Finite Group Databasebeta

← #14416758

Subgroups — #14416758

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#11754414C211yesmaximal, characteristic, complemented
3C2×#91418C211yesmaximal, characteristic, complemented
4C5×#5121C211yesmaximal, characteristic, complemented
5C3×M16×Dic5C211yesmaximal, characteristic, complemented
6#11748396C211yesmaximal, characteristic, complemented
7#11748394C211yesmaximal, characteristic, complemented
8#11747778C212yesmaximal, complemented
9#11747772C212yesmaximal, complemented
10#11747766C212yesmaximal, complemented
11#11747760C212yesmaximal, complemented
12#10610930—31nomaximal, Hall
13C2×#6817C2211yescharacteristic
14C22×D60C2211yescharacteristic
15C22×#1200—21no—
16C2×C3×C5×M16C2211yescharacteristic
17#91797C418yescomplemented
18C2×D120C2212yes—
19C2×#6706C2212yes—
20C5×#926C2211yescharacteristic
21C2×C5×C4⋊Dic3C2211yescharacteristic
22C2×C3×C4×Dic5C2211yescharacteristic
23C2×C3×#681C2211yescharacteristic
24#91552—21no—
25#91552—21no—
26C2×#6667—21no—
27#91418—21no—
28#91418—21no—
29#91418C2212yes—
30#91418C2212yes—
31#91418C2211yescharacteristic, complemented
32#91418C2211yescharacteristic, complemented
33C4×#1200—21no—
34#91386—21no—
35#91333C2211yescharacteristic, complemented
36#91333C2211yescharacteristic, complemented
37C5×#818C2212yescomplemented
38C5×#817C2212yescomplemented
39C3×#3947C2212yescomplemented
40C3×C8×Dic5C2212yescomplemented
41#90975C2212yescomplemented
42#90969C2212yes—
43#90969C2212yes—
44#90969C2212yescomplemented
45C4×#908—51nomaximal, Hall
46C2×#4054—31no—
47#65856—31no—
48C2×#4039—31no—
49C5×#428—31no—
50M16×Dic5D311yescharacteristic, complemented
51#65290—31no—
52#65255—31no—
53#65178—32no—
54#65161—32no—
55#65158—32no—
56#65141—32no—
57C23×D15—21no—
58C22×C3×C4×C5C2311yescharacteristic
59C2×#1220—21no—
60#6817—22no—
61#6817C2×C414yescomplemented
62C2×D60—21no—
63C2×D60—22no—
64C2×D60C2×C414yescomplemented
65C2×D60C2312yes—
66C2×D60C2311yescharacteristic
67C2×C4×D15—21no—
68C2×Dic30C2311yescharacteristic
69C22×C5×Dic3—21no—
70C22×C3×Dic5D411yescharacteristic
71C22×C3×Dic5D411yescharacteristic
72C2×#1200—21no—
73C2×#1200—21no—
74C2×#1200—22no—
75C2×#1200—22no—
76C2×#1200—44no—
77C3×C5×M16C2×C414yescomplemented
78C2×C3×C5×C8C2312yes—
79D120C2×C418yescomplemented
80#6706C2×C418yescomplemented
81C5×#217—21no—
82C5×C4⋊Dic3—21no—
83C5×C4⋊Dic3C2312yes—
84C5×C4⋊Dic3C2311yescharacteristic
85C5×C4⋊Dic3C2311yescharacteristic
86C4×C5×Dic3—21no—
87C3×C4×Dic5D411yescharacteristic
88C3×C4×Dic5D411yescharacteristic
89C3×C4×Dic5C2311yescharacteristic
90C3×C4×Dic5C2311yescharacteristic
91C3×#681—22no—
92C2×C3×#168C2312yes—
93#6669—21no—
94#6669—21no—
95#6667—22no—
96#6667—22no—
97#6667—21no—
98#6667—21no—
99Dic3×Dic5—21no—
100Dic3×Dic5—21no—
101C2×#908—51no—
102C4×#235—51no—
103C2×C4×D12—51no—
104C3×C4×M16—51no—
105#5133—51no—
106#5131—51no—
107#5121D511yescharacteristic, complemented
108#4711—52no—
109#4710—52no—
110C4×D24—52no—
111C4×#204—52no—
112C2×#709—31no—
113C22×D20—31no—
114C22×C4×D5—61no—
115C2×C5×M16C2×D311yescharacteristic
116C5×#117—31no—
