Finite Group Databasebeta

← C3×#10615591 #14424418

Subgroups — C3×#10615591 #14424418

Order 1920.

1464 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4147C211yesmaximal, characteristic, complemented
2C22×C3×C5×SD16C211yesmaximal, characteristic, complemented
3C3×#66139C211yesmaximal, characteristic, complemented
4C2×C3×#4078C211yesmaximal, characteristic, complemented
5C3×#65514C211yesmaximal, characteristic, complemented
6C3×#65461C214yesmaximal, complemented
7C3×#65459C214yesmaximal, complemented
8C3×#65402C211yesmaximal, characteristic, complemented
9C3×#65399C211yesmaximal, characteristic, complemented
10#10615591C311yesmaximal, characteristic, direct product factor, Hall
11C22×C3×C5×Q8C2211yescharacteristic
12C22×C3×C5×D4C2211yescharacteristic
13C3×#4147—24no—
14C3×#4147—24no—
15C2×C3×#713—21no—
16C2×C3×#713C2212yescomplemented
17C2×C3×Q8×D5—21no—
18C2×C3×Q8×D5C2212yescomplemented
19C2×C3×#709—22no—
20C2×C3×#709C2211yescharacteristic
21C2×C3×#709C2211yescharacteristic
22C2×C3×C5×SD16—24no—
23C2×C3×C5×SD16C2214yes—
24C22×C3×C5×C8C2211yescharacteristic
25C3×#4099—22no—
26C3×#4086—21no—
27C3×#4084C2214yes—
28C3×#4082—21no—
29C2×C3×#687C2212yes—
30C3×#4078—24no—
31C2×C3×#686C2212yes—
32C3×#4072C2211yescharacteristic
33C2×C3×#681C2211yescharacteristic
34C3×#3967C2214yescomplemented
35C3×#3953C2212yescomplemented
36C3×#3952C2214yescomplemented
37C3×#3948C2212yescomplemented
38C3×#3010—51nomaximal, Hall
39C2×#4147C2×C311yescharacteristic, complemented
40C22×C5×SD16C2×C311yescharacteristic, complemented
41#66139C2×C311yescharacteristic, complemented
42C2×#4078C2×C311yescharacteristic, complemented
43#65514C2×C311yescharacteristic, complemented
44#65461C2×C314yescomplemented
45#65459C2×C314yescomplemented
46#65402C2×C311yescharacteristic, complemented
47#65399C2×C311yescharacteristic, complemented
48C24×C3×C5—21no—
49C2×C3×C5×Q8—21no—
50C2×C3×C5×Q8—22no—
51C2×C3×C5×Q8D414yescomplemented
52C2×C3×C5×Q8C2312yes—
53C2×C3×C5×D4—22no—
54C2×C3×C5×D4—22no—
55C2×C3×C5×D4—21no—
56C2×C3×C5×D4C2312yes—
57C22×C3×C4×C5—21no—
58C22×C3×C4×C5C2311yescharacteristic
59C2×C3×C22⋊D5—21no—
60C2×C3×C22⋊D5—24no—
61C2×C3×C22⋊D5D412yes—
62C3×#713—28no—
63C3×#713—24no—
64C3×#713—24no—
65C3×Q8×D5—28no—
66C3×Q8×D5—24no—
67C3×Q8×D5—24no—
68C3×#709—216no—
69C3×#709—22no—
70C3×#709—22no—
71C3×#709—22no—
72C3×#709D414yescomplemented
73C2×C3×D20—22no—
74C2×C3×D20D411yescharacteristic
75C2×C3×D20C2311yescharacteristic
76C2×C3×C4×D5—22no—
77C2×C3×C4×D5—21no—
78C2×C3×C4×D5—22no—
79C2×C3×C4×D5C2312yes—
80C2×C3×Dic10—22no—
81C2×C3×Dic10D411yescharacteristic
82C2×C3×Dic10C2311yescharacteristic
83C3×C5×SD16—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×C3×2−1+4—51no—
95C2×C3×AGL(1,ℤ/8)—51no—
96C22×C3×SD16D511yescharacteristic, complemented
97C3×#535—51no—
