Finite Group Databasebeta

← #14416751

Subgroups — #14416751

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×#91782C211yesmaximal, characteristic, complemented
2#11754455C211yesmaximal, characteristic, complemented
3C2×#91305C211yesmaximal, characteristic
4C5×#4966C211yesmaximal, characteristic, complemented
5C3×#65253C211yesmaximal, characteristic, complemented
6#11748392C211yesmaximal, characteristic, complemented
7#11748328C211yesmaximal, characteristic, complemented
8#11747280C212yesmaximal, complemented
9#11747279C212yesmaximal, complemented
10#11747236C212yesmaximal, complemented
11#11747233C212yesmaximal, complemented
12#10610597—31nomaximal, Hall
13C22×D60C2211yescharacteristic
14C22×C4×D15—21no—
15C2×C5×#235C2211yescharacteristic
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17C2×#6714—21no—
18C2×C4×Dic15C2211yescharacteristic
19#91782—21no—
20#91782—21no—
21#91782C2212yes—
22#91782C2212yes—
23C2×C5×#208C2211yescharacteristic
24C3×#4072C2211yescharacteristic
25C2×C3×#681C2211yescharacteristic
26#67⋊Dic5—21no—
27#67⋊Dic5—21no—
28#91435C2211yescharacteristic, complemented
29#91435C2211yescharacteristic, complemented
30C4×D3×Dic5—21no—
31#91371—21no—
32#91344C2211yescharacteristic, complemented
33#91344C2211yescharacteristic, complemented
34C2×#6660C2212yes—
35#91305C418yescomplemented
36C2×#6654C2212yes—
37C5×#810C2212yescomplemented
38C5×#809C2212yescomplemented
39C3×#3940C2212yescomplemented
40C3×#3939C2212yescomplemented
41#90968C2212yes—
42#90968C2212yes—
43#90954C2212yescomplemented
44#90951C2212yescomplemented
45#79199—51nomaximal, Hall
46#41×Dic5—31no—
47C2×#4095—31no—
48C2×#4039—31no—
49C5×#417—31no—
50#65420—31no—
51M16×Dic5—31no—
52#65253D311yescharacteristic, complemented
53#65106—32no—
54#65077—32no—
55#65074—32no—
56#65045—32no—
57C23×D15—21no—
58C22×C3×C4×C5—21no—
59C22×C3×C4×C5C2311yescharacteristic
60C22×Dic15D411yescharacteristic
61C22×Dic15D411yescharacteristic
62C2×D60—21no—
63C2×D60—22no—
64C2×D60C2×C414yescomplemented
65C2×D60C2312yes—
66C2×C4×D15—22no—
67C2×C4×D15—22no—
68C2×C4×D15—44no—
69C2×C5×#67—21no—
70C5×#235—22no—
71C5×#235C2×C414yescomplemented
72C2×C5×D12C2311yescharacteristic
73C2×C4×C5×D3—21no—
74C2×C5×Dic6C2311yescharacteristic
75C2×D3×Dic5—21no—
76C2×D3×Dic5—21no—
77C3×C5×C4⋊C4—21no—
78C3×C5×C4⋊C4C2312yes—
79#6714—22no—
80#6714—22no—
81C4×Dic15D411yescharacteristic
82C4×Dic15D411yescharacteristic
83C4×Dic15C2311yescharacteristic
84C4×Dic15C2311yescharacteristic
85C5×#208—22no—
86C2×C5×C3⋊C8C2312yes—
87C3×C22⋊Dic5—21no—
88C3×C4⋊Dic5C2311yescharacteristic
89C3×C4⋊Dic5C2311yescharacteristic
90C3×C4×Dic5—21no—
91C3×#681C2×C414yescomplemented
92C2×C3×#168C2312yes—
93Dic3⋊Dic5—21no—
94Dic3⋊Dic5—21no—
95(C2×D3)⋊Dic5—21no—
