Finite Group Databasebeta

← C3×#10602891 #14307981

Subgroups — C3×#10602891 #14307981

Order 1920.

812 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×C5×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
2C3×#66167C211yesmaximal, characteristic, complemented
3C2×C3×#4078C211yesmaximal, characteristic, complemented
4C2×C3×#3969C211yesmaximal, characteristic, complemented
5C3×#65510C211yesmaximal, characteristic, complemented
6C2×C3×#3957C211yesmaximal, characteristic, complemented
7C3×#64774C211yesmaximal, characteristic, complemented
8#10602891C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C22×C3×C5×D4C2211yescharacteristic
11C2×C3×#709C2211yescharacteristic
12C3×C5×AGL(1,ℤ/8)—22no—
13C3×C5×AGL(1,ℤ/8)—22no—
14C2×C3×C5×SD16—21no—
15C2×C3×C5×D8—21no—
16C2×C3×C5×M16C2211yescharacteristic
17C2×C3×C22⋊Dic5—22no—
18C3×#4092—21no—
19C3×#4085—22no—
20C3×#4084—21no—
21C2×C3×#687—21no—
22C3×#4078—22no—
23C3×#4078—22no—
24C2×C3×#686—21no—
25C3×#4076—21no—
26C2×C3×C4⋊Dic5C2211yescharacteristic
27C3×#4070—21no—
28C2×C3×C4×Dic5C2211yescharacteristic
29C2×C3×#681C2211yescharacteristic
30C3×#3969—22no—
31C3×#3957—22no—
32C3×#2006—51nomaximal, Hall
33C2×C5×AGL(1,ℤ/8)C2×C311yescharacteristic, complemented
34#66167C2×C311yescharacteristic, complemented
35C2×#4078C2×C311yescharacteristic, complemented
36C2×#3969C2×C311yescharacteristic, complemented
37#65510C2×C311yescharacteristic, complemented
38C2×#3957C2×C311yescharacteristic, complemented
39#64774C2×C311yescharacteristic, complemented
40C24×C3×C5—21no—
41C3×C5×#41—21no—
42C3×C5×#41—21no—
43C3×C5×#41—21no—
44C3×C5×#41—21no—
45C2×C3×C5×Q8D411yescharacteristic
46C2×C3×C5×D4—21no—
47C2×C3×C5×D4—21no—
48C2×C3×C5×D4—21no—
49C2×C3×C5×D4—21no—
50C2×C3×C5×D4—21no—
51C2×C3×C5×D4—21no—
52C2×C3×C5×D4D411yescharacteristic
53C2×C3×C5×D4D412yescomplemented
54C22×C3×C4×C5—21no—
55C22×C3×C4×C5C2311yescharacteristic
56C2×C3×C22⋊D5—21no—
57C2×C3×C22⋊D5—22no—
58C22×C3×Dic5—21no—
59C22×C3×Dic5D412yes—
60C3×#709—21no—
61C3×#709—21no—
62C3×#709—21no—
63C3×#709—21no—
64C2×C3×D20D411yescharacteristic
65C2×C3×C4×D5—21no—
66C2×C3×Dic10D411yescharacteristic
67C3×C5×SD16—24no—
68C3×C5×D8—24no—
69C3×C5×M16—21no—
70C3×C5×M16D412yescomplemented
71C2×C3×C5×C8—21no—
72C3×C22⋊Dic5—22no—
73C3×C22⋊Dic5—44no—
74C3×#687—24no—
75C3×#686—24no—
76C3×#685—22no—
77C3×C4⋊Dic5—21no—
78C3×C4⋊Dic5—21no—
79C3×#683—22no—
80C3×C4×Dic5—21no—
81C3×C4×Dic5—21no—
