Finite Group Databasebeta

← #19890

Subgroups — #19890

Order 256.

359 conjugacy classes of subgroups.

The subgroup lattice has 315 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3384C211yesmaximal, characteristic, complemented
2#3335C211yesmaximal, characteristic, complemented
3D4×SD16C211yesmaximal, characteristic, complemented
4#3272C211yesmaximal, characteristic
5#1705C211yesmaximal, characteristic, complemented
6#1691C211yesmaximal, characteristic, complemented
7#1686C211yesmaximal, characteristic, complemented
8#1671C211yesmaximal, characteristic, complemented
9#1665C211yesmaximal, characteristic, complemented
10#1654C211yesmaximal, characteristic, complemented
11#1651C211yesmaximal, characteristic, complemented
12#1629C211yesmaximal, characteristic, complemented
13#1622C211yesmaximal, characteristic, complemented
14#1590C211yesmaximal, characteristic, complemented
15C8×D8C211yesmaximal, characteristic, complemented
16C2×AGL(1,ℤ/8)—21no—
17C22×SD16—21no—
18#554C2211yescharacteristic, complemented
19D4×Q8C2211yescharacteristic
20#547C2211yescharacteristic
21D42C2211yescharacteristic
22#501—21no—
23#498C2211yescharacteristic, complemented
24#498C2211yescharacteristic, complemented
25#498C2211yescharacteristic, complemented
26#492C2211yescharacteristic
27#492C2211yescharacteristic
28#492C2211yescharacteristic
29#483—21no—
30#481—21no—
31#478—21no—
32#476—21no—
33#476C2211yescharacteristic
34#476C2211yescharacteristic
35#475C2211yescharacteristic, complemented
36#474C2211yescharacteristic, complemented
37#468—21no—
38#465—21no—
39#465—21no—
40#461C2211yescharacteristic
41#461C2211yescharacteristic
42#460C2211yescharacteristic
43#459C2211yescharacteristic
44#450—21no—
45#450—21no—
46#449—21no—
47#448—21no—
48#446C2211yescharacteristic, complemented
49#442—21no—
50C4×SD16C2211yescharacteristic
51C4×D8C2211yescharacteristic
52#436C2211yescharacteristic
53C8×D4C2211yescharacteristic
54C8×D4C2211yescharacteristic, complemented
55C2×#106—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×Q8—21no—
69C22×D4—21no—
70C22×D4—21no—
71AGL(1,ℤ/8)—44no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×SD16—21no—
77C2×SD16—21no—
78C2×SD16—21no—
79C2×SD16—41no—
80C2×SD16—44no—
81C2×SD16D411yescharacteristic
82C2×D8—21no—
83C2×D8D411yescharacteristic
84C2×M16—21no—
85C22×C8—21no—
86C22×C8—21no—
87C4⋊Q8—21no—
88C4⋊Q8C2311yescharacteristic
89C4⋊Q8C2311yescharacteristic
90C4⋊Q8C2311yescharacteristic
91#127C2311yescharacteristic
92#125—21no—
93#123—41no—
94C22⋊Q8—21no—
95C22⋊Q8—21no—
96C22⋊Q8—21no—
97C22⋊Q8—41no—
98C22⋊Q8—41no—
99#121—21no—
100#121—21no—
101#121—21no—
102#120—41no—
103C4×Q8C2311yescharacteristic
104C4×D4—21no—
105C4×D4—21no—
106C4×D4C2311yescharacteristic
107C4×D4C2311yescharacteristic
108C4×D4C2311yescharacteristic
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111#107—21no—
112#107D411yescharacteristic
113#106—21no—
114#106—21no—
115#106—21no—
116#106—21no—
117#106—21no—
118#106—21no—
119#106—21no—
120#106—41no—
121#106D411yescharacteristic
122C4⋊C8—21no—
123C4⋊C8C2311yescharacteristic
124C4⋊C8C2311yescharacteristic
125C4⋊C8C2311yescharacteristic
126C4⋊C8C2311yescharacteristic
127#103—21no—
128#103—21no—
129#103—21no—
130#103D411yescharacteristic
131#102—21no—
132#102—21no—
133#102—21no—
134#102—21no—
135#102—21no—
136#102D411yescharacteristic
137#102D411yescharacteristic
138#102D411yescharacteristic
139C22⋊C8—21no—
140C22⋊C8—21no—
