Finite Group Databasebeta

← #14439128

Subgroups — #14439128

Order 1920.

1312 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
2#11756591C211yesmaximal, characteristic, complemented
3C2×#91838C211yesmaximal, characteristic, complemented
4C2×#91829C211yesmaximal, characteristic, complemented
5C2×#91821C211yesmaximal, characteristic, complemented
6C2×#91797C211yesmaximal, characteristic, complemented
7#11751277C212yesmaximal, complemented
8#11751276C212yesmaximal, complemented
9#11751255C212yesmaximal, complemented
10#11751252C212yesmaximal, complemented
11M16×Dic15C211yesmaximal, characteristic, complemented
12#10615668—31nomaximal, Hall
13C2×C3×C5×#41C2211yescharacteristic
14C22×C3×C5×D4C2211yescharacteristic
15C22×#1220—22no—
16C2×#6817C2211yescharacteristic
17C22×D60C2211yescharacteristic
18C3×C5×AGL(1,ℤ/8)—24no—
19C2×C3×C5×SD16C2212yescomplemented
20C2×C3×C5×D8C2212yescomplemented
21C2×C3×C5×M16C2211yescharacteristic
22#91838—24no—
23#91836C2211yescharacteristic, complemented
24#91833C2211yescharacteristic, complemented
25C2×#6717C2212yescomplemented
26#91829—22no—
27#91829C2212yes—
28#91829C2212yes—
29#91829C2211yescharacteristic, complemented
30#91828—22no—
31#91825C2211yescharacteristic, complemented
32C2×#6716C2212yescomplemented
33#91821—24no—
34C2×#6715C2212yescomplemented
35C2×#6715C2212yescomplemented
36C4×#1220—22no—
37C2×C4×Dic15C2211yescharacteristic
38C2×#6710C2211yescharacteristic
39#91797—24no—
40C2×D120C2212yescomplemented
41C2×#6706C2212yescomplemented
42#91099C2212yescomplemented
43C8×Dic15C2212yescomplemented
44#83937—51nomaximal, Hall
45C2×C5×AGL(1,ℤ/8)D311yescharacteristic, complemented
46#66166—31no—
47C2×#4095—31no—
48C2×#4086—31no—
49C2×#4078—31no—
50C2×#4054—31no—
51#65465—32no—
52#65464—32no—
53#65443—32no—
54#65440—32no—
55M16×Dic5—31no—
56C24×C3×C5—21no—
57C23×D15—21no—
58C3×C5×#41—22no—
59C3×C5×#41—22no—
60C2×C3×C5×Q8C2311yescharacteristic
61C2×C3×C5×D4—21no—
62C2×C3×C5×D4—22no—
63C2×C3×C5×D4—22no—
64C2×C3×C5×D4—21no—
65C2×C3×C5×D4C2311yescharacteristic
66C2×C3×C5×D4C2312yes—
67C22×C3×C4×C5—21no—
68C22×C3×C4×C5C2311yescharacteristic
69C2×#1220—21no—
70C2×#1220—22no—
71C2×#1220—24no—
72C2×#1220—24no—
73C2×#1220—48no—
74C22×Dic15D412yes—
75#6817—22no—
76#6817—22no—
77C2×D60—22no—
78C2×D60—22no—
79C2×D60—21no—
80C2×D60C2312yes—
81C2×D60C2311yescharacteristic
82C2×C4×D15—21no—
83C2×Dic30C2311yescharacteristic
84C3×C5×SD16—24no—
85C3×C5×D8—24no—
86C3×C5×M16D414yescomplemented
87C2×C3×C5×C8C2312yes—
88#6719—22no—
89#6717—24no—
90#6716—24no—
91#6715—24no—
92#6715—24no—
93#6714—22no—
94#6712—22no—
95C4×Dic15D412yes—
96C4×Dic15C2311yescharacteristic
97C4×Dic15C2311yescharacteristic
98#6710D414yescomplemented
99C2×C3⋊#168C2312yes—
100D120—24no—
101#6706—24no—
102C2×C3×AGL(1,ℤ/8)D511yescharacteristic, complemented
103#5832—51no—
104C2×#949—51no—
105C2×#940—51no—
106C2×#932—51no—
107C2×#908—51no—
