Finite Group Databasebeta

← #49133291

Subgroups — #49133291

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#31827111C211yesmaximal, characteristic
2#31826827C211yesmaximal, characteristic, complemented
3#31826806C211yesmaximal, characteristic, complemented
4#31826789C211yesmaximal, characteristic, complemented
5#31826752C211yesmaximal, characteristic, complemented
6#31826721C211yesmaximal, characteristic, complemented
7#31826696C211yesmaximal, characteristic, complemented
8#31826687C211yesmaximal, characteristic, complemented
9#31826625C211yesmaximal, characteristic, complemented
10#31826615C211yesmaximal, characteristic, complemented
11#31826563C211yesmaximal, characteristic, complemented
12#31826537C211yesmaximal, characteristic, complemented
13#31826469C211yesmaximal, characteristic, complemented
14#31826459C211yesmaximal, characteristic, complemented
15#556×C283C211yesmaximal, characteristic, complemented
16C2×#17985392C2211yescharacteristic
17C22×D1132C2211yescharacteristic
18C2×#17985382C2211yescharacteristic, complemented
19C2×#17985379C2211yescharacteristic
20C2×#17985379C2211yescharacteristic
21C2×#17985376C2211yescharacteristic
22C22×C283⋊C8C2211yescharacteristic, complemented
23#22583598C2211yescharacteristic
24#22583595—21no—
25#22583589C2211yescharacteristic
26#22583584C2211yescharacteristic
27#22583583—21no—
28#22583578—21no—
29C4×D1132C2211yescharacteristic
30C22⋊C4×D283—21no—
31#22583543C2211yescharacteristic, complemented
32#22583541C2211yescharacteristic, complemented
33#22583541C2211yescharacteristic, complemented
34C4⋊(C283⋊C8)C2211yescharacteristic, complemented
35#22583534C2211yescharacteristic, complemented
36#22583533C2211yescharacteristic, complemented
37#22583533C2211yescharacteristic, complemented
38#22583533C2211yescharacteristic, complemented
39#22583530C2211yescharacteristic
40#22583530C2211yescharacteristic
41#22583530C2211yescharacteristic
42#22583530C2211yescharacteristic
43#22583530C2211yescharacteristic
44#22583530C2211yescharacteristic
45C22⋊(C283⋊C8)C2211yescharacteristic
46C22⋊(C283⋊C8)C2211yescharacteristic, complemented
47#22583521C2211yescharacteristic, complemented
48C4⋊Q8×C283C2211yescharacteristic, complemented
49#126×C283—21no—
50#126×C283—21no—
51#124×C283—21no—
52C22⋊Q8×C283C2211yescharacteristic
53C22⋊Q8×C283C2211yescharacteristic, complemented
54C4×Q8×C283—21no—
55C4×Q8×C283C2211yescharacteristic, complemented
56C4×D4×C283C2211yescharacteristic
57#117×C283C2211yescharacteristic
58#117×C283C2211yescharacteristic, complemented
59C23×D283—21no—
60#17985392—22no—
61#17985392—41no—
62C2×Dic566C2311yescharacteristic
63C2×D1132—22no—
64C2×D1132—21no—
65C2×D1132—41no—
66C2×D1132C2311yescharacteristic
67C2×D1132C2311yescharacteristic
68C2×C22⋊D283—21no—
69C2×C22⋊D283—21no—
70C2×C4×D283—21no—
71C2×C4×D283—21no—
72#17985382—41no—
73#17985379—22no—
74#17985379—41no—
75#17985376—41no—
76C2×C283⋊C8—41no—
77C2×C283⋊C8C2311yescharacteristic
78C2×C283⋊C8C2311yescharacteristic
79C2×C283⋊C8C2311yescharacteristic
80C2×C283⋊C8C2311yescharacteristic
81C4⋊Dic283C2311yescharacteristic
82Dic283⋊C4—21no—
83C22⋊Dic283—21no—
84(C2×D283)⋊C4—21no—
85(C2×D283)⋊C4—21no—
86(C2×D283)⋊C4—21no—
87(C2×D283)⋊C4—21no—
88C2×Q8×C283—21no—
89C2×Q8×C283C2311yescharacteristic
90C2×D4×C283D411yescharacteristic
91C22×C4×C283C2311yescharacteristic
92C22×C4×C283C2311yescharacteristic
93C4⋊C4×C283—21no—
94C4⋊C4×C283—21no—
95C4⋊C4×C283—21no—
96C4⋊C4×C283—21no—
97C4⋊C4×C283D411yescharacteristic
98C4⋊C4×C283C2311yescharacteristic
99C4⋊C4×C283C2311yescharacteristic
100C4⋊C4×C283C2311yescharacteristic
101C22⋊C4×C283—21no—
102C22⋊C4×C283—21no—
103C22⋊C4×C283—21no—
104C22⋊C4×C283—21no—
105C22⋊C4×C283D411yescharacteristic
106C22⋊C4×C283D411yescharacteristic
107C42×C283—21no—
108C42×C283—21no—
109C42×C283—21no—
110C42×C283C2311yescharacteristic
111C22×D283—21no—
112C22×D283—21no—
113C22×D283—42no—
114C22×D283—42no—
115C22×D283—81no—
116Dic566—21no—
117Dic566—21no—
118D1132—21no—
119D1132—21no—
120D1132—21no—
121D1132—42no—
122D1132#4112yes—
123C22⋊D283—42no—
124C22⋊D283—81no—
125C2×Dic283—21no—
126C2×Dic283—21no—
127C4×D283—42no—
128C4×D283—81no—
129C283⋊C8—21no—
130C283⋊C8—21no—
131C283⋊C8—21no—
132C283⋊C8—21no—
133C23×C283C2×D411yescharacteristic
134C23×C283C2×D411yescharacteristic
135Q8×C283—21no—
136Q8×C283—21no—
137Q8×C283—42no—
