Finite Group Databasebeta

← #49130885

Subgroups — #49130885

Order 36224.

496 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 143 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×M16×D283C211yesmaximal, characteristic, complemented
2#31826972C211yesmaximal, characteristic
3C2×#22583586C211yesmaximal, characteristic, complemented
4#31826720C212yesmaximal, complemented
5#31826708C212yesmaximal, complemented
6#31826703C212yesmaximal, complemented
7#31826691C212yesmaximal, complemented
8#31826507C211yesmaximal, characteristic, complemented
9#31826474C211yesmaximal, characteristic, complemented
10#31826466C211yesmaximal, characteristic, complemented
11C2×#137×C283C211yesmaximal, characteristic, complemented
12C22×C4×D283C2211yescharacteristic
13C2×#17985376C2211yescharacteristic
14C2×#17985375C2212yes—
15M16×D283—24no—
16C2×C8×D283C2212yes—
17#22583591—21no—
18#22583586—21no—
19#22583586—21no—
20#22583586C2212yes—
21#22583586C2211yescharacteristic
22#22583586C2212yes—
23#22583586C2211yescharacteristic
24#22583581C2211yescharacteristic
25#22583581C2211yescharacteristic
26#22583578—21no—
27C4×C22⋊D283—21no—
28C4×C22⋊D283—21no—
29#22583569C2211yescharacteristic
30C2×C4⋊Dic283C2211yescharacteristic
31C2×Dic283⋊C4—21no—
32C2×(C2×D283)⋊C4—21no—
33#22583544C2212yescomplemented
34#22583542C2212yescomplemented
35#22583534C2212yescomplemented
36#22583534C2212yescomplemented
37#22583534C2212yescomplemented
38#22583531C2212yescomplemented
39C2×#41×C283C2211yescharacteristic
40C22×Q8×C283C2211yescharacteristic
41#137×C283—24no—
42C2×Q16×C283C2212yescomplemented
43C2×SD16×C283C2212yescomplemented
44C2×M16×C283C2211yescharacteristic
45C23×D283D411yescharacteristic
46C2×C22⋊D283—21no—
47C2×C22⋊D283—21no—
48C22×Dic283—21no—
49C22×Dic283D411yescharacteristic
50C2×C4×D283—22no—
51C2×C4×D283—22no—
52C2×C4×D283D411yescharacteristic
53C2×C4×D283D411yescharacteristic
54C2×C4×D283C2311yescharacteristic
55C2×C4×D283C2311yescharacteristic
56#17985376—22no—
57#17985375—24no—
58C2×C283⋊C8C2312yes—
59C8×D283—24no—
60C4⋊Dic283—21no—
61C4⋊Dic283C2312yes—
62C4⋊Dic283C2311yescharacteristic
63C4⋊Dic283C2311yescharacteristic
64Dic283⋊C4—22no—
65Dic283⋊C4—22no—
66Dic283⋊C4—21no—
67Dic283⋊C4—21no—
68C22⋊Dic283—21no—
69C22⋊Dic283—21no—
70C22⋊Dic283—21no—
71(C2×D283)⋊C4—22no—
72(C2×D283)⋊C4—22no—
73(C2×D283)⋊C4—21no—
74(C2×D283)⋊C4—21no—
75C4×Dic283—21no—
76#41×C283—22no—
77#41×C283—22no—
78C2×Q8×C283—21no—
79C2×Q8×C283—22no—
80C2×Q8×C283—22no—
81C2×Q8×C283C2312yes—
82C2×Q8×C283C2311yescharacteristic
83C2×D4×C283—21no—
84C2×D4×C283C2311yescharacteristic
85C22×C4×C283—21no—
86C22×C4×C283—21no—
87C22×C4×C283C2311yescharacteristic
88Q16×C283—24no—
89SD16×C283—24no—
90M16×C283D414yescomplemented
91C2×C8×C283C2312yes—
92C22×D283—22no—
93C22×D283—22no—
94C22×D283—41no—
