Finite Group Databasebeta

← C3×#10612206 #14426236

Subgroups — C3×#10612206 #14426236

Order 1920.

992 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4065C212yesmaximal, complemented
2C3×#66059C211yesmaximal, characteristic, complemented
3C3×C4×#713C211yesmaximal, characteristic, complemented
4C3×C5×#462C211yesmaximal, characteristic, complemented
5C3×#65474C212yesmaximal, complemented
6C3×#65372C211yesmaximal, characteristic, complemented
7C3×#65316C211yesmaximal, characteristic, complemented
8C3×#65132C211yesmaximal, characteristic, complemented
9C3×D5×C4⋊C8C211yesmaximal, characteristic, complemented
10C3×#65108C212yesmaximal, complemented
11C3×#65093C212yesmaximal, complemented
12#10612206C311yesmaximal, characteristic, direct product factor, Hall
13C2×C3×#713C2212yes—
14C2×C3×#713C2211yescharacteristic
15C2×C3×C5×Q16C2212yes—
16C3×C5×C4⋊Q8C2211yescharacteristic
17C3×C4×C5×Q8C2211yescharacteristic
18C3×#4093C2212yes—
19C3×Q8×Dic5C2211yescharacteristic
20C2×C3×#688C2212yes—
21C2×C3×#688C2212yes—
22C3×#4065—28no—
23C2×C3×D40C2212yes—
24C2×C3×C8×D5C2212yes—
25C3×#4041—21no—
26C3×#4041—21no—
27C3×#4039—21no—
28C3×#4039—21no—
29C3×#4039—21no—
30C3×#4039C2211yescharacteristic
31C3×#4038—21no—
32C3×#4038C2211yescharacteristic
33C3×#4020C2211yescharacteristic
34C3×C4×D20—21no—
35C3×C4×D20C2211yescharacteristic
36C3×C42×D5—21no—
37C3×C42×D5C2211yescharacteristic
38C3×C5×C4⋊C8C2211yescharacteristic
39C3×C5×#103C2212yes—
40C3×#3967C2212yes—
41C3×#3953C2212yes—
42C3×#3946C2212yes—
43C3×#3941C2212yes—
44C3×C4⋊#168C2211yescharacteristic
45C3×#3035—51nomaximal, Hall
46C2×#4065C2×C312yescomplemented
47#66059C2×C311yescharacteristic, complemented
48C4×#713C2×C311yescharacteristic, complemented
49C5×#462C2×C311yescharacteristic, complemented
50#65474C2×C312yescomplemented
51#65372C2×C311yescharacteristic, complemented
52#65316C2×C311yescharacteristic, complemented
53#65132C2×C311yescharacteristic, complemented
54D5×C4⋊C8C2×C311yescharacteristic, complemented
55#65108C2×C312yescomplemented
56#65093C2×C312yescomplemented
57C2×C3×C5×Q8C2312yes—
58C2×C3×C5×Q8C2311yescharacteristic
59C3×#713—22no—
60C3×#713—24no—
61C3×#713—24no—
62C3×#713D414yescomplemented
63C2×C3×D20—21no—
64C2×C3×D20—22no—
65C2×C3×D20—21no—
66C2×C3×D20—21no—
67C2×C3×D20C2312yes—
68C2×C3×D20C2311yescharacteristic
69C2×C3×C4×D5—21no—
70C2×C3×C4×D5—21no—
71C2×C3×C4×D5—22no—
72C2×C3×C4×D5—21no—
73C2×C3×C4×D5D411yescharacteristic
