Finite Group Databasebeta

← #20161

Subgroups — #20161

Order 256.

339 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3367C211yesmaximal, characteristic, complemented
2#3324C211yesmaximal, characteristic, complemented
3#3294C211yesmaximal, characteristic
4#3276C211yesmaximal, characteristic, complemented
5#1720C211yesmaximal, characteristic, complemented
6#1702C211yesmaximal, characteristic, complemented
7#1701C211yesmaximal, characteristic, complemented
8#1689C211yesmaximal, characteristic, complemented
9#1679C211yesmaximal, characteristic, complemented
10#1654C211yesmaximal, characteristic, complemented
11#1652C211yesmaximal, characteristic, complemented
12#1639C211yesmaximal, characteristic, complemented
13#1627C211yesmaximal, characteristic, complemented
14#1594C211yesmaximal, characteristic, complemented
15#1582C211yesmaximal, characteristic, complemented
16C2×#137—21no—
17#556C2211yescharacteristic
18D4×Q8C2211yescharacteristic, complemented
19#547C2211yescharacteristic
20D42C2211yescharacteristic
21#498C2211yescharacteristic, complemented
22#494C2211yescharacteristic, complemented
23#489—21no—
24#487C2211yescharacteristic, complemented
25#486C2211yescharacteristic
26#486C2211yescharacteristic
27#486C2211yescharacteristic
28#482—21no—
29#480—21no—
30#477—21no—
31#474C2211yescharacteristic
32#474C2211yescharacteristic
33#474C2211yescharacteristic
34#470—21no—
35#468—21no—
36#466—21no—
37#463—21no—
38#462C2211yescharacteristic, complemented
39#461C2211yescharacteristic
40#461C2211yescharacteristic, complemented
41#460C2211yescharacteristic
42#459C2211yescharacteristic
43#452—21no—
44#451—21no—
45#450—21no—
46#447—21no—
47#446C2211yescharacteristic, complemented
48#440—21no—
49C4×SD16C2211yescharacteristic
50C4×D8C2211yescharacteristic
51#436C2211yescharacteristic
52#436C2211yescharacteristic, complemented
53C8×D4C2211yescharacteristic
54#418—21no—
55C2×#102—21no—
56#334C2211yescharacteristic, complemented
57#334C2211yescharacteristic, 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—
71#137—44no—
72C2×Q16—21no—
73C2×Q16—21no—
74C2×Q16—21no—
75C2×SD16—21no—
76C2×SD16—21no—
77C2×SD16—21no—
78C2×SD16D411yescharacteristic
79C2×D8—21no—
80C2×D8D411yescharacteristic
81C2×M16—21no—
82C2×M16—21no—
83C22×C8—21no—
84C4⋊Q8—21no—
85C4⋊Q8C2311yescharacteristic
86C4⋊Q8C2311yescharacteristic
87C4⋊Q8C2311yescharacteristic
88#127C2311yescharacteristic
89#126—41no—
90#124—21no—
91#123—41no—
92C22⋊Q8—21no—
93C22⋊Q8—21no—
94C22⋊Q8—21no—
95C22⋊Q8—41no—
96#121—21no—
97#121—21no—
98#121—21no—
99#120—41no—
100C4×Q8—21no—
101C4×Q8C2311yescharacteristic
102C4×D4—21no—
103C4×D4C2311yescharacteristic
104C4×D4C2311yescharacteristic
105C4×D4C2311yescharacteristic
106#117—21no—
107C2×C4⋊C4—21no—
108#107—21no—
109#107—21no—
110#107—21no—
111#107D411yescharacteristic
112#106—41no—
113#106D411yescharacteristic
114C4⋊C8—21no—
115C4⋊C8C2311yescharacteristic
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119#103—21no—
120#103—21no—
121#103—21no—
122#103—21no—
123#103—21no—
124#103D411yescharacteristic
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#102—41no—
135#102D411yescharacteristic
136#102D411yescharacteristic
