Finite Group Databasebeta

← #20190

Subgroups — #20190

Order 256.

352 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#3367C211yesmaximal, characteristic, complemented
2#3353C211yesmaximal, characteristic, complemented
3D4×SD16C211yesmaximal, characteristic, complemented
4#3272C211yesmaximal, characteristic
5#1714C211yesmaximal, characteristic, complemented
6#1699C211yesmaximal, characteristic, complemented
7#1686C211yesmaximal, characteristic, complemented
8#1685C211yesmaximal, characteristic, complemented
9#1673C211yesmaximal, characteristic, complemented
10#1654C211yesmaximal, characteristic, complemented
11#1651C211yesmaximal, characteristic, complemented
12#1639C211yesmaximal, characteristic, complemented
13#1628C211yesmaximal, characteristic, complemented
14#1592C211yesmaximal, characteristic, complemented
15#1581C211yesmaximal, characteristic, complemented
16C2×AGL(1,ℤ/8)—21no—
17C22×SD16—21no—
18#556C2211yescharacteristic, complemented
19D4×Q8C2211yescharacteristic
20#547C2211yescharacteristic
21D42C2211yescharacteristic
22#500C2211yescharacteristic, complemented
23#498C2211yescharacteristic, complemented
24#492C2211yescharacteristic
25#492C2211yescharacteristic
26#489—21no—
27#487C2211yescharacteristic, complemented
28#486C2211yescharacteristic
29#484—21no—
30#482—21no—
31#477—21no—
32#475C2211yescharacteristic, complemented
33#474C2211yescharacteristic
34#474C2211yescharacteristic, complemented
35#468—21no—
36#468—21no—
37#465—21no—
38#463—21no—
39#461C2211yescharacteristic
40#461C2211yescharacteristic
41#461C2211yescharacteristic
42#460C2211yescharacteristic
43#459C2211yescharacteristic
44#450—21no—
45#450—21no—
46#449—21no—
47#448—21no—
48#446C2211yescharacteristic, complemented
49#440—21no—
50C4×SD16C2211yescharacteristic
51C4×D8C2211yescharacteristic
52#436C2211yescharacteristic
53#436C2211yescharacteristic, complemented
54C8×D4C2211yescharacteristic
55#428—21no—
56#335C2211yescharacteristic, complemented
57#335C2211yescharacteristic, complemented
58#332C2211yescharacteristic, complemented
59#332C2211yescharacteristic, complemented
60#332C2211yescharacteristic, complemented
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic, complemented
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic, complemented
66#322C2211yescharacteristic, 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—21no—
80C2×SD16—44no—
81C2×SD16D411yescharacteristic
82C2×D8—21no—
83C2×D8D411yescharacteristic
84C2×M16—21no—
85C2×M16—21no—
86C22×C8—21no—
87C4⋊Q8—21no—
88C4⋊Q8C2311yescharacteristic
89C4⋊Q8C2311yescharacteristic
90C4⋊Q8C2311yescharacteristic
91#127C2311yescharacteristic
92#126—41no—
93#124—21no—
94#123—41no—
95C22⋊Q8—21no—
96C22⋊Q8—21no—
97C22⋊Q8—21no—
98C22⋊Q8—41no—
99#121—21no—
100#121—21no—
101#121—21no—
102#120—41no—
103C4×Q8—21no—
104C4×Q8C2311yescharacteristic
105C4×D4—21no—
106C4×D4C2311yescharacteristic
107C4×D4C2311yescharacteristic
108C4×D4C2311yescharacteristic
109#117—21no—
110C2×C4⋊C4—21no—
111#107—21no—
112#107—21no—
113#107—21no—
114#107—21no—
115#107D411yescharacteristic
116#106—21no—
117#106—41no—
118#106D411yescharacteristic
119C4⋊C8—21no—
120C4⋊C8C2311yescharacteristic
121C4⋊C8C2311yescharacteristic
122C4⋊C8C2311yescharacteristic
123C4⋊C8C2311yescharacteristic
124#103—21no—
125#103—21no—
126#103—21no—
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—41no—
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—
174SD16—22no—
175SD16—42no—
176SD16—42no—
177SD16—42no—
178SD16—42no—
179SD16—42no—
180SD16—42no—
181SD16—81no—
182SD16—81no—
183D8—22no—
184D8—42no—
185M16—42no—
186M16—42no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—42no—
193C2×C8C2×D411yescharacteristic
194C2×C8C2×D411yescharacteristic
195C2×C8C2×D411yescharacteristic
196C2×C8C2×D411yescharacteristic
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4C2×D411yescharacteristic
208C4⋊C4C2×D411yescharacteristic
209C4⋊C4C2×D411yescharacteristic
210C4⋊C4C2×D411yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C22⋊C4—41no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C42—21no—
220C42—41no—
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—41no—
236Q8—41no—
237Q8—82no—
238D4—21no—
239D4—21no—
240D4—21no—
241D4—21no—
242D4—21no—
243D4—41no—
244D4—41no—
245D4—41no—
246D4—41no—
247D4—81no—
248D4—82no—
249D4—82no—
250D4—82no—
251D4AGL(1,ℤ/8)12yes—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
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—41no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4C22×D411yescharacteristic
269C2×C4C22×D411yescharacteristic
270C2×C4H(5,𝔽2)11yescharacteristic
271C8—21no—
272C8—21no—
273C8—21no—
274C8—41no—
275C8—41no—
276C8—41no—
277C8—41no—
278C8—41no—
279C8AGL(1,ℤ/8)12yescomplemented
280C22—22no—
281C22—41no—
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—
303C4C2×AGL(1,ℤ/8)11yescharacteristic
304C4C2×AGL(1,ℤ/8)11yescharacteristic
305C4#57711yescharacteristic
306C4#57711yescharacteristic
307C2—42no—
308C2—81no—
309C2—81no—
310C2—161no—
311C2#326511yescharacteristic
312C2#327211yescharacteristic
313C2#328911yescharacteristic
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. #3367C2semidirect
2. #3353C2semidirect
3. D4×SD16C2semidirect
5. #1714C2semidirect
6. #1699C2semidirect
7. #1686C2semidirect
8. #1685C2semidirect
9. #1673C2semidirect
10. #1654C2semidirect
11. #1651C2semidirect
12. #1639C2semidirect
13. #1628C2semidirect
14. #1592C2semidirect
15. #1581C2semidirect
18. #556C22semidirect
22. #500C22semidirect
23. #498C22semidirect
27. #487C22semidirect
32. #475C22semidirect
34. #474C22semidirect
48. #446C22semidirect
53. #436C22semidirect
56. #335C22semidirect
57. #335C22semidirect
58. #332C22semidirect
59. #332C22semidirect
60. #332C22semidirect
62. Q8⋊C8C22semidirect
65. D4⋊C8C22semidirect
66. #322C22semidirect
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
#33672
#33532
D4×SD162
#32722
#17142
#16992
#16862
#16852
#16732
#16542
#16512
#16392
#16282
#15922
#15812