Finite Group Databasebeta

← #49134645

Subgroups — #49134645

Order 36224.

1040 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827082C212yesmaximal, complemented
2#31827080C212yesmaximal, complemented
3#31827078C211yesmaximal, characteristic, complemented
4#31827077C211yesmaximal, characteristic, complemented
5#31827069C211yesmaximal, characteristic, complemented
6#31827058C214yesmaximal, complemented
7#31827047C214yesmaximal, complemented
8#31827021C211yesmaximal, characteristic, complemented
9#31827013C211yesmaximal, characteristic, complemented
10#31826996C214yesmaximal, complemented
11#31826992C214yesmaximal, complemented
12#31826975C212yesmaximal, complemented
13C4×#17985391C212yesmaximal, complemented
14#117×D283C211yesmaximal, characteristic, complemented
15#540×C283C211yesmaximal, characteristic, complemented
16C2×#17985391C2212yes—
17C22×C4×D283C2211yescharacteristic
18#22583610C2211yescharacteristic, complemented
19#22583609C2212yescomplemented
20#22583605C2214yes—
21#22583603C2214yes—
22#22583603C2214yescomplemented
23#22583603C2214yescomplemented
24#22583602C2211yescharacteristic, complemented
25#22583601C2214yescomplemented
26#22583600C2214yescomplemented
27#22583597C2214yes—
28#22583597C2212yescomplemented
29#22583596C2214yes—
30#22583593C2214yes—
31#22583592C2214yes—
32#22583591C2214yes—
33#22583591C2214yescomplemented
34#22583590C2212yes—
35#22583589C2212yes—
36#22583589C2214yescomplemented
37#22583588C2214yes—
38#22583587C2214yescomplemented
39#22583582C2214yes—
40#22583581C2212yes—
41#22583581C2214yes—
42#22583578C2214yes—
43C4×Dic566C2212yescomplemented
44#22583575C2214yescomplemented
45#22583575C2212yescomplemented
46D4×Dic283C2214yes—
47D4×Dic283C2214yescomplemented
48D4×Dic283C2212yescomplemented
49C4×C22⋊D283C2214yes—
50(C22⋊D283)⋊C4C2214yes—
51(C22⋊D283)⋊C4C2214yes—
52#22583569C2211yescharacteristic
53#22583568C2212yes—
54#22583568C2212yes—
55#22583567C2211yescharacteristic
56#22583567C2211yescharacteristic
57#22583566C2212yes—
58#22583566C2214yescomplemented
59C4⋊C4×D283C2212yes—
60C22⋊C4×D283C2212yes—
61C2×C4×Dic283C2214yescomplemented
62C42×D283C2211yescharacteristic
63C42×D283C2211yescharacteristic
64#127×C283C2211yescharacteristic, complemented
65#125×C283C2211yescharacteristic, complemented
66#123×C283C2214yescomplemented
67#121×C283C2212yes—
68#121×C283C2214yes—
69C4×D4×C283C2212yescomplemented
70#117×C283C2211yescharacteristic
71C23×D283C2311yescharacteristic
72#17985391—28no—
73C2×Dic566C2312yes—
74C2×Dic566C2314yes—
75C2×C22⋊D283C2314yes—
76C2×C22⋊D283C2314yes—
77C22×Dic283C2314yes—
78C22×Dic283C2311yescharacteristic
79C22×Dic283C2314yes—
80C2×C4×D283—24no—
81C2×C4×D283C2311yescharacteristic
82C2×C4×D283C2311yescharacteristic
83C2×C4×D283C2311yescharacteristic
84C2×C4×D283C2312yes—
85C2×C4×D283C2312yes—
86C2×C4×D283C2311yescharacteristic
87C4⋊Dic283C2312yes—
88C4⋊Dic283C2312yes—
89C4⋊Dic283C2314yes—
90C4⋊Dic283C2314yes—
91Dic283⋊C4C2312yes—
92Dic283⋊C4C2312yes—
93Dic283⋊C4C2314yes—
94Dic283⋊C4C2314yes—
95Dic283⋊C4C2314yes—
96Dic283⋊C4C2314yes—
97Dic283⋊C4C2314yes—
98C22⋊Dic283C2312yes—
99C22⋊Dic283C2314yes—
100C22⋊Dic283C2314yes—
101C22⋊Dic283C2314yes—
102C22⋊Dic283C2314yes—
103C22⋊Dic283C2314yes—
104(C2×D283)⋊C4C2312yes—
105(C2×D283)⋊C4C2312yes—
106(C2×D283)⋊C4C2314yes—
107(C2×D283)⋊C4C2314yes—
108C4×Dic283C2312yes—
109C4×Dic283C2312yes—
110C4×Dic283C2311yescharacteristic
111C4×Dic283C2311yescharacteristic
