Finite Group Databasebeta

← C3×#10610610 #14438650

Subgroups — C3×#10610610 #14438650

Order 1920.

784 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#66166C211yesmaximal, characteristic
2C2×C3×#4036C211yesmaximal, characteristic, complemented
3C3×C5×#417C211yesmaximal, characteristic, complemented
4C3×#65521C211yesmaximal, characteristic, complemented
5C3×#65422C211yesmaximal, characteristic, complemented
6C3×M16×Dic5C211yesmaximal, characteristic, complemented
7C3×#65255C211yesmaximal, characteristic, complemented
8C3×#65089C212yesmaximal, complemented
9C3×#65086C212yesmaximal, complemented
10C3×#65057C212yesmaximal, complemented
11C3×#65053C212yesmaximal, complemented
12#10610610C311yesmaximal, characteristic, direct product factor, Hall
13C2×C3×C5×#41C2211yescharacteristic
14C2×C3×#709C2211yescharacteristic
15C2×C3×C5×M16C2211yescharacteristic
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17C3×#4093C2211yescharacteristic
18C3×#4090C2211yescharacteristic
19C3×#4086C2211yescharacteristic
20C3×#4085—21no—
21C3×#4085—21no—
22C3×#4082C2211yescharacteristic
23C3×C4×C22⋊D5—21no—
24C3×C4×C22⋊D5—21no—
25C2×C3×C4⋊Dic5—21no—
26C2×C3×C4⋊Dic5C2211yescharacteristic
27C2×C3×#683—21no—
28C2×C3×C4×Dic5C2211yescharacteristic
29C2×C3×#681C2211yescharacteristic
30C3×#4036—21no—
31C3×#4036—21no—
32C3×#4036C2212yes—
33C3×#4036C2212yes—
34C3×C5×#103C2212yes—
35C3×C5×#102C2212yes—
36C3×#3967C2212yes—
37C3×#3964C2212yes—
38C3×#3953C2212yes—
39C3×#3948C2212yes—
40C3×#3947C2212yes—
41C3×C8×Dic5C2212yes—
42C3×#3942C2212yes—
43C3×#3941C2212yes—
44C3×#3142—51nomaximal, Hall
45#66166C2×C311yescharacteristic
46C2×#4036C2×C311yescharacteristic, complemented
47C5×#417C2×C311yescharacteristic, complemented
48#65521C2×C311yescharacteristic, complemented
49#65422C2×C311yescharacteristic, complemented
50M16×Dic5C2×C311yescharacteristic, complemented
51#65255C2×C311yescharacteristic, complemented
52#65089C2×C312yescomplemented
53#65086C2×C312yescomplemented
54#65057C2×C312yescomplemented
55#65053C2×C312yescomplemented
56C3×C5×#41—22no—
57C3×C5×#41—22no—
58C2×C3×C5×Q8C2311yescharacteristic
59C2×C3×C5×D4—21no—
60C2×C3×C5×D4C2311yescharacteristic
61C22×C3×C4×C5—21no—
62C22×C3×C4×C5—21no—
63C22×C3×C4×C5C2311yescharacteristic
64C2×C3×C22⋊D5—21no—
65C2×C3×C22⋊D5—21no—
66C2×C3×C22⋊D5—21no—
67C22×C3×Dic5—21no—
68C22×C3×Dic5D411yescharacteristic
69C22×C3×Dic5D411yescharacteristic
70C3×#709—22no—
71C3×#709—22no—
72C2×C3×D20C2311yescharacteristic
73C2×C3×C4×D5—21no—
74C2×C3×Dic10C2311yescharacteristic
75C3×C5×M16—22no—
76C2×C3×C5×C8C2312yes—
77C3×C5×C4⋊C4—21no—
78C3×C5×C4⋊C4C2312yes—
79C3×C22⋊Dic5—21no—
80C3×C22⋊Dic5—21no—
81C3×#685—21no—
82C3×#685—21no—
83C3×C4⋊Dic5—21no—
84C3×C4⋊Dic5—22no—
85C3×C4⋊Dic5—22no—
86C3×C4⋊Dic5C2312yes—
87C3×#683—21no—
88C3×#683—21no—
89C3×#683—22no—
90C3×#683—22no—
91C3×C4×Dic5D411yescharacteristic
