Finite Group Databasebeta

← #14416744

Subgroups — #14416744

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×#91838C211yesmaximal, characteristic
2C5×Dic3×#41C211yesmaximal, characteristic, complemented
3C2×#91428C211yesmaximal, characteristic, complemented
4C3×#65521C211yesmaximal, characteristic, complemented
5#11749235C212yesmaximal, complemented
6#11749232C212yesmaximal, complemented
7#11748989C212yesmaximal, complemented
8#11748986C212yesmaximal, complemented
9#11748470C211yesmaximal, characteristic, complemented
10#11748396C211yesmaximal, characteristic, complemented
11Dic3×#681C211yesmaximal, characteristic, complemented
12C4×#4095—31nomaximal, Hall
13C2×C3×C5×#41C2211yescharacteristic
14C22×D60C2211yescharacteristic
15C22×#1200—21no—
16#91838C418yescomplemented
17C2×#6717C2212yes—
18C2×#6715C2212yes—
19C2×#6710C2211yescharacteristic
20C5×Q8×Dic3C2211yescharacteristic, complemented
21C5×Q8×Dic3C2211yescharacteristic, complemented
22C5×D4×Dic3—21no—
23C5×D4×Dic3—21no—
24C5×D4×Dic3C2211yescharacteristic, complemented
25C5×D4×Dic3C2211yescharacteristic, complemented
26C5×#926—21no—
27C5×#926C2211yescharacteristic
28C2×C4×C5×Dic3—21no—
29C2×C4×C5×Dic3C2211yescharacteristic
30C2×C3×C4⋊Dic5C2211yescharacteristic
31C2×C3×#681C2211yescharacteristic
32C2×#6667—21no—
33#91428—21no—
34#91428—21no—
35#91428C2212yes—
36#91428C2212yes—
37C3×#3967C2212yescomplemented
38C3×#3964C2212yescomplemented
39#90989C2212yescomplemented
40#90986C2212yescomplemented
41#90969C2212yes—
42#90969C2212yes—
43#90953C2212yescomplemented
44Dic3×#168C2212yescomplemented
45#78866—51nomaximal, Hall
46C4×C5×#41—31no—
47C2×#4095—31no—
48C2×C4×D20—31no—
49#65521D311yescharacteristic, complemented
50#65315—32no—
51C4×#688—32no—
52#65303—32no—
53C4×#686—32no—
54#65290—31no—
55#65284—31no—
56C4×#681—31no—
57C23×D15—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
65C2×D60—21no—
66C2×D60—22no—
67C2×D60C2×C414yescomplemented
68C2×D60C2312yes—
69C22×C5×Dic3—21no—
70C22×C5×Dic3—21no—
71C22×C5×Dic3D411yescharacteristic
72C22×C5×Dic3D411yescharacteristic
73C22×C3×Dic5—21no—
74C2×#1200—22no—
75C2×#1200—22no—
76C2×#1200—44no—
77#6717C2×C418yescomplemented
78#6715C2×C418yescomplemented
79#6710C2×C414yescomplemented
80C2×C3⋊#168C2312yes—
81C5×#217—21no—
82C5×#217—21no—
83C5×#217—21no—
84C5×C4⋊Dic3—21no—
85C5×C4⋊Dic3—21no—
86C5×C4⋊Dic3C2311yescharacteristic
87C5×C4⋊Dic3C2311yescharacteristic
88C4×C5×Dic3—21no—
89C4×C5×Dic3—21no—
90C4×C5×Dic3—21no—
91C4×C5×Dic3D411yescharacteristic
92C4×C5×Dic3D411yescharacteristic
93C4×C5×Dic3C2311yescharacteristic
94C4×C5×Dic3C2311yescharacteristic
95C3×C4⋊Dic5—21no—
96C3×C4⋊Dic5C2312yes—
97C3×#681—22no—
98C2×C3×#168C2312yes—
99#6667—22no—
