Finite Group Databasebeta

← #14416756

Subgroups — #14416756

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#11754436C211yesmaximal, characteristic, complemented
3C2×#91428C211yesmaximal, characteristic, complemented
4C5×Dic3×M16C211yesmaximal, characteristic, complemented
5C3×#65410C211yesmaximal, characteristic, complemented
6#11748413C211yesmaximal, characteristic, complemented
7#11748392C211yesmaximal, characteristic, complemented
8#11747800C212yesmaximal, complemented
9#11747796C212yesmaximal, complemented
10#11747793C212yesmaximal, complemented
11#11747789C212yesmaximal, complemented
12C4×#4054—31nomaximal, Hall
13C2×#6817C2211yescharacteristic
14C22×D60C2211yescharacteristic
15C22×#1200—21no—
16C2×C3×C5×M16C2211yescharacteristic
17#91797C418yescomplemented
18C2×D120C2212yes—
19C2×#6706C2212yes—
20C2×C4×C5×Dic3C2211yescharacteristic
21C2×C5×#208C2211yescharacteristic
22C3×#4072C2211yescharacteristic
23C2×C3×C4⋊Dic5C2211yescharacteristic
24#91556—21no—
25#91556—21no—
26C2×#6667—21no—
27#91428—21no—
28#91428—21no—
29#91428C2212yes—
30#91428C2212yes—
31#91428C2211yescharacteristic, complemented
32#91428C2211yescharacteristic, complemented
33C4×#1200—21no—
34#91386—21no—
35#91340C2211yescharacteristic, complemented
36#91340C2211yescharacteristic, complemented
37C5×#815C2212yescomplemented
38C5×C8×Dic3C2212yescomplemented
39C3×#3950C2212yescomplemented
40C3×#3949C2212yescomplemented
41#90974C2212yescomplemented
42#90968C2212yes—
43#90968C2212yes—
44#90968C2212yescomplemented
45#79199—51nomaximal, Hall
46C2×#4054—31no—
47C4×#709—31no—
48C2×C4×D20—31no—
49C4×C5×M16—31no—
50#65422—31no—
51#65420—31no—
52#65410D311yescharacteristic, complemented
53#65000—32no—
54#64999—32no—
55C4×D40—32no—
56C4×#677—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×Dic3D411yescharacteristic
70C22×C5×Dic3D411yescharacteristic
71C22×C3×Dic5—21no—
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
81C4×C5×Dic3D411yescharacteristic
82C4×C5×Dic3D411yescharacteristic
83C4×C5×Dic3C2311yescharacteristic
84C4×C5×Dic3C2311yescharacteristic
85C5×#208—22no—
86C2×C5×C3⋊C8C2312yes—
87C3×C22⋊Dic5—21no—
88C3×C4⋊Dic5—21no—
89C3×C4⋊Dic5C2312yes—
90C3×C4⋊Dic5C2311yescharacteristic
91C3×C4⋊Dic5C2311yescharacteristic
92C3×C4×Dic5—21no—
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—
102#5522—51no—
103C2×#893—51no—
104C3×#428—51no—
105Dic3×M16D511yescharacteristic, complemented
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×M16—31no—
116C2×C5×M16C2×D311yescharacteristic
117C2×C42×C5—31no—
118C4×C22⋊D5—61no—
119C4×C22⋊D5—61no—
120C2×#685—61no—
121#4072C2×D311yescharacteristic
122C2×C4⋊Dic5C2×D311yescharacteristic
123#4054—38no—
124C2×D40—32no—
125C2×#677—32no—
126C4×D20—32no—
127C4×D20—32no—
128C4×D20—31no—
129C4×D20—31no—
130C4×D20—61no—
131C4×D20—61no—
132#4018—61no—
133C42×D5—61no—
134C4×Dic10—31no—
135C4×Dic10—31no—
136C5×#97—32no—
137C4×C5×C8—32no—
138#3953—32no—
139#3953—32no—
140#3953—32no—
141#3950C2×D312yescomplemented
142#3949C2×D312yescomplemented
143#3948—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—22no—
166C2×C5×Dic3—41no—
167C2×C5×Dic3C2×D411yescharacteristic
168C2×C5×Dic3C2×D411yescharacteristic
169C2×C5×Dic3C2×D411yescharacteristic
170C2×C5×Dic3C2×D411yescharacteristic
171C2×C3×Dic5—21no—
172C2×C3×Dic5—21no—
173C2×C3×Dic5—22no—
174C2×C3×Dic5—42no—
175#1200—42no—
176#1200—44no—
177#1200—42no—
178#1200—44no—
179C3×C5×C8C22×C414yescomplemented
180C5×C3⋊C8—22no—
181C2×#235—51no—
182C22×D12—51no—
183C22×C4×D3—101no—
184C2×C3×M16C2×D511yescharacteristic
185C3×#117—51no—
186C2×C3×C4⋊C4—51no—
187C2×(C2×D3)⋊C4—101no—