92C3×C4×Dic5D411yescharacteristic
93C3×C4×Dic5C2311yescharacteristic
94C3×C4×Dic5C2311yescharacteristic
95C3×#681—22no—
96C2×C3×#168C2312yes—
97C3×#532—51no—
98C2×C3×#125—51no—
99C3×#491—52no—
100C3×#490—52no—
101C3×#488—52no—
102C3×#488—52no—
103C3×#417—51no—
104C3×#417—51no—
105C3×#417—51no—
106C3×#417D511yescharacteristic, complemented
107C3×C4×M16—51no—
108C2×C5×#41C22×C311yescharacteristic
109C2×#709C22×C311yescharacteristic
110C2×C5×M16C22×C311yescharacteristic
111C2×C5×C4⋊C4C22×C311yescharacteristic
112#4093C22×C311yescharacteristic
113#4090C22×C311yescharacteristic
114#4086C22×C311yescharacteristic
115#4085—21no—
116#4085—21no—
117#4082C22×C311yescharacteristic
118C4×C22⋊D5—21no—
119C4×C22⋊D5—21no—
120C2×C4⋊Dic5—21no—
121C2×C4⋊Dic5C22×C311yescharacteristic
122C2×#683—21no—
123C2×C4×Dic5C22×C311yescharacteristic
124C2×#681C22×C311yescharacteristic
125#4036—21no—
126#4036—21no—
127#4036C22×C312yes—
128#4036C22×C312yes—
129C5×#103C22×C312yes—
130C5×#102C22×C312yes—
131#3967C22×C312yes—
132#3964C22×C312yes—
133#3953C22×C312yes—
134#3948C22×C312yes—
135#3947C22×C312yes—
136C8×Dic5C22×C312yes—
137#3942C22×C312yes—
138#3941C22×C312yes—
139#3142—51noSylow
140C23×C3×C5—21no—
141C23×C3×C5C2×D411yescharacteristic
142C22×C3×D5—21no—
143C3×C5×Q8—21no—
144C3×C5×Q8—21no—
145C3×C5×D4—21no—
146C3×C5×D4—21no—
147C3×C5×D4—42no—
148C2×C3×C4×C5—22no—
149C2×C3×C4×C5—21no—
150C2×C3×C4×C5—42no—
151C2×C3×C4×C5—42no—
152C2×C3×C4×C5C2×D411yescharacteristic
153C2×C3×C4×C5#4112yes—
154C2×C3×C4×C5#4112yes—
155C2×C3×C4×C5C2411yescharacteristic
156C3×C22⋊D5—42no—
157C3×C22⋊D5—42no—
158C3×C22⋊D5—42no—
159C2×C3×Dic5—22no—
160C2×C3×Dic5—21no—
161C2×C3×Dic5—41no—
162C2×C3×Dic5—41no—
163C2×C3×Dic5—42no—
164C2×C3×Dic5C2×D411yescharacteristic
165C2×C3×Dic5C2×D411yescharacteristic
166C2×C3×Dic5C2×D411yescharacteristic
167C2×C3×Dic5C2×D411yescharacteristic
168C3×D20—21no—
169C3×D20—21no—
170C3×C4×D5—42no—
171C3×Dic10—21no—
172C3×Dic10—21no—
173C3×C5×C8—22no—
174C3×#168—22no—
175C2×C3×#41—51no—
176C2×C3×#41C2×D511yescharacteristic
177C2×C3×M16—51no—
178C2×C3×M16C2×D511yescharacteristic
179C3×C4⋊Q8—51no—
180C3×#127—51no—
181C3×#125—52no—
182C3×#125—52no—
183C3×#125—101no—
184C3×#125—101no—
185C3×#124—51no—
186C3×#124—51no—
187C3×#121—101no—
188C3×#121—101no—
189C3×C4×D4—101no—
190C3×C4×D4—101no—
191C2×C3×C4⋊C4—51no—
192C2×C3×C4⋊C4—101no—
193C2×C3×C4⋊C4—101no—
194C2×C3×C4⋊C4C2×D511yescharacteristic
195C2×C3×C42—51no—
196C3×#103—52no—
197C3×#103—52no—
198C3×#103—52no—
199C3×#103C2×D512yes—
200C3×#102—52no—
201C3×#102—52no—
202C3×#102—52no—
203C3×#102C2×D512yes—
204C3×#97—52no—
205C3×C4×C8—52no—
206C5×#41—22no—
207C5×#41—22no—
208C2×C5×Q8C23×C311yescharacteristic
209C2×C5×D4—21no—
210C2×C5×D4C23×C311yescharacteristic
211C22×C4×C5—21no—
212C22×C4×C5—21no—
213C22×C4×C5C23×C311yescharacteristic
214C2×C22⋊D5—21no—
