Finite Group Databasebeta

← #49133277

Subgroups — #49133277

Order 36224.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827149C211yesmaximal, characteristic, complemented
2#31827099C211yesmaximal, characteristic
3#31826828C211yesmaximal, characteristic, complemented
4#31826786C211yesmaximal, characteristic, complemented
5#31826780C211yesmaximal, characteristic, complemented
6#31826768C211yesmaximal, characteristic, complemented
7#31826697C211yesmaximal, characteristic, complemented
8#31826663C211yesmaximal, characteristic, complemented
9#31826615C211yesmaximal, characteristic, complemented
10#31826614C211yesmaximal, characteristic, complemented
11#31826568C211yesmaximal, characteristic, complemented
12#31826534C211yesmaximal, characteristic, complemented
13#31826471C211yesmaximal, characteristic, complemented
14#102×D283C211yesmaximal, characteristic, complemented
15#468×C283C211yesmaximal, characteristic, complemented
16C22×D1132C2211yescharacteristic
17C2×D4×D283—21no—
18C2×D4×D283C2211yescharacteristic
19C2×#17985379C2211yescharacteristic
20C2×#17985379C2211yescharacteristic, complemented
21C2×C8×D283C2211yescharacteristic, complemented
22#22583606C2211yescharacteristic
23#22583603—21no—
24#22583601C2211yescharacteristic, complemented
25#22583600—21no—
26#22583592—21no—
27#22583586C2211yescharacteristic, complemented
28#22583585C2211yescharacteristic
29#22583583—21no—
30#22583580—21no—
31#22583578—21no—
32Q8×Dic283C2211yescharacteristic, complemented
33#22583572C2211yescharacteristic
34C4×C22⋊D283C2211yescharacteristic
35#22583569C2211yescharacteristic, complemented
36#22583568—21no—
37C4⋊C4×D283C2211yescharacteristic, complemented
38#22583543C2211yescharacteristic, complemented
39#22583541C2211yescharacteristic, complemented
40Dic283⋊C8C2211yescharacteristic, complemented
41#22583532C2211yescharacteristic
42#22583532C2211yescharacteristic
43#22583532C2211yescharacteristic, complemented
44#22583532C2211yescharacteristic, complemented
45#22583531C2211yescharacteristic, complemented
46#22583531C2211yescharacteristic, complemented
47#22583530C2211yescharacteristic
48#22583530C2211yescharacteristic, complemented
49C22⋊(C283⋊C8)C2211yescharacteristic, complemented
50(C2×D283)⋊C8C2211yescharacteristic, complemented
51#22583520C2211yescharacteristic, complemented
52C2×SD16×C283C2211yescharacteristic, complemented
53C2×M16×C283C2211yescharacteristic, complemented
54C22⋊Q8×C283C2211yescharacteristic, complemented
55#121×C283C2211yescharacteristic
56#107×C283C2211yescharacteristic, complemented
57#103×C283C2211yescharacteristic, complemented
58#102×C283C2211yescharacteristic, complemented
59C23×D283—21no—
60C23×D283—21no—
61C2×D1132—22no—
62C2×D1132—21no—
63C2×D1132—21no—
64C2×D1132—41no—
65C2×D1132C2311yescharacteristic
66C2×D1132C2311yescharacteristic
67C2×C22⋊D283—21no—
68C2×C22⋊D283—21no—
69C2×C22⋊D283—21no—
70C2×C22⋊D283D411yescharacteristic
71D4×D283—22no—
72D4×D283—41no—
73D4×D283—44no—
74C2×C4×D283—21no—
75C2×C4×D283—21no—
76C2×C4×D283C2311yescharacteristic
77#17985379—22no—
78#17985379—41no—
79C2×C283⋊C8C2311yescharacteristic
80C2×C283⋊C8C2311yescharacteristic
81C8×D283—41no—
82C4⋊Dic283—21no—
83C4⋊Dic283—21no—
84C4⋊Dic283C2311yescharacteristic
85C4⋊Dic283C2311yescharacteristic
86Dic283⋊C4—21no—
87Dic283⋊C4—21no—
88Dic283⋊C4—21no—
89Dic283⋊C4D411yescharacteristic
90C22⋊Dic283—21no—
91C22⋊Dic283D411yescharacteristic
92(C2×D283)⋊C4—21no—
93(C2×D283)⋊C4—21no—
94(C2×D283)⋊C4—21no—
95(C2×D283)⋊C4D411yescharacteristic
96C4×Dic283—21no—
97C4×Dic283—21no—
98C4×Dic283C2311yescharacteristic
99C2×Q8×C283C2311yescharacteristic
100C2×D4×C283—21no—
101C2×D4×C283C2311yescharacteristic