82C3×#681—21no—
83C3×#681D412yescomplemented
84C2×C3×#168—21no—
85C2×C3×AGL(1,ℤ/8)—51no—
86C2×C3×AGL(1,ℤ/8)D511yescharacteristic, complemented
87C3×#537—51no—
88C2×C3×#104—51no—
89C2×C3×#104—51no—
90C3×#390—51no—
91C3×#343—51no—
92C2×C5×#41C22×C311yescharacteristic
93C22×C5×D4C22×C311yescharacteristic
94C2×#709C22×C311yescharacteristic
95C5×AGL(1,ℤ/8)—22no—
96C5×AGL(1,ℤ/8)—22no—
97C2×C5×SD16—21no—
98C2×C5×D8—21no—
99C2×C5×M16C22×C311yescharacteristic
100C2×C22⋊Dic5—22no—
101#4092—21no—
102#4085—22no—
103#4084—21no—
104C2×#687—21no—
105#4078—22no—
106#4078—22no—
107C2×#686—21no—
108#4076—21no—
109C2×C4⋊Dic5C22×C311yescharacteristic
110#4070—21no—
111C2×C4×Dic5C22×C311yescharacteristic
112C2×#681C22×C311yescharacteristic
113#3969—22no—
114#3957—22no—
115#2006—51noSylow
116C23×C3×C5—21no—
117C23×C3×C5—22no—
118C23×C3×C5—42no—
119C23×C3×C5—42no—
120C23×C3×C5—41no—
121C23×C3×C5—41no—
122C23×C3×C5C2×D411yescharacteristic
123C22×C3×D5—21no—
124C3×C5×Q8—21no—
125C3×C5×Q8—21no—
126C3×C5×D4—22no—
127C3×C5×D4—21no—
128C3×C5×D4—22no—
129C3×C5×D4—21no—
130C3×C5×D4—41no—
131C3×C5×D4—41no—
132C3×C5×D4—41no—
133C3×C5×D4—41no—
134C2×C3×C4×C5—21no—
135C2×C3×C4×C5—41no—
136C2×C3×C4×C5—41no—
137C2×C3×C4×C5C2×D411yescharacteristic
138C2×C3×C4×C5C2×D411yescharacteristic
139C2×C3×C4×C5C2×D411yescharacteristic
140C2×C3×C4×C5C2×D411yescharacteristic
141C2×C3×C4×C5C2×D411yescharacteristic
142C2×C3×C4×C5#4111yescharacteristic
143C3×C22⋊D5—41no—
144C3×C22⋊D5—41no—
145C3×C22⋊D5—82no—
146C2×C3×Dic5—22no—
147C2×C3×Dic5—21no—
148C2×C3×Dic5—22no—
149C2×C3×Dic5—41no—
150C2×C3×Dic5—41no—
151C2×C3×Dic5—42no—
152C3×D20—21no—
153C3×D20—21no—
154C3×C4×D5—41no—
155C3×C4×D5—41no—
156C3×Dic10—21no—
157C3×Dic10—21no—
158C3×C5×C8—22no—
159C3×#168—22no—
160C2×C3×#41—51no—
161C2×C3×#41C2×D511yescharacteristic
162C22×C3×D4C2×D511yescharacteristic
163C3×AGL(1,ℤ/8)—22no—
164C3×AGL(1,ℤ/8)—22no—
165C3×AGL(1,ℤ/8)—102no—
166C3×AGL(1,ℤ/8)—102no—
167C2×C3×SD16—21no—
168C2×C3×SD16—101no—
169C2×C3×D8—21no—
170C2×C3×D8—101no—
171C2×C3×M16—51no—
172C2×C3×M16C2×D511yescharacteristic
173C3×C22⋊Q8—101no—
174C3×C22⋊Q8—101no—
175C3×#121—101no—
176C3×#121—102no—
177C3×#121—101no—
178C2×C3×C4⋊C4—51no—
179C2×C3×C22⋊C4—102no—
180C2×C3×C42—51no—
181C3×#104—102no—
182C3×#104—102no—
183C24×C5—21no—
184C5×#41—21no—
185C5×#41—21no—
186C5×#41—21no—
187C5×#41—21no—
188C2×C5×Q8C3×D411yescharacteristic
189C2×C5×D4—21no—
190C2×C5×D4—21no—
