Finite Group Databasebeta

← #14416654

Subgroups — #14416654

Order 1920.

1096 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91797C211yesmaximal, characteristic
2#11756281C211yesmaximal, characteristic, complemented
3C2×#91782C211yesmaximal, characteristic, complemented
4C3×C5×#428C211yesmaximal, characteristic, complemented
5M16×Dic15C211yesmaximal, characteristic, complemented
6#11751102C211yesmaximal, characteristic, complemented
7#11751067C211yesmaximal, characteristic, complemented
8#11750990C212yesmaximal, complemented
9#11750973C212yesmaximal, complemented
10#11750970C212yesmaximal, complemented
11#11750953C212yesmaximal, complemented
12#10610930—31nomaximal, Hall
13C2×#6817C2211yescharacteristic
14C22×D60C2211yescharacteristic
15C22×C4×D15—21no—
16C2×C3×C5×M16C2211yescharacteristic
17C3×C5×#117C2211yescharacteristic
18C2×C3×C5×C4⋊C4C2211yescharacteristic
19C2×#6714—21no—
20C2×C4×Dic15C2211yescharacteristic
21C2×#6710C2211yescharacteristic
22#91797C418yescomplemented
23C2×D120C2212yes—
24C2×#6706C2212yes—
25#91782—21no—
26#91782—21no—
27#91782C2212yes—
28#91782C2212yes—
29#91782C2211yescharacteristic, complemented
30#91782C2211yescharacteristic, complemented
31#91776C2211yescharacteristic, complemented
32#91776C2211yescharacteristic, complemented
33#91770—21no—
34#91770—21no—
35#91761—21no—
36C42×D15—21no—
37C3×C5×#107C2212yescomplemented
38C3×C5×#106C2212yescomplemented
39#91099C2212yescomplemented
40C8×Dic15C2212yescomplemented
41#91094C2212yescomplemented
42#91093C2212yes—
43#91093C2212yes—
44#91093C2212yescomplemented
45#79199—51nomaximal, Hall
46C2×#4054—31no—
47#65856—31no—
48C2×#4039—31no—
49C5×#428D311yescharacteristic, complemented
50M16×Dic5—31no—
51#65290—31no—
52#65255—31no—
53#65178—32no—
54#65161—32no—
55#65158—32no—
56#65141—32no—
57C23×D15—21no—
58C22×C3×C4×C5—21no—
59C22×C3×C4×C5C2311yescharacteristic
60C2×#1220—21no—
61C22×Dic15D411yescharacteristic
62C22×Dic15D411yescharacteristic
63#6817—22no—
64#6817C2×C414yescomplemented
65C2×D60—21no—
66C2×D60—22no—
67C2×D60C2×C414yescomplemented
68C2×D60C2312yes—
69C2×D60C2311yescharacteristic
70C2×C4×D15—21no—
71C2×C4×D15—21no—
72C2×C4×D15—21no—
73C2×C4×D15—22no—
74C2×C4×D15—22no—
75C2×C4×D15—44no—
76C2×Dic30C2311yescharacteristic
77C3×C5×M16C2×C414yescomplemented
78C2×C3×C5×C8C2312yes—
79C3×C5×C4⋊C4—21no—
80C3×C5×C4⋊C4C2311yescharacteristic
81C3×C5×C4⋊C4C2311yescharacteristic
82C3×C5×C4⋊C4C2312yes—
83C3×C5×C22⋊C4—21no—
84C3×C42×C5—21no—
85#6714—21no—
86#6714—21no—
87#6714—22no—
88#6714—22no—
89#6712—21no—
90#6712—21no—
91C4×Dic15—21no—
92C4×Dic15—21no—
93C4×Dic15D411yescharacteristic
94C4×Dic15D411yescharacteristic
95C4×Dic15C2311yescharacteristic
96C4×Dic15C2311yescharacteristic
97#6710—22no—
98C2×C3⋊#168C2312yes—
99D120C2×C418yescomplemented
100#6706C2×C418yescomplemented
101C2×#908—51no—
102#5522—51no—
103C2×#893—51no—
104C3×#428D511yescharacteristic, complemented