141C22⋊C8—21no—
142#97—21no—
143C4×C8C2311yescharacteristic
144C4×C8C2311yescharacteristic
145C4×C8C2311yescharacteristic
146C24—41no—
147#41—42no—
148#41—81no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—41no—
153C2×Q8—42no—
154C2×Q8—81no—
155C2×Q8C2×D411yescharacteristic
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—41no—
164C2×D4—81no—
165C2×D4—81no—
166C2×D4C2×D411yescharacteristic
167C2×D4C2×D411yescharacteristic
168C2×D4C2×D411yescharacteristic
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175SD16—22no—
176SD16—42no—
177SD16—42no—
178SD16—42no—
179SD16—42no—
180SD16—42no—
181SD16—81no—
182SD16—81no—
183SD16—82no—
184D8—42no—
185D8SD1614yescomplemented
186M16—42no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8C2×D411yescharacteristic
195C2×C8C2×D411yescharacteristic
196C2×C8C2×D411yescharacteristic
197C2×C8C2×D411yescharacteristic
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4C2×D411yescharacteristic
210C4⋊C4C2×D411yescharacteristic
211C4⋊C4C2×D411yescharacteristic
212C4⋊C4C2×D411yescharacteristic
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—41no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—41no—
220C42—21no—
221C42C2411yescharacteristic
222C23—21no—
223C23—21no—
224C23—21no—
225C23—41no—
226C23—82no—
227C23—82no—
228C23—161no—
229Q8—21no—
230Q8—21no—
231Q8—41no—
232Q8—41no—
233Q8—41no—
234Q8—41no—
235Q8—81no—
236Q8—82no—
237D4—21no—
238D4—21no—
239D4—21no—
240D4—21no—
241D4—41no—
242D4—41no—
243D4—41no—
244D4—41no—
245D4—81no—
246D4—82no—
247D4—82no—
248D4—82no—
249D4C2×SD1612yes—
250D4C2×SD1612yescomplemented
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—161no—
268C2×C4C22×D411yescharacteristic
269C2×C4C22×D411yescharacteristic
270C2×C4H(5,𝔽2)11yescharacteristic
271C8—22no—
272C8—21no—
273C8—21no—
274C8—41no—
275C8—41no—
276C8—41no—
277C8—41no—
278C8C2×SD1612yescomplemented
279C8AGL(1,ℤ/8)12yescomplemented
280C22—22no—
281C22—22no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—81no—
290C22—81no—
291C22—81no—
292C22—164no—
293C22D4211yescharacteristic
294C4—21no—
295C4—41no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—81no—
301C4—81no—
302C4—81no—
303C4C22×SD1611yescharacteristic
304C4C2×AGL(1,ℤ/8)11yescharacteristic
305C4#57711yescharacteristic
306C4#57711yescharacteristic
307C2—42no—
308C2—42no—
309C2—81no—
310C2—161no—
311C2#327211yescharacteristic
312C2D4×SD1611yescharacteristic
313C2#329111yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3272C2
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. #3384C2semidirect
2. #3335C2semidirect
3. D4×SD16C2semidirect
5. #1705C2semidirect
6. #1691C2semidirect
7. #1686C2semidirect
8. #1671C2semidirect
9. #1665C2semidirect
10. #1654C2semidirect
11. #1651C2semidirect
12. #1629C2semidirect
13. #1622C2semidirect
14. #1590C2semidirect
15. C8×D8C2semidirect
18. #554C22semidirect
23. #498C22semidirect
24. #498C22semidirect
25. #498C22semidirect
35. #475C22semidirect
36. #474C22semidirect
48. #446C22semidirect
54. C8×D4C22semidirect
56. #335C22semidirect
57. #334C22semidirect
58. #333C22semidirect
59. #332C22semidirect
60. #332C22semidirect
62. Q8⋊C8C22semidirect
65. D4⋊C8C22semidirect
66. C82C22semidirect
185. D8SD16semidirect
250. D4C2×SD16semidirect
278. C8C2×SD16semidirect
279. C8AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
#33842
#33352
D4×SD162
#32722
#17052
#16912
#16862
#16712
#16652
#16542
#16512
#16292
#16222
#15902
C8×D82