98C3×#449—54no—
99C3×#448—54no—
100C3×#419—51no—
101C3×#416—51no—
102C3×#408—51no—
103C22×C5×Q8C22×C311yescharacteristic
104C22×C5×D4C22×C311yescharacteristic
105#4147—24no—
106#4147—24no—
107C2×#713—21no—
108C2×#713C22×C312yescomplemented
109C2×Q8×D5—21no—
110C2×Q8×D5C22×C312yescomplemented
111C2×#709—22no—
112C2×#709C22×C311yescharacteristic
113C2×#709C22×C311yescharacteristic
114C2×C5×SD16—24no—
115C2×C5×SD16C22×C314yes—
116C22×C5×C8C22×C311yescharacteristic
117#4099—22no—
118#4086—21no—
119#4084C22×C314yes—
120#4082—21no—
121C2×#687C22×C312yes—
122#4078—24no—
123C2×#686C22×C312yes—
124#4072C22×C311yescharacteristic
125C2×#681C22×C311yescharacteristic
126#3967C22×C314yescomplemented
127#3953C22×C312yescomplemented
128#3952C22×C314yescomplemented
129#3948C22×C312yescomplemented
130#3010—51noSylow
131C23×C3×C5—22no—
132C23×C3×C5—42no—
133C23×C3×C5—82no—
134C23×C3×C5C2×D411yescharacteristic
135C22×C3×D5—21no—
136C22×C3×D5C2×D412yes—
137C3×C5×Q8—24no—
138C3×C5×Q8—22no—
139C3×C5×Q8C2×D414yes—
140C3×C5×D4—22no—
141C3×C5×D4—22no—
142C3×C5×D4—42no—
143C2×C3×C4×C5—24no—
144C2×C3×C4×C5—22no—
145C2×C3×C4×C5C2×D411yescharacteristic
146C2×C3×C4×C5C2×D412yes—
147C2×C3×C4×C5C2×D412yes—
148C2×C3×C4×C5C2411yescharacteristic
149C3×C22⋊D5—24no—
150C3×C22⋊D5—24no—
151C3×C22⋊D5—84no—
152C2×C3×Dic5—21no—
153C2×C3×Dic5—22no—
154C2×C3×Dic5C2×D412yes—
155C3×D20—28no—
156C3×D20—22no—
157C3×D20—21no—
158C3×D20C2×D412yes—
159C3×C4×D5—28no—
160C3×C4×D5—28no—
161C3×C4×D5—24no—
162C3×C4×D5—22no—
163C3×C4×D5—22no—
164C3×Dic10—28no—
165C3×Dic10—22no—
166C3×Dic10—21no—
167C3×Dic10C2×D412yes—
168C3×C5×C8—22no—
169C3×#168—22no—
170C3×2−1+4—104no—
171C3×2−1+4—104no—
172C2×C3×#41—51no—
173C2×C3×#41—52no—
174C2×C3×#41—51no—
175C2×C3×#41—102no—
176C2×C3×#41—101no—
177C22×C3×Q8—52no—
178C22×C3×Q8—101no—
179C22×C3×Q8C2×D511yescharacteristic
180C22×C3×D4C2×D511yescharacteristic
181C3×AGL(1,ℤ/8)—104no—
182C2×C3×SD16—24no—
183C2×C3×SD16—52no—
184C2×C3×SD16C2×D514yes—
185C2×C3×D8—52no—
186C2×C3×M16—51no—
187C22×C3×C8C2×D511yescharacteristic
188C3×#127—101no—
189C3×#124—101no—
190C3×#121—54no—
191C3×#120—102no—
192C3×#117—51no—
193C3×#103—54no—
194C3×#103—52no—
195C3×#102—52no—
196C3×C22⋊C8—54no—
197C24×C5—21no—
198C2×C5×Q8—21no—
199C2×C5×Q8—22no—
200C2×C5×Q8C3×D414yescomplemented
201C2×C5×Q8C23×C312yes—
202C2×C5×D4—22no—
203C2×C5×D4—22no—
204C2×C5×D4—21no—
205C2×C5×D4C23×C312yes—
206C22×C4×C5—21no—
207C22×C4×C5C23×C311yescharacteristic
208C2×C22⋊D5—21no—
209C2×C22⋊D5—24no—
210C2×C22⋊D5C3×D412yes—
211#713—28no—
212#713—24no—
213#713—24no—
214Q8×D5—28no—
215Q8×D5—24no—
216Q8×D5—24no—
217#709—216no—
218#709—22no—
219#709—22no—
220#709—22no—
221#709C3×D414yescomplemented
222C2×D20—22no—
223C2×D20C3×D411yescharacteristic
224C2×D20C23×C311yescharacteristic
225C2×C4×D5—22no—