215C2×C22⋊D5—21no—
216C2×C22⋊D5—21no—
217C22×Dic5—21no—
218C22×Dic5C3×D411yescharacteristic
219C22×Dic5C3×D411yescharacteristic
220#709—22no—
221#709—22no—
222C2×D20C23×C311yescharacteristic
223C2×C4×D5—21no—
224C2×Dic10C23×C311yescharacteristic
225C5×M16—22no—
226C2×C5×C8C23×C312yes—
227C5×C4⋊C4—21no—
228C5×C4⋊C4C23×C312yes—
229C22⋊Dic5—21no—
230C22⋊Dic5—21no—
231#685—21no—
232#685—21no—
233C4⋊Dic5—21no—
234C4⋊Dic5—22no—
235C4⋊Dic5—22no—
236C4⋊Dic5C23×C312yes—
237#683—22no—
238#683—22no—
239#683—21no—
240#683—21no—
241C4×Dic5C3×D411yescharacteristic
242C4×Dic5C3×D411yescharacteristic
243C4×Dic5C23×C311yescharacteristic
244C4×Dic5C23×C311yescharacteristic
245#681—22no—
246C2×#168C23×C312yes—
247#532—51no—
248C2×#125—51no—
249#491—52no—
250#490—52no—
251#488—52no—
252#488—52no—
253#417—51no—
254#417—51no—
255#417—51no—
256#417C3×D511yescharacteristic, complemented
257C4×M16—51no—
258C22×C3×C5—21no—
259C22×C3×C5—21no—
260C22×C3×C5—41no—
261C22×C3×C5—41no—
262C22×C3×C5—41no—
263C22×C3×C5#12412yes—
264C22×C3×C5C22×D411yescharacteristic
265C2×C3×D5—41no—
266C2×C3×D5—41no—
267C2×C3×D5—41no—
268C3×C4×C5—41no—
269C3×C4×C5—42no—
270C3×C4×C5#12412yes—
271C3×C4×C5C2×#4111yescharacteristic
272C3×C4×C5C2×#4111yescharacteristic
273C3×Dic5—21no—
274C3×Dic5—21no—
275C3×Dic5—21no—
276C3×Dic5—21no—
277C3×Dic5—41no—
278C3×Dic5—42no—
279C3×#41—22no—
280C3×#41—22no—
281C3×#41—102no—
282C3×#41—102no—
283C2×C3×Q8—51no—
284C2×C3×Q8C22×D511yescharacteristic
285C2×C3×D4—21no—
286C2×C3×D4—51no—
287C2×C3×D4—101no—
288C2×C3×D4—101no—
289C2×C3×D4—101no—
290C2×C3×D4C22×D511yescharacteristic
291C22×C3×C4—21no—
292C22×C3×C4—21no—
293C22×C3×C4—51no—
294C22×C3×C4—51no—
295C22×C3×C4—101no—
296C22×C3×C4—101no—
297C22×C3×C4C22×D511yescharacteristic
298C3×M16—22no—
299C3×M16—102no—
300C2×C3×C8—52no—
301C2×C3×C8C22×D512yes—
302C3×C4⋊C4—21no—
303C3×C4⋊C4—52no—
304C3×C4⋊C4—102no—
305C3×C4⋊C4—102no—
306C3×C4⋊C4—101no—
307C3×C4⋊C4—101no—
308C3×C4⋊C4—102no—
309C3×C4⋊C4—102no—
310C3×C4⋊C4—101no—
311C3×C4⋊C4C22×D512yes—
312C3×C22⋊C4—101no—
313C3×C22⋊C4—101no—
314C3×C22⋊C4—101no—
315C3×C22⋊C4—101no—
316C3×C42—51no—
317C3×C42—51no—
318C3×C42—51no—
319C3×C42—51no—
320C23×C5—21no—
321C23×C5C2×C3×D411yescharacteristic
322C22×D5—21no—
323C5×Q8—21no—
324C5×Q8—21no—
325C5×D4—21no—
326C5×D4—21no—
327C5×D4—42no—
328C2×C4×C5—22no—
329C2×C4×C5—21no—
330C2×C4×C5—42no—
331C2×C4×C5—42no—
332C2×C4×C5C2×C3×D411yescharacteristic
333C2×C4×C5C3×#4112yes—
334C2×C4×C5C3×#4112yes—
335C2×C4×C5C24×C311yescharacteristic
336C22⋊D5—42no—
337C22⋊D5—42no—
338C22⋊D5—42no—
339C2×Dic5—22no—
340C2×Dic5—21no—
341C2×Dic5—41no—
342C2×Dic5—41no—
343C2×Dic5—42no—
344C2×Dic5C2×C3×D411yescharacteristic
345C2×Dic5C2×C3×D411yescharacteristic
346C2×Dic5C2×C3×D411yescharacteristic
347C2×Dic5C2×C3×D411yescharacteristic
348D20—21no—
349D20—21no—
350C4×D5—42no—
