Finite Group Databasebeta

← #49134857

Subgroups — #49134857

Order 36224.

1112 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827198C212yesmaximal, complemented
2#31827193C211yesmaximal, characteristic, complemented
3#31827188C211yesmaximal, characteristic, complemented
4#31827184C212yesmaximal, complemented
5#31827165C212yesmaximal, complemented
6#31827160C214yesmaximal, complemented
7#31827135C211yesmaximal, characteristic, complemented
8#31827123C212yesmaximal, complemented
9#31827014C212yesmaximal, complemented
10#31827013C211yesmaximal, characteristic, complemented
11#31826994C212yesmaximal, complemented
12#31826992C214yesmaximal, complemented
13C4×#17985393C211yesmaximal, characteristic, complemented
14C4×#17985391C212yesmaximal, complemented
15C4×#17985391C212yesmaximal, complemented
16C4×Q8×D283C211yesmaximal, characteristic, complemented
17#559×C283C211yesmaximal, characteristic, complemented
18C2×#17985393C2211yescharacteristic
19C2×#17985391C2212yes—
20C2×#17985391C2212yes—
21C2×Q8×D283C2211yescharacteristic
22#22583610C2211yescharacteristic
23#22583608C2212yes—
24#22583606C2212yes—
25#22583605C2214yes—
26#22583603C2214yes—
27#22583603C2214yes—
28#22583602C2212yes—
29#22583600C2214yes—
30#22583598C2212yes—
31#22583597C2212yes—
32#22583596C2214yes—
33#22583593C2214yes—
34#22583592C2214yes—
35#22583587C2214yes—
36#22583586C2212yes—
37#22583583C2214yes—
38#22583583C2214yes—
39#22583582C2214yes—
40#22583581C2212yes—
41#22583581C2212yes—
42#22583581C2212yes—
43Q8×Dic283C2211yescharacteristic
44Q8×Dic283C2211yescharacteristic
45C4×Dic566C2212yes—
46C4×Dic566C2211yescharacteristic
47C4×Dic566C2212yes—
48C4×Dic566C2212yes—
49#22583575C2212yes—
50#22583575C2211yescharacteristic
51#22583575C2212yes—
52C4×D1132C2212yes—
53C4×D1132C2211yescharacteristic
54D4×Dic283C2212yes—
55D4×Dic283C2212yes—
56D4×Dic283C2212yes—
57#22583572C2212yes—
58#22583572C2211yescharacteristic
59C4×C22⋊D283C2214yes—
60C4×C22⋊D283C2214yes—
61(C22⋊D283)⋊C4C2214yes—
62(C22⋊D283)⋊C4C2214yes—
63(C22⋊D283)⋊C4C2214yes—
64#22583568C2212yes—
65#22583568C2212yes—
66#22583568C2211yescharacteristic
67#22583566C2214yes—
68#22583566C2214yes—
69C4⋊C4×D283C2212yes—
70C4⋊C4×D283C2211yescharacteristic
71C2×C4×Dic283C2214yes—
72C42×D283C2212yes—
73C42×D283C2211yescharacteristic
74C42×D283C2212yes—
75#127×C283C2211yescharacteristic
76#124×C283C2212yes—
77#124×C283C2212yes—
78#123×C283C2214yes—
79C4×Q8×C283C2211yescharacteristic
80C4×Q8×C283C2211yescharacteristic
81C4×D4×C283C2212yes—
82C4×D4×C283C2212yes—
83#17985393—24no—
84#17985391—28no—
85#17985391—28no—
86C2×Dic566C2312yes—
87C2×Dic566C2312yes—
88C2×Dic566C2311yescharacteristic
89Q8×D283—24no—
90C2×D1132C2312yes—
91C2×D1132C2311yescharacteristic
92C2×C22⋊D283C2314yes—
