Finite Group Databasebeta

← #49124398

Subgroups — #49124398

Order 36224.

688 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 173 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×#22583578C211yesmaximal, characteristic
2#31826365C212yesmaximal, complemented
3#31826356C212yesmaximal, complemented
4#31826355C212yesmaximal, complemented
5#31826353C212yesmaximal, complemented
6#31826345C211yesmaximal, characteristic, complemented
7#31826342C212yesmaximal, complemented
8#31826320C212yesmaximal, complemented
9#390×C283C211yesmaximal, characteristic, complemented
10C24×D283C2211yescharacteristic
11C22×C22⋊D283C2212yes—
12C23×Dic283C2211yescharacteristic, complemented
13#22583578C418yescomplemented
14C2×Dic283⋊C4C2212yescomplemented
15C2×C22⋊Dic283C2212yes—
16C2×C22⋊Dic283C2211yescharacteristic
17C2×C22⋊Dic283C2211yescharacteristic, complemented
18C2×(C2×D283)⋊C4C2212yes—
19C2×(C2×D283)⋊C4C2212yes—
20C2×(C2×D283)⋊C4C2212yes—
21C2×(C2×D283)⋊C4C2212yes—
22C2×(C2×D283)⋊C4C2212yes—
23C2×C4×Dic283C2212yescomplemented
24C283⋊#95C2212yescomplemented
25C283⋊#95C2212yescomplemented
26C283⋊#95C2212yescomplemented
27C22×D4×C283C2211yescharacteristic
28C2×C4⋊C4×C283C2211yescharacteristic, complemented
29C2×C22⋊C4×C283C2212yes—
30C2×C22⋊C4×C283C2212yescomplemented
31C2×C42×C283C2211yescharacteristic, complemented
32C23×D283—42no—
33C23×D283—41no—
34C23×D283—41no—
35C23×D283—42no—
36C23×D283C2×C414yescomplemented
37C23×D283C2312yes—
38C2×C22⋊D283—42no—
39C2×C22⋊D283—42no—
40C2×C22⋊D283C2×C418yes—
41C22×Dic283—41no—
42C22×Dic283—42no—
43C22×Dic283C2312yes—
44C22×Dic283C2312yes—
45C22×Dic283C2312yes—
46Dic283⋊C4—42no—
47C22⋊Dic283—42no—
48C22⋊Dic283—41no—
49C22⋊Dic283C2×C414yescomplemented
50(C2×D283)⋊C4—42no—
51(C2×D283)⋊C4—42no—
52(C2×D283)⋊C4—42no—
53(C2×D283)⋊C4—42no—
54(C2×D283)⋊C4C2×C418yescomplemented
55C4×Dic283—42no—
56C24×C283C2311yescharacteristic
57C24×C283C2311yescharacteristic
58C2×D4×C283—41no—
59C2×D4×C283—41no—
60C2×D4×C283C2×C414yes—
61C22×C4×C283C2312yes—
62C22×C4×C283C2312yes—
63C22×C4×C283C2311yescharacteristic
64C4⋊C4×C283—41no—
65C22⋊C4×C283—42no—
66C22⋊C4×C283—42no—
67C42×C283—41no—
68C22×D283—22no—
69C22×D283—22no—
70C22×D283—22no—
71C22×D283—22no—
72C22×D283—22no—
73C22×D283—22no—
74C22×D283—48no—
75C22×D283—44no—
76C22×D283—44no—
77C22×D283—48no—
78C22×D283—44no—
79C22×D283C22×C414yes—
80C22⋊D283—48no—
81C2×Dic283—22no—
82C2×Dic283—22no—
83C2×Dic283—22no—
84C2×Dic283—22no—
85C2×Dic283—22no—
86C2×Dic283—22no—
87C2×Dic283—22no—
88C2×Dic283—22no—
89C2×Dic283—44no—
90C2×Dic283C22×C414yes—
91C23×C283—22no—
92C23×C283—22no—
93C23×C283—21no—
94C23×C283—21no—
95C23×C283—21no—
96C23×C283—21no—
97C23×C283—22no—
98C23×C283—22no—
99C23×C283C22×C412yes—
100C23×C283C22×C412yes—
101C23×C283C2411yescharacteristic
102D4×C283—44no—
103C2×C4×C283—21no—
104C2×C4×C283—21no—
105C2×C4×C283—22no—
106C2×C4×C283—22no—
107C2×C4×C283—22no—
108C2×C4×C283—22no—
109C2×C4×C283—22no—
110C2×C4×C283—22no—
111C2×C4×C283C22×C412yes—
112C22×C283—22no—
113C22×C283—22no—
114C22×C283—24no—
115C22×C283—24no—
116C22×C283—41no—
117C22×C283—42no—
118C22×C283—41no—
119C22×C283—42no—
120C22×C283—41no—
121C22×C283—41no—
122C22×C283C23×C411yescharacteristic
123C22×C283H(5,𝔽2)12yes—
124C22×C283H(5,𝔽2)11yescharacteristic
125C22×C283H(5,𝔽2)11yescharacteristic
126C22×C283H(5,𝔽2)12yes—
127C2×D283—24no—
128C2×D283—24no—
129C2×D283—24no—
130C2×D283—416no—
131C2×D283—42no—
132C2×D283—42no—
133C2×D283—42no—
134C2×D283—42no—
135C4×C283—22no—
136C4×C283—42no—
137C4×C283—42no—
138Dic283—24no—
139Dic283—42no—
140Dic283—42no—
141C2×C283—42no—
142C2×C283—42no—
143C2×C283#51812yes—
144C2×C283#51811yescharacteristic
145C2×C283#56011yescharacteristic
146C2×C283#56012yes—
147C2×C283H(3,𝔽4)11yescharacteristic
148D283—44no—
149C283#237511yescharacteristic, complemented, Sylow
150#2375—2831nomaximal, Sylow
