Finite Group Databasebeta

← #14416742

Subgroups — #14416742

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
1#41×Dic15C211yesmaximal, characteristic, complemented
2C2×C5×#949C211yesmaximal, characteristic
3C2×#91435C211yesmaximal, characteristic, complemented
4C3×#65521C211yesmaximal, characteristic, complemented
5#11749242C212yesmaximal, complemented
6#11749240C212yesmaximal, complemented
7#11748996C212yesmaximal, complemented
8#11748994C212yesmaximal, complemented
9#11748453C211yesmaximal, characteristic, complemented
10#11748389C211yesmaximal, characteristic, complemented
11#11748328C211yesmaximal, characteristic, complemented
12Dic5×AGL(1,ℤ/8)—31nomaximal, Hall
13C2×C3×C5×#41C2211yescharacteristic
14C22×C5×D12C2211yescharacteristic
15C22×D3×Dic5—21no—
16Q8×Dic15C2211yescharacteristic, complemented
17Q8×Dic15C2211yescharacteristic, complemented
18D4×Dic15—21no—
19D4×Dic15—21no—
20D4×Dic15C2211yescharacteristic, complemented
21D4×Dic15C2211yescharacteristic, complemented
22#91815—21no—
23#91815C2211yescharacteristic
24C2×C4×Dic15—21no—
25C2×C4×Dic15C2211yescharacteristic
26C5×#949C418yescomplemented
27C2×C5×#215C2212yes—
28C2×C5×#213C2212yes—
29C2×C5×#208C2211yescharacteristic
30C2×C3×C4⋊Dic5C2211yescharacteristic
31C2×C3×#681C2211yescharacteristic
32C2×(C2×D3)⋊Dic5—21no—
33#91435—21no—
34#91435—21no—
35#91435C2212yes—
36#91435C2212yes—
37C3×#3967C2212yescomplemented
38C3×#3964C2212yescomplemented
39#90990C2212yescomplemented
40#90987C2212yescomplemented
41#90967C2212yes—
42#90967C2212yes—
43#90954C2212yescomplemented
44#90951C2212yescomplemented
45#78866—51nomaximal, Hall
46C2×C5×AGL(1,ℤ/8)—31no—
47#41×Dic5—31no—
48C2×D4×Dic5—31no—
49#65521D311yescharacteristic, complemented
50#65504—31no—
51#65453—32no—
52SD16×Dic5—32no—
53#65441—32no—
54D8×Dic5—32no—
55#65410—31no—
56M16×Dic5—31no—
57C23×C5×D3—21no—
58C3×C5×#41—22no—
59C3×C5×#41C2×C414yescomplemented
60C2×C3×C5×Q8C2311yescharacteristic
61C2×C3×C5×D4—21no—
62C2×C3×C5×D4C2311yescharacteristic
63C22×C3×C4×C5—21no—
64C22×C3×C4×C5C2311yescharacteristic
65C22×Dic15—21no—
66C22×Dic15—21no—
67C22×Dic15D411yescharacteristic
68C22×Dic15D411yescharacteristic
69C2×C5×D12—21no—
70C2×C5×D12—22no—
71C2×C5×D12C2×C414yescomplemented
72C2×C5×D12C2312yes—
73C22×C3×Dic5—21no—
74C2×D3×Dic5—22no—
75C2×D3×Dic5—22no—
76C2×D3×Dic5—44no—
77#6719—21no—
78#6719—21no—
79#6719—21no—
80C4⋊Dic15—21no—
81C4⋊Dic15—21no—
82C4⋊Dic15C2311yescharacteristic
