Finite Group Databasebeta

← #29646178

Subgroups — #29646178

Order 15616.

1012 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840362C211yesmaximal, characteristic
2#21454846C211yesmaximal, characteristic, complemented
3#21454845C211yesmaximal, characteristic, complemented
4#21444179C211yesmaximal, characteristic, complemented
5#21444177C212yesmaximal, complemented
6#21444168C211yesmaximal, characteristic, complemented
7#21444165C211yesmaximal, characteristic, complemented
8#21444164C212yesmaximal, complemented
9#21444158C212yesmaximal, complemented
10#21444150C212yesmaximal, complemented
11C61×#1471C211yesmaximal, characteristic, complemented
12#16840362—22no—
13#16840362—22no—
14#16840362—22no—
15#16840362—22no—
16C2×#14519773C411yescharacteristic
17C2×#14519767C411yescharacteristic
18C2×#14519762C411yescharacteristic
19C2×#14519760C2211yescharacteristic
20C2×#14519759C2211yescharacteristic
21C2×#14519758C411yescharacteristic
22C2×#14519750C2211yescharacteristic
23C2×#14519749C411yescharacteristic
24C2×#14519749C411yescharacteristic
25C2×#14519749C411yescharacteristic
26C2×#14519748C411yescharacteristic
27C2×C4×C22⋊D61C2211yescharacteristic
28C2×#14519737C2211yescharacteristic
29C2×#14519733C2211yescharacteristic
30#16839836—22no—
31#16839833—22no—
32#16839831—22no—
33#16839830—22no—
34#16839816C2212yes—
35#16839816C2212yes—
36#16839726C2211yescharacteristic
37#16839724C2212yes—
38C2×C22⋊#92418C2211yescharacteristic
39C22⋊C8×D61C2212yes—
40#16839487C2212yes—
41#16839487C2212yes—
42#16839486C2212yes—
43#16839486C2212yes—
44#16839485C2212yes—
45#16839485C2212yes—
46C2×#121×C61C2211yescharacteristic
47C61×#407C2211yescharacteristic
48C2×C22⋊C8×C61C2211yescharacteristic
49C61×#323C2212yes—
50C61×#323C2212yes—
51C22×D244C2×C411yescharacteristic
52C22×C22⋊D61C2×C411yescharacteristic
53C22×C22⋊D61C2×C411yescharacteristic
54C22×C22⋊D61C2×C411yescharacteristic
55C22×C22⋊D61C2×C411yescharacteristic
56C22×C22⋊D61C2×C411yescharacteristic
57C22×C22⋊D61C2×C411yescharacteristic
58C23×Dic61C2×C411yescharacteristic
59C23×Dic61C2×C411yescharacteristic
60C22×C4×D61C2×C411yescharacteristic
61C22×C4×D61C2311yescharacteristic
62C2×#11758194—21no—
63C2×#11758189—22no—
64C22×#92418—21no—
65C2×C8×D61—22no—
66#14519773—22no—
67#14519773—22no—
68#14519767—22no—
69#14519767—22no—
70#14519762—22no—
71#14519762—22no—
72#14519760—21no—
73#14519760—21no—
74#14519760—21no—
75#14519760—21no—
76#14519759—22no—
77#14519759—22no—
78#14519758—22no—
79#14519758—22no—
80#14519750—22no—
81#14519750—22no—
82#14519749—22no—
83#14519749—22no—
84#14519749—22no—
85#14519749—22no—
86#14519749—22no—
87#14519749—22no—
