Finite Group Databasebeta

← #49133303

Subgroups — #49133303

Order 36224.

402 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#31827186C211yesmaximal, characteristic, complemented
2#31827112C211yesmaximal, characteristic
3#31826823C211yesmaximal, characteristic, complemented
4#31826809C211yesmaximal, characteristic, complemented
5#31826786C211yesmaximal, characteristic, complemented
6#31826754C211yesmaximal, characteristic, complemented
7#31826722C211yesmaximal, characteristic, complemented
8#31826685C211yesmaximal, characteristic, complemented
9#31826625C211yesmaximal, characteristic, complemented
10#31826614C211yesmaximal, characteristic, complemented
11#31826568C211yesmaximal, characteristic, complemented
12#31826536C211yesmaximal, characteristic, complemented
13#31826468C211yesmaximal, characteristic, complemented
14#31826462C211yesmaximal, characteristic, complemented
15#465×C283C211yesmaximal, characteristic, complemented
16C2×#17985392C2211yescharacteristic
17C2×D4×D283C2211yescharacteristic
18C2×#17985382C2211yescharacteristic, complemented
19C2×#17985379C2211yescharacteristic
20C2×#17985375C2211yescharacteristic
21#22583609C2211yescharacteristic, complemented
22#22583605—21no—
23#22583603—21no—
24#22583597C2211yescharacteristic
25#22583596—21no—
26#22583592—21no—
27#22583590C2211yescharacteristic
28#22583586C2211yescharacteristic
29#22583581—21no—
30#22583578—21no—
31Q8×Dic283C2211yescharacteristic, complemented
32#22583575—21no—
33#22583572C2211yescharacteristic
34C4×C22⋊D283C2211yescharacteristic
35#22583569C2211yescharacteristic, complemented
36#22583568C2211yescharacteristic
37C22⋊C4×D283—21no—
38#22583543C2211yescharacteristic, complemented
39#22583541C2211yescharacteristic, complemented
40Dic283⋊C8C2211yescharacteristic, complemented
41#22583535C2211yescharacteristic, complemented
42#22583535C2211yescharacteristic, complemented
43#22583533C2211yescharacteristic, complemented
44#22583532C2211yescharacteristic
45#22583532C2211yescharacteristic
46#22583531C2211yescharacteristic
47#22583531C2211yescharacteristic
48#22583530C2211yescharacteristic
49C22⋊(C283⋊C8)C2211yescharacteristic, complemented
50(C2×D283)⋊C8C2211yescharacteristic
51#22583520C2211yescharacteristic, complemented
52C2×SD16×C283C2211yescharacteristic
53C22×C8×C283C2211yescharacteristic, complemented
54C22⋊Q8×C283C2211yescharacteristic, complemented
55#121×C283C2211yescharacteristic
56#106×C283C2211yescharacteristic, complemented
57#103×C283C2211yescharacteristic, complemented
58#102×C283C2211yescharacteristic
59C23×D283—21no—
60#17985392—22no—
61#17985392—41no—
62C2×Dic566—21no—
63C2×Dic566C2311yescharacteristic
64C2×D1132C2311yescharacteristic
65C2×C22⋊D283—21no—
66C2×C22⋊D283—21no—
67C2×C22⋊D283D411yescharacteristic
68D4×D283—22no—
69D4×D283—41no—
70C2×C4×D283—21no—
71C2×C4×D283—21no—
72C2×C4×D283C2311yescharacteristic
73#17985382—41no—
74#17985379—22no—
75#17985375—41no—
76C2×C283⋊C8C2311yescharacteristic
77C2×C283⋊C8C2311yescharacteristic
78C4⋊Dic283—21no—
79C4⋊Dic283—21no—
80C4⋊Dic283C2311yescharacteristic
81C4⋊Dic283C2311yescharacteristic
82Dic283⋊C4—21no—
83Dic283⋊C4—21no—
84Dic283⋊C4D411yescharacteristic
85C22⋊Dic283—21no—
86C22⋊Dic283—21no—
87C22⋊Dic283—21no—
88C22⋊Dic283D411yescharacteristic
