Finite Group Databasebeta

← #20078

Subgroups — #20078

Order 256.

320 conjugacy classes of subgroups.

The subgroup lattice has 301 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3392C211yesmaximal, characteristic, complemented
2#3374C211yesmaximal, characteristic, complemented
3#3368C211yesmaximal, characteristic, complemented
4#3287C211yesmaximal, characteristic
5#1717C211yesmaximal, characteristic, complemented
6#1688C211yesmaximal, characteristic, complemented
7#1686C211yesmaximal, characteristic, complemented
8#1682C211yesmaximal, characteristic, complemented
9#1679C211yesmaximal, characteristic, complemented
10#1642C211yesmaximal, characteristic, complemented
11#1641C211yesmaximal, characteristic, complemented
12#1626C211yesmaximal, characteristic, complemented
13#1626C211yesmaximal, characteristic, complemented
14#1587C211yesmaximal, characteristic, complemented
15#1585C211yesmaximal, characteristic, complemented
16C2×#135—21no—
17#557C2211yescharacteristic
18#557C2211yescharacteristic
19#550C2211yescharacteristic
20#550C2211yescharacteristic
21#493C2211yescharacteristic
22#492—21no—
23#492C2211yescharacteristic
24#491—21no—
25#490—21no—
26#487C2211yescharacteristic
27#486C2211yescharacteristic
28#486C2211yescharacteristic
29#486C2211yescharacteristic
30#479—21no—
31#478—21no—
32#475C2211yescharacteristic
33#474C2211yescharacteristic
34#466—21no—
35#462—21no—
36#462C2211yescharacteristic
37#461—21no—
38#461C2211yescharacteristic
39#460—21no—
40#460—21no—
41#460C2211yescharacteristic
42#460C2211yescharacteristic
43#460C2211yescharacteristic
44#460C2211yescharacteristic
45#449—21no—
46#449—21no—
47#446C2211yescharacteristic
48#446C2211yescharacteristic
49C8×Q8C2211yescharacteristic
50#441—21no—
51#440—21no—
52C4×Q16C2211yescharacteristic
53C4×SD16—21no—
54C4×SD16C2211yescharacteristic
55C8×D4C2211yescharacteristic
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#332C2211yescharacteristic
59#332C2211yescharacteristic
60#331C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64Q8⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×#41—21no—
69#135—44no—
70C2×Q16—21no—
71C2×Q16D411yescharacteristic
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×SD16—21no—
77C2×SD16—21no—
78C2×SD16—21no—
79C2×SD16D411yescharacteristic
80C2×D8—21no—
81C2×D8—21no—
82C22×C8—21no—
83C4⋊Q8—21no—
84C4⋊Q8—21no—
85C4⋊Q8C2311yescharacteristic
86C4⋊Q8C2311yescharacteristic
87#127—21no—
88#127—21no—
89#127C2311yescharacteristic
90#127C2311yescharacteristic
91#125—21no—
92#125—21no—
93#125—41no—
94#125—41no—
95#121—21no—
96#121—21no—
97#121—41no—
98#121—41no—
99C4×Q8—21no—
100C4×Q8—21no—
101C4×Q8—21no—
102C4×Q8—21no—
103C4×Q8C2311yescharacteristic
104C4×Q8C2311yescharacteristic
105C4×Q8C2311yescharacteristic
106C4×D4—21no—
107C4×D4—21no—
108C4×D4C2311yescharacteristic
109#107—21no—
110#107D411yescharacteristic
111#106—21no—
112#106D411yescharacteristic
113C4⋊C8—21no—
114C4⋊C8—21no—
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#103D411yescharacteristic
127#103D411yescharacteristic
128#102—21no—
129#102—21no—
130#102—21no—
131#102—21no—
132#102—21no—
133#102—21no—
134#102—21no—
135#102—41no—
136#102—41no—
137#102D411yescharacteristic
138C22⋊C8—21no—
139#97—21no—
140#97—21no—
141C4×C8—21no—
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144C4×C8C2311yescharacteristic
145#41—42no—
146#41—42no—
147#41—81no—
148#41—81no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8C2×D411yescharacteristic
152C2×Q8C2×D411yescharacteristic
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×D4C2×D411yescharacteristic
165C22×C4—21no—
166C22×C4—41no—
167C22×C4—41no—
168Q16—22no—
169Q16—42no—
170SD16—41no—
171SD16—42no—
172SD16—42no—
173SD16—42no—
174SD16—42no—
175SD16—81no—
176SD16—81no—
177SD16—81no—
178D8—42no—
179D8—81no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—42no—
186C2×C8C2×D411yescharacteristic
187C2×C8C2×D411yescharacteristic
188C2×C8C2×D411yescharacteristic
189C2×C8C2×D411yescharacteristic
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—41no—
198C4⋊C4—41no—
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—41no—
209C22⋊C4—41no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—41no—
214C42—41no—
215C42C2411yescharacteristic
216C23—21no—
217C23—41no—
218C23—41no—
219Q8—21no—
220Q8—21no—
221Q8—21no—
222Q8—21no—
223Q8—21no—
224Q8—41no—
225Q8—41no—
226Q8—41no—
227Q8—41no—
228Q8AGL(1,ℤ/8)12yes—
229D4—21no—
230D4—21no—
231D4—41no—
232D4—41no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—41no—
237D4—41no—
238D4—41no—
239D4—82no—
240D4—81no—
241D4—81no—
242D4—82no—
243D4—161no—
244D4—161no—
245C2×C4—21no—
246C2×C4—21no—
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—82no—
257C2×C4—82no—
258C2×C4C22×D411yescharacteristic
259C2×C4C22×D411yescharacteristic
260C2×C4H(5,𝔽2)11yescharacteristic
261C8—21no—
262C8—21no—
263C8—21no—
264C8—21no—
265C8—41no—
266C8—41no—
267C8—41no—
268C8—41no—
269C8—41no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—81no—
274C22—81no—
275C22—81no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22D4211yescharacteristic
280C4—22no—
281C4—21no—
282C4—41no—
283C4—41no—
284C4—41no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—81no—
289C4—81no—
290C4C2×AGL(1,ℤ/8)11yescharacteristic
291C4#57511yescharacteristic
292C4#57711yescharacteristic
293C4#57711yescharacteristic
294C2—81no—
295C2—161no—
296C2—161no—
297C2#326511yescharacteristic
298C2#327111yescharacteristic
299C2#329011yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3287C2
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. #3392C2semidirect
2. #3374C2semidirect
3. #3368C2semidirect
5. #1717C2semidirect
6. #1688C2semidirect
7. #1686C2semidirect
8. #1682C2semidirect
9. #1679C2semidirect
10. #1642C2semidirect
11. #1641C2semidirect
12. #1626C2semidirect
13. #1626C2semidirect
14. #1587C2semidirect
15. #1585C2semidirect

Maximal subgroups

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

subgroupindex
#33922
#33742
#33682
#32872
#17172
#16882
#16862
#16822
#16792
#16422
#16412
#16262
#15872
#15852