Finite Group Databasebeta

← #49135373

Subgroups — #49135373

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#31827181C213yesmaximal, complemented
3#31827168C213yesmaximal, complemented
4#31827153C213yesmaximal, complemented
5#31827137C211yesmaximal, characteristic, complemented
6#31827127C211yesmaximal, characteristic, complemented
7#31827127C211yesmaximal, characteristic, complemented
8#31827089C213yesmaximal, complemented
9#31827071C213yesmaximal, complemented
10#31827002C213yesmaximal, complemented
11#31827000C213yesmaximal, complemented
12#31826947C211yesmaximal, characteristic, complemented
13#31826941C211yesmaximal, characteristic, complemented
14C4×#17985393C211yesmaximal, characteristic, complemented
15#557×C283C211yesmaximal, characteristic, complemented
16C2×#17985393C2211yescharacteristic
17C2×#17985393C2211yescharacteristic
18C2×Q8×D283C2211yescharacteristic
19#22583607C2213yes—
20#22583605C2213yes—
21#22583605C2211yescharacteristic
22#22583605C2216yes—
23#22583601C2213yes—
24#22583599C2216yes—
25#22583599C2213yes—
26#22583598C2213yes—
27#22583598C2213yes—
28#22583595C2213yes—
29#22583595C2216yes—
30#22583595C2216yes—
31#22583595C2213yes—
32#22583590C2213yes—
33#22583590C2213yes—
34#22583586C2213yes—
35#22583585C2213yes—
36#22583585C2213yes—
37#22583585C2213yes—
38#22583585C2213yes—
39#22583585C2216yes—
40#22583585C2213yes—
41Q8×Dic283C2211yescharacteristic
42C4×Dic566C2213yes—
43#22583575C2213yes—
44C4×D1132C2213yes—
45C4×D1132C2213yes—
46C4×D1132C2213yes—
47C4×D1132C2213yes—
48C4×D1132C2213yes—
49#22583572C2216yes—
50#22583572C2213yes—
51#22583572C2213yes—
52#22583568C2216yes—
53#22583568C2213yes—
54#22583568C2213yes—
55#22583567C2213yes—
56#22583567C2213yes—
57C4⋊C4×D283C2213yes—
58C42×D283C2213yes—
59C4⋊Q8×C283C2213yes—
60#125×C283C2213yes—
61#125×C283C2213yes—
62C4×Q8×C283C2213yes—
63C4×Q8×C283C2211yescharacteristic
64C4×Q8×C283C2211yescharacteristic
65C4×Q8×C283C2211yescharacteristic
66#17985393—24no—
67#17985393—24no—
68C2×Dic566C2313yes—
69Q8×D283—24no—
70C2×D1132C2313yes—
71C2×D1132C2313yes—
72C2×D1132C2316yes—
73C2×D1132C2313yes—
74C2×D1132C2313yes—
75C2×D1132C2313yes—
76C2×C4×D283C2313yes—
77C2×C4×D283C2313yes—
78C2×C4×D283C2313yes—
79C2×C4×D283C2313yes—
80C2×C4×D283C2313yes—
81C2×C4×D283C2316yes—
82C4⋊Dic283C2313yes—
83C4⋊Dic283C2313yes—
84C4⋊Dic283C2311yescharacteristic
85C4⋊Dic283C2313yes—
86Dic283⋊C4C2313yes—
87Dic283⋊C4C2311yescharacteristic
88Dic283⋊C4C2313yes—
89Dic283⋊C4C2313yes—
90(C2×D283)⋊C4C2311yescharacteristic
91(C2×D283)⋊C4C2313yes—
92(C2×D283)⋊C4C2313yes—
93(C2×D283)⋊C4C2313yes—
94(C2×D283)⋊C4C2313yes—
95(C2×D283)⋊C4C2316yes—
96(C2×D283)⋊C4C2316yes—
97(C2×D283)⋊C4C2313yes—
98(C2×D283)⋊C4C2316yes—
99(C2×D283)⋊C4C2311yescharacteristic
100(C2×D283)⋊C4C2311yescharacteristic