100#6667—22no—
101Dic3×#41D511yescharacteristic, complemented
102C2×#949—51no—
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×D20—31no—
114C22×C4×D5—61no—
115C4×C5×Q8—31no—
116C4×C5×Q8—31no—
117C4×C5×D4—31no—
118C4×C5×D4—31no—
119C4×C5×D4—61no—
120C4×C5×D4—61no—
121C5×#117—31no—
122C5×#117—61no—
123C2×C42×C5—31no—
124C2×C42×C5—61no—
125#4095—38no—
126C2×#688—32no—
127C2×#686—32no—
128C2×#685—61no—
129C2×C4⋊Dic5C2×D311yescharacteristic
130C2×#681—31no—
131C2×#681C2×D311yescharacteristic
132C4×D20—32no—
133C4×D20—32no—
134C4×D20—61no—
135C4×D20—61no—
136#3967C2×D312yescomplemented
137#3964C2×D312yescomplemented
138#3941—32no—
139#3941—32no—
140#3940—32no—
141#3939—32no—
142#3935—32no—
143C4×#168—32no—
144C4×AGL(1,ℤ/8)—151noSylow
145C23×C3×C5—21no—
146C23×C3×C5C2×D411yescharacteristic
147C22×D15—24no—
148C22×D15—22no—
149C22×D15—42no—
150C22×D15—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
162D60—24no—
163D60—22no—
164D60C22×C414yes—
165C2×C5×Dic3—22no—
166C2×C5×Dic3—21no—
167C2×C5×Dic3—21no—
168C2×C5×Dic3—41no—
169C2×C5×Dic3—42no—
170C2×C5×Dic3—42no—
171C2×C5×Dic3C2×D411yescharacteristic
172C2×C5×Dic3C2×D411yescharacteristic
173C2×C5×Dic3C2×D411yescharacteristic
174C2×C5×Dic3C2×D411yescharacteristic
175C2×C3×Dic5—22no—
176C2×C3×Dic5—42no—
177#1200—44no—
178#1200—44no—
179C3⋊#168C22×C414yescomplemented
180C3×#168—22no—
181C2×C3×#41C2×D511yescharacteristic
182C22×D12—51no—
183C22×C4×D3—101no—
184C2×C3×M16—51no—
185C2×C3×C4⋊C4—51no—
186#949—58no—
187Q8×Dic3C2×D511yescharacteristic, complemented
188Q8×Dic3C2×D511yescharacteristic, complemented
189C2×#215—52no—
190D4×Dic3—21no—
191D4×Dic3—21no—
192D4×Dic3C2×D511yescharacteristic, complemented
193D4×Dic3C2×D511yescharacteristic, complemented
194C2×#213—52no—
195C2×(C2×D3)⋊C4—101no—
196#926—21no—
197#926C2×D511yescharacteristic
198C2×C4×Dic3—21no—
199C2×C4×Dic3C2×D511yescharacteristic
200C2×#208—51no—
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—
213C23×D5—61no—
214C5×#41—22no—
215C5×#41C4×D314yescomplemented
216C2×C5×Q8C22×D311yescharacteristic
217C2×C5×D4—21no—
218C2×C5×D4C22×D311yescharacteristic
219C22×C4×C5—21no—
220C22×C4×C5—31no—
221C22×C4×C5—31no—
222C22×C4×C5—61no—
223C22×C4×C5—61no—
224C22×C4×C5C22×D311yescharacteristic
225C22×Dic5—21no—
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×C4⋊C4—31no—
234C5×C4⋊C4—31no—
235C5×C4⋊C4—61no—
236C5×C4⋊C4—61no—
237C5×C22⋊C4—61no—
238C5×C22⋊C4—61no—
239C5×C22⋊C4—61no—
240C42×C5—31no—
241C42×C5—31no—
242C42×C5—31no—
243C42×C5—31no—
244C42×C5—61no—
245C42×C5—61no—
246C42×C5—61no—
247#688—38no—
248#686—38no—
249#685—62no—
250#685—62no—
251C4⋊Dic5—21no—
252C4⋊Dic5C22×D312yes—
253#681—22no—
254#681—34no—
255C2×#168—32no—
256C2×#168C22×D312yes—
