Finite Group Databasebeta

← #49132884

Subgroups — #49132884

Order 36224.

556 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 125 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×#22583634C212yesmaximal, complemented
2#31827135C211yesmaximal, characteristic, complemented
3D4×D1132C211yesmaximal, characteristic, complemented
4#31826841C211yesmaximal, characteristic, complemented
5#31826717C212yesmaximal, complemented
6#31826666C211yesmaximal, characteristic, complemented
7#31826611C212yesmaximal, complemented
8#31826597C212yesmaximal, complemented
9C4×#17985380C211yesmaximal, characteristic, complemented
10#31826542C211yesmaximal, characteristic, complemented
11#459×C283C211yesmaximal, characteristic, complemented
12C2×#17985391C2212yescomplemented
13C22×D1132—21no—
14C2×D4×D283C2212yes—
15#22583634—28no—
16C2×#17985381C2212yescomplemented
17C2×#17985380C2211yescharacteristic
18C2×#17985377C2212yescomplemented
19C2×#17985377C2212yescomplemented
20C2×#17985375C2212yescomplemented
21#22583607C2211yescharacteristic
22#22583606—21no—
23#22583585C2212yes—
24#22583584—21no—
25#22583582—22no—
26#22583581C2212yescomplemented
27#22583580—22no—
28C4×Dic566C2211yescharacteristic, complemented
29C4×D1132C2211yescharacteristic
30D4×Dic283—21no—
31#22583541C2211yescharacteristic, complemented
32#22583533C2211yescharacteristic, complemented
33#22583532C2212yescomplemented
34#22583531C2211yescharacteristic
35(C2×D283)⋊C8C2212yescomplemented
36C4×C283⋊C8C2211yescharacteristic, complemented
37C2×D8×C283C2212yes—
38#127×C283C2211yescharacteristic, complemented
39C4×D4×C283C2211yescharacteristic
40C4⋊C8×C283C2211yescharacteristic, complemented
41#102×C283C2212yescomplemented
42C23×D283—22no—
43#17985391—24no—
44#17985391—42no—
45C2×Dic566C2311yescharacteristic
46C2×D1132—22no—
47C2×D1132—21no—
48C2×D1132—21no—
49C2×D1132—44no—
50C2×D1132D411yescharacteristic
51C2×D1132C2312yes—
52C2×C22⋊D283—22no—
53C2×C22⋊D283—22no—
54C2×C22⋊D283—22no—
55D4×D283—24no—
56D4×D283—42no—
57C22×Dic283—22no—
58C2×C4×D283C2312yes—
59#17985381—24no—
60#17985380D414yescomplemented
61#17985377—24no—
62#17985377—42no—
63#17985375—24no—
64C2×C283⋊C8C2311yescharacteristic
65C2×C283⋊C8C2311yescharacteristic
66C4⋊Dic283D411yescharacteristic
67C4⋊Dic283C2311yescharacteristic
68Dic283⋊C4—21no—
69C22⋊Dic283—22no—
70(C2×D283)⋊C4—22no—
71(C2×D283)⋊C4D412yes—
72C4×Dic283—21no—
73C2×D4×C283—21no—
74C2×D4×C283—21no—
75C2×D4×C283C2312yes—
76C2×D4×C283C2311yescharacteristic
77C22×C4×C283—21no—
78D8×C283—24no—
79C2×C8×C283C2312yes—
80C4⋊C4×C283C2311yescharacteristic
81C22⋊C4×C283—21no—
82C42×C283C2311yescharacteristic
83C22×D283—22no—
84C22×D283—44no—
85C22×D283—44no—
86C22×D283—82no—
87C22×D283C2×D412yes—
88Dic566—21no—
89Dic566C2×D412yescomplemented
90D1132—22no—
91D1132—22no—
92D1132—22no—
93D1132—44no—
94D1132—42no—
95D1132—81no—
96C22⋊D283—44no—
97C22⋊D283—44no—
98C22⋊D283—82no—
99C2×Dic283—21no—
100C2×Dic283—21no—
101C2×Dic283—44no—
102C2×Dic283C2×D412yes—
103C4×D283—22no—
104C4×D283—22no—
105C283⋊C8—21no—
106C283⋊C8C2×D412yescomplemented
107C23×C283—21no—
108C23×C283—22no—
109D4×C283—22no—
110D4×C283—21no—
111D4×C283—22no—
112D4×C283—42no—
113D4×C283—81no—
114D4×C283C2×D412yes—
115C2×C4×C283—21no—
116C2×C4×C283—42no—
117C2×C4×C283C2×D411yescharacteristic
118C2×C4×C283C2×D411yescharacteristic
119C2×C4×C283C2411yescharacteristic
120C8×C283—22no—
121C22×C283—22no—
122C22×C283—41no—
123C22×C283—41no—
124C22×C283—42no—
125C22×C283—42no—
126C22×C283—42no—
127C22×C283C22×D411yescharacteristic
128C2×D283—22no—
129C2×D283—22no—
130C2×D283—22no—
131C2×D283—42no—
132C2×D283—42no—
133C2×D283—42no—
134C2×D283—88no—
135C4×C283—21no—
136C4×C283—41no—
137C4×C283AGL(1,ℤ/8)12yescomplemented
138C4×C283C22×D411yescharacteristic
139C4×C283H(5,𝔽2)11yescharacteristic
140Dic283—22no—
141Dic283—22no—
142Dic283—41no—
143C2×C283—42no—
144C2×C283—82no—
145C2×C283D4211yescharacteristic
146C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C283#57611yescharacteristic
148D283—42no—
149D283—82no—
150C283#327411yescharacteristic, complemented, Sylow
151#3274—2831nomaximal, Sylow
152C2×AGL(1,ℤ/8)—2832no—
