Finite Group Databasebeta

← #49135363

Subgroups — #49135363

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#31827197C213yesmaximal, complemented
2#31827191C211yesmaximal, characteristic, complemented
3#31827179C213yesmaximal, complemented
4#31827161C213yesmaximal, complemented
5#31827157C213yesmaximal, complemented
6#31827134C211yesmaximal, characteristic, complemented
7#31827126C211yesmaximal, characteristic, complemented
8#31827088C213yesmaximal, complemented
9#31827068C213yesmaximal, complemented
10#31826997C213yesmaximal, complemented
11#31826994C213yesmaximal, complemented
12#31826948C211yesmaximal, characteristic, complemented
13#31826943C211yesmaximal, characteristic, complemented
14C4×#17985393C211yesmaximal, characteristic, complemented
15#548×C283C211yesmaximal, characteristic, complemented
16C2×#17985393C2211yescharacteristic
17#22583609C2213yes—
18#22583606C2213yes—
19#22583605C2216yes—
20#22583603C2213yes—
21#22583603C2211yescharacteristic
22#22583602C2213yes—
23#22583601C2213yes—
24#22583601C2213yes—
25#22583599C2213yes—
26#22583597C2213yes—
27#22583596C2216yes—
28#22583594C2216yes—
29#22583592C2216yes—
30#22583592C2213yes—
31#22583592C2213yes—
32#22583590C2213yes—
33#22583586C2213yes—
34#22583584C2213yes—
35#22583584C2213yes—
36#22583582C2216yes—
37#22583581C2213yes—
38#22583581C2213yes—
39Q8×Dic283C2211yescharacteristic
40Q8×Dic283C2211yescharacteristic
41Q8×Dic283C2213yes—
42Q8×Dic283C2211yescharacteristic
43Q8×Dic283C2213yes—
44C4×Dic566C2213yes—
45C4×D1132C2213yes—
46D4×Dic283C2213yes—
47D4×Dic283C2213yes—
48#22583572C2213yes—
49C4×C22⋊D283C2213yes—
50C4×C22⋊D283C2213yes—
51(C22⋊D283)⋊C4C2216yes—
52#22583569C2213yes—
53#22583569C2213yes—
54#22583569C2213yes—
55#22583568C2213yes—
56#22583566C2216yes—
57C2×C4⋊Dic283C2213yes—
58C42×D283C2213yes—
59C2×#41×C283C2211yescharacteristic
60C22×Q8×C283C2211yescharacteristic
61#124×C283C2213yes—
62C22⋊Q8×C283C2213yes—
63#121×C283C2213yes—
64C4×Q8×C283C2211yescharacteristic
65C4×D4×C283C2213yes—
66#17985393—24no—
67C2×Dic566C2313yes—
68C2×D1132C2313yes—
69C2×C22⋊D283C2316yes—
70C2×C22⋊D283C2313yes—
71C2×C22⋊D283C2313yes—
72C22×Dic283C2316yes—
73C2×C4×D283C2313yes—
74C2×C4×D283C2313yes—
75C2×C4×D283C2313yes—
76C4⋊Dic283C2313yes—
77C4⋊Dic283C2313yes—
78C4⋊Dic283C2313yes—
79C4⋊Dic283C2311yescharacteristic
80C4⋊Dic283C2313yes—
81C4⋊Dic283C2313yes—
82C4⋊Dic283C2313yes—
83C4⋊Dic283C2313yes—
84Dic283⋊C4C2313yes—
85Dic283⋊C4C2316yes—
86Dic283⋊C4C2313yes—
87Dic283⋊C4C2311yescharacteristic
88Dic283⋊C4C2311yescharacteristic
89C22⋊Dic283C2313yes—
90C22⋊Dic283C2311yescharacteristic
91C22⋊Dic283C2313yes—
92C22⋊Dic283C2313yes—
93C22⋊Dic283C2313yes—
94C22⋊Dic283C2311yescharacteristic
95(C2×D283)⋊C4C2316yes—
96(C2×D283)⋊C4C2316yes—
97(C2×D283)⋊C4C2313yes—
98(C2×D283)⋊C4C2313yes—
99C4×Dic283C2311yescharacteristic
