Finite Group Databasebeta

← #20171

Subgroups — #20171

Order 256.

334 conjugacy classes of subgroups.

The subgroup lattice has 311 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3358C211yesmaximal, characteristic, complemented
2#3332C211yesmaximal, characteristic, complemented
3#3330C211yesmaximal, characteristic, complemented
4#3276C211yesmaximal, characteristic, complemented
5#1720C211yesmaximal, characteristic, complemented
6#1701C211yesmaximal, characteristic, complemented
7#1701C211yesmaximal, characteristic, complemented
8#1689C211yesmaximal, characteristic, complemented
9#1680C211yesmaximal, characteristic, complemented
10#1654C211yesmaximal, characteristic, complemented
11#1651C211yesmaximal, characteristic, complemented
12#1641C211yesmaximal, characteristic, complemented
13#1625C211yesmaximal, characteristic, complemented
14#1594C211yesmaximal, characteristic, complemented
15#1583C211yesmaximal, characteristic, complemented
16C2×#137—21no—
17#550C2211yescharacteristic
18D4×Q8C2211yescharacteristic
19#547C2211yescharacteristic
20#547C2211yescharacteristic
21#498C2211yescharacteristic
22#494C2211yescharacteristic
23#487C2211yescharacteristic
24#486—21no—
25#486C2211yescharacteristic
26#486C2211yescharacteristic
27#486C2211yescharacteristic
28#484—21no—
29#481—21no—
30#475C2211yescharacteristic
31#474—21no—
32#474C2211yescharacteristic
33#474C2211yescharacteristic
34#470—21no—
35#469—21no—
36#468—21no—
37#462C2211yescharacteristic
38#461C2211yescharacteristic
39#461C2211yescharacteristic
40#459—21no—
41#459C2211yescharacteristic
42#459C2211yescharacteristic
43#452—21no—
44#451—21no—
45#449—21no—
46#449—21no—
47C8×Q8C2211yescharacteristic
48C4×SD16C2211yescharacteristic
49C4×D8—21no—
50C4×D8C2211yescharacteristic
51#436C2211yescharacteristic
52#436C2211yescharacteristic
53#436C2211yescharacteristic
54#417—21no—
55#417—21no—
56#334C2211yescharacteristic
57#334C2211yescharacteristic
58#332C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×#41—21no—
69C2×#41—21no—
70C22×Q8—21no—
71#137—44no—
72C2×Q16—21no—
73C2×Q16—21no—
74C2×Q16—21no—
75C2×SD16—21no—
76C2×SD16—21no—
77C2×SD16D411yescharacteristic
78C2×D8—21no—
79C2×D8—21no—
80C2×D8D411yescharacteristic
81C2×M16—21no—
82C2×M16—21no—
83C2×M16—21no—
84C4⋊Q8—21no—
85C4⋊Q8C2311yescharacteristic
86C4⋊Q8C2311yescharacteristic
87C4⋊Q8C2311yescharacteristic
88#127—21no—
89#127C2311yescharacteristic
90#123—41no—
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#121—41no—
100C4×Q8C2311yescharacteristic
101C4×D4—21no—
102C4×D4—21no—
103C4×D4C2311yescharacteristic
104C4×D4C2311yescharacteristic
105C4×D4C2311yescharacteristic
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108#107—21no—
109#107—21no—
110#107D411yescharacteristic
111#106—21no—
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#103—21no—
125#103D411yescharacteristic
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—
141C4×C8—21no—
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144C4×C8C2311yescharacteristic
145#41—42no—
146#41—42no—
147#41—42no—
148#41—81no—
149#41—81no—
150#41—81no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—41no—
155C2×Q8—42no—
156C2×Q8—81no—
157C2×Q8C2×D411yescharacteristic
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4—41no—
165C2×D4C2×D411yescharacteristic
166C2×D4C2×D411yescharacteristic
167C2×D4C2×D411yescharacteristic
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175Q16—42no—
176Q16—42no—
177Q16—81no—
178SD16—22no—
179SD16—42no—
180SD16—81no—
181D8—41no—
182D8—42no—
183D8—81no—
184M16—42no—
185M16—81no—
186M16—81no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8C2×D411yescharacteristic
193C2×C8C2×D411yescharacteristic
194C2×C8C2×D411yescharacteristic
195C2×C8C2×D411yescharacteristic
196C4⋊C4—21no—
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⋊C4—41no—
206C4⋊C4C2×D411yescharacteristic
207C4⋊C4C2×D411yescharacteristic
208C4⋊C4C2×D411yescharacteristic
209C4⋊C4C2×D411yescharacteristic
210C22⋊C4—21no—
211C22⋊C4—21no—
212C22⋊C4—21no—
213C22⋊C4—41no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C22⋊C4—41no—
217C42—21no—
218C42C2411yescharacteristic
219C23—21no—
220C23—21no—
221C23—21no—
222C23—41no—
223Q8—21no—
224Q8—41no—
225Q8—41no—
226Q8—41no—
227Q8—41no—
228Q8—41no—
229Q8—41no—
230Q8—82no—
231Q8AGL(1,ℤ/8)12yes—
232D4—21no—
233D4—21no—
234D4—21no—
235D4—21no—
236D4—21no—
237D4—21no—
238D4—41no—
239D4—41no—
240D4—41no—
241D4—41no—
242D4—82no—
243D4—81no—
244D4—82no—
245D4—82no—
246D4—161no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—82no—
262C2×C4—82no—
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—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—81no—
287C22—81no—
288C22—81no—
289C22D4211yescharacteristic
290C4—22no—
291C4—21no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—81no—
297C4—81no—
298C4—81no—
299C4C2×AGL(1,ℤ/8)11yescharacteristic
300C4#57511yescharacteristic
301C4#57611yescharacteristic
302C4#57711yescharacteristic
303C2—81no—
304C2—81no—
305C2—81no—
306C2—161no—
307C2#326411yescharacteristic
308C2#327511yescharacteristic
309C2#328911yescharacteristic
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. #3358C2semidirect
2. #3332C2semidirect
3. #3330C2semidirect
4. #3276C2semidirect
5. #1720C2semidirect
6. #1701C2semidirect
7. #1701C2semidirect
8. #1689C2semidirect
9. #1680C2semidirect
10. #1654C2semidirect
11. #1651C2semidirect
12. #1641C2semidirect
13. #1625C2semidirect
14. #1594C2semidirect
15. #1583C2semidirect

Maximal subgroups

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

subgroupindex
#33582
#33322
#33302
#32762
#17202
#17012
#16892
#16802
#16542
#16512
#16412
#16252
#15942
#15832