Finite Group Databasebeta

← #20288

Subgroups — #20288

Order 256.

335 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#3398C211yesmaximal, characteristic, complemented
2#3368C211yesmaximal, characteristic, complemented
3#3324C211yesmaximal, characteristic, complemented
4#3274C211yesmaximal, characteristic
5#1709C211yesmaximal, characteristic, complemented
6#1702C211yesmaximal, characteristic, complemented
7#1701C211yesmaximal, characteristic, complemented
8#1689C211yesmaximal, characteristic, complemented
9#1688C211yesmaximal, characteristic, complemented
10#1641C211yesmaximal, characteristic, complemented
11#1639C211yesmaximal, characteristic, complemented
12#1628C211yesmaximal, characteristic, complemented
13#1626C211yesmaximal, characteristic, complemented
14#1590C211yesmaximal, characteristic, complemented
15#1590C211yesmaximal, characteristic, complemented
16C2×AGL(1,ℤ/8)—21no—
17#557C2211yescharacteristic, complemented
18#556C2211yescharacteristic
19#550C2211yescharacteristic
20D42C2211yescharacteristic
21#497—21no—
22#494C2211yescharacteristic, complemented
23#493C2211yescharacteristic
24#490—21no—
25#486C2211yescharacteristic
26#486C2211yescharacteristic
27#486C2211yescharacteristic, complemented
28#486C2211yescharacteristic, complemented
29#482—21no—
30#478—21no—
31#474C2211yescharacteristic
32#474C2211yescharacteristic
33#469—21no—
34#468—21no—
35#464—21no—
36#462—21no—
37#462C2211yescharacteristic, complemented
38#461C2211yescharacteristic
39#460—21no—
40#460C2211yescharacteristic
41#460C2211yescharacteristic
42#460C2211yescharacteristic, complemented
43#459C2211yescharacteristic
44#450—21no—
45#449—21no—
46#447—21no—
47#446C2211yescharacteristic, complemented
48#446C2211yescharacteristic, complemented
49#441—21no—
50#440—21no—
51C4×SD16C2211yescharacteristic
52C4×SD16C2211yescharacteristic
53#436C2211yescharacteristic
54#436C2211yescharacteristic
55#418—21no—
56#334C2211yescharacteristic, complemented
57#334C2211yescharacteristic, complemented
58#332C2211yescharacteristic, complemented
59#332C2211yescharacteristic, complemented
60#331C2211yescharacteristic
61#331C2211yescharacteristic, complemented
62Q8⋊C8C2211yescharacteristic, complemented
63Q8⋊C8C2211yescharacteristic, complemented
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66C82C2211yescharacteristic, complemented
67C2×#41—21no—
68C22×D4—21no—
69C22×D4—21no—
70AGL(1,ℤ/8)—44no—
71C2×Q16—21no—
72C2×Q16—21no—
73C2×Q16—21no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×SD16—21no—
77C2×SD16—21no—
78C2×SD16D411yescharacteristic
79C2×SD16D411yescharacteristic
80C2×D8—21no—
81C2×D8—21no—
82C2×D8—21no—
83C2×M16—21no—
84C2×M16—21no—
85C4⋊Q8—21no—
86C4⋊Q8C2311yescharacteristic
87C4⋊Q8C2311yescharacteristic
88#127—21no—
89#127C2311yescharacteristic
90#127C2311yescharacteristic
91#126—41no—
92#125—21no—
93#125—41no—
94#124—21no—
95C22⋊Q8—21no—
96#121—21no—
97#121—21no—
98#121—21no—
99#121—41no—
100#120—41no—
101C4×Q8—21no—
102C4×Q8—21no—
103C4×Q8—21no—
104C4×Q8C2311yescharacteristic
105C4×Q8C2311yescharacteristic
106C4×D4—21no—
107C4×D4C2311yescharacteristic
108C4×D4C2311yescharacteristic
109#117—21no—
110#107—21no—
111#107—21no—
112#106D411yescharacteristic
113#106D411yescharacteristic
114C4⋊C8—21no—
115C4⋊C8—21no—
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120#103—21no—
121#103—21no—
122#103D411yescharacteristic
123#103D411yescharacteristic
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#102—41no—
135#102—41no—
136#102D411yescharacteristic
137#102D411yescharacteristic
138C22⋊C8—21no—
139C22⋊C8—21no—
140#97—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×Q8C2×D411yescharacteristic
151C2×Q8C2×D411yescharacteristic
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—81no—
165C2×D4—81no—
166C2×D4C2×D411yescharacteristic
167C2×D4C2×D411yescharacteristic
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—41no—
171Q16—42no—
172Q16—81no—
173Q16—81no—
174SD16—22no—
175SD16—41no—
176SD16—42no—
177SD16—42no—
178SD16—81no—
179SD16—81no—
180D8—42no—
181D8—42no—
182D8—81no—
183M16—42no—
184M16—81no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
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—41no—
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—41no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C42—21no—
215C42—21no—
216C42—41no—
217C42—41no—
218C42C2411yescharacteristic
219C23—21no—
220C23—21no—
221C23—41no—
222C23—41no—
223C23—82no—
224C23—82no—
225C23—161no—
226Q8—21no—
227Q8—21no—
228Q8—21no—
229Q8—21no—
230Q8—41no—
231Q8—41no—
232Q8—41no—
233Q8—41no—
234D4—21no—
235D4—21no—
236D4—21no—
237D4—21no—
238D4—41no—
239D4—41no—
240D4—41no—
241D4—41no—
242D4—41no—
243D4—41no—
244D4—41no—
245D4—41no—
246D4—81no—
247D4—81no—
248D4—82no—
249D4—82no—
250D4—82no—
251D4—161no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—82no—
265C2×C4C22×D411yescharacteristic
266C2×C4C22×D411yescharacteristic
267C2×C4H(5,𝔽2)11yescharacteristic
268C8—21no—
269C8—21no—
270C8—21no—
271C8—41no—
272C8—41no—
273C8—41no—
274C8—41no—
275C8—41no—
276C8AGL(1,ℤ/8)12yes—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—81no—
284C22—81no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22—81no—
289C22—164no—
290C22D4211yescharacteristic
291C4—21no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—41no—
298C4—81no—
299C4—81no—
300C4C2×AGL(1,ℤ/8)11yescharacteristic
301C4#57511yescharacteristic
302C4#57611yescharacteristic
303C4#57711yescharacteristic
304C2—81no—
305C2—81no—
306C2—161no—
307C2—161no—
308C2#327411yescharacteristic
309C2#327511yescharacteristic
310C2#329011yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3274C2
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. #3398C2semidirect
2. #3368C2semidirect
3. #3324C2semidirect
5. #1709C2semidirect
6. #1702C2semidirect
7. #1701C2semidirect
8. #1689C2semidirect
9. #1688C2semidirect
10. #1641C2semidirect
11. #1639C2semidirect
12. #1628C2semidirect
13. #1626C2semidirect
14. #1590C2semidirect
15. #1590C2semidirect
17. #557C22semidirect
22. #494C22semidirect
27. #486C22semidirect
28. #486C22semidirect
37. #462C22semidirect
42. #460C22semidirect
47. #446C22semidirect
48. #446C22semidirect
56. #334C22semidirect
57. #334C22semidirect
58. #332C22semidirect
59. #332C22semidirect
61. #331C22semidirect
62. Q8⋊C8C22semidirect
63. Q8⋊C8C22semidirect
66. C82C22semidirect

Maximal subgroups

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

subgroupindex
#33982
#33682
#33242
#32742
#17092
#17022
#17012
#16892
#16882
#16412
#16392
#16282
#16262
#15902