Finite Group Databasebeta

← #49131396

Subgroups — #49131396

Order 36224.

450 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#31826996C211yesmaximal, characteristic, complemented
2C2×#22583572C211yesmaximal, characteristic, complemented
3#31826795C211yesmaximal, characteristic, complemented
4#31826768C212yesmaximal, complemented
5#31826679C212yesmaximal, complemented
6#31826618C211yesmaximal, characteristic, complemented
7#31826598C211yesmaximal, characteristic
8#31826574C212yesmaximal, complemented
9#31826564C212yesmaximal, complemented
10#31826519C211yesmaximal, characteristic, complemented
11#480×C283C211yesmaximal, characteristic, complemented
12C22×D1132C2211yescharacteristic
13C22×C4×D283—21no—
14C2×#17985381C2212yes—
15C2×#17985377C2212yes—
16#22583603—21no—
17#22583601C2211yescharacteristic, complemented
18#22583597C2211yescharacteristic, complemented
19#22583591—21no—
20#22583589C2211yescharacteristic
21#22583575C2211yescharacteristic, complemented
22D4×Dic283—21no—
23D4×Dic283C2211yescharacteristic, complemented
24#22583572—21no—
25#22583572—21no—
26#22583572C2212yes—
27#22583572C2212yes—
28#22583566—21no—
29C2×(C2×D283)⋊C4—21no—
30C2×C4×Dic283C2211yescharacteristic
31#22583541C2212yescomplemented
32Dic283⋊C8C2212yescomplemented
33#22583533C2212yescomplemented
34#22583532C2212yes—
35#22583530C2212yes—
36C22⋊(C283⋊C8)C2211yescharacteristic
37#22583520C2212yescomplemented
38#121×C283C2211yescharacteristic
39C2×C4⋊C4×C283C2211yescharacteristic
40#107×C283C2212yescomplemented
41#102×C283C2212yescomplemented
42C22⋊C8×C283C2211yescharacteristic
43C23×D283—21no—
44C2×Dic566C2311yescharacteristic
45C2×D1132—24no—
46C2×D1132—21no—
47C2×D1132C2312yes—
48C22×Dic283—21no—
49C22×Dic283D411yescharacteristic
50C22×Dic283D411yescharacteristic
51C2×C4×D283—22no—
52C2×C4×D283—22no—
53C2×C4×D283—44no—
54#17985381—24no—
55#17985377—24no—
56C2×C283⋊C8C2312yes—
57C4⋊Dic283—21no—
58C4⋊Dic283C2311yescharacteristic
59Dic283⋊C4—21no—
60Dic283⋊C4—21no—
61Dic283⋊C4—21no—
62C22⋊Dic283—21no—
63C22⋊Dic283—21no—
64C22⋊Dic283—21no—
65C22⋊Dic283—21no—
66(C2×D283)⋊C4—22no—
67(C2×D283)⋊C4—22no—
68C4×Dic283—21no—
69C4×Dic283D411yescharacteristic
70C4×Dic283D411yescharacteristic
71C4×Dic283C2311yescharacteristic
72C4×Dic283C2311yescharacteristic
73C2×D4×C283—21no—
74C2×D4×C283C2311yescharacteristic
75C22×C4×C283—21no—
76C22×C4×C283C2311yescharacteristic
77C2×C8×C283C2312yes—
78C4⋊C4×C283—21no—
79C4⋊C4×C283C2312yes—
80C4⋊C4×C283C2311yescharacteristic
81C22⋊C4×C283—21no—
82C22×D283—24no—
83C22×D283—22no—
84C22×D283—42no—
85C22×D283—42no—
86Dic566—22no—
87D1132—24no—
88D1132—22no—
89D1132#4114yes—
90C2×Dic283—22no—
91C2×Dic283—21no—
92C2×Dic283—21no—
93C2×Dic283—41no—
94C2×Dic283—42no—
95C2×Dic283C2×D411yescharacteristic
96C2×Dic283C2×D411yescharacteristic
97C2×Dic283C2×D411yescharacteristic
98C2×Dic283C2×D411yescharacteristic
99C4×D283—44no—
100C4×D283—44no—
101C283⋊C8—22no—
102C23×C283—21no—
103C23×C283C2×D411yescharacteristic
104D4×C283—22no—
105D4×C283—42no—
106C2×C4×C283—22no—
107C2×C4×C283—21no—
108C2×C4×C283—22no—
109C2×C4×C283—42no—
110C2×C4×C283C2×D411yescharacteristic
111C2×C4×C283C2411yescharacteristic
112C8×C283—22no—
113C22×C283—21no—
114C22×C283—21no—
115C22×C283—41no—
116C22×C283—41no—
117C22×C283—41no—
118C22×C283AGL(1,ℤ/8)12yes—
119C22×C283C22×D411yescharacteristic
120C2×D283—24no—
121C2×D283—48no—
122C2×D283—42no—
123C2×D283—42no—
124C4×C283—21no—
125C4×C283—42no—
126C4×C283—41no—
127C4×C283C2×#4112yes—
128Dic283—21no—
129Dic283—21no—
130Dic283—21no—
131Dic283—21no—
132Dic283—41no—
133Dic283—41no—
134C2×C283—22no—
135C2×C283—81no—
136C2×C283#52511yescharacteristic
