Finite Group Databasebeta

← #20069

Subgroups — #20069

Order 256.

328 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3366C211yesmaximal, characteristic, complemented
2#3335C211yesmaximal, characteristic, complemented
3#3332C211yesmaximal, characteristic, complemented
4#3283C211yesmaximal, characteristic, complemented
5#1720C211yesmaximal, characteristic, complemented
6#1691C211yesmaximal, characteristic, complemented
7#1690C211yesmaximal, characteristic, complemented
8#1681C211yesmaximal, characteristic, complemented
9#1677C211yesmaximal, characteristic, complemented
10#1643C211yesmaximal, characteristic, complemented
11#1642C211yesmaximal, characteristic, complemented
12#1635C211yesmaximal, characteristic, complemented
13#1625C211yesmaximal, characteristic, complemented
14#1586C211yesmaximal, characteristic, complemented
15#1585C211yesmaximal, characteristic, complemented
16C2×#135—21no—
17#554C2211yescharacteristic
18#550C2211yescharacteristic
19#547C2211yescharacteristic
20#547C2211yescharacteristic
21#498C2211yescharacteristic
22#494C2211yescharacteristic
23#490—21no—
24#487C2211yescharacteristic
25#486C2211yescharacteristic
26#486C2211yescharacteristic
27#486C2211yescharacteristic
28#483—21no—
29#481—21no—
30#481—21no—
31#478—21no—
32#476C2211yescharacteristic
33#476C2211yescharacteristic
34#475C2211yescharacteristic
35#475C2211yescharacteristic
36#469—21no—
37#467—21no—
38#465—21no—
39#461C2211yescharacteristic
40#461C2211yescharacteristic
41#459—21no—
42#459C2211yescharacteristic
43#459C2211yescharacteristic
44#452—21no—
45#452—21no—
46#449—21no—
47#446C2211yescharacteristic
48#442—21no—
49C4×SD16C2211yescharacteristic
50C4×D8C2211yescharacteristic
51#436C2211yescharacteristic
52C8×D4C2211yescharacteristic
53C8×D4C2211yescharacteristic
54C2×#106—21no—
55#417—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#333C2211yescharacteristic
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—
70#135—44no—
71C2×Q16—21no—
72C2×Q16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16D411yescharacteristic
76C2×D8—21no—
77C2×D8—21no—
78C2×D8D411yescharacteristic
79C2×M16—21no—
80C22×C8—21no—
81C22×C8—21no—
82C4⋊Q8C2311yescharacteristic
83C4⋊Q8C2311yescharacteristic
84C4⋊Q8C2311yescharacteristic
85#127—21no—
86#127C2311yescharacteristic
87#125—21no—
88#123—41no—
89#123—41no—
90C22⋊Q8—21no—
91C22⋊Q8—21no—
92C22⋊Q8—21no—
93C22⋊Q8—41no—
94#121—21no—
95#121—21no—
96#121—21no—
97#121—41no—
98C4×Q8C2311yescharacteristic
99C4×D4—21no—
100C4×D4—21no—
101C4×D4C2311yescharacteristic
102C4×D4C2311yescharacteristic
103C4×D4C2311yescharacteristic
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107#107D411yescharacteristic
108#106—21no—
109#106—21no—
110#106—21no—
111#106—21no—
112#106—21no—
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—41no—
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—
141#97—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×Q8C2×D411yescharacteristic
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4C2×D411yescharacteristic
163C2×D4C2×D411yescharacteristic
164C2×D4C2×D411yescharacteristic
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172Q16—42no—
173Q16—81no—
174SD16—41no—
175SD16—42no—
176SD16—42no—
177D8—22no—
178D8—42no—
179D8—81no—
180M16—81no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8C2×D411yescharacteristic
189C2×C8C2×D411yescharacteristic
190C2×C8C2×D411yescharacteristic
191C2×C8C2×D411yescharacteristic
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4C2×D411yescharacteristic
204C4⋊C4C2×D411yescharacteristic
205C4⋊C4C2×D411yescharacteristic
206C4⋊C4C2×D411yescharacteristic
207C22⋊C4—21no—
208C22⋊C4—21no—
209C22⋊C4—21no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C42—21no—
215C42C2411yescharacteristic
216C23—21no—
217C23—21no—
218C23—21no—
219C23—41no—
220Q8—21no—
221Q8—21no—
222Q8—41no—
223Q8—41no—
224Q8—41no—
225Q8—41no—
226Q8—81no—
227D4—21no—
228D4—21no—
229D4—21no—
230D4—21no—
231D4—21no—
232D4—41no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—81no—
237D4—82no—
238D4—82no—
239D4—82no—
240D4—161no—
241D4AGL(1,ℤ/8)12yes—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—161no—
260C2×C4C22×D411yescharacteristic
261C2×C4C22×D411yescharacteristic
262C2×C4H(5,𝔽2)11yescharacteristic
263C8—21no—
264C8—21no—
265C8—21no—
266C8—21no—
267C8—41no—
268C8—41no—
269C8—41no—
270C8—41no—
271C8—41no—
272C22—22no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22D4211yescharacteristic
285C4—21no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—81no—
292C4—81no—
293C4—81no—
294C4C2×AGL(1,ℤ/8)11yescharacteristic
295C4#57511yescharacteristic
296C4#57711yescharacteristic
297C4#57711yescharacteristic
298C2—42no—
299C2—81no—
300C2—81no—
301C2—161no—
302C2#326511yescharacteristic
303C2#326711yescharacteristic
304C2#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. #3366C2semidirect
2. #3335C2semidirect
3. #3332C2semidirect
4. #3283C2semidirect
5. #1720C2semidirect
6. #1691C2semidirect
7. #1690C2semidirect
8. #1681C2semidirect
9. #1677C2semidirect
10. #1643C2semidirect
11. #1642C2semidirect
12. #1635C2semidirect
13. #1625C2semidirect
14. #1586C2semidirect
15. #1585C2semidirect

Maximal subgroups

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

subgroupindex
#33662
#33352
#33322
#32832
#17202
#16912
#16902
#16812
#16772
#16432
#16422
#16352
#16252
#15862
#15852