95C22×D283C2×D411yescharacteristic
96C22×D283C2×D411yescharacteristic
97C22⋊D283—42no—
98C22⋊D283—42no—
99C2×Dic283—22no—
100C2×Dic283—21no—
101C2×Dic283—21no—
102C2×Dic283—22no—
103C2×Dic283—42no—
104C2×Dic283C2×D411yescharacteristic
105C2×Dic283C2×D411yescharacteristic
106C4×D283—22no—
107C4×D283—22no—
108C4×D283—21no—
109C4×D283—21no—
110C4×D283—21no—
111C4×D283—21no—
112C283⋊C8—22no—
113C23×C283—21no—
114C23×C283C2×D411yescharacteristic
115Q8×C283—22no—
116Q8×C283—21no—
117Q8×C283—22no—
118Q8×C283—21no—
119Q8×C283—42no—
120D4×C283—21no—
121D4×C283—21no—
122D4×C283—42no—
123C2×C4×C283—21no—
124C2×C4×C283—22no—
125C2×C4×C283—42no—
126C2×C4×C283—42no—
127C2×C4×C283C2×D411yescharacteristic
128C2×C4×C283C2×D412yes—
129C2×C4×C283#4112yes—
130C2×C4×C283C2411yescharacteristic
131C8×C283C2×D414yes—
132C22×C283—21no—
133C22×C283—21no—
134C22×C283—41no—
135C22×C283—41no—
136C22×C283—41no—
137C22×C283#12112yes—
138C22×C283C22×D411yescharacteristic
139C2×D283—21no—
140C2×D283—21no—
141C2×D283—21no—
142C2×D283—21no—
143C2×D283—21no—
144C2×D283—44no—
145C2×D283#13712yescomplemented
146C4×C283—41no—
147C4×C283—42no—
148C4×C283#12112yes—
149C4×C283C22×D411yescharacteristic
150C4×C283C2×#4111yescharacteristic
151Dic283—21no—
152Dic283—21no—
153Dic283—41no—
154Dic283—42no—
155Dic283—41no—
156C2×C283—22no—
157C2×C283—81no—
158C2×C283C2×#12111yescharacteristic
159C2×C283C2×#13711yescharacteristic
160C2×C283#57511yescharacteristic
161D283—22no—
162D283—41no—
163C283#305611yescharacteristic, complemented, Sylow
164#3056—2831nomaximal, Sylow
165C2×#137D28311yescharacteristic, complemented
166C22×M16—2831no—
167#525—2831no—
168C2×C22⋊Q8—2831no—
169#470—2832no—
170#468—2832no—
171#467—2832no—
172#465—2832no—
173#428—2831no—
174#419—2831no—
175#417—2831no—
176C2×#41C2×D28311yescharacteristic
177C22×Q8C2×D28311yescharacteristic
178C23×C4—2831no—
179#137—24no—
180C2×Q16C2×D28312yescomplemented
181C2×SD16C2×D28312yescomplemented
182C2×M16—2831no—
183C2×M16—2832no—
184C2×M16—5664no—
185C2×M16C2×D28311yescharacteristic
186C22×C8—2832no—
187#123—5661no—
188C22⋊Q8—2832no—
189C22⋊Q8—2832no—
190C22⋊Q8—2831no—
191C22⋊Q8—2831no—
192C22⋊Q8—5661no—
193C22⋊Q8—5661no—
194#121—2831no—
195#121—2831no—
196#120—5661no—
197C4×D4—5661no—
198C4×D4—5661no—
199#117—2831no—
200C2×C4⋊C4—2831no—
201C2×C4⋊C4—5661no—
202C2×C22⋊C4—5661no—
203#107—2832no—
204#106—2832no—
205#103—2832no—
206#103—2832no—
207#103—2832no—
208#102—2832no—
209C24—2831no—
210#41—22no—
211#41—22no—
212C2×Q8—21no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8C22×D28312yes—
216C2×Q8C22×D28311yescharacteristic
217C2×D4—21no—
218C2×D4—5661no—
219C2×D4—5661no—
220C2×D4C22×D28311yescharacteristic
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—2831no—
224C22×C4—2831no—
225C22×C4—2831no—
226C22×C4—2831no—
227C22×C4—2831no—
228C22×C4—5662no—
