Finite Group Databasebeta

← #49130489

Subgroups — #49130489

Order 36224.

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

normal subgroups only

This group has 994 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
1C22×D4×D283C211yesmaximal, characteristic, complemented
2C2×#22583637C212yesmaximal, complemented
3#31826978C211yesmaximal, characteristic, complemented
4#31826615C211yesmaximal, characteristic, complemented
5#31826614C212yesmaximal, complemented
6#31826601C211yesmaximal, characteristic, complemented
7#31826600C212yesmaximal, complemented
8#31826599C211yesmaximal, characteristic, complemented
9#31826468C212yesmaximal, complemented
10#31826427C211yesmaximal, characteristic, complemented
11#450×C283C211yesmaximal, characteristic, complemented
12C24×D283—21no—
13C2×#17985393C2211yescharacteristic, 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
22#22583637—28no—
23C2×#17985379C2212yescomplemented
24C2×D2264C2212yescomplemented
25C2×#17985377C2212yescomplemented
26C2×#17985375C2212yescomplemented
27#22583595—21no—
28#22583586C2211yescharacteristic, complemented
29#22583585C2211yescharacteristic, complemented
30#22583584C2211yescharacteristic, complemented
31#22583580—21no—
32#22583572—21no—
33(C22⋊D283)⋊C4—21no—
34#22583568C2211yescharacteristic, complemented
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
52#17985393—24no—
53C2×D1132—24no—
54C2×D1132—21no—
55C2×D1132—21no—
56C2×D1132—21no—
57C2×D1132C2312yes—
58C2×D1132C2311yescharacteristic, complemented
59C2×C22⋊D283—28no—
60C2×C22⋊D283—22no—
61C2×C22⋊D283—22no—
62C2×C22⋊D283—21no—
63D4×D283—28no—
64D4×D283—28no—
65D4×D283—24no—
66D4×D283—24no—
67D4×D283—24no—
68D4×D283D418yescomplemented
69C22×Dic283D411yescharacteristic
70C2×C4×D283—24no—
71C2×C4×D283—21no—
72C2×C4×D283D411yescharacteristic
73C2×C4×D283D411yescharacteristic
74C2×C4×D283C2311yescharacteristic
75C2×C4×D283C2311yescharacteristic
76#17985379—24no—
77D2264—24no—
78#17985377—24no—
79#17985375—24no—
80C2×C283⋊C8C2312yescomplemented
81C4⋊Dic283C2311yescharacteristic, complemented
82Dic283⋊C4—21no—
83(C2×D283)⋊C4—21no—
84(C2×D283)⋊C4—21no—
85(C2×D283)⋊C4—21no—
86(C2×D283)⋊C4—21no—
87C4×Dic283—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—22no—
102C22×D283—28no—
103C22×D283—24no—
104C22×D283—22no—
105C22×D283—28no—
106C22×D283—21no—
107C22×D283—432no—
108C22×D283—44no—
109C22×D283—44no—
110C22×D283—44no—
111C22×D283—42no—
112C22×D283—42no—
113C22×D283—44no—
114C22×D283C2×D411yescharacteristic
115C22×D283C2×D411yescharacteristic
116D1132—24no—
117D1132—22no—
118D1132—22no—
119D1132—42no—
120D1132—42no—
121D1132C2×D414yes—
122C22⋊D283—28no—
123C22⋊D283—28no—
124C22⋊D283—42no—
125C2×Dic283—22no—
126C2×Dic283—21no—
127C2×Dic283C2×D411yescharacteristic
128C2×Dic283C2×D411yescharacteristic
129C4×D283—22no—
130C4×D283—22no—
131C4×D283—22no—
132C4×D283—42no—
133C4×D283C2×D414yes—
134C283⋊C8—22no—
135C23×C283—24no—
136C23×C283—22no—
137C23×C283—42no—
138C23×C283—42no—
139C23×C283C2×D411yescharacteristic
