Finite Group Databasebeta

← C3×#10615564 #14424196

Subgroups — C3×#10615564 #14424196

Order 1920.

1568 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4144C211yesmaximal, characteristic, complemented
2C22×C3×C5×D8C211yesmaximal, characteristic, complemented
3C3×#66139C211yesmaximal, characteristic, complemented
4C2×C3×#4078C211yesmaximal, characteristic, complemented
5C3×#65504C211yesmaximal, characteristic, complemented
6C3×#65447C214yesmaximal, complemented
7C3×#65445C214yesmaximal, complemented
8C3×#65402C211yesmaximal, characteristic, complemented
9C3×#65399C211yesmaximal, characteristic, complemented
10#10615564C311yesmaximal, characteristic, direct product factor, Hall
11C22×C3×C5×D4C2211yescharacteristic
12C22×C3×C5×D4C2211yescharacteristic, complemented
13C22×C3×C22⋊D5—22no—
14C3×#4144—24no—
15C3×#4144—24no—
16C2×C3×#711—21no—
17C2×C3×#711C2212yescomplemented
18C2×C3×D4×D5—21no—
19C2×C3×D4×D5C2212yescomplemented
20C2×C3×#709C2211yescharacteristic
21C2×C3×#709C2211yescharacteristic
22C2×C3×C5×D8—24no—
23C2×C3×C5×D8C2214yescomplemented
24C22×C3×C5×C8C2211yescharacteristic, complemented
25C3×#4099—22no—
26C3×#4086—21no—
27C3×#4084C2214yescomplemented
28C3×#4082—21no—
29C2×C3×#687C2212yescomplemented
30C3×#4078—24no—
31C2×C3×#686C2212yescomplemented
32C3×#4072C2211yescharacteristic, complemented
33C2×C3×#681C2211yescharacteristic, complemented
34C3×#3964C2214yescomplemented
35C3×#3953C2212yescomplemented
36C3×#3952C2214yescomplemented
37C3×#3948C2212yescomplemented
38C3×#3008—51nomaximal, Hall
39C2×#4144C2×C311yescharacteristic, complemented
40C22×C5×D8C2×C311yescharacteristic, complemented
41#66139C2×C311yescharacteristic, complemented
42C2×#4078C2×C311yescharacteristic, complemented
43#65504C2×C311yescharacteristic, complemented
44#65447C2×C314yescomplemented
45#65445C2×C314yescomplemented
46#65402C2×C311yescharacteristic, complemented
47#65399C2×C311yescharacteristic, complemented
48C24×C3×C5—21no—
49C24×C3×C5—21no—
50C23×C3×D5—22no—
51C2×C3×C5×D4—21no—
52C2×C3×C5×D4—22no—
53C2×C3×C5×D4—22no—
54C2×C3×C5×D4—22no—
55C2×C3×C5×D4—21no—
56C2×C3×C5×D4D414yescomplemented
57C2×C3×C5×D4C2312yes—
58C2×C3×C5×D4C2312yes—
59C22×C3×C4×C5C2311yescharacteristic
60C2×C3×C22⋊D5—216no—
61C2×C3×C22⋊D5—21no—
62C2×C3×C22⋊D5—22no—
63C2×C3×C22⋊D5—24no—
64C2×C3×C22⋊D5—22no—
65C2×C3×C22⋊D5D412yes—
66C22×C3×Dic5—22no—
67C3×#711—28no—
68C3×#711—24no—
69C3×#711—24no—
70C3×D4×D5—28no—
71C3×D4×D5—24no—
72C3×D4×D5—24no—
73C3×#709—22no—
74C3×#709—22no—
75C3×#709—22no—
76C3×#709D414yescomplemented
77C2×C3×D20D411yescharacteristic
