Finite Group Databasebeta

← Dic3×#4078 #14416726

Subgroups — Dic3×#4078 #14416726

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×C5×D4×Dic3C211yesmaximal, characteristic, complemented
2C2×C3×#4078C211yesmaximal, characteristic
3Dic3×#709C211yesmaximal, characteristic, complemented
4#11751316C211yesmaximal, characteristic, complemented
5#11749010C212yesmaximal, complemented
6Dic3×#687C212yesmaximal, complemented
7#11748988C212yesmaximal, complemented
8Dic3×#686C212yesmaximal, complemented
9#11748459C211yesmaximal, characteristic, complemented
10#11748368C211yesmaximal, characteristic, complemented
11Dic3×#681C211yesmaximal, characteristic, complemented
12C4×#4078—31nomaximal, Hall
13C22×C3×C5×D4C2211yescharacteristic
14C23×C5×Dic3—21no—
15C2×C3×#709C2211yescharacteristic
16#91815C2211yescharacteristic
17C2×#6710C2211yescharacteristic
18C2×C5×#217—21no—
19C5×D4×Dic3—21no—
20C5×D4×Dic3—21no—
21C5×D4×Dic3C2212yes—
22C5×D4×Dic3C2212yes—
23C2×C5×C4⋊Dic3C2211yescharacteristic
24C2×C4×C5×Dic3C2211yescharacteristic
25C2×C3×#687C2212yes—
26C3×#4078C418yescomplemented
27C2×C3×#686C2212yes—
28C2×C3×#681C2211yescharacteristic
29Dic3×C22⋊D5—21no—
30Dic3×C22⋊D5—21no—
31Dic3×D20C2211yescharacteristic, complemented
32Dic3×D20C2211yescharacteristic, complemented
33C4×D5×Dic3—21no—
34#91357—21no—
35Dic3×Dic10C2211yescharacteristic, complemented
36Dic3×Dic10C2211yescharacteristic, complemented
37#91116C2212yes—
38#91116C2212yes—
39#90988C2212yescomplemented
40#90985C2212yescomplemented
41#90970C2212yescomplemented
42#90964C2212yescomplemented
43#90953C2212yescomplemented
44Dic3×#168C2212yescomplemented
45Dic3×AGL(1,ℤ/8)—51nomaximal, Hall
46C2×C4×C5×D4—31no—
47C2×#4078D311yescharacteristic
48C4×#709—31no—
49#65504—31no—
50#65307—32no—
51C4×#687—32no—
52#65303—32no—
53C4×#686—32no—
54#65255—31no—
55#65253—31no—
56C4×#681—31no—
57C24×C3×C5—21no—
58C2×C3×C5×D4—21no—
59C2×C3×C5×D4—22no—
60C2×C3×C5×D4C2×C414yescomplemented
61C2×C3×C5×D4C2312yes—
62C22×C3×C4×C5C2311yescharacteristic
63C22×C5×Dic3—21no—
64C22×C5×Dic3—22no—
65C22×C5×Dic3—22no—
66C22×C5×Dic3—44no—
67C22×C5×Dic3D411yescharacteristic
68C22×C5×Dic3D411yescharacteristic
69C2×C3×C22⋊D5—21no—
70C3×#709—22no—
71C3×#709C2×C414yescomplemented
72C2×C3×D20C2311yescharacteristic
73C2×C3×C4×D5—21no—
74C2×C3×Dic10C2311yescharacteristic
75C2×D5×Dic3—21no—
76C2×D5×Dic3—21no—
77#6719—21no—
78C4⋊Dic15C2311yescharacteristic
79C4⋊Dic15C2311yescharacteristic
