Finite Group Databasebeta

← #29646169

Subgroups — #29646169

Order 15616.

948 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840380C211yesmaximal, characteristic
2#21454846C211yesmaximal, characteristic, complemented
3#21454845C211yesmaximal, characteristic, complemented
4#21444184C211yesmaximal, characteristic, complemented
5#21444182C211yesmaximal, characteristic, complemented
6#21444181C212yesmaximal, complemented
7#21444167C211yesmaximal, characteristic, complemented
8#21444164C212yesmaximal, complemented
9#21444161C212yesmaximal, complemented
10#21444150C212yesmaximal, complemented
11C61×#1468C211yesmaximal, characteristic, complemented
12#16840380—22no—
13#16840380—22no—
14#16840380—22no—
15#16840380—22no—
16C2×#14519776C411yescharacteristic
17C2×#14519772C411yescharacteristic
18C2×#14519770C411yescharacteristic
19C2×#14519764C411yescharacteristic
20C2×#14519763C411yescharacteristic
21C2×#14519763C411yescharacteristic
22C2×#14519759C2211yescharacteristic
23C2×#14519756C2211yescharacteristic
24C2×#14519755C2211yescharacteristic
25C2×#14519754C411yescharacteristic
26C2×#14519749C411yescharacteristic
27C2×#14519742C2211yescharacteristic
28C2×C4×C22⋊D61C2211yescharacteristic
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#16839488C2212yes—
41#16839487C2212yes—
42#16839487C2212yes—
43#16839487C2212yes—
44#16839487C2212yes—
45#16839486C2212yes—
46C2×#123×C61C2211yescharacteristic
47C61×#407C2211yescharacteristic
48C2×C22⋊C8×C61C2211yescharacteristic
49C61×#324C2212yes—
50C61×#323C2212yes—
51C22×Dic122C2×C411yescharacteristic
52C22×Dic122C2×C411yescharacteristic
53C22×Dic122C2×C411yescharacteristic
54C22×C22⋊D61C2×C411yescharacteristic
55C22×C22⋊D61C2×C411yescharacteristic
56C23×Dic61C2×C411yescharacteristic
57C22×C4×D61C2311yescharacteristic
58C2×#11758194—21no—
59C2×#11758189—22no—
60C22×#92418—21no—
61C2×C8×D61—22no—
62#14519776—22no—
63#14519776—22no—
64#14519772—22no—
65#14519772—22no—
66#14519770—22no—
67#14519770—22no—
68#14519764—22no—
69#14519764—22no—
70#14519763—22no—
71#14519763—22no—
72#14519763—22no—
73#14519763—22no—
74#14519759—22no—
75#14519759—22no—
76#14519756—21no—
77#14519756—21no—
78#14519756—21no—
79#14519756—21no—
80#14519755—22no—
81#14519755—22no—
82#14519754—22no—
83#14519754—22no—
84#14519749—22no—
85#14519749—22no—
86#14519742—22no—
87#14519742—22no—
88C4×C22⋊D61D412yes—
89C4×C22⋊D61D412yes—
90C4×C22⋊D61D412yes—
91C4×C22⋊D61D412yes—
92#14519733—22no—
93#14519733—22no—
94C2×C4⋊Dic61C2×C411yescharacteristic
95C2×C4⋊Dic61C2×C411yescharacteristic
96C2×#11758196C2×C411yescharacteristic
97C2×#11758196C2×C411yescharacteristic
98C2×#11758196C2×C411yescharacteristic
99C2×#11758196C2×C411yescharacteristic
100C2×#11758196C2311yescharacteristic
101C2×#11758196C2311yescharacteristic
102C2×C22⋊Dic61C2×C411yescharacteristic
103C2×C22⋊Dic61C2×C411yescharacteristic