78C2×C3×D20C2311yescharacteristic
79C2×C3×C4×D5—21no—
80C2×C3×C4×D5C2312yes—
81C2×C3×Dic10D411yescharacteristic
82C2×C3×Dic10C2311yescharacteristic
83C3×C5×D8—28no—
84C2×C3×C5×C8—22no—
85C2×C3×C5×C8C2312yes—
86C3×C22⋊Dic5—21no—
87C3×C22⋊Dic5—24no—
88C3×#687—24no—
89C3×#686—24no—
90C3×C4⋊Dic5C2312yes—
91C3×C4×Dic5—21no—
92C3×#681—22no—
93C2×C3×#168C2312yes—
94C2×C3×H(5,𝔽2)—51no—
95C2×C3×AGL(1,ℤ/8)—51no—
96C22×C3×D8D511yescharacteristic, complemented
97C3×#535—51no—
98C3×#449—54no—
99C3×#447—54no—
100C3×#418—51no—
101C3×#416—51no—
102C3×#408—51no—
103C22×C5×D4C22×C311yescharacteristic
104C22×C5×D4C22×C311yescharacteristic, complemented
105C22×C22⋊D5—22no—
106#4144—24no—
107#4144—24no—
108C2×#711—21no—
109C2×#711C22×C312yescomplemented
110C2×D4×D5—21no—
111C2×D4×D5C22×C312yescomplemented
112C2×#709C22×C311yescharacteristic
113C2×#709C22×C311yescharacteristic
114C2×C5×D8—24no—
115C2×C5×D8C22×C314yescomplemented
116C22×C5×C8C22×C311yescharacteristic, complemented
117#4099—22no—
118#4086—21no—
119#4084C22×C314yescomplemented
120#4082—21no—
121C2×#687C22×C312yescomplemented
122#4078—24no—
123C2×#686C22×C312yescomplemented
124#4072C22×C311yescharacteristic, complemented
125C2×#681C22×C311yescharacteristic, complemented
126#3964C22×C314yescomplemented
127#3953C22×C312yescomplemented
128#3952C22×C314yescomplemented
129#3948C22×C312yescomplemented
130#3008—51noSylow
131C23×C3×C5—22no—
132C23×C3×C5—24no—
133C23×C3×C5—22no—
134C23×C3×C5—42no—
135C23×C3×C5—82no—
136C23×C3×C5—82no—
137C23×C3×C5C2×D411yescharacteristic
138C22×C3×D5—21no—
139C22×C3×D5—28no—
140C22×C3×D5—84no—
141C22×C3×D5C2×D412yes—
142C3×C5×D4—24no—
143C3×C5×D4—22no—
144C3×C5×D4—22no—
145C3×C5×D4—22no—
146C3×C5×D4—42no—
147C3×C5×D4C2×D414yes—
148C2×C3×C4×C5C2×D411yescharacteristic
149C2×C3×C4×C5C2×D412yes—
150C2×C3×C4×C5C2×D412yes—
151C2×C3×C4×C5C2411yescharacteristic
152C3×C22⋊D5—28no—
153C3×C22⋊D5—28no—
154C3×C22⋊D5—24no—
155C3×C22⋊D5—24no—
156C3×C22⋊D5—84no—
157C2×C3×Dic5—28no—
158C2×C3×Dic5—21no—
159C2×C3×Dic5—22no—
160C2×C3×Dic5C2×D412yes—
161C3×D20—22no—
162C3×D20—21no—
163C3×D20C2×D412yes—
164C3×C4×D5—24no—
165C3×C4×D5—22no—
166C3×C4×D5—22no—
167C3×Dic10—22no—
168C3×Dic10—21no—
169C3×Dic10C2×D412yes—
170C3×C5×C8—22no—
171C3×#168—22no—
172C3×H(5,𝔽2)—104no—
173C3×H(5,𝔽2)—104no—
174C2×C3×#41—51no—
175C2×C3×#41—52no—
176C2×C3×#41—51no—
177C2×C3×#41—101no—
178C22×C3×D4—52no—
179C22×C3×D4—101no—
180C22×C3×D4—102no—
181C22×C3×D4C2×D511yescharacteristic
182C22×C3×D4C2×D511yescharacteristic, complemented
183C3×AGL(1,ℤ/8)—104no—
184C2×C3×SD16—52no—
