Finite Group Databasebeta

← #49133336

Subgroups — #49133336

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#31827111C211yesmaximal, characteristic
2#31826827C211yesmaximal, characteristic, complemented
3#31826800C211yesmaximal, characteristic, complemented
4#31826778C211yesmaximal, characteristic, complemented
5#31826742C211yesmaximal, characteristic, complemented
6#31826713C211yesmaximal, characteristic, complemented
7#31826704C211yesmaximal, characteristic, complemented
8#31826687C211yesmaximal, characteristic, complemented
9#31826625C211yesmaximal, characteristic, complemented
10#31826615C211yesmaximal, characteristic, complemented
11#31826563C211yesmaximal, characteristic, complemented
12#31826537C211yesmaximal, characteristic, complemented
13#31826464C211yesmaximal, characteristic, complemented
14C2×#22583530C211yesmaximal, characteristic, complemented
15#554×C283C211yesmaximal, characteristic, complemented
16C2×#17985392C2211yescharacteristic
17C22×D1132C2211yescharacteristic
18C2×#17985382C2211yescharacteristic, complemented
19C2×#17985379C2211yescharacteristic
20C2×#17985379C2211yescharacteristic
21C2×#17985376C2211yescharacteristic, complemented
22C22×C283⋊C8C2211yescharacteristic
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#22583543C2211yescharacteristic, complemented
33#22583543C2211yescharacteristic, 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#125×C283—21no—
50C22⋊Q8×C283—21no—
51C22⋊Q8×C283—21no—
52C22⋊Q8×C283C2211yescharacteristic
53C22⋊Q8×C283C2211yescharacteristic, complemented
54C4×Q8×C283C2211yescharacteristic, complemented
55C4×D4×C283—21no—
56C4×D4×C283C2211yescharacteristic
57C2×C4⋊C4×C283C2211yescharacteristic
58C2×C4⋊C4×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×C283—21no—
92C22×C4×C283—21no—
93C22×C4×C283C2311yescharacteristic
94C22×C4×C283C2311yescharacteristic
95C4⋊C4×C283—21no—
96C4⋊C4×C283—21no—
97C4⋊C4×C283—21no—
98C4⋊C4×C283—21no—
99C4⋊C4×C283—21no—
100C4⋊C4×C283—21no—
101C4⋊C4×C283D411yescharacteristic
102C4⋊C4×C283C2311yescharacteristic
103C4⋊C4×C283C2311yescharacteristic
104C4⋊C4×C283C2311yescharacteristic
105C22⋊C4×C283—21no—
106C22⋊C4×C283—21no—
107C22⋊C4×C283D411yescharacteristic
108C22⋊C4×C283D411yescharacteristic
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—81no—
138D4×C283—22no—
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×C283—81no—
148C2×C4×C283—81no—
149C2×C4×C283C2×D411yescharacteristic
150C2×C4×C283C2×D411yescharacteristic
151C2×C4×C283C2×D411yescharacteristic
152C2×C4×C283C2×D411yescharacteristic
153C2×C4×C283C2411yescharacteristic
154C22×C283—21no—
155C22×C283—21no—
156C22×C283—21no—
157C22×C283—21no—
158C22×C283—21no—
159C22×C283#13712yes—
160C22×C283C22×D411yescharacteristic
161C2×D283—22no—
162C2×D283—41no—
163C2×D283—41no—
164C2×D283—41no—
165C2×D283—41no—
166C2×D283—41no—
167C2×D283—84no—
168C4×C283—21no—
169C4×C283—21no—
170C4×C283—21no—
171C4×C283—21no—
172C4×C283—41no—
173C4×C283—41no—
174C4×C283—41no—
175C4×C283—41no—
176C4×C283C2×#4111yescharacteristic
177C4×C283H(5,𝔽2)11yescharacteristic
178Dic283—41no—
179Dic283—41no—
180C2×C283—22no—
181C2×C283—41no—
182C2×C283#54611yescharacteristic
183C2×C283C2×#13711yescharacteristic
184C2×C283#57611yescharacteristic
185D283—42no—
186D283—81no—
187C283#331411yescharacteristic, complemented, Sylow
188#3314—2831nomaximal, Sylow
189#554D28311yescharacteristic, complemented
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#450—2831no—
200#440—2831no—
201#435—2831no—
202#417—2831no—
203C2×#102—2831no—
204C2×#41—2831no—
205C22×D4—2831no—
206C2×Q16—2831no—
207C2×SD16—2831no—
208C2×SD16—2831no—
209C2×M16—2831no—
210C22×C8—2831no—
211C4⋊Q8C2×D28311yescharacteristic, complemented
212#125—21no—
213#124—2831no—
214#123—5661no—
215C22⋊Q8—21no—
216C22⋊Q8—21no—
217C22⋊Q8—2831no—
218C22⋊Q8C2×D28311yescharacteristic