89(C2×D283)⋊C4—21no—
90(C2×D283)⋊C4—21no—
91(C2×D283)⋊C4—21no—
92(C2×D283)⋊C4—21no—
93(C2×D283)⋊C4D411yescharacteristic
94C4×Dic283—21no—
95C4×Dic283—21no—
96C4×Dic283—21no—
97C4×Dic283C2311yescharacteristic
98C2×Q8×C283C2311yescharacteristic
99C2×D4×C283—21no—
100C2×D4×C283C2311yescharacteristic
101C22×C4×C283C2311yescharacteristic
102SD16×C283—41no—
103C2×C8×C283—41no—
104C2×C8×C283C2311yescharacteristic
105C2×C8×C283C2311yescharacteristic
106C4⋊C4×C283—21no—
107C4⋊C4×C283C2311yescharacteristic
108C4⋊C4×C283C2311yescharacteristic
109C22⋊C4×C283—21no—
110C22⋊C4×C283—21no—
111C22×D283—21no—
112C22×D283—42no—
113C22×D283—42no—
114C22×D283—81no—
115C22×D283C2×D411yescharacteristic
116Dic566—21no—
117Dic566—21no—
118Dic566—42no—
119D1132—21no—
120D1132#4112yes—
121C22⋊D283—41no—
122C22⋊D283—42no—
123C22⋊D283—42no—
124C2×Dic283—21no—
125C2×Dic283—21no—
126C2×Dic283—21no—
127C2×Dic283C2×D411yescharacteristic
128C2×Dic283C2×D411yescharacteristic
129C2×Dic283C2×D411yescharacteristic
130C4×D283—21no—
131C4×D283—21no—
132C4×D283—42no—
133C4×D283—81no—
134C283⋊C8—21no—
135C283⋊C8—21no—
136C23×C283—21no—
137C23×C283C2×D411yescharacteristic
138Q8×C283—21no—
139Q8×C283—21no—
140D4×C283—21no—
141D4×C283—21no—
142D4×C283—81no—
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—21no—
155C22×C283—41no—
156C22×C283—41no—
157C22×C283—41no—
158C22×C283C22×D411yescharacteristic
159C2×D283—21no—
160C2×D283—22no—
161C2×D283—21no—
162C2×D283—21no—
163C2×D283—41no—
164C2×D283—41no—
165C2×D283—84no—
166C4×C283—21no—
167C4×C283—41no—
168C4×C283—41no—
169C4×C283—41no—
170C4×C283C2×#4111yescharacteristic
171C4×C283H(5,𝔽2)11yescharacteristic
172Dic283—21no—
173Dic283—21no—
174Dic283—21no—
175Dic283—41no—
176Dic283—41no—
177Dic283—41no—
178C2×C283—41no—
179C2×C283—81no—
180C2×C283#54611yescharacteristic
181C2×C283#57511yescharacteristic
182C2×C283#57711yescharacteristic
183D283—41no—
184D283—42no—
185C283#331111yescharacteristic, complemented, Sylow
186#3311—2831nomaximal, Sylow
187#556—2831no—
188#546—2831no—
189#489—2831no—
190#485—2831no—
191#482—2831no—
192#476—2831no—
193#470—2831no—
194#465D28311yescharacteristic, complemented
195#464—2831no—
196#452—2831no—
197#450—2831no—
198#440—2831no—
199#435—2831no—
200#418—2831no—
201#416—2831no—
202C2×#41—2831no—
203C22×D4—2831no—
204C2×Q16—2831no—
205C2×SD16—2831no—
206C2×SD16C2×D28311yescharacteristic
207C2×M16—2831no—
208C22×C8C2×D28311yescharacteristic, complemented
209C4⋊Q8—2831no—
210#126—5661no—
211#126—5661no—
212#124—2831no—
213#124—5661no—
214#123—5661no—
215C22⋊Q8—2831no—
216C22⋊Q8—2831no—
217C22⋊Q8C2×D28311yescharacteristic, complemented
218#121—5661no—
219#121C2×D28311yescharacteristic
220#120—5661no—
221C4×Q8—2831no—
222C4×Q8—5661no—
223C4×D4—2831no—
224C4×D4—2831no—
225#117—2831no—
226#117—2831no—
227C2×C22⋊C4—5661no—
228#107—2831no—
229#106—2831no—
230#106C2×D28311yescharacteristic, complemented
231C4⋊C8—2831no—
232#103—2831no—
233#103—2831no—
234#103—2831no—
235#103C2×D28311yescharacteristic, complemented
236#102—2831no—
237#102—2831no—
238#102—2831no—
239#102—2831no—
240#102—2831no—
241#102C2×D28311yescharacteristic
242C22⋊C8—2831no—
243C22⋊C8—2831no—
244#97—2831no—
245C24—5661no—
246#41—5662no—
247#41—11321no—
248C2×Q8—2831no—
249C2×Q8—5661no—