188C2×C4×Dic3C2×D511yescharacteristic
189C2×#208C2×D511yescharacteristic
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×#107—52no—
206C3×#106—52no—
207#815C2×D512yescomplemented
208C8×Dic3C2×D512yescomplemented
209#810—52no—
210#809—52no—
211#809—52no—
212#809—52no—
213C23×D5—61no—
214C22×C4×C5—31no—
215C22×C4×C5—31no—
216C22×C4×C5C22×D311yescharacteristic
217C2×C22⋊D5—61no—
218C22×Dic5—21no—
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×M16—62no—
234C5×M16C4×D314yescomplemented
235C2×C5×C8—32no—
236C2×C5×C8C22×D312yes—
237C42×C5—31no—
238C42×C5—31no—
239C42×C5—31no—
240C42×C5—31no—
241C22⋊Dic5—21no—
242#685—61no—
243#685—61no—
244#685—62no—
245#685—62no—
246C4⋊Dic5—21no—
247C4⋊Dic5C22×D312yes—
248C4⋊Dic5C22×D311yescharacteristic
249C4⋊Dic5C22×D311yescharacteristic
250#683—61no—
251#683—61no—
252C4×Dic5—21no—
253C4×Dic5—61no—
254C4×Dic5—61no—
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—51no—
265#418—151no—
266#417—151no—
267C4×M16—31no—
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—42no—
284C3×Dic5—41no—
285C3×Dic5—41no—
286C5×Dic3—21no—
287C5×Dic3—21no—
288C5×Dic3—21no—
289C5×Dic3AGL(1,ℤ/8)12yescomplemented
290C23×D3—101no—
291C22×C3×C4—101no—
292C22×C3×C4C22×D511yescharacteristic
293C2×#67—101no—
294C22×Dic3D2011yescharacteristic
295C22×Dic3D2011yescharacteristic
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—51no—
313C3×C4⋊C4—51no—
314C3×C4⋊C4—52no—
315C3×C4⋊C4—101no—
316C3×C22⋊C4—101no—
317C3×C42—101no—
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—101no—
325C4×Dic3—101no—
326C4×Dic3D2011yescharacteristic
327C4×Dic3D2011yescharacteristic
328C4×Dic3C22×D511yescharacteristic
329C4×Dic3C22×D511yescharacteristic
330#208—22no—
331C2×C3⋊C8C22×D512yes—
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—31no—
341C2×C4×C5—31no—
342C2×C4×C5—31no—
343C2×C4×C5—31no—
344C2×C4×C5—62no—
345C2×C4×C5—121no—
346C2×C4×C5C2×C4×D312yes—
347C2×C4×C5D3×D411yescharacteristic
348C2×C4×C5#23912yes—
349C2×C4×C5C23×D311yescharacteristic
350C22⋊D5—64no—
351C2×Dic5—21no—
352C2×Dic5—21no—
353C2×Dic5—22no—
354C2×Dic5—42no—
355C2×Dic5—61no—
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×C8—62no—
369C5×C8C2×C4×D314yescomplemented
370C2×#41—151no—
371C22×D4—151no—
372C23×C4—301no—
373AGL(1,ℤ/8)—158no—
374C2×SD16—152no—
375C2×D8—152no—
376C2×M16—31no—
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—51no—
389#117—301no—
390C2×C4⋊C4—51no—
391C2×C22⋊C4—301no—
392C2×C42—31no—
393C2×C42—301no—
394#107—52no—
395#106—52no—
396#103—152no—
397#102—152no—
398#102—152no—
399#102—152no—
400#97—32no—
401C4×C8—32no—
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×C3C2×D2011yescharacteristic
409C22×D3—101no—
410C22×D3—102no—
411C22×D3—104no—
412C22×D3—202no—
413C22×D3—202no—
414C2×C3×C4—101no—
415C2×C3×C4—101no—
416C2×C3×C4—102no—
417C2×C3×C4—202no—
418C2×C3×C4C2×C4×D512yes—
419C2×C3×C4C2×D2011yescharacteristic
420C2×C3×C4#70912yes—
421C2×C3×C4C23×D511yescharacteristic
422#67—104no—
423C2×Dic3—22no—
424C2×Dic3—41no—
425C2×Dic3—101no—
426C2×Dic3C2×D2011yescharacteristic
427C2×Dic3C2×D2011yescharacteristic
428C2×Dic3C2×D2011yescharacteristic
429C2×Dic3C2×D2011yescharacteristic
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—22no—
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—32no—
456C4×C5—61no—
457C4×C5—61no—
458C4×C5—61no—
459C4×C5#89312yes—
460C4×C5C22×C4×D311yescharacteristic
461C4×C5C2×#23911yescharacteristic
462Dic5—42no—
463Dic5—41no—
464Dic5—41no—
465Dic5—62no—
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—31no—
477C22×C4—31no—
478C22×C4—101no—
