Finite Group Databasebeta

← Dic3×#4054 #14416755

Subgroups — Dic3×#4054 #14416755

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×C3×#4054C211yesmaximal, characteristic
2Dic3×#709C211yesmaximal, characteristic, complemented
3C2×Dic3×D20C211yesmaximal, characteristic, complemented
4#11751222C211yesmaximal, characteristic, complemented
5C5×Dic3×M16C211yesmaximal, characteristic, complemented
6#11748369C211yesmaximal, characteristic, complemented
7#11748367C211yesmaximal, characteristic, complemented
8#11747799C212yesmaximal, complemented
9#11747795C212yesmaximal, complemented
10Dic3×D40C212yesmaximal, complemented
11Dic3×#677C212yesmaximal, complemented
12C4×#4054—31nomaximal, Hall
13C2×C3×#709C2211yescharacteristic
14C22×C3×D20C2211yescharacteristic
15C22×D5×Dic3—21no—
16C2×C3×C5×M16C2211yescharacteristic
17#91815C2211yescharacteristic
18C2×C4⋊Dic15C2211yescharacteristic
19C2×C4×C5×Dic3C2211yescharacteristic
20C2×C5×#208C2211yescharacteristic
21C3×#4054C418yescomplemented
22C2×C3×D40C2212yes—
23C2×C3×#677C2212yes—
24Dic3×C22⋊D5—21no—
25Dic3×C22⋊D5—21no—
26C2×#6665—21no—
27Dic3×D20—21no—
28Dic3×D20—21no—
29Dic3×D20C2212yes—
30Dic3×D20C2212yes—
31Dic3×D20C2211yescharacteristic, complemented
32Dic3×D20C2211yescharacteristic, complemented
33C4×D5×Dic3—21no—
34#91357—21no—
35Dic3×Dic10C2211yescharacteristic, complemented
36Dic3×Dic10C2211yescharacteristic, complemented
37#91102C2212yescomplemented
38#91101C2212yescomplemented
39C5×#815C2212yescomplemented
40C5×C8×Dic3C2212yescomplemented
41#90971C2212yescomplemented
42#90965C2212yes—
43#90965C2212yes—
44#90965C2212yescomplemented
45Dic3×AGL(1,ℤ/8)—51nomaximal, Hall
46C2×#4054D311yescharacteristic
47C4×#709—31no—
48C2×C4×D20—31no—
49C4×C5×M16—31no—
50#65422—31no—
51#65420—31no—
52#65410—31no—
53#65000—32no—
54#64999—32no—
55C4×D40—32no—
56C4×#677—32no—
57C23×C3×D5—21no—
58C22×C3×C4×C5C2311yescharacteristic
59C22×Dic15—21no—
60C22×C5×Dic3D411yescharacteristic
61C22×C5×Dic3D411yescharacteristic
62C2×C3×C22⋊D5—21no—
63C3×#709—22no—
64C3×#709C2×C414yescomplemented
65C2×C3×D20—21no—
66C2×C3×D20—22no—
67C2×C3×D20C2×C414yescomplemented
68C2×C3×D20C2312yes—
69C2×C3×D20C2311yescharacteristic
70C2×C3×C4×D5—21no—
71C2×C3×Dic10C2311yescharacteristic
72C2×D5×Dic3—22no—
73C2×D5×Dic3—22no—
74C2×D5×Dic3—21no—
75C2×D5×Dic3—21no—
76C2×D5×Dic3—44no—
77C3×C5×M16C2×C414yescomplemented
78C2×C3×C5×C8C2312yes—
79#6719—21no—
80C4⋊Dic15—21no—
81C4⋊Dic15C2312yes—