185C2×C3×D8—24no—
186C2×C3×D8—52no—
187C2×C3×D8C2×D514yescomplemented
188C2×C3×M16—51no—
189C22×C3×C8C2×D511yescharacteristic, complemented
190C3×#127—101no—
191C3×#124—101no—
192C3×#121—54no—
193C3×#120—102no—
194C3×#117—51no—
195C3×#103—52no—
196C3×#102—54no—
197C3×#102—52no—
198C3×C22⋊C8—54no—
199C24×C5—21no—
200C24×C5—21no—
201C23×D5—22no—
202C2×C5×D4—21no—
203C2×C5×D4—22no—
204C2×C5×D4—22no—
205C2×C5×D4—22no—
206C2×C5×D4—21no—
207C2×C5×D4C3×D414yescomplemented
208C2×C5×D4C23×C312yes—
209C2×C5×D4C23×C312yes—
210C22×C4×C5C23×C311yescharacteristic
211C2×C22⋊D5—216no—
212C2×C22⋊D5—21no—
213C2×C22⋊D5—22no—
214C2×C22⋊D5—24no—
215C2×C22⋊D5—22no—
216C2×C22⋊D5C3×D412yes—
217C22×Dic5—22no—
218#711—28no—
219#711—24no—
220#711—24no—
221D4×D5—28no—
222D4×D5—24no—
223D4×D5—24no—
224#709—22no—
225#709—22no—
226#709—22no—
227#709C3×D414yescomplemented
228C2×D20C3×D411yescharacteristic
229C2×D20C23×C311yescharacteristic
230C2×C4×D5—21no—
231C2×C4×D5C23×C312yes—
232C2×Dic10C3×D411yescharacteristic
233C2×Dic10C23×C311yescharacteristic
234C5×D8—28no—
235C2×C5×C8—22no—
236C2×C5×C8C23×C312yes—
237C22⋊Dic5—21no—
238C22⋊Dic5—24no—
239#687—24no—
240#686—24no—
241C4⋊Dic5C23×C312yes—
242C4×Dic5—21no—
243#681—22no—
244C2×#168C23×C312yes—
245C2×H(5,𝔽2)—51no—
246C2×AGL(1,ℤ/8)—51no—
247C22×D8C3×D511yescharacteristic, complemented
248#535—51no—
249#449—54no—
250#447—54no—
251#418—51no—
252#416—51no—
253#408—51no—
254C22×C3×C5—22no—
255C22×C3×C5—24no—
256C22×C3×C5—44no—
257C22×C3×C5—42no—
258C22×C3×C5—44no—
259C22×C3×C5—84no—
260C22×C3×C5—84no—
261C22×C3×C5#12012yes—
262C22×C3×C5C22×D411yescharacteristic
263C2×C3×D5—22no—
264C2×C3×D5—22no—
265C2×C3×D5—24no—
266C2×C3×D5—42no—
267C2×C3×D5—88no—
268C3×C4×C5#12012yes—
269C3×C4×C5C22×D411yescharacteristic
270C3×C4×C5C22×D411yescharacteristic
271C3×Dic5—22no—
272C3×Dic5—22no—
273C3×Dic5—42no—
274C24×C3—21no—
275C24×C3—21no—
276C24×C3—102no—
277C3×#41—54no—
278C3×#41—104no—
279C3×#41—108no—
280C3×#41—104no—
281C3×#41—102no—
282C3×#41—102no—
283C3×#41—102no—
284C2×C3×Q8—51no—
285C2×C3×Q8—51no—
286C2×C3×D4—21no—
287C2×C3×D4—22no—
288C2×C3×D4—22no—
289C2×C3×D4—21no—
290C2×C3×D4—22no—
291C2×C3×D4—52no—
292C2×C3×D4—51no—
293C2×C3×D4—51no—
294C2×C3×D4—102no—
295C2×C3×D4—104no—
296C2×C3×D4—108no—
297C2×C3×D4—102no—
298C2×C3×D4—1016no—
299C2×C3×D4—101no—
300C2×C3×D4—104no—
301C2×C3×D4—104no—
302C2×C3×D4C22⋊D514yescomplemented
303C2×C3×D4C22×D512yes—
304C2×C3×D4C22×D512yes—
305C22×C3×C4—52no—
306C22×C3×C4—102no—
307C22×C3×C4—101no—
308C22×C3×C4C22×D511yescharacteristic
309C3×SD16—104no—
310C3×D8—28no—
311C3×D8—104no—
