Finite Group Databasebeta

← #49133290

Subgroups — #49133290

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#31827109C211yesmaximal, characteristic
3#31826823C211yesmaximal, characteristic, complemented
4#31826807C211yesmaximal, characteristic, complemented
5#31826787C211yesmaximal, characteristic, complemented
6#31826753C211yesmaximal, characteristic, complemented
7#31826721C211yesmaximal, characteristic, complemented
8#31826691C211yesmaximal, characteristic, complemented
9#31826685C211yesmaximal, characteristic, complemented
10#31826624C211yesmaximal, characteristic, complemented
11#31826566C211yesmaximal, characteristic, complemented
12#31826539C211yesmaximal, characteristic, complemented
13#31826470C211yesmaximal, characteristic, complemented
14#31826458C211yesmaximal, characteristic, complemented
15#450×C283C211yesmaximal, characteristic, complemented
16C2×#17985391C2211yescharacteristic
17C2×#17985382C2211yescharacteristic, complemented
18C2×#17985376C2211yescharacteristic
19C2×C8×D283C2211yescharacteristic, complemented
20#22583609C2211yescharacteristic, complemented
21#22583605—21no—
22#22583603—21no—
23#22583597C2211yescharacteristic
24#22583596—21no—
25#22583591—21no—
26#22583589C2211yescharacteristic
27#22583586C2211yescharacteristic, complemented
28#22583582—21no—
29#22583581C2211yescharacteristic
30#22583579—21no—
31Q8×Dic283C2211yescharacteristic, complemented
32#22583575—21no—
33D4×Dic283C2211yescharacteristic
34C4×C22⋊D283C2211yescharacteristic
35#22583569C2211yescharacteristic
36#22583568C2211yescharacteristic, complemented
37C2×C22⋊Dic283—21no—
38#22583544C2211yescharacteristic, complemented
39#22583542C2211yescharacteristic, complemented
40#22583541C2211yescharacteristic, complemented
41Dic283⋊C8C2211yescharacteristic, complemented
42#22583535C2211yescharacteristic, complemented
43#22583535C2211yescharacteristic, complemented
44#22583534C2211yescharacteristic, complemented
45#22583533C2211yescharacteristic, complemented
46#22583531C2211yescharacteristic
47#22583531C2211yescharacteristic
48#22583531C2211yescharacteristic
49#22583531C2211yescharacteristic
50(C2×D283)⋊C8C2211yescharacteristic, complemented
51#22583520C2211yescharacteristic, complemented
52C22×D4×C283C2211yescharacteristic
53C2×SD16×C283C2211yescharacteristic
54C2×SD16×C283C2211yescharacteristic
55C22⋊Q8×C283C2211yescharacteristic
56#102×C283C2211yescharacteristic
57#102×C283C2211yescharacteristic
58C22⋊C8×C283C2211yescharacteristic
59#17985391—22no—
60#17985391—41no—
61C2×Dic566—21no—
62C2×Dic566C2311yescharacteristic
63C2×C22⋊D283—21no—
64C2×C22⋊D283—21no—
65C2×C22⋊D283D411yescharacteristic
66C22×Dic283—21no—
67C22×Dic283—21no—
68C2×C4×D283C2311yescharacteristic
69#17985382—41no—
70#17985376—41no—
71C2×C283⋊C8C2311yescharacteristic
72C2×C283⋊C8C2311yescharacteristic
73C8×D283—41no—
74C4⋊Dic283—21no—
75C4⋊Dic283C2311yescharacteristic
76C4⋊Dic283C2311yescharacteristic
77C4⋊Dic283C2311yescharacteristic
78Dic283⋊C4—21no—
79Dic283⋊C4—21no—
80Dic283⋊C4—21no—
81Dic283⋊C4D411yescharacteristic