108#5176—52no—
109#5175—52no—
110#5154—52no—
111#5151—52no—
112Dic3×M16—51no—
113C2×C5×#41C2×D311yescharacteristic
114C22×C5×D4C2×D311yescharacteristic
115C22×C22⋊D5—62no—
116C2×#709—31no—
117C22×D20—31no—
118C5×AGL(1,ℤ/8)—24no—
119C2×C5×SD16C2×D312yescomplemented
120C2×C5×D8C2×D312yescomplemented
121C2×C5×M16C2×D311yescharacteristic
122#4095—64no—
123#4093—31no—
124#4090—31no—
125C2×#688—32no—
126#4086—32no—
127#4086—32no—
128#4086—31no—
129#4086—62no—
130#4085—62no—
131#4082—31no—
132C2×#687—32no—
133#4078—64no—
134C2×#686—32no—
135C2×#686—32no—
136C4×C22⋊D5—62no—
137C2×C4×Dic5—31no—
138C2×#681—31no—
139#4054—64no—
140C2×D40—32no—
141C2×#677—32no—
142#3947—32no—
143C8×Dic5—32no—
144#3151—151noSylow
145C23×C3×C5—21no—
146C23×C3×C5—22no—
147C23×C3×C5—44no—
148C23×C3×C5—42no—
149C23×C3×C5C2×D411yescharacteristic
150C22×D15—21no—
151C22×D15—22no—
152C22×D15—44no—
153C22×D15—42no—
154C3×C5×Q8—21no—
155C3×C5×Q8—21no—
156C3×C5×D4—21no—
157C3×C5×D4—22no—
158C3×C5×D4—21no—
159C3×C5×D4—22no—
160C3×C5×D4—42no—
161C3×C5×D4—42no—
162C2×C3×C4×C5—21no—
163C2×C3×C4×C5—42no—
164C2×C3×C4×C5C2×D411yescharacteristic
165C2×C3×C4×C5C2×D412yes—
166C2×C3×C4×C5C2×D412yes—
167C2×C3×C4×C5C2411yescharacteristic
168#1220—48no—
169#1220—42no—
170#1220—44no—
171#1220—48no—
172C2×Dic15—24no—
173C2×Dic15—21no—
174C2×Dic15C2×D412yes—
175C2×Dic15C2×D412yes—
176D60—22no—
177D60—21no—
178D60—21no—
179D60—22no—
180D60—42no—
181C4×D15—42no—
182Dic30—21no—
183Dic30—21no—
184C3×C5×C8C2×D414yescomplemented
185C3⋊#168C2×D414yescomplemented
186C2×C3×#41C2×D511yescharacteristic
187C22×C3×D4C2×D511yescharacteristic
188C22×#67—102no—
189C2×#235—51no—
190C22×D12—51no—
191C3×AGL(1,ℤ/8)—24no—
192C2×C3×SD16C2×D512yescomplemented
193C2×C3×D8C2×D512yescomplemented
194C2×C3×M16C2×D511yescharacteristic
195#949—104no—
196#947—51no—
197#944—51no—
198C2×#215—52no—
199#940—51no—
200#940—52no—
201#940—52no—
202#940—102no—
203#939—102no—
204#936—51no—
205C2×#214—52no—
206#932—104no—
207C2×#213—52no—
208C2×#213—52no—
209C4×#67—102no—
210C2×C4×Dic3—51no—
211C2×#208—51no—
212#908—104no—
213C2×D24—52no—
214C2×#204—52no—
215#815—52no—
216C8×Dic3—52no—
217C24×C5—21no—
218C23×D5—61no—
219C5×#41—22no—
220C5×#41—22no—
221C2×C5×Q8C22×D311yescharacteristic
222C2×C5×D4—21no—
223C2×C5×D4—22no—
224C2×C5×D4—21no—
225C2×C5×D4—22no—
226C2×C5×D4C22×D311yescharacteristic
227C2×C5×D4C22×D312yes—
228C22×C4×C5—21no—
229C22×C4×C5C22×D311yescharacteristic
230C2×C22⋊D5—64no—
231C2×C22⋊D5—64no—
232C2×C22⋊D5—62no—
233C2×C22⋊D5—61no—
234C2×C22⋊D5—128no—
235C22×Dic5—32no—
236#709—62no—
237#709—62no—
238C2×D20—32no—
239C2×D20—31no—
240C2×D20—62no—
241C2×D20—62no—
242C2×D20—61no—
243C2×C4×D5—61no—
244C2×Dic10—31no—
245C5×SD16—24no—
246C5×D8—24no—
247C5×M16#6714yescomplemented
248C2×C5×C8C22×D312yes—
249C22⋊Dic5—62no—
250#688—64no—
251#687—64no—
252#686—64no—
253#686—64no—
254#685—62no—
255#683—62no—
