Finite Group Databasebeta

← #49135372

Subgroups — #49135372

Order 36224.

1030 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827200C213yesmaximal, complemented
2#31827191C211yesmaximal, characteristic, complemented
3#31827178C213yesmaximal, complemented
4#31827159C213yesmaximal, complemented
5#31827156C213yesmaximal, complemented
6#31827137C211yesmaximal, characteristic, complemented
7#31827126C211yesmaximal, characteristic, complemented
8#31827126C211yesmaximal, characteristic, complemented
9#31827087C213yesmaximal, complemented
10#31827070C213yesmaximal, complemented
11#31827002C213yesmaximal, complemented
12#31826993C213yesmaximal, complemented
13#31826947C211yesmaximal, characteristic, complemented
14C4×#17985393C211yesmaximal, characteristic, complemented
15#519×C283C211yesmaximal, characteristic, complemented
16C2×#17985393C2211yescharacteristic
17C2×#17985393C2211yescharacteristic
18#22583609C2213yes—
19#22583607C2213yes—
20#22583605C2213yes—
21#22583604C2216yes—
22#22583603C2211yescharacteristic
23#22583602C2213yes—
24#22583601C2213yes—
25#22583600C2213yes—
26#22583599C2213yes—
27#22583599C2213yes—
28#22583598C2216yes—
29#22583596C2213yes—
30#22583595C2213yes—
31#22583595C2216yes—
32#22583595C2216yes—
33#22583592C2213yes—
34#22583588C2213yes—
35#22583586C2213yes—
36#22583586C2213yes—
37#22583585C2213yes—
38#22583585C2213yes—
39#22583584C2213yes—
40#22583584C2213yes—
41#22583583C2213yes—
42#22583583C2216yes—
43Q8×Dic283C2211yescharacteristic
44Q8×Dic283C2211yescharacteristic
45C4×Dic566C2213yes—
46C4×D1132C2216yes—
47C4×D1132C2213yes—
48C4×D1132C2213yes—
49C4×D1132C2213yes—
50#22583572C2213yes—
51#22583572C2213yes—
52C4×C22⋊D283C2213yes—
53C4×C22⋊D283C2213yes—
54#22583568C2213yes—
55#22583567C2213yes—
56#22583567C2216yes—
57C4⋊C4×D283C2213yes—
58C42×D283C2213yes—
59C22×Q8×C283C2211yescharacteristic
60C4×Q8×C283C2213yes—
61C4×Q8×C283C2211yescharacteristic
62C4×Q8×C283C2213yes—
63C4×Q8×C283C2211yescharacteristic
64#117×C283C2213yes—
65#117×C283C2213yes—
66#17985393—24no—
67#17985393—24no—
68C2×Dic566C2313yes—
69C2×D1132C2313yes—
70C2×D1132C2313yes—
71C2×D1132C2316yes—
72C2×D1132C2313yes—
73C2×C22⋊D283C2313yes—
74C2×C22⋊D283C2313yes—
75C2×C4×D283C2313yes—
76C2×C4×D283C2313yes—
77C2×C4×D283C2313yes—
78C2×C4×D283C2313yes—
79C2×C4×D283C2316yes—
80C4⋊Dic283C2313yes—
81C4⋊Dic283C2313yes—
82C4⋊Dic283C2311yescharacteristic
83C4⋊Dic283C2313yes—
84Dic283⋊C4C2313yes—
85Dic283⋊C4C2311yescharacteristic
86Dic283⋊C4C2313yes—
87Dic283⋊C4C2316yes—
88Dic283⋊C4C2313yes—
89Dic283⋊C4C2311yescharacteristic
90Dic283⋊C4C2313yes—
91C22⋊Dic283C2311yescharacteristic
92C22⋊Dic283C2311yescharacteristic
93(C2×D283)⋊C4C2313yes—
94(C2×D283)⋊C4C2313yes—
95(C2×D283)⋊C4C2313yes—
96(C2×D283)⋊C4C2316yes—
97(C2×D283)⋊C4C2316yes—
98(C2×D283)⋊C4C2313yes—
99(C2×D283)⋊C4C2313yes—
100(C2×D283)⋊C4C2316yes—
101(C2×D283)⋊C4C2313yes—
102C4×Dic283C2311yescharacteristic
103C4×Dic283C2313yes—
104C4×Dic283C2313yes—