74C2×C3×C4×D5D411yescharacteristic
75C2×C3×C4×D5C2311yescharacteristic
76C3×C5×Q16—24no—
77C2×C3×C5×C8C2312yes—
78C3×C5×C4⋊C4—21no—
79C3×C5×C4⋊C4—21no—
80C3×C5×C4⋊C4—21no—
81C3×C5×C4⋊C4C2311yescharacteristic
82C3×C42×C5—21no—
83C3×C42×C5C2311yescharacteristic
84C3×#688—24no—
85C3×#688—24no—
86C3×#685—21no—
87C3×#685—21no—
88C3×#685—22no—
89C3×#685—22no—
90C3×#685—21no—
91C3×C4⋊Dic5—21no—
92C3×C4⋊Dic5C2311yescharacteristic
93C3×C4×Dic5—21no—
94C3×C4×Dic5—21no—
95C3×C4×Dic5D411yescharacteristic
96C3×C4×Dic5D411yescharacteristic
97C3×C4×Dic5C2311yescharacteristic
98C2×C3×#168C2312yes—
99C3×D40—24no—
100C3×C8×D5—24no—
101C2×C3×#135—52no—
102C3×#532—51no—
103C3×C4×#41—51no—
104C3×#464—52no—
105C3×#463—52no—
106C3×#462D511yescharacteristic, complemented
107C3×#461—51no—
108C3×#460—51no—
109C3×#459—51no—
110C2×C3×C4⋊C8—51no—
111C3×#416—52no—
112C2×#713C22×C312yes—
113C2×#713C22×C311yescharacteristic
114C2×C5×Q16C22×C312yes—
115C5×C4⋊Q8C22×C311yescharacteristic
116C4×C5×Q8C22×C311yescharacteristic
117#4093C22×C312yes—
118Q8×Dic5C22×C311yescharacteristic
119C2×#688C22×C312yes—
120C2×#688C22×C312yes—
121#4065—28no—
122C2×D40C22×C312yes—
123C2×C8×D5C22×C312yes—
124#4041—21no—
125#4041—21no—
126#4039—21no—
127#4039—21no—
128#4039—21no—
129#4039C22×C311yescharacteristic
130#4038—21no—
131#4038C22×C311yescharacteristic
132#4020C22×C311yescharacteristic
133C4×D20—21no—
134C4×D20C22×C311yescharacteristic
135C42×D5—21no—
136C42×D5C22×C311yescharacteristic
137C5×C4⋊C8C22×C311yescharacteristic
138C5×#103C22×C312yes—
139#3967C22×C312yes—
140#3953C22×C312yes—
141#3946C22×C312yes—
142#3941C22×C312yes—
143C4⋊#168C22×C311yescharacteristic
144#3035—51noSylow
145C22×C3×D5—21no—
146C22×C3×D5—22no—
147C22×C3×D5C2×D411yescharacteristic
148C3×C5×Q8—21no—
149C3×C5×Q8—22no—
150C3×C5×Q8—22no—
151C3×C5×Q8C2×D412yes—
152C2×C3×C4×C5—21no—
153C2×C3×C4×C5—21no—
154C2×C3×C4×C5—22no—
155C2×C3×C4×C5C2×D411yescharacteristic
156C2×C3×C4×C5C2×D411yescharacteristic
157C2×C3×C4×C5C2411yescharacteristic
158C2×C3×Dic5—21no—
159C2×C3×Dic5C2×D411yescharacteristic
160C2×C3×Dic5C2×D411yescharacteristic
161C2×C3×Dic5C2×D411yescharacteristic
162C3×D20—24no—
163C3×D20—22no—
164C3×D20—21no—
165C3×D20—22no—
166C3×D20—42no—
167C3×D20—44no—
168C3×D20—42no—
169C3×D20C2×D412yes—
170C3×C4×D5—22no—
171C3×C4×D5—24no—
172C3×C4×D5—41no—
173C3×C4×D5—42no—
174C3×C4×D5—42no—
175C3×C4×D5—44no—
176C3×C4×D5C2×D412yes—
177C3×C4×D5#4112yes—
178C3×C5×C8—22no—