104C2×C22⋊Dic61C2×C411yescharacteristic
105C2×C22⋊Dic61C2×C411yescharacteristic
106C2×C22⋊Dic61C2×C411yescharacteristic
107C2×#11758198C2×C411yescharacteristic
108C2×#11758198C2×C411yescharacteristic
109C2×#11758198C2×C411yescharacteristic
110C2×#11758198C2×C411yescharacteristic
111C2×C4×Dic61C2×C411yescharacteristic
112C2×C4×Dic61C2×C411yescharacteristic
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
126C23×C4×C61C2311yescharacteristic
127C2×M16×C61—21no—
128C22×C8×C61—21no—
129#123×C61—21no—
130#123×C61—21no—
131#123×C61—21no—
132#123×C61—21no—
133C2×C4⋊C4×C61C2×C411yescharacteristic
134C2×C4⋊C4×C61C2311yescharacteristic
135C2×C22⋊C4×C61C2×C411yescharacteristic
136C2×C22⋊C4×C61C2×C411yescharacteristic
137C2×C22⋊C4×C61C2311yescharacteristic
138C22⋊C8×C61—21no—
139C22⋊C8×C61—21no—
140C22⋊C8×C61C2312yes—
141C22⋊C8×C61C2312yes—
142C24×C61C22×C411yescharacteristic
143C24×C61C22×C411yescharacteristic
144C23×D61C22×C411yescharacteristic
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—81no—
148C22×C4×C61—81no—
149C22×C4×C61C22×C411yescharacteristic
150C22×C4×C61C22×C411yescharacteristic
151C22×C4×C61C22×C411yescharacteristic
152C22×C4×C61C22×C411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2411yescharacteristic
158C2×C22⋊D61—22no—
159C2×C22⋊D61—42no—
160C2×C22⋊D61—81no—
161C2×C22⋊D61C22⋊C412yes—
162C2×C22⋊D61C22⋊C412yes—
163C22×Dic61—22no—
164C22×Dic61—42no—
165C22×Dic61C22×C411yescharacteristic
166C22×Dic61C22×C411yescharacteristic
167C22×Dic61C22×C411yescharacteristic
168C22×Dic61C22×C411yescharacteristic
169C22×Dic61C22×C411yescharacteristic
170C22×Dic61C22×C411yescharacteristic
171C22×Dic61C22×C411yescharacteristic
172C2×C4×D61—81no—
173C2×C4×D61C2×D412yes—
174C2×C4×D61C2×D412yes—
175C2×Dic122—22no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—22no—
179C2×Dic122—42no—
180C2×Dic122—81no—
181C2×Dic122—81no—
182M16×C61—81no—
183C2×C8×C61—22no—
184C2×C8×C61—22no—
185C2×C8×C61—81no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C22⋊C4×C61—21no—
191C22⋊C4×C61—21no—
192C22⋊C4×C61—21no—
193C22⋊C4×C61—21no—
194C22⋊C4×C61—21no—
195C22⋊C4×C61—21no—
196C22⋊Dic61—21no—
197C22⋊Dic61—21no—
198C22⋊Dic61—21no—
199C22⋊Dic61—21no—
200C22⋊Dic61—22no—
201C22⋊Dic61—22no—
202C22⋊Dic61C22⋊C412yes—
203C22⋊Dic61C22⋊C412yes—
204#11758198—22no—
205#11758198—22no—
206#11758198—22no—
207#11758198C22⋊C412yes—
208#11758198C22⋊C412yes—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—22no—
212#11758196—21no—
213#11758196—21no—
214#11758196—21no—
215#11758196—21no—
216#11758196—22no—
217#11758196—22no—
218#11758196—22no—
219#11758196C22⋊C412yes—
220#11758196C22⋊C412yes—
221C4×Dic61—22no—
222C4×Dic61—22no—
223C4×Dic61—22no—
224C4×Dic61C2×D412yes—
225C4×Dic61C2×D412yes—
226#11758194—81no—
227C2×#92418—22no—
228C2×#92418—22no—
229C2×#92418—81no—
230#11758189—82no—