140Q8×C283—22no—
141D4×C283—24no—
142D4×C283—22no—
143D4×C283C2×D414yes—
144C2×C4×C283—22no—
145C2×C4×C283—21no—
146C2×C4×C283—21no—
147C2×C4×C283C2×D411yescharacteristic
148C2×C4×C283C2411yescharacteristic
149C8×C283—22no—
150C22×C283—21no—
151C22×C283—21no—
152C22×C283—24no—
153C22×C283—48no—
154C22×C283—42no—
155C22×C283—42no—
156C22×C283AGL(1,ℤ/8)12yescomplemented
157C22×C283C22×D411yescharacteristic
158C2×D283—28no—
159C2×D283—21no—
160C2×D283—21no—
161C2×D283—21no—
162C2×D283—21no—
163C2×D283—24no—
164C2×D283—416no—
165C2×D283—416no—
166C2×D283—48no—
167C2×D283—42no—
168C2×D283—42no—
169C2×D283—41no—
170C2×D283—41no—
171C2×D283—41no—
172C2×D283#12012yes—
173C2×D283AGL(1,ℤ/8)12yescomplemented
174C4×C283—21no—
175C4×C283—41no—
176C4×C283—41no—
177C4×C283C22×D412yes—
178Dic283—21no—
179Dic283—41no—
180Dic283#12012yes—
181C2×C283—22no—
182C2×C283—44no—
183C2×C283C2×#12011yescharacteristic
184C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
186D283—22no—
187D283—22no—
188D283—44no—
189D283—81no—
190C283#300311yescharacteristic, complemented, Sylow
191#3003—2831nomaximal, Sylow
192C23×D4—2831no—
193C2×AGL(1,ℤ/8)—2832no—
194#525—2831no—
195#450—2832no—
196#450—2831no—
197#450D28311yescharacteristic, complemented
198#447—2831no—
199#447—2832no—
200#447—2831no—
201#418—2832no—
202#407—2831no—
203C25—5661no—
204C2×#41—2831no—
205C22×D4—2832no—
206C22×D4—2832no—
207C22×D4—2831no—
208C22×D4—5661no—
209C22×D4—5661no—
210C22×D4—5661no—
211C22×D4—5668no—
212C22×D4C2×D28311yescharacteristic
213C23×C4—2831no—
214AGL(1,ℤ/8)—5668no—
215C2×SD16—2832no—
216C2×SD16C2×D28312yescomplemented
217C2×D8—2832no—
218C2×D8—2832no—
219C2×M16—2832no—
220#123—5661no—
221C22⋊Q8—2831no—
222C22⋊Q8C2×D28311yescharacteristic, complemented
223#121—2831no—
224#121—2831no—
225#120—5661no—
226C4×D4—5661no—
227C4×D4—5661no—
228#117—2831no—
229#102—2832no—
230#102—2832no—
231#102—2832no—
232#102C2×D28312yescomplemented
233C22⋊C8—2831no—
234C22⋊C8—2832no—
235C22⋊C8C2×D28311yescharacteristic, complemented
236C24—21no—
237C24—2831no—
238C24—5662no—
239C24—5662no—
240C24—5668no—
241C24—5661no—
242C24—11324no—
243C24—11324no—
244#41—5664no—
245C2×Q8C22×D28311yescharacteristic, complemented
246C2×D4—24no—
247C2×D4—21no—
248C2×D4—2838no—
249C2×D4—2832no—
250C2×D4—2831no—
251C2×D4—5664no—
252C2×D4—5661no—
253C2×D4—5661no—
254C2×D4—5664no—
255C2×D4—5662no—
256C2×D4—5662no—
257C2×D4—5664no—
258C2×D4—5668no—
259C2×D4—5668no—
260C2×D4—5668no—
261C2×D4—5664no—
262C2×D4—5661no—
263C2×D4—5661no—
264C2×D4C22×D28312yes—
265C22×C4—2831no—
266C22×C4—2831no—
267C22×C4—2831no—
268C22×C4—2831no—
269C22×C4—2831no—
270C22×C4—5664no—
271C22×C4—5661no—
272C22×C4C22×D28311yescharacteristic
273SD16—24no—
274SD16—5664no—
275D8—5664no—
276D8—5664no—
277M16—5664no—
278C2×C8—2832no—
279C2×C8C22×D28312yescomplemented
280C4⋊C4—21no—
281C4⋊C4—2831no—
282C4⋊C4—5661no—
283C4⋊C4C22×D28311yescharacteristic, complemented
284C22⋊C4—21no—
285C22⋊C4—5661no—
286C22⋊C4—5661no—
287C22⋊C4—5661no—
288C22⋊C4—5661no—
