Finite Group Databasebeta

← #14416746

Subgroups — #14416746

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
2C3×#41×Dic5C211yesmaximal, characteristic, complemented
3C2×#91418C211yesmaximal, characteristic, complemented
4C5×#5232C211yesmaximal, characteristic, complemented
5#11749225C212yesmaximal, complemented
6#11749221C212yesmaximal, complemented
7#11748979C212yesmaximal, complemented
8#11748975C212yesmaximal, complemented
9#11748472C211yesmaximal, characteristic, complemented
10#11748392C211yesmaximal, characteristic, complemented
11Dic5×#208C211yesmaximal, characteristic, complemented
12#10610597—31nomaximal, Hall
13C2×C3×C5×#41C2211yescharacteristic
14C22×D60C2211yescharacteristic
15C22×#1200—21no—
16#91838C418yescomplemented
17C2×#6717C2212yes—
18C2×#6715C2212yes—
19C2×#6710C2211yescharacteristic
20C2×C5×C4⋊Dic3C2211yescharacteristic
21C2×C5×#208C2211yescharacteristic
22C3×Q8×Dic5C2211yescharacteristic, complemented
23C3×Q8×Dic5C2211yescharacteristic, complemented
24C3×D4×Dic5—21no—
25C3×D4×Dic5—21no—
26C3×D4×Dic5C2211yescharacteristic, complemented
27C3×D4×Dic5C2211yescharacteristic, complemented
28C3×#4072—21no—
29C3×#4072C2211yescharacteristic
30C2×C3×C4×Dic5—21no—
31C2×C3×C4×Dic5C2211yescharacteristic
32C2×#6667—21no—
33#91418—21no—
34#91418—21no—
35#91418C2212yes—
36#91418C2212yes—
37C5×#835C2212yescomplemented
38C5×#832C2212yescomplemented
39#90989C2212yescomplemented
40#90986C2212yescomplemented
41#90968C2212yes—
42#90968C2212yes—
43#90952C2212yescomplemented
44Dic5×C3⋊C8C2212yescomplemented
45C4×#949—51nomaximal, Hall
46#41×Dic5D311yescharacteristic, complemented
47C2×#4095—31no—
48C2×#4039—31no—
49C5×#417—31no—
50#65420—31no—
51M16×Dic5—31no—
52#65253—31no—
53#65106—32no—
54#65077—32no—
55#65074—32no—
56#65045—32no—
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×C3×Dic5—21no—
71C22×C3×Dic5—21no—
72C22×C3×Dic5D411yescharacteristic
73C22×C3×Dic5D411yescharacteristic
74C2×#1200—22no—
75C2×#1200—22no—
76C2×#1200—44no—
77#6717C2×C418yescomplemented
78#6715C2×C418yescomplemented
79#6710C2×C414yescomplemented
80C2×C3⋊#168C2312yes—
81C5×C4⋊Dic3—21no—
82C5×C4⋊Dic3C2312yes—
83C5×#208—22no—
84C2×C5×C3⋊C8C2312yes—
85C3×C22⋊Dic5—21no—
86C3×C22⋊Dic5—21no—
87C3×C22⋊Dic5—21no—
88C3×C4⋊Dic5—21no—
89C3×C4⋊Dic5—21no—
90C3×C4⋊Dic5C2311yescharacteristic
91C3×C4⋊Dic5C2311yescharacteristic
92C3×C4×Dic5—21no—
93C3×C4×Dic5—21no—
94C3×C4×Dic5—21no—
95C3×C4×Dic5D411yescharacteristic
96C3×C4×Dic5D411yescharacteristic
97C3×C4×Dic5C2311yescharacteristic
98C3×C4×Dic5C2311yescharacteristic
99#6667—22no—
100#6667—22no—
101C3×C4×#41—51no—
102C2×#949—51no—
103C2×C4×D12—51no—
104#5232D511yescharacteristic, complemented
105#5026—52no—
106C4×#215—52no—
107#5014—52no—
108C4×#213—52no—
109#5001—51no—
110#4995—51no—
111C4×#208—51no—
112C2×C5×#41C2×D311yescharacteristic
113C22×D20—31no—
114C22×C4×D5—61no—
115C2×C5×M16—31no—
116C2×C5×C4⋊C4—31no—
117#4095—38no—
118Q8×Dic5C2×D311yescharacteristic, complemented
119Q8×Dic5C2×D311yescharacteristic, complemented
120C2×#688—32no—
121D4×Dic5—21no—
122D4×Dic5—21no—
123D4×Dic5C2×D311yescharacteristic, complemented