312C3×M16—102no—
313C2×C3×C8—22no—
314C2×C3×C8—52no—
315C2×C3×C8C22×D512yes—
316C3×C4⋊C4—52no—
317C3×C22⋊C4—104no—
318C3×C22⋊C4—101no—
319C3×C42—101no—
320C23×C5—22no—
321C23×C5—24no—
322C23×C5—22no—
323C23×C5—42no—
324C23×C5—82no—
325C23×C5—82no—
326C23×C5C2×C3×D411yescharacteristic
327C22×D5—21no—
328C22×D5—28no—
329C22×D5—84no—
330C22×D5C2×C3×D412yes—
331C5×D4—24no—
332C5×D4—22no—
333C5×D4—22no—
334C5×D4—22no—
335C5×D4—42no—
336C5×D4C2×C3×D414yes—
337C2×C4×C5C2×C3×D411yescharacteristic
338C2×C4×C5C2×C3×D412yes—
339C2×C4×C5C2×C3×D412yes—
340C2×C4×C5C24×C311yescharacteristic
341C22⋊D5—28no—
342C22⋊D5—28no—
343C22⋊D5—24no—
344C22⋊D5—24no—
345C22⋊D5—84no—
346C2×Dic5—28no—
347C2×Dic5—21no—
348C2×Dic5—22no—
349C2×Dic5C2×C3×D412yes—
350D20—22no—
351D20—21no—
352D20C2×C3×D412yes—
353C4×D5—24no—
354C4×D5—22no—
355C4×D5—22no—
356Dic10—22no—
357Dic10—21no—
358Dic10C2×C3×D412yes—
359C5×C8—22no—
360#168—22no—
361H(5,𝔽2)—104no—
362H(5,𝔽2)—104no—
363C2×#41—51no—
364C2×#41—52no—
365C2×#41—51no—
366C2×#41—101no—
367C22×D4—52no—
368C22×D4—101no—
369C22×D4—102no—
370C22×D4C2×C3×D511yescharacteristic
371C22×D4C2×C3×D511yescharacteristic, complemented
372AGL(1,ℤ/8)—104no—
373C2×SD16—52no—
374C2×D8—24no—
375C2×D8—52no—
376C2×D8C2×C3×D514yescomplemented
377C2×M16—51no—
378C22×C8C2×C3×D511yescharacteristic, complemented
379#127—101no—
380#124—101no—
381#121—54no—
382#120—102no—
383#117—51no—
384#103—52no—
385#102—54no—
386#102—52no—
387C22⋊C8—54no—
388C2×C3×C5—22no—
389C2×C3×C5—44no—
390C2×C3×C5—82no—
391C2×C3×C5C2×#12011yescharacteristic
392C2×C3×C5#57612yes—
393C3×D5—42no—
394C3×D5—42no—
395C23×C3—22no—
396C23×C3—24no—
397C23×C3—22no—
398C23×C3—42no—
399C23×C3—52no—
400C23×C3—82no—
401C23×C3—82no—
402C23×C3—108no—
403C23×C3—101no—
404C23×C3—404no—
405C23×C3D4×D511yescharacteristic
406C3×Q8—52no—
407C3×Q8—101no—
408C3×Q8—102no—
409C3×D4—22no—
410C3×D4—22no—
411C3×D4—22no—
412C3×D4—24no—
413C3×D4—42no—
414C3×D4—52no—
415C3×D4—108no—
416C3×D4—108no—
417C3×D4—104no—
418C3×D4—104no—
419C3×D4—101no—
420C3×D4—102no—
421C3×D4—404no—
422C3×D4C2×C22⋊D514yes—
423C2×C3×C4—52no—
424C2×C3×C4—108no—
425C2×C3×C4—101no—
426C2×C3×C4—102no—
427C2×C3×C4—102no—
428C2×C3×C4—104no—
429C2×C3×C4—102no—
430C2×C3×C4D4×D512yes—
431C2×C3×C4D4×D511yescharacteristic
432C2×C3×C4C2×C22⋊D512yes—
433C2×C3×C4C23×D511yescharacteristic
434C3×C8—22no—
435C3×C8—102no—
436C22×C5—22no—
437C22×C5—24no—
438C22×C5—44no—
439C22×C5—44no—
440C22×C5—42no—
441C22×C5—84no—
442C22×C5—84no—
443C22×C5C3×#12012yes—
444C22×C5C22×C3×D411yescharacteristic
445C2×D5—24no—
446C2×D5—22no—