80C4×Dic15—21no—
81#6710—22no—
82C2×C3⋊#168C2312yes—
83C5×#217—22no—
84C5×#217—22no—
85C5×C4⋊Dic3—21no—
86C5×C4⋊Dic3C2312yes—
87C4×C5×Dic3D411yescharacteristic
88C4×C5×Dic3D411yescharacteristic
89C4×C5×Dic3C2311yescharacteristic
90C4×C5×Dic3C2311yescharacteristic
91C3×#687C2×C418yescomplemented
92C3×#686C2×C418yescomplemented
93C3×#681C2×C414yescomplemented
94C2×C3×#168C2312yes—
95#6668—21no—
96#6668—21no—
97#6665—21no—
98#6665—21no—
99Dic3×Dic5—21no—
100Dic3×Dic5—21no—
101C2×C3×AGL(1,ℤ/8)—51no—
102Dic3×#41—51no—
103C2×D4×Dic3D511yescharacteristic, complemented
104#5232—51no—
105#5215—51no—
106#5164—52no—
107Dic3×SD16—52no—
108#5152—52no—
109Dic3×D8—52no—
110#5121—51no—
111Dic3×M16—51no—
112C22×C5×D4C2×D311yescharacteristic
113C23×C4×C5—61no—
114C2×#709C2×D311yescharacteristic
115C4×C5×D4—32no—
116C4×C5×D4—32no—
117C4×C5×D4—61no—
118C4×C5×D4—61no—
119C2×C5×C4⋊C4—31no—
120C2×C5×C22⋊C4—61no—
121C2×C42×C5—31no—
122C2×#687C2×D312yes—
123#4078Dic318yesdirect product factor
124C2×#686C2×D312yes—
125C4×C22⋊D5—61no—
126C4×C22⋊D5—61no—
127#4072—31no—
128C2×#681—31no—
129C2×#681C2×D311yescharacteristic
130C4×D20—31no—
131C4×D20—31no—
132#4018—61no—
133C42×D5—61no—
134C4×Dic10—31no—
135C4×Dic10—31no—
136#3964—32no—
137#3964—32no—
138#3942—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—24no—
146C23×C3×C5—22no—
147C23×C3×C5—42no—
148C23×C3×C5—42no—
149C23×C3×C5C2×D411yescharacteristic
150C22×C3×D5—21no—
151C3×C5×D4—24no—
152C3×C5×D4—22no—
153C3×C5×D4C22×C414yes—
154C2×C3×C4×C5C22×C412yes—
155C2×C3×C4×C5C2×D411yescharacteristic
156C2×C3×C4×C5#4112yes—
157C2×C3×C4×C5C2411yescharacteristic
158C2×Dic15—21no—
159C2×Dic15—21no—
160C2×C5×Dic3—22no—
161C2×C5×Dic3—22no—
162C2×C5×Dic3—41no—
163C2×C5×Dic3—44no—
164C2×C5×Dic3—42no—
165C2×C5×Dic3—44no—
166C2×C5×Dic3C2×D411yescharacteristic
167C2×C5×Dic3C2×D411yescharacteristic
168C2×C5×Dic3C2×D411yescharacteristic
169C2×C5×Dic3C2×D411yescharacteristic
170C3×C22⋊D5—24no—
171C2×C3×Dic5—21no—
172C3×D20—21no—
173C3×D20C22×C412yescomplemented
174C3×C4×D5—24no—
175C3×Dic10—21no—
176C3×Dic10C22×C412yescomplemented
177D5×Dic3—42no—
178D5×Dic3—42no—
179C3⋊#168—22no—
180C3×#168C22×C414yescomplemented
181C2×C3×#41—51no—
182C22×C3×D4C2×D511yescharacteristic
183C23×Dic3—21no—
184C3×AGL(1,ℤ/8)—58no—
185C2×C3×SD16—52no—
186C2×C3×D8—52no—
187C2×C3×M16—51no—
188C2×#217—21no—
189Q8×Dic3—51no—
190Q8×Dic3—51no—
191D4×Dic3—21no—
192D4×Dic3—21no—
193D4×Dic3—51no—
194D4×Dic3—51no—