100C4×Dic283C2313yes—
101C4×Dic283C2313yes—
102C4×Dic283C2316yes—
103C4×Dic283C2313yes—
104C4×Dic283C2313yes—
105#41×C283—24no—
106C2×Q8×C283—24no—
107C2×Q8×C283C2313yes—
108C2×Q8×C283C2311yescharacteristic
109C2×D4×C283C2313yes—
110C2×D4×C283C2313yes—
111C22×C4×C283C2313yes—
112C22×C4×C283C2313yes—
113C4⋊C4×C283C2313yes—
114C4⋊C4×C283C2313yes—
115C22⋊C4×C283C2316yes—
116C22⋊C4×C283C2313yes—
117C22⋊C4×C283C2311yescharacteristic
118C42×C283C2313yes—
119C22×D283C2413yes—
120Dic566—43no—
121D1132—26no—
122C22⋊D283—43no—
123C22⋊D283—43no—
124C22⋊D283—46no—
125C2×Dic283—46no—
126C2×Dic283C2413yes—
127C2×Dic283C2411yescharacteristic
128C2×Dic283C2411yescharacteristic
129C2×Dic283C2411yescharacteristic
130C2×Dic283C2413yes—
131C2×Dic283C2411yescharacteristic
132C2×Dic283C2413yes—
133C4×D283—26no—
134C4×D283—43no—
135C4×D283—43no—
136C23×C283C2413yes—
137C23×C283C2411yescharacteristic
138Q8×C283—26no—
139Q8×C283#4114yes—
140D4×C283—26no—
141D4×C283—43no—
142C2×C4×C283—26no—
143C2×C4×C283—26no—
144C2×C4×C283C2411yescharacteristic
145C2×C4×C283C2413yes—
146C2×C4×C283C2413yes—
147C2×C4×C283C2411yescharacteristic
148C2×C4×C283C2413yes—
149C22×C283—21no—
150C22×C283—23no—
151C22×C283—21no—
152C22×C283—23no—
153C22×C283—23no—
154C22×C283—21no—
155C22×C283C2511yescharacteristic
156C2×D283—23no—
157C2×D283—23no—
158C2×D283—23no—
159C4×C283—21no—
160C4×C283—23no—
161C4×C283—23no—
162C4×C283—21no—
163C4×C283C2×#4116yes—
164Dic283—21no—
165Dic283—21no—
166Dic283—23no—
167Dic283—23no—
168Dic283—21no—
169Dic283—23no—
170Dic283—21no—
171C2×C283—41no—
172C2×C283—43no—
173C2×C283C22×#4111yescharacteristic
174C2×C283#58511yescharacteristic
175C2×C283#58511yescharacteristic
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#548D28311yescharacteristic, complemented
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×#41C2×D28311yescharacteristic
196C22×Q8C2×D28311yescharacteristic
197C4⋊Q8—2833no—
198#127—2833no—
199#126—2836no—
200#126—2831no—
201#126—2833no—
202#125—2833no—
203#125—2833no—
204#125—2833no—
205#124—2833no—
206#124—2833no—
207#124—2836no—
208#124C2×D28313yes—
209#123—2836no—
210#123—2833no—
211#123—2833no—
212#123—2836no—
213C22⋊Q8—2833no—
214C22⋊Q8—2833no—
215C22⋊Q8C2×D28313yes—
216#121—2833no—
217#121—2833no—
218#121—2836no—
219#121—2833no—
220#121—2833no—
221#121C2×D28313yes—
222C4×Q8—2831no—
223C4×Q8—2833no—
224C4×Q8—2831no—
225C4×Q8—2833no—
226C4×Q8—2831no—
227C4×Q8—2833no—
228C4×Q8C2×D28311yescharacteristic
229C4×D4—2833no—
230C4×D4—2833no—
231C4×D4—2833no—
232C4×D4—2833no—
233C4×D4—2833no—
234C4×D4—2836no—
235C4×D4—2833no—
236C4×D4C2×D28313yes—
237#117—2833no—
238#117—2833no—
239#117—2833no—
240#117—2833no—
241#117—2836no—
242C2×C4⋊C4—2833no—
243C2×C42—2833no—
244#41—24no—
245#41—5664no—
246C2×Q8—24no—
247C2×Q8—2833no—
248C2×Q8C22×D28313yes—
249C2×Q8C22×D28311yescharacteristic