88#14519748—22no—
89#14519748—22no—
90C4×C22⋊D61D412yes—
91C4×C22⋊D61D412yes—
92C4×C22⋊D61D412yes—
93C4×C22⋊D61D412yes—
94#14519737—22no—
95#14519737—22no—
96#14519733—22no—
97#14519733—22no—
98C2×C4⋊Dic61C2×C411yescharacteristic
99C2×#11758196C2×C411yescharacteristic
100C2×#11758196C2×C411yescharacteristic
101C2×#11758196C2×C411yescharacteristic
102C2×#11758196C2×C411yescharacteristic
103C2×#11758196C2×C411yescharacteristic
104C2×#11758196C2311yescharacteristic
105C2×C22⋊Dic61C2×C411yescharacteristic
106C2×C22⋊Dic61C2×C411yescharacteristic
107C2×C22⋊Dic61C2×C411yescharacteristic
108C2×C22⋊Dic61C2×C411yescharacteristic
109C2×#11758198C2×C411yescharacteristic
110C2×#11758198C2×C411yescharacteristic
111C2×#11758198C2×C411yescharacteristic
112C2×#11758198C2311yescharacteristic
113C2×C4×Dic61C2×C411yescharacteristic
114C2×C4×Dic61C2311yescharacteristic
115#14519706—22no—
116#14519706—22no—
117C22⋊#92418—21no—
118C22⋊#92418—21no—
119C22⋊#92418C2312yes—
120C22⋊#92418C2312yes—
121#14519689—22no—
122#14519689—22no—
123#14519687—22no—
124#14519687—22no—
125C22×D4×C61C2×C411yescharacteristic
126C22×D4×C61C2×C411yescharacteristic
127C22×D4×C61C2×C411yescharacteristic
128C23×C4×C61C2311yescharacteristic
129C2×M16×C61—21no—
130C22×C8×C61—21no—
131#121×C61—21no—
132#121×C61—21no—
133#121×C61—21no—
134#121×C61—21no—
135C2×C4⋊C4×C61C2×C411yescharacteristic
136C2×C22⋊C4×C61C2311yescharacteristic
137C2×C22⋊C4×C61C2311yescharacteristic
138C22⋊C8×C61—21no—
139C22⋊C8×C61—21no—
140C22⋊C8×C61C2312yes—
141C22⋊C8×C61C2312yes—
142C24×C61C22×C411yescharacteristic
143C24×C61C22×C411yescharacteristic
144C24×C61C22×C411yescharacteristic
145C23×D61C22×C411yescharacteristic
146C23×D61C22×C411yescharacteristic
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—81no—
154C2×D4×C61—81no—
155C2×D4×C61—81no—
156C22×C4×C61—81no—
157C22×C4×C61C22×C411yescharacteristic
158C22×C4×C61C22×C411yescharacteristic
159C22×C4×C61C22×C411yescharacteristic
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2×D411yescharacteristic
164C22×C4×C61C2411yescharacteristic
165C2×C22⋊D61—21no—
166C2×C22⋊D61—21no—
167C2×C22⋊D61—22no—
168C2×C22⋊D61—22no—
169C2×C22⋊D61—22no—
170C2×C22⋊D61—22no—
171C2×C22⋊D61—42no—
172C2×C22⋊D61—42no—
173C2×C22⋊D61—81no—
174C2×C22⋊D61—81no—
175C2×C22⋊D61—81no—
176C2×C22⋊D61—81no—
177C2×C22⋊D61C22⋊C412yes—
178C2×C22⋊D61C22⋊C412yes—
179C22×Dic61—22no—
180C22×Dic61—22no—
181C22×Dic61—42no—
182C22×Dic61—81no—
183C22×Dic61C22×C411yescharacteristic
184C22×Dic61C22×C411yescharacteristic
185C22×Dic61C22×C411yescharacteristic
186C22×Dic61C22×C411yescharacteristic
187C22×Dic61C22×C411yescharacteristic
