Finite Group Databasebeta

← D3×#66034 #14495358

Subgroups — D3×#66034 #14495358

Order 1920.

2304 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756459C211yesmaximal, characteristic, complemented
2C5×D3×#125C211yesmaximal, characteristic, complemented
3C3×#66034C211yesmaximal, characteristic, complemented
4D3×#4042C214yesmaximal, complemented
5D3×#4041C214yesmaximal, complemented
6D3×#4040C214yesmaximal, complemented
7#11753820C214yesmaximal, complemented
8#11753648C214yesmaximal, complemented
9#11753639C214yesmaximal, complemented
10D3×#4021C212yesmaximal, complemented
11#11752638C212yesmaximal, complemented
12C2×#66034—31nomaximal, Hall
13C2×D3×D20C2212yes—
14C2×D3×D20C2214yes—
15C2×C4×D3×D5C2214yes—
16C2×D3×Dic10C2212yes—
17C3×C5×#125C2211yescharacteristic, complemented
18#91785C2214yescomplemented
19#91784C2214yescomplemented
20#91783C2214yescomplemented
21#91764C2212yescomplemented
22C5×D3×C4⋊C4C2212yes—
23C5×D3×C4⋊C4C2214yes—
24C5×#890C2212yescomplemented
25C5×#889C2214yescomplemented
26C42×C5×D3C2211yescharacteristic
27C5×#870C2211yescharacteristic, complemented
28C3×#4042C2214yescomplemented
29C3×#4041C2214yescomplemented
30C3×#4040C2214yescomplemented
31C3×#4021C2212yescomplemented
32D3×#685C2214yes—
33D3×#685C2214yes—
34D3×#685C2214yes—
35#91465C2214yescomplemented
36#91459C2214yescomplemented
37#91459C2214yescomplemented
38#91450C2214yescomplemented
39#91433C2214yescomplemented
40#91431C2214yescomplemented
41#91430C2214yescomplemented
42#91419C2214yescomplemented
43#91413C2214yescomplemented
44#91411C2214yescomplemented
45#91409C2214yescomplemented
46#91409C2214yescomplemented
47#91407C2214yescomplemented
48D3×#683C2214yes—
49#91377C2214yescomplemented
50#91374C2212yescomplemented
51#91364C2212yescomplemented
52#91356C2214yescomplemented
53#91354C2214yescomplemented
54#91353C2214yescomplemented
55#91353C2214yescomplemented
56#91349C2214yescomplemented
57#91348C2214yescomplemented
58D3×#562—51nomaximal, Hall
59C2×C5×#125—31no—
60#66034—34no—
61#66034—34no—
62#66034—34no—
63#66034—32no—
64#66034—31no—
65#66034D311yescharacteristic, direct product factor
66C2×#4042—34no—
67C2×#4041—34no—
68C2×#4040—34no—
69C2×#4021—32no—
70C22×D3×D5C2314yes—
71C2×D60C2314yes—
72C2×D60C2312yes—
73C2×C4×D15C2314yes—
74C2×Dic30C2312yes—
75C2×C4×C5×D3C2314yes—
76C2×C4×C5×D3C2311yescharacteristic
77C2×C4×C5×D3C2312yes—
78C2×C3×D20C2314yes—
79C2×C3×D20C2312yes—
80C2×C3×C4×D5C2314yes—
81C2×C3×Dic10C2312yes—
82C2×#1204C2314yes—
