Finite Group Databasebeta

← #49130529

Subgroups — #49130529

Order 36224.

784 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22583635C211yesmaximal, characteristic, complemented
2C22×#17985377C211yesmaximal, characteristic, complemented
3#31827029C211yesmaximal, characteristic, complemented
4#31826610C214yesmaximal, complemented
5#31826599C214yesmaximal, complemented
6#31826470C211yesmaximal, characteristic, complemented
7#31826461C211yesmaximal, characteristic, complemented
8#31826432C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)×C283C211yesmaximal, characteristic, complemented
10C22×D1132C2211yescharacteristic, complemented
11#22583635—24no—
12C2×#17985379C2212yescomplemented
13C2×#17985377—24no—
14C2×#17985377C2214yescomplemented
15C2×#17985377C2212yescomplemented
16C2×#17985376C2211yescharacteristic, complemented
17C22×C283⋊C8C2211yescharacteristic, complemented
18#22583606—21no—
19#22583599—21no—
20#22583584C2214yescomplemented
21#22583578—22no—
22#22583569C2211yescharacteristic, complemented
23#22583534C2212yescomplemented
24#22583531C2212yescomplemented
25#22583531C2214yescomplemented
26C22⋊(C283⋊C8)C2214yescomplemented
27H(5,𝔽2)×C283—24no—
28H(5,𝔽2)×C283—24no—
29C2×#41×C283—21no—
30C2×#41×C283C2211yescharacteristic
31C2×#41×C283C2211yescharacteristic
32C2×#41×C283C2212yescomplemented
33C22×D4×C283—21no—
34C22×D4×C283—22no—
35C22×D4×C283C2211yescharacteristic
36C22×D4×C283C2212yescomplemented
37C23×D283—21no—
38C2×D1132—22no—
39C2×D1132—22no—
40C2×D1132—21no—
41C2×D1132C2312yes—
42C2×C22⋊D283—24no—
43#17985379—24no—
44#17985377—28no—
45#17985377—24no—
46#17985376—22no—
47C2×C283⋊C8—22no—
48C2×C283⋊C8C2312yes—
49C2×C283⋊C8C2312yes—
50C4⋊Dic283C2312yes—
51C22⋊Dic283—21no—
52(C2×D283)⋊C4—24no—
53C4×Dic283—21no—
54C24×C283—21no—
55C24×C283—22no—
56#41×C283—28no—
57#41×C283—24no—
58#41×C283—24no—
59#41×C283—22no—
60#41×C283—22no—
61#41×C283—22no—
62#41×C283D414yescomplemented
63C2×Q8×C283D411yescharacteristic
64C2×Q8×C283C2311yescharacteristic
65C2×D4×C283—216no—
66C2×D4×C283—28no—
67C2×D4×C283—21no—
68C2×D4×C283—22no—
69C2×D4×C283—24no—
70C2×D4×C283—24no—
71C2×D4×C283—21no—
72C2×D4×C283—22no—
73C2×D4×C283—22no—
74C2×D4×C283D412yes—
75C2×D4×C283D411yescharacteristic
76C2×D4×C283D414yescomplemented
77C2×D4×C283C2312yes—
78C2×D4×C283C2311yescharacteristic
79C22×C4×C283—21no—
80C22×C4×C283—22no—
81C22×C4×C283C2312yes—
82C22×C4×C283C2311yescharacteristic
83C22×D283—22no—
84C22×D283—42no—
85C22×D283—82no—
86D1132—22no—
87D1132—22no—
88D1132—42no—
89C22⋊D283—84no—
90C2×Dic283—22no—
91C283⋊C8—22no—
92C283⋊C8—22no—
93C23×C283—28no—
94C23×C283—21no—
95C23×C283—22no—
96C23×C283—24no—
97C23×C283—82no—
98C23×C283—84no—
99C23×C283C2×D412yes—
100C23×C283C2×D411yescharacteristic
101Q8×C283—22no—
102Q8×C283—21no—
103Q8×C283C2×D412yes—
104D4×C283—28no—
105D4×C283—28no—
106D4×C283—24no—
107D4×C283—24no—
108D4×C283—22no—
109D4×C283—24no—
110D4×C283—22no—
111D4×C283—21no—
112D4×C283C2×D412yes—
113D4×C283C2×D414yes—
114C2×C4×C283—28no—
115C2×C4×C283—21no—
116C2×C4×C283—24no—
117C2×C4×C283—22no—
118C2×C4×C283—22no—
119C2×C4×C283C2×D412yes—
120C2×C4×C283C2×D411yescharacteristic
121C2×C4×C283C2×D412yes—
122C2×C4×C283C2×D412yes—
123C2×C4×C283C2411yescharacteristic
124C22×C283—22no—
125C22×C283—22no—
126C22×C283—24no—
127C22×C283—22no—
128C22×C283—24no—
129C22×C283—44no—
130C22×C283—42no—
131C22×C283—84no—
132C22×C283—88no—
133C22×C283#12012yes—
134C22×C283C22×D411yescharacteristic
135C2×D283—42no—
136C2×D283—44no—
137C2×D283—84no—
138C4×C283—22no—
139C4×C283—22no—
140C4×C283#12012yes—
141C4×C283C22×D411yescharacteristic
142C4×C283C22×D411yescharacteristic
143Dic283—42no—
144C2×C283—22no—
145C2×C283—42no—
146C2×C283—44no—
147C2×C283—42no—
148C2×C283C2×#12011yescharacteristic
149C2×C283#57612yes—
150D283—82no—
151C283#300811yescharacteristic, complemented, Sylow
152#3008—2831nomaximal, Sylow
153C2×H(5,𝔽2)D28311yescharacteristic, complemented
154C2×AGL(1,ℤ/8)—2831no—
155C22×D8—2831no—