83C4⋊Dic15C2311yescharacteristic
84C4×Dic15—21no—
85C4×Dic15—21no—
86C4×Dic15—21no—
87C4×Dic15D411yescharacteristic
88C4×Dic15D411yescharacteristic
89C4×Dic15C2311yescharacteristic
90C4×Dic15C2311yescharacteristic
91C5×#215C2×C418yescomplemented
92C5×#213C2×C418yescomplemented
93C5×#208C2×C414yescomplemented
94C2×C5×C3⋊C8C2312yes—
95C3×C4⋊Dic5—21no—
96C3×C4⋊Dic5C2312yes—
97C3×#681—22no—
98C2×C3×#168C2312yes—
99(C2×D3)⋊Dic5—22no—
100(C2×D3)⋊Dic5—22no—
101Dic3×#41—51no—
102C2×#949D511yescharacteristic
103C2×#893—51no—
104C3×#417—51no—
105#5131—51no—
106Dic3×M16—51no—
107#4964—51no—
108#4817—52no—
109#4788—52no—
110#4785—52no—
111#4756—52no—
112C2×C5×#41C2×D311yescharacteristic
113C22×C5×D4—31no—
114C23×Dic5—61no—
115C5×AGL(1,ℤ/8)—38no—
116C2×C5×SD16—32no—
117C2×C5×D8—32no—
118C2×C5×M16—31no—
119C2×C22⋊Dic5—61no—
120Q8×Dic5—31no—
121Q8×Dic5—31no—
122D4×Dic5—31no—
123D4×Dic5—31no—
124D4×Dic5—32no—
125D4×Dic5—32no—
126D4×Dic5—61no—
127D4×Dic5—61no—
128D4×Dic5—61no—
129D4×Dic5—61no—
130#4072—31no—
131#4072—61no—
132C2×C4⋊Dic5C2×D311yescharacteristic
133C2×C4×Dic5—31no—
134C2×C4×Dic5—61no—
135C2×#681C2×D311yescharacteristic
136#3967C2×D312yescomplemented
137#3964—32no—
138#3964—32no—
139#3964C2×D312yescomplemented
140#3950—32no—
141#3949—32no—
142#3947—32no—
143C8×Dic5—32no—
144C4×AGL(1,ℤ/8)—151noSylow
145C23×C3×C5—21no—
146C23×C3×C5C2×D411yescharacteristic
147C22×C5×D3—24no—
148C22×C5×D3—22no—
149C22×C5×D3—42no—
150C22×C5×D3—42no—
151C3×C5×Q8—21no—
152C3×C5×Q8C22×C412yescomplemented
153C3×C5×D4—24no—
154C3×C5×D4—21no—
155C3×C5×D4C22×C412yescomplemented
156C2×C3×C4×C5—21no—
157C2×C3×C4×C5—24no—
158C2×C3×C4×C5C22×C412yes—
159C2×C3×C4×C5C2×D411yescharacteristic
160C2×C3×C4×C5#4112yes—
161C2×C3×C4×C5C2411yescharacteristic
162C2×Dic15—22no—
163C2×Dic15—21no—
164C2×Dic15—21no—
165C2×Dic15—41no—
166C2×Dic15—42no—
167C2×Dic15—42no—
168C2×Dic15C2×D411yescharacteristic
169C2×Dic15C2×D411yescharacteristic
170C2×Dic15C2×D411yescharacteristic
171C2×Dic15C2×D411yescharacteristic
172C5×D12—24no—
173C5×D12—22no—
174C5×D12C22×C414yes—
175C2×C3×Dic5—22no—
176C2×C3×Dic5—42no—
177D3×Dic5—44no—
178D3×Dic5—44no—
179C3×#168—22no—
180C5×C3⋊C8C22×C414yescomplemented
181C2×C3×#41C2×D511yescharacteristic
182C22×D12C2×D511yescharacteristic
183C22×C4×D3—101no—
184C2×C3×M16—51no—
185C2×C3×C4⋊C4—51no—
186#949Dic518yescomplemented
187Q8×Dic3—51no—
188Q8×Dic3—51no—
189C2×#215C2×D512yes—
190D4×Dic3—51no—
191D4×Dic3—51no—
192D4×Dic3—101no—