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—62no—
489M16#120014yescomplemented
490C2×C8—32no—
491C2×C8C2×D3×D512yes—
492C4⋊C4—51no—
493C4⋊C4—51no—
494C4⋊C4—52no—
495C4⋊C4—101no—
496C4⋊C4—301no—
497C4⋊C4—301no—
498C22⋊C4—101no—
499C22⋊C4—302no—
500C22⋊C4—302no—
501C22⋊C4—301no—
502C22⋊C4—301no—
503C42—31no—
504C42—31no—
505C42—31no—
506C42—31no—
507C42—101no—
508C42—301no—
509C42—301no—
510C3×C5C4×AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
511C22×C3—21no—
512C22×C3—21no—
513C22×C3C4×D2012yes—
514C22×C3C22×D2011yescharacteristic
515C2×D3—102no—
516C2×D3—104no—
517C2×D3—208no—
518C2×D3—202no—
519C2×D3—201no—
520C2×D3—201no—
521C2×D3—202no—
522C3×C4—202no—
523C3×C4—201no—
524C3×C4—201no—
525C3×C4C4×D2012yes—
526C3×C4C22×C4×D511yescharacteristic
527C3×C4C2×#70911yescharacteristic
528Dic3—21no—
529Dic3—21no—
530Dic3—21no—
531Dic3—102no—
532Dic3#405412yescomplemented
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—
544C23D3×D2011yescharacteristic
545Q8—152no—
546Q8—301no—
547D4—152no—
548D4—154no—
549D4—304no—
550D4—304no—
551D4—302no—
552D4—301no—
553C2×C4—31no—
554C2×C4—31no—
555C2×C4—31no—
556C2×C4—31no—
557C2×C4—62no—
558C2×C4—101no—
559C2×C4—101no—
560C2×C4—102no—
561C2×C4—121no—
562C2×C4—202no—
563C2×C4—301no—
564C2×C4—304no—
565C2×C4—604no—
566C2×C4—602no—
567C2×C4—604no—
568C2×C4—602no—
569C2×C4#676912yes—
570C2×C4D3×D2011yescharacteristic
571C2×C4C2×#120012yes—
572C2×C4C22×D3×D511yescharacteristic
573C8—62no—
574C8C2×#120014yescomplemented
575C2×C3—22no—
576C2×C3C2×C4×D2011yescharacteristic
577C2×C3C2×#405411yescharacteristic
578C2×C3#6608211yescharacteristic
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#9142812yes—
592C22C2×D3×D2011yescharacteristic
593C4—32no—
594C4—61no—
595C4—61no—
596C4—61no—
597C4—202no—
598C4—201no—
599C4—201no—
600C4—302no—
601C4#9142812yes—
602C4C2×#676911yescharacteristic
603C4C22×#120011yescharacteristic
604C3C4×#405411yescharacteristic, complemented, Sylow
605C2—22no—
606C2—604no—
607C2—602no—
608C2D3×#405411yescharacteristic
609C2#1175402811yescharacteristic
610C2C2×#9142811yescharacteristic
subgroupG/H
centerC22C2×D3×D20
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. #11754436C2semidirect
3. C2×#91428C2semidirect
4. C5×Dic3×M16C2semidirect
5. C3×#65410C2semidirect
6. #11748413C2semidirect
7. #11748392C2semidirect
8. #11747800C2semidirect
9. #11747796C2semidirect
10. #11747793C2semidirect
11. #11747789C2semidirect
17. #91797C4semidirect
31. #91428C22semidirect
32. #91428C22semidirect
35. #91340C22semidirect
36. #91340C22semidirect
37. C5×#815C22semidirect
38. C5×C8×Dic3C22semidirect
39. C3×#3950C22semidirect
40. C3×#3949C22semidirect
41. #90974C22semidirect
44. #90968C22semidirect
52. #65410D3semidirect
61. #6817C2×C4semidirect
64. C2×D60C2×C4semidirect
77. C3×C5×M16C2×C4semidirect
79. D120C2×C4semidirect
80. #6706C2×C4semidirect
105. Dic3×M16D5semidirect
141. #3950C2×D3semidirect
142. #3949C2×D3semidirect
161. D60C22×C4semidirect
164. Dic30C22×C4semidirect
179. C3×C5×C8C22×C4semidirect
207. #815C2×D5semidirect
208. C8×Dic3C2×D5semidirect
234. C5×M16C4×D3semidirect
289. C5×Dic3AGL(1,ℤ/8)semidirect
310. C3×M16C4×D5semidirect
369. C5×C8C2×C4×D3semidirect
442. C3×C8C2×C4×D5semidirect
489. M16#1200semidirect
510. C3×C5C4×AGL(1,ℤ/8)semidirect
532. Dic3#4054semidirect
574. C8C2×#1200semidirect
581. C5#79199semidirect
604. C3C4×#4054semidirect

Maximal subgroups

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

subgroupindex
C2×#917972
#117544362
C2×#914282
C5×Dic3×M162
C3×#654102
#117484132
#117483922
#117478002
#117477962
#117477932
#117477892
C4×#40543
#791995