156#535—2831no—
157#449—2834no—
158#447—2834no—
159#418—2831no—
160#416—2831no—
161#408—2831no—
162H(5,𝔽2)—24no—
163H(5,𝔽2)—24no—
164C2×#41—21no—
165C2×#41C2×D28311yescharacteristic
166C2×#41C2×D28311yescharacteristic
167C2×#41C2×D28312yescomplemented
168C22×D4—21no—
169C22×D4—22no—
170C22×D4—2831no—
171C22×D4C2×D28311yescharacteristic
172C22×D4C2×D28312yescomplemented
173AGL(1,ℤ/8)—5664no—
174C2×SD16—2832no—
175C2×D8—2834no—
176C2×D8—2832no—
177C2×D8—5664no—
178C2×M16—2831no—
179C22×C8—2831no—
180#127—5661no—
181#124—5661no—
182#121—2834no—
183#120—5662no—
184#117—2831no—
185#103—2832no—
186#102—2832no—
187#102—2834no—
188C22⋊C8—2834no—
189C24—21no—
190C24—22no—
191C24—5661no—
192#41—28no—
193#41—24no—
194#41—24no—
195#41—22no—
196#41—22no—
197#41—22no—
198#41C22⋊D28314yescomplemented
199C2×Q8C22⋊D28311yescharacteristic
200C2×Q8C22×D28311yescharacteristic
201C2×D4—216no—
202C2×D4—28no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—24no—
206C2×D4—24no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—2832no—
211C2×D4—5664no—
212C2×D4—5662no—
213C2×D4—5662no—
214C2×D4—5661no—
215C2×D4C22⋊D28312yes—
216C2×D4C22⋊D28311yescharacteristic
217C2×D4C22⋊D28314yescomplemented
218C2×D4C22×D28312yes—
219C2×D4C22×D28311yescharacteristic
220C22×C4—21no—
221C22×C4—22no—
222C22×C4C22×D28312yes—
223C22×C4C22×D28311yescharacteristic
224SD16—5664no—
225D8—5668no—
226D8—5664no—
227M16—5662no—
228C2×C8—2832no—
229C2×C8—2832no—
230C2×C8—5662no—
231C4⋊C4—2832no—
232C22⋊C4—5661no—
233C22⋊C4—5664no—
234C42—5661no—
235C23—28no—
236C23—21no—
237C23—22no—
238C23—24no—
239C23—82no—
240C23—84no—
241C23—5662no—
242C23—11322no—
243C23—22642no—
244C23C2×C22⋊D28312yes—
245C23C2×C22⋊D28311yescharacteristic
246Q8—22no—
247Q8—21no—
248Q8C2×C22⋊D28312yes—
249D4—28no—
250D4—28no—
251D4—24no—
252D4—24no—
253D4—22no—
254D4—24no—
255D4—22no—
256D4—21no—
257D4—5662no—
258D4—5662no—
259D4—11322no—
260D4—22644no—
261D4C2×C22⋊D28312yes—
262D4C2×C22⋊D28314yes—
263C2×C4—28no—
264C2×C4—21no—
265C2×C4—24no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—5662no—
269C2×C4C2×C22⋊D28312yes—
270C2×C4C2×C22⋊D28311yescharacteristic
271C2×C4C2×C22⋊D28312yes—
272C2×C4C2×C22⋊D28312yes—
273C2×C4C23×D28311yescharacteristic
274C8—5662no—
275C8—5662no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—22no—
280C22—24no—
281C22—44no—
282C22—42no—
283C22—84no—
284C22—88no—
285C22—11322no—
286C22—11324no—
287C22—22644no—
288C22#2258357912yes—
289C22C22×C22⋊D28311yescharacteristic
290C4—22no—
291C4—22no—
292C4—11322no—
293C4#2258357912yes—
294C4C22×C22⋊D28311yescharacteristic
295C4C22×C22⋊D28311yescharacteristic
296C2—22no—
297C2—42no—
298C2—44no—
299C2—42no—
300C2—22642no—
301C2C2×#2258357911yescharacteristic
302C2#3182726812yes—
subgroupG/H
centerC22C22×C22⋊D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
FittingC2×H(5,𝔽2)×C283C2
socleC22×C283C22×D4
hypercenterC2×H(5,𝔽2)D283
hypocenterC283#3008
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#22583635C2semidirect
2. C22×#17985377C2semidirect
3. #31827029C2semidirect
4. #31826610C2semidirect
5. #31826599C2semidirect
6. #31826470C2semidirect
7. #31826461C2semidirect
8. #31826432C2semidirect
9. C2×H(5,𝔽2)×C283C2semidirect
10. C22×D1132C22semidirect
12. C2×#17985379C22semidirect
14. C2×#17985377C22semidirect
15. C2×#17985377C22semidirect
16. C2×#17985376C22semidirect
17. C22×C283⋊C8C22semidirect
20. #22583584C22semidirect
22. #22583569C22semidirect
23. #22583534C22semidirect
24. #22583531C22semidirect
25. #22583531C22semidirect
26. C22⋊(C283⋊C8)C22semidirect
32. C2×#41×C283C22semidirect
36. C22×D4×C283C22semidirect
62. #41×C283D4semidirect
76. C2×D4×C283D4semidirect
151. C283#3008semidirect
153. C2×H(5,𝔽2)D283semidirect
167. C2×#41C2×D283semidirect
172. C22×D4C2×D283semidirect
198. #41C22⋊D283semidirect
217. C2×D4C22⋊D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836352
C22×#179853772
#318270292
#318266102
#318265992
#318264702
#318264612
#318264322
C2×H(5,𝔽2)×C2832
#3008283