231C8×D61—82no—
232C23×C61—22no—
233C23×C61—21no—
234C23×C61—22no—
235C23×C61—41no—
236C23×C61—41no—
237C23×C61—41no—
238C23×C61—41no—
239C23×C61C2×C22⋊C411yescharacteristic
240C23×C61C2×C22⋊C411yescharacteristic
241C23×C61C23×C411yescharacteristic
242C22×D61—22no—
243C22×D61—41no—
244C22×D61—41no—
245C22×D61C2×C22⋊C412yes—
246D4×C61—81no—
247D4×C61—81no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—22no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61C2×C22⋊C411yescharacteristic
261C2×C4×C61C2×C22⋊C411yescharacteristic
262C2×C4×C61C22×D411yescharacteristic
263C2×C4×C61C22×D411yescharacteristic
264C22⋊D61—42no—
265C22⋊D61—42no—
266C22⋊D61—82no—
267C2×Dic61—22no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—22no—
273C2×Dic61—22no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—44no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61C2×C22⋊C412yes—
280C2×Dic61C2×C22⋊C412yes—
281C2×Dic61C2×C22⋊C412yes—
282C4×D61—42no—
283C4×D61—42no—
284Dic122—44no—
285Dic122—82no—
286Dic122—81no—
287Dic122—81no—
288C8×C61—42no—
289C8×C61—42no—
290#92418—42no—
291#92418—42no—
292#17407—611nomaximal, Sylow
293C22×C61—21no—
294C22×C61—21no—
295C22×C61—21no—
296C22×C61—21no—
297C22×C61—21no—
298C22×C61—21no—
299C22×C61—21no—
300C22×C61—21no—
301C22×C61—41no—
302C22×C61—41no—
303C22×C61—41no—
304C22×C61—82no—
305C22×C61—82no—
306C22×C61C22×C22⋊C411yescharacteristic
307C22×C61#56812yes—
308C2×D61—22no—
309C2×D61—22no—
310C2×D61—22no—
311C2×D61—82no—
312C4×C61—21no—
313C4×C61—21no—
314C4×C61—21no—
315C4×C61—21no—
316C4×C61—41no—
317C4×C61—41no—
318C4×C61—41no—
319Dic61—22no—
320Dic61—22no—
321Dic61—22no—
322Dic61—41no—
323Dic61—41no—
324Dic61—41no—
325Dic61—41no—
326C2×#542—611no—
327#2973—611no—
328#2973—611no—
329#1472—611no—
330#1472—611no—
331#1472—612no—
332#1468—612no—
333#1468—611no—
334#1468D6111yescharacteristic, complemented
335#1467—612no—
336#1464—612no—
337C2×C61—22no—
338C2×C61—41no—
339C2×C61—41no—
340C2×C61—81no—
341C2×C61#287911yescharacteristic
342C2×C61#288311yescharacteristic
343C2×C61#288911yescharacteristic
344D61—42no—
345#542—1222no—
346#542—1222no—
347#542—1222no—
348#542—1222no—
349C2×C4⋊Q8—611no—
350C2×#126—611no—
351C2×#126—611no—
352C2×#124—611no—
353C2×#124—611no—
354C2×#124—611no—
355C2×#123—611no—
356C2×#123C2×D6111yescharacteristic
357C2×C22⋊Q8—611no—
358C2×C22⋊Q8—611no—
359C2×C22⋊Q8—611no—
360C2×#121—611no—
361C2×C4×Q8—611no—
362C2×C4×D4—611no—
363C2×#117—611no—
364#435—1222no—
365#435—1222no—
366#435—1222no—
367#435—1222no—
368#432—612no—
369#432—612no—
370#407—611no—
371#407—612no—
372#407C2×D6111yescharacteristic
373C2×C22⋊C8—611no—
374C2×C22⋊C8—612no—
375C2×C22⋊C8C2×D6111yescharacteristic
376#324—612no—
377#324—612no—
378#324—612no—
379#324—612no—
380#324—612no—
381#324C2×D6112yes—
382#323—612no—
383#323C2×D6112yes—
