Finite Group Databasebeta

← #27366959

Subgroups — #27366959

Order 13568.

682 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285174C211yesmaximal, characteristic
2#20284746C211yesmaximal, characteristic
3C2×#16810908C211yesmaximal, characteristic, complemented
4#20274209C212yesmaximal, complemented
5#20274187C212yesmaximal, complemented
6#20274085C212yesmaximal, complemented
7#20274063C212yesmaximal, complemented
8#20273467C211yesmaximal, characteristic, complemented
9#20273393C211yesmaximal, characteristic
10#20273388C211yesmaximal, characteristic, complemented
11Q8×M16×C53C211yesmaximal, characteristic, complemented
12C2×#13133250—21no—
13C2×#13133250—21no—
14#16811173C2211yescharacteristic
15C2×#13133217C2211yescharacteristic
16#16811017—22no—
17#16811003C2212yes—
18#16810996C2212yes—
19#16810941—21no—
20#16810908—21no—
21#16810908—21no—
22#16810908C2212yes—
23#16810908C2211yescharacteristic
24#16810908C2212yes—
25#16810908C2211yescharacteristic
26#16810823C2211yescharacteristic
27#16810823C2211yescharacteristic
28#16810762—21no—
29C4×#11719924—21no—
30C4×#11719924—21no—
31#16810676C2211yescharacteristic
32C2×#13133148—21no—
33C2×C4⋊#88322C2211yescharacteristic
34C2×#13133141—21no—
35#16810376C2212yes—
36#16810374C2212yes—
37#16810370C2212yes—
38#16810370C2212yes—
39#16810370C2212yes—
40#16810366C2212yes—
41C2×C4×Q8×C53C2211yescharacteristic
42C53×#446—22no—
43C53×#446C2212yes—
44C8×Q8×C53C2212yes—
45C53×#423—21no—
46C53×#423C2211yescharacteristic
47C4×M16×C53—21no—
48C4×M16×C53C2211yescharacteristic
49C2×#11719945—21no—
50C22×Dic106—21no—
51#13133250—44no—
52#13133250—44no—
53C2×Dic212—22no—
54C2×Dic212—22no—
55C2×#11719913—22no—
56C2×#11719913—22no—
57C2×#11719924—21no—
58C2×#11719924—21no—
59C22×#88322—21no—
60#13133217—21no—
61#13133217—21no—
62#13133217—21no—
63#13133217C2312yes—
64#13133217C2311yescharacteristic
65#13133214C2311yescharacteristic
66#13133205—21no—
67#13133191—21no—
68#13133191—21no—
69C4×D212—21no—
70C4×D212—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73#13133148—22no—
74#13133148—22no—
75#13133148—21no—
76#13133148—21no—
77C4⋊#88322—21no—
78C4⋊#88322C2311yescharacteristic
79C4⋊#88322C2311yescharacteristic
80C4⋊#88322C2312yes—
81#13133141—21no—
82#13133141—21no—
83#13133141—22no—
84#13133141—22no—
85#13133137—21no—
86#13133137—21no—
87C22⋊#88322—21no—
88C4×#88322—21no—
89C22×Q8×C53D411yescharacteristic
90C2×M16×C53—21no—
91C2×M16×C53D411yescharacteristic
92C2×M16×C53D411yescharacteristic
93C4×Q8×C53—21no—
94C4×Q8×C53—21no—
95C4×Q8×C53D411yescharacteristic
96C4×Q8×C53D411yescharacteristic
97C4×Q8×C53C2311yescharacteristic
98C4×Q8×C53C2311yescharacteristic
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53D411yescharacteristic
101C2×C42×C53—21no—
102C2×C42×C53C2311yescharacteristic
103C4⋊C8×C53—22no—
104C4⋊C8×C53—22no—
