Finite Group Databasebeta

← #29779620

Subgroups — #29779620

Order 15616.

794 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21451654C211yesmaximal, characteristic
2#21448402C211yesmaximal, characteristic
3#21447231C211yesmaximal, characteristic, complemented
4#21447230C211yesmaximal, characteristic, complemented
5#21446969C211yesmaximal, characteristic, complemented
6#21446968C211yesmaximal, characteristic, complemented
7#21446509C211yesmaximal, characteristic, complemented
8#21446498C211yesmaximal, characteristic, complemented
9#21446494C211yesmaximal, characteristic, complemented
10#21445913C211yesmaximal, characteristic
11#21445912C211yesmaximal, characteristic
12#21445752C211yesmaximal, characteristic
13#21445752C211yesmaximal, characteristic
14#21445588C211yesmaximal, characteristic
15C61×#1935C211yesmaximal, characteristic, complemented
16C22×Q8×D61C2211yescharacteristic
17C2×#14519768C2211yescharacteristic
18C2×#14519735C2211yescharacteristic
19C2×#14519734C2211yescharacteristic
20C2×#14519702C2211yescharacteristic
21C2×#14519702C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519700C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519689C2211yescharacteristic
27C2×#14519689C2211yescharacteristic
28C2×C4×#92418C2211yescharacteristic
29#16839696—21no—
30#16839693—21no—
31#16839686—21no—
32#16839684—21no—
33#16839669C2211yescharacteristic
34#16839668C2211yescharacteristic
35#16839667C2211yescharacteristic
36#16839666—21no—
37#16839666—21no—
38#16839660C2211yescharacteristic
39#16839631C2211yescharacteristic
40#16839624—21no—
41#16839624—21no—
42#16839619C2211yescharacteristic
43#16839619C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#103×C61C2211yescharacteristic
53C2×#103×C61C2211yescharacteristic
54C2×#97×C61C2211yescharacteristic
55C61×#391C2211yescharacteristic
56C61×#389C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×Dic122—21no—
60C22×Dic122C2311yescharacteristic
61C2×Q8×D61—22no—
62C2×Q8×D61—22no—
63C2×Q8×D61—22no—
64C2×Q8×D61—22no—
65C2×Q8×D61—41no—
66C2×Q8×D61—41no—
67C22×C4×D61—21no—
68C22×C4×D61C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71#14519768—41no—
72#14519768—41no—
73#14519735—22no—
74#14519735—22no—
75#14519734—41no—
76#14519734—41no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196D411yescharacteristic
86C2×#11758196D411yescharacteristic
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198D411yescharacteristic
91C2×#11758198D411yescharacteristic
92C2×C4×Dic61—21no—
93C2×C4×Dic61C2311yescharacteristic
94#14519702—22no—
95#14519702—41no—
96#14519701—22no—
97#14519701—41no—
98#14519700—22no—
99#14519700—41no—
100#14519689—22no—
101#14519689—22no—
102C4×#92418—41no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112C22×Q8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C42×C61C2311yescharacteristic
120#103×C61—22no—
121#103×C61—41no—
122#97×C61—41no—
123#95×C61—21no—
124#95×C61—21no—
125#95×C61—21no—
126C23×D61C2×D411yescharacteristic
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61#4112yes—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2411yescharacteristic
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147Q8×D61—44no—
148Q8×D61—44no—
149Q8×D61—44no—
150Q8×D61—44no—
151C2×C4×D61—21no—
152C2×C4×D61—21no—
153C2×C4×D61—21no—
154C2×C4×D61—21no—
155C2×C4×D61—24no—
156C2×C4×D61—42no—
157C2×C4×D61—42no—
158C2×C4×D61#4112yes—
159C2×C4×D61#4112yes—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—24no—
165C2×Dic122—21no—
166C2×Dic122—42no—
167C2×Dic122—42no—
168C2×Dic122#4112yes—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—81no—
182C42×C61—21no—
183C42×C61—21no—
184#11758198—41no—
185#11758198—41no—
186#11758198—42no—
187#11758198—81no—
188#11758198—81no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—42no—
196#11758196—41no—
