Finite Group Databasebeta

← #49135359

Subgroups — #49135359

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#31827201C213yesmaximal, complemented
2#31827191C211yesmaximal, characteristic, complemented
3#31827179C213yesmaximal, complemented
4#31827167C213yesmaximal, complemented
5#31827155C213yesmaximal, complemented
6#31827122C211yesmaximal, characteristic, complemented
7#31827122C211yesmaximal, characteristic, complemented
8#31827082C213yesmaximal, complemented
9#31827069C213yesmaximal, complemented
10#31826996C213yesmaximal, complemented
11#31826995C213yesmaximal, complemented
12#31826948C211yesmaximal, characteristic, complemented
13#31826941C211yesmaximal, characteristic, complemented
14#41×Dic283C211yesmaximal, characteristic, complemented
15#550×C283C211yesmaximal, characteristic, complemented
16C2×Q8×D283C2211yescharacteristic
17#22583609C2213yes—
18#22583605C2211yescharacteristic
19#22583603C2213yes—
20#22583603C2216yes—
21#22583602C2213yes—
22#22583601C2213yes—
23#22583600C2213yes—
24#22583597C2213yes—
25#22583597C2213yes—
26#22583597C2216yes—
27#22583596C2213yes—
28#22583594C2213yes—
29#22583591C2213yes—
30#22583591C2216yes—
31#22583591C2216yes—
32#22583589C2213yes—
33#22583589C2213yes—
34#22583587C2213yes—
35#22583582C2213yes—
36#22583582C2216yes—
37#22583581C2213yes—
38#22583581C2213yes—
39Q8×Dic283C2211yescharacteristic
40Q8×Dic283C2211yescharacteristic
41Q8×Dic283C2211yescharacteristic
42C4×Dic566C2213yes—
43C4×Dic566C2213yes—
44#22583575C2213yes—
45D4×Dic283C2213yes—
46D4×Dic283C2216yes—
47D4×Dic283C2213yes—
48D4×Dic283C2213yes—
49D4×Dic283C2213yes—
50C4×C22⋊D283C2213yes—
51C4×C22⋊D283C2213yes—
52#22583569C2213yes—
53#22583569C2213yes—
54#22583568C2213yes—
55#22583567C2213yes—
56#22583566C2216yes—
57C2×C4⋊Dic283C2213yes—
58C2×C4×Dic283C2213yes—
59C2×#41×C283C2211yescharacteristic
60C2×#41×C283C2211yescharacteristic
61#127×C283C2213yes—
62#121×C283C2213yes—
63#121×C283C2213yes—
64C4×Q8×C283C2211yescharacteristic
65C4×D4×C283C2213yes—
66C2×Dic566C2313yes—
67C2×Dic566C2313yes—
68Q8×D283—24no—
69C2×C22⋊D283C2313yes—
70C2×C22⋊D283C2313yes—
71C22×Dic283C2313yes—
72C22×Dic283C2313yes—
73C22×Dic283C2316yes—
74C2×C4×D283C2313yes—
75C4⋊Dic283C2313yes—
76C4⋊Dic283C2313yes—
77C4⋊Dic283C2313yes—
78C4⋊Dic283C2311yescharacteristic
79C4⋊Dic283C2313yes—
80C4⋊Dic283C2313yes—
81Dic283⋊C4C2313yes—
82Dic283⋊C4C2316yes—
83Dic283⋊C4C2313yes—
84Dic283⋊C4C2313yes—
85Dic283⋊C4C2313yes—
86Dic283⋊C4C2311yescharacteristic
87Dic283⋊C4C2313yes—
88C22⋊Dic283C2316yes—
89C22⋊Dic283C2313yes—
90C22⋊Dic283C2316yes—
91C22⋊Dic283C2313yes—
92C22⋊Dic283C2313yes—
93C22⋊Dic283C2313yes—
94C22⋊Dic283C2316yes—
95(C2×D283)⋊C4C2311yescharacteristic
96(C2×D283)⋊C4C2313yes—
97(C2×D283)⋊C4C2311yescharacteristic
98(C2×D283)⋊C4C2311yescharacteristic