83C2×#1202C2314yes—
84C2×#1202C2314yes—
85C2×#1202C2314yes—
86C2×#1200C2314yes—
87C2×D3×Dic5C2314yes—
88C2×D5×Dic3C2314yes—
89D3×D20—44no—
90D3×D20—48no—
91C4×D3×D5—48no—
92D3×Dic10—44no—
93C3×C5×C4⋊C4C2314yes—
94C3×C5×C4⋊C4C2312yes—
95C3×C42×C5C2311yescharacteristic
96#6714C2314yes—
97#6714C2314yes—
98#6714C2314yes—
99#6712C2314yes—
100C5×C4⋊Dic3C2312yes—
101C5×C4⋊Dic3C2314yes—
102C5×Dic3⋊C4C2314yes—
103C5×Dic3⋊C4C2314yes—
104C5×Dic3⋊C4C2314yes—
105C4×C5×Dic3C2312yes—
106C4×C5×Dic3C2311yescharacteristic
107C3×#685C2314yes—
108C3×#685C2314yes—
109C3×#685C2314yes—
110C3×#683C2314yes—
111#6669C2314yes—
112#6668C2314yes—
113#6667C2314yes—
114#6667C2314yes—
115#6667C2314yes—
116#6665C2314yes—
117#6665C2314yes—
118#6665C2314yes—
119C3×#562—51no—
120#5700—51no—
121D3×#125D511yescharacteristic, complemented
122#5684—52no—
123D3×#124—52no—
124#5667—54no—
125D3×#123—54no—
126#5647—54no—
127D3×C22⋊Q8—54no—
128#5621—54no—
129D3×#121—54no—
130C22×D20—32no—
131C22×D20—34no—
132C22×C4×D5—34no—
133C22×Dic10—32no—
134C5×#125—34no—
135C5×#125—32no—
136C5×#125—31no—
137C5×#125C2×D311yescharacteristic, complemented
138C2×C5×C4⋊C4—32no—
139C2×C5×C4⋊C4—34no—
140C2×C42×C5—31no—
141C2×#685—34no—
142C2×#685—34no—
143C2×#685—34no—
144C2×#683—34no—
145#4042—34no—
146#4042—34no—
147#4042—34no—
148#4042—34no—
149#4042—34no—
150#4042—34no—
151#4042—34no—
152#4042C2×D314yescomplemented
153#4041—34no—
154#4041—34no—
155#4041—34no—
156#4041—34no—
157#4041—34no—
158#4041—34no—
159#4041—34no—
160#4041C2×D314yescomplemented
161#4040—34no—
162#4040—34no—
163#4040—34no—
164#4040—34no—
165#4040—34no—
166#4040—34no—
167#4040—34no—
168#4040C2×D314yescomplemented
169#4021—32no—
170#4021—32no—
171#4021—32no—
172#4021—34no—
173#4021—34no—
174#4021C2×D312yescomplemented
175C2×#562—151noSylow
176C22×D15C2414yes—
177C22×C5×D3C2411yescharacteristic
178C22×C3×D5C2414yes—
179C2×D3×D5—28no—
180C2×D3×D5—28no—
181C2×D3×D5—28no—
182C2×C3×C4×C5C2414yes—
183C2×C3×C4×C5C2411yescharacteristic
184C2×C3×C4×C5C2412yes—
185C2×Dic15C2414yes—
186D60—42no—
187D60—44no—
188C4×D15—44no—
189Dic30—42no—
190C2×C5×Dic3C2411yescharacteristic
191C2×C5×Dic3C2412yes—
192C2×C5×Dic3C2414yes—
193C4×C5×D3—24no—
194C4×C5×D3—28no—
195C4×C5×D3—22no—
196C2×C3×Dic5C2414yes—
197C3×D20—42no—
198C3×D20—44no—
199C3×C4×D5—44no—
200C3×Dic10—42no—
201#1204—44no—
202#1202—44no—
203#1202—44no—
204#1202—44no—
205#1200—44no—
206D3×Dic5—28no—
207D5×Dic3—44no—
208C2×D3×Q8—52no—
209C2×D3×D4—54no—
210C2×D3×D4—52no—