138D4×C283—41no—
139C2×C4×C283—21no—
140C2×C4×C283—21no—
141C2×C4×C283—21no—
142C2×C4×C283—21no—
143C2×C4×C283—21no—
144C2×C4×C283—21no—
145C2×C4×C283—21no—
146C2×C4×C283—21no—
147C2×C4×C283C2×D411yescharacteristic
148C2×C4×C283C2×D411yescharacteristic
149C2×C4×C283C2×D411yescharacteristic
150C2×C4×C283C2×D411yescharacteristic
151C2×C4×C283C2411yescharacteristic
152C22×C283—21no—
153C22×C283—21no—
154C22×C283—21no—
155C22×C283—21no—
156C22×C283—21no—
157C22×C283—21no—
158C22×C283C22×D411yescharacteristic
159C2×D283—22no—
160C2×D283—41no—
161C2×D283—41no—
162C2×D283—41no—
163C2×D283—41no—
164C2×D283—41no—
165C2×D283—84no—
166C4×C283—21no—
167C4×C283—21no—
168C4×C283—21no—
169C4×C283—21no—
170C4×C283—41no—
171C4×C283—41no—
172C4×C283—41no—
173C4×C283—41no—
174C4×C283C2×#4111yescharacteristic
175C4×C283H(5,𝔽2)11yescharacteristic
176Dic283—41no—
177Dic283—41no—
178C2×C283—41no—
179C2×C283—41no—
180C2×C283#54611yescharacteristic
181C2×C283#57511yescharacteristic
182C2×C283#57711yescharacteristic
183D283—42no—
184D283—81no—
185C283#331111yescharacteristic, complemented, Sylow
186#3311—2831nomaximal, Sylow
187#556D28311yescharacteristic, complemented
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#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×SD16—2831no—
207C2×M16—2831no—
208C22×C8—2831no—
209C4⋊Q8C2×D28311yescharacteristic, complemented
210#126—21no—
211#126—21no—
212#124—21no—
213#124—2831no—
214#123—5661no—
215C22⋊Q8—2831no—
216C22⋊Q8C2×D28311yescharacteristic
217C22⋊Q8C2×D28311yescharacteristic, complemented
218#121—2831no—
219#121—5661no—
220#120—5661no—
221C4×Q8—21no—
222C4×Q8C2×D28311yescharacteristic, complemented
223C4×D4—2831no—
224C4×D4C2×D28311yescharacteristic
225#117C2×D28311yescharacteristic
226#117C2×D28311yescharacteristic, complemented
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#102—2831no—
241#102—2831no—
242C22⋊C8—2831no—
243C22⋊C8—2831no—
244#97—2831no—
245C24—5661no—
246#41—5662no—
247#41—11321no—
248C2×Q8—21no—
249C2×Q8—2831no—
250C2×Q8C22×D28311yescharacteristic
251C2×D4—2831no—
252C2×D4—2831no—
253C2×D4—5662no—
254C2×D4—5661no—
255C2×D4—5661no—
256C2×D4—5661no—
257C2×D4—11321no—
258C2×D4C22⋊D28311yescharacteristic
259C22×C4—5661no—
260C22×C4—5661no—
261C22×C4C22×D28311yescharacteristic
262C22×C4C22×D28311yescharacteristic
263Q16—11321no—
264SD16—5662no—
265SD16—11321no—
266M16—11321no—
267C2×C8—2831no—
268C2×C8—2831no—
269C2×C8—2831no—
270C2×C8—2831no—
271C2×C8—11321no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—2831no—
277C4⋊C4—5661no—
278C4⋊C4C22⋊D28311yescharacteristic
279C4⋊C4C22×D28311yescharacteristic
280C4⋊C4C22×D28311yescharacteristic
281C4⋊C4C22×D28311yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—5661no—
287C22⋊C4—5661no—
288C22⋊C4—5661no—
289C22⋊C4—5661no—
290C22⋊C4—5661no—
291C22⋊C4C22⋊D28311yescharacteristic
292C22⋊C4C22⋊D28311yescharacteristic
293C42—21no—
294C42—21no—
295C42—21no—
296C42C22×D28311yescharacteristic
297C23—5661no—
298C23—5661no—
299C23—11322no—
300C23—11322no—
301C23—22641no—
302C23C2×C22⋊D28311yescharacteristic
303C23C2×C22⋊D28311yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—42no—
307Q8—5661no—
308Q8—5661no—
309D4—41no—
310D4—2832no—
311D4—5661no—
312D4—5661no—
313D4—5661no—
314D4—11322no—
315D4—11322no—
316D4—22641no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—5661no—
326C2×C4—5661no—
327C2×C4—11322no—
328C2×C4—22641no—
329C2×C4C2×C22⋊D28311yescharacteristic
330C2×C4C2×C22⋊D28311yescharacteristic
331C2×C4C2×C22⋊D28311yescharacteristic
332C2×C4C2×C22⋊D28311yescharacteristic
333C2×C4C23×D28311yescharacteristic
334C8—5661no—
335C8—5661no—
336C8—5661no—
337C8—5661no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—5662no—
345C22—11321no—
346C22—11321no—
347C22—11321no—
348C22—11321no—
349C22—11321no—
350C22—22644no—
351C22C22×C22⋊D28311yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—11321no—
361C4—11321no—
362C4#41×D28311yescharacteristic
363C4#2258365611yescharacteristic
364C2—41no—
365C2—41no—
366C2—11322no—
367C2—22641no—
368C2#3182710311yescharacteristic
369C2#3182726311yescharacteristic
370C2#3182727311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#556×C283C2
socleC22×C283C22×D4
prime radical#31827111C2
hypercenter#556D283
hypocenterC283#3311
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C4×D4 ◁ #556

