Finite Group Databasebeta

← #49131198

Subgroups — #49131198

Order 36224.

462 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 121 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#121×D283C211yesmaximal, characteristic
2C2×#22583568C211yesmaximal, characteristic, complemented
3#31826791C212yesmaximal, complemented
4#31826787C212yesmaximal, complemented
5#31826783C211yesmaximal, characteristic, complemented
6#31826779C211yesmaximal, characteristic, complemented
7#31826776C211yesmaximal, characteristic, complemented
8#31826504C212yesmaximal, complemented
9#31826468C212yesmaximal, complemented
10#31826427C211yesmaximal, characteristic, complemented
11#481×C283C211yesmaximal, characteristic, complemented
12C2×D4×D283—21no—
13C2×D4×D283C2211yescharacteristic
14C22×C4×D283C2211yescharacteristic
15C2×#17985375C2212yes—
16#22583585C2211yescharacteristic
17#22583584C2211yescharacteristic
18#22583583—21no—
19#22583582—21no—
20#22583581C2211yescharacteristic
21#22583568—21no—
22#22583568—21no—
23#22583568C2212yes—
24#22583568C2212yescomplemented
25C2×C4⋊Dic283C2211yescharacteristic, complemented
26C4⋊C4×D283C2211yescharacteristic
27C2×(C2×D283)⋊C4—21no—
28C22⋊C4×D283—21no—
29C2×C4×Dic283—21no—
30#22583544C2212yescomplemented
31#22583543C2212yescomplemented
32#22583541C2212yescomplemented
33#22583532C2212yescomplemented
34#22583531C2212yescomplemented
35#22583530C2212yescomplemented
36C22⋊(C283⋊C8)C2211yescharacteristic, complemented
37(C2×D283)⋊C8C2212yescomplemented
38#121×C283C2211yescharacteristic
39C2×C4⋊C4×C283C2211yescharacteristic, complemented
40#106×C283C2212yescomplemented
41#102×C283C2212yescomplemented
42C22⋊C8×C283C2211yescharacteristic, complemented
43C23×D283—21no—
44C23×D283D411yescharacteristic
45C2×D1132—21no—
46C2×D1132C2311yescharacteristic
47C2×C22⋊D283—21no—
48C2×C22⋊D283—21no—
49C2×C22⋊D283—21no—
50D4×D283—24no—
51D4×D283—44no—
52C22×Dic283—21no—
53C22×Dic283D411yescharacteristic
54C2×C4×D283—24no—
55C2×C4×D283—21no—
56C2×C4×D283D411yescharacteristic
57C2×C4×D283D411yescharacteristic
58C2×C4×D283C2311yescharacteristic
59C2×C4×D283C2311yescharacteristic
60#17985375—24no—
61C2×C283⋊C8C2312yes—
62C4⋊Dic283—21no—
63C4⋊Dic283C2311yescharacteristic
64C4⋊Dic283C2312yes—
65Dic283⋊C4—21no—
66C22⋊Dic283—21no—
67(C2×D283)⋊C4—22no—
68(C2×D283)⋊C4—22no—
69(C2×D283)⋊C4—21no—
70(C2×D283)⋊C4—21no—
71C4×Dic283—22no—
72C4×Dic283—22no—
73C2×D4×C283—21no—
74C2×D4×C283C2311yescharacteristic
75C22×C4×C283—21no—
76C22×C4×C283C2311yescharacteristic
77C2×C8×C283C2312yes—
78C4⋊C4×C283—21no—
79C4⋊C4×C283C2311yescharacteristic
80C4⋊C4×C283C2312yes—
81C22⋊C4×C283—21no—
82C22×D283—22no—
83C22×D283—22no—
84C22×D283—21no—
85C22×D283—41no—
86C22×D283—42no—
87C22×D283—42no—
88C22×D283—42no—