117C2×C5×C4⋊C4—31no—
118C2×#685—61no—
119C2×C4×Dic5C2×D311yescharacteristic
120C2×#681C2×D311yescharacteristic
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×#107—32no—
137C5×#106—32no—
138#3947C2×D312yescomplemented
139C8×Dic5C2×D312yescomplemented
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×C5C22×C412yes—
152C2×C3×C4×C5C2×D411yescharacteristic
153C2×C3×C4×C5#4112yes—
154C2×C3×C4×C5C2411yescharacteristic
155#1220—24no—
156C2×Dic15—21no—
157D60—24no—
158D60—22no—
159D60—21no—
160D60C22×C414yes—
161D60C22×C412yescomplemented
162C4×D15—24no—
163Dic30—21no—
164Dic30C22×C412yescomplemented
165C2×C5×Dic3—21no—
166C2×C5×Dic3—21no—
167C2×C5×Dic3—22no—
168C2×C5×Dic3—42no—
169C2×C3×Dic5—22no—
170C2×C3×Dic5—41no—
171C2×C3×Dic5C2×D411yescharacteristic
172C2×C3×Dic5C2×D411yescharacteristic
173C2×C3×Dic5C2×D411yescharacteristic
174C2×C3×Dic5C2×D411yescharacteristic
175#1200—42no—
176#1200—44no—
177#1200—42no—
178#1200—44no—
179C3×C5×C8C22×C414yescomplemented
180C3×#168—22no—
181C2×#235—51no—
182C22×D12—51no—
183C22×C4×D3—101no—
184C2×C3×M16—51no—
185C2×C3×M16C2×D511yescharacteristic
186C2×C3×C42—51no—
187C4×#67—101no—
188C4×#67—101no—
189C2×(C2×D3)⋊C4—101no—
190#926C2×D511yescharacteristic
191C2×C4⋊Dic3C2×D511yescharacteristic
192#908—58no—
193C2×D24—52no—
194C2×#204—52no—
195C4×D12—52no—
196C4×D12—52no—
197C4×D12—51no—
198C4×D12—51no—
199C4×D12—101no—
200C4×D12—101no—
201#872—101no—
202C42×D3—101no—
203C4×Dic6—51no—
204C4×Dic6—51no—
205C3×#97—52no—
206C3×C4×C8—52no—
207#821—52no—
208#821—52no—
209#821—52no—
210#818C2×D512yescomplemented
211#817C2×D512yescomplemented
212#816—52no—
213C23×D5—61no—
214C22×C4×C5—61no—
215C22×C4×C5C22×D311yescharacteristic
216C2×C22⋊D5—61no—
217C22×Dic5D1211yescharacteristic
218C22×Dic5D1211yescharacteristic
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—32no—
236C5×C4⋊C4—31no—
237C5×C4⋊C4—31no—
238C5×C4⋊C4—61no—
239C5×C22⋊C4—61no—
240C42×C5—61no—
241#685—61no—
242#685—61no—
243#685—62no—
244#685—62no—
245#683—61no—
246#683—61no—
247C4×Dic5—61no—
248C4×Dic5—61no—
249C4×Dic5D1211yescharacteristic
250C4×Dic5D1211yescharacteristic
251C4×Dic5C22×D311yescharacteristic
252C4×Dic5C22×D311yescharacteristic
253#681—22no—
254C2×#168C22×D312yes—
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#428—31no—
265#418—151no—
266#417—151no—
267C4×M16—51no—
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×C5C4×D412yes—
280C3×C4×C5C23×C411yescharacteristic
281C3×C4×C5C2×#4111yescharacteristic
282Dic15—22no—
283C3×Dic5—21no—
284C3×Dic5—21no—
285C3×Dic5—21no—
286C3×Dic5AGL(1,ℤ/8)12yescomplemented
287C5×Dic3—42no—
288C5×Dic3—41no—
289C5×Dic3—41no—
290C23×D3—101no—
291C22×C3×C4—51no—
292C22×C3×C4—51no—
293C22×C3×C4C22×D511yescharacteristic
294C2×#67—101no—
295C22×Dic3—21no—
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×M16—102no—
311C3×M16C4×D514yescomplemented
312C2×C3×C8—52no—
313C2×C3×C8C22×D512yes—
314C3×C42—51no—
315C3×C42—51no—
316C3×C42—51no—
317C3×C42—51no—
318#217—21no—
319(C2×D3)⋊C4—101no—
320(C2×D3)⋊C4—101no—
321(C2×D3)⋊C4—102no—
322(C2×D3)⋊C4—102no—
323C4⋊Dic3—21no—