289C42—5661no—
290C23—24no—
291C23—22no—
292C23—42no—
293C23—42no—
294C23—2831no—
295C23—2831no—
296C23—5662no—
297C23—5662no—
298C23—5668no—
299C23—5664no—
300C23—5668no—
301C23—5662no—
302C23—5662no—
303C23—5661no—
304C23—11322no—
305C23—11324no—
306C23—113232no—
307C23—11324no—
308C23—11324no—
309C23—11324no—
310C23—11322no—
311C23D4×D28311yescharacteristic
312Q8—22no—
313D4—24no—
314D4—22no—
315D4—2834no—
316D4—5664no—
317D4—5668no—
318D4—5668no—
319D4—5662no—
320D4—5662no—
321D4—11322no—
322D4—11322no—
323D4—11322no—
324D4D4×D28314yes—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—2831no—
329C2×C4—2834no—
330C2×C4—2831no—
331C2×C4—5662no—
332C2×C4—5662no—
333C2×C4—5662no—
334C2×C4—5662no—
335C2×C4—5661no—
336C2×C4—11322no—
337C2×C4D4×D28311yescharacteristic
338C2×C4C23×D28311yescharacteristic
339C8—22no—
340C8—5662no—
341C22—21no—
342C22—21no—
343C22—24no—
344C22—48no—
345C22—42no—
346C22—42no—
347C22—2832no—
348C22—2832no—
349C22—5668no—
350C22—5661no—
351C22—5661no—
352C22—5661no—
353C22—5661no—
354C22—5664no—
355C22—11328no—
356C22—113216no—
357C22—113216no—
358C22—11322no—
359C22—11322no—
360C22—11321no—
361C22—11321no—
362C22—11321no—
363C22#2258363712yescomplemented
364C22C2×D4×D28311yescharacteristic
365C4—21no—
366C4—41no—
367C4—41no—
368C4—2832no—
369C4—5661no—
370C4—11321no—
371C4C2×D4×D28312yes—
372C2—22no—
373C2—44no—
374C2—5662no—
375C2—5662no—
376C2—11324no—
377C2—22641no—
378C2#120×D28311yescharacteristic
379C2AGL(1,ℤ/8)×D28311yescharacteristic
380C2C2×#2258363711yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#450×C283C2
socleC22×C283C22×D4
hypercenter#450D283
hypocenterC283#3003
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
1. C22×D4×D283C2semidirect
2. C2×#22583637C2semidirect
3. #31826978C2semidirect
4. #31826615C2semidirect
5. #31826614C2semidirect
6. #31826601C2semidirect
7. #31826600C2semidirect
8. #31826599C2semidirect
9. #31826468C2semidirect
10. #31826427C2semidirect
11. #450×C283C2semidirect
13. C2×#17985393C22semidirect
23. C2×#17985379C22semidirect
24. C2×D2264C22semidirect
25. C2×#17985377C22semidirect
26. C2×#17985375C22semidirect
28. #22583586C22semidirect
29. #22583585C22semidirect
30. #22583584C22semidirect
34. #22583568C22semidirect
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
58. C2×D1132C23semidirect
68. D4×D283D4semidirect
80. C2×C283⋊C8C23semidirect
81. C4⋊Dic283C23semidirect
89. C2×Q8×C283C23semidirect
95. C2×C8×C283C23semidirect
97. C4⋊C4×C283C23semidirect
156. C22×C283AGL(1,ℤ/8)semidirect
173. C2×D283AGL(1,ℤ/8)semidirect
190. C283#3003semidirect
197. #450D283semidirect
216. C2×SD16C2×D283semidirect
222. C22⋊Q8C2×D283semidirect
232. #102C2×D283semidirect
235. C22⋊C8C2×D283semidirect
245. C2×Q8C22×D283semidirect
279. C2×C8C22×D283semidirect
283. C4⋊C4C22×D283semidirect
363. C22#22583637semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D2832
C2×#225836372
#318269782
#318266152
#318266142
#318266012
#318266002
#318265992
#318264682
#318264272
#450×C2832
#3003283