82C4⋊Dic15C2311yescharacteristic
83C4⋊Dic15C2311yescharacteristic
84C4×Dic15—21no—
85C4×C5×Dic3D411yescharacteristic
86C4×C5×Dic3D411yescharacteristic
87C4×C5×Dic3C2311yescharacteristic
88C4×C5×Dic3C2311yescharacteristic
89C5×#208—22no—
90C2×C5×C3⋊C8C2312yes—
91C3×D40C2×C418yescomplemented
92C3×#677C2×C418yescomplemented
93#6668—21no—
94#6668—21no—
95#6665—22no—
96#6665—22no—
97#6665—21no—
98#6665—21no—
99Dic3×Dic5—21no—
100Dic3×Dic5—21no—
101C2×C3×AGL(1,ℤ/8)—51no—
102Dic3×#41—51no—
103C2×D4×Dic3—51no—
104#5232—51no—
105#5215—51no—
106#5164—52no—
107Dic3×SD16—52no—
108#5152—52no—
109Dic3×D8—52no—
110#5121—51no—
111Dic3×M16D511yescharacteristic, complemented
112C2×#709C2×D311yescharacteristic
113C22×D20C2×D311yescharacteristic
114C22×C4×D5—61no—
115C2×C5×M16—31no—
116C2×C5×M16C2×D311yescharacteristic
117C2×C42×C5—31no—
118C4×C22⋊D5—61no—
119C4×C22⋊D5—61no—
120C2×#685—61no—
121#4072—31no—
122C2×C4⋊Dic5—31no—
123#4054Dic318yesdirect product factor
124C2×D40C2×D312yes—
125C2×#677C2×D312yes—
126C4×D20—32no—
127C4×D20—32no—
128C4×D20—31no—
129C4×D20—31no—
130C4×D20—61no—
131C4×D20—61no—
132#4018—61no—
133C42×D5—61no—
134C4×Dic10—31no—
135C4×Dic10—31no—
136C5×#97—32no—
137C4×C5×C8—32no—
138#3953—32no—
139#3953—32no—
140#3953—32no—
141#3950—32no—
142#3949—32no—
143#3948—32no—
144C4×AGL(1,ℤ/8)—151noSylow
145C23×C3×C5C2×D411yescharacteristic
146C22×C3×D5—24no—
147C22×C3×D5—21no—
148C22×C3×D5—22no—
149C22×C3×D5—42no—
150C22×C3×D5—42no—
151C2×C3×C4×C5C22×C412yes—
152C2×C3×C4×C5C2×D411yescharacteristic
153C2×C3×C4×C5#4112yes—
154C2×C3×C4×C5C2411yescharacteristic
155C2×Dic15—21no—
156C2×Dic15—21no—
157C2×Dic15—22no—
158C2×Dic15—42no—
159C2×C5×Dic3—22no—
160C2×C5×Dic3—41no—
161C2×C5×Dic3C2×D411yescharacteristic
162C2×C5×Dic3C2×D411yescharacteristic
163C2×C5×Dic3C2×D411yescharacteristic
164C2×C5×Dic3C2×D411yescharacteristic
165C3×C22⋊D5—24no—
166C2×C3×Dic5—21no—
167C3×D20—24no—
168C3×D20—22no—
169C3×D20—21no—
170C3×D20C22×C414yes—
171C3×D20C22×C412yescomplemented
172C3×C4×D5—24no—
173C3×Dic10—21no—
174C3×Dic10C22×C412yescomplemented
175D5×Dic3—42no—
176D5×Dic3—44no—
177D5×Dic3—42no—
178D5×Dic3—44no—
179C3×C5×C8C22×C414yescomplemented
180C5×C3⋊C8—22no—
181C2×C3×#41—51no—
182C22×C3×D4—51no—
183C23×Dic3—101no—
184C3×AGL(1,ℤ/8)—58no—
185C2×C3×SD16—52no—
186C2×C3×D8—52no—
187C2×C3×M16C2×D511yescharacteristic
188C2×#217—101no—
189Q8×Dic3—51no—
190Q8×Dic3—51no—
191D4×Dic3—51no—
192D4×Dic3—52no—
193D4×Dic3—51no—
194D4×Dic3—52no—
195D4×Dic3—101no—