101(C2×D283)⋊C4C2313yes—
102(C2×D283)⋊C4C2313yes—
103C4×Dic283C2311yescharacteristic
104C4×Dic283C2313yes—
105C4×Dic283C2313yes—
106C4×Dic283C2316yes—
107C2×Q8×C283C2313yes—
108C2×Q8×C283C2311yescharacteristic
109C4⋊C4×C283C2313yes—
110C4⋊C4×C283C2313yes—
111C4⋊C4×C283C2313yes—
112C4⋊C4×C283C2316yes—
113C4⋊C4×C283C2313yes—
114C4⋊C4×C283C2313yes—
115C4⋊C4×C283C2311yescharacteristic
116C42×C283C2313yes—
117C42×C283C2313yes—
118C42×C283C2313yes—
119C22×D283C2411yescharacteristic
120C22×D283C2413yes—
121C22×D283C2413yes—
122Dic566—26no—
123D1132—26no—
124D1132—26no—
125D1132—43no—
126D1132—46no—
127D1132—43no—
128D1132—43no—
129C2×Dic283C2411yescharacteristic
130C2×Dic283C2413yes—
131C2×Dic283C2411yescharacteristic
132C2×Dic283C2413yes—
133C2×Dic283C2411yescharacteristic
134C4×D283—26no—
135C4×D283—26no—
136C4×D283—26no—
137C4×D283—43no—
138C4×D283—43no—
139C4×D283—46no—
140Q8×C283—43no—
141Q8×C283#4114yes—
142C2×C4×C283C2411yescharacteristic
143C2×C4×C283C2411yescharacteristic
144C2×C4×C283C2411yescharacteristic
145C2×C4×C283C2413yes—
146C2×C4×C283C2413yes—
147C2×C4×C283C2413yes—
148C2×C4×C283C2413yes—
149C22×C283C2511yescharacteristic
150C2×D283—21no—
151C2×D283—23no—
152C2×D283—23no—
153C2×D283—23no—
154C2×D283—23no—
155C2×D283—23no—
156C2×D283—21no—
157C2×D283—23no—
158C2×D283—21no—
159C4×C283—21no—
160C4×C283—23no—
161C4×C283—23no—
162C4×C283—21no—
163C4×C283—23no—
164C4×C283—21no—
165C4×C283C2×#4116yes—
166Dic283—21no—
167Dic283—21no—
168Dic283—23no—
169Dic283—21no—
170Dic283—23no—
171C2×C283C22×#4111yescharacteristic
172C2×C283#58511yescharacteristic
173C2×C283#58511yescharacteristic
174D283—41no—
175D283—43no—
176D283—43no—
177C283#352411yescharacteristic, complemented, Sylow
178#3524—2831nomaximal, Sylow
179#559—2833no—
180#557D28311yescharacteristic, complemented
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#519—2831no—
193C4×#41—2831no—
194C2×#41—2831no—
195C2×#41—2831no—
196C22×Q8—2831no—
197C4⋊Q8C2×D28313yes—
198#127—2833no—
199#126—2831no—
200#126—2833no—
201#126—2836no—
202#125—2833no—
203#125C2×D28313yes—
204#125C2×D28313yes—
205#124—2833no—
206#124—2836no—
207#124—2833no—
208#124—2833no—
209#123—2836no—
210#123—2833no—
211#123—2833no—
212#123—2836no—
213C22⋊Q8—2833no—
214C22⋊Q8—2833no—
215C22⋊Q8—2833no—
216#121—2833no—
217#121—2833no—
218#121—2833no—
219#121—2833no—
220#121—2836no—
221#121—2833no—
222C4×Q8—2831no—
223C4×Q8—2833no—
224C4×Q8—2833no—
225C4×Q8C2×D28311yescharacteristic
226C4×Q8C2×D28311yescharacteristic
227C4×Q8C2×D28311yescharacteristic
228C4×Q8C2×D28313yes—
229C4×D4—2833no—
230C4×D4—2833no—
231C4×D4—2836no—
232C4×D4—2833no—
233C4×D4—2833no—
234C4×D4—2833no—
235C4×D4—2833no—
236C4×D4—2833no—
237#117—2833no—
238#117—2833no—
239#117—2836no—
240#117—2833no—
241#117—2833no—
242C2×C4⋊C4—2833no—
