Finite Group Databasebeta

← Dic6×#685 #14326133

Subgroups — Dic6×#685 #14326133

Order 1920.

1640 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D5×Dic6C211yesmaximal, characteristic
2C2×C4×C5×Dic6C211yesmaximal, characteristic, complemented
3C2×Dic5×Dic6C211yesmaximal, characteristic, complemented
4#11751030C211yesmaximal, characteristic, complemented
5C3×C4×#685C211yesmaximal, characteristic
6#11748934C212yesmaximal, complemented
7#11748878C212yesmaximal
8Dic3×#685C212yesmaximal
9#11748759C211yesmaximal, characteristic
10#11748631C212yesmaximal, complemented
11#11748587C211yesmaximal, characteristic, complemented
12Q8×#685—31nomaximal, Hall
13C22×Dic30C2211yescharacteristic
14C22×C5×Dic6C2211yescharacteristic
15C22×C3×C4×D5C2211yescharacteristic
16C22×#1204C2212yes—
17C22×D5×Dic3C2212yes—
18C2×D5×Dic6—24no—
19C2×D5×Dic6—24no—
20C2×D5×Dic6C418yescomplemented
21C2×C3×C42×C5C2211yescharacteristic
22C2×C4⋊Dic15C2211yescharacteristic
23C2×C5×C4⋊Dic3C2211yescharacteristic
24C2×C5×Dic3⋊C4C2212yes—
25C2×C4×C5×Dic3C2212yes—
26C4×C5×Dic6—24no—
27C2×C3×#685C2211yescharacteristic
28C2×C3×#685C2211yescharacteristic
29C2×C3×C4×Dic5C2211yescharacteristic
30C2×Dic3⋊Dic5C2212yes—
31C2×#6665C2212yes—
32C2×Dic3×Dic5C2212yes—
33Dic5×Dic6—24no—
34#91115C2212yes—
35#91115C2212yes—
36C3×#3963C2211yescharacteristic
37C3×#3963C2211yescharacteristic
38#90994C2212yes—
39#90994C2212yes—
40#90994C2212yes—
41C22⋊C4×Dic6—51nomaximal, Hall
42C22×Q8×D5—31no—
43C2×C4×C5×Q8—31no—
44C2×Q8×Dic5—31no—
45#65276—32no—
46#65276—31no—
47#65263—31no—
48#65263—32no—
49C4×#685—32no—
50C4×#685D311yescharacteristic
51#65218—32no—
52#65218—31no—
53C23×C3×D5C2311yescharacteristic
54C22×C3×C4×C5C2311yescharacteristic
55C22×C3×C4×C5C2311yescharacteristic
56C22×C3×C4×C5C2311yescharacteristic
57C22×Dic15C2312yes—
58C22×Dic15C2312yes—
59C2×Dic30—22no—
60C2×Dic30—22no—
61C2×Dic30C2×C414yes—
62C22×C5×Dic3C2312yes—
63C22×C5×Dic3C2312yes—
64C2×C5×Dic6C2×C414yes—
65C2×C5×Dic6D414yes—
66C2×C5×Dic6D414yes—
67C22×C3×Dic5C2311yescharacteristic
68C22×C3×Dic5C2311yescharacteristic
69C22×C3×Dic5C2311yescharacteristic
70C2×C3×C4×D5—22no—
71C2×C3×C4×D5—22no—
72C2×C3×C4×D5C2×C414yes—
73C2×#1204—24no—
74C2×#1204—24no—
75C2×#1204C2×C418yes—
76C2×D5×Dic3—24no—
77C2×D5×Dic3—24no—
78C2×D5×Dic3C2×C418yes—
79D5×Dic6—232no—
80C3×C42×C5—22no—
81C4⋊Dic15—22no—
82C5×C4⋊Dic3—22no—
83C5×Dic3⋊C4—24no—
84C4×C5×Dic3—24no—
85C3×#685—22no—
86C3×#685Q814yescomplemented