137C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C283#57511yescharacteristic
139D283—44no—
140C283#311411yescharacteristic, complemented, Sylow
141#3114—2831nomaximal, Sylow
142#529—2831no—
143C2×C4×D4—2831no—
144#483—2831no—
145#480D28311yescharacteristic, complemented
146#478—2832no—
147#463—2832no—
148#450—2831no—
149#447—2831no—
150#442—2832no—
151#440—2832no—
152#432—2831no—
153C22×D4—2831no—
154C23×C4—5661no—
155C2×SD16—2832no—
156C2×D8—2832no—
157#126—5661no—
158#125—2831no—
159#124—2831no—
160#123—5661no—
161C22⋊Q8—2831no—
162#121C2×D28311yescharacteristic
163C4×Q8—2831no—
164C4×D4—2831no—
165C4×D4—2832no—
166C4×D4—2832no—
167C4×D4—5661no—
168C4×D4—5661no—
169C4×D4—5661no—
170#117—5661no—
171C2×C4⋊C4C2×D28311yescharacteristic
172C2×C22⋊C4—5661no—
173C2×C42—2831no—
174#107C2×D28312yescomplemented
175#106—2832no—
176C4⋊C8—2832no—
177#103—2832no—
178#102—2832no—
179#102—2832no—
180#102C2×D28312yescomplemented
181C22⋊C8—2831no—
182C22⋊C8C2×D28311yescharacteristic
183#97—2832no—
184C24—5661no—
185C2×Q8—2831no—
186C2×D4—21no—
187C2×D4—2832no—
188C2×D4—5664no—
189C2×D4—5661no—
190C2×D4C22×D28311yescharacteristic
191C22×C4—21no—
192C22×C4—2831no—
193C22×C4—2831no—
194C22×C4—5662no—
195C22×C4—5662no—
196C22×C4—5661no—
197C22×C4—11324no—
198C22×C4C22×D28311yescharacteristic
199SD16—5664no—
200D8—5664no—
201C2×C8—2832no—
202C2×C8C22×D28312yes—
203C4⋊C4—21no—
204C4⋊C4—2831no—
205C4⋊C4—5661no—
206C4⋊C4—5661no—
207C4⋊C4—5661no—
208C4⋊C4—5661no—
209C4⋊C4C22×D28311yescharacteristic
210C4⋊C4C22×D28312yes—
211C22⋊C4—21no—
212C22⋊C4—5662no—
213C22⋊C4—5662no—
214C22⋊C4—5661no—
215C22⋊C4—5661no—
216C22⋊C4—5661no—
217C22⋊C4—5661no—
218C42—2831no—
219C42—2831no—
220C42—2831no—
221C42—2831no—
222C42—5661no—
223C23—21no—
224C23—5664no—
225C23—5662no—
226C23—11322no—
227C23—11322no—
228C23D4×D28311yescharacteristic
229Q8—5662no—
230D4—22no—
231D4—42no—
232D4—2834no—
233D4—5664no—
234D4—5662no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—2831no—
240C2×C4—2831no—
241C2×C4—2831no—
242C2×C4—2831no—
243C2×C4—5662no—
244C2×C4—5661no—
245C2×C4—5661no—
246C2×C4—11324no—
247C2×C4—11324no—
248C2×C4—11321no—
249C2×C4—11322no—
250C2×C4D4×D28311yescharacteristic
251C2×C4C23×D28311yescharacteristic
252C8—22no—
253C8—5662no—
254C22—21no—
255C22—21no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—5664no—
260C22—11328no—
261C22—11322no—
262C22—11322no—
263C22#2258363412yes—
264C22C2×D4×D28311yescharacteristic
265C4—21no—
266C4—42no—
267C4—41no—
268C4—5661no—
269C4—5661no—
270C4—5661no—
271C4—5661no—
272C4—11321no—
273C4—11321no—
274C4#41×D28312yes—
275C2—22no—
276C2—81no—
277C2—11324no—
278C2#3182697311yescharacteristic
279C2C2×#2258363411yescharacteristic
280C2#3182726411yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#480×C283C2
socleC22×C283C22×D4
prime radical#31826598C2
hypercenter#480D283
hypocenterC283#3114
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #31826996C2semidirect
2. C2×#22583572C2semidirect
3. #31826795C2semidirect
4. #31826768C2semidirect
5. #31826679C2semidirect
6. #31826618C2semidirect
8. #31826574C2semidirect
9. #31826564C2semidirect
10. #31826519C2semidirect
11. #480×C283C2semidirect
17. #22583601C22semidirect
18. #22583597C22semidirect
21. #22583575C22semidirect
23. D4×Dic283C22semidirect
31. #22583541C22semidirect
32. Dic283⋊C8C22semidirect
33. #22583533C22semidirect
37. #22583520C22semidirect
40. #107×C283C22semidirect
41. #102×C283C22semidirect
140. C283#3114semidirect
145. #480D283semidirect
174. #107C2×D283semidirect
180. #102C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318269962
C2×#225835722
#318267952
#318267682
#318266792
#318266182
#318265982
#318265742
#318265642
#318265192
#480×C2832
#3114283