226C2×C4×D5—21no—
227C2×C4×D5—22no—
228C2×C4×D5C23×C312yes—
229C2×Dic10—22no—
230C2×Dic10C3×D411yescharacteristic
231C2×Dic10C23×C311yescharacteristic
232C5×SD16—28no—
233C2×C5×C8—22no—
234C2×C5×C8C23×C312yes—
235C22⋊Dic5—21no—
236C22⋊Dic5—24no—
237#687—24no—
238#686—24no—
239C4⋊Dic5C23×C312yes—
240C4×Dic5—21no—
241#681—22no—
242C2×#168C23×C312yes—
243C2×2−1+4—51no—
244C2×AGL(1,ℤ/8)—51no—
245C22×SD16C3×D511yescharacteristic, complemented
246#535—51no—
247#449—54no—
248#448—54no—
249#419—51no—
250#416—51no—
251#408—51no—
252C22×C3×C5—22no—
253C22×C3×C5—42no—
254C22×C3×C5—44no—
255C22×C3×C5—84no—
256C22×C3×C5#12012yes—
257C22×C3×C5C22×D411yescharacteristic
258C2×C3×D5—22no—
259C2×C3×D5—22no—
260C2×C3×D5—24no—
261C2×C3×D5—42no—
262C3×C4×C5—24no—
263C3×C4×C5#12012yes—
264C3×C4×C5C22×D411yescharacteristic
265C3×C4×C5C22×D411yescharacteristic
266C3×Dic5—22no—
267C3×Dic5—22no—
268C3×Dic5—42no—
269C24×C3—21no—
270C3×#41—54no—
271C3×#41—1016no—
272C3×#41—104no—
273C3×#41—108no—
274C3×#41—104no—
275C3×#41—102no—
276C3×#41—102no—
277C3×#41—102no—
278C2×C3×Q8—21no—
279C2×C3×Q8—22no—
280C2×C3×Q8—51no—
281C2×C3×Q8—51no—
282C2×C3×Q8—102no—
283C2×C3×Q8—104no—
284C2×C3×Q8—108no—
285C2×C3×Q8—104no—
286C2×C3×Q8C22⋊D514yescomplemented
287C2×C3×Q8C22×D512yes—
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—101no—
296C2×C3×D4—104no—
297C2×C3×D4C22×D512yes—
298C22×C3×C4—21no—
299C22×C3×C4—52no—
300C22×C3×C4—102no—
301C22×C3×C4—102no—
302C22×C3×C4—101no—
303C22×C3×C4C22×D511yescharacteristic
304C3×SD16—28no—
305C3×SD16—104no—
306C3×D8—104no—
307C3×M16—102no—
308C2×C3×C8—22no—
309C2×C3×C8—52no—
310C2×C3×C8C22×D512yes—
311C3×C4⋊C4—52no—
312C3×C22⋊C4—104no—
313C3×C22⋊C4—101no—
314C3×C42—101no—
315C23×C5—22no—
316C23×C5—42no—
317C23×C5—82no—
318C23×C5C2×C3×D411yescharacteristic
319C22×D5—21no—
320C22×D5C2×C3×D412yes—
321C5×Q8—24no—
322C5×Q8—22no—
323C5×Q8C2×C3×D414yes—
324C5×D4—22no—
325C5×D4—22no—
326C5×D4—42no—
327C2×C4×C5—24no—
328C2×C4×C5—22no—
329C2×C4×C5C2×C3×D411yescharacteristic
330C2×C4×C5C2×C3×D412yes—
331C2×C4×C5C2×C3×D412yes—
332C2×C4×C5C24×C311yescharacteristic
333C22⋊D5—24no—
334C22⋊D5—24no—
335C22⋊D5—84no—
336C2×Dic5—21no—
337C2×Dic5—22no—
338C2×Dic5C2×C3×D412yes—
339D20—28no—
340D20—22no—
341D20—21no—
342D20C2×C3×D412yes—
343C4×D5—28no—
344C4×D5—28no—
345C4×D5—24no—
346C4×D5—22no—
347C4×D5—22no—
348Dic10—28no—
349Dic10—22no—
350Dic10—21no—
351Dic10C2×C3×D412yes—
352C5×C8—22no—
353#168—22no—
3542−1+4—104no—
3552−1+4—104no—
356C2×#41—51no—
357C2×#41—52no—
358C2×#41—51no—
359C2×#41—102no—
360C2×#41—101no—
361C22×Q8—52no—
362C22×Q8—101no—
363C22×Q8C2×C3×D511yescharacteristic
364C22×D4C2×C3×D511yescharacteristic
365AGL(1,ℤ/8)—104no—
366C2×SD16—24no—
367C2×SD16—52no—
368C2×SD16C2×C3×D514yes—
369C2×D8—52no—
370C2×M16—51no—