102C22×C4×C283C2311yescharacteristic
103SD16×C283—41no—
104M16×C283—41no—
105C2×C8×C283C2311yescharacteristic
106C2×C8×C283C2311yescharacteristic
107C4⋊C4×C283—21no—
108C4⋊C4×C283C2311yescharacteristic
109C4⋊C4×C283C2311yescharacteristic
110C22⋊C4×C283—21no—
111C22⋊C4×C283—21no—
112C22×D283—21no—
113C22×D283—21no—
114C22×D283—42no—
115C22×D283—42no—
116C22×D283—42no—
117C22×D283—42no—
118C22×D283—81no—
119C22×D283—81no—
120C22×D283C2×D411yescharacteristic
121D1132—21no—
122D1132—21no—
123D1132—21no—
124D1132—42no—
125D1132—42no—
126D1132#4112yes—
127C22⋊D283—22no—
128C22⋊D283—42no—
129C22⋊D283—42no—
130C22⋊D283—81no—
131C2×Dic283—21no—
132C2×Dic283—21no—
133C2×Dic283C2×D411yescharacteristic
134C2×Dic283C2×D411yescharacteristic
135C2×Dic283C2×D411yescharacteristic
136C4×D283—21no—
137C4×D283—21no—
138C4×D283—42no—
139C4×D283—42no—
140C283⋊C8—21no—
141C283⋊C8—21no—
142C23×C283—21no—
143C23×C283C2×D411yescharacteristic
144Q8×C283—21no—
145Q8×C283—21no—
146D4×C283—21no—
147D4×C283—21no—
148D4×C283—42no—
149C2×C4×C283—21no—
150C2×C4×C283—21no—
151C2×C4×C283—21no—
152C2×C4×C283—21no—
153C2×C4×C283—21no—
154C2×C4×C283C2×D411yescharacteristic
155C2×C4×C283C2411yescharacteristic
156C8×C283—21no—
157C8×C283—21no—
158C22×C283—21no—
159C22×C283—21no—
160C22×C283—21no—
161C22×C283—41no—
162C22×C283—41no—
163C22×C283—41no—
164C22×C283C22×D411yescharacteristic
165C2×D283—21no—
166C2×D283—21no—
167C2×D283—22no—
168C2×D283—41no—
169C2×D283—41no—
170C2×D283—41no—
171C2×D283—41no—
172C2×D283—41no—
173C2×D283—84no—
174C2×D283—84no—
175C2×D283AGL(1,ℤ/8)12yes—
176C4×C283—21no—
177C4×C283—41no—
178C4×C283—41no—
179C4×C283—41no—
180C4×C283C2×#4111yescharacteristic
181C4×C283H(5,𝔽2)11yescharacteristic
182Dic283—21no—
183Dic283—21no—
184Dic283—21no—
185Dic283—41no—
186Dic283—41no—
187C2×C283—41no—
188C2×C283—81no—
189C2×C283#54611yescharacteristic
190C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C283#57711yescharacteristic
192D283—22no—
193D283—42no—
194D283—81no—
195C283#331011yescharacteristic, complemented, Sylow
196#3310—2831nomaximal, Sylow
197#552—2831no—
198D42—2831no—
199#490—2831no—
200#482—2831no—
201#480—2831no—
202#478—2831no—
203#468D28311yescharacteristic, complemented
204#465—2831no—
205#460—2831no—
206#450—2831no—
207#450—2831no—
208#440—2831no—
209#435—2831no—
210#418—2831no—
211C2×#102—2831no—
212C22×D4—2831no—
213C22×D4—2831no—
214C22×D4—5661no—
215C2×SD16—2831no—
216C2×SD16—2831no—
217C2×SD16C2×D28311yescharacteristic, complemented
218C2×M16C2×D28311yescharacteristic, complemented
219C22×C8—2831no—
220#127—2831no—
221#126—5661no—
222#125—2831no—
223#125—5661no—
224#123—5661no—
225C22⋊Q8—2831no—
226C22⋊Q8C2×D28311yescharacteristic, complemented
227#121—2831no—
228#121—5661no—
229#121C2×D28311yescharacteristic
230#120—5661no—
231#120—5661no—
232C4×Q8—2831no—
233C4×D4—2831no—
234C4×D4—2831no—
235#117—2831no—
236#117—5661no—
237C2×C4⋊C4—2831no—
238#107—2831no—
239#107C2×D28311yescharacteristic, complemented
240#106—2831no—
241C4⋊C8—2831no—
242#103C2×D28311yescharacteristic, complemented
243#102—2831no—
244#102—2831no—
245#102—2831no—
246#102—2831no—
247#102—2831no—
248#102—2831no—
249#102—2831no—
250#102—2831no—
251#102C2×D28311yescharacteristic, complemented
252C22⋊C8—2831no—
253C22⋊C8—2831no—
254#97—2831no—
255C24—5661no—
256C24—5661no—
257C2×Q8C22×D28311yescharacteristic
258C2×D4—21no—
259C2×D4—2831no—
260C2×D4—2831no—
261C2×D4—2831no—
262C2×D4—5662no—
263C2×D4—5661no—
264C2×D4—5661no—
265C2×D4—5661no—
266C2×D4—5661no—
267C2×D4—5662no—
268C2×D4—5661no—
269C2×D4—11321no—
270C2×D4—11321no—
271C2×D4—11324no—
272C2×D4C22×D28311yescharacteristic