193D4×Dic3—101no—
194C2×#213C2×D512yes—
195C2×(C2×D3)⋊C4—101no—
196#926—51no—
197#926—101no—
198C2×C4×Dic3—51no—
199C2×C4×Dic3—101no—
200C2×#208C2×D511yescharacteristic
201#893—52no—
202#893—52no—
203#893—101no—
204#893—101no—
205C3×#103—52no—
206C3×#102—52no—
207#821—52no—
208#821—52no—
209#815—52no—
210C8×Dic3—52no—
211#808—52no—
212#807—52no—
213C24×C5—61no—
214C5×#41—22no—
215C5×#41C4×D314yescomplemented
216C2×C5×Q8C22×D311yescharacteristic
217C2×C5×D4—21no—
218C2×C5×D4—34no—
219C2×C5×D4—32no—
220C2×C5×D4—61no—
221C2×C5×D4—62no—
222C2×C5×D4C22×D311yescharacteristic
223C22×C4×C5—21no—
224C22×C4×C5C22×D311yescharacteristic
225C22×Dic5—21no—
226C22×Dic5—31no—
227C22×Dic5—31no—
228C22×Dic5—62no—
229C22×Dic5—62no—
230C22×Dic5—61no—
231C22×Dic5—61no—
232C22×Dic5—124no—
233C5×SD16—38no—
234C5×D8—38no—
235C5×M16—34no—
236C2×C5×C8—32no—
237C22⋊Dic5—62no—
238C22⋊Dic5—62no—
239C22⋊Dic5—61no—
240C22⋊Dic5—61no—
241C22⋊Dic5—61no—
242C4⋊Dic5—21no—
243C4⋊Dic5—31no—
244C4⋊Dic5—31no—
245C4⋊Dic5—61no—
246C4⋊Dic5—61no—
247C4⋊Dic5C22×D312yes—
248C4×Dic5—31no—
249C4×Dic5—31no—
250C4×Dic5—31no—
251C4×Dic5—31no—
252C4×Dic5—61no—
253C4×Dic5—61no—
254C4×Dic5—61no—
255#681—22no—
256C2×#168C22×D312yes—
257C2×AGL(1,ℤ/8)—31no—
258C4×#41—151no—
259C2×C4×D4—151no—
260#442—152no—
261#440—152no—
262C4×SD16—152no—
263C4×D8—152no—
264#428—151no—
265#418—151no—
266#417—51no—
267C4×M16—151no—
268C22×C3×C5—21no—
269C22×C3×C5—21no—
270C22×C3×C5—22no—
271C22×C3×C5—41no—
272C22×C3×C5—41no—
273C22×C3×C5C4×D412yes—
274C22×C3×C5C22×D411yescharacteristic
275C2×C5×D3—24no—
276C2×C5×D3—48no—
277C2×C5×D3—42no—
278C2×C5×D3—42no—
279C3×C4×C5—22no—
280C3×C4×C5C4×D412yes—
281C3×C4×C5C23×C411yescharacteristic
282C3×C4×C5C2×#4111yescharacteristic
283Dic15—21no—
284Dic15—21no—
285Dic15—21no—
286Dic15—41no—
287Dic15—41no—
288Dic15AGL(1,ℤ/8)12yescomplemented
289C3×Dic5—42no—
290C23×D3—21no—
291C3×#41—22no—
292C3×#41C2×Dic514yescomplemented
293C2×C3×Q8C22×D511yescharacteristic
294C2×C3×D4—21no—
295C2×C3×D4C22×D511yescharacteristic
296C22×C3×C4—21no—
297C22×C3×C4—101no—
298C22×C3×C4C22×D511yescharacteristic
299C22×Dic3—51no—
300C22×Dic3—51no—
301C22×Dic3—101no—
302C22×Dic3—101no—
303C2×D12—21no—
304C2×D12—22no—
305C2×D12C2×Dic514yescomplemented
306C2×D12C22×D512yes—
307C2×C4×D3—102no—
308C2×C4×D3—102no—
309C2×C4×D3—204no—
310C3×M16—102no—
311C2×C3×C8—52no—
312C3×C4⋊C4—52no—
313C3×C4⋊C4—101no—
314#217—101no—
315#217—101no—
316#217—101no—