351Dic10—21no—
352Dic10—21no—
353C5×C8—22no—
354#168—22no—
355C2×#41—51no—
356C2×#41C2×C3×D511yescharacteristic
357C2×M16—51no—
358C2×M16C2×C3×D511yescharacteristic
359C4⋊Q8—51no—
360#127—51no—
361#125—52no—
362#125—52no—
363#125—101no—
364#125—101no—
365#124—51no—
366#124—51no—
367#121—101no—
368#121—101no—
369C4×D4—101no—
370C4×D4—101no—
371C2×C4⋊C4—51no—
372C2×C4⋊C4—101no—
373C2×C4⋊C4—101no—
374C2×C4⋊C4C2×C3×D511yescharacteristic
375C2×C42—51no—
376#103—52no—
377#103—52no—
378#103—52no—
379#103C2×C3×D512yes—
380#102—52no—
381#102—52no—
382#102—52no—
383#102C2×C3×D512yes—
384#97—52no—
385C4×C8—52no—
386C2×C3×C5—22no—
387C2×C3×C5—81no—
388C2×C3×C5C2×#12411yescharacteristic
389C2×C3×C5#57511yescharacteristic
390C2×C3×C5#57511yescharacteristic
391C3×D5—81no—
392C23×C3—21no—
393C23×C3—101no—
394C23×C3D4×D511yescharacteristic
395C3×Q8—21no—
396C3×Q8—21no—
397C3×Q8—101no—
398C3×Q8—101no—
399C3×D4—21no—
400C3×D4—21no—
401C3×D4—42no—
402C3×D4—101no—
403C3×D4—101no—
404C3×D4—202no—
405C3×D4—202no—
406C3×D4—202no—
407C2×C3×C4—21no—
408C2×C3×C4—22no—
409C2×C3×C4—42no—
410C2×C3×C4—42no—
411C2×C3×C4—51no—
412C2×C3×C4—51no—
413C2×C3×C4—51no—
414C2×C3×C4—51no—
415C2×C3×C4—101no—
416C2×C3×C4—102no—
417C2×C3×C4—201no—
418C2×C3×C4—201no—
419C2×C3×C4—202no—
420C2×C3×C4—202no—
421C2×C3×C4#70912yes—
422C2×C3×C4D4×D511yescharacteristic
423C2×C3×C4#71112yes—
424C2×C3×C4C23×D511yescharacteristic
425C3×C8—22no—
426C3×C8—102no—
427C22×C5—21no—
428C22×C5—21no—
429C22×C5—41no—
430C22×C5—41no—
431C22×C5—41no—
432C22×C5C3×#12412yes—
433C22×C5C22×C3×D411yescharacteristic
434C2×D5—41no—
435C2×D5—41no—
436C2×D5—41no—
437C4×C5—42no—
438C4×C5—41no—
439C4×C5C3×#12412yes—
440C4×C5C2×C3×#4111yescharacteristic
441C4×C5C2×C3×#4111yescharacteristic
442Dic5—21no—
443Dic5—21no—
444Dic5—21no—
445Dic5—21no—
446Dic5—42no—
447Dic5—41no—
448#41—22no—
449#41—22no—
450#41—102no—
451#41—102no—
452C2×Q8—51no—
453C2×Q8C22×C3×D511yescharacteristic
454C2×D4—21no—
455C2×D4—51no—
456C2×D4—101no—
457C2×D4—101no—
458C2×D4—101no—
459C2×D4C22×C3×D511yescharacteristic
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—51no—
463C22×C4—51no—
464C22×C4—101no—
465C22×C4—101no—
466C22×C4C22×C3×D511yescharacteristic
467M16—22no—
468M16—102no—
469C2×C8—52no—
470C2×C8C22×C3×D512yes—
471C4⋊C4—21no—
472C4⋊C4—52no—
473C4⋊C4—102no—
474C4⋊C4—102no—
475C4⋊C4—102no—
476C4⋊C4—102no—
477C4⋊C4—101no—
478C4⋊C4—101no—
479C4⋊C4—101no—
480C4⋊C4C22×C3×D512yes—
481C22⋊C4—101no—
482C22⋊C4—101no—
483C22⋊C4—101no—
484C22⋊C4—101no—
485C42—51no—
486C42—51no—
487C42—51no—
488C42—51no—
489C3×C5#314211yescharacteristic, complemented, Hall
490C22×C3—21no—
491C22×C3—21no—
492C22×C3—41no—
493C22×C3—41no—
494C22×C3—41no—
495C22×C3—201no—
496C22×C3—201no—
497C22×C3—201no—
498C22×C3#403112yes—
499C22×C3C2×D4×D511yescharacteristic
500C3×C4—41no—
501C3×C4—42no—
502C3×C4—101no—