211C22×C4×D3—54no—
212C3×#125C2×D511yescharacteristic, complemented
213C3×#124—52no—
214C3×#123—54no—
215C3×C22⋊Q8—54no—
216C3×#121—54no—
217#947—52no—
218#946—54no—
219#939—54no—
220#939—54no—
221#938—54no—
222#936—52no—
223#935—54no—
224#930—54no—
225#929—54no—
226#924—54no—
227#897—54no—
228#896—54no—
229#896—54no—
230#895—54no—
231#894—54no—
232#894—54no—
233D3×C4⋊C4—54no—
234D3×C4⋊C4C2×D512yes—
235D3×C4⋊C4C2×D514yes—
236#890C2×D512yescomplemented
237#889C2×D514yescomplemented
238#886—54no—
239#885—54no—
240#885—54no—
241#884—54no—
242#884—54no—
243#883—54no—
244#883—54no—
245D3×C22⋊C4—54no—
246D3×C22⋊C4—54no—
247D3×C22⋊C4—54no—
248#878—54no—
249#878—54no—
250#875—52no—
251C42×D3C2×D511yescharacteristic
252#870C2×D511yescharacteristic, complemented
253C23×D5—34no—
254C22×C4×C5—32no—
255C22×C4×C5—31no—
256C22×C4×C5—34no—
257C22×Dic5—34no—
258C2×D20—34no—
259C2×D20—34no—
260C2×D20—34no—
261C2×D20—32no—
262C2×D20—34no—
263C2×D20—124no—
264C2×D20—128no—
265C2×D20C22×D312yes—
266C2×D20C22×D314yes—
267C2×C4×D5—34no—
268C2×C4×D5—34no—
269C2×C4×D5—34no—
270C2×C4×D5—128no—
271C2×C4×D5C22×D314yes—
272C2×Dic10—32no—
273C2×Dic10—34no—
274C2×Dic10—124no—
275C2×Dic10C22×D312yes—
276C5×C4⋊C4—34no—
277C5×C4⋊C4—34no—
278C5×C4⋊C4—34no—
279C5×C4⋊C4—32no—
280C5×C4⋊C4—34no—
281C5×C4⋊C4C22×D312yes—
282C5×C4⋊C4C22×D314yes—
283C42×C5—32no—
284C42×C5—31no—
285C42×C5C22×D311yescharacteristic
286#685—34no—
287#685—34no—
288#685—34no—
289#685—34no—
290#685—34no—
291#685—34no—
292#685—34no—
293#685—34no—
294#685—34no—
295#685C22×D314yes—
296#685C22×D314yes—
297#685C22×D314yes—
298#683—34no—
299#683—34no—
300#683—34no—
301#683C22×D314yes—
302#562—51no—
303#562—154no—
304#562—154no—
305#562—154no—
306#562—152no—
307#562—151no—
308C2×#125—31no—
309C2×#124—152no—
310C2×#123—154no—
311C2×C22⋊Q8—154no—
312C2×#121—154no—
313C22×C3×C5C2511yescharacteristic
314C2×D15—24no—
315C2×D15—24no—
316C2×D15—24no—
317C2×C5×D3H(5,𝔽2)14yes—
318C2×C5×D32−1+412yes—
319C2×C3×D5—24no—
320C2×C3×D5—24no—
321C2×C3×D5—24no—
322D3×D5—416no—
323C3×C4×C5—22no—
324C3×C4×C5—24no—
325C3×C4×C5—21no—
326Dic15—24no—
327C3×Dic5—24no—
328C5×Dic3—24no—
329C5×Dic3—22no—
330C5×Dic3—21no—
331C23×D3—54no—
332C2×C3×Q8—52no—
333C2×C3×D4—54no—
334C2×C3×D4—52no—
335C22×C3×C4—54no—
336C2×#67—54no—
337C2×#67—54no—
338C2×#67—54no—
339C22×Dic3—54no—
340D3×Q8—204no—
341D3×D4—208no—
342D3×D4—204no—
343C2×D12—52no—
344C2×D12—54no—
345C2×C4×D3—54no—
346C2×C4×D3—54no—
347C2×C4×D3—54no—
348C2×C4×D3—208no—
