Finite Group Databasebeta

← #49132881

Subgroups — #49132881

Order 36224.

556 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 133 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×#22583635C212yesmaximal, complemented
2D4×D1132C211yesmaximal, characteristic, complemented
3#31826841C211yesmaximal, characteristic, complemented
4#31826716C212yesmaximal, complemented
5#31826665C211yesmaximal, characteristic, complemented
6#31826657C211yesmaximal, characteristic, complemented
7#31826610C212yesmaximal, complemented
8#31826598C212yesmaximal, complemented
9C4×#17985379C211yesmaximal, characteristic, complemented
10#31826543C211yesmaximal, characteristic, complemented
11#550×C283C211yesmaximal, characteristic, complemented
12C22×D1132C2212yes—
13C2×D4×D283—21no—
14#22583635—28no—
15C2×#17985379C2211yescharacteristic
16C2×#17985379C2212yescomplemented
17C2×#17985377C2212yes—
18C2×#17985377C2212yescomplemented
19C2×#17985377C2212yescomplemented
20C2×#17985376C2212yescomplemented
21#22583607C2211yescharacteristic
22#22583585—21no—
23#22583584C2212yes—
24#22583580—22no—
25C4×D1132C2211yescharacteristic
26#22583541C2211yescharacteristic, complemented
27C4⋊(C283⋊C8)C2211yescharacteristic, complemented
28#22583534C2211yescharacteristic, complemented
29#22583531C2212yescomplemented
30#22583530C2211yescharacteristic
31#22583530C2212yescomplemented
32C22⋊(C283⋊C8)C2212yescomplemented
33C4×C283⋊C8C2211yescharacteristic, complemented
34C2×#41×C283C2212yescomplemented
35#127×C283—21no—
36#127×C283C2211yescharacteristic, complemented
37#121×C283—22no—
38#121×C283C2212yescomplemented
39C4×Q8×C283C2211yescharacteristic, complemented
40C4×D4×C283—21no—
41C4×D4×C283C2211yescharacteristic
42C23×D283—22no—
43C2×D1132—24no—
44C2×D1132—22no—
45C2×D1132—21no—
46C2×D1132—21no—
47C2×D1132—42no—
48C2×D1132C2312yes—
49C2×D1132C2311yescharacteristic
50C2×C22⋊D283—22no—
51D4×D283—44no—
52C2×C4×D283—21no—
53#17985379—24no—
54#17985379D414yescomplemented
55#17985377—24no—
56#17985377—24no—
57#17985377—42no—
58#17985376—24no—
59C2×C283⋊C8C2312yes—
60C2×C283⋊C8C2311yescharacteristic
61C2×C283⋊C8C2311yescharacteristic
62C4⋊Dic283C2311yescharacteristic
63(C2×D283)⋊C4—21no—
64(C2×D283)⋊C4—22no—
65#41×C283—24no—
66#41×C283—42no—
67C2×Q8×C283C2311yescharacteristic
68C2×D4×C283—22no—
69C2×D4×C283—22no—
70C2×D4×C283—21no—
71C2×D4×C283—21no—
72C2×D4×C283D411yescharacteristic
73C2×D4×C283C2312yes—
74C22×C4×C283—22no—
75C22×C4×C283C2312yes—
76C4⋊C4×C283—21no—
77C4⋊C4×C283D411yescharacteristic
78C4⋊C4×C283C2311yescharacteristic
79C22⋊C4×C283—22no—
80C22⋊C4×C283D412yes—
81C42×C283—21no—
82C42×C283C2311yescharacteristic
83C22×D283—21no—
84C22×D283—22no—
85C22×D283—44no—
86C22×D283—44no—
87C22×D283—82no—
88D1132—22no—
89D1132—21no—
90D1132—22no—
91D1132—44no—
92D1132—42no—
93D1132—81no—
94D1132C2×D412yes—
95C22⋊D283—44no—
96C2×Dic283—21no—
97C4×D283—42no—
98C283⋊C8—22no—
99C283⋊C8—21no—
100C283⋊C8C2×D412yescomplemented
101C23×C283—22no—
102C23×C283C2×D412yes—
103Q8×C283—21no—
104Q8×C283C2×D412yescomplemented
105D4×C283—22no—
106D4×C283—22no—
107D4×C283—22no—
108D4×C283—44no—
109D4×C283—42no—
110D4×C283—81no—
111D4×C283—82no—
112C2×C4×C283—21no—
113C2×C4×C283—22no—
114C2×C4×C283—22no—
115C2×C4×C283—21no—
116C2×C4×C283—44no—
117C2×C4×C283C2×D411yescharacteristic
118C2×C4×C283C2×D412yes—
119C2×C4×C283C2×D411yescharacteristic
120C2×C4×C283C2411yescharacteristic
121C22×C283—22no—
122C22×C283—22no—
123C22×C283—22no—
124C22×C283—42no—
125C22×C283—42no—
126C22×C283—42no—
127C22×C283C22×D411yescharacteristic
128C2×D283—22no—
129C2×D283—41no—
130C2×D283—41no—
131C2×D283—42no—
132C2×D283—42no—
133C2×D283—42no—
134C2×D283—88no—
135C4×C283—22no—
136C4×C283—22no—
137C4×C283—21no—
138C4×C283—41no—
139C4×C283AGL(1,ℤ/8)12yescomplemented
140C4×C283C22×D411yescharacteristic
141C4×C283H(5,𝔽2)11yescharacteristic
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#550D28311yescharacteristic, complemented
154D42—2831no—
155#493—2831no—
156#469—2832no—
157#460—2831no—
158#459—2831no—
159#449—2832no—