105C4×Dic283C2313yes—
106C2×Q8×C283—24no—
107C2×Q8×C283C2313yes—
108C2×Q8×C283C2311yescharacteristic
109C22×C4×C283C2313yes—
110C4⋊C4×C283C2313yes—
111C4⋊C4×C283C2313yes—
112C4⋊C4×C283C2313yes—
113C4⋊C4×C283C2313yes—
114C22⋊C4×C283C2311yescharacteristic
115C22⋊C4×C283C2313yes—
116C42×C283C2313yes—
117C42×C283C2316yes—
118C42×C283C2313yes—
119C22×D283C2413yes—
120C22×D283C2413yes—
121Dic566—43no—
122D1132—26no—
123D1132—26no—
124D1132—43no—
125D1132—46no—
126C22⋊D283—43no—
127C22⋊D283—43no—
128C2×Dic283C2411yescharacteristic
129C2×Dic283C2411yescharacteristic
130C2×Dic283C2411yescharacteristic
131C2×Dic283C2413yes—
132C2×Dic283C2413yes—
133C2×Dic283C2411yescharacteristic
134C4×D283—26no—
135C4×D283—26no—
136C4×D283—43no—
137C4×D283—46no—
138C4×D283—43no—
139C23×C283C2411yescharacteristic
140Q8×C283—26no—
141Q8×C283#4114yes—
142C2×C4×C283—26no—
143C2×C4×C283C2411yescharacteristic
144C2×C4×C283C2413yes—
145C2×C4×C283C2413yes—
146C2×C4×C283C2411yescharacteristic
147C2×C4×C283C2413yes—
148C2×C4×C283C2413yes—
149C22×C283—21no—
150C22×C283—21no—
151C22×C283—21no—
152C22×C283C2511yescharacteristic
153C2×D283—23no—
154C2×D283—23no—
155C2×D283—23no—
156C2×D283—23no—
157C2×D283—23no—
158C2×D283—23no—
159C4×C283—23no—
160C4×C283—23no—
161C4×C283—21no—
162C4×C283—21no—
163C4×C283—23no—
164C4×C283C2×#4116yes—
165Dic283—21no—
166Dic283—21no—
167Dic283—21no—
168Dic283—23no—
169Dic283—23no—
170Dic283—21no—
171C2×C283—41no—
172C2×C283C22×#4111yescharacteristic
173C2×C283#58511yescharacteristic
174C2×C283#58511yescharacteristic
175D283—43no—
176D283—43no—
177C283#352411yescharacteristic, complemented, Sylow
178#3524—2831nomaximal, Sylow
179#559—2833no—
180#557—2831no—
181#555—2833no—
182#553—2833no—
183#552—2833no—
184#550—2831no—
185#548—2831no—
186#548—2831no—
187#542—2833no—
188#540—2833no—
189#529—2833no—
190#529—2833no—
191#520—2831no—
192#519D28311yescharacteristic, complemented
193C4×#41—2831no—
194C2×#41—2831no—
195C2×#41—2831no—
196C22×Q8C2×D28311yescharacteristic
197C4⋊Q8—2833no—
198#127—2833no—
199#126—2833no—
200#126—2831no—
201#126—2836no—
202#125—2833no—
203#125—2833no—
204#125—2833no—
205#124—2833no—
206#124—2833no—
207#124—2833no—
208#124—2836no—
209#123—2836no—
210#123—2833no—
211#123—2836no—
212#123—2833no—
213C22⋊Q8—2833no—
214C22⋊Q8—2833no—
215C22⋊Q8—2833no—
216#121—2833no—
217#121—2833no—
218#121—2833no—
219#121—2833no—
220#121—2833no—
221#121—2836no—
222C4×Q8—2831no—
223C4×Q8—2831no—
224C4×Q8—2833no—
225C4×Q8C2×D28311yescharacteristic
226C4×Q8C2×D28313yes—
227C4×Q8C2×D28311yescharacteristic
228C4×Q8C2×D28313yes—
229C4×D4—2833no—
230C4×D4—2833no—
231C4×D4—2833no—
232C4×D4—2833no—
233C4×D4—2833no—
234C4×D4—2833no—
235C4×D4—2833no—
236C4×D4—2836no—
237#117—2833no—
238#117—2833no—
239#117—2836no—
240#117C2×D28313yes—
241#117C2×D28313yes—
242C2×C4⋊C4—2833no—
243C2×C42—2833no—
244#41—5664no—
245#41—5664no—
246C2×Q8—24no—
247C2×Q8—2833no—
248C2×Q8C22×D28313yes—