105Dic3×M16—51no—
106#5001—51no—
107#4966—51no—
108#4889—52no—
109#4872—52no—
110#4869—52no—
111#4852—52no—
112C2×#709—31no—
113C22×D20—31no—
114C22×C4×D5—61no—
115C2×C5×M16C2×D311yescharacteristic
116C5×#117C2×D311yescharacteristic
117C2×C5×C4⋊C4C2×D311yescharacteristic
118C2×#685—61no—
119C2×C4×Dic5—31no—
120C2×#681—31no—
121#4054—38no—
122C2×D40—32no—
123C2×#677—32no—
124#4039—32no—
125#4039—32no—
126#4039—31no—
127#4039—31no—
128#4039—61no—
129#4039—61no—
130#4033—31no—
131#4033—31no—
132#4027—61no—
133#4027—61no—
134#4018—61no—
135C42×D5—61no—
136C5×#107C2×D312yescomplemented
137C5×#106C2×D312yescomplemented
138#3947—32no—
139C8×Dic5—32no—
140#3942—32no—
141#3941—32no—
142#3941—32no—
143#3941—32no—
144C4×AGL(1,ℤ/8)—151noSylow
145C23×C3×C5C2×D411yescharacteristic
146C22×D15—24no—
147C22×D15—21no—
148C22×D15—22no—
149C22×D15—42no—
150C22×D15—42no—
151C2×C3×C4×C5—21no—
152C2×C3×C4×C5—21no—
153C2×C3×C4×C5—22no—
154C2×C3×C4×C5—42no—
155C2×C3×C4×C5C22×C412yes—
156C2×C3×C4×C5C2×D411yescharacteristic
157C2×C3×C4×C5#4112yes—
158C2×C3×C4×C5C2411yescharacteristic
159#1220—24no—
160C2×Dic15—21no—
161C2×Dic15—22no—
162C2×Dic15—41no—
163C2×Dic15C2×D411yescharacteristic
164C2×Dic15C2×D411yescharacteristic
165C2×Dic15C2×D411yescharacteristic
166C2×Dic15C2×D411yescharacteristic
167D60—24no—
168D60—22no—
169D60—21no—
170D60C22×C414yes—
171D60C22×C412yescomplemented
172C4×D15—24no—
173C4×D15—42no—
174C4×D15—44no—
175C4×D15—42no—
176C4×D15—44no—
177Dic30—21no—
178Dic30C22×C412yescomplemented
179C3×C5×C8C22×C414yescomplemented
180C3⋊#168—22no—
181C2×#235—51no—
182C22×D12—51no—
183C22×C4×D3—101no—
184C2×C3×M16C2×D511yescharacteristic
185C3×#117C2×D511yescharacteristic
186C2×C3×C4⋊C4C2×D511yescharacteristic
187C2×(C2×D3)⋊C4—101no—
188C2×C4×Dic3—51no—
189C2×#208—51no—
190#908—58no—
191C2×D24—52no—
192C2×#204—52no—
193#893—51no—
194#893—52no—
195#893—51no—
196#893—52no—
197#893—101no—
198#893—101no—
199#887—51no—
200#887—51no—
201#67⋊C4—101no—
202#67⋊C4—101no—
203#872—101no—
204C42×D3—101no—
205C3×#107C2×D512yescomplemented
206C3×#106C2×D512yescomplemented
207#815—52no—
208C8×Dic3—52no—
209#810—52no—
210#809—52no—
211#809—52no—
212#809—52no—
213C23×D5—61no—
214C22×C4×C5—21no—
215C22×C4×C5C22×D311yescharacteristic
216C2×C22⋊D5—61no—
217C22×Dic5—31no—
218C22×Dic5—31no—
219#709—34no—
220#709—62no—
221C2×D20—34no—
222C2×D20—32no—
223C2×D20—31no—
224C2×D20—61no—
225C2×D20—62no—
226C2×C4×D5—61no—
227C2×C4×D5—62no—
228C2×C4×D5—62no—
229C2×C4×D5—61no—
230C2×C4×D5—61no—
231C2×C4×D5—124no—
232C2×Dic10—31no—
233C5×M16C4×D314yescomplemented
234C2×C5×C8C22×D312yes—
235C5×C4⋊C4—21no—
236C5×C4⋊C4C22×D312yes—
237C5×C4⋊C4C22×D311yescharacteristic
238C5×C4⋊C4C22×D311yescharacteristic
239C5×C22⋊C4—21no—
240C42×C5—21no—
241#685—61no—
242#685—61no—
243#685—62no—
244#685—62no—
245#683—61no—
246#683—61no—
247C4×Dic5—31no—
248C4×Dic5—31no—
249C4×Dic5—31no—
250C4×Dic5—31no—