384C61#1740711yescharacteristic, complemented, Sylow
385C22×Q8—611no—
386C22×Q8—611no—
387C22×Q8—611no—
388C22×D4—611no—
389C22×D4—611no—
390C22×D4C4×D6111yescharacteristic
391C23×C4—611no—
392C23×C4—611no—
393C23×C4C22×D6111yescharacteristic
394C2×M16—21no—
395C2×M16—1222no—
396C2×M16—1221no—
397C22×C8—21no—
398C22×C8—1222no—
399C22×C8—1221no—
400C4⋊Q8—1222no—
401C4⋊Q8—1222no—
402#126—1222no—
403#126—1222no—
404#126—1222no—
405#126—1222no—
406#124—1222no—
407#124—1222no—
408#124—1222no—
409#124—1222no—
410#124—1222no—
411#124—1222no—
412#123—21no—
413#123—21no—
414#123—21no—
415#123—21no—
416#123—1222no—
417#123—1222no—
418C22⋊Q8—1222no—
419C22⋊Q8—1222no—
420C22⋊Q8—1222no—
421C22⋊Q8—1222no—
422C22⋊Q8—1221no—
423C22⋊Q8—1221no—
424C22⋊Q8—1221no—
425C22⋊Q8—1221no—
426#121—1222no—
427#121—1222no—
428C4×Q8—1222no—
429C4×Q8—1222no—
430C4×D4—612no—
431C4×D4—612no—
432C4×D4—612no—
433C4×D4—612no—
434#117—1222no—
435#117—1222no—
436C2×C4⋊C4—611no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—611no—
441C2×C4⋊C4—611no—
442C2×C4⋊C4—611no—
443C2×C4⋊C4—611no—
444C2×C4⋊C4C4×D6111yescharacteristic
445C2×C4⋊C4C22×D6111yescharacteristic
446C2×C22⋊C4—611no—
447C2×C22⋊C4—611no—
448C2×C22⋊C4—611no—
449C2×C22⋊C4—611no—
450C2×C22⋊C4—611no—
451C2×C22⋊C4—611no—
452C2×C22⋊C4—611no—
453C2×C22⋊C4—611no—
454C2×C22⋊C4—611no—
455C2×C22⋊C4C4×D6111yescharacteristic
456C2×C22⋊C4C4×D6111yescharacteristic
457C2×C22⋊C4C22×D6111yescharacteristic
458C2×C42—611no—
459C2×C42—611no—
460C2×C42—611no—
461C2×C42—611no—
462C4⋊C8—1222no—
463C4⋊C8—1222no—
464C22⋊C8—21no—
465C22⋊C8—21no—
466C22⋊C8—612no—
467C22⋊C8—612no—
468C22⋊C8—1222no—
469C22⋊C8—1222no—
470C22⋊C8—1221no—
471C22⋊C8—1221no—
472C22⋊C8C22×D6112yes—
473C22⋊C8C22×D6112yes—
474#97—1222no—
475#97—1222no—
476C24—611no—
477C24C2×C4×D6111yescharacteristic
478C24C2×C4×D6111yescharacteristic
479C2×Q8—1222no—
480C2×Q8—1222no—
481C2×Q8—1221no—
482C2×Q8—1221no—
483C2×Q8—4881no—
484C2×Q8—4881no—
485C2×Q8—2442no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—81no—
489C2×D4—612no—
490C2×D4—612no—
491C2×D4—1222no—
492C2×D4—4881no—
493C2×D4—2442no—
494C22×C4—81no—
495C22×C4—611no—
496C22×C4—611no—
497C22×C4—611no—
498C22×C4—612no—
499C22×C4—612no—
500C22×C4—611no—
501C22×C4—611no—
502C22×C4—611no—
503C22×C4—611no—
504C22×C4—1222no—
505C22×C4—4881no—
506C22×C4—2442no—
507C22×C4C2×C4×D6111yescharacteristic
508C22×C4C2×C4×D6111yescharacteristic
509C22×C4C2×C4×D6111yescharacteristic
510C22×C4C2×C4×D6111yescharacteristic
511C22×C4D4×D6111yescharacteristic
512C22×C4D4×D6111yescharacteristic
513C22×C4D4×D6111yescharacteristic
514C22×C4D4×D6111yescharacteristic
515C22×C4C23×D6111yescharacteristic
516M16—81no—
517M16—4882no—
518M16—4881no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—81no—
522C2×C8—1222no—
523C2×C8—1222no—
524C2×C8—4882no—
525C2×C8—4881no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—612no—
