Finite Group Databasebeta

← #14484835

Subgroups — #14484835

Order 1920.

2224 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6821C211yesmaximal, characteristic, complemented
2C5×#5569C211yesmaximal, characteristic, complemented
3C3×#65910C211yesmaximal, characteristic, complemented
4#11755251C211yesmaximal, characteristic, complemented
5Dic3×#713C211yesmaximal, characteristic, complemented
6Q8×D5×Dic3C211yesmaximal, characteristic, complemented
7#11755128C211yesmaximal, characteristic, complemented
8#11753769C213yesmaximal, complemented
9#11753529C213yesmaximal, complemented
10#11753511C213yesmaximal, complemented
11#11753415C213yesmaximal, complemented
12#11752620C213yesmaximal, complemented
13#11752606C213yesmaximal, complemented
14#11752577C213yesmaximal, complemented
15C4×#6766C213yesmaximal, complemented
16#10618917—31nomaximal, Hall
17C2×#6821C2211yescharacteristic
18C2×C3×#713C2211yescharacteristic
19C2×C3×Q8×D5C2211yescharacteristic
20C2×#6766C2213yes—
21C3×C4×C5×Q8C2211yescharacteristic
22Q8×Dic15C2211yescharacteristic
23#91782C2213yes—
24#91781C2213yes—
25C4×D60C2213yes—
26C42×D15C2213yes—
27C5×Q8×Dic3C2211yescharacteristic
28C5×Q8×Dic3C2211yescharacteristic
29C5×#890C2213yes—
30C5×#888C2213yes—
31C5×#870C2213yes—
32C4×C5×Dic6C2213yes—
33C3×#4043C2213yes—
34C3×#4041C2213yes—
35C3×#4021C2213yes—
36C3×C4×D20C2213yes—
37#91465C2213yes—
38#91458C2213yes—
39#91455C2216yes—
40#91450C2213yes—
41#91450C2213yes—
42C4×#1202C2213yes—
43C4×#1202C2213yes—
44#91434C2213yes—
45#91434C2213yes—
46Dic3×D20C2213yes—
47#91413C2216yes—
48D5×C4⋊Dic3C2213yes—
49#91409C2216yes—
50#91396C2216yes—
51#91396C2213yes—
52#91396C2213yes—
53#91394C2211yescharacteristic
54#91394C2213yes—
55C4×D5×Dic3C2213yes—
56C4×D5×Dic3C2213yes—
57#91390C2213yes—
58#91374C2213yes—
59#91366C2216yes—
60#91358C2213yes—
61#91354C2216yes—
62#91347C2213yes—
63#91345C2216yes—
64#91344C2213yes—
65#91331C2213yes—
66Dic3×Dic10C2213yes—
67#86711—51nomaximal, Hall
68C5×#557—31no—
69#66066—33no—
70#66059—33no—
71#66029—33no—
72#66029—33no—
73#66025—33no—
74#66025—33no—
75#65919—33no—
76#65911—31no—
77#65910—31no—
78#65910D311yescharacteristic, complemented
79C4×#713—33no—
80C4×#713—31no—
81C4×#713—31no—
82C4×Q8×D5—31no—
83C2×C3×C5×Q8C2311yescharacteristic
84#6821—24no—
85C2×D60C2313yes—
86C2×C4×D15C2313yes—
87C2×C4×D15C2313yes—
88C2×C4×D15C2313yes—
89C2×C5×Dic6C2313yes—
90C3×#713—24no—
91C3×Q8×D5—24no—
92C2×C3×D20C2313yes—
93C2×C3×D20C2313yes—
94C2×C3×C4×D5C2313yes—
95C2×C3×C4×D5C2313yes—
96C2×C3×Dic10C2313yes—
97C2×#1202C2313yes—