256C4×Dic5—32no—
257C4×Dic5—31no—
258C4×Dic5—31no—
259#681—34no—
260C2×#168—32no—
261D40—64no—
262#677—64no—
263C2×AGL(1,ℤ/8)—151no—
264C2×AGL(1,ℤ/8)—151no—
265C2×AGL(1,ℤ/8)—151no—
266C2×AGL(1,ℤ/8)D1511yescharacteristic, complemented
267#532—151no—
268C2×#127—151no—
269#496—152no—
270#496—152no—
271#493—152no—
272#492—152no—
273C4×M16—151no—
274C22×C3×C5—22no—
275C22×C3×C5—41no—
276C22×C3×C5—42no—
277C22×C3×C5—44no—
278C22×C3×C5—42no—
279C22×C3×C5—84no—
280C22×C3×C5#12712yes—
281C22×C3×C5C22×D411yescharacteristic
282C2×D15—41no—
283C2×D15—42no—
284C2×D15—44no—
285C2×D15—42no—
286C2×D15—84no—
287C3×C4×C5—41no—
288C3×C4×C5#12712yes—
289C3×C4×C5C22×D411yescharacteristic
290C3×C4×C5C22×D411yescharacteristic
291Dic15—22no—
292Dic15—41no—
293Dic15AGL(1,ℤ/8)14yescomplemented
294C24×C3—21no—
295C23×D3—101no—
296C3×#41—22no—
297C3×#41—22no—
298C2×C3×Q8C22×D511yescharacteristic
299C2×C3×D4—21no—
300C2×C3×D4—22no—
301C2×C3×D4—21no—
302C2×C3×D4—22no—
303C2×C3×D4C22×D511yescharacteristic
304C2×C3×D4C22×D512yes—
305C22×C3×C4—21no—
306C22×C3×C4C22×D511yescharacteristic
307C2×#67—102no—
308C2×#67—104no—
309C2×#67—104no—
310C2×#67—101no—
311C2×#67—208no—
312C22×Dic3—52no—
313#235—102no—
314#235—102no—
315C2×D12—52no—
316C2×D12—51no—
317C2×D12—102no—
318C2×D12—102no—
319C2×D12—101no—
320C2×C4×D3—101no—
321C2×Dic6—51no—
322C3×SD16—24no—
323C3×D8—24no—
324C3×M16C22⋊D514yescomplemented
325C2×C3×C8C22×D512yes—
326#217—102no—
327#215—104no—
328#214—104no—
329#213—104no—
330#213—104no—
331(C2×D3)⋊C4—102no—
332Dic3⋊C4—102no—
333C4×Dic3—52no—
334C4×Dic3—51no—
335C4×Dic3—51no—
336#208—54no—
337C2×C3⋊C8—52no—
338D24—104no—
339#204—104no—
340C23×C5—21no—
341C23×C5—22no—
342C23×C5—44no—
343C23×C5—42no—
344C23×C5D3×D411yescharacteristic
345C22×D5—61no—
346C22×D5—62no—
347C22×D5—124no—
348C22×D5—122no—
349C5×Q8—21no—
350C5×Q8—21no—
351C5×D4—21no—
352C5×D4—22no—
353C5×D4—21no—
354C5×D4—22no—
355C5×D4—42no—
356C5×D4—42no—
357C2×C4×C5—21no—
358C2×C4×C5—42no—
359C2×C4×C5D3×D411yescharacteristic
360C2×C4×C5D3×D412yes—
361C2×C4×C5C2×#6712yes—
362C2×C4×C5C23×D311yescharacteristic
363C22⋊D5—124no—
364C22⋊D5—128no—
365C22⋊D5—128no—
366C22⋊D5—122no—
367C2×Dic5—32no—
368C2×Dic5—32no—
369C2×Dic5—64no—
370C2×Dic5—61no—
371D20—61no—
372D20—62no—
373D20—62no—
374D20—61no—
375D20—122no—
376C4×D5—122no—
377Dic10—61no—
378Dic10—61no—
379C5×C8C2×#6714yescomplemented
380#168—34no—
381C2×#41—151no—
382C2×#41C2×D1511yescharacteristic
383C22×D4—151no—
384C22×D4—302no—
385C22×D4C2×D1511yescharacteristic
386AGL(1,ℤ/8)—24no—
387AGL(1,ℤ/8)—304no—
388AGL(1,ℤ/8)—304no—
389AGL(1,ℤ/8)—304no—
390C2×SD16—152no—
391C2×SD16—152no—
392C2×SD16—152no—
393C2×SD16C2×D1512yescomplemented
394C2×D8—152no—
395C2×D8—152no—
396C2×D8—152no—
397C2×D8C2×D1512yescomplemented
398C2×M16—151no—
399C2×M16C2×D1511yescharacteristic
400C4⋊Q8—151no—
401#127—152no—
402#127—152no—
403#127—151no—
404#127—302no—
