C22semidirect
59. #333C22semidirect
60. #332C22semidirect
62. Q8⋊C8C22semidirect
65. D4⋊C8C22semidirect
66. #322C22semidirect

Maximal subgroups

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

subgroupindex
#33802
#33312
#32952
#32942
#17192
#16972
#16832
#16822
#16752
#16442
#16352
#16292
#16222
#15812
SD16⋊C82