98C2×#1202C2313yes—
99C2×#1202C2316yes—
100C2×D5×Dic3C2313yes—
101C2×D5×Dic3C2313yes—
102C2×D5×Dic3C2313yes—
103C2×D5×Dic3C2311yescharacteristic
104#6766—212no—
105C3×C5×C4⋊C4C2313yes—
106C3×C42×C5C2313yes—
107#6714C2316yes—
108C4⋊Dic15C2313yes—
109C4×Dic15C2311yescharacteristic
110C4×Dic15C2313yes—
111C4×Dic15C2313yes—
112C5×C4⋊Dic3C2313yes—
113C5×C4⋊Dic3C2313yes—
114C5×C4⋊Dic3C2313yes—
115C5×Dic3⋊C4C2311yescharacteristic
116C5×Dic3⋊C4C2313yes—
117C5×Dic3⋊C4C2316yes—
118C4×C5×Dic3C2313yes—
119C4×C5×Dic3C2313yes—
120C3×#685C2311yescharacteristic
121C3×#685C2313yes—
122C3×#685C2316yes—
123C3×C4⋊Dic5C2313yes—
124Dic3⋊Dic5C2311yescharacteristic
125Dic3⋊Dic5C2313yes—
126#6668C2316yes—
127#6667C2313yes—
128#6667C2313yes—
129#6667C2316yes—
130#6665C2311yescharacteristic
131#6665C2313yes—
132#6665C2316yes—
133Dic3×Dic5C2313yes—
134Dic3×Dic5C2311yescharacteristic
135Dic3×Dic5C2313yes—
136Dic3×Dic5C2313yes—
137C3×#548—51no—
138#5834—51no—
139Dic3×#41—51no—
140#5817—51no—
141C2×Q8×Dic3—51no—
142#5681—53no—
143#5680—53no—
144#5641—53no—
145#5624—53no—
146#5618—53no—
147#5596—53no—
148C4×#239—51no—
149#5569D511yescharacteristic, complemented
150#5565—53no—
151C4×#237—53no—
152C2×#713—31no—
153C2×#713—33no—
154C2×#713C2×D311yescharacteristic
155C2×Q8×D5C2×D311yescharacteristic
156C5×C4⋊Q8—33no—
157C5×#125—33no—
158C5×#125—33no—
159C4×C5×Q8—31no—
160C4×C5×Q8—31no—
161C4×C5×Q8—33no—
162C4×C5×Q8C2×D311yescharacteristic
163#4093—36no—
164Q8×Dic5—31no—
165Q8×Dic5—33no—
166#4044—36no—
167#4044—36no—
168#4043—33no—
169#4043C2×D313yes—
170#4041—36no—
171#4041—33no—
172#4041—33no—
173#4041—33no—
174#4041C2×D313yes—
175#4040—36no—
176#4040—36no—
177#4039—33no—
178#4039—33no—
179#4039—33no—
180#4039—36no—
181#4039—33no—
182#4039—33no—
183#4038—33no—
184#4038—33no—
185#4038—31no—
186#4038—33no—
187#4038—33no—
188D5×C4⋊C4—33no—
189#4033—33no—
190#4021—33no—
191#4021C2×D313yes—
192#4020—33no—
193C4×D20—33no—
194C4×D20—33no—
195C4×D20—33no—
196C4×D20—33no—
197C4×D20C2×D313yes—
198C42×D5—33no—
199C42×D5—33no—
200C42×D5—33no—
201C4×Dic10—33no—
202#3480—151noSylow
203C22×D15C2413yes—
204C22×C3×D5C2413yes—
205C22×C3×D5C2411yescharacteristic
206C3×C5×Q8#4114yescomplemented
207C2×C3×C4×C5C2413yes—
208C2×C3×C4×C5C2411yescharacteristic
209C2×C3×C4×C5C2413yes—
210C2×Dic15C2413yes—
211C2×Dic15C2411yescharacteristic
212C2×Dic15C2411yescharacteristic
213D60—26no—
214C4×D15—26no—
215C4×D15—26no—
216C4×D15—43no—
217C2×C5×Dic3C2413yes—
218C2×C5×Dic3C2411yescharacteristic
219C2×C5×Dic3C2413yes—
220C2×C5×Dic3C2411yescharacteristic
221C5×Dic6—26no—