153#550—2831no—
154D42—2831no—
155#493—2831no—
156#469—2832no—
157#460—2831no—
158#459D28311yescharacteristic, complemented
159#449—2832no—
160#447—2832no—
161C4×SD16—2831no—
162#436—2831no—
163C2×#41—2832no—
164C22×D4—2832no—
165C22×D4—5661no—
166AGL(1,ℤ/8)—5668no—
167C2×SD16—2831no—
168C2×SD16—2832no—
169C2×D8—2832no—
170C2×D8—2832no—
171C2×D8C2×D28312yes—
172C2×M16—2832no—
173#127—2831no—
174#127—5661no—
175#127C2×D28311yescharacteristic, complemented
176#121—2832no—
177#121—2832no—
178#121—5662no—
179#121—5661no—
180#120—5662no—
181C4×Q8—2831no—
182C4×D4—2831no—
183C4×D4—5661no—
184C4×D4C2×D28311yescharacteristic
185#106—2831no—
186C4⋊C8C2×D28311yescharacteristic, complemented
187#103—2831no—
188#102—2831no—
189#102—2832no—
190#102C2×D28312yescomplemented
191C22⋊C8—2832no—
192C4×C8—2831no—
193C24—5662no—
194#41—5664no—
195#41—11322no—
196C2×Q8—2831no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—2831no—
200C2×D4—2832no—
201C2×D4—5662no—
202C2×D4—5662no—
203C2×D4—5664no—
204C2×D4—5662no—
205C2×D4—5662no—
206C2×D4—5661no—
207C2×D4—5661no—
208C2×D4—11324no—
209C2×D4—11322no—
210C2×D4C22×D28312yes—
211C2×D4C22×D28311yescharacteristic
212C22×C4—21no—
213C22×C4—2832no—
214C22×C4—5662no—
215SD16—2834no—
216SD16—5664no—
217D8—24no—
218D8—5664no—
219D8—11322no—
220M16—5664no—
221C2×C8—2831no—
222C2×C8—2831no—
223C2×C8C22×D28312yes—
224C4⋊C4—2831no—
225C4⋊C4—2831no—
226C4⋊C4—5661no—
227C4⋊C4C22×D28311yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—2832no—
230C22⋊C4—5662no—
231C22⋊C4—5662no—
232C42—5661no—
233C42C22×D28311yescharacteristic
234C23—21no—
235C23—22no—
236C23—2832no—
237C23—5662no—
238C23—11324no—
239C23—11324no—
240C23—22642no—
241Q8—2832no—
242Q8—5661no—
243D4—22no—
244D4—21no—
245D4—22no—
246D4—42no—
247D4—81no—
248D4—5662no—
249D4—5662no—
250D4—5662no—
251D4—11324no—
252D4—11324no—
253D4—11324no—
254D4—11322no—
255D4—22641no—
256D4—22642no—
257D4D4×D28312yes—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—2832no—
261C2×C4—5661no—
262C2×C4—5662no—
263C2×C4—5662no—
264C2×C4—5661no—
265C2×C4—11324no—
266C2×C4D4×D28311yescharacteristic
267C2×C4D4×D28311yescharacteristic
268C2×C4C23×D28311yescharacteristic
269C8—22no—
270C8—2832no—
271C8—5661no—
272C22—22no—
273C22—41no—
274C22—41no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—5662no—
279C22—5662no—
280C22—5662no—
281C22—11322no—
282C22—11322no—
283C22—11322no—
284C22—22648no—
285C22C2×D4×D28311yescharacteristic
286C4—21no—
287C4—41no—
288C4—5662no—
289C4—5662no—
290C4—11321no—
291C4#2258363412yescomplemented
292C4C2×D4×D28311yescharacteristic
293C4#2258365611yescharacteristic
294C2—42no—
295C2—82no—
296C2—11322no—
297C2—22642no—
298C2#3182709911yescharacteristic
299C2C2×#2258363411yescharacteristic
300C2#3182726911yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#459×C283C2
socleC22×C283C22×D4
hypercenter#459D283
hypocenterC283#3274
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#22583634C2semidirect
2. #31827135C2semidirect
3. D4×D1132C2semidirect
4. #31826841C2semidirect
5. #31826717C2semidirect
6. #31826666C2semidirect
7. #31826611C2semidirect
8. #31826597C2semidirect
9. C4×#17985380C2semidirect
10. #31826542C2semidirect
11. #459×C283C2semidirect
12. C2×#17985391C22semidirect
16. C2×#17985381C22semidirect
18. C2×#17985377C22semidirect
19. C2×#17985377C22semidirect
20. C2×#17985375C22semidirect
26. #22583581C22semidirect
28. C4×Dic566C22semidirect
31. #22583541C22semidirect
32. #22583533C22semidirect
33. #22583532C22semidirect
35. (C2×D283)⋊C8C22semidirect
36. C4×C283⋊C8C22semidirect
38. #127×C283C22semidirect
40. C4⋊C8×C283C22semidirect
41. #102×C283C22semidirect
60. #17985380D4semidirect
89. Dic566C2×D4semidirect
106. C283⋊C8C2×D4semidirect
137. C4×C283AGL(1,ℤ/8)semidirect
150. C283#3274semidirect
158. #459D283semidirect
175. #127C2×D283semidirect
186. C4⋊C8C2×D283semidirect
190. #102C2×D283semidirect
291. C4#22583634semidirect

Maximal subgroups

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

subgroupindex
C2×#225836342
#318271352
D4×D11322
#318268412
#318267172
#318266662
#318266112
#318265972
C4×#179853802
#318265422
#459×C2832
#3274283