349C2×C4×D3C22×D511yescharacteristic
350C2×C4×D3C22×D512yes—
351C2×C4×D3C22×D514yes—
352C2×Dic6—52no—
353C2×Dic6—54no—
354C3×C4⋊C4—54no—
355C3×C4⋊C4C22×D514yes—
356C3×C4⋊C4C22×D512yes—
357C3×C22⋊C4—54no—
358C3×C22⋊C4—54no—
359C3×C22⋊C4—54no—
360C3×C42C22×D511yescharacteristic
361#217—54no—
362#217—54no—
363#217—54no—
364(C2×D3)⋊C4—54no—
365(C2×D3)⋊C4—54no—
366(C2×D3)⋊C4—54no—
367(C2×D3)⋊C4—54no—
368(C2×D3)⋊C4—54no—
369(C2×D3)⋊C4—54no—
370C4⋊Dic3—54no—
371C4⋊Dic3C22×D512yes—
372C4⋊Dic3C22×D514yes—
373Dic3⋊C4—54no—
374Dic3⋊C4—54no—
375Dic3⋊C4C22×D514yes—
376Dic3⋊C4C22×D514yes—
377Dic3⋊C4C22×D514yes—
378C4×Dic3C22×D511yescharacteristic
379C4×Dic3C22×D512yes—
380C23×C5—31no—
381C22×D5—34no—
382C22×D5—68no—
383C22×D5—68no—
384C22×D5—68no—
385C22×D5C23×D314yes—
386C2×C4×C5—34no—
387C2×C4×C5—31no—
388C2×C4×C5—32no—
389C2×C4×C5—64no—
390C2×C4×C5—68no—
391C2×C4×C5—62no—
392C2×C4×C5C23×D314yes—
393C2×C4×C5C23×D311yescharacteristic
394C2×C4×C5C23×D312yes—
395C2×Dic5—34no—
396C2×Dic5—68no—
397C2×Dic5C23×D314yes—
398D20—42no—
399D20—44no—
400D20—122no—
401D20—124no—
402D20—124no—
403D20—124no—
404D20—124no—
405C4×D5—44no—
406C4×D5—124no—
407C4×D5—124no—
408C4×D5—124no—
409Dic10—42no—
410Dic10—124no—
411Dic10—122no—
412C22×Q8—152no—
413C22×D4—154no—
414C22×D4—152no—
415C23×C4—154no—
416#125—34no—
417#125—32no—
418#125—31no—
419#125D3×D511yescharacteristic, complemented
420#124—52no—
421#124—154no—
422#124—154no—
423#124—152no—
424#124—152no—
425#124—152no—
426#123—54no—
427#123—154no—
428#123—154no—
429#123—154no—
430#123—154no—
431#123—154no—
432#123—154no—
433#123—154no—
434C22⋊Q8—54no—
435C22⋊Q8—154no—
436C22⋊Q8—154no—
437C22⋊Q8—154no—
438C22⋊Q8—154no—
439C22⋊Q8—154no—
440C22⋊Q8—154no—
441C22⋊Q8—154no—
442#121—54no—
443#121—154no—
444#121—154no—
445#121—154no—
446#121—154no—
447#121—154no—
448#121—154no—
449#121—154no—
450C2×C4⋊C4—34no—
451C2×C4⋊C4—32no—
452C2×C4⋊C4—154no—
453C2×C22⋊C4—154no—
454C2×C22⋊C4—154no—
455C2×C22⋊C4—154no—
456C2×C42—31no—
457C2×C3×C5C2×H(5,𝔽2)12yes—
458C2×C3×C5C2×2−1+411yescharacteristic
459D15—44no—
460C3×D5—44no—
461C5×D3#56214yescomplemented
462C23×C3—54no—
463C22×D3—54no—
464C22×D3—108no—
465C22×D3—108no—
466C22×D3—108no—
467C22×D3C23×D511yescharacteristic
468C3×Q8—202no—
469C3×D4—202no—
470C3×D4—204no—
471C2×C3×C4—54no—
472C2×C3×C4—204no—
473C2×C3×C4C23×D512yes—
474C2×C3×C4C23×D511yescharacteristic
475C2×C3×C4C23×D514yes—
476#67—204no—
477#67—204no—
478#67—204no—
479C2×Dic3—54no—
480C2×Dic3—204no—