405#124—151no—
406#124—151no—
407#121—302no—
408C4×D4—302no—
409C2×C42—151no—
410#97—152no—
411C4×C8—152no—
412C2×C3×C5—22no—
413C2×C3×C5—81no—
414C2×C3×C5—82no—
415C2×C3×C5C2×#12711yescharacteristic
416C2×C3×C5C2×AGL(1,ℤ/8)12yes—
417D15—81no—
418D15—82no—
419C23×C3—21no—
420C23×C3—22no—
421C23×C3—44no—
422C23×C3—42no—
423C23×C3D4×D511yescharacteristic
424C22×D3—101no—
425C22×D3—102no—
426C22×D3—204no—
427C22×D3—202no—
428C3×Q8—21no—
429C3×Q8—21no—
430C3×D4—21no—
431C3×D4—22no—
432C3×D4—21no—
433C3×D4—22no—
434C3×D4—42no—
435C3×D4—42no—
436C2×C3×C4—21no—
437C2×C3×C4—42no—
438C2×C3×C4D4×D512yes—
439C2×C3×C4D4×D511yescharacteristic
440C2×C3×C4C2×C22⋊D512yes—
441C2×C3×C4C23×D511yescharacteristic
442#67—204no—
443#67—208no—
444#67—208no—
445#67—202no—
446C2×Dic3—52no—
447C2×Dic3—52no—
448C2×Dic3—104no—
449C2×Dic3—101no—
450D12—101no—
451D12—102no—
452D12—102no—
453D12—101no—
454D12—202no—
455C4×D3—202no—
456Dic6—101no—
457Dic6—101no—
458C3×C8C2×C22⋊D514yescomplemented
459C3⋊C8—54no—
460C22×C5—22no—
461C22×C5—44no—
462C22×C5—42no—
463C22×C5—41no—
464C22×C5—42no—
465C22×C5—84no—
466C22×C5#94012yes—
467C22×C5C2×D3×D411yescharacteristic
468C2×D5—124no—
469C2×D5—122no—
470C2×D5—121no—
471C2×D5—122no—
472C2×D5—244no—
473C4×C5—41no—
474C4×C5#94012yes—
475C4×C5C2×D3×D411yescharacteristic
476C4×C5C22×#6711yescharacteristic
477Dic5—34no—
478Dic5—62no—
479Dic5—121no—
480C24—21no—
481C24—301no—
482#41—22no—
483#41—22no—
484#41—302no—
485#41—302no—
486C2×Q8—151no—
487C2×Q8C22×D1511yescharacteristic
488C2×D4—21no—
489C2×D4—22no—
490C2×D4—21no—
491C2×D4—22no—
492C2×D4—151no—
493C2×D4—152no—
494C2×D4—302no—
495C2×D4—304no—
496C2×D4—304no—
497C2×D4—301no—
498C2×D4—302no—
499C2×D4—301no—
500C2×D4—302no—
501C2×D4—608no—
502C2×D4C22×D1511yescharacteristic
503C2×D4C22×D1512yes—
504C22×C4—21no—
505C22×C4—152no—
506C22×C4—301no—
507C22×C4C22×D1511yescharacteristic
508SD16—24no—
509SD16—304no—
510SD16—304no—
511SD16—304no—
512D8—24no—
513D8—304no—
514D8—304no—
515D8—304no—
516M16—154no—
517M16#122014yescomplemented
518C2×C8—152no—
519C2×C8C22×D1512yes—
520C4⋊C4—302no—
521C22⋊C4—302no—
522C22⋊C4—302no—
523C42—152no—
524C42—151no—
525C42—151no—
526C3×C5#315111yescharacteristic, complemented, Hall
527C22×C3—22no—
528C22×C3—44no—
529C22×C3—42no—
530C22×C3—41no—
531C22×C3—42no—
532C22×C3—84no—
533C22×C3#408612yes—
534C22×C3C2×D4×D511yescharacteristic
535C2×D3—204no—
536C2×D3—202no—
537C2×D3—201no—
538C2×D3—202no—
539C2×D3—404no—
540C3×C4—41no—
541C3×C4#408612yes—
542C3×C4C2×D4×D511yescharacteristic
543C3×C4C22×C22⋊D511yescharacteristic
544Dic3—54no—
545Dic3—102no—
546Dic3—201no—
547C2×C5—22no—
548C2×C5—82no—
549C2×C5—81no—
550C2×C5D3×AGL(1,ℤ/8)12yes—
551C2×C5C2×#94011yescharacteristic
552D5—242no—
553D5—241no—
554C23—21no—
555C23—22no—
556C23—44no—
557C23—42no—
558C23—301no—
559C23—302no—
560C23—604no—
561C23—602no—
562C23D4×D1511yescharacteristic
