Finite Group Databasebeta

← #49130480

Subgroups — #49130480

Order 36224.

764 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 141 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×#17985391C211yesmaximal, characteristic, complemented
2C2×#22583631C212yesmaximal, complemented
3#31826978C211yesmaximal, characteristic, complemented
4#31826624C212yesmaximal, complemented
5#31826620C211yesmaximal, characteristic, complemented
6#31826611C212yesmaximal, complemented
7#31826606C211yesmaximal, characteristic, complemented
8#31826599C211yesmaximal, characteristic, complemented
9#31826458C212yesmaximal, complemented
10C22⋊C8×D283C211yesmaximal, characteristic, complemented
11#450×C283C211yesmaximal, characteristic, complemented
12C2×#17985393C2211yescharacteristic, complemented
13C2×#17985391—28no—
14C2×#17985391—21no—
15C2×#17985391—21no—
16C2×#17985391C2212yes—
17C2×#17985391C2212yes—
18C22×Dic566C2211yescharacteristic
19C22×C22⋊D283—21no—
20C23×Dic283—21no—
21C22×C4×D283C2211yescharacteristic
22#22583631—28no—
23C2×#17985382C2212yescomplemented
24C2×#17985381C2212yescomplemented
25C2×#17985377C2212yescomplemented
26C2×C8×D283C2212yescomplemented
27#22583595—21no—
28#22583586C2211yescharacteristic, complemented
29#22583585C2211yescharacteristic, complemented
30#22583584C2211yescharacteristic
31#22583580—21no—
32#22583572—21no—
33(C22⋊D283)⋊C4—21no—
34#22583568C2211yescharacteristic, complemented
35#22583535C2212yescomplemented
36#22583533C2212yescomplemented
37#22583531C2212yescomplemented
38C22⋊(C283⋊C8)C2211yescharacteristic
39(C2×D283)⋊C8C2212yescomplemented
40C22×D4×C283C2211yescharacteristic
41C2×SD16×C283C2212yescomplemented
42C22⋊Q8×C283C2211yescharacteristic
43#102×C283C2212yescomplemented
44C22⋊C8×C283C2211yescharacteristic
45C23×D283D411yescharacteristic
46#17985393—24no—
47#17985391—28no—
48#17985391—28no—
49#17985391—24no—
50#17985391—24no—
51#17985391—24no—
52#17985391D418yescomplemented
53C2×Dic566—24no—
54C2×Dic566—21no—
55C2×Dic566C2312yes—
56C2×D1132—21no—
57C2×D1132—21no—
58C2×D1132C2311yescharacteristic
59C2×C22⋊D283—28no—
60C2×C22⋊D283—22no—
61C2×C22⋊D283—22no—
62C2×C22⋊D283—21no—
63C22×Dic283—28no—
64C22×Dic283—22no—
65C22×Dic283—22no—
66C22×Dic283—21no—
67C22×Dic283D411yescharacteristic
68C2×C4×D283—24no—
69C2×C4×D283—21no—
70C2×C4×D283D411yescharacteristic
71C2×C4×D283D411yescharacteristic
72C2×C4×D283C2311yescharacteristic
73C2×C4×D283C2311yescharacteristic
74#17985382—24no—
75#17985381—24no—
76#17985377—24no—
77C2×C283⋊C8C2312yes—
78C8×D283—24no—
79C4⋊Dic283C2311yescharacteristic
80Dic283⋊C4—21no—
81(C2×D283)⋊C4—21no—
82(C2×D283)⋊C4—21no—
83(C2×D283)⋊C4—21no—
84(C2×D283)⋊C4—21no—
85C4×Dic283—21no—
86C24×C283—21no—
87C2×Q8×C283C2311yescharacteristic
88C2×D4×C283—24no—
89C2×D4×C283—21no—
90C2×D4×C283C2312yes—
91C22×C4×C283C2311yescharacteristic
92SD16×C283—24no—
93C2×C8×C283C2312yes—
94C4⋊C4×C283—21no—
95C4⋊C4×C283C2311yescharacteristic
96C22⋊C4×C283—21no—
97C22×D283—22no—
98C22×D283—22no—
99C22×D283—21no—
100C22×D283—41no—
