Finite Group Databasebeta

← #49133102

Subgroups — #49133102

Order 36224.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827157C211yesmaximal, characteristic, complemented
2#31827106C211yesmaximal, characteristic
3#31826821C211yesmaximal, characteristic, complemented
4#31826793C211yesmaximal, characteristic, complemented
5#31826792C211yesmaximal, characteristic, complemented
6#31826768C211yesmaximal, characteristic, complemented
7#31826697C211yesmaximal, characteristic, complemented
8#31826689C211yesmaximal, characteristic, complemented
9#31826626C211yesmaximal, characteristic, complemented
10#31826615C211yesmaximal, characteristic, complemented
11#31826550C211yesmaximal, characteristic, complemented
12C8×C22⋊D283C211yesmaximal, characteristic, complemented
13#31826460C211yesmaximal, characteristic, complemented
14C2×#22583532C211yesmaximal, characteristic, complemented
15#467×C283C211yesmaximal, characteristic, complemented
16C2×#17985393C2211yescharacteristic
17C22×D1132C2211yescharacteristic
18C2×#17985379C2211yescharacteristic
19C2×#17985379C2211yescharacteristic
20C2×C8×D283C2211yescharacteristic, complemented
21#22583605—21no—
22#22583601—21no—
23#22583601C2211yescharacteristic, complemented
24#22583599C2211yescharacteristic
25#22583595—21no—
26#22583594—21no—
27#22583586C2211yescharacteristic, complemented
28#22583585—21no—
29#22583585C2211yescharacteristic
30#22583580—21no—
31Q8×Dic283C2211yescharacteristic, complemented
32#22583572C2211yescharacteristic
33C4×C22⋊D283C2211yescharacteristic
34#22583568C2211yescharacteristic, complemented
35#22583566—21no—
36C2×C4⋊Dic283C2211yescharacteristic
37C22⋊C4×D283—21no—
38#22583541C2211yescharacteristic, complemented
39#22583541C2211yescharacteristic, complemented
40Dic283⋊C8C2211yescharacteristic, complemented
41#22583534C2211yescharacteristic, complemented
42#22583534C2211yescharacteristic, complemented
43#22583532C2211yescharacteristic
44#22583532C2211yescharacteristic
45#22583532C2211yescharacteristic
46#22583532C2211yescharacteristic
47#22583530C2211yescharacteristic
48#22583530C2211yescharacteristic
49C22⋊(C283⋊C8)C2211yescharacteristic
50(C2×D283)⋊C8C2211yescharacteristic, complemented
51C8×Dic283C2211yescharacteristic, complemented
52C2×Q16×C283C2211yescharacteristic, complemented
53C22×C8×C283C2211yescharacteristic
54C22⋊Q8×C283C2211yescharacteristic
55C22⋊Q8×C283C2211yescharacteristic
56#107×C283C2211yescharacteristic, complemented
57#103×C283C2211yescharacteristic, complemented
58#103×C283C2211yescharacteristic, complemented
59C23×D283—21no—
60#17985393—22no—
61#17985393—41no—
62C2×D1132—22no—
63C2×D1132—21no—
64C2×D1132—21no—
65C2×D1132—21no—
66C2×D1132—41no—
67C2×D1132C2311yescharacteristic
68C2×D1132C2311yescharacteristic
69C2×C22⋊D283D411yescharacteristic
70C22×Dic283—21no—
71C2×C4×D283—21no—
72C2×C4×D283—21no—
73C2×C4×D283C2311yescharacteristic
74#17985379—22no—
75#17985379—41no—
76C2×C283⋊C8C2311yescharacteristic
77C2×C283⋊C8C2311yescharacteristic
78C8×D283—41no—
79C4⋊Dic283—21no—
80C4⋊Dic283—21no—
81C4⋊Dic283—21no—
82C4⋊Dic283C2311yescharacteristic
83C4⋊Dic283C2311yescharacteristic
84Dic283⋊C4—21no—
85Dic283⋊C4—21no—
86Dic283⋊C4D411yescharacteristic
87C22⋊Dic283D411yescharacteristic
88(C2×D283)⋊C4—21no—
89(C2×D283)⋊C4—21no—
90(C2×D283)⋊C4—21no—
91(C2×D283)⋊C4—21no—
92(C2×D283)⋊C4—21no—
93(C2×D283)⋊C4—21no—
94(C2×D283)⋊C4D411yescharacteristic