87C3×C4×Dic5—22no—
88Dic3⋊Dic5—24no—
89#6665—24no—
90Dic3×Dic5—24no—
91C23×Dic6—51no—
92C2×C4×Dic6—51no—
93C2×C4×Dic6D511yescharacteristic, complemented
94C3×C4×C22⋊C4—51no—
95#5212—51no—
96#4942—52no—
97Dic3×C22⋊C4—52no—
98#4929—51no—
99#4929—51no—
100#4646—52no—
101#4646—52no—
102C22×C5×Q8—31no—
103C2×Q8×D5—38no—
104C2×Q8×D5—64no—
105C2×Q8×D5—64no—
106C22×C4×D5—32no—
107C22×C4×D5C2×D311yescharacteristic
108C22×Dic10—31no—
109C22×Dic10—32no—
110C4×C5×Q8—64no—
111C2×C5×C4⋊C4—32no—
112C2×C5×C4⋊C4—31no—
113C2×C42×C5—32no—
114C2×C42×C5C2×D311yescharacteristic
115Q8×Dic5—64no—
116C2×#685—32no—
117C2×#685C2×D311yescharacteristic
118C2×#685C2×D311yescharacteristic
119C2×C4⋊Dic5—31no—
120C2×C4⋊Dic5—32no—
121C2×C4×Dic5—32no—
122C2×C4×Dic5C2×D311yescharacteristic
123#3963—32no—
124#3963—32no—
125#3963—32no—
126#3963—32no—
127#3963—32no—
128#3963C2×D311yescharacteristic
129#3963C2×D311yescharacteristic
130Q8×C22⋊C4—151noSylow
131C23×C3×C5C2411yescharacteristic
132C22×C3×D5—21no—
133C22×C3×D5—21no—
134C22×C3×D5—21no—
135C22×C3×D5—21no—
136C22×C3×D5C22×C412yes—
137C22×C3×D5C2×Q812yes—
138C22×C3×D5#4112yes—
139C2×C3×C4×C5—21no—
140C2×C3×C4×C5—21no—
141C2×C3×C4×C5—21no—
142C2×C3×C4×C5—21no—
143C2×C3×C4×C5—21no—
144C2×C3×C4×C5C22×C412yes—
145C2×C3×C4×C5C2×D412yes—
146C2×C3×C4×C5C2×D412yes—
147C2×C3×C4×C5C2×Q812yescomplemented
148C2×Dic15—22no—
149C2×Dic15—22no—
150C2×Dic15—22no—
151C2×Dic15—22no—
152C2×Dic15—22no—
153C2×Dic15C22×C414yes—
154Dic30—28no—
155C2×C5×Dic3—22no—
156C2×C5×Dic3—22no—
157C2×C5×Dic3—22no—
158C2×C5×Dic3C22×C414yes—
159C2×C5×Dic3C2×D414yes—
160C2×C5×Dic3C2×D414yes—
161C5×Dic6C22⋊C4116yescomplemented
162C2×C3×Dic5—21no—
163C2×C3×Dic5—21no—
164C2×C3×Dic5—21no—
165C2×C3×Dic5—21no—
166C2×C3×Dic5—21no—
167C2×C3×Dic5—21no—
168C2×C3×Dic5—21no—
169C2×C3×Dic5C22×C412yes—
170C2×C3×Dic5C2×Q812yescomplemented
171C3×C4×D5—28no—
172#1204—216no—
173D5×Dic3—216no—
174C23×C3×C4—51no—
175C23×Dic3—52no—
176C22×Dic6—51no—
177C22×Dic6—52no—
178C22×Dic6—58no—
179C22×Dic6—104no—
180C22×Dic6—104no—
181C22×Dic6C2×D511yescharacteristic
182C2×C3×C22⋊C4—51no—
183C2×C3×C22⋊C4—51no—
184C2×C3×C42—51no—
185C2×C3×C42C2×D511yescharacteristic
186C2×#217—52no—
187C2×C4⋊Dic3—51no—
188C2×C4⋊Dic3C2×D511yescharacteristic
189C2×Dic3⋊C4—52no—
190C2×Dic3⋊C4C2×D512yes—
191C2×C4×Dic3—52no—
192C2×C4×Dic3C2×D512yes—
193C4×Dic6—24no—
194C4×Dic6—104no—
195C3×#95—51no—
196C3×#95—51no—
197C3⋊#95—52no—
198C3⋊#95—52no—