195D4×Dic3—101no—
196D4×Dic3—101no—
197D4×Dic3C2×D512yes—
198D4×Dic3C2×D512yes—
199#926—51no—
200#926—101no—
201C2×C4⋊Dic3C2×D511yescharacteristic
202C2×C4×Dic3—101no—
203C2×C4×Dic3C2×D511yescharacteristic
204C2×#208—51no—
205#835—52no—
206#832—52no—
207#832—52no—
208#832—52no—
209#818—52no—
210#817—52no—
211#815—52no—
212C8×Dic3—52no—
213C24×C5—21no—
214C2×C5×D4—21no—
215C2×C5×D4—22no—
216C2×C5×D4C2×Dic314yescomplemented
217C2×C5×D4C22×D312yes—
218C22×C4×C5—31no—
219C22×C4×C5—31no—
220C22×C4×C5—62no—
221C22×C4×C5—62no—
222C22×C4×C5—61no—
223C22×C4×C5—124no—
224C22×C4×C5C22×D311yescharacteristic
225C2×C22⋊D5—21no—
226#709—22no—
227#709C2×Dic314yescomplemented
228C2×D20C22×D311yescharacteristic
229C2×C4×D5—21no—
230C2×C4×D5—61no—
231C2×C4×D5—61no—
232C2×Dic10C22×D311yescharacteristic
233C5×C4⋊C4—32no—
234C5×C4⋊C4—61no—
235C5×C22⋊C4—62no—
236C5×C22⋊C4—62no—
237C42×C5—31no—
238C42×C5—31no—
239C42×C5—31no—
240C42×C5—31no—
241C22⋊Dic5—61no—
242#687C2×Dic318yescomplemented
243#686C2×Dic318yescomplemented
244#685—61no—
245#685—61no—
246C4⋊Dic5—31no—
247C4⋊Dic5—31no—
248#683—61no—
249#683—61no—
250C4×Dic5—61no—
251C4×Dic5—61no—
252C4×Dic5—61no—
253#681—62no—
254#681C2×Dic314yescomplemented
255C2×#168—32no—
256C2×#168C22×D312yes—
257C2×AGL(1,ℤ/8)—51no—
258C4×#41—151no—
259C2×C4×D4—31no—
260#442—152no—
261#440—152no—
262C4×SD16—152no—
263C4×D8—152no—
264#428—151no—
265#418—151no—
266#417—151no—
267C4×M16—151no—
268C22×C3×C5—21no—
269C22×C3×C5—21no—
270C22×C3×C5—24no—
271C22×C3×C5—48no—
272C22×C3×C5—42no—
273C22×C3×C5—42no—
274C22×C3×C5C4×D412yes—
275C22×C3×C5C22×D411yescharacteristic
276C2×C3×D5—22no—
277C2×C3×D5—41no—
278C2×C3×D5—41no—
279C3×C4×C5C4×D412yes—
280C3×C4×C5C23×C411yescharacteristic
281C3×C4×C5C2×#4111yescharacteristic
282Dic15—41no—
283Dic15—41no—
284C3×Dic5—22no—
285C5×Dic3—21no—
286C5×Dic3—21no—
287C5×Dic3—21no—
288C5×Dic3—42no—
289C5×Dic3AGL(1,ℤ/8)12yescomplemented
290C24×C3—21no—
291C3×#41—54no—
292C3×#41—102no—
293C2×C3×Q8—51no—
294C2×C3×D4—22no—
295C2×C3×D4—21no—
296C2×C3×D4—51no—
297C2×C3×D4—101no—
298C2×C3×D4C4×D514yescomplemented
299C2×C3×D4C22×D512yes—
300C22×C3×C4—101no—
301C22×C3×C4C22×D511yescharacteristic
302C22×Dic3—21no—
303C22×Dic3—22no—
304C22×Dic3—22no—
305C22×Dic3—44no—
306C22×Dic3—101no—
307C22×Dic3—101no—
308C22×Dic3C22⋊D511yescharacteristic
309C22×Dic3C22⋊D511yescharacteristic
310C3×SD16—58no—
311C3×D8—58no—
312C3×M16—54no—
313C2×C3×C8—52no—
314#217—22no—