124D4×Dic5C2×D311yescharacteristic, complemented
125C2×#686—32no—
126C2×#685—61no—
127#4072—21no—
128#4072C2×D311yescharacteristic
129C2×C4×Dic5—21no—
130C2×C4×Dic5C2×D311yescharacteristic
131C2×#681—31no—
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#3940—32no—
143#3939—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—42no—
167C2×C3×Dic5—22no—
168C2×C3×Dic5—21no—
169C2×C3×Dic5—21no—
170C2×C3×Dic5—41no—
171C2×C3×Dic5—42no—
172C2×C3×Dic5—42no—
173C2×C3×Dic5C2×D411yescharacteristic
174C2×C3×Dic5C2×D411yescharacteristic
175C2×C3×Dic5C2×D411yescharacteristic
176C2×C3×Dic5C2×D411yescharacteristic
177#1200—44no—
178#1200—44no—
179C3⋊#168C22×C414yescomplemented
180C5×C3⋊C8—22no—
181C2×C3×#41C2×D511yescharacteristic
182C22×D12—51no—
183C22×C4×D3—101no—
184C3×C4×Q8—51no—
185C3×C4×Q8—51no—
186C3×C4×D4—51no—
187C3×C4×D4—51no—
188C3×C4×D4—101no—
189C3×C4×D4—101no—
190C3×#117—51no—
191C3×#117—101no—
192C2×C3×C42—51no—
193C2×C3×C42—101no—
194#949—58no—
195C2×#215—52no—
196C2×#213—52no—
197C2×(C2×D3)⋊C4—101no—
198C2×C4⋊Dic3C2×D511yescharacteristic
199C2×#208—51no—
200C2×#208C2×D511yescharacteristic
201C4×D12—52no—
202C4×D12—52no—
203C4×D12—101no—
204C4×D12—101no—
205#835C2×D512yescomplemented
206#832C2×D512yescomplemented
207#809—52no—
208#809—52no—
209#808—52no—
210#807—52no—
211#803—52no—
212C4×C3⋊C8—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—61no—
221C22×C4×C5C22×D311yescharacteristic
222C22×Dic5—21no—
223C22×Dic5—21no—
224C22×Dic5#6711yescharacteristic
225C22×Dic5#6711yescharacteristic
226C2×D20—34no—
227C2×D20—32no—
228C2×D20—62no—
229C2×D20—61no—
230C2×C4×D5—62no—
231C2×C4×D5—62no—
232C2×C4×D5—124no—
233C5×M16—62no—
234C2×C5×C8—32no—
235C5×C4⋊C4—32no—
236C5×C4⋊C4—61no—
237C22⋊Dic5—21no—
238C22⋊Dic5—21no—
239C22⋊Dic5—21no—
240#688—38no—
241#686—38no—
242#685—62no—
243#685—62no—
244C4⋊Dic5—21no—
245C4⋊Dic5—21no—
246C4⋊Dic5C22×D311yescharacteristic
247C4⋊Dic5C22×D311yescharacteristic
248C4×Dic5—21no—
249C4×Dic5—21no—
250C4×Dic5—21no—
251C4×Dic5#6711yescharacteristic
252C4×Dic5#6711yescharacteristic
253C4×Dic5C22×D311yescharacteristic
254C4×Dic5C22×D311yescharacteristic
255#681—34no—
256C2×#168—32no—
257C2×AGL(1,ℤ/8)—151no—
258C4×#41—51no—
259C2×C4×D4—151no—
260#442—152no—
261#440—152no—
262C4×SD16—152no—
263C4×D8—152no—
264#428—151no—
265#418—151no—
266#417—31no—
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—21no—
284C3×Dic5—21no—
285C3×Dic5—21no—
286C3×Dic5—41no—
287C3×Dic5—41no—
288C3×Dic5AGL(1,ℤ/8)12yescomplemented
289C5×Dic3—42no—
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—51no—
298C22×C3×C4—51no—
299C22×C3×C4—101no—
300C22×C3×C4—101no—
301C22×C3×C4C22×D511yescharacteristic
302C22×Dic3—21no—
303C2×D12—54no—
304C2×D12—52no—
305C2×D12—101no—
306C2×D12—102no—
307C2×C4×D3—102no—
308C2×C4×D3—102no—
309C2×C4×D3—204no—
310C3×C4⋊C4—51no—
311C3×C4⋊C4—51no—
312C3×C4⋊C4—101no—
313C3×C4⋊C4—101no—
314C3×C22⋊C4—101no—
315C3×C22⋊C4—101no—
316C3×C22⋊C4—101no—
317C3×C42—51no—
318C3×C42—51no—
319C3×C42—51no—
320C3×C42—51no—
321C3×C42—101no—
322C3×C42—101no—
323C3×C42—101no—
324#215—58no—