371C22×C8C2×C3×D511yescharacteristic
372#127—101no—
373#124—101no—
374#121—54no—
375#120—102no—
376#117—51no—
377#103—54no—
378#103—52no—
379#102—52no—
380C22⋊C8—54no—
381C2×C3×C5—22no—
382C2×C3×C5—82no—
383C2×C3×C5C2×#12011yescharacteristic
384C2×C3×C5#57712yes—
385C3×D5—42no—
386C3×D5—42no—
387C23×C3—22no—
388C23×C3—42no—
389C23×C3—52no—
390C23×C3—82no—
391C23×C3—101no—
392C23×C3D4×D511yescharacteristic
393C3×Q8—22no—
394C3×Q8—24no—
395C3×Q8—52no—
396C3×Q8—108no—
397C3×Q8—101no—
398C3×Q8—102no—
399C3×Q8C2×C22⋊D514yes—
400C3×D4—22no—
401C3×D4—22no—
402C3×D4—42no—
403C3×D4—52no—
404C3×D4—108no—
405C3×D4—104no—
406C3×D4—104no—
407C3×D4—101no—
408C3×D4—102no—
409C3×D4—404no—
410C2×C3×C4—22no—
411C2×C3×C4—24no—
412C2×C3×C4—52no—
413C2×C3×C4—108no—
414C2×C3×C4—108no—
415C2×C3×C4—101no—
416C2×C3×C4—102no—
417C2×C3×C4—102no—
418C2×C3×C4—104no—
419C2×C3×C4—102no—
420C2×C3×C4D4×D512yes—
421C2×C3×C4D4×D511yescharacteristic
422C2×C3×C4C2×C22⋊D512yes—
423C2×C3×C4C23×D511yescharacteristic
424C3×C8—22no—
425C3×C8—102no—
426C22×C5—22no—
427C22×C5—44no—
428C22×C5—42no—
429C22×C5—84no—
430C22×C5C3×#12012yes—
431C22×C5C22×C3×D411yescharacteristic
432C2×D5—24no—
433C2×D5—22no—
434C2×D5—22no—
435C2×D5—42no—
436C4×C5—24no—
437C4×C5C3×#12012yes—
438C4×C5C22×C3×D411yescharacteristic
439C4×C5C22×C3×D411yescharacteristic
440Dic5—22no—
441Dic5—22no—
442Dic5—42no—
443C24—21no—
444#41—54no—
445#41—1016no—
446#41—104no—
447#41—108no—
448#41—104no—
449#41—102no—
450#41—102no—
451#41—102no—
452C2×Q8—21no—
453C2×Q8—22no—
454C2×Q8—51no—
455C2×Q8—51no—
456C2×Q8—102no—
457C2×Q8—104no—
458C2×Q8—108no—
459C2×Q8—104no—
460C2×Q8C3×C22⋊D514yescomplemented
461C2×Q8C22×C3×D512yes—
462C2×D4—22no—
463C2×D4—21no—
464C2×D4—22no—
465C2×D4—52no—
466C2×D4—51no—
467C2×D4—51no—
468C2×D4—102no—
469C2×D4—101no—
470C2×D4—104no—
471C2×D4C22×C3×D512yes—
472C22×C4—21no—
473C22×C4—52no—
474C22×C4—102no—
475C22×C4—102no—
476C22×C4—101no—
477C22×C4C22×C3×D511yescharacteristic
478SD16—28no—
479SD16—104no—
480D8—104no—
481M16—102no—
482C2×C8—22no—
483C2×C8—52no—
484C2×C8C22×C3×D512yes—
485C4⋊C4—52no—
486C22⋊C4—104no—
487C22⋊C4—101no—
488C42—101no—
489C3×C5#301011yescharacteristic, complemented, Hall
490C22×C3—22no—
491C22×C3—44no—
492C22×C3—42no—
493C22×C3—84no—
494C22×C3—104no—
495C22×C3—102no—
496C22×C3—102no—
497C22×C3—202no—
498C22×C3#408312yes—
499C22×C3C2×D4×D511yescharacteristic
500C3×C4—24no—
501C3×C4—102no—
502C3×C4—102no—
503C3×C4—202no—
504C3×C4#408312yes—
505C3×C4C2×D4×D511yescharacteristic
506C3×C4C22×C22⋊D511yescharacteristic
507C2×C5—22no—
508C2×C5—82no—
509C2×C5C2×C3×#12011yescharacteristic
510C2×C5C3×#57712yes—
511D5—42no—
512D5—42no—
513C23—22no—
514C23—42no—
515C23—52no—
516C23—82no—
517C23—101no—
518C23C3×D4×D511yescharacteristic
519Q8—24no—
520Q8—22no—
521Q8—52no—