273C22×C4—2831no—
274C22×C4—5661no—
275C22×C4—5661no—
276C22×C4C22×D28311yescharacteristic
277SD16—41no—
278SD16—5662no—
279SD16—11321no—
280M16—41no—
281C2×C8—2831no—
282C2×C8—2831no—
283C2×C8—11321no—
284C2×C8C22×D28311yescharacteristic
285C2×C8C22×D28311yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—2831no—
288C4⋊C4—2831no—
289C4⋊C4—2831no—
290C4⋊C4—5661no—
291C4⋊C4—5661no—
292C4⋊C4—5661no—
293C4⋊C4—5661no—
294C4⋊C4—5661no—
295C4⋊C4C22×D28311yescharacteristic
296C4⋊C4C22×D28311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—2831no—
300C22⋊C4—2831no—
301C22⋊C4—5661no—
302C22⋊C4—5661no—
303C22⋊C4—5661no—
304C22⋊C4—5661no—
305C42—2831no—
306C42—5661no—
307C42—5661no—
308C23—21no—
309C23—2831no—
310C23—5661no—
311C23—5661no—
312C23—11322no—
313C23—11322no—
314C23—11322no—
315C23—11322no—
316C23—22641no—
317C23—22641no—
318C23D4×D28311yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—21no—
322D4—21no—
323D4—42no—
324D4—2832no—
325D4—5662no—
326D4—5661no—
327D4—5661no—
328D4—5661no—
329D4—11322no—
330D4—11322no—
331D4—11322no—
332D4—11322no—
333D4—22641no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—2831no—
340C2×C4—2831no—
341C2×C4—2831no—
342C2×C4—5661no—
343C2×C4—5661no—
344C2×C4—5661no—
345C2×C4—5661no—
346C2×C4—11322no—
347C2×C4—11322no—
348C2×C4D4×D28311yescharacteristic
349C2×C4C23×D28311yescharacteristic
350C8—21no—
351C8—21no—
352C8—5661no—
353C8—5661no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—2832no—
361C22—5661no—
362C22—5661no—
363C22—5662no—
364C22—11321no—
365C22—11321no—
366C22—11321no—
367C22—11321no—
368C22—11321no—
369C22—22644no—
370C22—22644no—
371C22C2×D4×D28311yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—5661no—
377C4—5661no—
378C4—5661no—
379C4—11321no—
380C4—11321no—
381C4#41×D28311yescharacteristic
382C4#2258365511yescharacteristic
383C2—41no—
384C2—81no—
385C2—5662no—
386C2—11322no—
387C2—22641no—
388C2#3182710411yescharacteristic
389C2AGL(1,ℤ/8)×D28311yescharacteristic
390C2#3182727411yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#468×C283C2
socleC22×C283C22×D4
prime radical#31827099C2
hypercenter#468D283
hypocenterC283#3310
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #31827149C2semidirect
3. #31826828C2semidirect
4. #31826786C2semidirect
5. #31826780C2semidirect
6. #31826768C2semidirect
7. #31826697C2semidirect
8. #31826663C2semidirect
9. #31826615C2semidirect
10. #31826614C2semidirect
11. #31826568C2semidirect
12. #31826534C2semidirect
13. #31826471C2semidirect
14. #102×D283C2semidirect
15. #468×C283C2semidirect
20. C2×#17985379C22semidirect
21. C2×C8×D283C22semidirect
24. #22583601C22semidirect
27. #22583586C22semidirect
32. Q8×Dic283C22semidirect
35. #22583569C22semidirect
37. C4⋊C4×D283C22semidirect
38. #22583543C22semidirect
39. #22583541C22semidirect
40. Dic283⋊C8C22semidirect
43. #22583532C22semidirect
44. #22583532C22semidirect
45. #22583531C22semidirect
46. #22583531C22semidirect
48. #22583530C22semidirect
49. C22⋊(C283⋊C8)C22semidirect
50. (C2×D283)⋊C8C22semidirect
51. #22583520C22semidirect
52. C2×SD16×C283C22semidirect
53. C2×M16×C283C22semidirect
54. C22⋊Q8×C283C22semidirect
56. #107×C283C22semidirect
57. #103×C283C22semidirect
58. #102×C283C22semidirect
195. C283#3310semidirect
203. #468D283semidirect
217. C2×SD16C2×D283semidirect
218. C2×M16C2×D283semidirect
226. C22⋊Q8C2×D283semidirect
239. #107C2×D283semidirect
242. #103C2×D283semidirect
251. #102C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318271492
#318270992
#318268282
#318267862
#318267802
#318267682
#318266972
#318266632
#318266152
#318266142
#318265682
#318265342
#318264712
#102×D2832
#468×C2832
#3310283