96(C2×D3)⋊Dic5—21no—
97Dic3×Dic5—21no—
98Dic3×Dic5—21no—
99#6660C2×C418yescomplemented
100#6654C2×C418yescomplemented
101C2×#908—51no—
102#5522—51no—
103C2×#893—51no—
104C3×#428—51no—
105Dic3×M16—51no—
106#5001—51no—
107#4966D511yescharacteristic, complemented
108#4889—52no—
109#4872—52no—
110#4869—52no—
111#4852—52no—
112C2×C5×#41—31no—
113C22×D20—31no—
114C22×C4×D5—61no—
115C2×C5×M16—31no—
116C2×C5×C4⋊C4C2×D311yescharacteristic
117#4095—38no—
118Q8×Dic5—31no—
119Q8×Dic5—31no—
120C2×#688—32no—
121D4×Dic5—31no—
122D4×Dic5—31no—
123D4×Dic5—61no—
124D4×Dic5—61no—
125C2×#686—32no—
126C2×#685—61no—
127#4072—61no—
128#4072C2×D311yescharacteristic
129C2×C4×Dic5—31no—
130C2×C4×Dic5—61no—
131C2×#681C2×D311yescharacteristic
132#4039—32no—
133#4039—32no—
134#4039—61no—
135#4039—61no—
136C5×#103—32no—
137C5×#102—32no—
138#3953—32no—
139#3953—32no—
140#3947—32no—
141C8×Dic5—32no—
142#3940C2×D312yescomplemented
143#3939C2×D312yescomplemented
144C4×AGL(1,ℤ/8)—151noSylow
145C23×C3×C5C2×D411yescharacteristic
146C22×D15—24no—
147C22×D15—22no—
148C22×D15—42no—
149C22×D15—42no—
150C22×C5×D3—21no—
151C2×C3×C4×C5—22no—
152C2×C3×C4×C5—42no—
153C2×C3×C4×C5C22×C412yes—
154C2×C3×C4×C5C2×D411yescharacteristic
155C2×C3×C4×C5#4112yes—
156C2×C3×C4×C5C2411yescharacteristic
157C2×Dic15—22no—
158C2×Dic15—41no—
159C2×Dic15C2×D411yescharacteristic
160C2×Dic15C2×D411yescharacteristic
161C2×Dic15C2×D411yescharacteristic
162C2×Dic15C2×D411yescharacteristic
163D60—24no—
164D60—22no—
165D60C22×C414yes—
166C4×D15—44no—
167C4×D15—44no—
168C5×#67—24no—
169C2×C5×Dic3—21no—
170C5×D12—21no—
171C5×D12C22×C412yescomplemented
172C4×C5×D3—24no—
173C5×Dic6—21no—
174C5×Dic6C22×C412yescomplemented
175C2×C3×Dic5—21no—
176C2×C3×Dic5—21no—
177D3×Dic5—42no—
178D3×Dic5—42no—
179C3×#168C22×C414yescomplemented
180C5×C3⋊C8—22no—
181C2×#235C2×D511yescharacteristic
182C22×D12—51no—
183C22×C4×D3—101no—
184C2×C3×M16—51no—
185C3×#117—51no—
186C2×C3×C4⋊C4C2×D511yescharacteristic
187C2×(C2×D3)⋊C4—101no—
188C2×C4×Dic3—51no—
189C2×#208C2×D511yescharacteristic
190#908—58no—
191C2×D24—52no—
192C2×#204—52no—
193#893—51no—
194#893—51no—
195#893—52no—
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#815—52no—
208C8×Dic3—52no—
209#810C2×D512yescomplemented
210#809—52no—
211#809—52no—
212#809C2×D512yescomplemented
213C23×D5—61no—
214C5×#41—34no—
215C5×#41—62no—
216C2×C5×Q8—31no—
217C2×C5×D4—31no—
218C2×C5×D4—61no—
219C22×C4×C5—21no—
220C22×C4×C5—61no—
221C22×C4×C5C22×D311yescharacteristic
222C22×Dic5—31no—
223C22×Dic5—31no—
224C22×Dic5—61no—
225C22×Dic5—61no—
226C2×D20—34no—
227C2×D20—32no—
228C2×D20—61no—
229C2×D20—62no—
230C2×C4×D5—62no—
231C2×C4×D5—62no—
232C2×C4×D5—124no—
233C5×M16—62no—
234C2×C5×C8—32no—
235C5×C4⋊C4—21no—
236C5×C4⋊C4C22×D312yes—