222C2×C3×Dic5C2411yescharacteristic
223C2×C3×Dic5C2413yes—
224C3×D20—26no—
225C3×D20—26no—
226C3×C4×D5—26no—
227C3×C4×D5—26no—
228C3×Dic10—26no—
229#1202—26no—
230#1202—26no—
231#1202—46no—
232D5×Dic3—26no—
233D5×Dic3—43no—
234D5×Dic3—43no—
235D5×Dic3#4114yescomplemented
236C2×C3×#41—51no—
237C22×C3×Q8—51no—
238C2×#239—51no—
239C2×#237—53no—
240C3×#124—53no—
241C3×C22⋊Q8—53no—
242C3×#121—53no—
243C3×C4×Q8C2×D511yescharacteristic
244C3×C4×D4—53no—
245#947—53no—
246#946—53no—
247Q8×Dic3—51no—
248Q8×Dic3—53no—
249Q8×Dic3—53no—
250Q8×Dic3C2×D511yescharacteristic
251Q8×Dic3C2×D511yescharacteristic
252#940—53no—
253#939—56no—
254#938—53no—
255D4×Dic3—53no—
256D4×Dic3—53no—
257D4×Dic3—53no—
258#930—53no—
259#930—53no—
260C4×#67—53no—
261C4×#67—53no—
262#926—53no—
263C2×C4⋊Dic3—53no—
264C2×C4×Dic3—53no—
265C2×C4×Dic3—53no—
266#898—56no—
267#893—53no—
268#892—53no—
269#892—53no—
270#890C2×D513yes—
271#888C2×D513yes—
272#887—53no—
273#886—56no—
274#885—56no—
275#883—56no—
276#67⋊C4—56no—
277#67⋊C4—53no—
278#67⋊C4—53no—
279#879—56no—
280#877—53no—
281#877—51no—
282C4×D12—53no—
283C42×D3—53no—
284#870C2×D513yes—
285C4×Dic6C2×D513yes—
286C2×C5×Q8—33no—
287C2×C5×Q8C22×D311yescharacteristic
288#713—24no—
289#713—612no—
290#713—64no—
291Q8×D5—24no—
292C2×D20—33no—
293C2×D20—33no—
294C2×D20—33no—
295C2×D20—36no—
296C2×D20C22×D313yes—
297C2×D20C22×D313yes—
298C2×C4×D5—31no—
299C2×C4×D5—33no—
300C2×C4×D5—33no—
301C2×C4×D5—33no—
302C2×C4×D5—33no—
303C2×C4×D5—33no—
304C2×C4×D5—33no—
305C2×C4×D5C22×D313yes—
306C2×C4×D5C22×D313yes—
307C2×Dic10C22×D313yes—
308C5×C4⋊C4—31no—
309C5×C4⋊C4—33no—
310C5×C4⋊C4—33no—
311C5×C4⋊C4—33no—
312C5×C4⋊C4—33no—
313C5×C4⋊C4—36no—
314C5×C4⋊C4C22×D313yes—
315C42×C5—33no—
316C42×C5—33no—
317C42×C5C22×D313yes—
318#685—31no—
319#685—36no—
320#685—33no—
321#685—36no—
322#685—33no—
323#685—33no—
324#685—36no—
325#685C22×D311yescharacteristic
326#685C22×D313yes—
327#685C22×D316yes—
328C4⋊Dic5—33no—
329C4⋊Dic5—31no—
330C4⋊Dic5—33no—
331C4⋊Dic5C22×D313yes—
332#683—36no—
333C4×Dic5—33no—
334C4×Dic5—31no—
335C4×Dic5—33no—
336C4×Dic5—33no—
337C4×Dic5—31no—
338C4×Dic5—33no—
339C4×Dic5—33no—
340#559—153no—
341#557—31no—
342#556—153no—
343#553—153no—
344#553—153no—
345#550—151no—
346#548—51no—
347#548—151no—
348#532—153no—
349#529—153no—
350#529—153no—
351C4×#41—153no—
352C4×#41—151no—
353C4×#41—151no—
354C2×C4×Q8—151no—
355C22×C3×C5C2511yescharacteristic
356C2×D15—23no—
357C2×D15—23no—
358C2×D15—23no—
359C2×C3×D5—21no—
360C2×C3×D5—21no—
361C2×C3×D5—23no—
362C2×C3×D5—23no—
363C2×C3×D5—23no—
364C2×C3×D5C2×#4112yes—