82C22⋊Dic283—21no—
83C22⋊Dic283—21no—
84C22⋊Dic283—21no—
85C22⋊Dic283—21no—
86C22⋊Dic283—21no—
87C22⋊Dic283—21no—
88C22⋊Dic283D411yescharacteristic
89(C2×D283)⋊C4—21no—
90(C2×D283)⋊C4—21no—
91(C2×D283)⋊C4D411yescharacteristic
92C4×Dic283—21no—
93C4×Dic283—21no—
94C4×Dic283—21no—
95C4×Dic283C2311yescharacteristic
96C24×C283—21no—
97C2×Q8×C283C2311yescharacteristic
98C2×D4×C283—22no—
99C2×D4×C283—21no—
100C2×D4×C283—41no—
101C2×D4×C283C2311yescharacteristic
102C2×D4×C283C2311yescharacteristic
103C22×C4×C283C2311yescharacteristic
104SD16×C283—22no—
105SD16×C283—41no—
106C2×C8×C283C2311yescharacteristic
107C2×C8×C283C2311yescharacteristic
108C4⋊C4×C283—21no—
109C4⋊C4×C283C2311yescharacteristic
110C22⋊C4×C283—21no—
111C22×D283C2×D411yescharacteristic
112Dic566—21no—
113Dic566—21no—
114Dic566—42no—
115C22⋊D283—41no—
116C22⋊D283—42no—
117C22⋊D283—81no—
118C2×Dic283—21no—
119C2×Dic283—21no—
120C2×Dic283—21no—
121C2×Dic283—21no—
122C2×Dic283—42no—
123C2×Dic283—81no—
124C2×Dic283C2×D411yescharacteristic
125C2×Dic283C2×D411yescharacteristic
126C2×Dic283C2×D411yescharacteristic
127C4×D283—21no—
128C4×D283—21no—
129C283⋊C8—21no—
130C283⋊C8—21no—
131C23×C283—21no—
132C23×C283—21no—
133C23×C283—42no—
134C23×C283—42no—
135C23×C283—81no—
136C23×C283C2×D411yescharacteristic
137Q8×C283—21no—
138Q8×C283—21no—
139D4×C283—21no—
140D4×C283—21no—
141D4×C283—21no—
142D4×C283—42no—
143D4×C283#4112yes—
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—22no—
154C22×C283—21no—
155C22×C283—21no—
156C22×C283—41no—
157C22×C283—41no—
158C22×C283—41no—
159C22×C283—41no—
160C22×C283—41no—
161C22×C283—84no—
162C22×C283C22×D411yescharacteristic
163C2×D283—21no—
164C2×D283—21no—
165C2×D283—21no—
166C4×C283—21no—
167C4×C283—41no—
168C4×C283—41no—
169C4×C283C2×#4111yescharacteristic
170C4×C283H(5,𝔽2)11yescharacteristic
171Dic283—21no—
172Dic283—21no—
173Dic283—21no—
174Dic283—41no—
175Dic283—41no—
176Dic283—41no—
177Dic283—41no—
178C2×C283—41no—
179C2×C283—42no—
180C2×C283—81no—
181C2×C283#54611yescharacteristic
182C2×C283#57511yescharacteristic
183C2×C283#57711yescharacteristic
184D283—41no—
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#465—2831no—
195#464—2831no—
196#452—2831no—
197#450D28311yescharacteristic, complemented
198#440—2831no—
199#435—2831no—
200#418—2831no—
201#416—2831no—
202C2×#41—2831no—
203C22×D4C2×D28311yescharacteristic
204C2×Q16—2831no—
205C2×SD16C2×D28311yescharacteristic
206C2×SD16C2×D28311yescharacteristic
207C2×M16—2831no—
208C22×C8—2831no—
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
218#121—2831no—
219#121—5661no—
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#106—2831no—
231C4⋊C8—2831no—
232#103—2831no—
233#103—2831no—
234#103—2831no—
235#103—2831no—
236#102—2831no—
237#102—2831no—
238#102—2831no—
239#102—2831no—
240#102C2×D28311yescharacteristic
241#102C2×D28311yescharacteristic
242C22⋊C8—2831no—
243C22⋊C8C2×D28311yescharacteristic
244#97—2831no—
245C24—21no—
246#41—5662no—
247#41—11321no—
248C2×Q8—2831no—
249C2×Q8—5661no—
250C2×Q8C22×D28311yescharacteristic