197#11758196—41no—
198#11758196—42no—
199#11758196—81no—
200#11758196—81no—
201#11758196—81no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—42no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C23×C61C22×D411yescharacteristic
212C22×D61—21no—
213C22×D61—21no—
214C22×D61—21no—
215C22×D61—21no—
216C22×D61—21no—
217C22×D61—21no—
218C22×D61#12412yes—
219Q8×C61—22no—
220Q8×C61—22no—
221Q8×C61—41no—
222Q8×C61—41no—
223C2×C4×C61—22no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61H(5,𝔽2)11yescharacteristic
246C2×C4×C61H(5,𝔽2)11yescharacteristic
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—22no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61#12412yes—
268C4×D61—22no—
269C4×D61—22no—
270C4×D61—22no—
271C4×D61—22no—
272C4×D61—48no—
273Dic122—22no—
274Dic122—22no—
275Dic122—48no—
276Dic122—41no—
277Dic122—41no—
278C8×C61—41no—
279C8×C61—41no—
280#92418—41no—
281#92418—41no—
282#29117—611nomaximal, Sylow
283C22×C61#52511yescharacteristic
284C22×C61C2×#12411yescharacteristic
285C22×C61#53411yescharacteristic
286C22×C61#57711yescharacteristic
287C22×C61#57711yescharacteristic
288C22×C61#57811yescharacteristic
289C22×C61#57811yescharacteristic
290C2×D61—22no—
291C2×D61—22no—
292C2×D61—22no—
293C2×D61—41no—
294C2×D61—41no—
295C2×D61—41no—
296C2×D61—41no—
297C4×C61—42no—
298C4×C61—41no—
299C4×C61—41no—
300C4×C61—81no—
301C4×C61—81no—
302C4×C61—81no—
303C4×C61#55111yescharacteristic
304C4×C61#55111yescharacteristic
305C4×C61#55111yescharacteristic
306C4×C61#55111yescharacteristic
307Dic61—22no—
308Dic61—42no—
309Dic61—41no—
310Dic61—41no—
311Dic61—81no—
312Dic61—81no—
313Dic61—81no—
314#2629—611no—
315#2310—611no—
316#2072—611no—
317#2072—611no—
318#2026—611no—
319#2026—611no—
320#1935—611no—
321#1935D6111yescharacteristic, complemented
322#1933—611no—
323#1933—611no—
324#1825—611no—
325#1825—611no—
326#1787—611no—
327#1787—611no—
328#1750—611no—
329C2×C61#256111yescharacteristic
330C2×C61#312611yescharacteristic
331C2×C61#312711yescharacteristic
332C2×C61#313711yescharacteristic
333C2×C61#313711yescharacteristic
334C2×C61#320111yescharacteristic
335C2×C61#320111yescharacteristic
336D61—42no—
337C23×Q8—611no—
338C2×#125—611no—
339C2×#117—611no—
340C2×#117—611no—
341C2×#103—611no—
342C2×#103—611no—
343C2×#103—611no—
344C2×#103—611no—
345C2×#103—611no—
346C2×#103—611no—
347C2×#103C2×D6111yescharacteristic
348C2×#103C2×D6111yescharacteristic
349C2×C22⋊C8—611no—
350C2×C22⋊C8—611no—
351C2×#97C2×D6111yescharacteristic
352C2×C4×C8—611no—
353#398—1221no—
354#398—1221no—
355#396—1221no—
356#396—1221no—
357#391—611no—
358#391—611no—
359#391—611no—
360#391—1221no—
361#391—1221no—
362#391C2×D6111yescharacteristic
363#389—611no—
364#389C2×D6111yescharacteristic
365#384—611no—
366#381—1221no—
367#381—1221no—
368#380—611no—
369#380—611no—
370#340—611no—
371#340—611no—
372#340—611no—
373#340—611no—
374#340—611no—
375#340—611no—
376#340C2×D6111yescharacteristic
377#340C2×D6111yescharacteristic
378#336—611no—
379#336—611no—
380C61#2911711yescharacteristic, complemented, Sylow
381C22×Q8—611no—
382C22×Q8—1221no—
383C22×Q8—1222no—
384C22×Q8—1222no—
385C22×Q8—1222no—
386C22×Q8—1222no—
387C22×Q8—2441no—
388C22×Q8—2441no—
389C22×Q8C22×D6111yescharacteristic
390C23×C4—611no—
391C23×C4—1221no—
392C22×C8—611no—
393C22×C8—611no—
394C22×C8C22×D6111yescharacteristic
395C22×C8C22×D6111yescharacteristic
396#125—2441no—
397#125—2441no—
398#117—1222no—
399#117—1222no—
400#117—2441no—
401#117—2441no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4C22×D6111yescharacteristic
414C2×C4⋊C4C22×D6111yescharacteristic
415C2×C4⋊C4C22×D6111yescharacteristic
416C2×C22⋊C4—611no—
417C2×C22⋊C4—611no—
418C2×C22⋊C4—1221no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1221no—
421C2×C42—611no—
422C2×C42—1221no—
423C2×C42C22×D6111yescharacteristic
424#103—22no—
425#103—41no—
