Finite Group Databasebeta

← #450×D283 #49130440

Subgroups — #450×D283 #49130440

Order 36224.

996 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 155 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
1C22×D4×D283C211yesmaximal, characteristic
2C2×SD16×D283C212yesmaximal, complemented
3C22⋊Q8×D283C211yesmaximal, characteristic, complemented
4#31826618C211yesmaximal, characteristic, complemented
5#31826617C212yesmaximal, complemented
6#31826616C211yesmaximal, characteristic, complemented
7#31826615C211yesmaximal, characteristic, complemented
8#31826614C212yesmaximal, complemented
9#102×D283C212yesmaximal, complemented
10C22⋊C8×D283C211yesmaximal, characteristic, complemented
11#450×C283C211yesmaximal, characteristic, complemented
12C24×D283—21no—
13C2×Q8×D283C2211yescharacteristic, complemented
14C22×D1132C2211yescharacteristic
15C22×C22⋊D283—21no—
16C2×D4×D283—28no—
17C2×D4×D283—21no—
18C2×D4×D283—21no—
19C2×D4×D283C2212yes—
20C2×D4×D283C2212yes—
21C22×C4×D283C2211yescharacteristic
22C2×#17985381C2212yescomplemented
23C2×#17985380C2212yescomplemented
24C2×#17985379C2212yescomplemented
25SD16×D283—28no—
26C2×C8×D283C2212yescomplemented
27#22583590C2211yescharacteristic, complemented
28#22583589C2211yescharacteristic, complemented
29#22583588—21no—
30#22583587—21no—
31#22583586C2211yescharacteristic, complemented
32C4⋊C4×D283—21no—
33C4⋊C4×D283C2211yescharacteristic, complemented
34C22⋊C4×D283—21no—
35#22583532C2212yescomplemented
36#22583531C2212yescomplemented
37#22583530C2212yescomplemented
38C22⋊(C283⋊C8)C2211yescharacteristic, complemented
39(C2×D283)⋊C8C2212yescomplemented
40C22×D4×C283C2211yescharacteristic
41C2×SD16×C283C2212yescomplemented
42C22⋊Q8×C283C2211yescharacteristic, complemented
43#102×C283C2212yescomplemented
44C22⋊C8×C283C2211yescharacteristic, complemented
45C23×D283—28no—
46C23×D283—22no—
47C23×D283—22no—
48C23×D283—21no—
49C23×D283—44no—
50C23×D283—44no—
51C23×D283D411yescharacteristic
52C2×Dic566—21no—
53C2×Dic566C2311yescharacteristic, complemented
54Q8×D283—24no—
55C2×D1132—24no—
56C2×D1132—21no—
57C2×D1132C2312yes—
58C2×C22⋊D283—28no—
59C2×C22⋊D283—22no—
60C2×C22⋊D283—22no—
61D4×D283—28no—
62D4×D283—28no—
63D4×D283—24no—
64D4×D283—24no—
65D4×D283—24no—
66D4×D283D418yes—
67C22×Dic283D411yescharacteristic
68C2×C4×D283—24no—
69C2×C4×D283—21no—
70C2×C4×D283—21no—
71C2×C4×D283D411yescharacteristic
72C2×C4×D283D411yescharacteristic
73C2×C4×D283C2311yescharacteristic
74C2×C4×D283C2311yescharacteristic
75#17985381—24no—
76#17985380—24no—
77#17985379—24no—
78C2×C283⋊C8C2312yescomplemented
79C8×D283—24no—
80C4⋊Dic283—21no—
81C4⋊Dic283C2311yescharacteristic, complemented
82Dic283⋊C4—21no—
83Dic283⋊C4—21no—
84Dic283⋊C4—21no—
85C22⋊Dic283—21no—
86(C2×D283)⋊C4—21no—
87(C2×D283)⋊C4—21no—
88C24×C283—21no—
89C2×Q8×C283C2311yescharacteristic, complemented
90C2×D4×C283—24no—
91C2×D4×C283—21no—
92C2×D4×C283C2312yes—
93C22×C4×C283C2311yescharacteristic
94SD16×C283—24no—
95C2×C8×C283C2312yescomplemented
96C4⋊C4×C283—21no—
97C4⋊C4×C283C2311yescharacteristic, complemented
98C22⋊C4×C283—21no—
99C22×D283—22no—
100C22×D283—22no—
101C22×D283—28no—
102C22×D283—24no—
103C22×D283—22no—
104C22×D283—28no—
105C22×D283—44no—
106C22×D283—432no—
107C22×D283—42no—
108C22×D283—42no—
109C22×D283—44no—
110C22×D283—44no—
111C22×D283—44no—
112C22×D283SD1614yescomplemented
113C22×D283C2×D411yescharacteristic