237C22⋊Dic5—21no—
238C22⋊Dic5—61no—
239C22⋊Dic5—61no—
240#688—38no—
241#686—38no—
242#685—62no—
243#685—62no—
244C4⋊Dic5—61no—
245C4⋊Dic5—61no—
246C4⋊Dic5C22×D311yescharacteristic
247C4⋊Dic5C22×D311yescharacteristic
248C4×Dic5—21no—
249C4×Dic5—31no—
250C4×Dic5—31no—
251C4×Dic5—31no—
252C4×Dic5—31no—
253C4×Dic5—61no—
254C4×Dic5—61no—
255#681C4×D314yescomplemented
256C2×#168C22×D312yes—
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—31no—
267C4×M16—151no—
268C22×C3×C5—21no—
269C22×C3×C5—21no—
270C22×C3×C5C4×D412yes—
271C22×C3×C5C22×D411yescharacteristic
272C2×D15—24no—
273C2×D15—48no—
274C2×D15—42no—
275C2×D15—42no—
276C2×C5×D3—22no—
277C2×C5×D3—41no—
278C2×C5×D3—41no—
279C3×C4×C5—42no—
280C3×C4×C5C4×D412yes—
281C3×C4×C5C23×C411yescharacteristic
282C3×C4×C5C2×#4111yescharacteristic
283Dic15—21no—
284Dic15—21no—
285Dic15—21no—
286Dic15AGL(1,ℤ/8)12yescomplemented
287C3×Dic5—41no—
288C3×Dic5—41no—
289C5×Dic3—22no—
290C23×D3—101no—
291C22×C3×C4—21no—
292C22×C3×C4C22×D511yescharacteristic
293C2×#67—21no—
294C22×Dic3—51no—
295C22×Dic3—51no—
296#235—22no—
297#235C4×D514yescomplemented
298C2×D12—54no—
299C2×D12—52no—
300C2×D12—101no—
301C2×D12—102no—
302C2×D12C22×D511yescharacteristic
303C2×C4×D3—21no—
304C2×C4×D3—102no—
305C2×C4×D3—102no—
306C2×C4×D3—101no—
307C2×C4×D3—101no—
308C2×C4×D3—204no—
309C2×Dic6C22×D511yescharacteristic
310C3×M16—54no—
311C2×C3×C8—52no—
312C3×C4⋊C4—21no—
313C3×C4⋊C4—51no—
314C3×C4⋊C4—51no—
315C3×C4⋊C4C22×D512yes—
316C3×C22⋊C4—101no—
317C3×C42—101no—
318(C2×D3)⋊C4—102no—
319(C2×D3)⋊C4—102no—
320(C2×D3)⋊C4—101no—
321(C2×D3)⋊C4—101no—
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—22no—
331C2×C3⋊C8C22×D512yes—
332D24—58no—
333#204—58no—
334C23×C5—61no—
335C23×C5D3×D411yescharacteristic
336C22×D5—64no—
337C22×D5—62no—
338C22×D5—122no—
339C22×D5—122no—
340C5×Q8—32no—
341C5×Q8—61no—
342C5×D4—32no—
343C5×D4—64no—
344C5×D4—61no—
345C2×C4×C5—22no—
346C2×C4×C5—42no—
347C2×C4×C5—61no—
348C2×C4×C5—64no—
349C2×C4×C5C2×C4×D312yes—
350C2×C4×C5D3×D411yescharacteristic
351C2×C4×C5#23912yes—
352C2×C4×C5C23×D311yescharacteristic
353C2×Dic5—21no—
354C2×Dic5—21no—
355C2×Dic5—31no—
356C2×Dic5—31no—
357C2×Dic5—31no—
358C2×Dic5—31no—
359C2×Dic5—62no—
360C2×Dic5—122no—
361C2×Dic5—122no—
362C2×Dic5—121no—
363D20—34no—
364D20—64no—
365D20—62no—
366C4×D5—124no—
367C4×D5—124no—
368C5×C8—62no—
369#168C2×C4×D314yescomplemented
370C2×#41—31no—
371C22×D4—151no—
372C23×C4—301no—
373AGL(1,ℤ/8)—158no—
374C2×SD16—152no—
375C2×D8—152no—
376C2×M16—31no—
377C2×M16—51no—
378C4×Q8—151no—
379C4×Q8—151no—
380C4×D4—151no—
381C4×D4—151no—
382C4×D4—152no—
383C4×D4—152no—
384C4×D4—301no—
385C4×D4—301no—
386C4×D4—301no—
387C4×D4—301no—
388#117—51no—
389#117—301no—
390C2×C4⋊C4D3×D511yescharacteristic