317#215C2×Dic518yescomplemented
318#213C2×Dic518yescomplemented
319(C2×D3)⋊C4—102no—
320(C2×D3)⋊C4—102no—
321C4⋊Dic3—51no—
322C4⋊Dic3—51no—
323C4⋊Dic3—101no—
324C4⋊Dic3—101no—
325C4×Dic3—51no—
326C4×Dic3—51no—
327C4×Dic3—51no—
328C4×Dic3—51no—
329C4×Dic3—101no—
330C4×Dic3—101no—
331C4×Dic3—101no—
332#208C2×Dic514yescomplemented
333C2×C3⋊C8C22×D512yes—
334C23×C5—21no—
335C23×C5—64no—
336C23×C5—62no—
337C23×C5—122no—
338C23×C5—122no—
339C23×C5D3×D411yescharacteristic
340C5×Q8—21no—
341C5×Q8C2×C4×D312yescomplemented
342C5×D4—24no—
343C5×D4—21no—
344C5×D4—34no—
345C5×D4—64no—
346C5×D4—62no—
347C5×D4C2×C4×D312yescomplemented
348C2×C4×C5—21no—
349C2×C4×C5—24no—
350C2×C4×C5C2×C4×D312yes—
351C2×C4×C5D3×D411yescharacteristic
352C2×C4×C5#23712yes—
353C2×C4×C5C23×D311yescharacteristic
354C2×Dic5—22no—
355C2×Dic5—31no—
356C2×Dic5—31no—
357C2×Dic5—31no—
358C2×Dic5—31no—
359C2×Dic5—42no—
360C2×Dic5—62no—
361C2×Dic5—61no—
362C2×Dic5—61no—
363C2×Dic5—124no—
364C2×Dic5—122no—
365C2×Dic5—124no—
366C2×Dic5—121no—
367C2×Dic5—122no—
368C5×C8—34no—
369#168—22no—
370C2×#41D3×D511yescharacteristic
371C22×D4—31no—
372C23×C4—301no—
373AGL(1,ℤ/8)—38no—
374C2×SD16—32no—
375C2×D8—32no—
376C2×M16—31no—
377C2×M16—51no—
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—151no—
389#117—301no—
390C2×C4⋊C4—51no—
391C2×C22⋊C4—301no—
392C2×C42—151no—
393C2×C42—301no—
394#107—152no—
395#106—152no—
396#103—52no—
397#102—52no—
398#102—152no—
399#102—152no—
400#97—152no—
401C4×C8—152no—
402C2×C3×C5—22no—
403C2×C3×C5—42no—
404C2×C3×C5C2×C4×D411yescharacteristic
405C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
406C2×C3×C5#57511yescharacteristic
407C5×D3—44no—
408C23×C3—21no—
409C23×C3D4×D511yescharacteristic
410C22×D3—22no—
411C22×D3—24no—
412C22×D3—42no—
413C22×D3—42no—
414C3×Q8—21no—
415C3×Q8C22×Dic512yescomplemented
416C3×D4—24no—
417C3×D4—21no—
418C3×D4C22×Dic512yescomplemented
419C2×C3×C4—21no—
420C2×C3×C4—24no—
421C2×C3×C4—102no—
422C2×C3×C4—202no—
423C2×C3×C4D4×D511yescharacteristic
424C2×C3×C4#71112yes—
425C2×C3×C4C22×Dic512yes—
426C2×C3×C4C23×D511yescharacteristic
427C2×Dic3—51no—
428C2×Dic3—51no—
429C2×Dic3—51no—
430C2×Dic3—51no—
431C2×Dic3—102no—
432C2×Dic3—101no—
433C2×Dic3—101no—
434C2×Dic3—202no—
435C2×Dic3—201no—
436C2×Dic3—202no—
437D12—22no—
438D12—24no—
439D12C22×Dic514yes—
440C4×D3—204no—
441C4×D3—204no—
442C3×C8—102no—
443C3⋊C8C22×Dic514yescomplemented
444C22×C5—21no—
445C22×C5—21no—
446C22×C5—22no—