249C2×Q8C22×D28311yescharacteristic
250C2×D4—2833no—
251C2×D4—2833no—
252C2×D4—2833no—
253C2×D4—2833no—
254C2×D4—2836no—
255C2×D4—2833no—
256C22×C4—2833no—
257C22×C4—2833no—
258C22×C4—2833no—
259C22×C4—2833no—
260C22×C4—2836no—
261C22×C4C22×D28313yes—
262C4⋊C4—2833no—
263C4⋊C4—2831no—
264C4⋊C4—2833no—
265C4⋊C4—2836no—
266C4⋊C4—2833no—
267C4⋊C4—2831no—
268C4⋊C4—2833no—
269C4⋊C4—2833no—
270C4⋊C4—2833no—
271C4⋊C4—2831no—
272C4⋊C4—2833no—
273C4⋊C4C22×D28313yes—
274C4⋊C4C22×D28313yes—
275C4⋊C4C22×D28313yes—
276C4⋊C4C22×D28313yes—
277C22⋊C4—2831no—
278C22⋊C4—2831no—
279C22⋊C4—2833no—
280C22⋊C4—2833no—
281C22⋊C4—2836no—
282C22⋊C4—2833no—
283C22⋊C4—2836no—
284C22⋊C4—2833no—
285C22⋊C4—2833no—
286C22⋊C4—2836no—
287C22⋊C4—2833no—
288C22⋊C4C22×D28311yescharacteristic
289C22⋊C4C22×D28313yes—
290C42—2831no—
291C42—2833no—
292C42—2833no—
293C42—2833no—
294C42C22×D28313yes—
295C42C22×D28316yes—
296C42C22×D28313yes—
297C23—2833no—
298C23—2833no—
299C23C23×D28311yescharacteristic
300Q8—26no—
301Q8—11323no—
302Q8#1798539214yes—
303D4—5666no—
304D4—5666no—
305D4—11323no—
306D4—11326no—
307D4—11323no—
308D4—11323no—
309C2×C4—26no—
310C2×C4—2831no—
311C2×C4—2831no—
312C2×C4—2831no—
313C2×C4—2833no—
314C2×C4—2833no—
315C2×C4—2831no—
316C2×C4—5666no—
317C2×C4—5666no—
318C2×C4—11323no—
319C2×C4—11326no—
320C2×C4—11323no—
321C2×C4C23×D28311yescharacteristic
322C2×C4C23×D28313yes—
323C2×C4C23×D28313yes—
324C2×C4C23×D28311yescharacteristic
325C2×C4C23×D28313yes—
326C2×C4C23×D28313yes—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—5663no—
331C22—5663no—
332C22—5663no—
333C22—5663no—
334C22—5663no—
335C22—5663no—
336C22C24×D28311yescharacteristic
337C4—23no—
338C4—23no—
339C4—21no—
340C4—21no—
341C4—23no—
342C4—5661no—
343C4—5661no—
344C4—5661no—
345C4—5663no—
346C4—5663no—
347C4—5661no—
348C4C2×#1798539216yes—
349C2—41no—
350C2—11323no—
351C2—11323no—
352C2C22×#1798539211yescharacteristic
353C2#3182730211yescharacteristic
354C2#3182730211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D283
commutator, socle, cosocleC22×C283C25
Fitting#519×C283C2
hypercenter#519D283
hypocenterC283#3524
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ #519

Complements

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

NH ≅ G/Ntype
1. #31827200C2semidirect
2. #31827191C2semidirect
3. #31827178C2semidirect
4. #31827159C2semidirect
5. #31827156C2semidirect
6. #31827137C2semidirect
7. #31827126C2semidirect
8. #31827126C2semidirect
9. #31827087C2semidirect
10. #31827070C2semidirect
11. #31827002C2semidirect
12. #31826993C2semidirect
13. #31826947C2semidirect
14. C4×#17985393C2semidirect
15. #519×C283C2semidirect
177. C283#3524semidirect
192. #519D283semidirect

Maximal subgroups

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

subgroupindex
#318272002
#318271912
#318271782
#318271592
#318271562
#318271372
#318271262
#318270872
#318270702
#318270022
#318269932
#318269472
C4×#179853932
#519×C2832
#3524283