188C22×Dic61C22×C411yescharacteristic
189C2×D244—22no—
190C2×D244—81no—
191C2×C4×D61—22no—
192C2×C4×D61—81no—
193C2×C4×D61—81no—
194C2×C4×D61C2×D412yes—
195C2×C4×D61C2×D412yes—
196M16×C61—81no—
197C2×C8×C61—22no—
198C2×C8×C61—22no—
199C2×C8×C61—81no—
200C4⋊C4×C61—21no—
201C4⋊C4×C61—21no—
202C22⋊C4×C61—21no—
203C22⋊C4×C61—21no—
204C22⋊C4×C61—21no—
205C22⋊C4×C61—21no—
206C22⋊Dic61—21no—
207C22⋊Dic61—21no—
208C22⋊Dic61—22no—
209C22⋊Dic61—22no—
210C22⋊Dic61C22⋊C412yes—
211C22⋊Dic61C22⋊C412yes—
212#11758198—22no—
213#11758198—21no—
214#11758198—21no—
215#11758198—21no—
216#11758198—21no—
217#11758198C22⋊C412yes—
218#11758198C22⋊C412yes—
219C4⋊Dic61—22no—
220#11758196—21no—
221#11758196—21no—
222#11758196—21no—
223#11758196—21no—
224#11758196—22no—
225#11758196—22no—
226#11758196—22no—
227#11758196C22⋊C412yes—
228#11758196C22⋊C412yes—
229C4×Dic61—22no—
230C4×Dic61C2×D412yes—
231C4×Dic61C2×D412yes—
232#11758194—81no—
233C2×#92418—22no—
234C2×#92418—22no—
235C2×#92418—81no—
236#11758189—82no—
237C8×D61—82no—
238C23×C61—22no—
239C23×C61—21no—
240C23×C61—21no—
241C23×C61—22no—
242C23×C61—22no—
243C23×C61—41no—
244C23×C61—41no—
245C23×C61—41no—
246C23×C61—41no—
247C23×C61—41no—
248C23×C61—41no—
249C23×C61C2×C22⋊C411yescharacteristic
250C23×C61C2×C22⋊C411yescharacteristic
251C23×C61C23×C411yescharacteristic
252C22×D61—22no—
253C22×D61—21no—
254C22×D61—22no—
255C22×D61—41no—
256C22×D61—41no—
257C22×D61—41no—
258C22×D61—41no—
259C22×D61C2×C22⋊C412yes—
260D4×C61—81no—
261D4×C61—81no—
262D4×C61—81no—
263D4×C61—81no—
264D4×C61—81no—
265D4×C61—81no—
266C2×C4×C61—21no—
267C2×C4×C61—21no—
268C2×C4×C61—41no—
269C2×C4×C61—41no—
270C2×C4×C61—41no—
271C2×C4×C61—41no—
272C2×C4×C61—41no—
273C2×C4×C61—41no—
274C2×C4×C61—41no—
275C2×C4×C61—41no—
276C2×C4×C61C2×C22⋊C411yescharacteristic
277C2×C4×C61C2×C22⋊C411yescharacteristic
278C2×C4×C61C22×D411yescharacteristic
279C2×C4×C61C22×D411yescharacteristic
280C22⋊D61—42no—
281C22⋊D61—42no—
282C22⋊D61—44no—
283C22⋊D61—82no—
284C22⋊D61—82no—
285C22⋊D61—82no—
286C22⋊D61—81no—
287C22⋊D61—81no—
288C2×Dic61—22no—
289C2×Dic61—21no—
290C2×Dic61—21no—
291C2×Dic61—21no—
292C2×Dic61—22no—
293C2×Dic61—41no—
294C2×Dic61—41no—
295C2×Dic61—44no—
296C2×Dic61—41no—
297C2×Dic61—41no—
298C2×Dic61—82no—
299C2×Dic61C2×C22⋊C412yes—
300C2×Dic61C2×C22⋊C412yes—
301C2×Dic61C2×C22⋊C412yes—
302D244—82no—
303C4×D61—42no—
304C4×D61—42no—
305C4×D61—82no—
306C8×C61—42no—
307C8×C61—42no—
308#92418—42no—
309#92418—42no—
310#17408—611nomaximal, Sylow
311C22×C61—21no—
312C22×C61—21no—
313C22×C61—21no—
314C22×C61—21no—
315C22×C61—21no—
316C22×C61—21no—
317C22×C61—21no—
318C22×C61—21no—
319C22×C61—41no—