179C3×#168—22no—
180C2×C3×#41—51no—
181C2×C3×#41—52no—
182C3×#135—108no—
183C2×C3×Q16C2×D512yes—
184C2×C3×SD16—52no—
185C2×C3×SD16—52no—
186C2×C3×D8—52no—
187C22×C3×C8—52no—
188C3×C4⋊Q8C2×D511yescharacteristic
189C3×#127—51no—
190C3×#124—52no—
191C3×#121—101no—
192C3×#121—101no—
193C3×C4×Q8—51no—
194C3×C4×Q8C2×D511yescharacteristic
195C3×C4×D4—51no—
196C3×C4×D4—51no—
197C3×C4×D4—101no—
198C3×C4×D4—101no—
199C3×C4×D4—101no—
200C3×C4×D4—101no—
201C3×#117—51no—
202C3×#117—101no—
203C2×C3×C42—51no—
204C2×C3×C42—101no—
205C3×C4⋊C8—52no—
206C3×C4⋊C8—51no—
207C3×C4⋊C8C2×D511yescharacteristic
208C3×#103—52no—
209C3×#103C2×D512yes—
210C3×#102—52no—
211C3×#102—52no—
212C2×C5×Q8C23×C312yes—
213C2×C5×Q8C23×C311yescharacteristic
214#713—22no—
215#713—24no—
216#713—24no—
217#713C3×D414yescomplemented
218C2×D20—21no—
219C2×D20—22no—
220C2×D20—21no—
221C2×D20—21no—
222C2×D20C23×C312yes—
223C2×D20C23×C311yescharacteristic
224C2×C4×D5—21no—
225C2×C4×D5—21no—
226C2×C4×D5—22no—
227C2×C4×D5—21no—
228C2×C4×D5C3×D411yescharacteristic
229C2×C4×D5C3×D411yescharacteristic
230C2×C4×D5C23×C311yescharacteristic
231C5×Q16—24no—
232C2×C5×C8C23×C312yes—
233C5×C4⋊C4—21no—
234C5×C4⋊C4—21no—
235C5×C4⋊C4—21no—
236C5×C4⋊C4C23×C311yescharacteristic
237C42×C5—21no—
238C42×C5C23×C311yescharacteristic
239#688—24no—
240#688—24no—
241#685—21no—
242#685—21no—
243#685—21no—
244#685—22no—
245#685—22no—
246C4⋊Dic5—21no—
247C4⋊Dic5C23×C311yescharacteristic
248C4×Dic5—21no—
249C4×Dic5—21no—
250C4×Dic5C3×D411yescharacteristic
251C4×Dic5C3×D411yescharacteristic
252C4×Dic5C23×C311yescharacteristic
253C2×#168C23×C312yes—
254D40—24no—
255C8×D5—24no—
256C2×#135—52no—
257#532—51no—
258C4×#41—51no—
259#464—52no—
260#463—52no—
261#462C3×D511yescharacteristic, complemented
262#461—51no—
263#460—51no—
264#459—51no—
265C2×C4⋊C8—51no—
266#416—52no—
267C22×C3×C5C22×D411yescharacteristic
268C2×C3×D5—21no—
269C2×C3×D5—21no—
270C2×C3×D5—22no—
271C2×C3×D5—41no—
272C2×C3×D5—41no—
273C2×C3×D5—42no—
274C2×C3×D5—42no—
275C2×C3×D5—42no—
276C2×C3×D5#12112yes—
277C3×C4×C5—22no—
278C3×C4×C5—21no—
279C3×C4×C5—42no—
280C3×C4×C5—41no—
281C3×C4×C5#13512yescomplemented
282C3×C4×C5C22×D411yescharacteristic
283C3×C4×C5C2×#4111yescharacteristic
284C3×Dic5—21no—
285C3×Dic5—21no—
286C3×Dic5—41no—
287C3×Dic5#12112yes—
288C3×#41—54no—
289C3×#41—102no—
290C3×#41—104no—
291C3×#41—104no—
292C2×C3×Q8C22×D512yes—
293C2×C3×Q8C22×D511yescharacteristic
294C2×C3×D4—51no—
295C2×C3×D4—52no—