503C3×C4—101no—
504C3×C4—101no—
505C3×C4—101no—
506C3×C4—201no—
507C3×C4—202no—
508C3×C4#403112yes—
509C3×C4C2×#70911yescharacteristic
510C3×C4C2×#71111yescharacteristic
511C2×C5—22no—
512C2×C5—81no—
513C2×C5C2×C3×#12411yescharacteristic
514C2×C5C3×#57511yescharacteristic
515C2×C5C3×#57511yescharacteristic
516D5—81no—
517C23—21no—
518C23—101no—
519C23C3×D4×D511yescharacteristic
520Q8—21no—
521Q8—21no—
522Q8—101no—
523Q8—101no—
524D4—21no—
525D4—21no—
526D4—42no—
527D4—101no—
528D4—101no—
529D4—202no—
530D4—202no—
531D4—202no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—51no—
537C2×C4—51no—
538C2×C4—51no—
539C2×C4—51no—
540C2×C4—101no—
541C2×C4—102no—
542C2×C4—201no—
543C2×C4—201no—
544C2×C4—202no—
545C2×C4—202no—
546C2×C4C3×#70912yes—
547C2×C4C3×D4×D511yescharacteristic
548C2×C4C3×#71112yes—
549C2×C4C23×C3×D511yescharacteristic
550C8—22no—
551C8—102no—
552C2×C3—22no—
553C2×C3—81no—
554C2×C3—401no—
555C2×C3C2×#403111yescharacteristic
556C2×C3#6610711yescharacteristic
557C2×C3#6611111yescharacteristic
558C5C3×#314211yescharacteristic, complemented, Sylow
559C22—21no—
560C22—21no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—201no—
565C22—201no—
566C22—201no—
567C22C3×#403112yes—
568C22C2×C3×D4×D511yescharacteristic
569C4—42no—
570C4—41no—
571C4—101no—
572C4—101no—
573C4—101no—
574C4—101no—
575C4—202no—
576C4—201no—
577C4C3×#403112yes—
578C4C2×C3×#70911yescharacteristic
579C4C2×C3×#71111yescharacteristic
580C3#1061061011yescharacteristic, direct product factor, Sylow
581C2—22no—
582C2—81no—
583C2—401no—
584C2C2×C3×#403111yescharacteristic
585C2C3×#6610711yescharacteristic
586C2C3×#6611111yescharacteristic
subgroupG/H
centerC22×C3C2×D4×D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC3×C5×#417C2
socleC22×C3×C5C22×D4
prime radicalC3×#66166C2
hypercenterC3×#417D5
hypocenterC5C3×#3142
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#417

Complements

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

NH ≅ G/Ntype
2. C2×C3×#4036C2semidirect
3. C3×C5×#417C2semidirect
4. C3×#65521C2semidirect
5. C3×#65422C2semidirect
6. C3×M16×Dic5C2semidirect
7. C3×#65255C2semidirect
8. C3×#65089C2semidirect
9. C3×#65086C2semidirect
10. C3×#65057C2semidirect
11. C3×#65053C2semidirect
12. #10610610C3direct (N × H)
46. C2×#4036C2×C3semidirect
47. C5×#417C2×C3semidirect
48. #65521C2×C3semidirect
49. #65422C2×C3semidirect
50. M16×Dic5C2×C3semidirect
51. #65255C2×C3semidirect
52. #65089C2×C3semidirect
53. #65086C2×C3semidirect
54. #65057C2×C3semidirect
55. #65053C2×C3semidirect
106. C3×#417D5semidirect
256. #417C3×D5semidirect
489. C3×C5#3142semidirect
558. C5C3×#3142semidirect
580. C3#10610610direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#661662
C2×C3×#40362
C3×C5×#4172
C3×#655212
C3×#654222
C3×M16×Dic52
C3×#652552
C3×#650892
C3×#650862
C3×#650572
C3×#650532
#106106103
C3×#31425