Finite Group Databasebeta

← #20026

Subgroups — #20026

Order 256.

347 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3354C211yesmaximal, characteristic, complemented
2#3303C211yesmaximal, characteristic, complemented
3#3294C211yesmaximal, characteristic, complemented
4#3272C211yesmaximal, characteristic, complemented
5#1714C211yesmaximal, characteristic, complemented
6#1703C211yesmaximal, characteristic, complemented
7#1701C211yesmaximal, characteristic, complemented
8#1671C211yesmaximal, characteristic, complemented
9#1665C211yesmaximal, characteristic, complemented
10#1647C211yesmaximal, characteristic, complemented
11#1647C211yesmaximal, characteristic, complemented
12#1629C211yesmaximal, characteristic, complemented
13#1624C211yesmaximal, characteristic, complemented
14#1586C211yesmaximal, characteristic, complemented
15#1586C211yesmaximal, characteristic, complemented
16C2×AGL(1,ℤ/8)—21no—
17#555C2211yescharacteristic, complemented
18#547C2211yescharacteristic
19#547C2211yescharacteristic
20D42C2211yescharacteristic
21#500C2211yescharacteristic, complemented
22#498C2211yescharacteristic, complemented
23#492C2211yescharacteristic
24#492C2211yescharacteristic
25#487C2211yescharacteristic, complemented
26#486C2211yescharacteristic
27#485—21no—
28#482—21no—
29#481—21no—
30#480—21no—
31#476C2211yescharacteristic
32#476C2211yescharacteristic
33#474C2211yescharacteristic
34#474C2211yescharacteristic, complemented
35#469—21no—
36#468—21no—
37#466—21no—
38#465—21no—
39#461C2211yescharacteristic
40#461C2211yescharacteristic, complemented
41#459C2211yescharacteristic
42#459C2211yescharacteristic
43#450—21no—
44#449—21no—
45#449—21no—
46#447—21no—
47C4×D8C2211yescharacteristic
48C4×D8C2211yescharacteristic
49#436C2211yescharacteristic
50#436C2211yescharacteristic, complemented
51C8×D4C2211yescharacteristic
52C8×D4C2211yescharacteristic, complemented
53#428—21no—
54#416—21no—
55C2×#102—21no—
56#334C2211yescharacteristic, complemented
57#334C2211yescharacteristic, complemented
58#333C2211yescharacteristic, complemented
59#332C2211yescharacteristic, complemented
60#332C2211yescharacteristic, complemented
61#331C2211yescharacteristic
62D4⋊C8C2211yescharacteristic
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic, complemented
65D4⋊C8C2211yescharacteristic, complemented
66#322C2211yescharacteristic, complemented
67C2×#41—21no—
68C2×#41—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—41no—
77C2×D8—21no—
78C2×D8—21no—
79C2×D8D411yescharacteristic
80C2×D8D411yescharacteristic
81C2×M16—21no—
82C2×M16—21no—
83C22×C8—21no—
84C22×C8—21no—
85C4⋊Q8C2311yescharacteristic
86C4⋊Q8C2311yescharacteristic
87C4⋊Q8C2311yescharacteristic
88#127C2311yescharacteristic
89#124—41no—
90#123—41no—
91#123—41no—
92C22⋊Q8—21no—
93C22⋊Q8—21no—
94#121—21no—
95#121—21no—
96#121—21no—
97#121—21no—
98#121—21no—
99#121—21no—
100#120—41no—
101C4×D4C2311yescharacteristic
102C4×D4C2311yescharacteristic
103C4×D4C2311yescharacteristic
104C4×D4C2311yescharacteristic
105#117—21no—
106#117—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109#107—21no—
110#107—21no—
111#107—41no—
112#107D411yescharacteristic
113#107D411yescharacteristic
114#106—21no—
115#106—21no—
116#106—21no—
117#106—21no—
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121C4⋊C8C2311yescharacteristic
122#103—21no—
123#103—21no—
124#103—21no—
125#103—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
137#102D411yescharacteristic
138C22⋊C8—21no—
139C22⋊C8—21no—
140C22⋊C8—21no—
141C22⋊C8—21no—
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144C4×C8C2311yescharacteristic
145C24—41no—
146#41—42no—
147#41—42no—
148#41—81no—
149#41—81no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—81no—
164C2×D4—81no—
165C2×D4C2×D411yescharacteristic
166C2×D4C2×D411yescharacteristic
167C2×D4C2×D411yescharacteristic
168C2×D4C2×D411yescharacteristic
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—41no—
174C22×C4—41no—
175SD16—42no—
176SD16—42no—
177SD16—81no—
178SD16—82no—
179SD16—81no—
180D8—22no—
181D8—41no—
182D8—42no—
183D8—81no—
184M16—42no—
185M16—42no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—81no—
192C2×C8—81no—
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—41no—
204C4⋊C4—41no—
205C4⋊C4C2×D411yescharacteristic
206C4⋊C4C2×D411yescharacteristic
207C4⋊C4C2×D411yescharacteristic
208C4⋊C4C2×D411yescharacteristic
209C22⋊C4—21no—
210C22⋊C4—21no—
211C22⋊C4—21no—
212C22⋊C4—21no—
213C22⋊C4—41no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C22⋊C4—41no—
217C22⋊C4—41no—
218C42—41no—
219C42C2411yescharacteristic
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—41no—
225C23—82no—
226C23—82no—
227C23—161no—
228Q8—41no—
229Q8—41no—
230Q8—41no—
231Q8—41no—
232Q8—81no—
233D4—21no—
234D4—21no—
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—
248D4—82no—
249D4—161no—
250D4AGL(1,ℤ/8)12yes—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
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—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—22no—
272C8—21no—
273C8—21no—
274C8—21no—
275C8—21no—
276C8—41no—
277C8—41no—
278C8—41no—
279C8—41no—
280C22—22no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—81no—
293C22—81no—
294C22—81no—
295C22—164no—
296C22D4211yescharacteristic
297C4—21no—
298C4—41no—
299C4—41no—
300C4—41no—
301C4—41no—
302C4—81no—
303C4—81no—
304C4—81no—
305C4C2×AGL(1,ℤ/8)11yescharacteristic
306C4#57511yescharacteristic
307C4#57611yescharacteristic
308C4#57711yescharacteristic
309C2—42no—
310C2—81no—
311C2—81no—
312C2—81no—
313C2—161no—
314C2#326411yescharacteristic
315C2#326711yescharacteristic
316C2#329111yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
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. #3354C2semidirect
2. #3303C2semidirect
3. #3294C2semidirect
4. #3272C2semidirect
5. #1714C2semidirect
6. #1703C2semidirect
7. #1701C2semidirect
8. #1671C2semidirect
9. #1665C2semidirect
10. #1647C2semidirect
11. #1647C2semidirect
12. #1629C2semidirect
13. #1624C2semidirect
14. #1586C2semidirect
15. #1586C2semidirect
17. #555C22semidirect
21. #500C22semidirect
22. #498C22semidirect
25. #487C22semidirect
34. #474C22semidirect
40. #461C22semidirect
50. #436C22semidirect
52. C8×D4C22semidirect
56. #334C22semidirect
57. #334C22semidirect
58. #333C22semidirect
59. #332C22semidirect
60. #332C22semidirect
64. D4⋊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
#33542
#33032
#32942
#32722
#17142
#17032
#17012
#16712
#16652
#16472
#16292
#16242
#15862