99C4×Dic283C2311yescharacteristic
100C4×Dic283C2313yes—
101C4×Dic283C2313yes—
102C4×Dic283C2313yes—
103C4×Dic283C2316yes—
104C4×Dic283C2313yes—
105#41×C283—24no—
106#41×C283—24no—
107C2×Q8×C283C2311yescharacteristic
108C2×D4×C283C2313yes—
109C2×D4×C283C2313yes—
110C2×D4×C283C2316yes—
111C2×D4×C283C2313yes—
112C22×C4×C283C2313yes—
113C22×C4×C283C2313yes—
114C4⋊C4×C283C2313yes—
115C4⋊C4×C283C2311yescharacteristic
116C22⋊C4×C283C2313yes—
117C22⋊C4×C283C2313yes—
118C42×C283C2313yes—
119C22×D283C2411yescharacteristic
120Dic566—26no—
121Dic566—43no—
122C22⋊D283—43no—
123C22⋊D283—43no—
124C2×Dic283—43no—
125C2×Dic283—46no—
126C2×Dic283—43no—
127C2×Dic283C2411yescharacteristic
128C2×Dic283C2411yescharacteristic
129C2×Dic283C2413yes—
130C2×Dic283C2411yescharacteristic
131C2×Dic283C2413yes—
132C2×Dic283C2413yes—
133C2×Dic283C2413yes—
134C4×D283—26no—
135C23×C283C2413yes—
136C23×C283C2413yes—
137Q8×C283#4114yes—
138D4×C283—26no—
139D4×C283—26no—
140D4×C283—43no—
141D4×C283—46no—
142C2×C4×C283—26no—
143C2×C4×C283—26no—
144C2×C4×C283C2411yescharacteristic
145C2×C4×C283C2413yes—
146C2×C4×C283C2411yescharacteristic
147C2×C4×C283C2411yescharacteristic
148C2×C4×C283C2413yes—
149C22×C283—23no—
150C22×C283—23no—
151C22×C283—23no—
152C22×C283—23no—
153C22×C283—23no—
154C22×C283—23no—
155C22×C283C2511yescharacteristic
156C2×D283—21no—
157C2×D283—21no—
158C2×D283—21no—
159C4×C283—21no—
160C4×C283—21no—
161C4×C283—21no—
162C4×C283—23no—
163C4×C283C2×#4116yes—
164Dic283—23no—
165Dic283—21no—
166Dic283—21no—
167Dic283—23no—
168Dic283—23no—
169Dic283—23no—
170Dic283—21no—
171C2×C283—43no—
172C2×C283—43no—
173C2×C283C22×#4111yescharacteristic
174C2×C283#58511yescharacteristic
175C2×C283#58511yescharacteristic
176D283—41no—
177C283#352411yescharacteristic, complemented, Sylow
178#3524—2831nomaximal, Sylow
179#559—2833no—
180#557—2831no—
181#555—2833no—
182#553—2833no—
183#552—2833no—
184#550D28311yescharacteristic, complemented
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×#41C2×D28311yescharacteristic
195C2×#41C2×D28311yescharacteristic
196C22×Q8—2831no—
197C4⋊Q8—2833no—
198#127C2×D28313yes—
199#126—2831no—
200#126—2833no—
201#126—2836no—
202#125—2833no—
203#125—2833no—
204#125—2833no—
205#124—2833no—
206#124—2833no—
207#124—2836no—
208#124—2833no—
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—2836no—
220#121C2×D28313yes—
221#121C2×D28313yes—
222C4×Q8—2831no—
223C4×Q8—2831no—
224C4×Q8—2831no—
225C4×Q8—2833no—
226C4×Q8—2833no—
227C4×Q8—2833no—
228C4×Q8C2×D28311yescharacteristic
229C4×D4—2833no—
230C4×D4—2833no—
231C4×D4—2833no—
232C4×D4—2833no—
233C4×D4—2836no—
234C4×D4—2833no—
235C4×D4—2833no—
236C4×D4C2×D28313yes—
237#117—2833no—
238#117—2833no—
239#117—2833no—
240#117—2836no—
241#117—2833no—
242C2×C4⋊C4—2833no—
243C2×C42—2833no—
244#41—24no—