105C4⋊C8×C53—21no—
106C4⋊C8×C53C2312yes—
107#97×C53—22no—
108#97×C53D412yes—
109C4×C8×C53—22no—
110C4×C8×C53C2312yes—
111C2×Q8×C53—21no—
112C2×Q8×C53—22no—
113C2×Q8×C53—41no—
114C2×Q8×C53C2×D411yescharacteristic
115C2×Q8×C53C2×D411yescharacteristic
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53C2×D411yescharacteristic
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53#4111yescharacteristic
121C2×C22⋊D53—41no—
122C22×Dic53—41no—
123#11719945—42no—
124#11719945—42no—
125C2×D212—21no—
126C2×D212—21no—
127C2×D212—21no—
128C2×C4×D53—41no—
129C2×Dic106—21no—
130C2×Dic106—22no—
131C2×Dic106—42no—
132C2×Dic106—42no—
133C2×Dic106—41no—
134M16×C53—22no—
135M16×C53—42no—
136M16×C53SD1614yescomplemented
137C2×C8×C53—22no—
138C2×C8×C53C2×D412yes—
139C2×C8×C53C2×D412yes—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53C2×D411yescharacteristic
146C4⋊C4×C53C2×D411yescharacteristic
147C42×C53—21no—
148C42×C53—21no—
149C42×C53—21no—
150C42×C53—21no—
151C42×C53C2×D411yescharacteristic
152C42×C53C2×D411yescharacteristic
153C42×C53#4111yescharacteristic
154C42×C53C2411yescharacteristic
155#11719924—42no—
156#11719924—42no—
157#11719921—41no—
158#11719921—41no—
159C4⋊Dic53—22no—
160C4⋊Dic53—21no—
161C4⋊Dic53—22no—
162C4⋊Dic53—21no—
163C4⋊Dic53—41no—
164C4⋊Dic53—41no—
165C2×#88322—21no—
166C2×#88322—21no—
167C2×#88322—22no—
168C2×#88322—42no—
169Dic212—44no—
170Dic212—44no—
171#11719913—44no—
172#11719913—44no—
173C23×C53#12111yescharacteristic
174C22×D53—41no—
175Q8×C53—21no—
176Q8×C53—21no—
177Q8×C53—21no—
178Q8×C53—42no—
179Q8×C53#13712yescomplemented
180C2×C4×C53—21no—
181C2×C4×C53—21no—
182C2×C4×C53—21no—
183C2×C4×C53—21no—
184C2×C4×C53—21no—
185C2×C4×C53—21no—
186C2×C4×C53—41no—
187C2×C4×C53—42no—
188C2×C4×C53—81no—
189C2×C4×C53#12111yescharacteristic
190C2×C4×C53#12111yescharacteristic
191C2×C4×C53#12111yescharacteristic
192C2×C4×C53C2×SD1612yes—
193C2×C4×C53AGL(1,ℤ/8)12yes—
194C2×C4×C53C22×D411yescharacteristic
195C2×C4×C53C22×D411yescharacteristic
196C2×C4×C53C2×#4111yescharacteristic
197C22⋊D53—82no—
198C2×Dic53—42no—
199C2×Dic53—41no—
200C2×Dic53—82no—
201D212—41no—
202D212—41no—
203D212—41no—
204D212—41no—
205D212—81no—
206C4×D53—82no—
207Dic106—42no—
208Dic106—41no—
209Dic106—42no—
210Dic106—41no—
211Dic106—82no—
212C8×C53—22no—
213C8×C53—42no—
214C8×C53C2×SD1614yes—
215#88322—41no—
216#88322—42no—
217#88322—41no—
218#19171—531nomaximal, Sylow
219C22×C53—21no—
220C22×C53—21no—
221C22×C53#46012yes—
222C22×C53C2×#12111yescharacteristic
223C2×D53—81no—
224C2×D53—81no—
225C2×D53—81no—
226C4×C53—21no—
227C4×C53—22no—
228C4×C53—21no—
229C4×C53—21no—
230C4×C53—41no—
231C4×C53—41no—
232C4×C53—41no—
233C4×C53#46012yes—
234C4×C53C22×SD1611yescharacteristic
235C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
236C4×C53C2×#13711yescharacteristic
237C4×C53#57511yescharacteristic