315#217—22no—
316#217—101no—
317#217—101no—
318#217—101no—
319C4⋊Dic3—21no—
320C4⋊Dic3—51no—
321C4⋊Dic3—51no—
322C4⋊Dic3—101no—
323C4⋊Dic3—101no—
324C4⋊Dic3C22×D512yes—
325C4×Dic3—101no—
326C4×Dic3—101no—
327C4×Dic3—101no—
328C4×Dic3C22⋊D511yescharacteristic
329C4×Dic3C22⋊D511yescharacteristic
330C4×Dic3C22×D511yescharacteristic
331C4×Dic3C22×D511yescharacteristic
332#208—102no—
333C2×C3⋊C8—52no—
334C23×C5—24no—
335C23×C5—22no—
336C23×C5—42no—
337C23×C5—42no—
338C23×C5D3×D411yescharacteristic
339C22×D5—21no—
340C5×D4—24no—
341C5×D4—22no—
342C5×D4C22×Dic314yes—
343C2×C4×C5—31no—
344C2×C4×C5—31no—
345C2×C4×C5—31no—
346C2×C4×C5—31no—
347C2×C4×C5—62no—
348C2×C4×C5—62no—
349C2×C4×C5—124no—
350C2×C4×C5—121no—
351C2×C4×C5—124no—
352C2×C4×C5—122no—
353C2×C4×C5D3×D411yescharacteristic
354C2×C4×C5#23712yes—
355C2×C4×C5C22×Dic312yes—
356C2×C4×C5C23×D311yescharacteristic
357C22⋊D5—24no—
358C2×Dic5—21no—
359C2×Dic5—61no—
360C2×Dic5—61no—
361D20—21no—
362D20C22×Dic312yescomplemented
363C4×D5—24no—
364C4×D5—122no—
365C4×D5—122no—
366Dic10—21no—
367Dic10C22×Dic312yescomplemented
368#168—62no—
369#168C22×Dic314yescomplemented
370C2×#41—51no—
371C22×D4D3×D511yescharacteristic
372C23×C4—61no—
373AGL(1,ℤ/8)—58no—
374C2×SD16—52no—
375C2×D8—52no—
376C2×M16—51no—
377C2×M16—151no—
378C4×Q8—151no—
379C4×Q8—151no—
380C4×D4—32no—
381C4×D4—32no—
382C4×D4—61no—
383C4×D4—61no—
384C4×D4—151no—
385C4×D4—151no—
386C4×D4—301no—
387C4×D4—301no—
388#117—151no—
389#117—301no—
390C2×C4⋊C4—31no—
391C2×C22⋊C4—61no—
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—152no—
401C4×C8—152no—
402C2×C3×C5—22no—
403C2×C3×C5—44no—
404C2×C3×C5C2×C4×D411yescharacteristic
405C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
406C2×C3×C5#57511yescharacteristic
407C3×D5—42no—
408C23×C3—22no—
409C23×C3—24no—
410C23×C3—42no—
411C23×C3—42no—
412C23×C3—101no—
413C23×C3C2×C22⋊D511yescharacteristic
414C3×Q8—52no—
415C3×Q8—101no—
416C3×D4—22no—
417C3×D4—24no—
418C3×D4—52no—
419C3×D4—104no—
420C3×D4—101no—
421C3×D4C2×C4×D514yes—
422C2×C3×C4—101no—
423C2×C3×C4—104no—
424C2×C3×C4C2×C4×D512yes—
425C2×C3×C4#70912yes—
426C2×C3×C4C2×C22⋊D511yescharacteristic
427C2×C3×C4C23×D511yescharacteristic
428C2×Dic3—22no—
429C2×Dic3—22no—
430C2×Dic3—41no—
431C2×Dic3—44no—
432C2×Dic3—42no—
433C2×Dic3—44no—
434C2×Dic3—101no—
435C2×Dic3—101no—
436C2×Dic3—202no—
437C2×Dic3—202no—
438C2×Dic3C2×C22⋊D511yescharacteristic
439C2×Dic3C2×C22⋊D511yescharacteristic