391C2×C22⋊C4—301no—
392C2×C42—151no—
393C2×C42—301no—
394#107—52no—
395#106—52no—
396#103—32no—
397#102—32no—
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—44no—
407C5×D3—42no—
408C23×C3D4×D511yescharacteristic
409C22×D3—21no—
410C22×D3—102no—
411C22×D3—104no—
412C22×D3—202no—
413C22×D3—202no—
414C2×C3×C4—22no—
415C2×C3×C4—42no—
416C2×C3×C4—101no—
417C2×C3×C4—101no—
418C2×C3×C4C2×C4×D512yes—
419C2×C3×C4D4×D511yescharacteristic
420C2×C3×C4#71112yes—
421C2×C3×C4C23×D511yescharacteristic
422#67—24no—
423C2×Dic3—21no—
424C2×Dic3—51no—
425C2×Dic3—51no—
426C2×Dic3—51no—
427C2×Dic3—51no—
428C2×Dic3—102no—
429C2×Dic3—201no—
430D12—21no—
431D12—54no—
432D12—102no—
433D12—104no—
434D12C2×C4×D512yescomplemented
435C4×D3—24no—
436C4×D3—204no—
437C4×D3—202no—
438C4×D3—204no—
439C4×D3—202no—
440Dic6—21no—
441Dic6C2×C4×D512yescomplemented
442C3×C8—54no—
443C3⋊C8—22no—
444C22×C5—21no—
445C22×C5—21no—
446C22×C5—62no—
447C22×C5—121no—
448C22×C5—121no—
449C22×C5#89312yes—
450C22×C5C2×D3×D411yescharacteristic
451C2×D5—64no—
452C2×D5—128no—
453C2×D5—122no—
454C2×D5—122no—
455C4×C5—42no—
456C4×C5—62no—
457C4×C5#89312yes—
458C4×C5C22×C4×D311yescharacteristic
459C4×C5C2×#23911yescharacteristic
460Dic5—32no—
461Dic5—41no—
462Dic5—41no—
463Dic5—61no—
464Dic5—61no—
465Dic5—61no—
466C24—301no—
467#41—34no—
468#41—62no—
469C2×Q8—31no—
470C2×D4—31no—
471C2×D4—61no—
472C2×D4—154no—
473C2×D4—152no—
474C2×D4—302no—
475C2×D4—301no—
476C22×C4—21no—
477C22×C4—61no—
478C22×C4—151no—
479C22×C4—151no—
480C22×C4—302no—
481C22×C4—302no—
482C22×C4—301no—
483C22×C4—301no—
484C22×C4—604no—
485C22×C4C2×D3×D511yescharacteristic
486SD16—158no—
487D8—158no—
488M16—54no—
489M16—62no—
490C2×C8—32no—
491C2×C8—52no—
492C4⋊C4—21no—
493C4⋊C4—51no—
494C4⋊C4—51no—
495C4⋊C4—301no—
496C4⋊C4—301no—
497C4⋊C4C2×D3×D512yes—
498C22⋊C4—101no—
499C22⋊C4—302no—
500C22⋊C4—302no—
501C22⋊C4—301no—
502C22⋊C4—301no—
503C42—101no—
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#402712yes—
514C22×C3C2×D4×D511yescharacteristic
515C2×D3—22no—
516C2×D3—41no—
517C2×D3—41no—
518C2×D3—104no—
519C2×D3—208no—
520C2×D3—202no—
521C2×D3—202no—
522C3×C4—42no—
523C3×C4—201no—
524C3×C4—201no—
525C3×C4#402712yes—
526C3×C4C22×C4×D511yescharacteristic
527C3×C4C2×#71111yescharacteristic
528Dic3—22no—
529Dic3—52no—
530Dic3—101no—
531Dic3—101no—
532Dic3—101no—
533C2×C5—22no—
534C2×C5—122no—
535C2×C5C2×#89311yescharacteristic
536C2×C5D3×AGL(1,ℤ/8)11yescharacteristic
537C2×C5#577711yescharacteristic
538D5—124no—
539C23—61no—
540C23—304no—
541C23—302no—
542C23—602no—
543C23—602no—
544C23#679011yescharacteristic
545Q8—32no—
546Q8—61no—
547D4—32no—
548D4—64no—
549D4—61no—
550D4—154no—
551D4—304no—
552D4—302no—
553C2×C4—22no—
554C2×C4—42no—
555C2×C4—61no—
556C2×C4—64no—