196D4×Dic3—101no—
197D4×Dic3—101no—
198D4×Dic3—101no—
199#926—51no—
200#926—101no—
201C2×C4⋊Dic3—51no—
202C2×C4×Dic3—101no—
203C2×C4×Dic3C2×D511yescharacteristic
204C2×#208C2×D511yescharacteristic
205#835—52no—
206#832—52no—
207#832—52no—
208#832—52no—
209#818—52no—
210#817—52no—
211#815C2×D512yescomplemented
212C8×Dic3C2×D512yescomplemented
213C23×D5—21no—
214C22×C4×C5—31no—
215C22×C4×C5—31no—
216C22×C4×C5C22×D311yescharacteristic
217C2×C22⋊D5—21no—
218C22×Dic5—61no—
219#709—22no—
220#709C2×Dic314yescomplemented
221C2×D20—21no—
222C2×D20—22no—
223C2×D20C2×Dic314yescomplemented
224C2×D20C22×D312yes—
225C2×D20C22×D311yescharacteristic
226C2×C4×D5—21no—
227C2×C4×D5—62no—
228C2×C4×D5—62no—
229C2×C4×D5—61no—
230C2×C4×D5—61no—
231C2×C4×D5—124no—
232C2×Dic10C22×D311yescharacteristic
233C5×M16—62no—
234C5×M16C2×Dic314yescomplemented
235C2×C5×C8—32no—
236C2×C5×C8C22×D312yes—
237C42×C5—31no—
238C42×C5—31no—
239C42×C5—31no—
240C42×C5—31no—
241C22⋊Dic5—61no—
242#685—61no—
243#685—61no—
244#685—62no—
245#685—62no—
246C4⋊Dic5—32no—
247C4⋊Dic5—31no—
248C4⋊Dic5—31no—
249C4⋊Dic5—61no—
250#683—61no—
251#683—61no—
252C4×Dic5—61no—
253C4×Dic5—61no—
254C4×Dic5—61no—
255D40C2×Dic318yescomplemented
256#677C2×Dic318yescomplemented
257C2×AGL(1,ℤ/8)—51no—
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—151no—
267C4×M16—31no—
268C22×C3×C5—21no—
269C22×C3×C5—21no—
270C22×C3×C5C4×D412yes—
271C22×C3×C5C22×D411yescharacteristic
272C2×C3×D5—22no—
273C2×C3×D5—24no—
274C2×C3×D5—48no—
275C2×C3×D5—42no—
276C2×C3×D5—41no—
277C2×C3×D5—41no—
278C2×C3×D5—42no—
279C3×C4×C5C4×D412yes—
280C3×C4×C5C23×C411yescharacteristic
281C3×C4×C5C2×#4111yescharacteristic
282Dic15—42no—
283Dic15—41no—
284Dic15—41no—
285C3×Dic5—22no—
286C5×Dic3—21no—
287C5×Dic3—21no—
288C5×Dic3—21no—
289C5×Dic3AGL(1,ℤ/8)12yescomplemented
290C24×C3—101no—
291C3×#41—54no—
292C3×#41—102no—
293C2×C3×Q8—51no—
294C2×C3×D4—54no—
295C2×C3×D4—51no—
296C2×C3×D4—52no—
297C2×C3×D4—101no—
298C2×C3×D4—101no—
299C2×C3×D4—102no—
300C22×C3×C4—101no—
301C22×C3×C4C22×D511yescharacteristic
302C22×Dic3—101no—
303C22×Dic3—101no—
304C22×Dic3—101no—
305C22×Dic3—102no—
306C22×Dic3—102no—
307C22×Dic3—204no—
308C22×Dic3D2011yescharacteristic
309C22×Dic3D2011yescharacteristic
310C3×SD16—58no—
311C3×D8—58no—
312C3×M16C4×D514yescomplemented
313C2×C3×C8C22×D512yes—
314#217—101no—
315#217—101no—
316#217—101no—
317#217—102no—
318#217—102no—
319C4⋊Dic3—51no—
320C4⋊Dic3—52no—
321C4⋊Dic3—51no—