137#102D411yescharacteristic
138C22⋊C8—21no—
139C22⋊C8—21no—
140C22⋊C8—21no—
141#97—21no—
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144C4×C8C2311yescharacteristic
145C24—41no—
146#41—42no—
147#41—81no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—41no—
152C2×Q8—42no—
153C2×Q8—81no—
154C2×Q8C2×D411yescharacteristic
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—41no—
163C2×D4—81no—
164C2×D4—81no—
165C2×D4C2×D411yescharacteristic
166C2×D4C2×D411yescharacteristic
167C2×D4C2×D411yescharacteristic
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—41no—
172C22×C4—41no—
173Q16—42no—
174Q16—42no—
175Q16—81no—
176SD16—22no—
177SD16—42no—
178SD16—42no—
179SD16—81no—
180D8—41no—
181D8—81no—
182M16—42no—
183M16—81no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—81no—
190C2×C8C2×D411yescharacteristic
191C2×C8C2×D411yescharacteristic
192C2×C8C2×D411yescharacteristic
193C2×C8C2×D411yescharacteristic
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4C2×D411yescharacteristic
205C4⋊C4C2×D411yescharacteristic
206C4⋊C4C2×D411yescharacteristic
207C4⋊C4C2×D411yescharacteristic
208C22⋊C4—21no—
209C22⋊C4—21no—
210C22⋊C4—21no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C42—21no—
217C42—41no—
218C42C2411yescharacteristic
219C23—21no—
220C23—21no—
221C23—21no—
222C23—41no—
223C23—82no—
224C23—82no—
225C23—161no—
226Q8—21no—
227Q8—21no—
228Q8—41no—
229Q8—41no—
230Q8—41no—
231Q8—41no—
232Q8—41no—
233Q8—41no—
234Q8—82no—
235D4—21no—
236D4—21no—
237D4—21no—
238D4—21no—
239D4—21no—
240D4—41no—
241D4—41no—
242D4—41no—
243D4—41no—
244D4—81no—
245D4—82no—
246D4—82no—
247D4—82no—
248D4AGL(1,ℤ/8)12yes—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4C22×D411yescharacteristic
266C2×C4C22×D411yescharacteristic
267C2×C4H(5,𝔽2)11yescharacteristic
268C8—21no—
269C8—21no—
270C8—21no—
271C8—21no—
272C8—41no—
273C8—41no—
274C8—41no—
275C8—41no—
276C8—41no—
277C22—22no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—81no—
287C22—81no—
288C22—81no—
289C22—164no—
290C22D4211yescharacteristic
291C4—21no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—81no—
298C4—81no—
299C4—81no—
300C4C2×AGL(1,ℤ/8)11yescharacteristic
301C4#57511yescharacteristic
302C4#57611yescharacteristic
303C4#57711yescharacteristic
304C2—42no—
305C2—81no—
306C2—81no—
307C2—161no—
308C2#326411yescharacteristic
309C2#327511yescharacteristic
310C2#328911yescharacteristic
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. #3367C2semidirect
2. #3324C2semidirect
4. #3276C2semidirect
5. #1720C2semidirect
6. #1702C2semidirect
7. #1701C2semidirect
8. #1689C2semidirect
9. #1679C2semidirect
10. #1654C2semidirect
11. #1652C2semidirect
12. #1639C2semidirect
13. #1627C2semidirect
14. #1594C2semidirect
15. #1582C2semidirect
18. D4×Q8C22semidirect
21. #498C22semidirect
22. #494C22semidirect
24. #487C22semidirect
38. #462C22semidirect
40. #461C22semidirect
47. #446C22semidirect
52. #436C22semidirect
56. #334C22semidirect
57. #334C22semidirect
58. #332C22semidirect
59. #332C22semidirect
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
#33672
#33242
#32942
#32762
#17202
#17022
#17012
#16892
#16792
#16542
#16522
#16392
#16272
#15942
#15822