365C3×C4×C5—23no—
366C3×C4×C5—21no—
367C3×C4×C5C2×#4116yes—
368Dic15—21no—
369Dic15—23no—
370Dic15C2×#4112yes—
371C3×Dic5—23no—
372C3×Dic5—21no—
373C5×Dic3—23no—
374C5×Dic3—21no—
375C5×Dic3—23no—
376C5×Dic3C2×#4112yes—
377C3×#41—104no—
378C2×C3×Q8—53no—
379C2×C3×Q8—104no—
380C2×C3×Q8C22×D511yescharacteristic
381C2×C3×D4—53no—
382C2×C3×D4—53no—
383C22×C3×C4—53no—
384C22×C3×C4—53no—
385C2×#67—53no—
386C2×#67—53no—
387C2×#67—56no—
388C22×Dic3—51no—
389C22×Dic3—53no—
390C22×Dic3—53no—
391C22×Dic3—53no—
392#239—104no—
393#237—1012no—
394C2×D12—53no—
395C2×C4×D3—53no—
396C2×C4×D3—53no—
397C2×C4×D3—53no—
398C2×Dic6C22×D513yes—
399C3×C4⋊C4—53no—
400C3×C4⋊C4C22×D513yes—
401C3×C22⋊C4—51no—
402C3×C22⋊C4—53no—
403C3×C22⋊C4—56no—
404C3×C42C22×D513yes—
405#217—53no—
406#217—51no—
407#217—56no—
408(C2×D3)⋊C4—56no—
409(C2×D3)⋊C4—56no—
410(C2×D3)⋊C4—53no—
411(C2×D3)⋊C4—53no—
412C4⋊Dic3—53no—
413C4⋊Dic3—56no—
414C4⋊Dic3C22×D513yes—
415C4⋊Dic3C22×D513yes—
416C4⋊Dic3C22×D513yes—
417Dic3⋊C4—53no—
418Dic3⋊C4—51no—
419Dic3⋊C4C22×D511yescharacteristic
420Dic3⋊C4C22×D513yes—
421Dic3⋊C4C22×D516yes—
422C4×Dic3—51no—
423C4×Dic3—53no—
424C4×Dic3—53no—
425C4×Dic3—53no—
426C4×Dic3—51no—
427C4×Dic3—53no—
428C4×Dic3—53no—
429C4×Dic3C22×D513yes—
430C4×Dic3C22×D513yes—
431C22×D5—33no—
432C22×D5C23×D313yes—
433C22×D5C23×D311yescharacteristic
434C5×Q8—66no—
435C5×Q8#23714yescomplemented
436C2×C4×C5—31no—
437C2×C4×C5—33no—
438C2×C4×C5—33no—
439C2×C4×C5—31no—
440C2×C4×C5C23×D313yes—
441C2×C4×C5C23×D311yescharacteristic
442C2×C4×C5C23×D313yes—
443C2×Dic5—31no—
444C2×Dic5—33no—
445C2×Dic5—31no—
446C2×Dic5C23×D311yescharacteristic
447C2×Dic5C23×D313yes—
448D20—26no—
449D20—26no—
450D20—66no—
451D20—66no—
452D20—66no—
453D20—126no—
454C4×D5—26no—
455C4×D5—26no—
456C4×D5—34no—
457C4×D5—66no—
458C4×D5—66no—
459C4×D5—66no—
460C4×D5—123no—
461C4×D5—123no—
462C4×D5—123no—
463Dic10—26no—
464C2×#41—51no—
465C2×#41—153no—
466C2×#41—151no—
467C22×Q8—51no—
468C4⋊Q8—33no—
469#127—153no—
470#126—156no—
471#126—156no—
472#125—33no—
473#125—33no—
474#124—53no—
475#124—153no—
476#124—156no—
477#123—156no—
478#123—156no—
479C22⋊Q8—53no—
480C22⋊Q8—153no—
481#121—53no—
482#121—153no—
483#121—153no—
484#121—153no—
485#121—156no—
486C4×Q8—31no—
487C4×Q8—31no—
488C4×Q8—33no—
489C4×Q8—153no—
490C4×Q8—153no—
491C4×Q8—153no—
492C4×Q8—151no—
493C4×Q8D3×D511yescharacteristic
494C4×D4—53no—
495C4×D4—153no—
496C4×D4—153no—
497C4×D4—153no—
498C4×D4—153no—
499C4×D4—153no—
500C4×D4—156no—
501C4×D4—153no—
502C4×D4—153no—
503C4×D4—153no—
504C4×D4—153no—
505#117—151no—