101C22×D283C2×D411yescharacteristic
102C22×D283C2×D411yescharacteristic
103Dic566—24no—
104Dic566—22no—
105Dic566C2×D414yes—
106D1132—22no—
107D1132—42no—
108D1132—42no—
109C22⋊D283—28no—
110C22⋊D283—28no—
111C22⋊D283—42no—
112C2×Dic283—24no—
113C2×Dic283—28no—
114C2×Dic283—22no—
115C2×Dic283—22no—
116C2×Dic283—28no—
117C2×Dic283—21no—
118C2×Dic283C2×D411yescharacteristic
119C2×Dic283C2×D411yescharacteristic
120C4×D283—22no—
121C4×D283—22no—
122C4×D283—22no—
123C4×D283—42no—
124C4×D283C2×D414yes—
125C283⋊C8—22no—
126C23×C283—24no—
127C23×C283—22no—
128C23×C283—42no—
129C23×C283—42no—
130C23×C283C2×D411yescharacteristic
131Q8×C283—22no—
132D4×C283—24no—
133D4×C283—22no—
134D4×C283C2×D414yes—
135C2×C4×C283—22no—
136C2×C4×C283—21no—
137C2×C4×C283—21no—
138C2×C4×C283C2×D411yescharacteristic
139C2×C4×C283C2411yescharacteristic
140C8×C283—22no—
141C22×C283—21no—
142C22×C283—21no—
143C22×C283—24no—
144C22×C283—48no—
145C22×C283—42no—
146C22×C283—42no—
147C22×C283#13512yescomplemented
148C22×C283C22×D411yescharacteristic
149C2×D283—21no—
150C2×D283—21no—
151C2×D283—21no—
152C2×D283—21no—
153C2×D283—21no—
154C2×D283—44no—
155C2×D283—41no—
156C2×D283—41no—
157C2×D283—41no—
158C2×D283#12012yes—
159C4×C283—21no—
160C4×C283—41no—
161C4×C283—41no—
162C4×C283C22×D412yes—
163Dic283—21no—
164Dic283—24no—
165Dic283—41no—
166Dic283#12012yes—
167C2×C283—22no—
168C2×C283—44no—
169C2×C283C2×#12011yescharacteristic
170C2×C283C2×#13511yescharacteristic
171C2×C283#57511yescharacteristic
172D283—22no—
173D283—41no—
174D283—81no—
175C283#300211yescharacteristic, complemented, Sylow
176#3002—2831nomaximal, Sylow
177C22×#41—2831no—
178C2×#135—2832no—
179#525—2831no—
180#452—2832no—
181#451—2831no—
182#450D28311yescharacteristic, complemented
183#449—2832no—
184#448—2831no—
185#447—2831no—
186#416—2832no—
187C2×C22⋊C8—2831no—
188C2×#41—2831no—
189C2×#41—2832no—
190C2×#41—2832no—
191C2×#41—5661no—
192C2×#41—5661no—
193C2×#41—5668no—
194C22×Q8—2831no—
195C22×D4—5661no—
196C22×D4C2×D28311yescharacteristic
197C23×C4—2831no—
198C23×C4—5661no—
199#135—5668no—
200C2×Q16—2832no—
201C2×SD16—2832no—
202C2×SD16C2×D28312yescomplemented
203C2×D8—2832no—
204C22×C8—2832no—
205#123—5661no—
206C22⋊Q8—2831no—
207C22⋊Q8C2×D28311yescharacteristic
208#121—2831no—
209#121—2831no—
210#120—5661no—
211C4×D4—5661no—
212C4×D4—5661no—
213#117—2831no—
214#103—2832no—
215#103—2832no—
216#102—2832no—
217#102C2×D28312yescomplemented
218C22⋊C8—2831no—
219C22⋊C8—2832no—
220C22⋊C8C2×D28311yescharacteristic
221C24—21no—
222C24—2831no—
223#41—2838no—
224#41—5664no—
225#41—5664no—
226#41—5664no—
227#41—5668no—
228#41—5668no—
229#41—5664no—
230C2×Q8—2832no—
231C2×Q8—5664no—
232C2×Q8—5661no—
233C2×Q8C22×D28311yescharacteristic
234C2×D4—24no—
235C2×D4—21no—
236C2×D4—2831no—
237C2×D4—5661no—
238C2×D4—5661no—
239C2×D4—5662no—
240C2×D4—5662no—
241C2×D4—5668no—
242C2×D4—5661no—
243C2×D4C22×D28312yes—
244C22×C4—2831no—
245C22×C4—2831no—
246C22×C4—2831no—
247C22×C4—2831no—
248C22×C4—2831no—
249C22×C4—5664no—