114C22×D283C2×D411yescharacteristic
115Dic566—22no—
116Dic566—42no—
117D1132—24no—
118D1132—22no—
119D1132C2×D414yes—
120C22⋊D283—28no—
121C22⋊D283—28no—
122C2×Dic283—22no—
123C2×Dic283—21no—
124C2×Dic283—21no—
125C2×Dic283C2×D411yescharacteristic
126C2×Dic283C2×D411yescharacteristic
127C4×D283—22no—
128C4×D283—22no—
129C4×D283—22no—
130C4×D283—42no—
131C4×D283—42no—
132C4×D283C2×D414yes—
133C283⋊C8—22no—
134C23×C283—24no—
135C23×C283—22no—
136C23×C283—42no—
137C23×C283—42no—
138C23×C283C2×D411yescharacteristic
139Q8×C283—22no—
140D4×C283—24no—
141D4×C283—22no—
142D4×C283C2×D414yes—
143C2×C4×C283—22no—
144C2×C4×C283—21no—
145C2×C4×C283—21no—
146C2×C4×C283C2×D411yescharacteristic
147C2×C4×C283C2411yescharacteristic
148C8×C283—22no—
149C22×C283—21no—
150C22×C283—21no—
151C22×C283—24no—
152C22×C283—48no—
153C22×C283—42no—
154C22×C283—42no—
155C22×C283C2×SD1612yescomplemented
156C22×C283C22×D411yescharacteristic
157C2×D283—21no—
158C2×D283—21no—
159C2×D283—28no—
160C2×D283—24no—
161C2×D283—416no—
162C2×D283—48no—
163C2×D283—42no—
164C2×D283—42no—
165C2×D283—416no—
166C2×D283#12012yes—
167C2×D283C2×SD1612yes—
168C2×D283C2×SD1612yescomplemented
169C2×D283AGL(1,ℤ/8)12yes—
170C4×C283—21no—
171C4×C283—41no—
172C4×C283—41no—
173C4×C283C22×D412yes—
174Dic283—21no—
175Dic283—41no—
176Dic283—41no—
177Dic283#12012yes—
178C2×C283—22no—
179C2×C283—44no—
180C2×C283C2×#12011yescharacteristic
181C2×C283C22×SD1611yescharacteristic
182C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
183D283—22no—
184D283—44no—
185D283#45014yesdirect product factor
186C283C2×#45011yescharacteristic, complemented, Sylow
187C2×#450—2831nomaximal, Sylow
188C23×D4—2831no—
189C22×SD16—2832no—
190C2×C22⋊Q8—2831no—
191#450—2831no—
192#450—2832no—
193#450—2831no—
194#450—2832no—
195#450—2831no—
196#450D28311yescharacteristic, direct product factor
197C2×#102—2832no—
198C2×C22⋊C8—2831no—
199C25—5661no—
200C22×Q8—2831no—
201C22×D4—2832no—
202C22×D4—2832no—
203C22×D4—2831no—
204C22×D4—5661no—
205C22×D4—5661no—
206C22×D4—5661no—
207C22×D4—5668no—
208C22×D4C2×D28311yescharacteristic
209C23×C4—2831no—
210C2×SD16—2832no—
211C2×SD16—2832no—
212C2×SD16—2832no—
213C2×SD16—5668no—
214C2×SD16C2×D28312yescomplemented
215C22×C8—2832no—
216C22⋊Q8—2831no—
217C22⋊Q8—2831no—
218C22⋊Q8—2831no—
219C22⋊Q8—5661no—
220C22⋊Q8—5661no—
221C22⋊Q8C2×D28311yescharacteristic, complemented
222C2×C4⋊C4—2831no—
223C2×C4⋊C4—5661no—
224C2×C22⋊C4—5661no—
225#102—2832no—
226#102—2832no—
227#102—2832no—
228#102C2×D28312yescomplemented
229C22⋊C8—2831no—
230C22⋊C8—2832no—
231C22⋊C8C2×D28311yescharacteristic, complemented
232C24—21no—
233C24—2831no—
234C24—5662no—
235C24—5662no—
236C24—5668no—
237C24—5661no—
238C24—11324no—
239C24—11324no—
240C2×Q8—2831no—
241C2×Q8—5661no—
242C2×Q8—5664no—
243C2×Q8C22×D28311yescharacteristic, complemented
244C2×D4—24no—
245C2×D4—21no—
246C2×D4—2838no—
247C2×D4—2832no—
248C2×D4—5664no—
249C2×D4—5664no—
250C2×D4—5662no—
251C2×D4—5662no—
252C2×D4—5664no—
253C2×D4—5668no—
254C2×D4—5668no—
255C2×D4—5668no—
256C2×D4—5664no—
257C2×D4—5661no—
258C2×D4C22×D28312yes—
259C22×C4—2831no—
260C22×C4—2831no—
261C22×C4—2831no—
262C22×C4—2831no—
263C22×C4—2831no—
264C22×C4—5664no—
265C22×C4—5661no—
266C22×C4—5661no—
267C22×C4C22×D28311yescharacteristic
268SD16—24no—
269SD16—5664no—
270SD16—5664no—
271SD16—5664no—
272C2×C8—2832no—
273C2×C8—5664no—
274C2×C8C22×D28312yescomplemented
275C4⋊C4—21no—