322C4⋊Dic3—101no—
323C4⋊Dic3—101no—
324C4⋊Dic3—101no—
325C4×Dic3—101no—
326C4×Dic3—101no—
327C4×Dic3—101no—
328C4×Dic3D2011yescharacteristic
329C4×Dic3D2011yescharacteristic
330C4×Dic3C22×D511yescharacteristic
331C4×Dic3C22×D511yescharacteristic
332#208—22no—
333C2×C3⋊C8C22×D512yes—
334C23×C5D3×D411yescharacteristic
335C22×D5—24no—
336C22×D5—21no—
337C22×D5—22no—
338C22×D5—42no—
339C22×D5—42no—
340C2×C4×C5—31no—
341C2×C4×C5—31no—
342C2×C4×C5—31no—
343C2×C4×C5—31no—
344C2×C4×C5—62no—
345C2×C4×C5—121no—
346C2×C4×C5D3×D411yescharacteristic
347C2×C4×C5#23712yes—
348C2×C4×C5C22×Dic312yes—
349C2×C4×C5C23×D311yescharacteristic
350C22⋊D5—24no—
351C2×Dic5—21no—
352C2×Dic5—61no—
353C2×Dic5—61no—
354C2×Dic5—62no—
355C2×Dic5—122no—
356D20—24no—
357D20—22no—
358D20—21no—
359D20C22×Dic314yes—
360D20C22×Dic312yescomplemented
361C4×D5—24no—
362C4×D5—122no—
363C4×D5—124no—
364C4×D5—122no—
365C4×D5—124no—
366Dic10—21no—
367Dic10C22×Dic312yescomplemented
368C5×C8—62no—
369C5×C8C22×Dic314yescomplemented
370C2×#41—51no—
371C22×D4—51no—
372C23×C4—301no—
373AGL(1,ℤ/8)—58no—
374C2×SD16—52no—
375C2×D8—52no—
376C2×M16—31no—
377C2×M16D3×D511yescharacteristic
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—151no—
391C2×C22⋊C4—301no—
392C2×C42—31no—
393C2×C42—301no—
394#107—152no—
395#106—152no—
396#103—152no—
397#102—152no—
398#102—152no—
399#102—152no—
400#97—32no—
401C4×C8—32no—
402C2×C3×C5—22no—
403C2×C3×C5C2×C4×D411yescharacteristic
404C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
405C2×C3×C5#57511yescharacteristic
406C3×D5—42no—
407C3×D5—44no—
408C23×C3—101no—
409C23×C3—102no—
410C23×C3—104no—
411C23×C3—202no—
412C23×C3—202no—
413C23×C3C2×D2011yescharacteristic
414C3×Q8—52no—
415C3×Q8—101no—
416C3×D4—54no—
417C3×D4—52no—
418C3×D4—104no—
419C3×D4—102no—
420C3×D4—101no—
421C3×D4—104no—
422C2×C3×C4—101no—
423C2×C3×C4—104no—
424C2×C3×C4C2×C4×D512yes—
425C2×C3×C4C2×D2011yescharacteristic
426C2×C3×C4#70912yes—
427C2×C3×C4C23×D511yescharacteristic
428C2×Dic3—22no—
429C2×Dic3—41no—
430C2×Dic3—101no—
431C2×Dic3—101no—
432C2×Dic3—102no—
433C2×Dic3—202no—
434C2×Dic3—204no—
435C2×Dic3—202no—
436C2×Dic3—202no—
437C2×Dic3—204no—
438C2×Dic3C2×D2011yescharacteristic
439C2×Dic3C2×D2011yescharacteristic
440C2×Dic3C2×D2011yescharacteristic
441C2×Dic3C2×D2011yescharacteristic
442C3×C8C2×C4×D514yescomplemented
443C3⋊C8—22no—
444C22×C5—21no—
445C22×C5—21no—
446C22×C5D4×Dic312yes—
447C22×C5C2×D3×D411yescharacteristic
448C2×D5—22no—
