Finite Group Databasebeta

← #49130516

Subgroups — #49130516

Order 36224.

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

normal subgroups only

This group has 754 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
2C2×#22583639C212yesmaximal, complemented
3C22⋊Q8×D283C211yesmaximal, characteristic, complemented
4#31826628C212yesmaximal, complemented
5#31826624C212yesmaximal, complemented
6#31826623C211yesmaximal, characteristic, complemented
7#31826620C211yesmaximal, characteristic, complemented
8#31826616C211yesmaximal, characteristic, complemented
9#31826463C212yesmaximal, complemented
10#31826427C211yesmaximal, characteristic, complemented
11#450×C283C211yesmaximal, characteristic, complemented
12C2×#17985391—28no—
13C2×#17985391—21no—
14C2×#17985391—21no—
15C2×#17985391C2212yes—
16C2×#17985391C2212yes—
17C22×Dic566C2211yescharacteristic
18C2×Q8×D283C2211yescharacteristic, complemented
19C22×C22⋊D283—21no—
20C23×Dic283—21no—
21C22×C4×D283C2211yescharacteristic
22#22583639—28no—
23C2×Dic1132C2212yescomplemented
24C2×#17985382C2212yescomplemented
25C2×#17985380C2212yescomplemented
26C2×#17985375C2212yescomplemented
27#22583590C2211yescharacteristic, complemented
28#22583589C2211yescharacteristic
29#22583588—21no—
30#22583587—21no—
31#22583586C2211yescharacteristic, complemented
32C4⋊C4×D283—21no—
33C4⋊C4×D283C2211yescharacteristic, complemented
34C22⋊C4×D283—21no—
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#17985391—28no—
47#17985391—28no—
48#17985391—24no—
49#17985391—24no—
50#17985391—24no—
51#17985391D418yes—
52C2×Dic566—24no—
53C2×Dic566—21no—
54C2×Dic566—21no—
55C2×Dic566C2311yescharacteristic
56C2×Dic566C2312yes—
57Q8×D283—24no—
58C2×C22⋊D283—28no—
59C2×C22⋊D283—22no—
60C2×C22⋊D283—22no—
61C22×Dic283—28no—
62C22×Dic283—22no—
63C22×Dic283—22no—
64C22×Dic283—21no—
65C22×Dic283D411yescharacteristic
66C2×C4×D283—24no—
67C2×C4×D283—21no—
68C2×C4×D283—21no—
69C2×C4×D283D411yescharacteristic
70C2×C4×D283D411yescharacteristic
71C2×C4×D283C2311yescharacteristic
72C2×C4×D283C2311yescharacteristic
73Dic1132—24no—
74#17985382—24no—
75#17985380—24no—
76#17985375—24no—
77C2×C283⋊C8C2312yes—
78C4⋊Dic283—21no—
79C4⋊Dic283C2311yescharacteristic
80Dic283⋊C4—21no—
81Dic283⋊C4—21no—
82Dic283⋊C4—21no—
83C22⋊Dic283—21no—
84(C2×D283)⋊C4—21no—
85(C2×D283)⋊C4—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—41no—
100C22×D283C2×D411yescharacteristic
101C22×D283C2×D411yescharacteristic
102Dic566—24no—
103Dic566—22no—
104Dic566—22no—
105Dic566—42no—
106Dic566C2×D414yes—
107C22⋊D283—28no—
108C22⋊D283—28no—
109C2×Dic283—24no—
110C2×Dic283—28no—
111C2×Dic283—22no—
112C2×Dic283—22no—
113C2×Dic283—28no—
114C2×Dic283—21no—
115C2×Dic283—21no—
116C2×Dic283C2×D411yescharacteristic
117C2×Dic283C2×D411yescharacteristic