93C2×C22⋊D283C2314yes—
94C2×C22⋊D283C2314yes—
95C22×Dic283C2314yes—
96C22×Dic283C2314yes—
97C2×C4×D283C2312yes—
98C2×C4×D283C2311yescharacteristic
99C2×C4×D283C2312yes—
100C2×C4×D283C2312yes—
101C2×C4×D283C2312yes—
102C2×C4×D283C2311yescharacteristic
103C2×C4×D283C2312yes—
104C2×C4×D283C2311yescharacteristic
105C4⋊Dic283C2311yescharacteristic
106C4⋊Dic283C2312yes—
107C4⋊Dic283C2311yescharacteristic
108C4⋊Dic283C2312yes—
109C4⋊Dic283C2312yes—
110C4⋊Dic283C2312yes—
111Dic283⋊C4C2312yes—
112Dic283⋊C4C2312yes—
113Dic283⋊C4C2312yes—
114Dic283⋊C4C2314yes—
115Dic283⋊C4C2314yes—
116Dic283⋊C4C2314yes—
117Dic283⋊C4C2314yes—
118C22⋊Dic283C2314yes—
119C22⋊Dic283C2314yes—
120C22⋊Dic283C2314yes—
121(C2×D283)⋊C4C2312yes—
122(C2×D283)⋊C4C2314yes—
123(C2×D283)⋊C4C2312yes—
124(C2×D283)⋊C4C2312yes—
125(C2×D283)⋊C4C2314yes—
126C4×Dic283C2311yescharacteristic
127C4×Dic283C2312yes—
128C4×Dic283C2312yes—
129C4×Dic283C2312yes—
130C4×Dic283C2312yes—
131C4×Dic283C2311yescharacteristic
132C4×Dic283C2311yescharacteristic
133C4×Dic283C2312yes—
134C4×Dic283C2314yes—
135C4×Dic283C2314yes—
136C2×Q8×C283C2311yescharacteristic
137C2×Q8×C283C2311yescharacteristic
138C2×D4×C283C2312yes—
139C2×D4×C283C2312yes—
140C2×D4×C283C2312yes—
141C22×C4×C283C2314yes—
142C4⋊C4×C283C2311yescharacteristic
143C4⋊C4×C283C2312yes—
144C4⋊C4×C283C2311yescharacteristic
145C4⋊C4×C283C2312yes—
146C22⋊C4×C283C2314yes—
147C22⋊C4×C283C2314yes—
148C22⋊C4×C283C2314yes—
149C42×C283C2312yes—
150C42×C283C2311yescharacteristic
151C42×C283C2312yes—
152C22×D283C2412yes—
153C22×D283C2411yescharacteristic
154Dic566—24no—
155Dic566—24no—
156Dic566#4114yescomplemented
157D1132—24no—
158D1132—22no—
159C22⋊D283—28no—
160C22⋊D283—28no—
161C22⋊D283—44no—
162C2×Dic283—28no—
163C2×Dic283—28no—
164C2×Dic283C2412yes—
165C2×Dic283C2411yescharacteristic
166C2×Dic283C2411yescharacteristic
167C2×Dic283C2414yes—
168C2×Dic283C2412yes—
169C2×Dic283C2411yescharacteristic
170C2×Dic283C2412yes—
171C4×D283—22no—
172C4×D283—24no—
173C4×D283—24no—
174C4×D283—24no—
175C4×D283—42no—
176C4×D283—41no—
177C4×D283—42no—
178C4×D283#4114yescomplemented
179C23×C283C2414yes—
180Q8×C283—22no—
181Q8×C283—22no—
182D4×C283—24no—
183D4×C283—24no—
184D4×C283—42no—
185C2×C4×C283—44no—
186C2×C4×C283C2412yes—
187C2×C4×C283C2412yes—
188C2×C4×C283C2412yes—
189C2×C4×C283C2412yes—
190C2×C4×C283C2411yescharacteristic
191C2×C4×C283C2411yescharacteristic
192C2×C4×C283C2411yescharacteristic
193C22×C283—24no—
194C22×C283—24no—
195C22×C283—24no—
196C22×C283C2511yescharacteristic
197C2×D283—21no—
198C2×D283—21no—
199C2×D283—22no—
200C2×D283—22no—
201C2×D283—22no—
202C2×D283C2×#4112yes—
203C4×C283—22no—
204C4×C283—22no—
205C4×C283—22no—
206C4×C283—22no—
207C4×C283—21no—
208C4×C283C2×#4112yes—
209C4×C283C2×#4112yes—
210Dic283—22no—