324C4⋊Dic3C22×D512yes—
325C4⋊Dic3C22×D511yescharacteristic
326C4⋊Dic3C22×D511yescharacteristic
327Dic3⋊C4—101no—
328Dic3⋊C4—101no—
329C4×Dic3—21no—
330C4×Dic3—101no—
331C4×Dic3—101no—
332D24—58no—
333#204—58no—
334C23×C5C2×D1211yescharacteristic
335C22×D5—64no—
336C22×D5—61no—
337C22×D5—62no—
338C22×D5—122no—
339C22×D5—122no—
340C2×C4×C5—61no—
341C2×C4×C5—61no—
342C2×C4×C5—62no—
343C2×C4×C5—122no—
344C2×C4×C5C2×C4×D312yes—
345C2×C4×C5C2×D1211yescharacteristic
346C2×C4×C5#23512yes—
347C2×C4×C5C23×D311yescharacteristic
348C22⋊D5—64no—
349C2×Dic5—22no—
350C2×Dic5—41no—
351C2×Dic5—61no—
352C2×Dic5C2×D1211yescharacteristic
353C2×Dic5C2×D1211yescharacteristic
354C2×Dic5C2×D1211yescharacteristic
355C2×Dic5C2×D1211yescharacteristic
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—22no—
370C2×#41—151no—
371C22×D4—151no—
372C23×C4—301no—
373AGL(1,ℤ/8)—158no—
374C2×SD16—152no—
375C2×D8—152no—
376C2×M16—51no—
377C2×M16D3×D511yescharacteristic
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—31no—
389#117—301no—
390C2×C4⋊C4—31no—
391C2×C22⋊C4—301no—
392C2×C42—51no—
393C2×C42—301no—
394#107—32no—
395#106—32no—
396#103—152no—
397#102—152no—
398#102—152no—
399#102—152no—
400#97—52no—
401C4×C8—52no—
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—51no—
415C2×C3×C4—51no—
416C2×C3×C4—51no—
417C2×C3×C4—51no—
418C2×C3×C4—102no—
419C2×C3×C4—201no—
420C2×C3×C4C2×C4×D512yes—
421C2×C3×C4D4×D511yescharacteristic
422C2×C3×C4#71312yes—
423C2×C3×C4C23×D511yescharacteristic
424#67—104no—
425C2×Dic3—21no—
426C2×Dic3—21no—
427C2×Dic3—22no—
428C2×Dic3—42no—
429C2×Dic3—101no—
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×C8—102no—
443C3×C8C2×C4×D514yescomplemented
444C22×C5—21no—
445C22×C5—21no—
446C22×C5C4×D1212yes—
447C22×C5C22×D1211yescharacteristic
448C2×D5—62no—
449C2×D5—64no—
450C2×D5—128no—
451C2×D5—122no—
452C2×D5—121no—
453C2×D5—121no—
454C2×D5—122no—
455C4×C5—122no—
456C4×C5—121no—
457C4×C5—121no—
458C4×C5C4×D1212yes—
459C4×C5C22×C4×D311yescharacteristic
460C4×C5C2×#23511yescharacteristic
461Dic5—21no—
462Dic5—21no—
463Dic5—21no—
464Dic5—62no—
465Dic5#90812yescomplemented
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—302no—
475C2×D4—301no—
476C22×C4—51no—
477C22×C4—51no—
478C22×C4—61no—
479C22×C4—301no—
480C22×C4—301no—
481C22×C4—302no—
482C22×C4—302no—
483C22×C4—301no—
484C22×C4—604no—
485C22×C4C2×D3×D511yescharacteristic
486SD16—158no—
487D8—158no—
488M16—102no—
489M16#120014yescomplemented
490C2×C8—52no—
491C2×C8C2×D3×D512yes—
492C4⋊C4—31no—
493C4⋊C4—31no—
494C4⋊C4—32no—
495C4⋊C4—61no—
496C4⋊C4—301no—
497C4⋊C4—301no—
498C22⋊C4—61no—
499C22⋊C4—302no—
500C22⋊C4—302no—
501C22⋊C4—301no—
502C22⋊C4—301no—
503C42—51no—
504C42—51no—
505C42—51no—
506C42—51no—
507C42—61no—
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—52no—
523C3×C4—101no—
524C3×C4—101no—
525C3×C4—101no—
526C3×C4#403912yes—
527C3×C4C22×C4×D511yescharacteristic
528C3×C4C2×#71311yescharacteristic
529Dic3—42no—
530Dic3—41no—
531Dic3—41no—
532Dic3—102no—
533C2×C5—22no—
534C2×C5C2×C4×D1211yescharacteristic
535C2×C5C2×#90811yescharacteristic
