Finite Group Databasebeta

← #29779617

Subgroups — #29779617

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#21451653C211yesmaximal, characteristic
2#21448403C211yesmaximal, characteristic
3#21447231C211yesmaximal, characteristic, complemented
4#21447231C211yesmaximal, characteristic, complemented
5#21446969C211yesmaximal, characteristic, complemented
6#21446969C211yesmaximal, characteristic, complemented
7#21446510C211yesmaximal, characteristic, complemented
8#21446507C211yesmaximal, characteristic, complemented
9#21446496C211yesmaximal, characteristic, complemented
10#21446493C211yesmaximal, characteristic, complemented
11#21445914C211yesmaximal, characteristic
12#21445755C211yesmaximal, characteristic
13#21445753C211yesmaximal, characteristic
14#21445587C211yesmaximal, characteristic
15C61×#1825C211yesmaximal, characteristic
16C23×Dic122C2211yescharacteristic
17C2×#14519768C2211yescharacteristic
18C2×#14519736C2211yescharacteristic
19C2×#14519702C2211yescharacteristic
20C2×#14519702C2211yescharacteristic
21C2×#14519702C2211yescharacteristic
22C2×#14519702C2211yescharacteristic
23C2×#14519700C2211yescharacteristic
24C2×#14519700C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×C22⋊#92418C2211yescharacteristic
28C2×#14519687C2211yescharacteristic
29C2×C8×Dic61C2211yescharacteristic
30#16839696—21no—
31#16839694—21no—
32#16839685—21no—
33#16839685—21no—
34#16839668—21no—
35#16839668—21no—
36#16839667C2211yescharacteristic
37#16839667C2211yescharacteristic
38#16839667C2211yescharacteristic
39#16839667C2211yescharacteristic
40#16839661C2211yescharacteristic
41#16839658C2211yescharacteristic
42#16839631C2211yescharacteristic
43#16839623—21no—
44#16839623—21no—
45#16839622C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#117×C61C2211yescharacteristic
53C2×C22⋊C8×C61C2211yescharacteristic
54C61×#380C2211yescharacteristic
55C61×#340C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×Dic122—21no—
60C22×Dic122—22no—
61C22×Dic122—22no—
62C22×Dic122—22no—
63C22×Dic122—22no—
64C22×Dic122—41no—
65C22×Dic122—41no—
66C22×Dic122C2311yescharacteristic
67C22×Dic122C2311yescharacteristic
68C23×Dic61—21no—
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71#14519768—41no—
72#14519768—41no—
73#14519736—41no—
74#14519736—41no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61D411yescharacteristic
78C2×C4⋊Dic61D411yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61D411yescharacteristic
87C2×C22⋊Dic61D411yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89C2×C4×Dic61C2311yescharacteristic
90#14519702—22no—
91#14519702—22no—
92#14519702—41no—
93#14519702—41no—
94#14519700—22no—
95#14519700—22no—
96#14519700—41no—
97#14519700—41no—
98C22⋊#92418—22no—
99#14519687—41no—
100C8×Dic61—41no—
101#14519684—21no—
102#14519684—21no—
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—
112C23×C4×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115#117×C61—22no—
116#117×C61—22no—
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C22⋊C4×C61—21no—
122C2×C22⋊C4×C61—21no—
123C2×C42×C61—21no—
124C22⋊C8×C61—22no—
125#95×C61—21no—
126C24×C61C2×D411yescharacteristic
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61#4112yes—
137C22×C4×C61#4112yes—
138C22×C4×C61C2411yescharacteristic
139C22×Dic61—24no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—42no—
145C22×Dic61—42no—
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—24no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—44no—
162C2×Dic122—44no—
163C2×Dic122—44no—
164C2×Dic122—44no—
165C2×Dic122—42no—
166C2×Dic122—42no—
167C2×Dic122#4112yes—
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—21no—
182C4⋊C4×C61—21no—
183C22⋊C4×C61—42no—
184C22⋊C4×C61—81no—
185C42×C61—42no—