245#41—24no—
246C2×Q8—2833no—
247C2×Q8—2833no—
248C2×Q8—5664no—
249C2×Q8C22×D28311yescharacteristic
250C2×D4—2833no—
251C2×D4—2833no—
252C2×D4C22×D28313yes—
253C2×D4C22×D28313yes—
254C2×D4C22×D28316yes—
255C2×D4C22×D28313yes—
256C22×C4—2833no—
257C22×C4—2833no—
258C22×C4—2836no—
259C22×C4—2833no—
260C22×C4C22×D28313yes—
261C22×C4C22×D28313yes—
262C4⋊C4—2833no—
263C4⋊C4—2833no—
264C4⋊C4—2833no—
265C4⋊C4—2836no—
266C4⋊C4—2833no—
267C4⋊C4—2831no—
268C4⋊C4—2833no—
269C4⋊C4—2833no—
270C4⋊C4—2833no—
271C4⋊C4—2833no—
272C4⋊C4—2831no—
273C4⋊C4—2833no—
274C4⋊C4—2833no—
275C4⋊C4C22×D28313yes—
276C4⋊C4C22×D28311yescharacteristic
277C22⋊C4—2833no—
278C22⋊C4—2836no—
279C22⋊C4—2833no—
280C22⋊C4—2836no—
281C22⋊C4—2833no—
282C22⋊C4—2836no—
283C22⋊C4—2833no—
284C22⋊C4—2831no—
285C22⋊C4—2833no—
286C22⋊C4—2831no—
287C22⋊C4—2831no—
288C22⋊C4C22×D28313yes—
289C22⋊C4C22×D28313yes—
290C42—2831no—
291C42—2833no—
292C42—2833no—
293C42—2833no—
294C42—2836no—
295C42—2833no—
296C42C22×D28313yes—
297C23—2831no—
298C23C23×D28313yes—
299C23C23×D28313yes—
300Q8—5666no—
301Q8—11323no—
302Q8#1798539114yes—
303D4—26no—
304D4—26no—
305D4—46no—
306D4—43no—
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—2833no—
317C2×C4—2833no—
318C2×C4—5666no—
319C2×C4—11323no—
320C2×C4—11323no—
321C2×C4—11326no—
322C2×C4C23×D28311yescharacteristic
323C2×C4C23×D28313yes—
324C2×C4C23×D28311yescharacteristic
325C2×C4C23×D28311yescharacteristic
326C2×C4C23×D28313yes—
327C22—23no—
328C22—23no—
329C22—23no—
330C22—23no—
331C22—23no—
332C22—23no—
333C22—5661no—
334C22—5661no—
335C22—5661no—
336C22C24×D28311yescharacteristic
337C4—21no—
338C4—21no—
339C4—23no—
340C4—21no—
341C4—5663no—
342C4—5661no—
343C4—5663no—
344C4—5661no—
345C4—5661no—
346C4—5663no—
347C4—5663no—
348C4C2×#1798539116yes—
349C2—43no—
350C2—43no—
351C2—11321no—
352C2C22×#1798539111yescharacteristic
353C2#3182730011yescharacteristic
354C2#3182730111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D283
commutator, socle, cosocleC22×C283C25
Fitting#550×C283C2
hypercenter#550D283
hypocenterC283#3524
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #31827201C2semidirect
2. #31827191C2semidirect
3. #31827179C2semidirect
4. #31827167C2semidirect
5. #31827155C2semidirect
6. #31827122C2semidirect
7. #31827122C2semidirect
8. #31827082C2semidirect
9. #31827069C2semidirect
10. #31826996C2semidirect
11. #31826995C2semidirect
12. #31826948C2semidirect
13. #31826941C2semidirect
14. #41×Dic283C2semidirect
15. #550×C283C2semidirect
177. C283#3524semidirect
184. #550D283semidirect

Maximal subgroups

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

subgroupindex
#318272012
#318271912
#318271792
#318271672
#318271552
#318271222
#318270822
#318270692
#318269962
#318269952
#318269482
#318269412
#41×Dic2832
#550×C2832
#3524283