481C2×Dic3C23×D511yescharacteristic
482C2×Dic3C23×D512yes—
483C2×Dic3C23×D514yes—
484D12—202no—
485D12—204no—
486C4×D3—22no—
487C4×D3—28no—
488C4×D3—24no—
489C4×D3—108no—
490C4×D3—204no—
491C4×D3—204no—
492Dic6—204no—
493Dic6—202no—
494C22×C5—34no—
495C22×C5—32no—
496C22×C5C24×D311yescharacteristic
497C2×D5—24no—
498C2×D5—24no—
499C2×D5—24no—
500C2×D5—64no—
501C2×D5—64no—
502C2×D5—64no—
503C2×D5—1216no—
504C4×C5—22no—
505C4×C5—24no—
506C4×C5—21no—
507C4×C5—62no—
508C4×C5—64no—
509C4×C5—61no—
510Dic5—24no—
511Dic5—64no—
512C24—154no—
513C2×Q8—52no—
514C2×Q8—154no—
515C2×Q8—152no—
516C2×Q8—604no—
517C2×D4—54no—
518C2×D4—52no—
519C2×D4—154no—
520C2×D4—154no—
521C2×D4—154no—
522C2×D4—154no—
523C2×D4—152no—
524C2×D4—608no—
525C2×D4—604no—
526C22×C4—31no—
527C22×C4—32no—
528C22×C4—34no—
529C22×C4—54no—
530C22×C4—154no—
531C22×C4—154no—
532C22×C4—154no—
533C22×C4—154no—
534C22×C4—608no—
535C4⋊C4—34no—
536C4⋊C4—34no—
537C4⋊C4—34no—
538C4⋊C4—34no—
539C4⋊C4—32no—
540C4⋊C4—54no—
541C4⋊C4—154no—
542C4⋊C4—154no—
543C4⋊C4—154no—
544C4⋊C4C2×D3×D514yes—
545C4⋊C4C2×D3×D512yes—
546C22⋊C4—54no—
547C22⋊C4—54no—
548C22⋊C4—54no—
549C22⋊C4—154no—
550C22⋊C4—154no—
551C22⋊C4—154no—
552C22⋊C4—154no—
553C22⋊C4—154no—
554C22⋊C4—154no—
555C22⋊C4—154no—
556C22⋊C4—154no—
557C22⋊C4—154no—
558C42—32no—
559C42—31no—
560C42C2×D3×D511yescharacteristic
561C3×C5C2×#56211yescharacteristic, complemented, Hall
562C22×C3—104no—
563C22×C3—104no—
564C22×C3—104no—
565C22×C3C24×D511yescharacteristic
566C2×D3—104no—
567C2×D3—104no—
568C2×D3—104no—
569C2×D3—2016no—
570C2×D3#414712yes—
571C2×D3#414914yes—
572C3×C4—21no—
573C3×C4—22no—
574C3×C4—24no—
575C3×C4—104no—
576Dic3—21no—
577Dic3—22no—
578Dic3—24no—
579Dic3—104no—
580C2×C5—34no—
581C2×C5D3×H(5,𝔽2)12yes—
582C2×C5D3×2−1+411yescharacteristic
583D5—44no—
584D5—124no—
585C23—31no—
586C23—54no—
587C23—154no—
588C23—308no—
589C23—308no—
590C23—308no—
591Q8—202no—
592Q8—602no—
593Q8—604no—
594D4—204no—
595D4—202no—
596D4—604no—
597D4—602no—
598D4—604no—
599D4—604no—
600D4—604no—
601C2×C4—31no—
602C2×C4—32no—
603C2×C4—34no—
604C2×C4—54no—
605C2×C4—68no—
606C2×C4—64no—
607C2×C4—62no—
608C2×C4—154no—
609C2×C4—204no—
610C2×C4—308no—
611C2×C4—604no—
612C2×C4—604no—
613C2×C4—604no—
614C2×C4C22×D3×D511yescharacteristic
615C2×C4C22×D3×D512yes—
616C2×C4C22×D3×D514yes—
617C2×C3—204no—
618C2×C3C2×#414711yescharacteristic
619C2×C3C2×#414912yes—
620D3—204no—
621D3#6603414yesdirect product factor
622C5D3×#56211yescharacteristic, complemented, Sylow