95C4×Dic283—21no—
96C4×Dic283—21no—
97C4×Dic283C2311yescharacteristic
98C2×Q8×C283C2311yescharacteristic
99C2×Q8×C283C2311yescharacteristic
100C22×C4×C283C2311yescharacteristic
101Q16×C283—41no—
102C2×C8×C283—41no—
103C2×C8×C283C2311yescharacteristic
104C2×C8×C283C2311yescharacteristic
105C4⋊C4×C283—21no—
106C4⋊C4×C283—21no—
107C4⋊C4×C283C2311yescharacteristic
108C4⋊C4×C283C2311yescharacteristic
109C22⋊C4×C283—21no—
110C22⋊C4×C283—21no—
111C22×D283—21no—
112C22×D283—21no—
113C22×D283—42no—
114C22×D283—42no—
115C22×D283—81no—
116C22×D283C2×D411yescharacteristic
117D1132—21no—
118D1132—21no—
119D1132—21no—
120D1132—42no—
121D1132—42no—
122D1132—81no—
123D1132#4112yes—
124C22⋊D283—22no—
125C2×Dic283—21no—
126C2×Dic283—21no—
127C2×Dic283—42no—
128C2×Dic283C2×D411yescharacteristic
129C2×Dic283C2×D411yescharacteristic
130C2×Dic283C2×D411yescharacteristic
131C4×D283—21no—
132C4×D283—21no—
133C4×D283—42no—
134C4×D283—81no—
135C283⋊C8—21no—
136C283⋊C8—21no—
137C23×C283C2×D411yescharacteristic
138Q8×C283—21no—
139Q8×C283—21no—
140Q8×C283—21no—
141Q8×C283—21no—
142C2×C4×C283—21no—
143C2×C4×C283—21no—
144C2×C4×C283—21no—
145C2×C4×C283—21no—
146C2×C4×C283—21no—
147C2×C4×C283—21no—
148C2×C4×C283C2×D411yescharacteristic
149C2×C4×C283C2411yescharacteristic
150C8×C283—21no—
151C8×C283—21no—
152C22×C283—21no—
153C22×C283—21no—
154C22×C283#13512yes—
155C22×C283C22×D411yescharacteristic
156C2×D283—21no—
157C2×D283—22no—
158C2×D283—21no—
159C2×D283—21no—
160C2×D283—41no—
161C2×D283—41no—
162C2×D283—41no—
163C2×D283—41no—
164C2×D283—41no—
165C2×D283—84no—
166C4×C283—21no—
167C4×C283—41no—
168C4×C283—41no—
169C4×C283—41no—
170C4×C283—41no—
171C4×C283C2×#4111yescharacteristic
172C4×C283H(5,𝔽2)11yescharacteristic
173Dic283—21no—
174Dic283—21no—
175Dic283—21no—
176Dic283—41no—
177Dic283—41no—
178C2×C283—22no—
179C2×C283#54611yescharacteristic
180C2×C283C2×#13511yescharacteristic
181C2×C283#57711yescharacteristic
182D283—41no—
183D283—42no—
184D283—81no—
185C283#329611yescharacteristic, complemented, Sylow
186#3296—2831nomaximal, Sylow
187#552—2831no—
188#546—2831no—
189#488—2831no—
190#483—2831no—
191#482—2831no—
192#478—2831no—
193#467D28311yescharacteristic, complemented
194#465—2831no—
195#464—2831no—
196#452—2831no—
197#450—2831no—
198C4×SD16—2831no—
199C8×D4—2831no—
200#416—2831no—
201C2×#102—2831no—
202C2×#41—2831no—
203C22×D4—2831no—
204C2×Q16C2×D28311yescharacteristic, complemented
205C2×SD16—2831no—
206C2×SD16—2831no—
207C22×C8—2831no—
208C22×C8C2×D28311yescharacteristic
209#126—5661no—
210#125—2831no—
211#125—5661no—
212#124—2831no—
213#123—5661no—
214#123—5661no—
215C22⋊Q8—2831no—
216C22⋊Q8C2×D28311yescharacteristic
217C22⋊Q8C2×D28311yescharacteristic
218#121—2831no—
219#121—5661no—
220#120—5661no—
221C4×Q8—2831no—
222C4×D4—2831no—
223C4×D4—2831no—
224#117—2831no—
225#117—5661no—
226C2×C4⋊C4—2831no—
227C2×C22⋊C4—5661no—
228#107C2×D28311yescharacteristic, complemented
229#106—2831no—
230#106—2831no—
231C4⋊C8—2831no—
232#103—2831no—
233#103—2831no—
234#103C2×D28311yescharacteristic, complemented
235#103C2×D28311yescharacteristic, complemented
236#102—2831no—
237#102—2831no—
238#102—2831no—
239#102—2831no—
240#102—2831no—
241#102—2831no—
242C22⋊C8—2831no—
243C22⋊C8—2831no—
244C4×C8—2831no—
245C24—5661no—
246#41—5662no—
247#41—11321no—
248C2×Q8C22×D28311yescharacteristic
249C2×Q8C22×D28311yescharacteristic
250C2×D4—2831no—
251C2×D4—2831no—
