Finite Group Databasebeta

← #49133346

Subgroups — #49133346

Order 36224.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827173C211yesmaximal, characteristic, complemented
2#31827109C211yesmaximal, characteristic
3#31826823C211yesmaximal, characteristic, complemented
4#31826801C211yesmaximal, characteristic, complemented
5#31826777C211yesmaximal, characteristic, complemented
6#31826743C211yesmaximal, characteristic, complemented
7#31826708C211yesmaximal, characteristic, complemented
8#31826704C211yesmaximal, characteristic, complemented
9#31826685C211yesmaximal, characteristic, complemented
10#31826624C211yesmaximal, characteristic, complemented
11#31826566C211yesmaximal, characteristic, complemented
12#31826533C211yesmaximal, characteristic, complemented
13#31826463C211yesmaximal, characteristic, complemented
14C2×#22583531C211yesmaximal, characteristic, complemented
15#450×C283C211yesmaximal, characteristic, complemented
16C2×#17985391C2211yescharacteristic
17C2×#17985382C2211yescharacteristic, complemented
18C2×#17985375C2211yescharacteristic, complemented
19C22×C283⋊C8C2211yescharacteristic
20#22583609C2211yescharacteristic, complemented
21#22583600—21no—
22#22583597C2211yescharacteristic
23#22583591—21no—
24#22583590—21no—
25#22583589C2211yescharacteristic
26#22583587—21no—
27#22583586C2211yescharacteristic, complemented
28#22583582—21no—
29#22583581C2211yescharacteristic
30#22583579—21no—
31Q8×Dic283C2211yescharacteristic, complemented
32D4×Dic283C2211yescharacteristic
33C4×C22⋊D283C2211yescharacteristic
34(C22⋊D283)⋊C4—21no—
35C2×C4⋊Dic283C2211yescharacteristic
36C4⋊C4×D283C2211yescharacteristic, complemented
37C2×C22⋊Dic283—21no—
38#22583544C2211yescharacteristic, complemented
39#22583544C2211yescharacteristic, complemented
40#22583543C2211yescharacteristic, 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—
68C22×Dic283—21no—
69C2×C4×D283—21no—
70C2×C4×D283C2311yescharacteristic
71#17985382—41no—
72#17985375—41no—
73C2×C283⋊C8—41no—
74C2×C283⋊C8C2311yescharacteristic
75C2×C283⋊C8C2311yescharacteristic
76C4⋊Dic283—21no—
77C4⋊Dic283—21no—
78C4⋊Dic283C2311yescharacteristic
79C4⋊Dic283C2311yescharacteristic
80C4⋊Dic283C2311yescharacteristic
81Dic283⋊C4—21no—
82Dic283⋊C4—21no—
83Dic283⋊C4—21no—
84Dic283⋊C4—21no—
85Dic283⋊C4D411yescharacteristic
86C22⋊Dic283—21no—
87C22⋊Dic283—21no—
88C22⋊Dic283—21no—
89C22⋊Dic283—21no—
90C22⋊Dic283—21no—
91C22⋊Dic283D411yescharacteristic
92(C2×D283)⋊C4—21no—
93(C2×D283)⋊C4D411yescharacteristic
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—81no—
115C22⋊D283—22no—
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×Dic283—81no—
125C2×Dic283C2×D411yescharacteristic
126C2×Dic283C2×D411yescharacteristic
127C2×Dic283C2×D411yescharacteristic
128C4×D283—21no—
129C4×D283—21no—
130C4×D283—81no—
131C283⋊C8—21no—
132C283⋊C8—21no—
133C23×C283—21no—
134C23×C283—21no—
135C23×C283—42no—
136C23×C283—42no—