199C3⋊#95—52no—
200C3⋊#95—52no—
201C3⋊#95—52no—
202C23×D5C22×D311yescharacteristic
203C2×C5×Q8—34no—
204C2×C5×Q8—34no—
205C2×C5×Q8—34no—
206C22×C4×C5—32no—
207C22×C4×C5—32no—
208C22×C4×C5C22×D311yescharacteristic
209C22×C4×C5C22×D311yescharacteristic
210C22×C4×C5C22×D311yescharacteristic
211C22×Dic5—32no—
212C22×Dic5—32no—
213C22×Dic5C22×D311yescharacteristic
214C22×Dic5C22×D311yescharacteristic
215C22×Dic5C22×D311yescharacteristic
216Q8×D5—632no—
217C2×C4×D5—22no—
218C2×C4×D5—22no—
219C2×C4×D5—38no—
220C2×C4×D5—64no—
221C2×C4×D5—64no—
222C2×C4×D5C4×D314yes—
223C2×Dic10—34no—
224C2×Dic10—38no—
225C2×Dic10—64no—
226C2×Dic10—64no—
227C2×Dic10—62no—
228C2×Dic10—62no—
229C5×C4⋊C4—64no—
230C5×C4⋊C4—62no—
231C42×C5—22no—
232C42×C5—64no—
233#685—22no—
234#685—64no—
235#685Dic614yesdirect product factor
236C4⋊Dic5—64no—
237C4⋊Dic5—62no—
238C4×Dic5—22no—
239C4×Dic5—64no—
240C23×Q8—151no—
241C2×C4×Q8—31no—
242C2×C4×Q8—151no—
243#391—152no—
244#391—152no—
245#391—151no—
246#391—151no—
247#380—152no—
248#380—151no—
249C4×C22⋊C4—51no—
250C4×C22⋊C4—152no—
251C22×C3×C5C23×C411yescharacteristic
252C22×C3×C5C22×D411yescharacteristic
253C22×C3×C5C22×D411yescharacteristic
254C22×C3×C5C22×Q811yescharacteristic
255C22×C3×C5C2×#4111yescharacteristic
256C22×C3×C52−1+411yescharacteristic
257C22×C3×C52−1+411yescharacteristic
258C2×C3×D5—22no—
259C2×C3×D5—22no—
260C2×C3×D5—22no—
261C2×C3×D5—22no—
262C2×C3×D5—22no—
263C2×C3×D5—22no—
264C2×C3×D5C4×Q814yescomplemented
265C3×C4×C5—22no—
266C3×C4×C5—41no—
267C3×C4×C5C2×C22⋊C414yes—
268Dic15—24no—
269Dic15—42no—
270C3×Dic5—22no—
271C3×Dic5—22no—
272C3×Dic5—41no—
273C5×Dic3—42no—
274C5×Dic3C2×C22⋊C418yes—
275C24×C3—51no—
276C22×C3×C4—51no—
277C22×C3×C4—51no—
278C22×C3×C4—54no—
279C22×C3×C4—51no—
280C22×C3×C4—102no—
281C22×C3×C4—102no—
282C22×C3×C4C22×D511yescharacteristic
283C22×C3×C4C22×D511yescharacteristic
284C22×C3×C4C22×D511yescharacteristic
285C22×Dic3—52no—
286C22×Dic3—58no—
287C22×Dic3—52no—
288C22×Dic3—104no—
289C22×Dic3—104no—
290C22×Dic3C22×D512yes—
291C22×Dic3C22×D512yes—
292C2×Dic6—54no—
293C2×Dic6—58no—
294C2×Dic6—102no—
295C2×Dic6—102no—
296C2×Dic6—104no—
297C2×Dic6—104no—
298C2×Dic6—1032no—
299C2×Dic6C4×D514yes—
300C2×Dic6D2014yes—
301C2×Dic6C22⋊D514yes—
302C3×C22⋊C4—54no—
303C3×C22⋊C4—102no—
304C3×C42—22no—
305C3×C42—102no—
306#217—104no—
307C4⋊Dic3—22no—
308C4⋊Dic3—102no—
309Dic3⋊C4—24no—
310Dic3⋊C4—104no—
311C4×Dic3—24no—
312C4×Dic3—104no—