191C2×C5×D4—21no—
192C2×C5×D4—21no—
193C2×C5×D4—21no—
194C2×C5×D4—21no—
195C2×C5×D4C3×D411yescharacteristic
196C2×C5×D4C3×D412yescomplemented
197C22×C4×C5—21no—
198C22×C4×C5C23×C311yescharacteristic
199C2×C22⋊D5—21no—
200C2×C22⋊D5—22no—
201C22×Dic5—21no—
202C22×Dic5C3×D412yes—
203#709—21no—
204#709—21no—
205#709—21no—
206#709—21no—
207C2×D20C3×D411yescharacteristic
208C2×C4×D5—21no—
209C2×Dic10C3×D411yescharacteristic
210C5×SD16—24no—
211C5×D8—24no—
212C5×M16—21no—
213C5×M16C3×D412yescomplemented
214C2×C5×C8—21no—
215C22⋊Dic5—22no—
216C22⋊Dic5—44no—
217#687—24no—
218#686—24no—
219#685—22no—
220C4⋊Dic5—21no—
221C4⋊Dic5—21no—
222#683—22no—
223C4×Dic5—21no—
224C4×Dic5—21no—
225#681—21no—
226#681C3×D412yescomplemented
227C2×#168—21no—
228C2×AGL(1,ℤ/8)—51no—
229C2×AGL(1,ℤ/8)C3×D511yescharacteristic, complemented
230#537—51no—
231C2×#104—51no—
232C2×#104—51no—
233#390—51no—
234#343—51no—
235C22×C3×C5—21no—
236C22×C3×C5—21no—
237C22×C3×C5—44no—
238C22×C3×C5—41no—
239C22×C3×C5—42no—
240C22×C3×C5—41no—
241C22×C3×C5—41no—
242C22×C3×C5—82no—
243C22×C3×C5—81no—
244C22×C3×C5—81no—
245C22×C3×C5#12011yescharacteristic
246C22×C3×C5#12111yescharacteristic
247C22×C3×C5#12711yescharacteristic
248C2×C3×D5—41no—
249C2×C3×D5—41no—
250C2×C3×D5—41no—
251C3×C4×C5—41no—
252C3×C4×C5#12011yescharacteristic
253C3×C4×C5#12011yescharacteristic
254C3×C4×C5#12111yescharacteristic
255C3×C4×C5#12111yescharacteristic
256C3×Dic5—42no—
257C3×Dic5—41no—
258C3×Dic5—42no—
259C24×C3—21no—
260C3×#41—21no—
261C3×#41—21no—
262C3×#41—21no—
263C3×#41—21no—
264C3×#41—101no—
265C3×#41—101no—
266C3×#41—101no—
267C3×#41—101no—
268C2×C3×Q8—51no—
269C2×C3×Q8C22⋊D511yescharacteristic
270C2×C3×D4—21no—
271C2×C3×D4—21no—
272C2×C3×D4—21no—
273C2×C3×D4—21no—
274C2×C3×D4—21no—
275C2×C3×D4—21no—
276C2×C3×D4—51no—
277C2×C3×D4—101no—
278C2×C3×D4—102no—
279C2×C3×D4C22⋊D511yescharacteristic
280C2×C3×D4C22⋊D512yescomplemented
281C22×C3×C4—21no—
282C22×C3×C4—52no—
283C22×C3×C4—101no—
284C22×C3×C4—101no—
285C22×C3×C4C22×D511yescharacteristic
286C3×SD16—24no—
287C3×SD16—104no—
288C3×D8—24no—
289C3×D8—104no—
290C3×M16—21no—
291C3×M16—52no—
292C3×M16—101no—
293C3×M16C22⋊D512yescomplemented
294C2×C3×C8—21no—
295C2×C3×C8—101no—
296C3×C4⋊C4—102no—
297C3×C4⋊C4—101no—
298C3×C4⋊C4—101no—
299C3×C22⋊C4—102no—
300C3×C22⋊C4—102no—
301C3×C22⋊C4—204no—
302C3×C42—101no—
303C3×C42—101no—
304C23×C5—21no—
305C23×C5—22no—
306C23×C5—42no—
307C23×C5—42no—