118C4×D283—22no—
119C4×D283—22no—
120C4×D283—22no—
121C4×D283—42no—
122C4×D283—42no—
123C4×D283C2×D414yes—
124C283⋊C8—22no—
125C23×C283—24no—
126C23×C283—22no—
127C23×C283—42no—
128C23×C283—42no—
129C23×C283C2×D411yescharacteristic
130Q8×C283—22no—
131D4×C283—24no—
132D4×C283—22no—
133D4×C283C2×D414yes—
134C2×C4×C283—22no—
135C2×C4×C283—21no—
136C2×C4×C283—21no—
137C2×C4×C283C2×D411yescharacteristic
138C2×C4×C283C2411yescharacteristic
139C8×C283—22no—
140C22×C283—21no—
141C22×C283—21no—
142C22×C283—24no—
143C22×C283—48no—
144C22×C283—42no—
145C22×C283—42no—
146C22×C283#13712yescomplemented
147C22×C283C22×D411yescharacteristic
148C2×D283—21no—
149C2×D283—21no—
150C2×D283—21no—
151C2×D283—21no—
152C2×D283—21no—
153C2×D283—44no—
154C2×D283#12012yes—
155C4×C283—21no—
156C4×C283—41no—
157C4×C283—41no—
158C4×C283C22×D412yes—
159Dic283—21no—
160Dic283—24no—
161Dic283—41no—
162Dic283—41no—
163Dic283#12012yes—
164C2×C283—22no—
165C2×C283—44no—
166C2×C283C2×#12011yescharacteristic
167C2×C283C2×#13711yescharacteristic
168C2×C283#57511yescharacteristic
169D283—22no—
170D283—41no—
171C283#300711yescharacteristic, complemented, Sylow
172#3007—2831nomaximal, Sylow
173C22×#41—2831no—
174C2×#137—2832no—
175C2×C22⋊Q8—2831no—
176#452—2832no—
177#452—2832no—
178#451—2831no—
179#451—2831no—
180#450—2831no—
181#450D28311yescharacteristic, complemented
182#417—2832no—
183#407—2831no—
184C2×#41—2832no—
185C2×#41—2832no—
186C2×#41—5661no—
187C2×#41—5661no—
188C2×#41—5668no—
189C22×Q8—2831no—
190C22×Q8—2831no—
191C22×D4—5661no—
192C22×D4C2×D28311yescharacteristic
193C23×C4—2831no—
194C23×C4—5661no—
195#137—5668no—
196C2×Q16—2832no—
197C2×Q16—2832no—
198C2×SD16—2832no—
199C2×SD16C2×D28312yescomplemented
200C2×M16—2832no—
201C22⋊Q8—2831no—
202C22⋊Q8—2831no—
203C22⋊Q8—2831no—
204C22⋊Q8—5661no—
205C22⋊Q8—5661no—
206C22⋊Q8C2×D28311yescharacteristic
207C2×C4⋊C4—2831no—
208C2×C4⋊C4—5661no—
209C2×C22⋊C4—5661no—
210#103—2832no—
211#103—2832no—
212#102—2832no—
213#102C2×D28312yescomplemented
214C22⋊C8—2831no—
215C22⋊C8—2832no—
216C22⋊C8C2×D28311yescharacteristic
217C24—21no—
218C24—2831no—
219#41—2838no—
220#41—5664no—
221#41—5664no—
222#41—5668no—
223#41—5668no—
224#41—5664no—
225C2×Q8—2832no—
226C2×Q8—2831no—
227C2×Q8—5661no—
228C2×Q8—5664no—
229C2×Q8—5664no—
230C2×Q8—5661no—
231C2×Q8C22×D28311yescharacteristic
232C2×D4—24no—
233C2×D4—21no—
234C2×D4—5662no—
235C2×D4—5662no—
236C2×D4—5668no—
237C2×D4C22×D28312yes—
238C22×C4—2831no—
239C22×C4—2831no—
240C22×C4—2831no—
241C22×C4—2831no—
242C22×C4—2831no—
243C22×C4—5664no—
244C22×C4—5661no—
245C22×C4—5661no—
246C22×C4—5662no—
247C22×C4—5662no—
248C22×C4—5668no—
249C22×C4—5661no—
250C22×C4C22×D28311yescharacteristic
251Q16—5664no—
252Q16—5664no—
253SD16—24no—
254SD16—5664no—