276C4⋊C4—2831no—
277C4⋊C4—5661no—
278C4⋊C4—5661no—
279C4⋊C4—5661no—
280C4⋊C4—5661no—
281C4⋊C4C22×D28311yescharacteristic, complemented
282C22⋊C4—21no—
283C22⋊C4—5661no—
284C22⋊C4—5661no—
285C22⋊C4—5661no—
286C23—24no—
287C23—22no—
288C23—42no—
289C23—42no—
290C23—2831no—
291C23—2834no—
292C23—2831no—
293C23—5662no—
294C23—5662no—
295C23—5668no—
296C23—5664no—
297C23—5668no—
298C23—5662no—
299C23—11322no—
300C23—11324no—
301C23—11324no—
302C23—11324no—
303C23—113232no—
304C23—11324no—
305C23—11322no—
306C23D4×D28311yescharacteristic
307Q8—22no—
308Q8—5662no—
309Q8—11322no—
310D4—24no—
311D4—22no—
312D4—2834no—
313D4—5664no—
314D4—5668no—
315D4—5668no—
316D4—5662no—
317D4D4×D28314yes—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—2831no—
322C2×C4—2834no—
323C2×C4—2831no—
324C2×C4—5662no—
325C2×C4—5662no—
326C2×C4—5662no—
327C2×C4—5662no—
328C2×C4—5661no—
329C2×C4—5661no—
330C2×C4—11322no—
331C2×C4—11322no—
332C2×C4D4×D28311yescharacteristic
333C2×C4C23×D28311yescharacteristic
334C8—22no—
335C8—5662no—
336C22—21no—
337C22—21no—
338C22—24no—
339C22—48no—
340C22—42no—
341C22—42no—
342C22—2832no—
343C22—2832no—
344C22—2832no—
345C22—2832no—
346C22—5668no—
347C22—5661no—
348C22—5661no—
349C22—5664no—
350C22—11328no—
351C22—113216no—
352C22—113216no—
353C22—11322no—
354C22—11322no—
355C22SD16×D28312yescomplemented
356C22C2×D4×D28311yescharacteristic
357C4—21no—
358C4—41no—
359C4—41no—
360C4—2832no—
361C4—5661no—
362C4—11321no—
363C4—11321no—
364C4C2×D4×D28312yes—
365C2—22no—
366C2—44no—
367C2—2834no—
368C2—5662no—
369C2—11324no—
370C2#120×D28311yescharacteristic
371C2C2×SD16×D28311yescharacteristic
372C2AGL(1,ℤ/8)×D28311yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#450×C283C2
socleC22×C283C22×D4
prime radicalC22×D4×D283C2
hypercenter#450D283
hypocenterC283C2×#450
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. C2×SD16×D283C2semidirect
3. C22⋊Q8×D283C2semidirect
4. #31826618C2semidirect
5. #31826617C2semidirect
6. #31826616C2semidirect
7. #31826615C2semidirect
8. #31826614C2semidirect
9. #102×D283C2semidirect
10. C22⋊C8×D283C2semidirect
11. #450×C283C2semidirect
13. C2×Q8×D283C22semidirect
22. C2×#17985381C22semidirect
23. C2×#17985380C22semidirect
24. C2×#17985379C22semidirect
26. C2×C8×D283C22semidirect
27. #22583590C22semidirect
28. #22583589C22semidirect
31. #22583586C22semidirect
33. C4⋊C4×D283C22semidirect
35. #22583532C22semidirect
36. #22583531C22semidirect
37. #22583530C22semidirect
38. C22⋊(C283⋊C8)C22semidirect
39. (C2×D283)⋊C8C22semidirect
41. C2×SD16×C283C22semidirect
42. C22⋊Q8×C283C22semidirect
43. #102×C283C22semidirect
44. C22⋊C8×C283C22semidirect
53. C2×Dic566C23semidirect
78. C2×C283⋊C8C23semidirect
81. C4⋊Dic283C23semidirect
89. C2×Q8×C283C23semidirect
95. C2×C8×C283C23semidirect
97. C4⋊C4×C283C23semidirect
112. C22×D283SD16semidirect
155. C22×C283C2×SD16semidirect
168. C2×D283C2×SD16semidirect
185. D283#450direct (N × H)
186. C283C2×#450semidirect
196. #450D283direct (N × H)
214. C2×SD16C2×D283semidirect
221. C22⋊Q8C2×D283semidirect
228. #102C2×D283semidirect
231. C22⋊C8C2×D283semidirect
243. C2×Q8C22×D283semidirect
274. C2×C8C22×D283semidirect
281. C4⋊C4C22×D283semidirect
355. C22SD16×D283semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D2832
C2×SD16×D2832
C22⋊Q8×D2832
#318266182
#318266172
#318266162
#318266152
#318266142
#102×D2832
C22⋊C8×D2832
#450×C2832
C2×#450283