89C22×D283C2×D411yescharacteristic
90C22×D283C2×D411yescharacteristic
91D1132—22no—
92D1132—42no—
93C22⋊D283—42no—
94C22⋊D283—42no—
95C22⋊D283—42no—
96C2×Dic283—22no—
97C2×Dic283—22no—
98C2×Dic283—21no—
99C2×Dic283—42no—
100C2×Dic283C2×D411yescharacteristic
101C2×Dic283C2×D411yescharacteristic
102C4×D283—22no—
103C4×D283—22no—
104C4×D283—22no—
105C4×D283—42no—
106C4×D283#4114yes—
107C283⋊C8—22no—
108C23×C283—21no—
109C23×C283C2×D411yescharacteristic
110D4×C283—22no—
111D4×C283—42no—
112C2×C4×C283—22no—
113C2×C4×C283—21no—
114C2×C4×C283—22no—
115C2×C4×C283—42no—
116C2×C4×C283C2×D411yescharacteristic
117C2×C4×C283C2411yescharacteristic
118C8×C283—22no—
119C22×C283—21no—
120C22×C283—21no—
121C22×C283—41no—
122C22×C283—41no—
123C22×C283—41no—
124C22×C283AGL(1,ℤ/8)12yes—
125C22×C283C22×D411yescharacteristic
126C2×D283—21no—
127C2×D283—21no—
128C2×D283—21no—
129C2×D283—21no—
130C2×D283—21no—
131C2×D283—44no—
132C2×D283—41no—
133C2×D283—41no—
134C2×D283—41no—
135C2×D283—84no—
136C2×D283#12312yes—
137C4×C283—21no—
138C4×C283—41no—
139C4×C283—42no—
140C4×C283C2×#4112yes—
141Dic283—21no—
142Dic283—41no—
143Dic283—42no—
144Dic283#12312yes—
145C2×C283—22no—
146C2×C283—81no—
147C2×C283C2×#12311yescharacteristic
148C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
149C2×C283#57511yescharacteristic
150D283—22no—
151D283—41no—
152D283—81no—
153C283#309411yescharacteristic, complemented, Sylow
154#3094—2831nomaximal, Sylow
155C2×#121—2831no—
156C2×#117—2831no—
157#482—2832no—
158#482—2832no—
159#481—2831no—
160#481D28311yescharacteristic, complemented
161#480—2831no—
162#480—2831no—
163#428—2832no—
164#418—2832no—
165#407—2831no—
166C22×D4—2831no—
167C22×D4—5661no—
168C23×C4—2831no—
169C2×M16—2832no—
170#121—2831no—
171#121—2831no—
172#121—2831no—
173#121—5661no—
174#121—5661no—
175#121C2×D28311yescharacteristic
176#117—2832no—
177#117—2832no—
178#117—5661no—
179#117—5661no—
180C2×C4⋊C4—2831no—
181C2×C4⋊C4—2831no—
182C2×C4⋊C4C2×D28311yescharacteristic, complemented
183C2×C22⋊C4—5661no—
184C2×C22⋊C4—5661no—
185C2×C42—5661no—
186#107—2832no—
187#107—2832no—
188#106—2832no—
189#106C2×D28312yescomplemented
190#102—2832no—
191#102—2832no—
192#102—2832no—
193#102C2×D28312yescomplemented
194C22⋊C8—2831no—
195C22⋊C8—2832no—
196C22⋊C8C2×D28311yescharacteristic, complemented
197C24—2831no—
198C24—5661no—
199C2×D4—21no—
200C2×D4—2831no—
201C2×D4—5661no—
202C2×D4—5664no—
203C2×D4—5661no—
204C2×D4—5661no—
205C2×D4—5661no—
206C2×D4—11324no—
207C2×D4C22×D28311yescharacteristic
208C22×C4—21no—
209C22×C4—2831no—
210C22×C4—2831no—
211C22×C4—2831no—
212C22×C4—2831no—
213C22×C4—2831no—
214C22×C4—5664no—
215C22×C4—5661no—
216C22×C4—5661no—
217C22×C4C22×D28311yescharacteristic
218M16—5664no—
219C2×C8—2832no—