211Dic283—21no—
212Dic283—24no—
213Dic283—21no—
214Dic283—22no—
215Dic283C2×#4112yes—
216Dic283C2×#4114yes—
217C2×C283—44no—
218C2×C283C22×#4111yescharacteristic
219C2×C283C22×#4111yescharacteristic
220C2×C283#58511yescharacteristic
221D283—22no—
222D283—42no—
223C283#348011yescharacteristic, complemented, Sylow
224#3480—2831nomaximal, Sylow
225#559—2832no—
226#559D28311yescharacteristic, complemented
227#557—2831no—
228#556—2832no—
229#556—2831no—
230#553—2834no—
231#553—2832no—
232#550—2831no—
233#548—2832no—
234#532—2831no—
235#532—2832no—
236#529—2832no—
237#529—2834no—
238C4×#41—2831no—
239C4×#41—2832no—
240C4×#41—2832no—
241C2×C4×Q8—2831no—
242C2×#41—2831no—
243C2×#41—2832no—
244C2×#41—2832no—
245C22×Q8—2831no—
246C4⋊Q8—2832no—
247C4⋊Q8—2831no—
248#127—2832no—
249#127C2×D28311yescharacteristic
250#126—2834no—
251#126—2834no—
252#126—2834no—
253#125—2834no—
254#125—2832no—
255#124—2832no—
256#124—2834no—
257#124—2832no—
258#124C2×D28312yes—
259#124C2×D28312yes—
260#123—2834no—
261#123—2834no—
262#123C2×D28314yes—
263C22⋊Q8—2832no—
264C22⋊Q8—2834no—
265#121—2832no—
266#121—2834no—
267#121—2832no—
268#121—2832no—
269#121—2834no—
270#121—2834no—
271C4×Q8—2831no—
272C4×Q8—2831no—
273C4×Q8—2831no—
274C4×Q8—2832no—
275C4×Q8—2832no—
276C4×Q8—2831no—
277C4×Q8—2832no—
278C4×Q8—2832no—
279C4×Q8—2832no—
280C4×Q8C2×D28311yescharacteristic
281C4×Q8C2×D28311yescharacteristic
282C4×D4—2832no—
283C4×D4—2832no—
284C4×D4—2832no—
285C4×D4—2834no—
286C4×D4—2834no—
287C4×D4—2832no—
288C4×D4—2831no—
289C4×D4—2834no—
290C4×D4—2834no—
291C4×D4—2834no—
292C4×D4—2832no—
293C4×D4—2831no—
294C4×D4C2×D28312yes—
295C4×D4C2×D28312yes—
296#117—2831no—
297#117—2832no—
298#117—2832no—
299#117—2834no—
300#117—2834no—
301C2×C4⋊C4—2832no—
302C2×C4⋊C4—2831no—
303C2×C42—2834no—
304C2×C42—2831no—
305C2×C42—2832no—
306C2×C42—2832no—
307#41—5668no—
308#41—5664no—
309#41—5668no—
310C2×Q8—2831no—
311C2×Q8—2832no—
312C2×Q8—2832no—
313C2×Q8—5664no—
314C2×Q8C22×D28311yescharacteristic
315C2×Q8C22×D28311yescharacteristic
316C2×D4—2834no—
317C2×D4—2834no—
318C2×D4—2834no—
319C2×D4—2832no—
320C2×D4—2831no—
321C2×D4C22×D28312yes—
322C2×D4C22×D28312yes—
323C2×D4C22×D28312yes—
324C22×C4—2831no—
325C22×C4—2834no—
326C22×C4—2834no—
327C22×C4—2832no—
328C22×C4—2831no—
329C22×C4—2832no—
330C22×C4—2832no—
331C22×C4—2831no—
332C22×C4—2832no—
333C22×C4—2832no—
334C22×C4C22×D28314yes—
335C4⋊C4—2832no—
336C4⋊C4—2832no—
337C4⋊C4—2832no—
338C4⋊C4—2834no—
339C4⋊C4—2834no—
340C4⋊C4—2834no—
341C4⋊C4—2834no—
342C4⋊C4—2831no—
343C4⋊C4—2832no—
344C4⋊C4—2831no—
345C4⋊C4—2832no—
346C4⋊C4—2832no—
347C4⋊C4—2832no—
348C4⋊C4C22×D28311yescharacteristic
349C4⋊C4C22×D28312yes—
350C4⋊C4C22×D28311yescharacteristic
351C4⋊C4C22×D28312yes—
352C22⋊C4—2834no—
353C22⋊C4—2834no—