Complements

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

NH ≅ G/Ntype
2. #31826827C2semidirect
3. #31826806C2semidirect
4. #31826789C2semidirect
5. #31826752C2semidirect
6. #31826721C2semidirect
7. #31826696C2semidirect
8. #31826687C2semidirect
9. #31826625C2semidirect
10. #31826615C2semidirect
11. #31826563C2semidirect
12. #31826537C2semidirect
13. #31826469C2semidirect
14. #31826459C2semidirect
15. #556×C283C2semidirect
18. C2×#17985382C22semidirect
22. C22×C283⋊C8C22semidirect
31. #22583543C22semidirect
32. #22583541C22semidirect
33. #22583541C22semidirect
34. C4⋊(C283⋊C8)C22semidirect
35. #22583534C22semidirect
36. #22583533C22semidirect
37. #22583533C22semidirect
38. #22583533C22semidirect
46. C22⋊(C283⋊C8)C22semidirect
47. #22583521C22semidirect
48. C4⋊Q8×C283C22semidirect
53. C22⋊Q8×C283C22semidirect
55. C4×Q8×C283C22semidirect
58. #117×C283C22semidirect
185. C283#3311semidirect
187. #556D283semidirect
209. C4⋊Q8C2×D283semidirect
217. C22⋊Q8C2×D283semidirect
222. C4×Q8C2×D283semidirect
226. #117C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318271112
#318268272
#318268062
#318267892
#318267522
#318267212
#318266962
#318266872
#318266252
#318266152
#318265632
#318265372
#318264692
#318264592
#556×C2832
#3311283