137C23×C283—81no—
138C23×C283C2×D411yescharacteristic
139Q8×C283—21no—
140Q8×C283—21no—
141D4×C283—21no—
142D4×C283—21no—
143D4×C283—21no—
144D4×C283—42no—
145D4×C283#4112yes—
146C2×C4×C283—21no—
147C2×C4×C283—21no—
148C2×C4×C283—21no—
149C2×C4×C283—21no—
150C2×C4×C283C2×D411yescharacteristic
151C2×C4×C283C2411yescharacteristic
152C8×C283—21no—
153C8×C283—21no—
154C22×C283—21no—
155C22×C283—21no—
156C22×C283—22no—
157C22×C283—41no—
158C22×C283—41no—
159C22×C283—41no—
160C22×C283—41no—
161C22×C283—41no—
162C22×C283—84no—
163C22×C283#13712yes—
164C22×C283C22×D411yescharacteristic
165C2×D283—21no—
166C2×D283—21no—
167C2×D283—21no—
168C4×C283—21no—
169C4×C283—41no—
170C4×C283—41no—
171C4×C283C2×#4111yescharacteristic
172C4×C283H(5,𝔽2)11yescharacteristic
173Dic283—21no—
174Dic283—21no—
175Dic283—21no—
176Dic283—41no—
177Dic283—41no—
178Dic283—41no—
179Dic283—41no—
180C2×C283—22no—
181C2×C283—42no—
182C2×C283—81no—
183C2×C283#54611yescharacteristic
184C2×C283C2×#13711yescharacteristic
185C2×C283#57611yescharacteristic
186D283—41no—
187C283#331411yescharacteristic, complemented, Sylow
188#3314—2831nomaximal, Sylow
189#554—2831no—
190#546—2831no—
191#489—2831no—
192#484—2831no—
193#480—2831no—
194#474—2831no—
195#468—2831no—
196#467—2831no—
197#464—2831no—
198#452—2831no—
199#450D28311yescharacteristic, complemented
200#440—2831no—
201#435—2831no—
202#417—2831no—
203C2×#102—2831no—
204C2×#41—2831no—
205C22×D4C2×D28311yescharacteristic
206C2×Q16—2831no—
207C2×SD16C2×D28311yescharacteristic
208C2×SD16C2×D28311yescharacteristic
209C2×M16—2831no—
210C22×C8—2831no—
211C4⋊Q8—2831no—
212#125—5661no—
213#124—2831no—
214#123—5661no—
215C22⋊Q8—2831no—
216C22⋊Q8—2831no—
217C22⋊Q8—5661no—
218C22⋊Q8—5661no—
219C22⋊Q8C2×D28311yescharacteristic
220#121—2831no—
221#121—5661no—
222#120—5661no—
223C4×Q8—2831no—
224C4×D4—2831no—
225C4×D4—2831no—
226C4×D4—5661no—
227C2×C4⋊C4—2831no—
228C2×C4⋊C4—2831no—
229C2×C22⋊C4—5661no—
230#107—2831no—
231#107—2831no—
232#107—2831no—
233C4⋊C8—2831no—
234#103—2831no—
235#103—2831no—
236#103—2831no—
237#103—2831no—
238#102—2831no—
239#102—2831no—
240#102—2831no—
241#102—2831no—
242#102C2×D28311yescharacteristic
243#102C2×D28311yescharacteristic
244C22⋊C8—2831no—
245C22⋊C8C2×D28311yescharacteristic
246#97—2831no—
247C24—21no—
248#41—5662no—
249#41—11321no—
250C2×Q8—2831no—
251C2×Q8—5661no—
252C2×Q8C22×D28311yescharacteristic
253C2×D4—22no—
254C2×D4—21no—
255C2×D4—41no—
256C2×D4—2831no—
257C2×D4—5661no—
258C2×D4—5661no—
259C2×D4C22×D28311yescharacteristic
260C2×D4C22×D28311yescharacteristic
261C22×C4—2831no—
262C22×C4—5661no—
263C22×C4—5661no—
264C22×C4—5661no—
265C22×C4—5661no—
266C22×C4C22×D28311yescharacteristic
267Q16—11321no—
268SD16—22no—
269SD16—41no—
270M16—11321no—
271C2×C8—2831no—
272C2×C8—2831no—
273C2×C8—11321no—
274C2×C8C22×D28311yescharacteristic
275C2×C8C22×D28311yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—2831no—