440C2×Dic3C2×C22⋊D511yescharacteristic
441C2×Dic3C2×C22⋊D511yescharacteristic
442C3×C8—54no—
443C3⋊C8—102no—
444C22×C5—21no—
445C22×C5—21no—
446C22×C5—24no—
447C22×C5—48no—
448C22×C5—42no—
449C22×C5—42no—
450C22×C5D4×Dic312yes—
451C22×C5C2×D3×D411yescharacteristic
452C2×D5—22no—
453C2×D5—41no—
454C2×D5—41no—
455C4×C5—32no—
456C4×C5—61no—
457C4×C5—61no—
458C4×C5—61no—
459C4×C5—122no—
460C4×C5D4×Dic312yes—
461C4×C5C2×#23711yescharacteristic
462C4×C5C23×Dic311yescharacteristic
463Dic5—22no—
464Dic5—121no—
465Dic5—121no—
466C24—21no—
467#41—54no—
468#41—102no—
469C2×Q8—51no—
470C2×D4—22no—
471C2×D4—21no—
472C2×D4—51no—
473C2×D4—101no—
474C2×D4D5×Dic314yescomplemented
475C2×D4C2×D3×D512yes—
476C22×C4—31no—
477C22×C4—31no—
478C22×C4—62no—
479C22×C4—62no—
480C22×C4—61no—
481C22×C4—101no—
482C22×C4—124no—
483C22×C4—301no—
484C22×C4—301no—
485C22×C4C2×D3×D511yescharacteristic
486SD16—58no—
487D8—58no—
488M16—54no—
489M16—302no—
490C2×C8—52no—
491C2×C8—152no—
492C4⋊C4—32no—
493C4⋊C4—61no—
494C4⋊C4—151no—
495C4⋊C4—151no—
496C4⋊C4—301no—
497C4⋊C4—301no—
498C22⋊C4—62no—
499C22⋊C4—62no—
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—24no—
514C22×C3—48no—
515C22×C3—42no—
516C22×C3—42no—
517C22×C3—102no—
518C22×C3—201no—
519C22×C3—201no—
520C22×C3C4×C22⋊D512yes—
521C22×C3C22×C22⋊D511yescharacteristic
522C3×C4—102no—
523C3×C4C4×C22⋊D512yes—
524C3×C4C22×C4×D511yescharacteristic
525C3×C4C2×#70911yescharacteristic
526Dic3—21no—
527Dic3—21no—
528Dic3—21no—
529Dic3—42no—
530Dic3—201no—
531Dic3—201no—
532Dic3#407812yesdirect product factor
533C2×C5—22no—
534C2×C5—44no—
535C2×C5D3×AGL(1,ℤ/8)11yescharacteristic
536C2×C5#577311yescharacteristic
537C2×C5C2×D4×Dic311yescharacteristic
538D5—42no—
539C23—22no—
540C23—24no—
541C23—42no—
542C23—42no—
543C23—101no—
544C23D3×C22⋊D511yescharacteristic
545Q8—52no—
546Q8—101no—
547D4—22no—
548D4—24no—
549D4—52no—
550D4—104no—
551D4—101no—
552D4C2×D5×Dic314yes—
553C2×C4—31no—
554C2×C4—31no—
555C2×C4—31no—
556C2×C4—31no—
557C2×C4—62no—
558C2×C4—62no—
559C2×C4—101no—
560C2×C4—104no—
561C2×C4—124no—
562C2×C4—121no—
563C2×C4—124no—
564C2×C4—122no—
565C2×C4—301no—
566C2×C4—301no—
567C2×C4—602no—
568C2×C4—602no—
569C2×C4C2×D5×Dic312yes—
570C2×C4#677912yes—
571C2×C4D3×C22⋊D511yescharacteristic
572C2×C4C22×D3×D511yescharacteristic
573C8—54no—
574C8—302no—
575C2×C3—22no—
576C2×C3—44no—
577C2×C3—202no—
578C2×C3C2×C4×C22⋊D511yescharacteristic
579C2×C3C2×#407811yescharacteristic