531C4⋊C4—612no—
532C4⋊C4—1222no—
533C4⋊C4—1222no—
534C4⋊C4—1222no—
535C4⋊C4—1222no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C22⋊C4—21no—
543C22⋊C4—21no—
544C22⋊C4—21no—
545C22⋊C4—21no—
546C22⋊C4—21no—
547C22⋊C4—21no—
548C22⋊C4—612no—
549C22⋊C4—612no—
550C22⋊C4—612no—
551C22⋊C4—612no—
552C22⋊C4—1222no—
553C22⋊C4—1222no—
554C22⋊C4—1222no—
555C22⋊C4—1222no—
556C22⋊C4—1222no—
557C22⋊C4—1221no—
558C22⋊C4—1221no—
559C22⋊C4—1221no—
560C22⋊C4—1221no—
561C42—612no—
562C42—612no—
563C42—1222no—
564C42—1222no—
565C42—1222no—
566C23—22no—
567C23—21no—
568C23—22no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—612no—
574C23—1222no—
575C23—2441no—
576C23—2441no—
577C23C22⋊C4×D6111yescharacteristic
578C23C22⋊C4×D6111yescharacteristic
579C23C22×C4×D6111yescharacteristic
580Q8—4881no—
581Q8—4881no—
582Q8—4882no—
583Q8—2444no—
584D4—81no—
585D4—81no—
586D4—4882no—
587D4—2442no—
588D4—2442no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—612no—
602C2×C4—612no—
603C2×C4—612no—
604C2×C4—1222no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1222no—
610C2×C4—1222no—
611C2×C4—2442no—
612C2×C4—2442no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2444no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4C22⋊C4×D6111yescharacteristic
619C2×C4C22⋊C4×D6111yescharacteristic
620C2×C4C2×D4×D6111yescharacteristic
621C2×C4C2×D4×D6111yescharacteristic
622C8—42no—
623C8—42no—
624C8—2442no—
625C8—2442no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—82no—
638C22—82no—
639C22—1222no—
640C22—1222no—
641C22—1222no—
642C22—4882no—
643C22C2×C22⋊C4×D6111yescharacteristic
644C22#1684052712yes—
645C4—21no—
646C4—21no—
647C4—21no—
648C4—21no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—1222no—
653C4—1222no—
654C4—1222no—
655C4—2441no—
656C4—2441no—
657C4—2441no—
658C4—2441no—
659C2—22no—
660C2—41no—
661C2—41no—
662C2—81no—
663C2—2442no—
664C2#2145426111yescharacteristic
665C2#2145427711yescharacteristic
666C2#2145430011yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1468C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#14519749C4
cosocleC22×C4×C61C24
hypercenter#1468D61
hypocenterC61#17407
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1468

Complements

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

NH ≅ G/Ntype
2. #21454846C2semidirect
3. #21454845C2semidirect
4. #21444184C2semidirect
5. #21444182C2semidirect
6. #21444181C2semidirect
7. #21444167C2semidirect
8. #21444164C2semidirect
9. #21444161C2semidirect
10. #21444150C2semidirect
11. C61×#1468C2semidirect
334. #1468D61semidirect
384. C61#17407semidirect

Maximal subgroups

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

subgroupindex
C2×#168403802
#214548462
#214548452
#214441842
#214441822
#214441812
#214441672
#214441642
#214441612
#214441502
C61×#14682
#1740761