449C2×D5—24no—
450C2×D5—48no—
451C2×D5—42no—
452C2×D5—41no—
453C2×D5—41no—
454C2×D5—42no—
455C4×C5—32no—
456C4×C5—61no—
457C4×C5—61no—
458C4×C5—61no—
459C4×C5D4×Dic312yes—
460C4×C5C2×#23711yescharacteristic
461C4×C5C23×Dic311yescharacteristic
462Dic5—22no—
463Dic5—122no—
464Dic5—121no—
465Dic5—121no—
466C24—101no—
467#41—54no—
468#41—102no—
469C2×Q8—51no—
470C2×D4—54no—
471C2×D4—51no—
472C2×D4—52no—
473C2×D4—101no—
474C2×D4—102no—
475C2×D4—101no—
476C22×C4—31no—
477C22×C4—31no—
478C22×C4—101no—
479C22×C4—301no—
480C22×C4—301no—
481C22×C4—302no—
482C22×C4—302no—
483C22×C4—301no—
484C22×C4—604no—
485C22×C4C2×D3×D511yescharacteristic
486SD16—58no—
487D8—58no—
488M16—62no—
489M16D5×Dic314yescomplemented
490C2×C8—32no—
491C2×C8C2×D3×D512yes—
492C4⋊C4—151no—
493C4⋊C4—151no—
494C4⋊C4—152no—
495C4⋊C4—301no—
496C4⋊C4—301no—
497C4⋊C4—301no—
498C22⋊C4—302no—
499C22⋊C4—302no—
500C22⋊C4—301no—
501C22⋊C4—301no—
502C22⋊C4—301no—
503C42—31no—
504C42—31no—
505C42—31no—
506C42—31no—
507C42—301no—
508C42—301no—
509C42—301no—
510C3×C5C4×AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
511C22×C3—21no—
512C22×C3—21no—
513C22×C3—102no—
514C22×C3—104no—
515C22×C3—208no—
516C22×C3—202no—
517C22×C3—201no—
518C22×C3—201no—
519C22×C3—202no—
520C22×C3C4×D2012yes—
521C22×C3C22×D2011yescharacteristic
522C3×C4—102no—
523C3×C4C4×D2012yes—
524C3×C4C22×C4×D511yescharacteristic
525C3×C4C2×#70911yescharacteristic
526Dic3—21no—
527Dic3—21no—
528Dic3—21no—
529Dic3—201no—
530Dic3—202no—
531Dic3—201no—
532Dic3#405412yesdirect product factor
533C2×C5—22no—
534C2×C5D3×AGL(1,ℤ/8)11yescharacteristic
535C2×C5#577311yescharacteristic
536C2×C5C2×D4×Dic311yescharacteristic
537D5—44no—
538D5—42no—
539C23—104no—
540C23—101no—
541C23—102no—
542C23—202no—
543C23—202no—
544C23D3×D2011yescharacteristic
545Q8—52no—
546Q8—101no—
547D4—52no—
548D4—54no—
549D4—104no—
550D4—104no—
551D4—102no—
552D4—101no—
553C2×C4—31no—
554C2×C4—31no—
555C2×C4—31no—
556C2×C4—31no—
557C2×C4—62no—
558C2×C4—101no—
559C2×C4—104no—
560C2×C4—121no—
561C2×C4—301no—
562C2×C4—301no—
563C2×C4—302no—
564C2×C4—604no—
565C2×C4—602no—
566C2×C4—604no—
567C2×C4—602no—
568C2×C4—602no—
569C2×C4#676512yes—
570C2×C4D3×D2011yescharacteristic
571C2×C4C2×D5×Dic312yes—
572C2×C4C22×D3×D511yescharacteristic
573C8—62no—
574C8C2×D5×Dic314yescomplemented
575C2×C3—22no—
576C2×C3—204no—
577C2×C3—202no—
578C2×C3C2×C4×D2011yescharacteristic
579C2×C3C2×#405411yescharacteristic
580C2×C3#6608211yescharacteristic
