Finite Group Databasebeta

← #19906

Subgroups — #19906

Order 256.

304 conjugacy classes of subgroups.

The subgroup lattice has 293 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3360C211yesmaximal, characteristic, complemented
2#3358C211yesmaximal, characteristic
3#3333C211yesmaximal, characteristic, complemented
4#3328C211yesmaximal, characteristic, complemented
5#1711C211yesmaximal, characteristic, complemented
6#1699C211yesmaximal, characteristic, complemented
7#1697C211yesmaximal, characteristic, complemented
8#1684C211yesmaximal, characteristic, complemented
9#1682C211yesmaximal, characteristic, complemented
10#1644C211yesmaximal, characteristic, complemented
11#1642C211yesmaximal, characteristic, complemented
12#1637C211yesmaximal, characteristic, complemented
13#1628C211yesmaximal, characteristic, complemented
14#1589C211yesmaximal, characteristic, complemented
15C8×Q16C211yesmaximal, characteristic, complemented
16#556C2211yescharacteristic
17#554C2211yescharacteristic
18#550C2211yescharacteristic
19D4×Q8C2211yescharacteristic
20#488—21no—
21#487C2211yescharacteristic
22#487C2211yescharacteristic
23#487C2211yescharacteristic
24#486—21no—
25#486C2211yescharacteristic
26#486C2211yescharacteristic
27#486C2211yescharacteristic
28#485—21no—
29#480—21no—
30#479—21no—
31#477C2211yescharacteristic
32#476C2211yescharacteristic
33#475C2211yescharacteristic
34#474—21no—
35#474C2211yescharacteristic
36#470—21no—
37#468—21no—
38#463—21no—
39#462C2211yescharacteristic
40#461C2211yescharacteristic
41#460C2211yescharacteristic
42#459—21no—
43#459C2211yescharacteristic
44#452—21no—
45#451—21no—
46C8×Q8C2211yescharacteristic
47C8×Q8C2211yescharacteristic
48C4×Q16C2211yescharacteristic
49C4×SD16—21no—
50C4×D8—21no—
51C4×D8C2211yescharacteristic
52#436C2211yescharacteristic
53#436C2211yescharacteristic
54#419—21no—
55#417—21no—
56#335C2211yescharacteristic
57#335C2211yescharacteristic
58#333C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66C82C2211yescharacteristic
67C2×#41—21no—
68C22×Q8—21no—
69C2×Q16—21no—
70C2×Q16D411yescharacteristic
71C2×SD16—21no—
72C2×SD16—21no—
73C2×D8—21no—
74C2×D8D411yescharacteristic
75C2×M16—21no—
76C2×M16—21no—
77C4⋊Q8—21no—
78C4⋊Q8C2311yescharacteristic
79C4⋊Q8C2311yescharacteristic
80C4⋊Q8C2311yescharacteristic
81#127—21no—
82#127C2311yescharacteristic
83#126—41no—
84#125—21no—
85#124—21no—
86C22⋊Q8—21no—
87C22⋊Q8—21no—
88C22⋊Q8—21no—
89C22⋊Q8—41no—
90C22⋊Q8—41no—
91#121—21no—
92#121—41no—
93C4×Q8—21no—
94C4×Q8C2311yescharacteristic
95C4×Q8C2311yescharacteristic
96C4×D4—21no—
97C4×D4—21no—
98C4×D4—21no—
99C4×D4C2311yescharacteristic
100C4×D4C2311yescharacteristic
101#117—21no—
102C2×C4⋊C4—21no—
103#107—21no—
104#107—21no—
105#107D411yescharacteristic
106#107D411yescharacteristic
107#106—21no—
108#106—21no—
109C4⋊C8—21no—
110C4⋊C8—21no—
111C4⋊C8C2311yescharacteristic
112C4⋊C8C2311yescharacteristic
113C4⋊C8C2311yescharacteristic
114C4⋊C8C2311yescharacteristic
115#103—21no—
116#103—21no—
117#103—21no—
118#103—21no—
119#103—21no—
120#103—21no—
121#103—21no—
122#103—21no—
123#103—41no—
124#103D411yescharacteristic
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
137C22⋊C8—21no—
138C22⋊C8—21no—
139C4×C8—21no—
140C4×C8—21no—
141C4×C8C2311yescharacteristic
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144#41—42no—
145#41—81no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—41no—
150C2×Q8—42no—
151C2×Q8—81no—
152C2×Q8C2×D411yescharacteristic
153C2×Q8C2×D411yescharacteristic
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4C2×D411yescharacteristic
160C2×D4C2×D411yescharacteristic
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166Q16—22no—
167Q16—81no—
168SD16—42no—
169SD16—81no—
170D8—41no—
171D8—42no—
172M16—81no—
173M16—81no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8C2×D411yescharacteristic
180C2×C8C2×D411yescharacteristic
181C2×C8C2×D411yescharacteristic
182C2×C8C2×D411yescharacteristic
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4C2×D411yescharacteristic
196C4⋊C4C2×D411yescharacteristic
197C4⋊C4C2×D411yescharacteristic
198C4⋊C4C2×D411yescharacteristic
199C22⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—41no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C42—21no—
207C42—21no—
208C42—41no—
209C42C2411yescharacteristic
210C23—21no—
211C23—21no—
212C23—41no—
213Q8—21no—
214Q8—21no—
215Q8—21no—
216Q8—41no—
217Q8—41no—
218Q8—41no—
219Q8—41no—
220Q8—82no—
221Q8—81no—
222Q8#13512yes—
223D4—21no—
224D4—21no—
225D4—21no—
226D4—21no—
227D4—41no—
228D4—41no—
229D4—41no—
230D4—41no—
231D4—82no—
232D4—81no—
233D4—161no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—161no—
250C2×C4C22×D411yescharacteristic
251C2×C4C22×D411yescharacteristic
252C2×C4H(5,𝔽2)11yescharacteristic
253C8—21no—
254C8—21no—
255C8—21no—
256C8—21no—
257C8—41no—
258C8—41no—
259C8—41no—
260C8—41no—
261C8—41no—
262C22—41no—
263C22—41no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—81no—
269C22—81no—
270C22—81no—
271C22D4211yescharacteristic
272C4—22no—
273C4—21no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—41no—
279C4—81no—
280C4—81no—
281C4—81no—
282C4C2×#13511yescharacteristic
283C4#57511yescharacteristic
284C4#57611yescharacteristic
285C4#57811yescharacteristic
286C2—81no—
287C2—81no—
288C2—161no—
289C2#327711yescharacteristic
290C2#328011yescharacteristic
291C2#329011yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3358C2
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. #3360C2semidirect
3. #3333C2semidirect
4. #3328C2semidirect
5. #1711C2semidirect
6. #1699C2semidirect
7. #1697C2semidirect
8. #1684C2semidirect
9. #1682C2semidirect
10. #1644C2semidirect
11. #1642C2semidirect
12. #1637C2semidirect
13. #1628C2semidirect
14. #1589C2semidirect
15. C8×Q16C2semidirect

Maximal subgroups

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

subgroupindex
#33602
#33582
#33332
#33282
#17112
#16992
#16972
#16842
#16822
#16442
#16422
#16372
#16282
#15892
C8×Q162