257C2×AGL(1,ℤ/8)—151no—
258C4×#41—31no—
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×D15—24no—
276C2×D15—48no—
277C2×D15—42no—
278C2×D15—42no—
279C3×C4×C5—22no—
280C3×C4×C5C4×D412yes—
281C3×C4×C5C23×C411yescharacteristic
282C3×C4×C5C2×#4111yescharacteristic
283C3×Dic5—42no—
284C5×Dic3—21no—
285C5×Dic3—21no—
286C5×Dic3—21no—
287C5×Dic3—41no—
288C5×Dic3—41no—
289C5×Dic3AGL(1,ℤ/8)12yescomplemented
290C23×D3—101no—
291C3×#41—22no—
292C3×#41C4×D514yescomplemented
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—21no—
300C22×Dic3—21no—
301C22×Dic3C22⋊D511yescharacteristic
302C22×Dic3C22⋊D511yescharacteristic
303C2×D12—54no—
304C2×D12—52no—
305C2×D12—102no—
306C2×D12—101no—
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—21no—
315#217—21no—
316#217—21no—
317#215—58no—
318#213—58no—
319(C2×D3)⋊C4—102no—
320(C2×D3)⋊C4—102no—
321C4⋊Dic3—21no—
322C4⋊Dic3—21no—
323C4⋊Dic3C22×D511yescharacteristic
324C4⋊Dic3C22×D511yescharacteristic
325C4×Dic3—21no—
326C4×Dic3—21no—
327C4×Dic3—21no—
328C4×Dic3C22⋊D511yescharacteristic
329C4×Dic3C22⋊D511yescharacteristic
330C4×Dic3C22×D511yescharacteristic
331C4×Dic3C22×D511yescharacteristic
332#208—54no—
333C2×C3⋊C8—52no—
334C23×C5—21no—
335C23×C5D3×D411yescharacteristic
336C22×D5—64no—
337C22×D5—62no—
338C22×D5—122no—
339C22×D5—122no—
340C5×Q8—21no—
341C5×Q8C2×C4×D312yescomplemented
342C5×D4—24no—
343C5×D4—21no—
344C5×D4C2×C4×D312yescomplemented
345C2×C4×C5—21no—
346C2×C4×C5—24no—
347C2×C4×C5—31no—
348C2×C4×C5—31no—
349C2×C4×C5—31no—
350C2×C4×C5—31no—
351C2×C4×C5—62no—
352C2×C4×C5—61no—
353C2×C4×C5—61no—
354C2×C4×C5—122no—
355C2×C4×C5—121no—
356C2×C4×C5—122no—
357C2×C4×C5C2×C4×D312yes—
358C2×C4×C5D3×D411yescharacteristic
359C2×C4×C5#23712yes—
360C2×C4×C5C23×D311yescharacteristic
361C2×Dic5—22no—
362C2×Dic5—42no—
363D20—34no—
364D20—64no—
365D20—62no—
366C4×D5—124no—
367C4×D5—124no—
368#168—22no—
369#168—34no—
370C2×#41D3×D511yescharacteristic
371C22×D4—151no—
372C23×C4—301no—
373AGL(1,ℤ/8)—158no—
374C2×SD16—152no—
375C2×D8—152no—
376C2×M16—51no—
377C2×M16—151no—
378C4×Q8—31no—
379C4×Q8—31no—
380C4×D4—31no—
381C4×D4—31no—
382C4×D4—61no—
383C4×D4—61no—
384C4×D4—152no—
385C4×D4—152no—
386C4×D4—301no—
387C4×D4—301no—
388#117—31no—
389#117—61no—
390C2×C4⋊C4—51no—
391C2×C22⋊C4—301no—
392C2×C42—31no—
393C2×C42—61no—
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
407D15—44no—
408C23×C3—21no—
409C23×C3C2×C22⋊D511yescharacteristic
410C22×D3—102no—
411C22×D3—104no—
412C22×D3—202no—
413C22×D3—202no—
414C3×Q8—21no—
415C3×Q8C2×C4×D512yescomplemented
416C3×D4—24no—
417C3×D4—21no—
418C3×D4C2×C4×D512yescomplemented
419C2×C3×C4—21no—
420C2×C3×C4—24no—
421C2×C3×C4—102no—
422C2×C3×C4—202no—
423C2×C3×C4C2×C4×D512yes—