229C22×C4—5662no—
230C22×C4—5661no—
231C22×C4C22×D28311yescharacteristic
232Q16—24no—
233SD16—24no—
234M16—5662no—
235M16—5664no—
236M16C22⋊D28314yescomplemented
237C2×C8—2832no—
238C2×C8—5664no—
239C2×C8C22×D28312yes—
240C4⋊C4—2832no—
241C4⋊C4—2831no—
242C4⋊C4—2831no—
243C4⋊C4—5662no—
244C4⋊C4—5662no—
245C4⋊C4—5661no—
246C4⋊C4—5661no—
247C4⋊C4—5661no—
248C22⋊C4—5662no—
249C22⋊C4—5662no—
250C22⋊C4—5661no—
251C22⋊C4—5661no—
252C22⋊C4—5661no—
253C22⋊C4—5661no—
254C22⋊C4—5661no—
255C42—5661no—
256C23—21no—
257C23—2831no—
258C23—2831no—
259C23—5662no—
260C23—5662no—
261C23—11321no—
262C23D4×D28311yescharacteristic
263Q8—22no—
264Q8—21no—
265Q8—22no—
266Q8—21no—
267Q8—42no—
268D4—21no—
269D4—21no—
270D4—42no—
271D4—11322no—
272D4—11322no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—2831no—
278C2×C4—2831no—
279C2×C4—5662no—
280C2×C4—5662no—
281C2×C4—5662no—
282C2×C4—5661no—
283C2×C4—5661no—
284C2×C4—5661no—
285C2×C4—5661no—
286C2×C4—5661no—
287C2×C4—5661no—
288C2×C4—5662no—
289C2×C4—11322no—
290C2×C4D4×D28311yescharacteristic
291C2×C4C2×C22⋊D28312yes—
292C2×C4#1798539112yes—
293C2×C4C23×D28311yescharacteristic
294C8—5662no—
295C8C2×C22⋊D28314yes—
296C22—21no—
297C22—21no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—2832no—
302C22—5661no—
303C22—5661no—
304C22—5661no—
305C22—5661no—
306C22—5661no—
307C22—11324no—
308C22#2258358112yes—
309C22C2×D4×D28311yescharacteristic
310C4—41no—
311C4—42no—
312C4—5661no—
313C4—5661no—
314C4—11321no—
315C4—11322no—
316C4—11321no—
317C4#2258358112yes—
318C4C22×C22⋊D28311yescharacteristic
319C4C2×#1798539111yescharacteristic
320C2—22no—
321C2—81no—
322C2—5662no—
323C2—11321no—
324C2C2×#2258358111yescharacteristic
325C2#137×D28311yescharacteristic
326C2#3182726411yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
FittingC2×#137×C283C2
socleC22×C283C22×D4
prime radical#31826972C2
hypercenterC2×#137D283
hypocenterC283#3056
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×M16×D283C2semidirect
3. C2×#22583586C2semidirect
4. #31826720C2semidirect
5. #31826708C2semidirect
6. #31826703C2semidirect
7. #31826691C2semidirect
8. #31826507C2semidirect
9. #31826474C2semidirect
10. #31826466C2semidirect
11. C2×#137×C283C2semidirect
33. #22583544C22semidirect
34. #22583542C22semidirect
35. #22583534C22semidirect
36. #22583534C22semidirect
37. #22583534C22semidirect
38. #22583531C22semidirect
42. C2×Q16×C283C22semidirect
43. C2×SD16×C283C22semidirect
90. M16×C283D4semidirect
145. C2×D283#137semidirect
163. C283#3056semidirect
165. C2×#137D283semidirect
180. C2×Q16C2×D283semidirect
181. C2×SD16C2×D283semidirect
236. M16C22⋊D283semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D2832
#318269722
C2×#225835862
#318267202
#318267082
#318267032
#318266912
#318265072
#318264742
#318264662
C2×#137×C2832
#3056283