522Q8—108no—
523Q8—102no—
524Q8—101no—
525Q8C2×C3×C22⋊D514yes—
526D4—22no—
527D4—22no—
528D4—42no—
529D4—52no—
530D4—108no—
531D4—104no—
532D4—104no—
533D4—102no—
534D4—101no—
535D4—404no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—52no—
539C2×C4—108no—
540C2×C4—108no—
541C2×C4—101no—
542C2×C4—104no—
543C2×C4—102no—
544C2×C4—102no—
545C2×C4—102no—
546C2×C4C3×D4×D511yescharacteristic
547C2×C4C3×D4×D512yes—
548C2×C4C2×C3×C22⋊D512yes—
549C2×C4C23×C3×D511yescharacteristic
550C8—22no—
551C8—102no—
552C2×C3—22no—
553C2×C3—82no—
554C2×C3—202no—
555C2×C3—202no—
556C2×C3#6609612yes—
557C2×C3C2×#408311yescharacteristic
558C5C3×#301011yescharacteristic, complemented, Sylow
559C22—22no—
560C22—42no—
561C22—44no—
562C22—84no—
563C22—102no—
564C22—102no—
565C22—104no—
566C22—202no—
567C22C3×#408312yes—
568C22C2×C3×D4×D511yescharacteristic
569C4—24no—
570C4—102no—
571C4—102no—
572C4—202no—
573C4C3×#408312yes—
574C4C2×C3×D4×D511yescharacteristic
575C4C22×C3×C22⋊D511yescharacteristic
576C3#1061559111yescharacteristic, direct product factor, Sylow
577C2—22no—
578C2—82no—
579C2—202no—
580C2—202no—
581C2C3×#6609612yes—
582C2C2×C3×#408311yescharacteristic
subgroupG/H
centerC22×C3C2×D4×D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC22×C3×C5×SD16C2
socleC22×C3×C5C22×D4
hypercenterC22×C3×SD16D5
hypocenterC5C3×#3010
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×SD16

Complements

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

NH ≅ G/Ntype
1. C2×C3×#4147C2semidirect
2. C22×C3×C5×SD16C2semidirect
3. C3×#66139C2semidirect
4. C2×C3×#4078C2semidirect
5. C3×#65514C2semidirect
6. C3×#65461C2semidirect
7. C3×#65459C2semidirect
8. C3×#65402C2semidirect
9. C3×#65399C2semidirect
10. #10615591C3direct (N × H)
16. C2×C3×#713C22semidirect
18. C2×C3×Q8×D5C22semidirect
34. C3×#3967C22semidirect
35. C3×#3953C22semidirect
36. C3×#3952C22semidirect
37. C3×#3948C22semidirect
39. C2×#4147C2×C3semidirect
40. C22×C5×SD16C2×C3semidirect
41. #66139C2×C3semidirect
42. C2×#4078C2×C3semidirect
43. #65514C2×C3semidirect
44. #65461C2×C3semidirect
45. #65459C2×C3semidirect
46. #65402C2×C3semidirect
47. #65399C2×C3semidirect
51. C2×C3×C5×Q8D4semidirect
72. C3×#709D4semidirect
96. C22×C3×SD16D5semidirect
108. C2×#713C22×C3semidirect
110. C2×Q8×D5C22×C3semidirect
126. #3967C22×C3semidirect
127. #3953C22×C3semidirect
128. #3952C22×C3semidirect
129. #3948C22×C3semidirect
200. C2×C5×Q8C3×D4semidirect
221. #709C3×D4semidirect
245. C22×SD16C3×D5semidirect
286. C2×C3×Q8C22⋊D5semidirect
460. C2×Q8C3×C22⋊D5semidirect
489. C3×C5#3010semidirect
558. C5C3×#3010semidirect
576. C3#10615591direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#41472
C22×C3×C5×SD162
C3×#661392
C2×C3×#40782
C3×#655142
C3×#654612
C3×#654592
C3×#654022
C3×#653992
#106155913
C3×#30105