296C2×C3×D4—101no—
297C2×C3×D4—102no—
298C2×C3×D4—101no—
299C2×C3×D4—101no—
300C22×C3×C4—51no—
301C22×C3×C4—51no—
302C22×C3×C4—51no—
303C22×C3×C4—101no—
304C22×C3×C4—101no—
305C22×C3×C4—102no—
306C22×C3×C4—101no—
307C3×Q16—24no—
308C3×SD16—104no—
309C3×SD16—104no—
310C3×D8—104no—
311C2×C3×C8—52no—
312C2×C3×C8—104no—
313C2×C3×C8C22×D512yes—
314C3×C4⋊C4—21no—
315C3×C4⋊C4—21no—
316C3×C4⋊C4—21no—
317C3×C4⋊C4—51no—
318C3×C4⋊C4—101no—
319C3×C4⋊C4C22×D511yescharacteristic
320C3×C22⋊C4—101no—
321C3×C22⋊C4—102no—
322C3×C22⋊C4—102no—
323C3×C22⋊C4—101no—
324C3×C22⋊C4—101no—
325C3×C42—21no—
326C3×C42—51no—
327C3×C42—51no—
328C3×C42—51no—
329C3×C42—101no—
330C3×C42—101no—
331C3×C42C22×D511yescharacteristic
332C22×D5—21no—
333C22×D5—22no—
334C22×D5C2×C3×D411yescharacteristic
335C5×Q8—21no—
336C5×Q8—22no—
337C5×Q8—22no—
338C5×Q8C2×C3×D412yes—
339C2×C4×C5—21no—
340C2×C4×C5—21no—
341C2×C4×C5—22no—
342C2×C4×C5C2×C3×D411yescharacteristic
343C2×C4×C5C2×C3×D411yescharacteristic
344C2×C4×C5C24×C311yescharacteristic
345C2×Dic5—21no—
346C2×Dic5C2×C3×D411yescharacteristic
347C2×Dic5C2×C3×D411yescharacteristic
348C2×Dic5C2×C3×D411yescharacteristic
349D20—24no—
350D20—22no—
351D20—21no—
352D20—22no—
353D20—42no—
354D20—44no—
355D20—42no—
356D20C2×C3×D412yes—
357C4×D5—22no—
358C4×D5—24no—
359C4×D5—41no—
360C4×D5—42no—
361C4×D5—42no—
362C4×D5—44no—
363C4×D5C2×C3×D412yes—
364C4×D5C3×#4112yes—
365C5×C8—22no—
366#168—22no—
367C2×#41—51no—
368C2×#41—52no—
369#135—108no—
370C2×Q16C2×C3×D512yes—
371C2×SD16—52no—
372C2×SD16—52no—
373C2×D8—52no—
374C22×C8—52no—
375C4⋊Q8C2×C3×D511yescharacteristic
376#127—51no—
377#124—52no—
378#121—101no—
379#121—101no—
380C4×Q8—51no—
381C4×Q8C2×C3×D511yescharacteristic
382C4×D4—51no—
383C4×D4—51no—
384C4×D4—101no—
385C4×D4—101no—
386C4×D4—101no—
387C4×D4—101no—
388#117—51no—
389#117—101no—
390C2×C42—51no—
391C2×C42—101no—
392C4⋊C8—52no—
393C4⋊C8—51no—
394C4⋊C8C2×C3×D511yescharacteristic
395#103—52no—
396#103C2×C3×D512yes—
397#102—52no—
398#102—52no—
399C2×C3×C5C2×#12111yescharacteristic
400C2×C3×C5C2×#13511yescharacteristic
401C2×C3×C5#57511yescharacteristic
402C3×D5—22no—
403C3×D5—42no—
404C3×D5—82no—
405C23×C3—51no—
406C23×C3—101no—
407C23×C3—102no—
408C3×Q8—21no—
409C3×Q8—22no—
410C3×Q8—22no—
411C3×Q8D4×D512yes—
412C3×D4—52no—
413C3×D4—104no—
414C3×D4—102no—
415C3×D4—101no—
416C3×D4—102no—
417C3×D4—202no—
418C3×D4—204no—
419C3×D4—202no—
420C2×C3×C4—21no—
421C2×C3×C4—21no—
422C2×C3×C4—22no—
