Finite Group Databasebeta

← #20063

Subgroups — #20063

Order 256.

344 conjugacy classes of subgroups.

The subgroup lattice has 321 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3391C211yesmaximal, characteristic, complemented
2#3332C211yesmaximal, characteristic, complemented
3#3330C211yesmaximal, characteristic, complemented
4#3284C211yesmaximal, characteristic, complemented
5#1717C211yesmaximal, characteristic, complemented
6#1687C211yesmaximal, characteristic, complemented
7#1685C211yesmaximal, characteristic, complemented
8#1681C211yesmaximal, characteristic, complemented
9#1680C211yesmaximal, characteristic, complemented
10#1642C211yesmaximal, characteristic, complemented
11#1641C211yesmaximal, characteristic, complemented
12#1625C211yesmaximal, characteristic, complemented
13#1625C211yesmaximal, characteristic, complemented
14#1588C211yesmaximal, characteristic, complemented
15#1586C211yesmaximal, characteristic, complemented
16C2×#135—21no—
17#550C2211yescharacteristic
18#550C2211yescharacteristic
19#547C2211yescharacteristic
20#547C2211yescharacteristic
21#493—21no—
22#493C2211yescharacteristic
23#492C2211yescharacteristic
24#487C2211yescharacteristic
25#486C2211yescharacteristic
26#486C2211yescharacteristic
27#486C2211yescharacteristic
28#484—21no—
29#481—21no—
30#475C2211yescharacteristic
31#474C2211yescharacteristic
32#469—21no—
33#468—21no—
34#465—21no—
35#462C2211yescharacteristic
36#461C2211yescharacteristic
37#459—21no—
38#459—21no—
39#459C2211yescharacteristic
40#459C2211yescharacteristic
41#459C2211yescharacteristic
42#459C2211yescharacteristic
43#452—21no—
44#449—21no—
45#449—21no—
46#449—21no—
47C8×Q8C2211yescharacteristic
48C4×SD16C2211yescharacteristic
49C4×D8—21no—
50C4×D8C2211yescharacteristic
51#436C2211yescharacteristic
52#436C2211yescharacteristic
53C8×D4C2211yescharacteristic
54#417—21no—
55#417—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#332C2211yescharacteristic
59#332C2211yescharacteristic
60#331C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×#41—21no—
69C2×#41—21no—
70C2×#41—21no—
71#135—44no—
72C2×Q16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×SD16D411yescharacteristic
77C2×D8—21no—
78C2×D8—21no—
79C2×D8—21no—
80C2×D8—21no—
81C2×D8—21no—
82C2×D8—21no—
83C2×D8D411yescharacteristic
84C2×M16—21no—
85C2×M16—21no—
86C22×C8—21no—
87C4⋊Q8C2311yescharacteristic
88C4⋊Q8C2311yescharacteristic
89#127—21no—
90#127—21no—
91#127C2311yescharacteristic
92#127C2311yescharacteristic
93#123—41no—
94#123—41no—
95C22⋊Q8—21no—
96C22⋊Q8—21no—
97#121—21no—
98#121—21no—
99#121—21no—
100#121—21no—
101#121—41no—
102#121—41no—
103C4×Q8C2311yescharacteristic
104C4×D4—21no—
105C4×D4—21no—
106C4×D4C2311yescharacteristic
107C4×D4C2311yescharacteristic
108C4×D4C2311yescharacteristic
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111#107—21no—
112#107D411yescharacteristic
113#106—21no—
114#106D411yescharacteristic
115C4⋊C8—21no—
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
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—41no—
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—42no—
149#41—81no—
150#41—81no—
151#41—81no—
152#41—81no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8C2×D411yescharacteristic
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4—41no—
165C2×D4—41no—
166C2×D4—41no—
167C2×D4—41no—
168C2×D4C2×D411yescharacteristic
169C2×D4C2×D411yescharacteristic
170C2×D4C2×D411yescharacteristic
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178Q16—42no—
179SD16—22no—
180SD16—42no—
181SD16—81no—
182SD16—81no—
183D8—41no—
184D8—42no—
185D8—42no—
186D8—81no—
187D8—81no—
188D8—81no—
189D8—81no—
190M16—81no—
191M16—81no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—42no—
198C2×C8C2×D411yescharacteristic
199C2×C8C2×D411yescharacteristic
200C2×C8C2×D411yescharacteristic
201C2×C8C2×D411yescharacteristic
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4C2×D411yescharacteristic
210C4⋊C4C2×D411yescharacteristic
211C4⋊C4C2×D411yescharacteristic
212C4⋊C4C2×D411yescharacteristic
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—41no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—41no—
220C42—21no—
221C42C2411yescharacteristic
222C23—21no—
223C23—21no—
224C23—21no—
225C23—41no—
226C23—41no—
227Q8—21no—
228Q8—41no—
229Q8—41no—
230Q8—41no—
231Q8—41no—
232Q8AGL(1,ℤ/8)12yes—
233D4—21no—
234D4—21no—
235D4—21no—
236D4—21no—
237D4—21no—
238D4—21no—
239D4—41no—
240D4—41no—
241D4—41no—
242D4—41no—
243D4—41no—
244D4—41no—
245D4—41no—
246D4—41no—
247D4—82no—
248D4—82no—
249D4—81no—
250D4—81no—
251D4—82no—
252D4—82no—
253D4—161no—
254D4—161no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4C22×D411yescharacteristic
273C2×C4C22×D411yescharacteristic
274C2×C4H(5,𝔽2)11yescharacteristic
275C8—21no—
276C8—21no—
277C8—21no—
278C8—21no—
279C8—41no—
280C8—41no—
281C8—41no—
282C8—41no—
283C8—41no—
284C22—41no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22—81no—
299C22D4211yescharacteristic
300C4—22no—
301C4—21no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—81no—
307C4—81no—
308C4C2×AGL(1,ℤ/8)11yescharacteristic
309C4#57511yescharacteristic
310C4#57611yescharacteristic
311C4#57611yescharacteristic
312C2—81no—
313C2—81no—
314C2—81no—
315C2—161no—
316C2—161no—
317C2#326411yescharacteristic
318C2#326811yescharacteristic
319C2#329011yescharacteristic
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. #3391C2semidirect
2. #3332C2semidirect
3. #3330C2semidirect
4. #3284C2semidirect
5. #1717C2semidirect
6. #1687C2semidirect
7. #1685C2semidirect
8. #1681C2semidirect
9. #1680C2semidirect
10. #1642C2semidirect
11. #1641C2semidirect
12. #1625C2semidirect
13. #1625C2semidirect
14. #1588C2semidirect
15. #1586C2semidirect

Maximal subgroups

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

subgroupindex
#33912
#33322
#33302
#32842
#17172
#16872
#16852
#16812
#16802
#16422
#16412
#16252
#15882
#15862