238Dic53—82no—
239Dic53—81no—
240#3152—531no—
241#3103—531no—
242C2×#475—531no—
243Q8×M16D5311yescharacteristic, complemented
244#1690—532no—
245#1686—532no—
246#1670—532no—
247#1666—532no—
248#1565—531no—
249#1547—531no—
250#1546—531no—
251C2×C53—22no—
252C2×C53C2×#46011yescharacteristic
253C2×C53#305611yescharacteristic
254C2×C53#306911yescharacteristic
255D53—161no—
256C2×#137—1061no—
257C2×#137—1061no—
258#532—531no—
259C2×C4⋊Q8—531no—
260C2×C4×Q8C2×D5311yescharacteristic
261#497—1062no—
262#494—532no—
263#492—532no—
264#481—1061no—
265#475—532no—
266#475—532no—
267#475—531no—
268#475—531no—
269#475—1061no—
270#475—1061no—
271#460—531no—
272#460—531no—
273#448—1061no—
274#446—22no—
275#446C2×D5312yes—
276C8×Q8C2×D5312yes—
277C4×SD16—1061no—
278C4×SD16—1061no—
279#431—531no—
280C2×#106—1061no—
281#423—21no—
282#423C2×D5311yescharacteristic
283C2×C4⋊C8—531no—
284C2×#103—1061no—
285C4×M16—21no—
286C4×M16C2×D5311yescharacteristic
287#335—532no—
288#334—532no—
289#332—532no—
290#332—532no—
291#332—532no—
292#331—532no—
293C53#1917111yescharacteristic, complemented, Sylow
294C2×#41—1061no—
295C22×Q8—1061no—
296C22×Q8D21211yescharacteristic
297#137—2124no—
298#137—2124no—
299C2×Q16—1062no—
300C2×Q16—1062no—
301C2×SD16—1061no—
302C2×SD16—1061no—
303C2×SD16—1062no—
304C2×SD16—1062no—
305C2×M16—21no—
306C2×M16C22⋊D5311yescharacteristic
307C2×M16C22⋊D5311yescharacteristic
308C22×C8—1061no—
309C4⋊Q8—532no—
310C4⋊Q8—531no—
311C4⋊Q8—1061no—
312C4⋊Q8—1061no—
313C4⋊Q8—1061no—
314#127—531no—
315#124—1061no—
316#121—1061no—
317#121—1061no—
318C4×Q8—21no—
319C4×Q8—21no—
320C4×Q8D21211yescharacteristic
321C4×Q8D21211yescharacteristic
322C4×Q8C22×D5311yescharacteristic
323C4×Q8C22×D5311yescharacteristic
324C4×D4—1061no—
325C4×D4—1061no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—1061no—
328C2×C4⋊C4—1061no—
329C2×C4⋊C4D21211yescharacteristic
330C2×C42—21no—
331C2×C42C22×D5311yescharacteristic
332#106—1062no—
333#106—1062no—
334#106—1061no—
335#106—1061no—
336C4⋊C8—22no—
337C4⋊C8—22no—
338C4⋊C8—21no—
339C4⋊C8—532no—
340C4⋊C8—531no—
341C4⋊C8—531no—
342C4⋊C8—1061no—
343C4⋊C8C22×D5312yes—
344#103—1062no—
345#103—1062no—
346#103—1061no—
347#103—1061no—
348#102—1061no—
349#102—1061no—
350C22⋊C8—1061no—
351#97—22no—
352#97C22⋊D5312yes—
353C4×C8—22no—
354C4×C8—1061no—
355C4×C8C22×D5312yes—
356#41—2122no—
357#41—2122no—
358C2×Q8—21no—
359C2×Q8—22no—
360C2×Q8—41no—
361C2×Q8—1061no—
362C2×Q8—1062no—
363C2×Q8—2122no—
364C2×Q8—2122no—
365C2×Q8—2121no—
366C2×Q8C2×D21211yescharacteristic
367C2×Q8C2×D21211yescharacteristic
368C2×D4—1061no—
369C2×D4—1061no—
370C2×D4—1061no—
371C2×D4—2121no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—2121no—
375C22×C4—2121no—
376C22×C4C2×D21211yescharacteristic
377C22×C4#1171994511yescharacteristic
378C22×C4C2×C22⋊D5311yescharacteristic