251C2×D4—22no—
252C2×D4—21no—
253C2×D4—41no—
254C2×D4—2831no—
255C2×D4—5661no—
256C2×D4—5661no—
257C2×D4C22×D28311yescharacteristic
258C2×D4C22×D28311yescharacteristic
259C22×C4—2831no—
260C22×C4—5661no—
261C22×C4—5661no—
262C22×C4C22×D28311yescharacteristic
263Q16—11321no—
264SD16—22no—
265SD16—41no—
266M16—11321no—
267C2×C8—2831no—
268C2×C8—2831no—
269C2×C8—11321no—
270C2×C8C22×D28311yescharacteristic
271C2×C8C22×D28311yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—2831no—
274C4⋊C4—2831no—
275C4⋊C4—2831no—
276C4⋊C4—2831no—
277C4⋊C4—5661no—
278C4⋊C4—5661no—
279C4⋊C4—5661no—
280C4⋊C4—5661no—
281C4⋊C4C22×D28311yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—2831no—
284C22⋊C4—2831no—
285C22⋊C4—5661no—
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—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23—2831no—
303C23D4×D28311yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—5661no—
307Q8—5661no—
308Q8—11322no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—42no—
313D4—11321no—
314D4—11322no—
315D4—22641no—
316D4#1798539112yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—2831no—
322C2×C4—2831no—
323C2×C4—2831no—
324C2×C4—5661no—
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—22no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—84no—
348C22—5661no—
349C22—5661no—
350C22—5661no—
351C22C2×D4×D28311yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—5661no—
356C4—5661no—
357C4—5661no—
358C4—11321no—
359C4—11321no—
360C4—11321no—
361C4—11321no—
362C4C2×#1798539111yescharacteristic
363C4#2258365511yescharacteristic
364C2—41no—
365C2—42no—
366C2—81no—
367C2—11321no—
368C2#3182710211yescharacteristic
369C2#3182726111yescharacteristic
370C2#3182727111yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#450×C283C2
socleC22×C283C22×D4
prime radical#31827109C2
hypercenter#450D283
hypocenterC283#3311
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. #31827186C2semidirect
3. #31826823C2semidirect
4. #31826807C2semidirect
5. #31826787C2semidirect
6. #31826753C2semidirect
7. #31826721C2semidirect
8. #31826691C2semidirect
9. #31826685C2semidirect
10. #31826624C2semidirect
11. #31826566C2semidirect
12. #31826539C2semidirect
13. #31826470C2semidirect
14. #31826458C2semidirect
15. #450×C283C2semidirect
17. C2×#17985382C22semidirect
19. C2×C8×D283C22semidirect
20. #22583609C22semidirect
27. #22583586C22semidirect
31. Q8×Dic283C22semidirect
36. #22583568C22semidirect
38. #22583544C22semidirect
39. #22583542C22semidirect
40. #22583541C22semidirect
41. Dic283⋊C8C22semidirect
42. #22583535C22semidirect
43. #22583535C22semidirect
44. #22583534C22semidirect
45. #22583533C22semidirect
50. (C2×D283)⋊C8C22semidirect
51. #22583520C22semidirect
185. C283#3311semidirect
197. #450D283semidirect

Maximal subgroups

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

subgroupindex
#318271862
#318271092
#318268232
#318268072
#318267872
#318267532
#318267212
#318266912
#318266852
#318266242
#318265662
#318265392
#318264702
#318264582
#450×C2832
#3311283