278C4⋊C4—2831no—
279C4⋊C4—2831no—
280C4⋊C4—2831no—
281C4⋊C4—5661no—
282C4⋊C4—5661no—
283C4⋊C4—5661no—
284C4⋊C4—5661no—
285C4⋊C4—5661no—
286C4⋊C4—5661no—
287C4⋊C4C22×D28311yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—2831no—
290C22⋊C4—2831no—
291C22⋊C4—5661no—
292C22⋊C4—5661no—
293C22⋊C4—5661no—
294C22⋊C4—5661no—
295C22⋊C4—5661no—
296C22⋊C4—5661no—
297C42—2831no—
298C42—5661no—
299C23—21no—
300C23—21no—
301C23—42no—
302C23—42no—
303C23—81no—
304C23—2831no—
305C23D4×D28311yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—5661no—
309Q8—5661no—
310Q8—22641no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—42no—
315D4—5662no—
316D4—11322no—
317D4—22641no—
318D4#1798539112yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—2831no—
324C2×C4—2831no—
325C2×C4—2831no—
326C2×C4—5661no—
327C2×C4—5661no—
328C2×C4—5661no—
329C2×C4—5661no—
330C2×C4—5661no—
331C2×C4—5661no—
332C2×C4—11322no—
333C2×C4—22641no—
334C2×C4—22641no—
335C2×C4—22641no—
336C2×C4D4×D28311yescharacteristic
337C2×C4C23×D28311yescharacteristic
338C8—21no—
339C8—21no—
340C8—5661no—
341C8—5661no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—84no—
351C22—5661no—
352C22—5661no—
353C22—5661no—
354C22#2258363912yes—
355C22C2×D4×D28311yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—5661no—
360C4—5661no—
361C4—5661no—
362C4—11321no—
363C4—11321no—
364C4—11321no—
365C4—11321no—
366C4C2×#1798539111yescharacteristic
367C4#2258365511yescharacteristic
368C2—22no—
369C2—42no—
370C2—81no—
371C2—11321no—
372C2#3182710211yescharacteristic
373C2C2×#2258363911yescharacteristic
374C2#3182726911yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#450×C283C2
socleC22×C283C22×D4
prime radical#31827109C2
hypercenter#450D283
hypocenterC283#3314
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. #31827173C2semidirect
3. #31826823C2semidirect
4. #31826801C2semidirect
5. #31826777C2semidirect
6. #31826743C2semidirect
7. #31826708C2semidirect
8. #31826704C2semidirect
9. #31826685C2semidirect
10. #31826624C2semidirect
11. #31826566C2semidirect
12. #31826533C2semidirect
13. #31826463C2semidirect
14. C2×#22583531C2semidirect
15. #450×C283C2semidirect
17. C2×#17985382C22semidirect
18. C2×#17985375C22semidirect
20. #22583609C22semidirect
27. #22583586C22semidirect
31. Q8×Dic283C22semidirect
36. C4⋊C4×D283C22semidirect
38. #22583544C22semidirect
39. #22583544C22semidirect
40. #22583543C22semidirect
41. Dic283⋊C8C22semidirect
42. #22583535C22semidirect
43. #22583535C22semidirect
44. #22583534C22semidirect
45. #22583533C22semidirect
50. (C2×D283)⋊C8C22semidirect
51. #22583520C22semidirect
187. C283#3314semidirect
199. #450D283semidirect

Maximal subgroups

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

subgroupindex
#318271732
#318271092
#318268232
#318268012
#318267772
#318267432
#318267082
#318267042
#318266852
#318266242
#318265662
#318265332
#318264632
C2×#225835312
#450×C2832
#3314283