219C22⋊Q8C2×D28311yescharacteristic, complemented
220#121—2831no—
221#121—5661no—
222#120—5661no—
223C4×Q8C2×D28311yescharacteristic, complemented
224C4×D4—21no—
225C4×D4—2831no—
226C4×D4C2×D28311yescharacteristic
227C2×C4⋊C4C2×D28311yescharacteristic
228C2×C4⋊C4C2×D28311yescharacteristic, complemented
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#102—2831no—
243#102—2831no—
244C22⋊C8—2831no—
245C22⋊C8—2831no—
246#97—2831no—
247C24—5661no—
248#41—5662no—
249#41—11321no—
250C2×Q8—21no—
251C2×Q8—2831no—
252C2×Q8C22×D28311yescharacteristic
253C2×D4—2831no—
254C2×D4—2831no—
255C2×D4—5662no—
256C2×D4—5661no—
257C2×D4—5661no—
258C2×D4—5661no—
259C2×D4—11321no—
260C2×D4C22⋊D28311yescharacteristic
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—5661no—
264C22×C4—5661no—
265C22×C4C22×D28311yescharacteristic
266C22×C4C22×D28311yescharacteristic
267Q16—11321no—
268SD16—5662no—
269SD16—11321no—
270M16—11321no—
271C2×C8—2831no—
272C2×C8—2831no—
273C2×C8—2831no—
274C2×C8—2831no—
275C2×C8—11321no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—2831no—
283C4⋊C4—5661no—
284C4⋊C4C22⋊D28311yescharacteristic
285C4⋊C4C22×D28311yescharacteristic
286C4⋊C4C22×D28311yescharacteristic
287C4⋊C4C22×D28311yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—5661no—
291C22⋊C4—5661no—
292C22⋊C4—5661no—
293C22⋊C4—5661no—
294C22⋊C4—5661no—
295C22⋊C4C22⋊D28311yescharacteristic
296C22⋊C4C22⋊D28311yescharacteristic
297C42—21no—
298C42C22×D28311yescharacteristic
299C23—5661no—
300C23—5661no—
301C23—11322no—
302C23—11322no—
303C23—22641no—
304C23C2×C22⋊D28311yescharacteristic
305C23C2×C22⋊D28311yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—81no—
309Q8—5661no—
310Q8—5661no—
311D4—22no—
312D4—2832no—
313D4—5661no—
314D4—5661no—
315D4—5661no—
316D4—11322no—
317D4—11322no—
318D4—22641no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—5661no—
330C2×C4—5661no—
331C2×C4—11322no—
332C2×C4—22641no—
333C2×C4C2×C22⋊D28311yescharacteristic
334C2×C4C2×C22⋊D28311yescharacteristic
335C2×C4C2×C22⋊D28311yescharacteristic
336C2×C4C2×C22⋊D28311yescharacteristic
337C2×C4C23×D28311yescharacteristic
338C8—5661no—
339C8—5661no—
340C8—5661no—
341C8—5661no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—5662no—
348C22—11321no—
349C22—11321no—
350C22—11321no—
351C22—11321no—
352C22—11321no—
353C22—22644no—
354C22#2258364112yes—
355C22C22×C22⋊D28311yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—11321no—
365C4—11321no—
366C4#41×D28311yescharacteristic
367C4#2258365611yescharacteristic
368C2—22no—
369C2—41no—
370C2—11322no—
371C2—22641no—
372C2#3182710311yescharacteristic
373C2C2×#2258364111yescharacteristic
374C2#3182726811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#554×C283C2
socleC22×C283C22×D4
prime radical#31827111C2
hypercenter#554D283
hypocenterC283#3314
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
2. #31826827C2semidirect
3. #31826800C2semidirect
4. #31826778C2semidirect
5. #31826742C2semidirect
6. #31826713C2semidirect
7. #31826704C2semidirect
8. #31826687C2semidirect
9. #31826625C2semidirect
10. #31826615C2semidirect
11. #31826563C2semidirect
12. #31826537C2semidirect
13. #31826464C2semidirect
14. C2×#22583530C2semidirect
15. #554×C283C2semidirect
18. C2×#17985382C22semidirect
21. C2×#17985376C22semidirect
31. #22583543C22semidirect
32. #22583543C22semidirect
33. #22583543C22semidirect
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
54. C4×Q8×C283C22semidirect
58. C2×C4⋊C4×C283C22semidirect
187. C283#3314semidirect
189. #554D283semidirect
211. C4⋊Q8C2×D283semidirect
219. C22⋊Q8C2×D283semidirect
223. C4×Q8C2×D283semidirect
228. C2×C4⋊C4C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318271112
#318268272
#318268002
#318267782
#318267422
#318267132
#318267042
#318266872
#318266252
#318266152
#318265632
#318265372
#318264642
C2×#225835302
#554×C2832
#3314283