186C22⋊Dic61—41no—
187C22⋊Dic61—41no—
188C22⋊Dic61—81no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—41no—
194C4⋊Dic61—41no—
195C4⋊Dic61—42no—
196C4⋊Dic61—81no—
197#11758196—42no—
198#11758196—81no—
199#11758196—81no—
200#11758196—81no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C23×C61—21no—
212C23×C61—21no—
213C23×C61—21no—
214C23×C61—21no—
215C23×C61—21no—
216C23×C61—21no—
217C23×C61#12412yes—
218C23×C61C22×D411yescharacteristic
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
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#12412yes—
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61H(5,𝔽2)11yescharacteristic
243C2×C4×C61H(5,𝔽2)11yescharacteristic
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—48no—
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—41no—
268C2×Dic61—41no—
269Dic122—22no—
270Dic122—22no—
271Dic122—22no—
272Dic122—22no—
273Dic122—48no—
274Dic122—41no—
275Dic122—41no—
276Dic122—41no—
277Dic122—41no—
278C8×C61—41no—
279C8×C61—41no—
280#92418—41no—
281#92418—41no—
282#29117—611nomaximal, Sylow
283C22×C61—22no—
284C22×C61—22no—
285C22×C61—22no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61—41no—
290C22×C61#52511yescharacteristic
291C22×C61C2×#12411yescharacteristic
292C22×C61#53411yescharacteristic
293C22×C61#57711yescharacteristic
294C22×C61#57711yescharacteristic
295C22×C61#57811yescharacteristic
296C22×C61#57811yescharacteristic
297C4×C61—22no—
298C4×C61—81no—
299C4×C61—81no—
300C4×C61—81no—
301C4×C61—81no—
302C4×C61#55111yescharacteristic
303C4×C61#55111yescharacteristic
304C4×C61#55111yescharacteristic
305C4×C61#55111yescharacteristic
306Dic61—42no—
307Dic61—42no—
308Dic61—41no—
309Dic61—41no—
310Dic61—41no—
311Dic61—41no—
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#1935—611no—
322#1933—611no—
323#1933—611no—
324#1825—611no—
325#1825D6111yescharacteristic
326#1787—611no—
327#1787—611no—
328#1750—611no—
329C2×C61—42no—
330C2×C61#256111yescharacteristic
331C2×C61#312611yescharacteristic
332C2×C61#312711yescharacteristic
333C2×C61#313711yescharacteristic
334C2×C61#313711yescharacteristic
335C2×C61#320111yescharacteristic
336C2×C61#320111yescharacteristic
337C23×Q8—611no—
338C2×#125—611no—
339C2×#117—611no—
340C2×#117C2×D6111yescharacteristic
341C2×#103—611no—
342C2×#103—611no—
343C2×#103—611no—
344C2×#103—611no—
345C2×#103—611no—
346C2×#103—611no—
347C2×#103—611no—
348C2×#103—611no—
349C2×C22⋊C8—611no—
350C2×C22⋊C8C2×D6111yescharacteristic
351C2×#97—611no—
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—611no—
361#391—1221no—
362#391—1221no—
363#389—611no—
364#389—611no—
365#384—611no—
366#381—1221no—
367#381—1221no—
368#380—611no—
369#380C2×D6111yescharacteristic
370#340—611no—
371#340—611no—
372#340—611no—
373#340—611no—
374#340C2×D6111yescharacteristic
375#340C2×D6111yescharacteristic
376#340C2×D6111yescharacteristic
377#340C2×D6111yescharacteristic
378#336—611no—
379#336—611no—
380C61#2911711yescharacteristic, complemented, Sylow
381C22×Q8—611no—
382C22×Q8—611no—
383C22×Q8—1222no—
384C22×Q8—1222no—
385C22×Q8—1221no—
386C22×Q8—1222no—
387C22×Q8—1222no—
388C22×Q8—2441no—
389C22×Q8—2441no—
390C23×C4—1221no—
391C23×C4C22×D6111yescharacteristic
392C22×C8—611no—
393C22×C8—611no—
394C22×C8C22×D6111yescharacteristic
395C22×C8C22×D6111yescharacteristic
396#125—2441no—
397#125—2441no—
398#117—22no—
399#117—22no—
400#117—2441no—
401#117—2441no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4C22×D6111yescharacteristic
413C2×C4⋊C4C22×D6111yescharacteristic
414C2×C4⋊C4C22×D6111yescharacteristic
415C2×C4⋊C4C22×D6111yescharacteristic
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—611no—
419C2×C22⋊C4—611no—
420C2×C22⋊C4—1221no—
421C2×C42—21no—
422C2×C42—611no—