424C2×C3×C4#70912yes—
425C2×C3×C4C2×C22⋊D511yescharacteristic
426C2×C3×C4C23×D511yescharacteristic
427C2×Dic3—22no—
428C2×Dic3—21no—
429C2×Dic3—21no—
430C2×Dic3—41no—
431C2×Dic3—42no—
432C2×Dic3—42no—
433C2×Dic3C2×C22⋊D511yescharacteristic
434C2×Dic3C2×C22⋊D511yescharacteristic
435C2×Dic3C2×C22⋊D511yescharacteristic
436C2×Dic3C2×C22⋊D511yescharacteristic
437D12—54no—
438D12—102no—
439D12—104no—
440C4×D3—204no—
441C4×D3—204no—
442C3×C8—102no—
443C3⋊C8—54no—
444C22×C5—21no—
445C22×C5—21no—
446C22×C5—22no—
447C22×C5—41no—
448C22×C5—41no—
449C22×C5#67⋊C412yes—
450C22×C5C2×D3×D411yescharacteristic
451C2×D5—64no—
452C2×D5—128no—
453C2×D5—122no—
454C2×D5—122no—
455C4×C5—22no—
456C4×C5—32no—
457C4×C5—61no—
458C4×C5—61no—
459C4×C5—61no—
460C4×C5—121no—
461C4×C5—121no—
462C4×C5#67⋊C412yes—
463C4×C5C22×C4×D311yescharacteristic
464C4×C5C2×#23711yescharacteristic
465Dic5—42no—
466C24—301no—
467#41—22no—
468#41#120014yescomplemented
469C2×Q8C2×D3×D511yescharacteristic
470C2×D4—21no—
471C2×D4—154no—
472C2×D4—152no—
473C2×D4—302no—
474C2×D4—301no—
475C2×D4C2×D3×D511yescharacteristic
476C22×C4—21no—
477C22×C4—31no—
478C22×C4—31no—
479C22×C4—61no—
480C22×C4—61no—
481C22×C4—101no—
482C22×C4—302no—
483C22×C4—302no—
484C22×C4—604no—
485C22×C4C2×D3×D511yescharacteristic
486SD16—158no—
487D8—158no—
488M16—102no—
489M16—154no—
490C2×C8—52no—
491C2×C8—152no—
492C4⋊C4—31no—
493C4⋊C4—31no—
494C4⋊C4—52no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—101no—
498C22⋊C4—61no—
499C22⋊C4—61no—
500C22⋊C4—61no—
501C22⋊C4—302no—
502C22⋊C4—302no—
503C42—31no—
504C42—31no—
505C42—31no—
506C42—31no—
507C42—61no—
508C42—61no—
509C42—61no—
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×C3C4×C22⋊D512yes—
517C22×C3C22×C22⋊D511yescharacteristic
518C2×D3—104no—
519C2×D3—208no—
520C2×D3—202no—
521C2×D3—202no—
522C3×C4—22no—
523C3×C4—202no—
524C3×C4C4×C22⋊D512yes—
525C3×C4C22×C4×D511yescharacteristic
526C3×C4C2×#70911yescharacteristic
527Dic3—21no—
528Dic3—21no—
529Dic3—21no—
530Dic3—41no—
531Dic3—41no—
532Dic3#409512yescomplemented
533C2×C5—22no—
534C2×C5—42no—
535C2×C5C2×#67⋊C411yescharacteristic
536C2×C5D3×AGL(1,ℤ/8)11yescharacteristic
537C2×C5#577711yescharacteristic
538D5—124no—
539C23—21no—
540C23—302no—
541C23—304no—
542C23—602no—
543C23—602no—
544C23D3×C22⋊D511yescharacteristic
545Q8—21no—
546Q8C2×#120012yescomplemented
547D4—24no—
548D4—21no—
549D4—154no—
550D4—302no—
551D4—304no—
552D4C2×#120012yescomplemented
553C2×C4—21no—
554C2×C4—24no—
555C2×C4—31no—
556C2×C4—31no—
557C2×C4—31no—
558C2×C4—31no—
559C2×C4—62no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—102no—
563C2×C4—122no—
564C2×C4—121no—
565C2×C4—122no—
566C2×C4—202no—
567C2×C4—604no—
568C2×C4—604no—