220C2×C8C22×D28312yes—
221C4⋊C4—21no—
222C4⋊C4—2832no—
223C4⋊C4—2831no—
224C4⋊C4—5661no—
225C4⋊C4—5661no—
226C4⋊C4C22×D28312yes—
227C4⋊C4C22×D28311yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—5661no—
230C22⋊C4—5662no—
231C22⋊C4—5662no—
232C22⋊C4—5661no—
233C22⋊C4—5661no—
234C42—5662no—
235C42—5662no—
236C23—21no—
237C23—2831no—
238C23—2831no—
239C23—5662no—
240C23—5662no—
241C23—5661no—
242C23—11321no—
243C23—11322no—
244C23—11322no—
245C23—11322no—
246C23D4×D28311yescharacteristic
247D4—22no—
248D4—42no—
249D4—5662no—
250D4—11322no—
251D4—11322no—
252D4—11322no—
253D4—11322no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—2831no—
259C2×C4—2834no—
260C2×C4—2831no—
261C2×C4—5662no—
262C2×C4—5662no—
263C2×C4—5662no—
264C2×C4—5662no—
265C2×C4—5662no—
266C2×C4—5661no—
267C2×C4—11322no—
268C2×C4—11322no—
269C2×C4D4×D28311yescharacteristic
270C2×C4C23×D28311yescharacteristic
271C8—22no—
272C8—5662no—
273C22—21no—
274C22—21no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—2832no—
279C22—5661no—
280C22—5661no—
281C22—5661no—
282C22—5661no—
283C22—5661no—
284C22—11324no—
285C22—11321no—
286C22—11321no—
287C22—11321no—
288C22—22644no—
289C22#2258363712yes—
290C22C2×D4×D28311yescharacteristic
291C4—21no—
292C4—41no—
293C4—42no—
294C4—2832no—
295C4—5661no—
296C4—11321no—
297C4—11322no—
298C4#41×D28312yes—
299C2—22no—
300C2—81no—
301C2—5662no—
302C2—11321no—
303C2—22641no—
304C2#123×D28311yescharacteristic
305C2C2×#2258363711yescharacteristic
306C2#3182726111yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#481×C283C2
socleC22×C283C22×D4
prime radical#121×D283C2
hypercenter#481D283
hypocenterC283#3094
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. C2×#22583568C2semidirect
3. #31826791C2semidirect
4. #31826787C2semidirect
5. #31826783C2semidirect
6. #31826779C2semidirect
7. #31826776C2semidirect
8. #31826504C2semidirect
9. #31826468C2semidirect
10. #31826427C2semidirect
11. #481×C283C2semidirect
24. #22583568C22semidirect
25. C2×C4⋊Dic283C22semidirect
30. #22583544C22semidirect
31. #22583543C22semidirect
32. #22583541C22semidirect
33. #22583532C22semidirect
34. #22583531C22semidirect
35. #22583530C22semidirect
36. C22⋊(C283⋊C8)C22semidirect
37. (C2×D283)⋊C8C22semidirect
39. C2×C4⋊C4×C283C22semidirect
40. #106×C283C22semidirect
41. #102×C283C22semidirect
42. C22⋊C8×C283C22semidirect
153. C283#3094semidirect
160. #481D283semidirect
182. C2×C4⋊C4C2×D283semidirect
189. #106C2×D283semidirect
193. #102C2×D283semidirect
196. C22⋊C8C2×D283semidirect

Maximal subgroups

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

subgroupindex
#121×D2832
C2×#225835682
#318267912
#318267872
#318267832
#318267792
#318267762
#318265042
#318264682
#318264272
#481×C2832
#3094283