506#117—153no—
507#117—153no—
508#117—153no—
509#117—153no—
510C2×C4⋊C4—153no—
511C2×C42—153no—
512C2×C42—153no—
513C2×C42—153no—
514C2×C3×C5C22×#4111yescharacteristic
515C2×C3×C5C22×#4111yescharacteristic
516C2×C3×C5#58511yescharacteristic
517D15—43no—
518C3×D5—22no—
519C3×D5—43no—
520C23×C3—51no—
521C23×C3—53no—
522C22×D3—53no—
523C3×Q8—106no—
524C3×Q8#71314yescomplemented
525C3×D4—106no—
526C3×D4—106no—
527C2×C3×C4—51no—
528C2×C3×C4—53no—
529C2×C3×C4—106no—
530C2×C3×C4—106no—
531C2×C3×C4C23×D511yescharacteristic
532C2×C3×C4C23×D513yes—
533C2×C3×C4C23×D513yes—
534#67—106no—
535#67—106no—
536#67—206no—
537C2×Dic3—51no—
538C2×Dic3—54no—
539C2×Dic3—51no—
540C2×Dic3—53no—
541C2×Dic3—106no—
542C2×Dic3—203no—
543C2×Dic3—203no—
544C2×Dic3C23×D511yescharacteristic
545C2×Dic3C23×D513yes—
546C2×Dic3C23×D513yes—
547C2×Dic3C23×D511yescharacteristic
548D12—106no—
549C4×D3—106no—
550C4×D3—106no—
551C4×D3—203no—
552Dic6—26no—
553C22×C5C24×D311yescharacteristic
554C2×D5—21no—
555C2×D5—21no—
556C2×D5—23no—
557C2×D5—23no—
558C2×D5—23no—
559C2×D5—63no—
560C2×D5—63no—
561C2×D5—63no—
562C2×D5D3×#4112yes—
563C4×C5—23no—
564C4×C5—21no—
565C4×C5—32no—
566C4×C5—63no—
567C4×C5—61no—
568C4×C5—63no—
569C4×C5C2×#23716yes—
570Dic5—23no—
571Dic5—21no—
572Dic5—32no—
573Dic5—61no—
574Dic5—63no—
575#41—104no—
576#41—3012no—
577#41—304no—
578C2×Q8—33no—
579C2×Q8—53no—
580C2×Q8—104no—
581C2×Q8C2×D3×D511yescharacteristic
582C2×D4—53no—
583C2×D4—53no—
584C2×D4—153no—
585C2×D4—153no—
586C2×D4—156no—
587C2×D4—153no—
588C22×C4—53no—
589C22×C4—53no—
590C22×C4—151no—
591C22×C4—153no—
592C22×C4—153no—
593C22×C4—153no—
594C22×C4—153no—
595C22×C4—153no—
596C22×C4—153no—
597C4⋊C4—33no—
598C4⋊C4—31no—
599C4⋊C4—33no—
600C4⋊C4—33no—
601C4⋊C4—33no—
602C4⋊C4—36no—
603C4⋊C4—53no—
604C4⋊C4—156no—
605C4⋊C4—151no—
606C4⋊C4—153no—
607C4⋊C4—153no—
608C4⋊C4C2×D3×D513yes—
609C22⋊C4—51no—
610C22⋊C4—53no—
611C22⋊C4—56no—
612C22⋊C4—153no—
613C22⋊C4—153no—
614C22⋊C4—156no—
615C22⋊C4—151no—
616C22⋊C4—153no—
617C22⋊C4—156no—
618C22⋊C4—156no—
619C42—33no—
620C42—33no—
621C42—151no—
622C42—153no—
623C42—153no—
624C42—153no—
625C42—151no—
626C42—153no—
627C42—153no—
628C42C2×D3×D513yes—
629C3×C5#348011yescharacteristic, complemented, Hall
630C22×C3—52no—
631C22×C3—101no—
632C22×C3—101no—
633C22×C3—103no—
634C22×C3—103no—
635C22×C3—103no—
636C22×C3C24×D511yescharacteristic
637C2×D3—103no—
638C2×D3—103no—
639C2×D3—103no—
640C3×C4—23no—
641C3×C4—21no—
642C3×C4—101no—
643C3×C4—103no—
644C3×C4C2×#71316yes—
645Dic3—23no—
646Dic3—21no—
647Dic3—23no—
648Dic3—52no—