569C2×C4#677912yes—
570C2×C4C2×#120012yes—
571C2×C4D3×C22⋊D511yescharacteristic
572C2×C4C22×D3×D511yescharacteristic
573C8—102no—
574C8—154no—
575C2×C3—22no—
576C2×C3—42no—
577C2×C3C2×C4×C22⋊D511yescharacteristic
578C2×C3C2×#409511yescharacteristic
579C2×C3#6615611yescharacteristic
580D3—204no—
581C5#7886611yescharacteristic, complemented, Sylow
582C22—21no—
583C22—21no—
584C22—22no—
585C22—41no—
586C22—41no—
587C22—304no—
588C22—608no—
589C22—602no—
590C22—602no—
591C22#9155612yes—
592C22C2×D3×C22⋊D511yescharacteristic
593C4—22no—
594C4—32no—
595C4—61no—
596C4—61no—
597C4—61no—
598C4—121no—
599C4—121no—
600C4—202no—
601C4#9155612yes—
602C4C2×#677911yescharacteristic
603C4C22×#120011yescharacteristic
604C3C4×#409511yescharacteristic, complemented, Sylow
605C2—22no—
606C2—42no—
607C2—604no—
608C2D3×#409511yescharacteristic
609C2#1175534811yescharacteristic
610C2C2×#9155611yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊D5
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×#41C22
socleC22×C3×C5C22×D4
prime radicalC2×#91838C2
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
2. C5×Dic3×#41C2semidirect
3. C2×#91428C2semidirect
4. C3×#65521C2semidirect
5. #11749235C2semidirect
6. #11749232C2semidirect
7. #11748989C2semidirect
8. #11748986C2semidirect
9. #11748470C2semidirect
10. #11748396C2semidirect
11. Dic3×#681C2semidirect
16. #91838C4semidirect
20. C5×Q8×Dic3C22semidirect
21. C5×Q8×Dic3C22semidirect
24. C5×D4×Dic3C22semidirect
25. C5×D4×Dic3C22semidirect
37. C3×#3967C22semidirect
38. C3×#3964C22semidirect
39. #90989C22semidirect
40. #90986C22semidirect
43. #90953C22semidirect
44. Dic3×#168C22semidirect
49. #65521D3semidirect
59. C3×C5×#41C2×C4semidirect
67. C2×D60C2×C4semidirect
77. #6717C2×C4semidirect
78. #6715C2×C4semidirect
79. #6710C2×C4semidirect
101. Dic3×#41D5semidirect
136. #3967C2×D3semidirect
137. #3964C2×D3semidirect
152. C3×C5×Q8C22×C4semidirect
155. C3×C5×D4C22×C4semidirect
179. C3⋊#168C22×C4semidirect
187. Q8×Dic3C2×D5semidirect
188. Q8×Dic3C2×D5semidirect
192. D4×Dic3C2×D5semidirect
193. D4×Dic3C2×D5semidirect
215. C5×#41C4×D3semidirect
289. C5×Dic3AGL(1,ℤ/8)semidirect
292. C3×#41C4×D5semidirect
341. C5×Q8C2×C4×D3semidirect
344. C5×D4C2×C4×D3semidirect
415. C3×Q8C2×C4×D5semidirect
418. C3×D4C2×C4×D5semidirect
468. #41#1200semidirect
510. C3×C5C4×AGL(1,ℤ/8)semidirect
532. Dic3#4095semidirect
546. Q8C2×#1200semidirect
552. D4C2×#1200semidirect
581. C5#78866semidirect
604. C3C4×#4095semidirect

Maximal subgroups

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

subgroupindex
C2×#918382
C5×Dic3×#412
C2×#914282
C3×#655212
#117492352
#117492322
#117489892
#117489862
#117484702
#117483962
Dic3×#6812
C4×#40953
#788665