243C2×C42—2833no—
244#41—5664no—
245#41—5664no—
246C2×Q8—2833no—
247C2×Q8—5664no—
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—2833no—
261C22×C4—2836no—
262C4⋊C4—2833no—
263C4⋊C4—2831no—
264C4⋊C4—2833no—
265C4⋊C4—2833no—
266C4⋊C4—2833no—
267C4⋊C4—2833no—
268C4⋊C4—2831no—
269C4⋊C4—2833no—
270C4⋊C4C22×D28313yes—
271C4⋊C4C22×D28313yes—
272C4⋊C4C22×D28313yes—
273C4⋊C4C22×D28316yes—
274C4⋊C4C22×D28313yes—
275C4⋊C4C22×D28311yescharacteristic
276C4⋊C4C22×D28313yes—
277C22⋊C4—2831no—
278C22⋊C4—2833no—
279C22⋊C4—2833no—
280C22⋊C4—2833no—
281C22⋊C4—2836no—
282C22⋊C4—2833no—
283C22⋊C4—2836no—
284C22⋊C4—2831no—
285C22⋊C4—2833no—
286C22⋊C4—2831no—
287C22⋊C4—2833no—
288C22⋊C4—2836no—
289C22⋊C4—2833no—
290C42—2831no—
291C42—2833no—
292C42—2833no—
293C42—2836no—
294C42C22×D28313yes—
295C42C22×D28313yes—
296C42C22×D28313yes—
297C23—2831no—
298C23—2833no—
299C23—2833no—
300Q8—43no—
301Q8—5666no—
302Q8#1798539314yes—
303D4—5666no—
304D4—5666no—
305D4—11323no—
306D4—11326no—
307D4—11323no—
308D4—11323no—
309C2×C4—2831no—
310C2×C4—2831no—
311C2×C4—2833no—
312C2×C4—2833no—
313C2×C4—2831no—
314C2×C4—5666no—
315C2×C4—5666no—
316C2×C4—5666no—
317C2×C4—11323no—
318C2×C4—11323no—
319C2×C4—11326no—
320C2×C4C23×D28313yes—
321C2×C4C23×D28311yescharacteristic
322C2×C4C23×D28313yes—
323C2×C4C23×D28311yescharacteristic
324C2×C4C23×D28311yescharacteristic
325C2×C4C23×D28313yes—
326C2×C4C23×D28313yes—
327C22—5661no—
328C22—5663no—
329C22—5663no—
330C22—5663no—
331C22—5661no—
332C22—5663no—
333C22—5663no—
334C22—5663no—
335C22—5661no—
336C22C24×D28311yescharacteristic
337C4—23no—
338C4—23no—
339C4—21no—
340C4—21no—
341C4—23no—
342C4—21no—
343C4—5661no—
344C4—5661no—
345C4—5663no—
346C4—5661no—
347C4—5663no—
348C4C2×#1798539316yes—
349C2—11321no—
350C2—11323no—
351C2—11323no—
352C2C22×#1798539311yescharacteristic
353C2#3182730111yescharacteristic
354C2#3182730211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D283
commutator, socle, cosocleC22×C283C25
Fitting#557×C283C2
hypercenter#557D283
hypocenterC283#3524
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. #31827200C2semidirect
2. #31827181C2semidirect
3. #31827168C2semidirect
4. #31827153C2semidirect
5. #31827137C2semidirect
6. #31827127C2semidirect
7. #31827127C2semidirect
8. #31827089C2semidirect
9. #31827071C2semidirect
10. #31827002C2semidirect
11. #31827000C2semidirect
12. #31826947C2semidirect
13. #31826941C2semidirect
14. C4×#17985393C2semidirect
15. #557×C283C2semidirect
177. C283#3524semidirect
180. #557D283semidirect

Maximal subgroups

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

subgroupindex
#318272002
#318271812
#318271682
#318271532
#318271372
#318271272
#318270892
#318270712
#318270022
#318270002
#318269472
#318269412
C4×#179853932
#557×C2832
#3524283