649Dic3—101no—
650Dic3—103no—
651Dic3D5×#4112yes—
652C2×C5C22×#23711yescharacteristic
653C2×C5C2×D3×#4111yescharacteristic
654C2×C5#596811yescharacteristic
655D5—22no—
656D5—43no—
657D5—123no—
658C23—51no—
659C23—53no—
660C23—153no—
661Q8—66no—
662Q8—106no—
663Q8#676614yescomplemented
664D4—106no—
665D4—106no—
666D4—306no—
667D4—306no—
668D4—306no—
669D4—606no—
670C2×C4—31no—
671C2×C4—31no—
672C2×C4—33no—
673C2×C4—33no—
674C2×C4—51no—
675C2×C4—53no—
676C2×C4—106no—
677C2×C4—106no—
678C2×C4—154no—
679C2×C4—151no—
680C2×C4—151no—
681C2×C4—153no—
682C2×C4—306no—
683C2×C4—306no—
684C2×C4—306no—
685C2×C4—603no—
686C2×C4—603no—
687C2×C4—603no—
688C2×C4C22×D3×D511yescharacteristic
689C2×C4C22×D3×D513yes—
690C2×C4C22×D3×D513yes—
691C2×C3—102no—
692C2×C3—203no—
693C2×C3C22×#71311yescharacteristic
694C2×C3C2×D5×#4111yescharacteristic
695C2×C3#6628711yescharacteristic
696D3—203no—
697C5#8671111yescharacteristic, complemented, Sylow
698C22—52no—
699C22—101no—
700C22—101no—
701C22—103no—
702C22—103no—
703C22—103no—
704C22—303no—
705C22—303no—
706C22—303no—
707C22C23×D3×D511yescharacteristic
708C4—21no—
709C4—23no—
710C4—32no—
711C4—61no—
712C4—63no—
713C4—63no—
714C4—101no—
715C4—103no—
716C4—152no—
717C4—301no—
718C4—303no—
719C4C2×#676616yes—
720C3#1061891711yescharacteristic, complemented, Sylow
721C2—102no—
722C2—203no—
723C2—603no—
724C2C22×#676611yescharacteristic
725C2#1175705011yescharacteristic
726C2D3×D5×#4111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D3×D5
commutator, socle, cosocleC22×C3×C5C25
FittingC3×C4×C5×Q8C22
hypercenterC4×Q8D3×D5
hypocenterC3×C5#3480
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. C4×#6821C2semidirect
2. C5×#5569C2semidirect
3. C3×#65910C2semidirect
4. #11755251C2semidirect
5. Dic3×#713C2semidirect
6. Q8×D5×Dic3C2semidirect
7. #11755128C2semidirect
8. #11753769C2semidirect
9. #11753529C2semidirect
10. #11753511C2semidirect
11. #11753415C2semidirect
12. #11752620C2semidirect
13. #11752606C2semidirect
14. #11752577C2semidirect
15. C4×#6766C2semidirect
78. #65910D3semidirect
149. #5569D5semidirect
206. C3×C5×Q8#41semidirect
235. D5×Dic3#41semidirect
435. C5×Q8#237semidirect
524. C3×Q8#713semidirect
629. C3×C5#3480semidirect
663. Q8#6766semidirect
697. C5#86711semidirect
720. C3#10618917semidirect

Maximal subgroups

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

subgroupindex
C4×#68212
C5×#55692
C3×#659102
#117552512
Dic3×#7132
Q8×D5×Dic32
#117551282
#117537692
#117535292
#117535112
#117534152
#117526202
#117526062
#117525772
C4×#67662
#106189173
#867115