313C23×C5C23×D311yescharacteristic
314C22×D5—21no—
315C22×D5—21no—
316C22×D5—21no—
317C22×D5—21no—
318C22×D5C2×Dic612yes—
319C22×D5C2×C4×D312yes—
320C22×D5#23512yes—
321C5×Q8—316no—
322C2×C4×C5—21no—
323C2×C4×C5—21no—
324C2×C4×C5—21no—
325C2×C4×C5—21no—
326C2×C4×C5—21no—
327C2×C4×C5—34no—
328C2×C4×C5—34no—
329C2×C4×C5—34no—
330C2×C4×C5—62no—
331C2×C4×C5—62no—
332C2×C4×C5—62no—
333C2×C4×C5C2×Dic612yescomplemented
334C2×C4×C5C2×C4×D312yes—
335C2×C4×C5D3×D412yes—
336C2×C4×C5D3×D412yes—
337C2×Dic5—21no—
338C2×Dic5—21no—
339C2×Dic5—21no—
340C2×Dic5—21no—
341C2×Dic5—21no—
342C2×Dic5—21no—
343C2×Dic5—21no—
344C2×Dic5—34no—
345C2×Dic5—62no—
346C2×Dic5—62no—
347C2×Dic5—62no—
348C2×Dic5—62no—
349C2×Dic5—62no—
350C2×Dic5C2×Dic612yescomplemented
351C2×Dic5C2×C4×D312yes—
352C4×D5—28no—
353C4×D5—616no—
354Dic10—616no—
355Dic10—68no—
356C22×Q8—31no—
357C22×Q8—152no—
358C22×Q8—158no—
359C22×Q8—151no—
360C22×Q8—304no—
361C22×Q8—304no—
362C23×C4—51no—
363C23×C4—152no—
364C4×Q8—64no—
365C4×Q8—304no—
366C2×C4⋊C4—32no—
367C2×C4⋊C4—31no—
368C2×C4⋊C4—152no—
369C2×C4⋊C4—151no—
370C2×C22⋊C4—51no—
371C2×C22⋊C4—51no—
372C2×C22⋊C4—152no—
373C2×C42—32no—
374C2×C42—51no—
375C2×C42—152no—
376C2×C42D3×D511yescharacteristic
377#95—51no—
378#95—51no—
379#95—152no—
380#95—152no—
381#95—152no—
382#95—152no—
383#95—152no—
384C2×C3×C5C22×C22⋊C411yescharacteristic
385C2×C3×C5C2×C4×Q811yescharacteristic
386C2×C3×C5#51911yescharacteristic
387C2×C3×C5#54811yescharacteristic
388C2×C3×C5#54811yescharacteristic
389C2×C3×C5D4×Q811yescharacteristic
390C2×C3×C5D4×Q811yescharacteristic
391C3×D5—24no—
392C23×C3—52no—
393C23×C3—52no—
394C23×C3—52no—
395C23×C3—101no—
396C23×C3—101no—
397C23×C3—101no—
398C23×C3—101no—
399C23×C3C23×D511yescharacteristic
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—52no—
406C2×C3×C4—52no—
407C2×C3×C4—101no—
408C2×C3×C4—101no—
409C2×C3×C4—101no—
410C2×C3×C4—101no—
411C2×C3×C4—108no—
412C2×C3×C4—101no—
413C2×C3×C4—101no—
414C2×C3×C4—101no—
415C2×C3×C4C2×C4×D512yes—
416C2×C3×C4C2×D2012yes—
417C2×C3×C4Q8×D512yescomplemented
418C2×C3×C4C2×C22⋊D512yes—
419C2×Dic3—22no—
420C2×Dic3—22no—
421C2×Dic3—22no—
422C2×Dic3—54no—
423C2×Dic3—102no—
424C2×Dic3—102no—
425C2×Dic3—1016no—
426C2×Dic3—102no—
427C2×Dic3—102no—
428C2×Dic3—102no—
429C2×Dic3C2×C4×D514yes—
430C2×Dic3C2×D2014yes—
431C2×Dic3C2×C22⋊D514yes—
432Dic6—108no—
433Dic6—1016no—
434Dic6#685116yesdirect product factor
435C22×C5C22×Dic611yescharacteristic