423C2×C3×C4—51no—
424C2×C3×C4—52no—
425C2×C3×C4—52no—
426C2×C3×C4—51no—
427C2×C3×C4—51no—
428C2×C3×C4—104no—
429C2×C3×C4—102no—
430C2×C3×C4—101no—
431C2×C3×C4—201no—
432C2×C3×C4—204no—
433C2×C3×C4—202no—
434C2×C3×C4—202no—
435C2×C3×C4D4×D511yescharacteristic
436C2×C3×C4D4×D511yescharacteristic
437C2×C3×C4C23×D511yescharacteristic
438C3×C8—22no—
439C3×C8—102no—
440C22×C5C22×C3×D411yescharacteristic
441C2×D5—21no—
442C2×D5—21no—
443C2×D5—22no—
444C2×D5—41no—
445C2×D5—41no—
446C2×D5—42no—
447C2×D5—42no—
448C2×D5—42no—
449C2×D5C3×#12112yes—
450C4×C5—21no—
451C4×C5—22no—
452C4×C5—41no—
453C4×C5—42no—
454C4×C5C3×#13512yescomplemented
455C4×C5C22×C3×D411yescharacteristic
456C4×C5C2×C3×#4111yescharacteristic
457Dic5—21no—
458Dic5—21no—
459Dic5—41no—
460Dic5C3×#12112yes—
461#41—54no—
462#41—102no—
463#41—104no—
464#41—104no—
465C2×Q8C22×C3×D512yes—
466C2×Q8C22×C3×D511yescharacteristic
467C2×D4—51no—
468C2×D4—52no—
469C2×D4—101no—
470C2×D4—102no—
471C2×D4—101no—
472C2×D4—101no—
473C22×C4—51no—
474C22×C4—51no—
475C22×C4—51no—
476C22×C4—101no—
477C22×C4—101no—
478C22×C4—102no—
479C22×C4—101no—
480Q16—24no—
481SD16—104no—
482SD16—104no—
483D8—104no—
484C2×C8—52no—
485C2×C8—104no—
486C2×C8C22×C3×D512yes—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—51no—
491C4⋊C4—101no—
492C4⋊C4C22×C3×D511yescharacteristic
493C22⋊C4—101no—
494C22⋊C4—102no—
495C22⋊C4—102no—
496C22⋊C4—101no—
497C22⋊C4—101no—
498C42—21no—
499C42—51no—
500C42—51no—
501C42—51no—
502C42—101no—
503C42—101no—
504C42C22×C3×D511yescharacteristic
505C3×C5#303511yescharacteristic, complemented, Hall
506C22×C3—52no—
507C22×C3—101no—
508C22×C3—101no—
509C22×C3—102no—
510C22×C3—201no—
511C22×C3—201no—
512C22×C3—202no—
513C22×C3—202no—
514C22×C3—202no—
515C22×C3C2×D4×D511yescharacteristic
516C3×C4—21no—
517C3×C4—22no—
518C3×C4—41no—
519C3×C4—42no—
520C3×C4—52no—
521C3×C4—101no—
522C3×C4—101no—
523C3×C4—201no—
524C3×C4#406512yescomplemented
525C3×C4C2×D4×D511yescharacteristic
526C3×C4D5×#4111yescharacteristic
527C2×C5C2×C3×#12111yescharacteristic
528C2×C5C2×C3×#13511yescharacteristic
529C2×C5C3×#57511yescharacteristic
530D5—22no—
531D5—42no—
532D5—82no—
533C23—51no—
534C23—101no—
535C23—102no—
536Q8—21no—
537Q8—22no—
538Q8—22no—
539Q8C3×D4×D512yes—
540D4—52no—
541D4—104no—
542D4—102no—
543D4—101no—
544D4—102no—
545D4—202no—
546D4—204no—
547D4—202no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—51no—
552C2×C4—52no—
553C2×C4—52no—