320C22×C61—41no—
321C22×C61—41no—
322C22×C61—41no—
323C22×C61—41no—
324C22×C61—41no—
325C22×C61—82no—
326C22×C61—82no—
327C22×C61—82no—
328C22×C61C22×C22⋊C411yescharacteristic
329C22×C61#56812yes—
330C2×D61—22no—
331C2×D61—22no—
332C2×D61—22no—
333C2×D61—41no—
334C2×D61—41no—
335C2×D61—41no—
336C2×D61—82no—
337C2×D61—82no—
338C4×C61—21no—
339C4×C61—21no—
340C4×C61—21no—
341C4×C61—21no—
342C4×C61—41no—
343C4×C61—41no—
344Dic61—22no—
345Dic61—22no—
346Dic61—22no—
347Dic61—41no—
348Dic61—41no—
349Dic61—41no—
350C2×#540—611no—
351#2973—611no—
352#2973—611no—
353#1471—612no—
354#1471—611no—
355#1471D6111yescharacteristic, complemented
356#1468—611no—
357#1468—611no—
358#1468—612no—
359#1466—612no—
360#1464—612no—
361C2×C61—22no—
362C2×C61—41no—
363C2×C61—41no—
364C2×C61—81no—
365C2×C61—81no—
366C2×C61#287911yescharacteristic
367C2×C61#288311yescharacteristic
368C2×C61#288811yescharacteristic
369D61—42no—
370D61—81no—
371#540—1222no—
372#540—1222no—
373#540—1222no—
374#540—1222no—
375C2×#127—611no—
376C2×#125—611no—
377C2×#123—611no—
378C2×#123—611no—
379C2×#123—611no—
380C2×#123—611no—
381C2×#121—611no—
382C2×#121—611no—
383C2×#121—611no—
384C2×#121—611no—
385C2×#121—611no—
386C2×#121C2×D6111yescharacteristic
387C2×C4×D4—611no—
388C2×C4×D4—611no—
389C2×#117—611no—
390#435—1222no—
391#435—1222no—
392#435—1222no—
393#435—1222no—
394#432—612no—
395#432—612no—
396#407—611no—
397#407—612no—
398#407C2×D6111yescharacteristic
399C2×C22⋊C8—611no—
400C2×C22⋊C8—612no—
401C2×C22⋊C8C2×D6111yescharacteristic
402#324—612no—
403#324—612no—
404#323—612no—
405#323—612no—
406#323—612no—
407#323—612no—
408#323C2×D6112yes—
409#323C2×D6112yes—
410C61#1740811yescharacteristic, complemented, Sylow
411C22×D4—611no—
412C22×D4—611no—
413C22×D4—611no—
414C22×D4—611no—
415C22×D4—611no—
416C22×D4—611no—
417C22×D4—611no—
418C22×D4C4×D6111yescharacteristic
419C22×D4C4×D6111yescharacteristic
420C22×D4C4×D6111yescharacteristic
421C23×C4—611no—
422C23×C4—611no—
423C23×C4—611no—
424C23×C4—611no—
425C23×C4C22×D6111yescharacteristic
426C2×M16—21no—
427C2×M16—1222no—
428C2×M16—1221no—
429C22×C8—21no—
430C22×C8—1222no—
431C22×C8—1221no—
432#127—1222no—
433#127—1222no—
434#125—1222no—
435#125—1222no—
436#123—1222no—
437#123—1222no—
438#123—1222no—
439#123—1222no—
440#123—1222no—
441#123—1222no—
442#123—1221no—
443#123—1221no—
444#123—1221no—
445#123—1221no—
446#121—21no—
447#121—21no—
448#121—21no—
449#121—21no—
450#121—1222no—
451#121—1222no—
452#121—1222no—
453#121—1222no—
454#121—1222no—
455#121—1222no—
456#121—1222no—
457#121—1222no—
458#121—1222no—
459#121—1222no—
460C4×D4—612no—
461C4×D4—612no—
462C4×D4—612no—