308C23×C5—41no—
309C23×C5—41no—
310C23×C5C2×C3×D411yescharacteristic
311C22×D5—21no—
312C5×Q8—21no—
313C5×Q8—21no—
314C5×D4—22no—
315C5×D4—21no—
316C5×D4—22no—
317C5×D4—21no—
318C5×D4—41no—
319C5×D4—41no—
320C5×D4—41no—
321C5×D4—41no—
322C2×C4×C5—21no—
323C2×C4×C5—41no—
324C2×C4×C5—41no—
325C2×C4×C5C2×C3×D411yescharacteristic
326C2×C4×C5C2×C3×D411yescharacteristic
327C2×C4×C5C2×C3×D411yescharacteristic
328C2×C4×C5C2×C3×D411yescharacteristic
329C2×C4×C5C2×C3×D411yescharacteristic
330C2×C4×C5C3×#4111yescharacteristic
331C22⋊D5—41no—
332C22⋊D5—41no—
333C22⋊D5—82no—
334C2×Dic5—22no—
335C2×Dic5—21no—
336C2×Dic5—22no—
337C2×Dic5—41no—
338C2×Dic5—41no—
339C2×Dic5—42no—
340D20—21no—
341D20—21no—
342C4×D5—41no—
343C4×D5—41no—
344Dic10—21no—
345Dic10—21no—
346C5×C8—22no—
347#168—22no—
348C2×#41—51no—
349C2×#41C2×C3×D511yescharacteristic
350C22×D4C2×C3×D511yescharacteristic
351AGL(1,ℤ/8)—22no—
352AGL(1,ℤ/8)—22no—
353AGL(1,ℤ/8)—102no—
354AGL(1,ℤ/8)—102no—
355C2×SD16—21no—
356C2×SD16—101no—
357C2×D8—21no—
358C2×D8—101no—
359C2×M16—51no—
360C2×M16C2×C3×D511yescharacteristic
361C22⋊Q8—101no—
362C22⋊Q8—101no—
363#121—101no—
364#121—102no—
365#121—101no—
366C2×C4⋊C4—51no—
367C2×C22⋊C4—102no—
368C2×C42—51no—
369#104—102no—
370#104—102no—
371C2×C3×C5—21no—
372C2×C3×C5—21no—
373C2×C3×C5—81no—
374C2×C3×C5—82no—
375C2×C3×C5#39211yescharacteristic
376C2×C3×C5#45311yescharacteristic
377C2×C3×C5#45411yescharacteristic
378C3×D5—81no—
379C23×C3—21no—
380C23×C3—22no—
381C23×C3—42no—
382C23×C3—42no—
383C23×C3—41no—
384C23×C3—41no—
385C23×C3—101no—
386C23×C3D4×D511yescharacteristic
387C3×Q8—21no—
388C3×Q8—21no—
389C3×Q8—101no—
390C3×Q8—101no—
391C3×D4—21no—
392C3×D4—22no—
393C3×D4—22no—
394C3×D4—21no—
395C3×D4—41no—
396C3×D4—41no—
397C3×D4—41no—
398C3×D4—41no—
399C3×D4—101no—
400C3×D4—101no—
401C3×D4—201no—
402C3×D4—201no—
403C3×D4—402no—
404C2×C3×C4—21no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—102no—
408C2×C3×C4—102no—
409C2×C3×C4—101no—
410C2×C3×C4—201no—
411C2×C3×C4—201no—
412C2×C3×C4—201no—
413C2×C3×C4—201no—
414C2×C3×C4—202no—
415C2×C3×C4D4×D511yescharacteristic
416C2×C3×C4D4×D511yescharacteristic
417C2×C3×C4#71111yescharacteristic
418C2×C3×C4C2×C22⋊D511yescharacteristic
419C2×C3×C4C2×C22⋊D511yescharacteristic
420C2×C3×C4C2×C22⋊D511yescharacteristic
421C3×C8—22no—
422C3×C8—102no—
423C22×C5—21no—
424C22×C5—21no—
425C22×C5—44no—
426C22×C5—41no—
427C22×C5—42no—
428C22×C5—41no—