447C22×C5—41no—
448C22×C5—41no—
449C22×C5—64no—
450C22×C5—128no—
451C22×C5—122no—
452C22×C5—122no—
453C22×C5#67⋊C412yes—
454C22×C5C2×D3×D411yescharacteristic
455C4×C5—22no—
456C4×C5#67⋊C412yes—
457C4×C5C22×C4×D311yescharacteristic
458C4×C5C2×#23711yescharacteristic
459Dic5—32no—
460Dic5—42no—
461Dic5—61no—
462Dic5—61no—
463Dic5—61no—
464Dic5—121no—
465Dic5—121no—
466C24—61no—
467#41—22no—
468#41D3×Dic514yescomplemented
469C2×Q8C2×D3×D511yescharacteristic
470C2×D4—21no—
471C2×D4—34no—
472C2×D4—32no—
473C2×D4—62no—
474C2×D4—61no—
475C2×D4C2×D3×D511yescharacteristic
476C22×C4—21no—
477C22×C4—101no—
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—38no—
487D8—38no—
488M16—34no—
489M16—102no—
490C2×C8—32no—
491C2×C8—52no—
492C4⋊C4—52no—
493C4⋊C4—101no—
494C4⋊C4—151no—
495C4⋊C4—151no—
496C4⋊C4—301no—
497C4⋊C4—301no—
498C22⋊C4—302no—
499C22⋊C4—302no—
500C22⋊C4—301no—
501C22⋊C4—301no—
502C22⋊C4—301no—
503C42—151no—
504C42—151no—
505C42—151no—
506C42—151no—
507C42—301no—
508C42—301no—
509C42—301no—
510C3×C5C4×AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
511C22×C3—21no—
512C22×C3—21no—
513C22×C3—22no—
514C22×C3—41no—
515C22×C3—41no—
516C22×C3D4×Dic512yes—
517C22×C3C2×D4×D511yescharacteristic
518C2×D3—24no—
519C2×D3—48no—
520C2×D3—42no—
521C2×D3—42no—
522C3×C4—22no—
523C3×C4—202no—
524C3×C4D4×Dic512yes—
525C3×C4C2×#71111yescharacteristic
526C3×C4C23×Dic511yescharacteristic
527Dic3—52no—
528Dic3—101no—
529Dic3—101no—
530Dic3—101no—
531Dic3—201no—
532Dic3—201no—
533C2×C5—22no—
534C2×C5—42no—
535C2×C5—124no—
536C2×C5C2×#67⋊C411yescharacteristic
537C2×C5D3×AGL(1,ℤ/8)11yescharacteristic
538C2×C5#577711yescharacteristic
539C23—21no—
540C23—62no—
541C23—64no—
542C23—122no—
543C23—122no—
544C23#679011yescharacteristic
545Q8—21no—
546Q8C2×D3×Dic512yescomplemented
547D4—24no—
548D4—21no—
549D4—34no—
550D4—62no—
551D4—64no—
552D4C2×D3×Dic512yescomplemented
553C2×C4—21no—
554C2×C4—24no—
555C2×C4—102no—
556C2×C4—151no—
557C2×C4—151no—
558C2×C4—151no—
559C2×C4—151no—
560C2×C4—202no—
561C2×C4—302no—
562C2×C4—301no—
563C2×C4—301no—
564C2×C4—604no—
565C2×C4—602no—
566C2×C4—604no—
567C2×C4—601no—
568C2×C4—602no—
569C2×C4#678012yes—
570C2×C4C2×D3×Dic512yes—
571C2×C4#679011yescharacteristic
572C2×C4C22×D3×D511yescharacteristic
573C8—34no—
574C8—102no—
575C2×C3—22no—
576C2×C3—42no—
577C2×C3D5×AGL(1,ℤ/8)11yescharacteristic
578C2×C3#6610711yescharacteristic
579C2×C3C2×D4×Dic511yescharacteristic
580D3—44no—