423C2×C42—611no—
424#103—1222no—
425#103—1222no—
426#103—1222no—
427#103—1222no—
428#103—2441no—
429#103—2441no—
430#103—2441no—
431#103—2441no—
432C22⋊C8—22no—
433C22⋊C8—1222no—
434#97—2441no—
435C4×C8—2441no—
436#95—21no—
437#95—1221no—
438#95—1221no—
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—
448C24D4×D6111yescharacteristic
449C2×Q8—612no—
450C2×Q8—612no—
451C2×Q8—1224no—
452C2×Q8—1221no—
453C2×Q8—1221no—
454C2×Q8—1221no—
455C2×Q8—1221no—
456C2×Q8—1221no—
457C2×Q8—1221no—
458C2×Q8—1221no—
459C2×Q8—1221no—
460C2×Q8—1221no—
461C2×Q8—1221no—
462C2×Q8—2444no—
463C2×Q8—2444no—
464C2×Q8—2444no—
465C2×Q8—2444no—
466C2×Q8—2442no—
467C2×Q8—2442no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—611no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—1224no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—2442no—
486C22×C4—2442no—
487C22×C4#1175822212yes—
488C22×C4D4×D6111yescharacteristic
489C22×C4#1175822412yes—
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—21no—
510C4⋊C4—21no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—4881no—
516C4⋊C4—4881no—
517C4⋊C4—4881no—
518C4⋊C4—4881no—
519C4⋊C4—2441no—
520C4⋊C4—2441no—
521C4⋊C4—2442no—
522C4⋊C4—2442no—
523C22⋊C4—42no—
524C22⋊C4—81no—
525C22⋊C4—4881no—
526C22⋊C4—2441no—
527C22⋊C4—2441no—
528C42—42no—
529C42—1221no—
530C42—1221no—
531C42—1221no—
532C42—1221no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23#1451976312yes—
540C23C2×D4×D6111yescharacteristic
541Q8—1222no—
542Q8—1222no—
543Q8—1222no—
544Q8—1222no—
545Q8—2448no—
546Q8—2441no—
547Q8—2441no—
548Q8—2441no—
549Q8—2441no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—41no—
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—1222no—
569C2×C4—1222no—
570C2×C4—1221no—
571C2×C4—1221no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—1221no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2448no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4#1451976312yes—
594C2×C4#41×D6111yescharacteristic
595C2×C4#41×D6111yescharacteristic
596C2×C4C2×#1175822411yescharacteristic
597C2×C4C2×#1175822211yescharacteristic
598C2×C4#1451982311yescharacteristic
599C2×C4#1451982311yescharacteristic
600C8—41no—
601C8—41no—
602C8—2441no—
603C8—2441no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22#1684027011yescharacteristic
612C22C2×#1451976311yescharacteristic
613C22#1684032311yescharacteristic
614C22#1684056911yescharacteristic
615C22#1684057211yescharacteristic
616C22#1684057611yescharacteristic
617C22#1684057811yescharacteristic
618C4—22no—
619C4—81no—
620C4—81no—
621C4—81no—
622C4—81no—
623C4—4881no—
624C4—4881no—
625C4—2441no—
626C4—2441no—
627C4—2441no—
628C4—2441no—
629C4—2442no—
630C4—2442no—
631C4#1684043511yescharacteristic
632C4#1684043511yescharacteristic
633C4#1684043611yescharacteristic
634C4#1684043611yescharacteristic
635C2—42no—
636C2#2145097611yescharacteristic
637C2#2145610611yescharacteristic
638C2#2145611511yescharacteristic
639C2#2145618711yescharacteristic
640C2#2145618811yescharacteristic
641C2#2145669911yescharacteristic
642C2#2145670711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1825C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#1825D61
hypocenterC61#29117
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
3. #21447231C2semidirect
4. #21447231C2semidirect
5. #21446969C2semidirect
6. #21446969C2semidirect
7. #21446510C2semidirect
8. #21446507C2semidirect
9. #21446496C2semidirect
10. #21446493C2semidirect
380. C61#29117semidirect

Maximal subgroups

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

subgroupindex
#214516532
#214484032
#214472312
#214469692
#214465102
#214465072
#214464962
#214464932
#214459142
#214457552
#214457532
#214455872
C61×#18252
#2911761