426#103—1222no—
427#103—1222no—
428#103—1222no—
429#103—2441no—
430#103—2441no—
431#103—2441no—
432C22⋊C8—1222no—
433C22⋊C8—1222no—
434#97—41no—
435C4×C8—2441no—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—1221no—
440#95—1221no—
441#95—1221no—
442#95—1221no—
443#95—1221no—
444#95—1221no—
445#95—1221no—
446#95—1221no—
447#95—1221no—
448C24—611no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—612no—
455C2×Q8—1224no—
456C2×Q8—1221no—
457C2×Q8—1221no—
458C2×Q8—1221no—
459C2×Q8—1221no—
460C2×Q8—1221no—
461C2×Q8—2442no—
462C2×Q8—2442no—
463C2×Q8—2444no—
464C2×Q8—2444no—
465C2×Q8—2444no—
466C2×Q8—2444no—
467C2×Q8#1175822212yes—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—611no—
473C22×C4—612no—
474C22×C4—612no—
475C22×C4—611no—
476C22×C4—611no—
477C22×C4—1224no—
478C22×C4—1221no—
479C22×C4—1221no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—2442no—
487C22×C4—2442no—
488C22×C4D4×D6111yescharacteristic
489C22×C4D4×D6111yescharacteristic
490C22×C4C23×D6111yescharacteristic
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—81no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—4881no—
517C4⋊C4—4881no—
518C4⋊C4—4881no—
519C4⋊C4—2441no—
520C4⋊C4—2441no—
521C4⋊C4—2442no—
522C4⋊C4—2442no—
523C22⋊C4—4881no—
524C22⋊C4—4881no—
525C22⋊C4—2441no—
526C22⋊C4—2441no—
527C22⋊C4—2442no—
528C42—21no—
529C42—21no—
530C42—1221no—
531C42—1221no—
532C42—2442no—
533C23—612no—
534C23—1221no—
535C23—1221no—
536C23—1221no—
537C23—1221no—
538C23—1221no—
539C23—1221no—
540C23C2×D4×D6111yescharacteristic
541Q8—22no—
542Q8—22no—
543Q8—41no—
544Q8—41no—
545Q8—1222no—
546Q8—1222no—
547Q8—2448no—
548Q8—2441no—
549Q8—2441no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—612no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—1222no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—2448no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4#41×D6111yescharacteristic
595C2×C4#41×D6111yescharacteristic
596C2×C4#41×D6111yescharacteristic
597C2×C4C2×#1175822211yescharacteristic
598C2×C4#1451982211yescharacteristic
599C2×C4#1451982311yescharacteristic
600C8—41no—
601C8—41no—
602C8—2441no—
603C8—2441no—
604C22—1222no—
605C22—1222no—
606C22—1222no—
607C22—2441no—
608C22—2441no—
609C22—2441no—
610C22—2441no—
611C22#1684027111yescharacteristic
612C22#124×D6111yescharacteristic
613C22#1684032011yescharacteristic
614C22#1684057111yescharacteristic
615C22#1684057211yescharacteristic
616C22#1684057811yescharacteristic
617C22#1684057811yescharacteristic
618C4—41no—
619C4—41no—
620C4—42no—
621C4—81no—
622C4—81no—
623C4—81no—
624C4—1222no—
625C4—4881no—
626C4—4881no—
627C4—4881no—
628C4—2441no—
629C4—2441no—
630C4—2442no—
631C4#1684043611yescharacteristic
632C4#1684043711yescharacteristic
633C4#1684044011yescharacteristic
634C4#1684044111yescharacteristic
635C2—2442no—
636C2#2145097511yescharacteristic
637C2#2145610911yescharacteristic
638C2#2145611811yescharacteristic
639C2#2145618111yescharacteristic
640C2#2145618511yescharacteristic
641C2#2145669611yescharacteristic
642C2#2145670211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1935C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#1935D61
hypocenterC61#29117
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1935

Complements

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

NH ≅ G/Ntype
3. #21447231C2semidirect
4. #21447230C2semidirect
5. #21446969C2semidirect
6. #21446968C2semidirect
7. #21446509C2semidirect
8. #21446498C2semidirect
9. #21446494C2semidirect
15. C61×#1935C2semidirect
321. #1935D61semidirect
380. C61#29117semidirect

Maximal subgroups

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

subgroupindex
#214516542
#214484022
#214472312
#214472302
#214469692
#214469682
#214465092
#214464982
#214464942
#214459132
#214459122
#214457522
#214455882
C61×#19352
#2911761