379Q16—2124no—
380Q16—2124no—
381SD16—2124no—
382SD16—2122no—
383SD16—2122no—
384SD16—2124no—
385M16—22no—
386M16—42no—
387M16#1171992414yescomplemented
388C2×C8—22no—
389C2×C8—1061no—
390C2×C8—1061no—
391C2×C8—1062no—
392C2×C8—2122no—
393C2×C8C2×C22⋊D5312yes—
394C2×C8C2×C22⋊D5312yes—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—1062no—
401C4⋊C4—1061no—
402C4⋊C4—1062no—
403C4⋊C4—1061no—
404C4⋊C4—2121no—
405C4⋊C4—2121no—
406C4⋊C4C2×D21211yescharacteristic
407C4⋊C4C2×D21211yescharacteristic
408C22⋊C4—2121no—
409C22⋊C4—2121no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42C2×D21211yescharacteristic
415C42#1171994511yescharacteristic
416C42C2×C22⋊D5311yescharacteristic
417C42C23×D5311yescharacteristic
418C23—2121no—
419C23#1313319111yescharacteristic
420Q8—21no—
421Q8—21no—
422Q8—21no—
423Q8—42no—
424Q8—4242no—
425Q8—2122no—
426Q8—2121no—
427Q8—2122no—
428Q8—2121no—
429Q8#1313325012yescomplemented
430D4—4241no—
431D4—4242no—
432D4—2121no—
433D4—2121no—
434D4—2121no—
435D4—2121no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—81no—
445C2×C4—4242no—
446C2×C4—4242no—
447C2×C4—2121no—
448C2×C4—2122no—
449C2×C4#1313319111yescharacteristic
450C2×C4#1313319111yescharacteristic
451C2×C4#1313319111yescharacteristic
452C2×C4C2×#1171992412yes—
453C2×C4#1313324212yes—
454C2×C4C22×C22⋊D5311yescharacteristic
455C2×C4C22×D21211yescharacteristic
456C2×C4C2×#1171994511yescharacteristic
457C8—22no—
458C8—42no—
459C8—2121no—
460C8—2122no—
461C8—2121no—
462C8C2×#1171992414yes—
463C22—21no—
464C22—21no—
465C22—4241no—
466C22—4241no—
467C22—4241no—
468C22#1681082312yes—
469C22C2×#1313319111yescharacteristic
470C4—21no—
471C4—21no—
472C4—21no—
473C4—22no—
474C4—41no—
475C4—41no—
476C4—41no—
477C4—4241no—
478C4—4242no—
479C4#1681082312yes—
480C4C22×#1171992411yescharacteristic
481C4C2×#1313324211yescharacteristic
482C4C2×#1313325011yescharacteristic
483C4#1681142311yescharacteristic
484C2—22no—
485C2—8481no—
486C2C2×#1681082311yescharacteristic
487C2#2028434511yescharacteristic
488C2#2028445211yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingQ8×M16×C53C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenterQ8×M16D53
hypocenterC53#19171
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
3. C2×#16810908C2semidirect
4. #20274209C2semidirect
5. #20274187C2semidirect
6. #20274085C2semidirect
7. #20274063C2semidirect
8. #20273467C2semidirect
10. #20273388C2semidirect
11. Q8×M16×C53C2semidirect
136. M16×C53SD16semidirect
179. Q8×C53#137semidirect
243. Q8×M16D53semidirect
293. C53#19171semidirect
387. M16#11719924semidirect
429. Q8#13133250semidirect

Maximal subgroups

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

subgroupindex
#202851742
#202847462
C2×#168109082
#202742092
#202741872
#202740852
#202740632
#202734672
#202733932
#202733882
Q8×M16×C532
#1917153