112C4×Dic283C2314yes—
113C4×Dic283C2314yes—
114C4×Dic283C2314yes—
115C2×D4×C283C2312yes—
116C2×D4×C283C2314yes—
117C2×D4×C283C2314yes—
118C22×C4×C283C2311yescharacteristic
119C22×C4×C283C2314yes—
120C4⋊C4×C283C2312yes—
121C4⋊C4×C283C2312yes—
122C4⋊C4×C283C2314yes—
123C22⋊C4×C283C2312yes—
124C22⋊C4×C283C2314yes—
125C22⋊C4×C283C2314yes—
126C42×C283C2311yescharacteristic
127C42×C283C2311yescharacteristic
128C22×D283—22no—
129C22×D283—42no—
130C22×D283C2411yescharacteristic
131C22×D283C2411yescharacteristic
132Dic566—24no—
133Dic566—44no—
134C22⋊D283—28no—
135C22⋊D283—44no—
136C2×Dic283—22no—
137C2×Dic283—28no—
138C2×Dic283—44no—
139C2×Dic283C2412yes—
140C2×Dic283C2412yes—
141C2×Dic283C2414yes—
142C2×Dic283C2414yes—
143C2×Dic283C2411yescharacteristic
144C2×Dic283C2411yescharacteristic
145C4×D283—22no—
146C4×D283—22no—
147C4×D283—22no—
148C4×D283—42no—
149C4×D283—42no—
150C4×D283#4114yes—
151C23×C283C2411yescharacteristic
152C23×C283C2414yes—
153D4×C283—24no—
154D4×C283—44no—
155D4×C283—44no—
156C2×C4×C283—22no—
157C2×C4×C283—44no—
158C2×C4×C283C2412yes—
159C2×C4×C283C2412yes—
160C2×C4×C283C2411yescharacteristic
161C2×C4×C283C2414yes—
162C2×C4×C283C2411yescharacteristic
163C22×C283—21no—
164C22×C283—24no—
165C22×C283—24no—
166C22×C283—24no—
167C22×C283—22no—
168C22×C283C2511yescharacteristic
169C2×D283—22no—
170C2×D283—21no—
171C2×D283—22no—
172C2×D283—44no—
173C2×D283C2×#4112yes—
174C4×C283—21no—
175C4×C283—22no—
176C4×C283—24no—
177C4×C283—22no—
178C4×C283C2×#4112yes—
179Dic283—21no—
180Dic283—24no—
181Dic283—22no—
182Dic283—22no—
183Dic283—24no—
184Dic283C2×#4112yes—
185C2×C283—41no—
186C2×C283—44no—
187C2×C283C22×#4111yescharacteristic
188C2×C283#58512yes—
189D283—22no—
190D283—41no—
191C283#345511yescharacteristic, complemented, Sylow
192#3455—2831nomaximal, Sylow
193#542—2832no—
194#542—2832no—
195#541—2831no—
196#541—2831no—
197#540—2831no—
198#540D28311yescharacteristic, complemented
199#539—2834no—
200#538—2834no—
201#533—2831no—
202#532—2831no—
203#529—2834no—
204#529—2834no—
205#525—2832no—
206C4×#41—2832no—
207C2×#117—2831no—
208C2×#41—2832no—
209C23×C4—2831no—
210C4⋊Q8—2832no—
211C4⋊Q8—2831no—
212#127C2×D28311yescharacteristic, complemented
213#126—2834no—
214#126—2834no—
215#126—2834no—
216#126—2834no—
217#125—2834no—
218#125—2834no—
219#125—2831no—
220#125C2×D28311yescharacteristic, complemented
221#124—2834no—
222#124—2832no—
223#124—2834no—
224#123—2834no—
225#123—2834no—
226#123—2834no—
227#123—2834no—
228#123C2×D28314yescomplemented
229C22⋊Q8—2832no—
230C22⋊Q8—2834no—
231C22⋊Q8—2832no—
232C22⋊Q8—2834no—
233C22⋊Q8—2834no—
234#121—2832no—
235#121—2834no—
236#121—2834no—
237#121C2×D28312yes—
238#121C2×D28314yes—
239#120—2834no—
240C4×Q8—2832no—
241C4×Q8—2834no—
242C4×Q8—2832no—
243C4×D4—2832no—
244C4×D4—2834no—
245C4×D4—2834no—
246C4×D4—2834no—
247C4×D4—2834no—
248C4×D4—2834no—
249C4×D4C2×D28312yescomplemented
250#117—2831no—
251#117—2832no—
252#117—2832no—
253#117—2832no—
254#117—2834no—
255#117—2831no—
256#117—2831no—
257#117C2×D28311yescharacteristic
258C2×C4⋊C4—2832no—
259C2×C22⋊C4—2832no—
260C2×C42—2834no—
261C2×C42—2831no—
262C2×C42—2831no—
263C24—2831no—
264#41—5668no—
265C2×Q8—2832no—
266C2×Q8—2834no—
267C2×D4—2834no—
268C2×D4—2834no—
269C2×D4C22×D28312yes—
270C2×D4C22×D28314yes—
271C2×D4C22×D28314yes—
272C22×C4—2831no—
273C22×C4—2834no—
274C22×C4—2831no—
275C22×C4—2834no—
276C22×C4—2831no—
277C22×C4—2831no—
278C22×C4—2831no—