436C22×C5C22×C4×D311yescharacteristic
437C22×C5C2×#23511yescharacteristic
438C22×C5C2×D3×D411yescharacteristic
439C22×C5C2×D3×D411yescharacteristic
440C22×C5#101011yescharacteristic
441C22×C5#101011yescharacteristic
442C2×D5—22no—
443C2×D5—22no—
444C2×D5—22no—
445C2×D5—22no—
446C2×D5—22no—
447C2×D5—22no—
448C2×D5C4×Dic614yescomplemented
449C4×C5—22no—
450C4×C5—38no—
451C4×C5—41no—
452C4×C5—122no—
453C4×C5D3×C22⋊C414yes—
454Dic5—22no—
455Dic5—22no—
456Dic5—41no—
457Dic5—64no—
458Dic5—122no—
459C24—51no—
460C2×Q8—34no—
461C2×Q8—34no—
462C2×Q8—34no—
463C2×Q8—158no—
464C2×Q8—154no—
465C2×Q8—304no—
466C2×Q8—304no—
467C2×Q8—3032no—
468C2×Q8—302no—
469C2×Q8—302no—
470C22×C4—32no—
471C22×C4—32no—
472C22×C4—51no—
473C22×C4—51no—
474C22×C4—54no—
475C22×C4—51no—
476C22×C4—102no—
477C22×C4—102no—
478C22×C4—158no—
479C22×C4—152no—
480C22×C4—152no—
481C22×C4—304no—
482C22×C4—304no—
483C22×C4C2×D3×D511yescharacteristic
484C22×C4C2×D3×D511yescharacteristic
485C22×C4C2×D3×D511yescharacteristic
486C4⋊C4—64no—
487C4⋊C4—62no—
488C4⋊C4—304no—
489C4⋊C4—302no—
490C22⋊C4—54no—
491C22⋊C4—102no—
492C22⋊C4—304no—
493C42—22no—
494C42—64no—
495C42—102no—
496C42—304no—
497C3×C5Q8×C22⋊C411yescharacteristic, complemented, Hall
498C22×C3—54no—
499C22×C3—102no—
500C22×C3—102no—
501C22×C3—102no—
502C22×C3—102no—
503C22×C3—102no—
504C22×C3—102no—
505C22×C3C22×C4×D511yescharacteristic
506C22×C3C22×D2011yescharacteristic
507C22×C3C2×Q8×D511yescharacteristic
508C22×C3#414711yescharacteristic
509C22×C3D5×#4111yescharacteristic
510C22×C3#415011yescharacteristic
511C22×C3C22×C22⋊D511yescharacteristic
512C3×C4—22no—
513C3×C4—41no—
514C3×C4—102no—
515C3×C4—102no—
516C3×C4—201no—
517C3×C4C2×#68514yes—
518Dic3—42no—
519Dic3—104no—
520Dic3—202no—
521Dic3C2×#68518yes—
522C2×C5C2×C4×Dic611yescharacteristic
523C2×C5C2×D3×C22⋊C411yescharacteristic
524C2×C5#551611yescharacteristic
525C2×C5D4×Dic611yescharacteristic
526C2×C5D4×Dic611yescharacteristic
527C2×C5#555211yescharacteristic
528C2×C5#555211yescharacteristic
529D5—24no—
530C23—52no—
531C23—52no—
532C23—52no—
533C23—101no—
534C23—101no—
535C23—101no—
536C23—101no—
537C23C22×D3×D511yescharacteristic
538Q8—316no—
539Q8—3016no—
540Q8—308no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—34no—
547C2×C4—34no—
548C2×C4—34no—
549C2×C4—52no—
550C2×C4—52no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—101no—
555C2×C4—101no—
556C2×C4—101no—
557C2×C4—101no—
558C2×C4—108no—
559C2×C4—101no—