151C2×#120—2831no—
152#390—2832no—
153#390D28311yescharacteristic, complemented
154#388—2832no—
155#388—2832no—
156#388—2832no—
157#386—2831no—
158#386—2832no—
159#379—2832no—
160C25—2831no—
161C22×D4—2832no—
162C22×D4C2×D28311yescharacteristic
163C23×C4—2831no—
164#120—2838no—
165C2×C4⋊C4—2832no—
166C2×C4⋊C4C2×D28311yescharacteristic, complemented
167C2×C22⋊C4—2831no—
168C2×C22⋊C4—2832no—
169C2×C22⋊C4—2832no—
170C2×C22⋊C4—2831no—
171C2×C22⋊C4—2832no—
172C2×C22⋊C4—2832no—
173C2×C22⋊C4—2832no—
174C2×C22⋊C4—2832no—
175C2×C22⋊C4C2×D28312yes—
176C2×C22⋊C4C2×D28312yescomplemented
177C2×C42—2832no—
178C2×C42C2×D28311yescharacteristic, complemented
179#95—2832no—
180#95—2832no—
181#95—2832no—
182C24—2832no—
183C24—2834no—
184C24—11322no—
185C24—11321no—
186C24—11321no—
187C24—11322no—
188C24C22×D28311yescharacteristic
189C24C22×D28311yescharacteristic
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—2838no—
193C2×D4—11322no—
194C2×D4—11322no—
195C2×D4C4×D28314yes—
196C22×C4—2832no—
197C22×C4—2832no—
198C22×C4—2832no—
199C22×C4—11321no—
200C22×C4—11322no—
201C22×C4C22×D28312yes—
202C22×C4C22×D28312yes—
203C22×C4C22×D28311yescharacteristic
204C4⋊C4—41no—
205C4⋊C4—11322no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—2834no—
209C22⋊C4—2838no—
210C22⋊C4—11322no—
211C22⋊C4—11322no—
212C22⋊C4—11322no—
213C22⋊C4—11322no—
214C22⋊C4—11321no—
215C22⋊C4—11322no—
216C42—41no—
217C42—11322no—
218C23—22no—
219C23—22no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—22no—
225C23—22no—
226C23—2834no—
227C23—5662no—
228C23—5662no—
229C23—5662no—
230C23—5662no—
231C23—5662no—
232C23—5662no—
233C23—11328no—
234C23—11324no—
235C23—11324no—
236C23—11328no—
237C23—11324no—
238C23C2×C4×D28312yes—
239C23C2×C4×D28312yes—
240C23C23×D28311yescharacteristic
241D4—44no—
242D4—11328no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—2834no—
252C2×C4—5662no—
253C2×C4—5662no—
254C2×C4—5662no—
255C2×C4—5662no—
256C2×C4—5662no—
257C2×C4—5662no—
258C2×C4—5662no—
259C2×C4—5662no—
260C2×C4—11324no—
261C2×C4C2×C4×D28312yes—
262C22—24no—
263C22—24no—
264C22—22no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—42no—
269C22—42no—
270C22—41no—
271C22—41no—
272C22—5664no—
273C22—5664no—
274C22—5664no—
275C22—113216no—
276C22—11322no—
277C22—11322no—
278C22—11322no—
279C22—11322no—
280C22C22×C4×D28311yescharacteristic
281C22#2258365512yes—
282C22#2258365511yescharacteristic
283C22#2258365511yescharacteristic
284C22#2258365612yes—
285C4—22no—
286C4—42no—
287C4—42no—
288C4—5664no—
289C4—11322no—
290C4—11322no—
291C2—42no—
292C2—42no—
293C2—11324no—
294C2#3182693712yes—
295C2#3182693811yescharacteristic
296C2#3182720211yescharacteristic
297C2#3182720312yes—
298C2#3182720711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D283
commutatorC22×C283C23×C4
Fitting#390×C283C2
socle, cosocleC23×C283C24
prime radicalC2×#22583578C2
hypercenter#390D283
hypocenterC283#2375
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #31826365C2semidirect
3. #31826356C2semidirect
4. #31826355C2semidirect
5. #31826353C2semidirect
6. #31826345C2semidirect
7. #31826342C2semidirect
8. #31826320C2semidirect
9. #390×C283C2semidirect
12. C23×Dic283C22semidirect
13. #22583578C4semidirect
14. C2×Dic283⋊C4C22semidirect
17. C2×C22⋊Dic283C22semidirect
23. C2×C4×Dic283C22semidirect
24. C283⋊#95C22semidirect
25. C283⋊#95C22semidirect
26. C283⋊#95C22semidirect
28. C2×C4⋊C4×C283C22semidirect
30. C2×C22⋊C4×C283C22semidirect
31. C2×C42×C283C22semidirect
36. C23×D283C2×C4semidirect
49. C22⋊Dic283C2×C4semidirect
54. (C2×D283)⋊C4C2×C4semidirect
149. C283#2375semidirect
153. #390D283semidirect
166. C2×C4⋊C4C2×D283semidirect
176. C2×C22⋊C4C2×D283semidirect
178. C2×C42C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225835782
#318263652
#318263562
#318263552
#318263532
#318263452
#318263422
#318263202
#390×C2832
#2375283