279C22×C4—2832no—
280C22×C4—2832no—
281C22×C4—5664no—
282C22×C4C22×D28311yescharacteristic
283C22×C4C22×D28314yes—
284C4⋊C4—2832no—
285C4⋊C4—2832no—
286C4⋊C4—2834no—
287C4⋊C4—2834no—
288C4⋊C4—2834no—
289C4⋊C4—2834no—
290C4⋊C4—2834no—
291C4⋊C4—2832no—
292C4⋊C4—2832no—
293C4⋊C4—2834no—
294C4⋊C4—2834no—
295C4⋊C4C22×D28312yes—
296C4⋊C4C22×D28312yes—
297C4⋊C4C22×D28314yes—
298C22⋊C4—2832no—
299C22⋊C4—2834no—
300C22⋊C4—2834no—
301C22⋊C4—2834no—
302C22⋊C4—2834no—
303C22⋊C4—2834no—
304C22⋊C4—2832no—
305C22⋊C4—2832no—
306C22⋊C4—2834no—
307C22⋊C4—2834no—
308C22⋊C4C22×D28312yes—
309C22⋊C4C22×D28314yes—
310C22⋊C4C22×D28314yes—
311C42—2832no—
312C42—2832no—
313C42—2834no—
314C42—2831no—
315C42—2831no—
316C42—2834no—
317C42—2834no—
318C42C22×D28311yescharacteristic
319C42C22×D28311yescharacteristic
320C23—2831no—
321C23—2831no—
322C23—5662no—
323C23—11322no—
324C23C23×D28311yescharacteristic
325C23C23×D28314yes—
326Q8—5664no—
327Q8—11324no—
328D4—24no—
329D4—44no—
330D4—44no—
331D4—5668no—
332D4—11324no—
333C2×C4—22no—
334C2×C4—44no—
335C2×C4—2831no—
336C2×C4—2834no—
337C2×C4—2832no—
338C2×C4—2832no—
339C2×C4—2834no—
340C2×C4—2834no—
341C2×C4—2831no—
342C2×C4—5662no—
343C2×C4—5662no—
344C2×C4—5662no—
345C2×C4—5668no—
346C2×C4—5662no—
347C2×C4—11322no—
348C2×C4—11322no—
349C2×C4—11324no—
350C2×C4C23×D28312yes—
351C2×C4C23×D28314yes—
352C2×C4C23×D28312yes—
353C2×C4C23×D28311yescharacteristic
354C2×C4C23×D28311yescharacteristic
355C22—21no—
356C22—24no—
357C22—24no—
358C22—24no—
359C22—22no—
360C22—2832no—
361C22—5662no—
362C22—5661no—
363C22—5662no—
364C22—11324no—
365C22C24×D28311yescharacteristic
366C4—21no—
367C4—22no—
368C4—24no—
369C4—22no—
370C4—2832no—
371C4—5661no—
372C4—5664no—
373C4—5662no—
374C4—5664no—
375C4—5662no—
376C4#41×D28312yes—
377C2—41no—
378C2—44no—
379C2—5662no—
380C2—11321no—
381C2C2×#41×D28311yescharacteristic
382C2#3182730012yes—
subgroupG/H
center, FrattiniC22C24×D283
commutator, socle, cosocleC22×C283C25
Fitting#540×C283C2
hypercenter#540D283
hypocenterC283#3455
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #31827082C2semidirect
2. #31827080C2semidirect
3. #31827078C2semidirect
4. #31827077C2semidirect
5. #31827069C2semidirect
6. #31827058C2semidirect
7. #31827047C2semidirect
8. #31827021C2semidirect
9. #31827013C2semidirect
10. #31826996C2semidirect
11. #31826992C2semidirect
12. #31826975C2semidirect
13. C4×#17985391C2semidirect
14. #117×D283C2semidirect
15. #540×C283C2semidirect
18. #22583610C22semidirect
19. #22583609C22semidirect
22. #22583603C22semidirect
23. #22583603C22semidirect
24. #22583602C22semidirect
25. #22583601C22semidirect
26. #22583600C22semidirect
28. #22583597C22semidirect
33. #22583591C22semidirect
36. #22583589C22semidirect
38. #22583587C22semidirect
43. C4×Dic566C22semidirect
44. #22583575C22semidirect
45. #22583575C22semidirect
47. D4×Dic283C22semidirect
48. D4×Dic283C22semidirect
58. #22583566C22semidirect
61. C2×C4×Dic283C22semidirect
64. #127×C283C22semidirect
65. #125×C283C22semidirect
66. #123×C283C22semidirect
69. C4×D4×C283C22semidirect
191. C283#3455semidirect
198. #540D283semidirect
212. #127C2×D283semidirect
220. #125C2×D283semidirect
228. #123C2×D283semidirect
249. C4×D4C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318270822
#318270802
#318270782
#318270772
#318270692
#318270582
#318270472
#318270212
#318270132
#318269962
#318269922
#318269752
C4×#179853912
#117×D2832
#540×C2832
#3455283