557C2×C4—101no—
558C2×C4—101no—
559C2×C4—151no—
560C2×C4—151no—
561C2×C4—151no—
562C2×C4—151no—
563C2×C4—302no—
564C2×C4—604no—
565C2×C4—604no—
566C2×C4—602no—
567C2×C4—602no—
568C2×C4—601no—
569C2×C4#676812yes—
570C2×C4C4×D3×D512yes—
571C2×C4#679011yescharacteristic
572C2×C4C22×D3×D511yescharacteristic
573C8—54no—
574C8—62no—
575C2×C3—22no—
576C2×C3C2×#402711yescharacteristic
577C2×C3D5×AGL(1,ℤ/8)11yescharacteristic
578C2×C3#6611111yescharacteristic
579D3—42no—
580D3—204no—
581C5#7919911yescharacteristic, complemented, Sylow
582C22—21no—
583C22—21no—
584C22—62no—
585C22—121no—
586C22—121no—
587C22—304no—
588C22—608no—
589C22—602no—
590C22—602no—
591C22#9140612yes—
592C22C2×#679011yescharacteristic
593C4—42no—
594C4—62no—
595C4—152no—
596C4—201no—
597C4—201no—
598C4—301no—
599C4—301no—
600C4—301no—
601C4#9140612yes—
602C4C2×#676811yescharacteristic
603C4C2×C4×D3×D511yescharacteristic
604C3#1061059711yescharacteristic, complemented, Sylow
605C2—22no—
606C2—122no—
607C2—604no—
608C2C2×#9140611yescharacteristic
609C2#1175470111yescharacteristic
610C2#1175509611yescharacteristic
subgroupG/H
centerC22C2×#6790
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×C4⋊C4C22
socleC22×C3×C5C22×D4
prime radicalC2×#91305C2
cosocleC2×C3×C4×C5C24
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5C4×AGL(1,ℤ/8)
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#91782C2semidirect
2. #11754455C2semidirect
4. C5×#4966C2semidirect
5. C3×#65253C2semidirect
6. #11748392C2semidirect
7. #11748328C2semidirect
8. #11747280C2semidirect
9. #11747279C2semidirect
10. #11747236C2semidirect
11. #11747233C2semidirect
28. #91435C22semidirect
29. #91435C22semidirect
32. #91344C22semidirect
33. #91344C22semidirect
35. #91305C4semidirect
37. C5×#810C22semidirect
38. C5×#809C22semidirect
39. C3×#3940C22semidirect
40. C3×#3939C22semidirect
43. #90954C22semidirect
44. #90951C22semidirect
52. #65253D3semidirect
64. C2×D60C2×C4semidirect
71. C5×#235C2×C4semidirect
91. C3×#681C2×C4semidirect
99. #6660C2×C4semidirect
100. #6654C2×C4semidirect
107. #4966D5semidirect
142. #3940C2×D3semidirect
143. #3939C2×D3semidirect
171. C5×D12C22×C4semidirect
174. C5×Dic6C22×C4semidirect
179. C3×#168C22×C4semidirect
209. #810C2×D5semidirect
212. #809C2×D5semidirect
255. #681C4×D3semidirect
286. Dic15AGL(1,ℤ/8)semidirect
297. #235C4×D5semidirect
369. #168C2×C4×D3semidirect
434. D12C2×C4×D5semidirect
441. Dic6C2×C4×D5semidirect
510. C3×C5C4×AGL(1,ℤ/8)semidirect
581. C5#79199semidirect
604. C3#10610597semidirect

Maximal subgroups

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

subgroupindex
C2×#917822
#117544552
C2×#913052
C5×#49662
C3×#652532
#117483922
#117483282
#117472802
#117472792
#117472362
#117472332
#106105973
#791995