447C2×D5—22no—
448C2×D5—42no—
449C2×D5—88no—
450C4×C5C3×#12012yes—
451C4×C5C22×C3×D411yescharacteristic
452C4×C5C22×C3×D411yescharacteristic
453Dic5—22no—
454Dic5—22no—
455Dic5—42no—
456C24—21no—
457C24—21no—
458C24—102no—
459#41—54no—
460#41—104no—
461#41—108no—
462#41—104no—
463#41—102no—
464#41—102no—
465#41—102no—
466C2×Q8—51no—
467C2×Q8—51no—
468C2×D4—21no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—21no—
472C2×D4—22no—
473C2×D4—52no—
474C2×D4—51no—
475C2×D4—51no—
476C2×D4—102no—
477C2×D4—104no—
478C2×D4—108no—
479C2×D4—102no—
480C2×D4—1016no—
481C2×D4—101no—
482C2×D4—104no—
483C2×D4—104no—
484C2×D4C3×C22⋊D514yescomplemented
485C2×D4C22×C3×D512yes—
486C2×D4C22×C3×D512yes—
487C22×C4—52no—
488C22×C4—102no—
489C22×C4—101no—
490C22×C4C22×C3×D511yescharacteristic
491SD16—104no—
492D8—28no—
493D8—104no—
494M16—102no—
495C2×C8—22no—
496C2×C8—52no—
497C2×C8C22×C3×D512yes—
498C4⋊C4—52no—
499C22⋊C4—104no—
500C22⋊C4—101no—
501C42—101no—
502C3×C5#300811yescharacteristic, complemented, Hall
503C22×C3—22no—
504C22×C3—24no—
505C22×C3—44no—
506C22×C3—44no—
507C22×C3—42no—
508C22×C3—84no—
509C22×C3—84no—
510C22×C3—104no—
511C22×C3—102no—
512C22×C3—102no—
513C22×C3—202no—
514C22×C3—408no—
515C22×C3#408312yes—
516C22×C3C2×D4×D511yescharacteristic
517C3×C4—102no—
518C3×C4—102no—
519C3×C4—202no—
520C3×C4#408312yes—
521C3×C4C2×D4×D511yescharacteristic
522C3×C4C22×C22⋊D511yescharacteristic
523C2×C5—22no—
524C2×C5—44no—
525C2×C5—82no—
526C2×C5C2×C3×#12011yescharacteristic
527C2×C5C3×#57612yes—
528D5—42no—
529D5—42no—
530C23—22no—
531C23—24no—
532C23—22no—
533C23—42no—
534C23—52no—
535C23—82no—
536C23—82no—
537C23—108no—
538C23—101no—
539C23—404no—
540C23C3×D4×D511yescharacteristic
541Q8—52no—
542Q8—102no—
543Q8—101no—
544D4—24no—
545D4—22no—
546D4—22no—
547D4—22no—
548D4—42no—
549D4—52no—
550D4—108no—
551D4—108no—
552D4—104no—
553D4—104no—
554D4—102no—
555D4—101no—
556D4—404no—
557D4C2×C3×C22⋊D514yes—
558C2×C4—52no—
559C2×C4—108no—
560C2×C4—101no—
561C2×C4—104no—
562C2×C4—102no—
563C2×C4—102no—
564C2×C4—102no—
565C2×C4C3×D4×D511yescharacteristic
566C2×C4C3×D4×D512yes—
567C2×C4C2×C3×C22⋊D512yes—
568C2×C4C23×C3×D511yescharacteristic
569C8—22no—
570C8—102no—
571C2×C3—22no—
572C2×C3—44no—
573C2×C3—82no—
574C2×C3—202no—
575C2×C3—202no—
576C2×C3#6609112yes—
577C2×C3C2×#408311yescharacteristic
578C5C3×#300811yescharacteristic, complemented, Sylow
579C22—22no—
580C22—24no—
581C22—44no—
582C22—42no—
583C22—44no—
584C22—84no—
585C22—84no—
586C22—102no—
587C22—102no—
588C22—104no—
589C22—202no—
590C22—408no—
591C22C3×#408312yes—