554C2×C4—51no—
555C2×C4—51no—
556C2×C4—104no—
557C2×C4—102no—
558C2×C4—101no—
559C2×C4—201no—
560C2×C4—204no—
561C2×C4—202no—
562C2×C4—202no—
563C2×C4C3×D4×D511yescharacteristic
564C2×C4C3×D4×D511yescharacteristic
565C2×C4C23×C3×D511yescharacteristic
566C8—22no—
567C8—102no—
568C2×C3—102no—
569C2×C3—202no—
570C2×C3—402no—
571C2×C3D5×#12111yescharacteristic
572C2×C3C2×#406511yescharacteristic
573C2×C3#6611111yescharacteristic
574C5C3×#303511yescharacteristic, complemented, Sylow
575C22—52no—
576C22—101no—
577C22—101no—
578C22—102no—
579C22—201no—
580C22—201no—
581C22—202no—
582C22—202no—
583C22—202no—
584C22C2×C3×D4×D511yescharacteristic
585C4—21no—
586C4—22no—
587C4—41no—
588C4—42no—
589C4—52no—
590C4—101no—
591C4—101no—
592C4—201no—
593C4C3×#406512yescomplemented
594C4C2×C3×D4×D511yescharacteristic
595C4C3×D5×#4111yescharacteristic
596C3#1061220611yescharacteristic, direct product factor, Sylow
597C2—102no—
598C2—202no—
599C2—402no—
600C2C3×D5×#12111yescharacteristic
601C2C2×C3×#406511yescharacteristic
602C2C3×#6611111yescharacteristic
subgroupG/H
centerC22×C3C2×D4×D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC3×C5×#462C2
socleC22×C3×C5C22×D4
hypercenterC3×#462D5
hypocenterC5C3×#3035
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#462

Complements

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

NH ≅ G/Ntype
1. C2×C3×#4065C2semidirect
2. C3×#66059C2semidirect
3. C3×C4×#713C2semidirect
4. C3×C5×#462C2semidirect
5. C3×#65474C2semidirect
6. C3×#65372C2semidirect
7. C3×#65316C2semidirect
8. C3×#65132C2semidirect
9. C3×D5×C4⋊C8C2semidirect
10. C3×#65108C2semidirect
11. C3×#65093C2semidirect
12. #10612206C3direct (N × H)
46. C2×#4065C2×C3semidirect
47. #66059C2×C3semidirect
48. C4×#713C2×C3semidirect
49. C5×#462C2×C3semidirect
50. #65474C2×C3semidirect
51. #65372C2×C3semidirect
52. #65316C2×C3semidirect
53. #65132C2×C3semidirect
54. D5×C4⋊C8C2×C3semidirect
55. #65108C2×C3semidirect
56. #65093C2×C3semidirect
62. C3×#713D4semidirect
106. C3×#462D5semidirect
217. #713C3×D4semidirect
261. #462C3×D5semidirect
281. C3×C4×C5#135semidirect
454. C4×C5C3×#135semidirect
505. C3×C5#3035semidirect
524. C3×C4#4065semidirect
574. C5C3×#3035semidirect
593. C4C3×#4065semidirect
596. C3#10612206direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#40652
C3×#660592
C3×C4×#7132
C3×C5×#4622
C3×#654742
C3×#653722
C3×#653162
C3×#651322
C3×D5×C4⋊C82
C3×#651082
C3×#650932
#106122063
C3×#30355