580C2×C3#6615611yescharacteristic
581C5Dic3×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
582C22—21no—
583C22—21no—
584C22—24no—
585C22—48no—
586C22—42no—
587C22—42no—
588C22—102no—
589C22—201no—
590C22—201no—
591C22Dic3×C22⋊D512yes—
592C22C2×D3×C22⋊D511yescharacteristic
593C4—32no—
594C4—61no—
595C4—61no—
596C4—61no—
597C4—102no—
598C4—122no—
599C4—601no—
600C4—601no—
601C4Dic3×C22⋊D512yes—
602C4C22×D5×Dic311yescharacteristic
603C4C2×#677911yescharacteristic
604C3C4×#407811yescharacteristic, complemented, Sylow
605C2—22no—
606C2—44no—
607C2—202no—
608C2D3×#407811yescharacteristic
609C2#1175469611yescharacteristic
610C2C2×Dic3×C22⋊D511yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊D5
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC22×C3×C5×D4C22
socleC22×C3×C5C22×D4
prime radicalC2×C3×#4078C2
cosocleC2×C3×C4×C5C24
hypercenterC22×D4D3×D5
hypocenterC3×C5C4×AGL(1,ℤ/8)
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×C5×D4×Dic3C2semidirect
3. Dic3×#709C2semidirect
4. #11751316C2semidirect
5. #11749010C2semidirect
6. Dic3×#687C2semidirect
7. #11748988C2semidirect
8. Dic3×#686C2semidirect
9. #11748459C2semidirect
10. #11748368C2semidirect
11. Dic3×#681C2semidirect
26. C3×#4078C4semidirect
31. Dic3×D20C22semidirect
32. Dic3×D20C22semidirect
35. Dic3×Dic10C22semidirect
36. Dic3×Dic10C22semidirect
39. #90988C22semidirect
40. #90985C22semidirect
41. #90970C22semidirect
42. #90964C22semidirect
43. #90953C22semidirect
44. Dic3×#168C22semidirect
60. C2×C3×C5×D4C2×C4semidirect
71. C3×#709C2×C4semidirect
91. C3×#687C2×C4semidirect
92. C3×#686C2×C4semidirect
93. C3×#681C2×C4semidirect
103. C2×D4×Dic3D5semidirect
123. #4078Dic3direct (N × H)
173. C3×D20C22×C4semidirect
176. C3×Dic10C22×C4semidirect
180. C3×#168C22×C4semidirect
216. C2×C5×D4C2×Dic3semidirect
227. #709C2×Dic3semidirect
242. #687C2×Dic3semidirect
243. #686C2×Dic3semidirect
254. #681C2×Dic3semidirect
289. C5×Dic3AGL(1,ℤ/8)semidirect
298. C2×C3×D4C4×D5semidirect
362. D20C22×Dic3semidirect
367. Dic10C22×Dic3semidirect
369. #168C22×Dic3semidirect
474. C2×D4D5×Dic3semidirect
510. C3×C5C4×AGL(1,ℤ/8)semidirect
532. Dic3#4078direct (N × H)
581. C5Dic3×AGL(1,ℤ/8)semidirect
604. C3C4×#4078semidirect

Maximal subgroups

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

subgroupindex
C2×C5×D4×Dic32
C2×C3×#40782
Dic3×#7092
#117513162
#117490102
Dic3×#6872
#117489882
Dic3×#6862
#117484592
#117483682
Dic3×#6812
C4×#40783
Dic3×AGL(1,ℤ/8)5