429C22×C5—41no—
430C22×C5—82no—
431C22×C5—81no—
432C22×C5—81no—
433C22×C5C3×#12011yescharacteristic
434C22×C5C3×#12111yescharacteristic
435C22×C5C3×#12711yescharacteristic
436C2×D5—41no—
437C2×D5—41no—
438C2×D5—41no—
439C4×C5—41no—
440C4×C5C3×#12011yescharacteristic
441C4×C5C3×#12011yescharacteristic
442C4×C5C3×#12111yescharacteristic
443C4×C5C3×#12111yescharacteristic
444Dic5—42no—
445Dic5—41no—
446Dic5—42no—
447C24—21no—
448#41—21no—
449#41—21no—
450#41—21no—
451#41—21no—
452#41—101no—
453#41—101no—
454#41—101no—
455#41—101no—
456C2×Q8—51no—
457C2×Q8C3×C22⋊D511yescharacteristic
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—51no—
465C2×D4—101no—
466C2×D4—102no—
467C2×D4C3×C22⋊D511yescharacteristic
468C2×D4C3×C22⋊D512yescomplemented
469C22×C4—21no—
470C22×C4—52no—
471C22×C4—101no—
472C22×C4—101no—
473C22×C4C22×C3×D511yescharacteristic
474SD16—24no—
475SD16—104no—
476D8—24no—
477D8—104no—
478M16—21no—
479M16—52no—
480M16—101no—
481M16C3×C22⋊D512yescomplemented
482C2×C8—21no—
483C2×C8—101no—
484C4⋊C4—102no—
485C4⋊C4—101no—
486C4⋊C4—101no—
487C22⋊C4—102no—
488C22⋊C4—102no—
489C22⋊C4—204no—
490C42—101no—
491C42—101no—
492C3×C5#200611yescharacteristic, complemented, Hall
493C22×C3—21no—
494C22×C3—21no—
495C22×C3—44no—
496C22×C3—41no—
497C22×C3—41no—
498C22×C3—41no—
499C22×C3—42no—
500C22×C3—82no—
501C22×C3—81no—
502C22×C3—81no—
503C22×C3—201no—
504C22×C3—201no—
505C22×C3—201no—
506C22×C3#408311yescharacteristic
507C22×C3#408511yescharacteristic
508C22×C3#408611yescharacteristic
509C3×C4—41no—
510C3×C4—202no—
511C3×C4—202no—
512C3×C4—201no—
513C3×C4#408311yescharacteristic
514C3×C4#408411yescharacteristic
515C3×C4#408511yescharacteristic
516C3×C4#409911yescharacteristic
517C2×C5—21no—
518C2×C5—21no—
519C2×C5—82no—
520C2×C5—81no—
521C2×C5C3×#39211yescharacteristic
522C2×C5C3×#45311yescharacteristic
523C2×C5C3×#45411yescharacteristic
524D5—81no—
525C23—21no—
526C23—22no—
527C23—42no—
528C23—42no—
529C23—41no—
530C23—41no—
531C23—101no—
532C23C3×D4×D511yescharacteristic
533Q8—21no—
534Q8—21no—
535Q8—101no—
536Q8—101no—
537D4—22no—
538D4—21no—
539D4—21no—
540D4—22no—
541D4—41no—
542D4—41no—
543D4—41no—
544D4—41no—
545D4—101no—
546D4—101no—
547D4—201no—
548D4—201no—
549D4—402no—
550C2×C4—21no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—102no—
554C2×C4—102no—
555C2×C4—101no—
556C2×C4—201no—
557C2×C4—201no—
558C2×C4—201no—
559C2×C4—201no—
560C2×C4—202no—