250C22×C4—5661no—
251C22×C4—5662no—
252C22×C4—5662no—
253C22×C4—5668no—
254C22×C4—5661no—
255C22×C4C22×D28311yescharacteristic
256Q16—5664no—
257SD16—24no—
258SD16—5664no—
259D8—5664no—
260C2×C8—2832no—
261C2×C8—5664no—
262C2×C8C22×D28312yes—
263C4⋊C4—21no—
264C4⋊C4—2831no—
265C4⋊C4—5661no—
266C4⋊C4C22×D28311yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—5661no—
269C22⋊C4—5661no—
270C22⋊C4—5661no—
271C22⋊C4—5661no—
272C42—5661no—
273C23—24no—
274C23—22no—
275C23—42no—
276C23—42no—
277C23—2831no—
278C23—2831no—
279C23—5662no—
280C23—5662no—
281C23—5661no—
282C23—11321no—
283C23D4×D28311yescharacteristic
284Q8—22no—
285Q8—2834no—
286Q8—5664no—
287Q8—5662no—
288D4—24no—
289D4—22no—
290D4—5668no—
291D4—5668no—
292D4—5662no—
293D4—11322no—
294D4—11322no—
295D4—11322no—
296D4D4×D28314yes—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—2831no—
301C2×C4—2834no—
302C2×C4—2831no—
303C2×C4—5662no—
304C2×C4—5662no—
305C2×C4—5662no—
306C2×C4—5668no—
307C2×C4—5664no—
308C2×C4—5668no—
309C2×C4—5662no—
310C2×C4—5662no—
311C2×C4—5661no—
312C2×C4—11322no—
313C2×C4D4×D28311yescharacteristic
314C2×C4C23×D28311yescharacteristic
315C8—22no—
316C8—5662no—
317C22—21no—
318C22—21no—
319C22—24no—
320C22—48no—
321C22—42no—
322C22—42no—
323C22—2832no—
324C22—5661no—
325C22—5661no—
326C22—5661no—
327C22—5661no—
328C22—5661no—
329C22—11324no—
330C22—11321no—
331C22—11321no—
332C22—11321no—
333C22#2258363112yescomplemented
334C22C2×D4×D28311yescharacteristic
335C4—21no—
336C4—41no—
337C4—41no—
338C4—2832no—
339C4—5661no—
340C4—5664no—
341C4—11321no—
342C4C2×D4×D28312yes—
343C2—22no—
344C2—44no—
345C2—5662no—
346C2—11321no—
347C2—22641no—
348C2#120×D28311yescharacteristic
349C2C2×#2258363111yescharacteristic
350C2#3182726111yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#450×C283C2
socleC22×C283C22×D4
hypercenter#450D283
hypocenterC283#3002
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×#17985391C2semidirect
2. C2×#22583631C2semidirect
3. #31826978C2semidirect
4. #31826624C2semidirect
5. #31826620C2semidirect
6. #31826611C2semidirect
7. #31826606C2semidirect
8. #31826599C2semidirect
9. #31826458C2semidirect
10. C22⋊C8×D283C2semidirect
11. #450×C283C2semidirect
12. C2×#17985393C22semidirect
23. C2×#17985382C22semidirect
24. C2×#17985381C22semidirect
25. C2×#17985377C22semidirect
26. C2×C8×D283C22semidirect
28. #22583586C22semidirect
29. #22583585C22semidirect
34. #22583568C22semidirect
35. #22583535C22semidirect
36. #22583533C22semidirect
37. #22583531C22semidirect
39. (C2×D283)⋊C8C22semidirect
41. C2×SD16×C283C22semidirect
43. #102×C283C22semidirect
52. #17985391D4semidirect
147. C22×C283#135semidirect
175. C283#3002semidirect
182. #450D283semidirect
202. C2×SD16C2×D283semidirect
217. #102C2×D283semidirect
333. C22#22583631semidirect

Maximal subgroups

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

subgroupindex
C22×#179853912
C2×#225836312
#318269782
#318266242
#318266202
#318266112
#318266062
#318265992
#318264582
C22⋊C8×D2832
#450×C2832
#3002283