252C2×D4—2831no—
253C2×D4—5662no—
254C2×D4—5661no—
255C2×D4—5661no—
256C2×D4—5661no—
257C2×D4—11321no—
258C22×C4—2831no—
259C22×C4—5661no—
260C22×C4—5661no—
261C22×C4—5661no—
262C22×C4C22×D28311yescharacteristic
263Q16—41no—
264SD16—5662no—
265SD16—11321no—
266C2×C8—41no—
267C2×C8—2831no—
268C2×C8—2831no—
269C2×C8—11321no—
270C2×C8C22×D28311yescharacteristic
271C2×C8C22×D28311yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—2831no—
275C4⋊C4—2831no—
276C4⋊C4—2831no—
277C4⋊C4—5661no—
278C4⋊C4—5661no—
279C4⋊C4—5661no—
280C4⋊C4—5661no—
281C4⋊C4—5661no—
282C4⋊C4C22×D28311yescharacteristic
283C4⋊C4C22×D28311yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—2831no—
287C22⋊C4—2831no—
288C22⋊C4—5661no—
289C22⋊C4—5661no—
290C22⋊C4—5661no—
291C22⋊C4—5661no—
292C22⋊C4—5661no—
293C22⋊C4—5661no—
294C42—2831no—
295C42—5661no—
296C42—5661no—
297C23—2831no—
298C23—5661no—
299C23—5661no—
300C23—11322no—
301C23—11322no—
302C23—22641no—
303C23D4×D28311yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308D4—2832no—
309D4—5662no—
310D4—5661no—
311D4—5661no—
312D4—5661no—
313D4—11322no—
314D4—11322no—
315D4—22641no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—2831no—
323C2×C4—2831no—
324C2×C4—2831no—
325C2×C4—5661no—
326C2×C4—5661no—
327C2×C4—5661no—
328C2×C4—5661no—
329C2×C4—11322no—
330C2×C4—11322no—
331C2×C4—22641no—
332C2×C4D4×D28311yescharacteristic
333C2×C4C23×D28311yescharacteristic
334C8—21no—
335C8—21no—
336C8—5661no—
337C8—5661no—
338C22—21no—
339C22—21no—
340C22—5661no—
341C22—5661no—
342C22—5661no—
343C22—5662no—
344C22—11321no—
345C22—11321no—
346C22—11321no—
347C22—11321no—
348C22—11321no—
349C22—22644no—
350C22#2258363312yes—
351C22C2×D4×D28311yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—5661no—
358C4—5661no—
359C4—5661no—
360C4—11321no—
361C4—11321no—
362C4#41×D28311yescharacteristic
363C4#2258365511yescharacteristic
364C2—22no—
365C2—11321no—
366C2—11322no—
367C2—22641no—
368C2#3182710411yescharacteristic
369C2C2×#2258363311yescharacteristic
370C2#3182727211yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#467×C283C2
socleC22×C283C22×D4
prime radical#31827106C2
hypercenter#467D283
hypocenterC283#3296
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #31827157C2semidirect
3. #31826821C2semidirect
4. #31826793C2semidirect
5. #31826792C2semidirect
6. #31826768C2semidirect
7. #31826697C2semidirect
8. #31826689C2semidirect
9. #31826626C2semidirect
10. #31826615C2semidirect
11. #31826550C2semidirect
12. C8×C22⋊D283C2semidirect
13. #31826460C2semidirect
14. C2×#22583532C2semidirect
15. #467×C283C2semidirect
20. C2×C8×D283C22semidirect
23. #22583601C22semidirect
27. #22583586C22semidirect
31. Q8×Dic283C22semidirect
34. #22583568C22semidirect
38. #22583541C22semidirect
39. #22583541C22semidirect
40. Dic283⋊C8C22semidirect
41. #22583534C22semidirect
42. #22583534C22semidirect
50. (C2×D283)⋊C8C22semidirect
51. C8×Dic283C22semidirect
52. C2×Q16×C283C22semidirect
56. #107×C283C22semidirect
57. #103×C283C22semidirect
58. #103×C283C22semidirect
185. C283#3296semidirect
193. #467D283semidirect
204. C2×Q16C2×D283semidirect
228. #107C2×D283semidirect
234. #103C2×D283semidirect
235. #103C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318271572
#318271062
#318268212
#318267932
#318267922
#318267682
#318266972
#318266892
#318266262
#318266152
#318265502
C8×C22⋊D2832
#318264602
C2×#225835322
#467×C2832
#3296283