592C22C2×C3×D4×D511yescharacteristic
593C4—102no—
594C4—102no—
595C4—202no—
596C4C3×#408312yes—
597C4C2×C3×D4×D511yescharacteristic
598C4C22×C3×C22⋊D511yescharacteristic
599C3#1061556411yescharacteristic, direct product factor, Sylow
600C2—22no—
601C2—44no—
602C2—82no—
603C2—202no—
604C2—202no—
605C2C3×#6609112yes—
606C2C2×C3×#408311yescharacteristic
subgroupG/H
centerC22×C3C2×D4×D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC22×C3×C5×D8C2
socleC22×C3×C5C22×D4
hypercenterC22×C3×D8D5
hypocenterC5C3×#3008
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×D8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#4144C2semidirect
2. C22×C3×C5×D8C2semidirect
3. C3×#66139C2semidirect
4. C2×C3×#4078C2semidirect
5. C3×#65504C2semidirect
6. C3×#65447C2semidirect
7. C3×#65445C2semidirect
8. C3×#65402C2semidirect
9. C3×#65399C2semidirect
10. #10615564C3direct (N × H)
12. C22×C3×C5×D4C22semidirect
17. C2×C3×#711C22semidirect
19. C2×C3×D4×D5C22semidirect
23. C2×C3×C5×D8C22semidirect
24. C22×C3×C5×C8C22semidirect
27. C3×#4084C22semidirect
29. C2×C3×#687C22semidirect
31. C2×C3×#686C22semidirect
32. C3×#4072C22semidirect
33. C2×C3×#681C22semidirect
34. C3×#3964C22semidirect
35. C3×#3953C22semidirect
36. C3×#3952C22semidirect
37. C3×#3948C22semidirect
39. C2×#4144C2×C3semidirect
40. C22×C5×D8C2×C3semidirect
41. #66139C2×C3semidirect
42. C2×#4078C2×C3semidirect
43. #65504C2×C3semidirect
44. #65447C2×C3semidirect
45. #65445C2×C3semidirect
46. #65402C2×C3semidirect
47. #65399C2×C3semidirect
56. C2×C3×C5×D4D4semidirect
76. C3×#709D4semidirect
96. C22×C3×D8D5semidirect
104. C22×C5×D4C22×C3semidirect
109. C2×#711C22×C3semidirect
111. C2×D4×D5C22×C3semidirect
115. C2×C5×D8C22×C3semidirect
116. C22×C5×C8C22×C3semidirect
119. #4084C22×C3semidirect
121. C2×#687C22×C3semidirect
123. C2×#686C22×C3semidirect
124. #4072C22×C3semidirect
125. C2×#681C22×C3semidirect
126. #3964C22×C3semidirect
127. #3953C22×C3semidirect
128. #3952C22×C3semidirect
129. #3948C22×C3semidirect
182. C22×C3×D4C2×D5semidirect
187. C2×C3×D8C2×D5semidirect
189. C22×C3×C8C2×D5semidirect
207. C2×C5×D4C3×D4semidirect
227. #709C3×D4semidirect
247. C22×D8C3×D5semidirect
302. C2×C3×D4C22⋊D5semidirect
371. C22×D4C2×C3×D5semidirect
376. C2×D8C2×C3×D5semidirect
378. C22×C8C2×C3×D5semidirect
484. C2×D4C3×C22⋊D5semidirect
502. C3×C5#3008semidirect
578. C5C3×#3008semidirect
599. C3#10615564direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#41442
C22×C3×C5×D82
C3×#661392
C2×C3×#40782
C3×#655042
C3×#654472
C3×#654452
C3×#654022
C3×#653992
#106155643
C3×#30085