623C22—32no—
624C22—34no—
625C22—104no—
626C22—104no—
627C22—104no—
628C22—304no—
629C22—304no—
630C22—304no—
631C22—6016no—
632C22C23×D3×D511yescharacteristic
633C4—21no—
634C4—24no—
635C4—22no—
636C4—61no—
637C4—64no—
638C4—62no—
639C4—104no—
640C4—304no—
641C3C2×#6603411yescharacteristic, complemented, Sylow
642C2—34no—
643C2—204no—
644C2—604no—
645C2D3×#414711yescharacteristic
646C2D3×#414912yes—
subgroupG/H
center, FrattiniC22C23×D3×D5
commutator, socle, cosocleC22×C3×C5C25
FittingC3×C5×#125C22
hypercenter#125D3×D5
hypocenterC3×C5C2×#562
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ #125

Complements

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

NH ≅ G/Ntype
1. #11756459C2semidirect
2. C5×D3×#125C2semidirect
3. C3×#66034C2semidirect
4. D3×#4042C2semidirect
5. D3×#4041C2semidirect
6. D3×#4040C2semidirect
7. #11753820C2semidirect
8. #11753648C2semidirect
9. #11753639C2semidirect
10. D3×#4021C2semidirect
11. #11752638C2semidirect
17. C3×C5×#125C22semidirect
18. #91785C22semidirect
19. #91784C22semidirect
20. #91783C22semidirect
21. #91764C22semidirect
24. C5×#890C22semidirect
25. C5×#889C22semidirect
27. C5×#870C22semidirect
28. C3×#4042C22semidirect
29. C3×#4041C22semidirect
30. C3×#4040C22semidirect
31. C3×#4021C22semidirect
35. #91465C22semidirect
36. #91459C22semidirect
37. #91459C22semidirect
38. #91450C22semidirect
39. #91433C22semidirect
40. #91431C22semidirect
41. #91430C22semidirect
42. #91419C22semidirect
43. #91413C22semidirect
44. #91411C22semidirect
45. #91409C22semidirect
46. #91409C22semidirect
47. #91407C22semidirect
49. #91377C22semidirect
50. #91374C22semidirect
51. #91364C22semidirect
52. #91356C22semidirect
53. #91354C22semidirect
54. #91353C22semidirect
55. #91353C22semidirect
56. #91349C22semidirect
57. #91348C22semidirect
65. #66034D3direct (N × H)
121. D3×#125D5semidirect
137. C5×#125C2×D3semidirect
152. #4042C2×D3semidirect
160. #4041C2×D3semidirect
168. #4040C2×D3semidirect
174. #4021C2×D3semidirect
212. C3×#125C2×D5semidirect
236. #890C2×D5semidirect
237. #889C2×D5semidirect
252. #870C2×D5semidirect
419. #125D3×D5semidirect
461. C5×D3#562semidirect
561. C3×C5C2×#562semidirect
621. D3#66034direct (N × H)
622. C5D3×#562semidirect
641. C3C2×#66034semidirect

Maximal subgroups

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

subgroupindex
#117564592
C5×D3×#1252
C3×#660342
D3×#40422
D3×#40412
D3×#40402
#117538202
#117536482
#117536392
D3×#40212
#117526382
C2×#660343
D3×#5625