561C2×C4C3×D4×D511yescharacteristic
562C2×C4C3×D4×D511yescharacteristic
563C2×C4C3×#71111yescharacteristic
564C2×C4C2×C3×C22⋊D511yescharacteristic
565C2×C4C2×C3×C22⋊D511yescharacteristic
566C2×C4C2×C3×C22⋊D511yescharacteristic
567C8—22no—
568C8—102no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—82no—
572C2×C3—81no—
573C2×C3—401no—
574C2×C3#6548711yescharacteristic
575C2×C3#6549011yescharacteristic
576C2×C3#6551211yescharacteristic
577C5C3×#200611yescharacteristic, complemented, Sylow
578C22—21no—
579C22—21no—
580C22—44no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—42no—
585C22—82no—
586C22—81no—
587C22—81no—
588C22—201no—
589C22—201no—
590C22—201no—
591C22C3×#408311yescharacteristic
592C22C3×#408511yescharacteristic
593C22C3×#408611yescharacteristic
594C4—41no—
595C4—202no—
596C4—202no—
597C4—201no—
598C4C3×#408311yescharacteristic
599C4C3×#408411yescharacteristic
600C4C3×#408511yescharacteristic
601C4C3×#409911yescharacteristic
602C3#1060289111yescharacteristic, direct product factor, Sylow
603C2—21no—
604C2—21no—
605C2—82no—
606C2—81no—
607C2—401no—
608C2C3×#6548711yescharacteristic
609C2C3×#6549011yescharacteristic
610C2C3×#6551211yescharacteristic
subgroupG/H
centerC22×C3#4083
commutator, cosocleC22×C4×C5C23×C3
FrattiniC22×C4C22×C3×D5
FittingC2×C3×C5×AGL(1,ℤ/8)C2
socleC22×C3×C5#120
hypercenterC2×C3×AGL(1,ℤ/8)D5
hypocenterC5C3×#2006
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×AGL(1,ℤ/8)C2semidirect
2. C3×#66167C2semidirect
3. C2×C3×#4078C2semidirect
4. C2×C3×#3969C2semidirect
5. C3×#65510C2semidirect
6. C2×C3×#3957C2semidirect
7. C3×#64774C2semidirect
8. #10602891C3direct (N × H)
33. C2×C5×AGL(1,ℤ/8)C2×C3semidirect
34. #66167C2×C3semidirect
35. C2×#4078C2×C3semidirect
36. C2×#3969C2×C3semidirect
37. #65510C2×C3semidirect
38. C2×#3957C2×C3semidirect
39. #64774C2×C3semidirect
53. C2×C3×C5×D4D4semidirect
70. C3×C5×M16D4semidirect
83. C3×#681D4semidirect
86. C2×C3×AGL(1,ℤ/8)D5semidirect
196. C2×C5×D4C3×D4semidirect
213. C5×M16C3×D4semidirect
226. #681C3×D4semidirect
229. C2×AGL(1,ℤ/8)C3×D5semidirect
280. C2×C3×D4C22⋊D5semidirect
293. C3×M16C22⋊D5semidirect
468. C2×D4C3×C22⋊D5semidirect
481. M16C3×C22⋊D5semidirect
492. C3×C5#2006semidirect
577. C5C3×#2006semidirect
602. C3#10602891direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×AGL(1,ℤ/8)2
C3×#661672
C2×C3×#40782
C2×C3×#39692
C3×#655102
C2×C3×#39572
C3×#647742
#106028913
C3×#20065