536C2×C5#574811yescharacteristic
537D5—124no—
538D5—122no—
539C23—304no—
540C23—301no—
541C23—302no—
542C23—602no—
543C23—602no—
544C23D5×D1211yescharacteristic
545Q8—152no—
546Q8—301no—
547D4—152no—
548D4—154no—
549D4—304no—
550D4—304no—
551D4—302no—
552D4—301no—
553C2×C4—51no—
554C2×C4—51no—
555C2×C4—51no—
556C2×C4—51no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—62no—
560C2×C4—102no—
561C2×C4—122no—
562C2×C4—201no—
563C2×C4—301no—
564C2×C4—304no—
565C2×C4—604no—
566C2×C4—602no—
567C2×C4—604no—
568C2×C4—602no—
569C2×C4#677312yes—
570C2×C4D5×D1211yescharacteristic
571C2×C4C2×#120012yes—
572C2×C4C22×D3×D511yescharacteristic
573C8—102no—
574C8C2×#120014yescomplemented
575C2×C3—22no—
576C2×C3C2×#403911yescharacteristic
577C2×C3D5×AGL(1,ℤ/8)11yescharacteristic
578C2×C3#6611111yescharacteristic
579D3—204no—
580D3—202no—
581C5C4×#90811yescharacteristic, complemented, Sylow
582C22—21no—
583C22—21no—
584C22—302no—
585C22—304no—
586C22—608no—
587C22—602no—
588C22—601no—
589C22—601no—
590C22—602no—
591C22#9141812yes—
592C22C2×D5×D1211yescharacteristic
593C4—52no—
594C4—101no—
595C4—101no—
596C4—101no—
597C4—122no—
598C4—121no—
599C4—121no—
600C4—302no—
601C4#9141812yes—
602C4C2×#677311yescharacteristic
603C4C22×#120011yescharacteristic
604C3#1061093011yescharacteristic, complemented, Sylow
605C2—22no—
606C2—604no—
607C2—602no—
608C2D5×#90811yescharacteristic
609C2#1175402111yescharacteristic
610C2C2×#9141811yescharacteristic
subgroupG/H
centerC22C2×D5×D12
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×M16C22
socleC22×C3×C5C22×D4
prime radicalC2×#91797C2
cosocleC2×C3×C4×C5C24
hypercenterC2×M16D3×D5
hypocenterC3×C5C4×AGL(1,ℤ/8)
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. #11754414C2semidirect
3. C2×#91418C2semidirect
4. C5×#5121C2semidirect
5. C3×M16×Dic5C2semidirect
6. #11748396C2semidirect
7. #11748394C2semidirect
8. #11747778C2semidirect
9. #11747772C2semidirect
10. #11747766C2semidirect
11. #11747760C2semidirect
17. #91797C4semidirect
31. #91418C22semidirect
32. #91418C22semidirect
35. #91333C22semidirect
36. #91333C22semidirect
37. C5×#818C22semidirect
38. C5×#817C22semidirect
39. C3×#3947C22semidirect
40. C3×C8×Dic5C22semidirect
41. #90975C22semidirect
44. #90969C22semidirect
50. M16×Dic5D3semidirect
61. #6817C2×C4semidirect
64. C2×D60C2×C4semidirect
77. C3×C5×M16C2×C4semidirect
79. D120C2×C4semidirect
80. #6706C2×C4semidirect
107. #5121D5semidirect
138. #3947C2×D3semidirect
139. C8×Dic5C2×D3semidirect
161. D60C22×C4semidirect
164. Dic30C22×C4semidirect
179. C3×C5×C8C22×C4semidirect
210. #818C2×D5semidirect
211. #817C2×D5semidirect
233. C5×M16C4×D3semidirect
286. C3×Dic5AGL(1,ℤ/8)semidirect
311. C3×M16C4×D5semidirect
368. C5×C8C2×C4×D3semidirect
443. C3×C8C2×C4×D5semidirect
465. Dic5#908semidirect
489. M16#1200semidirect
510. C3×C5C4×AGL(1,ℤ/8)semidirect
574. C8C2×#1200semidirect
581. C5C4×#908semidirect
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
#117544142
C2×#914182
C5×#51212
C3×M16×Dic52
#117483962
#117483942
#117477782
#117477722
#117477662
#117477602
#106109303
C4×#9085