250C2×Q8C22×D28311yescharacteristic
251C2×D4—21no—
252C2×D4—2831no—
253C2×D4—2831no—
254C2×D4—5661no—
255C2×D4—5662no—
256C2×D4—5661no—
257C2×D4—11321no—
258C2×D4C22×D28311yescharacteristic
259C22×C4—2831no—
260C22×C4—5661no—
261C22×C4—5661no—
262C22×C4C22×D28311yescharacteristic
263Q16—11321no—
264SD16—41no—
265SD16—5662no—
266M16—11321no—
267C2×C8—41no—
268C2×C8—2831no—
269C2×C8—2831no—
270C2×C8C22×D28311yescharacteristic
271C2×C8C22×D28311yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—2831no—
274C4⋊C4—2831no—
275C4⋊C4—2831no—
276C4⋊C4—5661no—
277C4⋊C4—5661no—
278C4⋊C4—5661no—
279C4⋊C4—5661no—
280C4⋊C4C22×D28311yescharacteristic
281C4⋊C4C22×D28311yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—2831no—
285C22⋊C4—2831no—
286C22⋊C4—5661no—
287C22⋊C4—5661no—
288C22⋊C4—5661no—
289C22⋊C4—5661no—
290C22⋊C4—5661no—
291C22⋊C4—5661no—
292C22⋊C4—5661no—
293C42—2831no—
294C42—5661no—
295C42—5661no—
296C42—5661no—
297C23—21no—
298C23—2831no—
299C23—5661no—
300C23—11322no—
301C23—11322no—
302C23—22641no—
303C23D4×D28311yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—5661no—
307Q8—5661no—
308Q8—11322no—
309D4—21no—
310D4—21no—
311D4—81no—
312D4—2832no—
313D4—5661no—
314D4—11321no—
315D4—11322no—
316D4—11322no—
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—5661no—
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—21no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—5661no—
345C22—5661no—
346C22—5661no—
347C22—5662no—
348C22—11321no—
349C22—11321no—
350C22—22644no—
351C22C2×D4×D28311yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—5661no—
357C4—5661no—
358C4—5661no—
359C4—11321no—
360C4—11321no—
361C4—11321no—
362C4#41×D28311yescharacteristic
363C4#2258365511yescharacteristic
364C2—41no—
365C2—81no—
366C2—11321no—
367C2—11322no—
368C2#3182710411yescharacteristic
369C2#3182726211yescharacteristic
370C2#3182727111yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#465×C283C2
socleC22×C283C22×D4
prime radical#31827112C2
hypercenter#465D283
hypocenterC283#3311
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #31827186C2semidirect
3. #31826823C2semidirect
4. #31826809C2semidirect
5. #31826786C2semidirect
6. #31826754C2semidirect
7. #31826722C2semidirect
8. #31826685C2semidirect
9. #31826625C2semidirect
10. #31826614C2semidirect
11. #31826568C2semidirect
12. #31826536C2semidirect
13. #31826468C2semidirect
14. #31826462C2semidirect
15. #465×C283C2semidirect
18. C2×#17985382C22semidirect
21. #22583609C22semidirect
31. Q8×Dic283C22semidirect
35. #22583569C22semidirect
38. #22583543C22semidirect
39. #22583541C22semidirect
40. Dic283⋊C8C22semidirect
41. #22583535C22semidirect
42. #22583535C22semidirect
43. #22583533C22semidirect
49. C22⋊(C283⋊C8)C22semidirect
51. #22583520C22semidirect
53. C22×C8×C283C22semidirect
54. C22⋊Q8×C283C22semidirect
56. #106×C283C22semidirect
57. #103×C283C22semidirect
185. C283#3311semidirect
194. #465D283semidirect
208. C22×C8C2×D283semidirect
217. C22⋊Q8C2×D283semidirect
230. #106C2×D283semidirect
235. #103C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318271862
#318271122
#318268232
#318268092
#318267862
#318267542
#318267222
#318266852
#318266252
#318266142
#318265682
#318265362
#318264682
#318264622
#465×C2832
#3311283