560C2×C4—101no—
561C2×C4—101no—
562C2×C4—154no—
563C2×C4—3016no—
564C2×C4—302no—
565C2×C4—302no—
566C2×C4—302no—
567C2×C4—302no—
568C2×C4—302no—
569C2×C4D5×Dic612yescomplemented
570C2×C4C4×D3×D512yes—
571C2×C4D3×D2012yes—
572C2×C4D3×C22⋊D512yes—
573C2×C3—104no—
574C2×C3C4×Q8×D511yescharacteristic
575C2×C3#6590611yescharacteristic
576C2×C3Q8×D2011yescharacteristic
577C2×C3#6591011yescharacteristic
578C2×C3C22×#68511yescharacteristic
579C2×C3Q8×C22⋊D511yescharacteristic
580C2×C3#6616511yescharacteristic
581C5C22⋊C4×Dic611yescharacteristic, complemented, Sylow
582C22—54no—
583C22—102no—
584C22—102no—
585C22—102no—
586C22—102no—
587C22—102no—
588C22—102no—
589C22C2×D5×Dic611yescharacteristic
590C22#9200011yescharacteristic
591C22#9200111yescharacteristic
592C22C2×C4×D3×D511yescharacteristic
593C22C2×D3×D2011yescharacteristic
594C22D5×#23511yescharacteristic
595C22C2×D3×C22⋊D511yescharacteristic
596C4—22no—
597C4—38no—
598C4—41no—
599C4—102no—
600C4—102no—
601C4—122no—
602C4—201no—
603C4—304no—
604C4—602no—
605C4D3×#68514yes—
606C3Q8×#68511yescharacteristic, complemented, Sylow
607C2—104no—
608C2C4×D5×Dic611yescharacteristic
609C2#1175251611yescharacteristic
610C2Dic6×D2011yescharacteristic
611C2#1175254111yescharacteristic
612C2Dic6×C22⋊D511yescharacteristic
613C2#1175433811yescharacteristic
614C2C2×D3×#68511yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C42×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC23×C3×D5C23
hypercenterC2×C42D3×D5
hypocenterC3×C5Q8×C22⋊C4
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
2. C2×C4×C5×Dic6C2semidirect
3. C2×Dic5×Dic6C2semidirect
4. #11751030C2semidirect
6. #11748934C2semidirect
10. #11748631C2semidirect
11. #11748587C2semidirect
20. C2×D5×Dic6C4semidirect
86. C3×#685Q8semidirect
93. C2×C4×Dic6D5semidirect
147. C2×C3×C4×C5C2×Q8semidirect
161. C5×Dic6C22⋊C4semidirect
170. C2×C3×Dic5C2×Q8semidirect
235. #685Dic6direct (N × H)
264. C2×C3×D5C4×Q8semidirect
333. C2×C4×C5C2×Dic6semidirect
350. C2×Dic5C2×Dic6semidirect
417. C2×C3×C4Q8×D5semidirect
434. Dic6#685direct (N × H)
448. C2×D5C4×Dic6semidirect
497. C3×C5Q8×C22⋊C4semidirect
569. C2×C4D5×Dic6semidirect
581. C5C22⋊C4×Dic6semidirect
606. C3Q8×#685semidirect

Maximal subgroups

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

subgroupindex
C22×D5×Dic62
C2×C4×C5×Dic62
C2×Dic5×Dic62
#117510302
C3×C4×#6852
#117489342
#117488782
Dic3×#6852
#117487592
#117486312
#117485872
Q8×#6853
C22⋊C4×Dic65