325#213—58no—
326(C2×D3)⋊C4—102no—
327(C2×D3)⋊C4—102no—
328C4⋊Dic3—21no—
329C4⋊Dic3C22×D512yes—
330#208—22no—
331#208—54no—
332C2×C3⋊C8—52no—
333C2×C3⋊C8C22×D512yes—
334C23×C5—21no—
335C23×C5C2×#6711yescharacteristic
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—62no—
348C2×C4×C5—122no—
349C2×C4×C5C2×C4×D312yes—
350C2×C4×C5#23512yes—
351C2×C4×C5C2×#6711yescharacteristic
352C2×C4×C5C23×D311yescharacteristic
353C2×Dic5—22no—
354C2×Dic5—21no—
355C2×Dic5—21no—
356C2×Dic5—41no—
357C2×Dic5—42no—
358C2×Dic5—42no—
359C2×Dic5C2×#6711yescharacteristic
360C2×Dic5C2×#6711yescharacteristic
361C2×Dic5C2×#6711yescharacteristic
362C2×Dic5C2×#6711yescharacteristic
363D20—34no—
364D20—64no—
365D20—62no—
366C4×D5—124no—
367C4×D5—124no—
368C5×C8—62no—
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—31no—
377C2×M16—151no—
378C4×Q8—51no—
379C4×Q8—51no—
380C4×D4—51no—
381C4×D4—51no—
382C4×D4—101no—
383C4×D4—101no—
384C4×D4—152no—
385C4×D4—152no—
386C4×D4—301no—
387C4×D4—301no—
388#117—51no—
389#117—101no—
390C2×C4⋊C4—31no—
391C2×C22⋊C4—301no—
392C2×C42—51no—
393C2×C42—101no—
394#107—152no—
395#106—152no—
396#103—32no—
397#102—32no—
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×C3D4×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—51no—
422C2×C3×C4—51no—
423C2×C3×C4—51no—
424C2×C3×C4—51no—
425C2×C3×C4—102no—
426C2×C3×C4—101no—
427C2×C3×C4—101no—
428C2×C3×C4—202no—
429C2×C3×C4—201no—
430C2×C3×C4—202no—
431C2×C3×C4C2×C4×D512yes—
432C2×C3×C4D4×D511yescharacteristic
433C2×C3×C4#71112yes—
434C2×C3×C4C23×D511yescharacteristic
435C2×Dic3—22no—
436C2×Dic3—42no—
437D12—54no—
438D12—102no—
439D12—104no—
440C4×D3—204no—
441C4×D3—204no—
442C3⋊C8—22no—
443C3⋊C8—54no—
444C22×C5—21no—
445C22×C5—21no—
446C22×C5—22no—
447C22×C5—41no—
448C22×C5—41no—
449C22×C5C4×#6712yes—
450C22×C5C22×#6711yescharacteristic
451C2×D5—64no—
452C2×D5—128no—
453C2×D5—122no—
454C2×D5—122no—
455C4×C5—22no—
456C4×C5—122no—
457C4×C5C4×#6712yes—
458C4×C5C22×C4×D311yescharacteristic
459C4×C5C2×#23511yescharacteristic
460Dic5—21no—
461Dic5—21no—
462Dic5—21no—
463Dic5—41no—
464Dic5—41no—
465Dic5#94912yescomplemented
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—51no—
478C22×C4—51no—
479C22×C4—61no—
480C22×C4—101no—
481C22×C4—101no—
482C22×C4—302no—
483C22×C4—302no—
484C22×C4—604no—
485C22×C4C2×D3×D511yescharacteristic
486SD16—158no—
487D8—158no—
488M16—62no—
489M16—154no—
490C2×C8—32no—
491C2×C8—152no—
492C4⋊C4—32no—
493C4⋊C4—51no—
494C4⋊C4—51no—
495C4⋊C4—61no—
496C4⋊C4—101no—
497C4⋊C4—101no—
498C22⋊C4—101no—
499C22⋊C4—101no—
500C22⋊C4—101no—
501C22⋊C4—302no—
502C22⋊C4—302no—
503C42—51no—
504C42—51no—
505C42—51no—
506C42—51no—
507C42—101no—
508C42—101no—
509C42—101no—
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×C3#402712yes—
517C22×C3C2×D4×D511yescharacteristic
518C2×D3—104no—
519C2×D3—208no—
520C2×D3—202no—
521C2×D3—202no—
522C3×C4—22no—
523C3×C4—52no—
524C3×C4—101no—
525C3×C4—101no—
526C3×C4—101no—
527C3×C4—201no—
528C3×C4—201no—
529C3×C4#402712yes—
530C3×C4C22×C4×D511yescharacteristic
531C3×C4C2×#71111yescharacteristic
532Dic3—42no—
533C2×C5—22no—