354C22⋊C4—2834no—
355C22⋊C4—2832no—
356C22⋊C4—2832no—
357C22⋊C4—2834no—
358C22⋊C4—2832no—
359C22⋊C4—2834no—
360C22⋊C4C22×D28314yes—
361C22⋊C4C22×D28314yes—
362C22⋊C4C22×D28314yes—
363C42—2832no—
364C42—2832no—
365C42—2831no—
366C42—2831no—
367C42—2832no—
368C42—2832no—
369C42—2834no—
370C42—2834no—
371C42—2831no—
372C42—2832no—
373C42C22×D28312yes—
374C42C22×D28311yescharacteristic
375C42C22×D28312yes—
376C23—2831no—
377C23—2832no—
378C23C23×D28314yes—
379Q8—22no—
380Q8—22no—
381Q8—2834no—
382Q8—5664no—
383Q8—5664no—
384D4—24no—
385D4—24no—
386D4—42no—
387D4—5664no—
388D4—5668no—
389D4—5668no—
390D4—5662no—
391D4—11324no—
392C2×C4—44no—
393C2×C4—2831no—
394C2×C4—2832no—
395C2×C4—2834no—
396C2×C4—2832no—
397C2×C4—2831no—
398C2×C4—2831no—
399C2×C4—2834no—
400C2×C4—2832no—
401C2×C4—5664no—
402C2×C4—5664no—
403C2×C4—5668no—
404C2×C4—5668no—
405C2×C4—5662no—
406C2×C4—5664no—
407C2×C4—11322no—
408C2×C4—11322no—
409C2×C4—11321no—
410C2×C4C23×D28312yes—
411C2×C4C23×D28312yes—
412C2×C4C23×D28312yes—
413C2×C4C23×D28311yescharacteristic
414C2×C4C23×D28312yes—
415C2×C4C23×D28311yescharacteristic
416C2×C4C23×D28311yescharacteristic
417C22—24no—
418C22—24no—
419C22—24no—
420C22—2832no—
421C22—5661no—
422C22—5661no—
423C22—5662no—
424C22—5662no—
425C22—5662no—
426C22C24×D28311yescharacteristic
427C4—22no—
428C4—22no—
429C4—22no—
430C4—22no—
431C4—21no—
432C4—2832no—
433C4—2834no—
434C4—5662no—
435C4—5664no—
436C4—5661no—
437C4—5661no—
438C4—5662no—
439C4#41×D28312yes—
440C4#41×D28312yes—
441C2—44no—
442C2—5662no—
443C2—11322no—
444C2C2×#41×D28311yescharacteristic
445C2C2×#41×D28311yescharacteristic
446C2#3182730011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D283
commutator, socle, cosocleC22×C283C25
Fitting#559×C283C2
hypercenter#559D283
hypocenterC283#3480
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ #559

Complements

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

NH ≅ G/Ntype
1. #31827198C2semidirect
2. #31827193C2semidirect
3. #31827188C2semidirect
4. #31827184C2semidirect
5. #31827165C2semidirect
6. #31827160C2semidirect
7. #31827135C2semidirect
8. #31827123C2semidirect
9. #31827014C2semidirect
10. #31827013C2semidirect
11. #31826994C2semidirect
12. #31826992C2semidirect
13. C4×#17985393C2semidirect
14. C4×#17985391C2semidirect
15. C4×#17985391C2semidirect
16. C4×Q8×D283C2semidirect
17. #559×C283C2semidirect
156. Dic566#41semidirect
178. C4×D283#41semidirect
223. C283#3480semidirect
226. #559D283semidirect

Maximal subgroups

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

subgroupindex
#318271982
#318271932
#318271882
#318271842
#318271652
#318271602
#318271352
#318271232
#318270142
#318270132
#318269942
#318269922
C4×#179853932
C4×#179853912
C4×Q8×D2832
#559×C2832
#3480283