251C4×Dic5—61no—
252C4×Dic5—61no—
253#681—62no—
254C2×#168—32no—
255D40—38no—
256#677—38no—
257C2×AGL(1,ℤ/8)—151no—
258C4×#41—151no—
259C2×C4×D4—151no—
260#442—152no—
261#440—152no—
262C4×SD16—152no—
263C4×D8—152no—
264#428D1511yescharacteristic, complemented
265#418—151no—
266#417—151no—
267C4×M16—151no—
268C22×C3×C5—21no—
269C22×C3×C5—21no—
270C22×C3×C5C4×D412yes—
271C22×C3×C5C22×D411yescharacteristic
272C2×D15—22no—
273C2×D15—24no—
274C2×D15—48no—
275C2×D15—42no—
276C2×D15—41no—
277C2×D15—41no—
278C2×D15—42no—
279C3×C4×C5—42no—
280C3×C4×C5—41no—
281C3×C4×C5—41no—
282C3×C4×C5C4×D412yes—
283C3×C4×C5C23×C411yescharacteristic
284C3×C4×C5C2×#4111yescharacteristic
285Dic15—22no—
286Dic15—21no—
287Dic15—21no—
288Dic15—21no—
289Dic15AGL(1,ℤ/8)12yescomplemented
290C23×D3—101no—
291C22×C3×C4—21no—
292C22×C3×C4C22×D511yescharacteristic
293C2×#67—101no—
294C22×Dic3—51no—
295C22×Dic3—51no—
296#235—54no—
297#235—102no—
298C2×D12—54no—
299C2×D12—52no—
300C2×D12—51no—
301C2×D12—101no—
302C2×D12—102no—
303C2×C4×D3—101no—
304C2×C4×D3—101no—
305C2×C4×D3—101no—
306C2×C4×D3—102no—
307C2×C4×D3—102no—
308C2×C4×D3—204no—
309C2×Dic6—51no—
310C3×M16C4×D514yescomplemented
311C2×C3×C8C22×D512yes—
312C3×C4⋊C4—21no—
313C3×C4⋊C4C22×D511yescharacteristic
314C3×C4⋊C4C22×D511yescharacteristic
315C3×C4⋊C4C22×D512yes—
316C3×C22⋊C4—21no—
317C3×C42—21no—
318(C2×D3)⋊C4—101no—
319(C2×D3)⋊C4—101no—
320(C2×D3)⋊C4—102no—
321(C2×D3)⋊C4—102no—
322Dic3⋊C4—101no—
323Dic3⋊C4—101no—
324C4×Dic3—51no—
325C4×Dic3—51no—
326C4×Dic3—51no—
327C4×Dic3—51no—
328C4×Dic3—101no—
329C4×Dic3—101no—
330#208—102no—
331C2×C3⋊C8—52no—
332D24—58no—
333#204—58no—
334C23×C5D3×D411yescharacteristic
335C22×D5—64no—
336C22×D5—61no—
337C22×D5—62no—
338C22×D5—122no—
339C22×D5—122no—
340C2×C4×C5—21no—
341C2×C4×C5—21no—
342C2×C4×C5—22no—
343C2×C4×C5—42no—
344C2×C4×C5C2×C4×D312yes—
345C2×C4×C5D3×D411yescharacteristic
346C2×C4×C5#23912yes—
347C2×C4×C5C23×D311yescharacteristic
348C22⋊D5—64no—
349C2×Dic5—31no—
350C2×Dic5—31no—
351C2×Dic5—31no—
352C2×Dic5—31no—
353C2×Dic5—62no—
354C2×Dic5—61no—
355C2×Dic5—121no—
356D20—34no—
357D20—32no—
358D20—64no—
359D20—62no—
360D20—61no—
361C4×D5—64no—
362C4×D5—122no—
363C4×D5—124no—
364C4×D5—122no—
365C4×D5—124no—
366Dic10—32no—
367Dic10—61no—
368C5×C8C2×C4×D314yescomplemented
369#168—62no—
370C2×#41—151no—
371C22×D4—151no—
372C23×C4—301no—
373AGL(1,ℤ/8)—158no—
374C2×SD16—152no—
375C2×D8—152no—
376C2×M16—151no—
377C2×M16C2×D1511yescharacteristic
378C4×Q8—151no—
379C4×Q8—151no—
380C4×D4—152no—
381C4×D4—152no—
382C4×D4—151no—
383C4×D4—151no—
384C4×D4—301no—
385C4×D4—301no—
386C4×D4—301no—
387C4×D4—301no—
388#117—301no—
389#117C2×D1511yescharacteristic
390C2×C4⋊C4C2×D1511yescharacteristic
391C2×C22⋊C4—301no—
392C2×C42—151no—
393C2×C42—301no—
394#107C2×D1512yescomplemented
395#106C2×D1512yescomplemented
396#103—152no—
397#102—152no—
398#102—152no—
399#102—152no—
400#97—152no—