255M16—5664no—
256C2×C8—2832no—
257C2×C8C22×D28312yes—
258C4⋊C4—21no—
259C4⋊C4—2831no—
260C4⋊C4—5661no—
261C4⋊C4—5661no—
262C4⋊C4—5661no—
263C4⋊C4—5661no—
264C4⋊C4C22×D28311yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—5661no—
267C22⋊C4—5661no—
268C22⋊C4—5661no—
269C23—24no—
270C23—22no—
271C23—42no—
272C23—42no—
273C23—2831no—
274C23—2831no—
275C23—5662no—
276C23—5662no—
277C23—11321no—
278C23D4×D28311yescharacteristic
279Q8—22no—
280Q8—2834no—
281Q8—5664no—
282Q8—5662no—
283Q8—5662no—
284Q8—11322no—
285D4—24no—
286D4—22no—
287D4—5668no—
288D4—5668no—
289D4D4×D28314yes—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—2831no—
294C2×C4—2834no—
295C2×C4—2831no—
296C2×C4—5662no—
297C2×C4—5662no—
298C2×C4—5662no—
299C2×C4—5668no—
300C2×C4—5664no—
301C2×C4—5668no—
302C2×C4—5662no—
303C2×C4—5662no—
304C2×C4—5661no—
305C2×C4—5661no—
306C2×C4—11322no—
307C2×C4—11322no—
308C2×C4D4×D28311yescharacteristic
309C2×C4C23×D28311yescharacteristic
310C8—22no—
311C8—5662no—
312C22—21no—
313C22—21no—
314C22—24no—
315C22—48no—
316C22—42no—
317C22—42no—
318C22—2832no—
319C22—5661no—
320C22—5661no—
321C22—5661no—
322C22—5661no—
323C22—5661no—
324C22—11324no—
325C22#2258363912yescomplemented
326C22C2×D4×D28311yescharacteristic
327C4—21no—
328C4—41no—
329C4—41no—
330C4—2832no—
331C4—5661no—
332C4—5664no—
333C4—11321no—
334C4—11321no—
335C4C2×D4×D28312yes—
336C2—22no—
337C2—44no—
338C2—5662no—
339C2—11321no—
340C2#120×D28311yescharacteristic
341C2C2×#2258363911yescharacteristic
342C2#3182726111yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#450×C283C2
socleC22×C283C22×D4
prime radicalC22×#17985391C2
hypercenter#450D283
hypocenterC283#3007
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×#22583639C2semidirect
3. C22⋊Q8×D283C2semidirect
4. #31826628C2semidirect
5. #31826624C2semidirect
6. #31826623C2semidirect
7. #31826620C2semidirect
8. #31826616C2semidirect
9. #31826463C2semidirect
10. #31826427C2semidirect
11. #450×C283C2semidirect
18. C2×Q8×D283C22semidirect
23. C2×Dic1132C22semidirect
24. C2×#17985382C22semidirect
25. C2×#17985380C22semidirect
26. C2×#17985375C22semidirect
27. #22583590C22semidirect
31. #22583586C22semidirect
33. C4⋊C4×D283C22semidirect
35. #22583535C22semidirect
36. #22583533C22semidirect
37. #22583531C22semidirect
39. (C2×D283)⋊C8C22semidirect
41. C2×SD16×C283C22semidirect
43. #102×C283C22semidirect
146. C22×C283#137semidirect
171. C283#3007semidirect
181. #450D283semidirect
199. C2×SD16C2×D283semidirect
213. #102C2×D283semidirect
325. C22#22583639semidirect

Maximal subgroups

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

subgroupindex
C22×#179853912
C2×#225836392
C22⋊Q8×D2832
#318266282
#318266242
#318266232
#318266202
#318266162
#318264632
#318264272
#450×C2832
#3007283