250C2×D4—2833no—
251C2×D4—2836no—
252C2×D4—2833no—
253C2×D4—2833no—
254C2×D4C22×D28313yes—
255C2×D4C22×D28313yes—
256C22×C4—2836no—
257C22×C4—2833no—
258C22×C4—2833no—
259C22×C4—2833no—
260C22×C4C22×D28313yes—
261C22×C4C22×D28313yes—
262C4⋊C4—2833no—
263C4⋊C4—2836no—
264C4⋊C4—2831no—
265C4⋊C4—2833no—
266C4⋊C4—2831no—
267C4⋊C4—2833no—
268C4⋊C4—2833no—
269C4⋊C4—2833no—
270C4⋊C4—2831no—
271C4⋊C4—2833no—
272C4⋊C4—2833no—
273C4⋊C4—2833no—
274C4⋊C4—2833no—
275C4⋊C4C22×D28313yes—
276C4⋊C4C22×D28313yes—
277C22⋊C4—2831no—
278C22⋊C4—2833no—
279C22⋊C4—2833no—
280C22⋊C4—2833no—
281C22⋊C4—2833no—
282C22⋊C4—2831no—
283C22⋊C4—2836no—
284C22⋊C4—2833no—
285C22⋊C4—2836no—
286C22⋊C4—2833no—
287C22⋊C4C22×D28316yes—
288C22⋊C4C22×D28313yes—
289C22⋊C4C22×D28311yescharacteristic
290C42—2831no—
291C42—2833no—
292C42—2833no—
293C42—2836no—
294C42—2833no—
295C42—2833no—
296C42C22×D28313yes—
297C23—2833no—
298C23C23×D28311yescharacteristic
299C23C23×D28313yes—
300Q8—26no—
301Q8—11323no—
302Q8#1798539114yes—
303D4—26no—
304D4—43no—
305D4—5666no—
306D4—11326no—
307D4—11323no—
308D4—11323no—
309C2×C4—26no—
310C2×C4—26no—
311C2×C4—2831no—
312C2×C4—2831no—
313C2×C4—2833no—
314C2×C4—2831no—
315C2×C4—2833no—
316C2×C4—2831no—
317C2×C4—2833no—
318C2×C4—5666no—
319C2×C4—11323no—
320C2×C4—11323no—
321C2×C4—11326no—
322C2×C4C23×D28313yes—
323C2×C4C23×D28311yescharacteristic
324C2×C4C23×D28313yes—
325C2×C4C23×D28311yescharacteristic
326C2×C4C23×D28313yes—
327C22—21no—
328C22—23no—
329C22—21no—
330C22—23no—
331C22—23no—
332C22—21no—
333C22—5663no—
334C22—5663no—
335C22—5663no—
336C22C24×D28311yescharacteristic
337C4—21no—
338C4—23no—
339C4—23no—
340C4—21no—
341C4—5661no—
342C4—5661no—
343C4—5663no—
344C4—5661no—
345C4—5663no—
346C4—5663no—
347C4—5661no—
348C4C2×#1798539116yes—
349C2—41no—
350C2—43no—
351C2—11323no—
352C2C22×#1798539111yescharacteristic
353C2#3182730111yescharacteristic
354C2#3182730211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D283
commutator, socle, cosocleC22×C283C25
Fitting#548×C283C2
hypercenter#548D283
hypocenterC283#3524
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #31827197C2semidirect
2. #31827191C2semidirect
3. #31827179C2semidirect
4. #31827161C2semidirect
5. #31827157C2semidirect
6. #31827134C2semidirect
7. #31827126C2semidirect
8. #31827088C2semidirect
9. #31827068C2semidirect
10. #31826997C2semidirect
11. #31826994C2semidirect
12. #31826948C2semidirect
13. #31826943C2semidirect
14. C4×#17985393C2semidirect
15. #548×C283C2semidirect
177. C283#3524semidirect
186. #548D283semidirect

Maximal subgroups

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

subgroupindex
#318271972
#318271912
#318271792
#318271612
#318271572
#318271342
#318271262
#318270882
#318270682
#318269972
#318269942
#318269482
#318269432
C4×#179853932
#548×C2832
#3524283