534C2×C5—42no—
535C2×C5C2×C4×#6711yescharacteristic
536C2×C5C2×#94911yescharacteristic
537C2×C5#582211yescharacteristic
538D5—124no—
539C23—21no—
540C23—302no—
541C23—304no—
542C23—602no—
543C23—602no—
544C23D5×#6711yescharacteristic
545Q8—21no—
546Q8C2×#120012yescomplemented
547D4—24no—
548D4—21no—
549D4—154no—
550D4—302no—
551D4—304no—
552D4C2×#120012yescomplemented
553C2×C4—21no—
554C2×C4—24no—
555C2×C4—51no—
556C2×C4—51no—
557C2×C4—51no—
558C2×C4—51no—
559C2×C4—62no—
560C2×C4—102no—
561C2×C4—101no—
562C2×C4—101no—
563C2×C4—122no—
564C2×C4—202no—
565C2×C4—201no—
566C2×C4—202no—
567C2×C4—604no—
568C2×C4—604no—
569C2×C4#678212yes—
570C2×C4C2×#120012yes—
571C2×C4D5×#6711yescharacteristic
572C2×C4C22×D3×D511yescharacteristic
573C8—62no—
574C8—154no—
575C2×C3—22no—
576C2×C3—42no—
577C2×C3C2×#402711yescharacteristic
578C2×C3D5×AGL(1,ℤ/8)11yescharacteristic
579C2×C3#6611111yescharacteristic
580D3—204no—
581C5C4×#94911yescharacteristic, complemented, Sylow
582C22—21no—
583C22—21no—
584C22—22no—
585C22—41no—
586C22—41no—
587C22—304no—
588C22—608no—
589C22—602no—
590C22—602no—
591C22#9155212yes—
592C22C2×D5×#6711yescharacteristic
593C4—22no—
594C4—52no—
595C4—101no—
596C4—101no—
597C4—101no—
598C4—122no—
599C4—201no—
600C4—201no—
601C4#9155212yes—
602C4C2×#678211yescharacteristic
603C4C22×#120011yescharacteristic
604C3#1061059711yescharacteristic, complemented, Sylow
605C2—22no—
606C2—42no—
607C2—604no—
608C2D5×#94911yescharacteristic
609C2#1175534411yescharacteristic
610C2C2×#9155211yescharacteristic
subgroupG/H
centerC22C2×D5×#67
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. C3×#41×Dic5C2semidirect
3. C2×#91418C2semidirect
4. C5×#5232C2semidirect
5. #11749225C2semidirect
6. #11749221C2semidirect
7. #11748979C2semidirect
8. #11748975C2semidirect
9. #11748472C2semidirect
10. #11748392C2semidirect
11. Dic5×#208C2semidirect
16. #91838C4semidirect
22. C3×Q8×Dic5C22semidirect
23. C3×Q8×Dic5C22semidirect
26. C3×D4×Dic5C22semidirect
27. C3×D4×Dic5C22semidirect
37. C5×#835C22semidirect
38. C5×#832C22semidirect
39. #90989C22semidirect
40. #90986C22semidirect
43. #90952C22semidirect
44. Dic5×C3⋊C8C22semidirect
46. #41×Dic5D3semidirect
59. C3×C5×#41C2×C4semidirect
67. C2×D60C2×C4semidirect
77. #6717C2×C4semidirect
78. #6715C2×C4semidirect
79. #6710C2×C4semidirect
104. #5232D5semidirect
118. Q8×Dic5C2×D3semidirect
119. Q8×Dic5C2×D3semidirect
123. D4×Dic5C2×D3semidirect
124. D4×Dic5C2×D3semidirect
152. C3×C5×Q8C22×C4semidirect
155. C3×C5×D4C22×C4semidirect
179. C3⋊#168C22×C4semidirect
205. #835C2×D5semidirect
206. #832C2×D5semidirect
215. C5×#41C4×D3semidirect
288. C3×Dic5AGL(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
465. Dic5#949semidirect
468. #41#1200semidirect
510. C3×C5C4×AGL(1,ℤ/8)semidirect
546. Q8C2×#1200semidirect
552. D4C2×#1200semidirect
581. C5C4×#949semidirect
604. C3#10610597semidirect

Maximal subgroups

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

subgroupindex
C2×#918382
C3×#41×Dic52
C2×#914182
C5×#52322
#117492252
#117492212
#117489792
#117489752
#117484722
#117483922
Dic5×#2082
#106105973
C4×#9495