401C4×C8—152no—
402C2×C3×C5—22no—
403C2×C3×C5C2×C4×D411yescharacteristic
404C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
405C2×C3×C5#57511yescharacteristic
406D15—42no—
407D15—44no—
408C23×C3D4×D511yescharacteristic
409C22×D3—101no—
410C22×D3—102no—
411C22×D3—104no—
412C22×D3—202no—
413C22×D3—202no—
414C2×C3×C4—21no—
415C2×C3×C4—21no—
416C2×C3×C4—22no—
417C2×C3×C4—42no—
418C2×C3×C4C2×C4×D512yes—
419C2×C3×C4D4×D511yescharacteristic
420C2×C3×C4#71312yes—
421C2×C3×C4C23×D511yescharacteristic
422#67—104no—
423C2×Dic3—51no—
424C2×Dic3—51no—
425C2×Dic3—51no—
426C2×Dic3—51no—
427C2×Dic3—102no—
428C2×Dic3—101no—
429C2×Dic3—201no—
430D12—54no—
431D12—52no—
432D12—102no—
433D12—101no—
434D12—104no—
435C4×D3—104no—
436C4×D3—202no—
437C4×D3—204no—
438C4×D3—202no—
439C4×D3—204no—
440Dic6—52no—
441Dic6—101no—
442C3×C8C2×C4×D514yescomplemented
443C3⋊C8—102no—
444C22×C5—21no—
445C22×C5—21no—
446C22×C5#89312yes—
447C22×C5C2×D3×D411yescharacteristic
448C2×D5—62no—
449C2×D5—64no—
450C2×D5—128no—
451C2×D5—122no—
452C2×D5—121no—
453C2×D5—121no—
454C2×D5—122no—
455C4×C5—42no—
456C4×C5—41no—
457C4×C5—41no—
458C4×C5#89312yes—
459C4×C5C22×C4×D311yescharacteristic
460C4×C5C2×#23911yescharacteristic
461Dic5—32no—
462Dic5—62no—
463Dic5—61no—
464Dic5—61no—
465Dic5—61no—
466C24—301no—
467#41—154no—
468#41—302no—
469C2×Q8—151no—
470C2×D4—154no—
471C2×D4—151no—
472C2×D4—152no—
473C2×D4—301no—
474C2×D4—301no—
475C2×D4—302no—
476C22×C4—21no—
477C22×C4—151no—
478C22×C4—151no—
479C22×C4—301no—
480C22×C4—301no—
481C22×C4—302no—
482C22×C4—302no—
483C22×C4—301no—
484C22×C4—604no—
485C22×C4C22×D1511yescharacteristic
486SD16—158no—
487D8—158no—
488M16—302no—
489M16C4×D1514yescomplemented
490C2×C8—152no—
491C2×C8C22×D1512yes—
492C4⋊C4—21no—
493C4⋊C4—301no—
494C4⋊C4—301no—
495C4⋊C4C22×D1511yescharacteristic
496C4⋊C4C22×D1511yescharacteristic
497C4⋊C4C22×D1512yes—
498C22⋊C4—21no—
499C22⋊C4—302no—
500C22⋊C4—302no—
501C22⋊C4—301no—
502C22⋊C4—301no—
503C42—21no—
504C42—151no—
505C42—151no—
506C42—151no—
507C42—151no—
508C42—301no—
509C42—301no—
510C3×C5C4×AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
511C22×C3—21no—
512C22×C3—21no—
513C22×C3#403912yes—
514C22×C3C2×D4×D511yescharacteristic
515C2×D3—102no—
516C2×D3—104no—
517C2×D3—208no—
518C2×D3—202no—
519C2×D3—201no—
520C2×D3—201no—
521C2×D3—202no—
522C3×C4—42no—
523C3×C4—41no—
524C3×C4—41no—
525C3×C4#403912yes—
526C3×C4C22×C4×D511yescharacteristic
527C3×C4C2×#71311yescharacteristic
528Dic3—52no—
529Dic3—102no—
530Dic3—101no—
531Dic3—101no—
532Dic3—101no—
533C2×C5—22no—
534C2×C5C2×#89311yescharacteristic
535C2×C5D3×AGL(1,ℤ/8)11yescharacteristic
536C2×C5#577711yescharacteristic
537D5—124no—
538D5—122no—
539C23—304no—
540C23—301no—
541C23—302no—
542C23—602no—
543C23—602no—
544C23D4×D1511yescharacteristic
545Q8—152no—
546Q8—301no—
547D4—152no—
548D4—154no—
549D4—304no—
550D4—304no—
551D4—302no—