563Q8—21no—
564Q8—21no—
565Q8—301no—
566Q8—301no—
567D4—21no—
568D4—22no—
569D4—21no—
570D4—22no—
571D4—42no—
572D4—42no—
573D4—301no—
574D4—302no—
575D4—301no—
576D4—302no—
577D4—608no—
578D4—604no—
579D4—608no—
580D4—602no—
581D4—602no—
582C2×C4—21no—
583C2×C4—42no—
584C2×C4—152no—
585C2×C4—152no—
586C2×C4—304no—
587C2×C4—301no—
588C2×C4—602no—
589C2×C4D4×D1511yescharacteristic
590C2×C4D4×D1512yes—
591C2×C4C2×#122012yes—
592C2×C4C23×D1511yescharacteristic
593C8—154no—
594C8C2×#122014yescomplemented
595C2×C3—22no—
596C2×C3—82no—
597C2×C3—81no—
598C2×C3D5×AGL(1,ℤ/8)12yes—
599C2×C3C2×#408611yescharacteristic
600D3—402no—
601D3—401no—
602C5#8393711yescharacteristic, complemented, Sylow
603C22—22no—
604C22—41no—
605C22—42no—
606C22—44no—
607C22—42no—
608C22—84no—
609C22—601no—
610C22—602no—
611C22—604no—
612C22—602no—
613C22—1204no—
614C22#9182912yes—
615C22C2×D4×D1511yescharacteristic
616C4—41no—
617C4—154no—
618C4—302no—
619C4—601no—
620C4#9182912yes—
621C4C2×D4×D1511yescharacteristic
622C4C22×#122011yescharacteristic
623C3#1061566811yescharacteristic, complemented, Sylow
624C2—22no—
625C2—82no—
626C2—81no—
627C2—1202no—
628C2—1201no—
629C2D15×AGL(1,ℤ/8)12yes—
630C2C2×#9182911yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC2×C3×C5×AGL(1,ℤ/8)C2
socleC22×C3×C5C22×D4
hypercenterC2×AGL(1,ℤ/8)D15
hypocenterC3×C5#3151
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×AGL(1,ℤ/8)C2semidirect
2. #11756591C2semidirect
3. C2×#91838C2semidirect
4. C2×#91829C2semidirect
5. C2×#91821C2semidirect
6. C2×#91797C2semidirect
7. #11751277C2semidirect
8. #11751276C2semidirect
9. #11751255C2semidirect
10. #11751252C2semidirect
11. M16×Dic15C2semidirect
19. C2×C3×C5×SD16C22semidirect
20. C2×C3×C5×D8C22semidirect
23. #91836C22semidirect
24. #91833C22semidirect
25. C2×#6717C22semidirect
29. #91829C22semidirect
31. #91825C22semidirect
32. C2×#6716C22semidirect
34. C2×#6715C22semidirect
35. C2×#6715C22semidirect
40. C2×D120C22semidirect
41. C2×#6706C22semidirect
42. #91099C22semidirect
43. C8×Dic15C22semidirect
45. C2×C5×AGL(1,ℤ/8)D3semidirect
86. C3×C5×M16D4semidirect
98. #6710D4semidirect
102. C2×C3×AGL(1,ℤ/8)D5semidirect
119. C2×C5×SD16C2×D3semidirect
120. C2×C5×D8C2×D3semidirect
184. C3×C5×C8C2×D4semidirect
185. C3⋊#168C2×D4semidirect
192. C2×C3×SD16C2×D5semidirect
193. C2×C3×D8C2×D5semidirect
247. C5×M16#67semidirect
266. C2×AGL(1,ℤ/8)D15semidirect
293. Dic15AGL(1,ℤ/8)semidirect
324. C3×M16C22⋊D5semidirect
379. C5×C8C2×#67semidirect
393. C2×SD16C2×D15semidirect
397. C2×D8C2×D15semidirect
458. C3×C8C2×C22⋊D5semidirect
517. M16#1220semidirect
526. C3×C5#3151semidirect
594. C8C2×#1220semidirect
602. C5#83937semidirect
623. C3#10615668semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C5×AGL(1,ℤ/8)2
#117565912
C2×#918382
C2×#918292
C2×#918212
C2×#917972
#117512772
#117512762
#117512552
#117512522
M16×Dic152
#106156683
#839375