Finite Group Databasebeta

← #19875

Subgroups — #19875

Order 256.

341 conjugacy classes of subgroups.

The subgroup lattice has 317 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3391C211yesmaximal, characteristic, complemented
2#3354C211yesmaximal, characteristic, complemented
3#3305C211yesmaximal, characteristic, complemented
4#3284C211yesmaximal, characteristic, complemented
5#1715C211yesmaximal, characteristic, complemented
6#1687C211yesmaximal, characteristic, complemented
7#1677C211yesmaximal, characteristic, complemented
8#1672C211yesmaximal, characteristic, complemented
9#1665C211yesmaximal, characteristic, complemented
10#1641C211yesmaximal, characteristic, complemented
11#1631C211yesmaximal, characteristic, complemented
12#1625C211yesmaximal, characteristic, complemented
13#1624C211yesmaximal, characteristic, complemented
14#1583C211yesmaximal, characteristic, complemented
15#1580C211yesmaximal, characteristic, complemented
16C2×#135—21no—
17#555C2211yescharacteristic
18#550C2211yescharacteristic
19#550C2211yescharacteristic
20#547C2211yescharacteristic
21#498C2211yescharacteristic
22#493—21no—
23#493C2211yescharacteristic
24#492C2211yescharacteristic
25#492C2211yescharacteristic
26#486C2211yescharacteristic
27#486C2211yescharacteristic
28#482—21no—
29#482—21no—
30#481—21no—
31#477C2211yescharacteristic
32#476C2211yescharacteristic
33#474C2211yescharacteristic
34#469—21no—
35#466—21no—
36#465—21no—
37#462C2211yescharacteristic
38#459—21no—
39#459—21no—
40#459C2211yescharacteristic
41#459C2211yescharacteristic
42#459C2211yescharacteristic
43#459C2211yescharacteristic
44#452—21no—
45#449—21no—
46#449—21no—
47C8×Q8C2211yescharacteristic
48C4×SD16C2211yescharacteristic
49C4×D8—21no—
50C4×D8C2211yescharacteristic
51#436C2211yescharacteristic
52C8×D4C2211yescharacteristic
53C8×D4C2211yescharacteristic
54#428—21no—
55#416—21no—
56#334C2211yescharacteristic
57#334C2211yescharacteristic
58#333C2211yescharacteristic
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—
70#135—44no—
71C2×Q16—21no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—41no—
75C2×SD16D411yescharacteristic
76C2×D8—21no—
77C2×D8—21no—
78C2×D8—21no—
79C2×D8—21no—
80C2×D8—21no—
81C2×D8—21no—
82C2×D8D411yescharacteristic
83C2×M16—21no—
84C22×C8—21no—
85C22×C8—21no—
86C4⋊Q8C2311yescharacteristic
87C4⋊Q8C2311yescharacteristic
88#127—21no—
89#127—21no—
90#127C2311yescharacteristic
91#127C2311yescharacteristic
92#124—41no—
93#123—41no—
94C22⋊Q8—21no—
95#121—21no—
96#121—21no—
97#121—21no—
98#121—21no—
99#121—21no—
100#121—41no—
101#121—41no—
102C4×Q8C2311yescharacteristic
103C4×D4—21no—
104C4×D4—21no—
105C4×D4C2311yescharacteristic
106C4×D4C2311yescharacteristic
107C4×D4C2311yescharacteristic
108#117—21no—
109#117—21no—
110C2×C4⋊C4—21no—
111#107—21no—
112#107—21no—
113#107D411yescharacteristic
114#106—21no—
115#106—21no—
116#106—21no—
117#106D411yescharacteristic
118C4⋊C8—21no—
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121C4⋊C8C2311yescharacteristic
122C4⋊C8C2311yescharacteristic
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×Q8C2×D411yescharacteristic
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4—41no—
165C2×D4—41no—
166C2×D4C2×D411yescharacteristic
167C2×D4C2×D411yescharacteristic
168C2×D4C2×D411yescharacteristic
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175Q16—42no—
176SD16—22no—
177SD16—42no—
178SD16—81no—
179SD16—82no—
180D8—22no—
181D8—42no—
182D8—42no—
183D8—81no—
184D8—81no—
185D8—81no—
186D8—81no—
187M16—42no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—42no—
194C2×C8—81no—
195C2×C8C2×D411yescharacteristic
196C2×C8C2×D411yescharacteristic
197C2×C8C2×D411yescharacteristic
198C2×C8C2×D411yescharacteristic
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
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—41no—
213C22⋊C4—41no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C22⋊C4—41no—
217C42—21no—
218C42—41no—
219C42C2411yescharacteristic
220C23—21no—
221C23—21no—
222C23—21no—
223C23—41no—
224C23—41no—
225Q8—21no—
226Q8—41no—
227Q8—41no—
228Q8—81no—
229Q8AGL(1,ℤ/8)12yes—
230D4—21no—
231D4—21no—
232D4—21no—
233D4—21no—
234D4—21no—
235D4—21no—
236D4—41no—
237D4—41no—
238D4—41no—
239D4—41no—
240D4—41no—
241D4—41no—
242D4—41no—
243D4—41no—
244D4—82no—
245D4—82no—
246D4—81no—
247D4—81no—
248D4—82no—
249D4—161no—
250D4—161no—
251D4—161no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
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—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4C22×D411yescharacteristic
269C2×C4C22×D411yescharacteristic
270C2×C4H(5,𝔽2)11yescharacteristic
271C8—21no—
272C8—21no—
273C8—21no—
274C8—41no—
275C8—41no—
276C8—41no—
277C8—41no—
278C8—41no—
279C8#13512yescomplemented
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—81no—
290C22—81no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22—81no—
295C22D4211yescharacteristic
296C4—22no—
297C4—21no—
298C4—41no—
299C4—41no—
300C4—41no—
301C4—41no—
302C4—81no—
303C4—81no—
304C4C2×#13511yescharacteristic
305C4C2×AGL(1,ℤ/8)11yescharacteristic
306C4#57611yescharacteristic
307C4#57711yescharacteristic
308C2—81no—
309C2—81no—
310C2—81no—
311C2—161no—
312C2—161no—
313C2#326411yescharacteristic
314C2#328411yescharacteristic
315C2#328811yescharacteristic
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. #3354C2semidirect
3. #3305C2semidirect
4. #3284C2semidirect
5. #1715C2semidirect
6. #1687C2semidirect
7. #1677C2semidirect
8. #1672C2semidirect
9. #1665C2semidirect
10. #1641C2semidirect
11. #1631C2semidirect
12. #1625C2semidirect
13. #1624C2semidirect
14. #1583C2semidirect
15. #1580C2semidirect
279. C8#135semidirect

Maximal subgroups

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

subgroupindex
#33912
#33542
#33052
#32842
#17152
#16872
#16772
#16722
#16652
#16412
#16312
#16252
#16242
#15832
#15802