160#447—2832no—
161C4×SD16—2831no—
162#436—2831no—
163C2×#41C2×D28312yescomplemented
164C22×D4—2832no—
165C22×D4—5661no—
166AGL(1,ℤ/8)—5668no—
167C2×SD16—2831no—
168C2×SD16—2832no—
169C2×D8—2832no—
170C2×D8—2832no—
171C2×D8—2832no—
172C2×M16—2832no—
173#127—21no—
174#127—2831no—
175#127C2×D28311yescharacteristic, complemented
176#121—22no—
177#121—2832no—
178#121—5661no—
179#121C2×D28312yescomplemented
180#120—5662no—
181C4×Q8C2×D28311yescharacteristic, complemented
182C4×D4—21no—
183C4×D4—2831no—
184C4×D4C2×D28311yescharacteristic
185#106—2831no—
186C4⋊C8—2831no—
187#103—2831no—
188#102—2832no—
189#102—2831no—
190#102—2832no—
191C22⋊C8—2832no—
192C4×C8—2831no—
193C24—5662no—
194#41—24no—
195#41—42no—
196C2×Q8C22×D28311yescharacteristic
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—2831no—
202C2×D4—2832no—
203C2×D4—5664no—
204C2×D4—5662no—
205C2×D4—5662no—
206C2×D4—5661no—
207C2×D4—5661no—
208C2×D4—11322no—
209C2×D4—11324no—
210C2×D4C22⋊D28311yescharacteristic
211C2×D4C22×D28312yes—
212C22×C4—22no—
213C22×C4—5661no—
214C22×C4C22×D28312yes—
215SD16—2834no—
216SD16—5664no—
217D8—5664no—
218D8—5664no—
219D8—11322no—
220M16—5664no—
221C2×C8—2831no—
222C2×C8—2831no—
223C2×C8—2832no—
224C4⋊C4—21no—
225C4⋊C4—2831no—
226C4⋊C4C22⋊D28311yescharacteristic
227C4⋊C4C22×D28311yescharacteristic
228C22⋊C4—22no—
229C22⋊C4—5661no—
230C22⋊C4—5662no—
231C22⋊C4C22⋊D28312yes—
232C42—21no—
233C42C22×D28311yescharacteristic
234C23—22no—
235C23—5661no—
236C23—5662no—
237C23—11324no—
238C23—11324no—
239C23—22642no—
240C23C2×C22⋊D28312yes—
241Q8—21no—
242Q8D4×D28312yescomplemented
243D4—22no—
244D4—22no—
245D4—22no—
246D4—44no—
247D4—42no—
248D4—81no—
249D4—82no—
250D4—2832no—
251D4—5662no—
252D4—5661no—
253D4—5662no—
254D4—11324no—
255D4—11324no—
256D4—11322no—
257D4—22641no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—44no—
263C2×C4—5661no—
264C2×C4—11322no—
265C2×C4C2×C22⋊D28311yescharacteristic
266C2×C4C2×C22⋊D28312yes—
267C2×C4C2×C22⋊D28311yescharacteristic
268C2×C4C23×D28311yescharacteristic
269C8—2832no—
270C8—5662no—
271C8—5661no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—5662no—
279C22—11321no—
280C22—11321no—
281C22—11322no—
282C22—11322no—
283C22—11322no—
284C22—22648no—
285C22C22×C22⋊D28311yescharacteristic
286C4—21no—
287C4—22no—
288C4—22no—
289C4—41no—
290C4—11321no—
291C4#2258363512yescomplemented
292C4C2×D4×D28311yescharacteristic
293C4#2258365511yescharacteristic
294C2—42no—
295C2—82no—
296C2—11322no—
297C2—22642no—
298C2D4×C22⋊D28311yescharacteristic
299C2C2×#2258363511yescharacteristic
300C2#3182726811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#550×C283C2
socleC22×C283C22×D4
hypercenter#550D283
hypocenterC283#3274
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#22583635C2semidirect
2. D4×D1132C2semidirect
3. #31826841C2semidirect
4. #31826716C2semidirect
5. #31826665C2semidirect
6. #31826657C2semidirect
7. #31826610C2semidirect
8. #31826598C2semidirect
9. C4×#17985379C2semidirect
10. #31826543C2semidirect
11. #550×C283C2semidirect
16. C2×#17985379C22semidirect
18. C2×#17985377C22semidirect
19. C2×#17985377C22semidirect
20. C2×#17985376C22semidirect
26. #22583541C22semidirect
27. C4⋊(C283⋊C8)C22semidirect
28. #22583534C22semidirect
29. #22583531C22semidirect
31. #22583530C22semidirect
32. C22⋊(C283⋊C8)C22semidirect
33. C4×C283⋊C8C22semidirect
34. C2×#41×C283C22semidirect
36. #127×C283C22semidirect
38. #121×C283C22semidirect
39. C4×Q8×C283C22semidirect
54. #17985379D4semidirect
100. C283⋊C8C2×D4semidirect
104. Q8×C283C2×D4semidirect
139. C4×C283AGL(1,ℤ/8)semidirect
150. C283#3274semidirect
153. #550D283semidirect
163. C2×#41C2×D283semidirect
175. #127C2×D283semidirect
179. #121C2×D283semidirect
181. C4×Q8C2×D283semidirect
242. Q8D4×D283semidirect
291. C4#22583635semidirect

Maximal subgroups

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

subgroupindex
C2×#225836352
D4×D11322
#318268412
#318267162
#318266652
#318266572
#318266102
#318265982
C4×#179853792
#318265432
#550×C2832
#3274283