463C4×D4—612no—
464C4×D4—1222no—
465C4×D4—1222no—
466#117—1222no—
467#117—1222no—
468C2×C4⋊C4—611no—
469C2×C4⋊C4—611no—
470C2×C4⋊C4—611no—
471C2×C4⋊C4—611no—
472C2×C4⋊C4—611no—
473C2×C4⋊C4—611no—
474C2×C4⋊C4—611no—
475C2×C4⋊C4C4×D6111yescharacteristic
476C2×C22⋊C4—611no—
477C2×C22⋊C4—611no—
478C2×C22⋊C4—611no—
479C2×C22⋊C4—611no—
480C2×C22⋊C4—611no—
481C2×C22⋊C4—611no—
482C2×C22⋊C4—611no—
483C2×C22⋊C4—611no—
484C2×C22⋊C4C22×D6111yescharacteristic
485C2×C22⋊C4C22×D6111yescharacteristic
486C2×C42—611no—
487C2×C42—611no—
488C4⋊C8—1222no—
489C4⋊C8—1222no—
490C22⋊C8—21no—
491C22⋊C8—21no—
492C22⋊C8—612no—
493C22⋊C8—612no—
494C22⋊C8—1222no—
495C22⋊C8—1222no—
496C22⋊C8—1221no—
497C22⋊C8—1221no—
498C22⋊C8C22×D6112yes—
499C22⋊C8C22×D6112yes—
500#97—1222no—
501#97—1222no—
502C24—611no—
503C24—611no—
504C24C2×C4×D6111yescharacteristic
505C24C2×C4×D6111yescharacteristic
506C24C2×C4×D6111yescharacteristic
507C2×D4—21no—
508C2×D4—21no—
509C2×D4—21no—
510C2×D4—21no—
511C2×D4—21no—
512C2×D4—21no—
513C2×D4—81no—
514C2×D4—81no—
515C2×D4—81no—
516C2×D4—612no—
517C2×D4—612no—
518C2×D4—1222no—
519C2×D4—1222no—
520C2×D4—1222no—
521C2×D4—1222no—
522C2×D4—1222no—
523C2×D4—1221no—
524C2×D4—1221no—
525C2×D4—4881no—
526C2×D4—4881no—
527C2×D4—4881no—
528C2×D4—4881no—
529C2×D4—4881no—
530C2×D4—2442no—
531C2×D4—2442no—
532C22×C4—81no—
533C22×C4—611no—
534C22×C4—611no—
535C22×C4—611no—
536C22×C4—612no—
537C22×C4—612no—
538C22×C4—611no—
539C22×C4—611no—
540C22×C4—611no—
541C22×C4—1222no—
542C22×C4—1222no—
543C22×C4—1222no—
544C22×C4—4881no—
545C22×C4—4881no—
546C22×C4—4881no—
547C22×C4—2442no—
548C22×C4C2×C4×D6111yescharacteristic
549C22×C4C2×C4×D6111yescharacteristic
550C22×C4C2×C4×D6111yescharacteristic
551C22×C4D4×D6111yescharacteristic
552C22×C4D4×D6111yescharacteristic
553C22×C4D4×D6111yescharacteristic
554C22×C4D4×D6111yescharacteristic
555C22×C4C23×D6111yescharacteristic
556M16—81no—
557M16—4882no—
558M16—4881no—
559C2×C8—22no—
560C2×C8—22no—
561C2×C8—81no—
562C2×C8—1222no—
563C2×C8—1222no—
564C2×C8—4882no—
565C2×C8—4881no—
566C4⋊C4—21no—
567C4⋊C4—21no—
568C4⋊C4—612no—
569C4⋊C4—612no—
570C4⋊C4—1222no—
571C4⋊C4—1222no—
572C4⋊C4—1222no—
573C4⋊C4—1222no—
574C4⋊C4—1221no—
575C4⋊C4—1221no—
576C4⋊C4—1221no—
577C4⋊C4—1221no—
578C22⋊C4—21no—
579C22⋊C4—21no—
580C22⋊C4—21no—
581C22⋊C4—21no—
582C22⋊C4—612no—
583C22⋊C4—612no—
584C22⋊C4—612no—
585C22⋊C4—612no—
586C22⋊C4—1222no—
587C22⋊C4—1222no—
588C22⋊C4—1222no—
589C22⋊C4—1221no—
590C22⋊C4—1221no—
591C22⋊C4—1221no—
592C22⋊C4—1221no—
593C22⋊C4—1221no—
594C22⋊C4—1221no—
595C42—612no—
596C42—612no—
597C42—1222no—
598C23—22no—
599C23—21no—
600C23—21no—
601C23—22no—