552D4—301no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—42no—
557C2×C4—151no—
558C2×C4—151no—
559C2×C4—151no—
560C2×C4—151no—
561C2×C4—302no—
562C2×C4—301no—
563C2×C4—304no—
564C2×C4—604no—
565C2×C4—602no—
566C2×C4—604no—
567C2×C4—602no—
568C2×C4—601no—
569C2×C4C2×C4×D1512yes—
570C2×C4D4×D1511yescharacteristic
571C2×C4#682112yes—
572C2×C4C23×D1511yescharacteristic
573C8—302no—
574C8C2×C4×D1514yescomplemented
575C2×C3—22no—
576C2×C3C2×#403911yescharacteristic
577C2×C3D5×AGL(1,ℤ/8)11yescharacteristic
578C2×C3#6611111yescharacteristic
579D3—204no—
580D3—202no—
581C5#7919911yescharacteristic, complemented, Sylow
582C22—21no—
583C22—21no—
584C22—302no—
585C22—304no—
586C22—608no—
587C22—602no—
588C22—601no—
589C22—601no—
590C22—602no—
591C22#9178212yes—
592C22C2×D4×D1511yescharacteristic
593C4—42no—
594C4—41no—
595C4—41no—
596C4—152no—
597C4—302no—
598C4—301no—
599C4—301no—
600C4—301no—
601C4#9178212yes—
602C4C22×C4×D1511yescharacteristic
603C4C2×#682111yescharacteristic
604C3#1061093011yescharacteristic, complemented, Sylow
605C2—22no—
606C2—604no—
607C2—602no—
608C2C2×#9178211yescharacteristic
609C2D15×AGL(1,ℤ/8)11yescharacteristic
610C2#1175653611yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C23×D15
FittingC3×C5×#428C2
socleC22×C3×C5C22×D4
prime radicalC2×#91797C2
cosocleC2×C3×C4×C5C24
hypercenter#428D15
hypocenterC3×C5C4×AGL(1,ℤ/8)
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. #11756281C2semidirect
3. C2×#91782C2semidirect
4. C3×C5×#428C2semidirect
5. M16×Dic15C2semidirect
6. #11751102C2semidirect
7. #11751067C2semidirect
8. #11750990C2semidirect
9. #11750973C2semidirect
10. #11750970C2semidirect
11. #11750953C2semidirect
22. #91797C4semidirect
29. #91782C22semidirect
30. #91782C22semidirect
31. #91776C22semidirect
32. #91776C22semidirect
37. C3×C5×#107C22semidirect
38. C3×C5×#106C22semidirect
39. #91099C22semidirect
40. C8×Dic15C22semidirect
41. #91094C22semidirect
44. #91093C22semidirect
49. C5×#428D3semidirect
64. #6817C2×C4semidirect
67. C2×D60C2×C4semidirect
77. C3×C5×M16C2×C4semidirect
99. D120C2×C4semidirect
100. #6706C2×C4semidirect
104. C3×#428D5semidirect
136. C5×#107C2×D3semidirect
137. C5×#106C2×D3semidirect
171. D60C22×C4semidirect
178. Dic30C22×C4semidirect
179. C3×C5×C8C22×C4semidirect
205. C3×#107C2×D5semidirect
206. C3×#106C2×D5semidirect
233. C5×M16C4×D3semidirect
264. #428D15semidirect
289. Dic15AGL(1,ℤ/8)semidirect
310. C3×M16C4×D5semidirect
368. C5×C8C2×C4×D3semidirect
394. #107C2×D15semidirect
395. #106C2×D15semidirect
442. C3×C8C2×C4×D5semidirect
489. M16C4×D15semidirect
510. C3×C5C4×AGL(1,ℤ/8)semidirect
574. C8C2×C4×D15semidirect
581. C5#79199semidirect
604. C3#10610930semidirect

Maximal subgroups

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

subgroupindex
C2×#917972
#117562812
C2×#917822
C3×C5×#4282
M16×Dic152
#117511022
#117510672
#117509902
#117509732
#117509702
#117509532
#106109303
#791995