581C5#7886611yescharacteristic, complemented, Sylow
582C22—21no—
583C22—21no—
584C22—22no—
585C22—41no—
586C22—41no—
587C22—64no—
588C22—128no—
589C22—122no—
590C22—122no—
591C22#67⋊Dic512yes—
592C22C2×#679011yescharacteristic
593C4—22no—
594C4—152no—
595C4—202no—
596C4—301no—
597C4—301no—
598C4—301no—
599C4—601no—
600C4—601no—
601C4#67⋊Dic512yes—
602C4C2×#678011yescharacteristic
603C4C22×D3×Dic511yescharacteristic
604C3Dic5×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
605C2—22no—
606C2—42no—
607C2—124no—
608C2#1175532711yescharacteristic
609C2#1175535211yescharacteristic
610C2C2×#67⋊Dic511yescharacteristic
subgroupG/H
centerC22C2×#6790
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×#41C22
socleC22×C3×C5C22×D4
prime radicalC2×C5×#949C2
cosocleC2×C3×C4×C5C24
hypercenterC2×#41D3×D5
hypocenterC3×C5C4×AGL(1,ℤ/8)
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #41×Dic15C2semidirect
3. C2×#91435C2semidirect
4. C3×#65521C2semidirect
5. #11749242C2semidirect
6. #11749240C2semidirect
7. #11748996C2semidirect
8. #11748994C2semidirect
9. #11748453C2semidirect
10. #11748389C2semidirect
11. #11748328C2semidirect
16. Q8×Dic15C22semidirect
17. Q8×Dic15C22semidirect
20. D4×Dic15C22semidirect
21. D4×Dic15C22semidirect
26. C5×#949C4semidirect
37. C3×#3967C22semidirect
38. C3×#3964C22semidirect
39. #90990C22semidirect
40. #90987C22semidirect
43. #90954C22semidirect
44. #90951C22semidirect
49. #65521D3semidirect
59. C3×C5×#41C2×C4semidirect
71. C2×C5×D12C2×C4semidirect
91. C5×#215C2×C4semidirect
92. C5×#213C2×C4semidirect
93. C5×#208C2×C4semidirect
136. #3967C2×D3semidirect
139. #3964C2×D3semidirect
152. C3×C5×Q8C22×C4semidirect
155. C3×C5×D4C22×C4semidirect
180. C5×C3⋊C8C22×C4semidirect
186. #949Dic5semidirect
215. C5×#41C4×D3semidirect
288. Dic15AGL(1,ℤ/8)semidirect
292. C3×#41C2×Dic5semidirect
305. C2×D12C2×Dic5semidirect
317. #215C2×Dic5semidirect
318. #213C2×Dic5semidirect
332. #208C2×Dic5semidirect
341. C5×Q8C2×C4×D3semidirect
347. C5×D4C2×C4×D3semidirect
415. C3×Q8C22×Dic5semidirect
418. C3×D4C22×Dic5semidirect
443. C3⋊C8C22×Dic5semidirect
468. #41D3×Dic5semidirect
510. C3×C5C4×AGL(1,ℤ/8)semidirect
546. Q8C2×D3×Dic5semidirect
552. D4C2×D3×Dic5semidirect
581. C5#78866semidirect
604. C3Dic5×AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
#41×Dic152
C2×C5×#9492
C2×#914352
C3×#655212
#117492422
#117492402
#117489962
#117489942
#117484532
#117483892
#117483282
Dic5×AGL(1,ℤ/8)3
#788665