602C23—22no—
603C23—41no—
604C23—41no—
605C23—41no—
606C23—41no—
607C23—41no—
608C23—41no—
609C23—612no—
610C23—1222no—
611C23—1222no—
612C23—1221no—
613C23—2441no—
614C23—2441no—
615C23—2441no—
616C23—2441no—
617C23C22⋊C4×D6111yescharacteristic
618C23C22⋊C4×D6111yescharacteristic
619C23C22×C4×D6111yescharacteristic
620D4—81no—
621D4—81no—
622D4—81no—
623D4—81no—
624D4—81no—
625D4—81no—
626D4—4882no—
627D4—4882no—
628D4—4881no—
629D4—4881no—
630D4—4882no—
631D4—4882no—
632D4—2442no—
633D4—2442no—
634D4—2444no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—612no—
646C2×C4—612no—
647C2×C4—612no—
648C2×C4—1222no—
649C2×C4—1221no—
650C2×C4—1221no—
651C2×C4—1221no—
652C2×C4—1222no—
653C2×C4—4882no—
654C2×C4—4882no—
655C2×C4—2442no—
656C2×C4—2442no—
657C2×C4—2441no—
658C2×C4—2441no—
659C2×C4—2444no—
660C2×C4—2441no—
661C2×C4—2441no—
662C2×C4C22⋊C4×D6111yescharacteristic
663C2×C4C22⋊C4×D6111yescharacteristic
664C2×C4C2×D4×D6111yescharacteristic
665C2×C4C2×D4×D6111yescharacteristic
666C8—42no—
667C8—42no—
668C8—2442no—
669C8—2442no—
670C22—21no—
671C22—21no—
672C22—21no—
673C22—21no—
674C22—21no—
675C22—21no—
676C22—21no—
677C22—21no—
678C22—41no—
679C22—41no—
680C22—41no—
681C22—41no—
682C22—41no—
683C22—41no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—1222no—
688C22—1222no—
689C22—1222no—
690C22—4882no—
691C22—4882no—
692C22—2441no—
693C22—2441no—
694C22—2441no—
695C22C2×C22⋊C4×D6111yescharacteristic
696C22#1684052712yes—
697C4—21no—
698C4—21no—
699C4—21no—
700C4—21no—
701C4—41no—
702C4—41no—
703C4—1222no—
704C4—1222no—
705C4—1222no—
706C4—2441no—
707C4—2441no—
708C4—2441no—
709C2—22no—
710C2—41no—
711C2—41no—
712C2—81no—
713C2—81no—
714C2—4881no—
715C2—2442no—
716C2#2145426111yescharacteristic
717C2#2145427711yescharacteristic
718C2#2145429511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1471C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840362C2
cosocleC22×C4×C61C24
hypercenter#1471D61
hypocenterC61#17408
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1471

Complements

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

NH ≅ G/Ntype
2. #21454846C2semidirect
3. #21454845C2semidirect
4. #21444179C2semidirect
5. #21444177C2semidirect
6. #21444168C2semidirect
7. #21444165C2semidirect
8. #21444164C2semidirect
9. #21444158C2semidirect
10. #21444150C2semidirect
11. C61×#1471C2semidirect
355. #1471D61semidirect
410. C61#17408semidirect

Maximal subgroups

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

subgroupindex
C2×#168403622
#214548462
#214548452
#214441792
#214441772
#214441682
#214441652
#214441642
#214441582
#214441502
C61×#14712
#1740861