581C5Dic3×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
582C22—21no—
583C22—21no—
584C22—102no—
585C22—104no—
586C22—208no—
587C22—202no—
588C22—201no—
589C22—201no—
590C22—202no—
591C22Dic3×D2012yes—
592C22C2×D3×D2011yescharacteristic
593C4—32no—
594C4—61no—
595C4—61no—
596C4—61no—
597C4—102no—
598C4—602no—
599C4—601no—
600C4—601no—
601C4Dic3×D2012yes—
602C4C2×#676511yescharacteristic
603C4C22×D5×Dic311yescharacteristic
604C3C4×#405411yescharacteristic, complemented, Sylow
605C2—22no—
606C2—204no—
607C2—202no—
608C2D3×#405411yescharacteristic
609C2#1175402911yescharacteristic
610C2C2×Dic3×D2011yescharacteristic
subgroupG/H
centerC22C2×D3×D20
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×M16C22
socleC22×C3×C5C22×D4
prime radicalC2×C3×#4054C2
cosocleC2×C3×C4×C5C24
hypercenterC2×M16D3×D5
hypocenterC3×C5C4×AGL(1,ℤ/8)
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. Dic3×#709C2semidirect
3. C2×Dic3×D20C2semidirect
4. #11751222C2semidirect
5. C5×Dic3×M16C2semidirect
6. #11748369C2semidirect
7. #11748367C2semidirect
8. #11747799C2semidirect
9. #11747795C2semidirect
10. Dic3×D40C2semidirect
11. Dic3×#677C2semidirect
21. C3×#4054C4semidirect
31. Dic3×D20C22semidirect
32. Dic3×D20C22semidirect
35. Dic3×Dic10C22semidirect
36. Dic3×Dic10C22semidirect
37. #91102C22semidirect
38. #91101C22semidirect
39. C5×#815C22semidirect
40. C5×C8×Dic3C22semidirect
41. #90971C22semidirect
44. #90965C22semidirect
64. C3×#709C2×C4semidirect
67. C2×C3×D20C2×C4semidirect
77. C3×C5×M16C2×C4semidirect
91. C3×D40C2×C4semidirect
92. C3×#677C2×C4semidirect
111. Dic3×M16D5semidirect
123. #4054Dic3direct (N × H)
171. C3×D20C22×C4semidirect
174. C3×Dic10C22×C4semidirect
179. C3×C5×C8C22×C4semidirect
211. #815C2×D5semidirect
212. C8×Dic3C2×D5semidirect
220. #709C2×Dic3semidirect
223. C2×D20C2×Dic3semidirect
234. C5×M16C2×Dic3semidirect
255. D40C2×Dic3semidirect
256. #677C2×Dic3semidirect
289. C5×Dic3AGL(1,ℤ/8)semidirect
312. C3×M16C4×D5semidirect
360. D20C22×Dic3semidirect
367. Dic10C22×Dic3semidirect
369. C5×C8C22×Dic3semidirect
442. C3×C8C2×C4×D5semidirect
489. M16D5×Dic3semidirect
510. C3×C5C4×AGL(1,ℤ/8)semidirect
532. Dic3#4054direct (N × H)
574. C8C2×D5×Dic3semidirect
581. C5Dic3×AGL(1,ℤ/8)semidirect
604. C3C4×#4054semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#40542
Dic3×#7092
C2×Dic3×D202
#117512222
C5×Dic3×M162
#117483692
#117483672
#117477992
#117477952
Dic3×D402
Dic3×#6772
C4×#40543
Dic3×AGL(1,ℤ/8)5