Finite Group Databasebeta

← #29779768

Subgroups — #29779768

Order 15616.

754 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21451795C211yesmaximal, characteristic, complemented
2#21448545C211yesmaximal, characteristic
3#21447388C211yesmaximal, characteristic, complemented
4#21447387C211yesmaximal, characteristic, complemented
5#21447050C211yesmaximal, characteristic
6#21446518C211yesmaximal, characteristic, complemented
7#21446513C211yesmaximal, characteristic, complemented
8#21446504C211yesmaximal, characteristic, complemented
9#21446499C211yesmaximal, characteristic, complemented
10#21445827C211yesmaximal, characteristic, complemented
11#21445827C211yesmaximal, characteristic, complemented
12#21445817C211yesmaximal, characteristic, complemented
13#21445817C211yesmaximal, characteristic, complemented
14#21445675C211yesmaximal, characteristic, complemented
15C61×#2041C211yesmaximal, characteristic
16C2×D4×Dic61C2211yescharacteristic
17C2×D4×Dic61C2211yescharacteristic
18C2×#14519706C2211yescharacteristic
19C2×#14519706C2211yescharacteristic
20C2×#14519698C2211yescharacteristic
21C2×#14519698C2211yescharacteristic
22C2×#14519698C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519687C2211yescharacteristic
25C2×C8×Dic61C2211yescharacteristic
26#16839703—21no—
27#16839702—21no—
28#16839695—21no—
29#16839693—21no—
30#16839665C2211yescharacteristic
31#16839665C2211yescharacteristic
32#16839661C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839658C2211yescharacteristic
35#16839642—21no—
36#16839642—21no—
37#16839642—21no—
38#16839642—21no—
39#16839631C2211yescharacteristic
40#16839628C2211yescharacteristic
41#16839627C2211yescharacteristic
42#16839608C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52C2×#127×C61C2211yescharacteristic
53C2×#102×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C61×#389C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C23×Dic61—21no—
60C23×Dic61—21no—
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63D4×Dic61—22no—
64D4×Dic61—22no—
65D4×Dic61—22no—
66D4×Dic61—22no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C4×Dic61C2311yescharacteristic
83C2×C4×Dic61C2311yescharacteristic
84#14519706—22no—
85#14519706—22no—
86#14519698—22no—
87#14519698—22no—
88#14519698—41no—
89#14519698—41no—
90#14519687—41no—
91C8×Dic61—41no—
92#14519684—21no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684D411yescharacteristic
100#14519684D411yescharacteristic
101#14519684D411yescharacteristic
102#14519684D411yescharacteristic
103C22×D4×C61—21no—
104C22×D4×C61—21no—
105C22×D4×C61C2311yescharacteristic
106C22×D4×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109#127×C61—22no—
110#127×C61—22no—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C42×C61C2311yescharacteristic
115#102×C61—22no—
116#102×C61—22no—
117#102×C61—41no—
118#102×C61—41no—
119#95×C61—21no—
120C24×C61—21no—
121C24×C61—21no—
122C2×D4×C61—24no—
123C2×D4×C61—24no—
124C2×D4×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—81no—
135C2×D4×C61—81no—
136C2×D4×C61—81no—
137C2×D4×C61—81no—
138C2×D4×C61#4112yes—
139C2×D4×C61#4112yes—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2411yescharacteristic
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—42no—
150C22×Dic61—42no—
151C22×Dic61—81no—
152C22×Dic61—81no—
153C22×Dic61—81no—
154C22×Dic61—81no—
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—81no—
170C42×C61—21no—
171C42×C61—21no—
172C22⋊Dic61—42no—
173C22⋊Dic61—42no—
174C22⋊Dic61—81no—
175C22⋊Dic61—81no—
176C22⋊Dic61—81no—
177C22⋊Dic61—81no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—81no—
187#11758196—81no—
188#11758196—81no—
189#11758196—81no—
190#11758196—81no—
191C4×Dic61—21no—
192C4×Dic61—21no—
193C4×Dic61#4112yes—
194C4×Dic61#4112yes—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C23×C61—22no—
202C23×C61—22no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—41no—
212C23×C61—41no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61C22×D411yescharacteristic
216D4×C61—22no—
217D4×C61—22no—
218D4×C61—22no—
219D4×C61—22no—
220D4×C61—41no—
221D4×C61—41no—
222D4×C61—41no—
223D4×C61—41no—
224D4×C61—84no—
225D4×C61—84no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
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×C61C2×#4111yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61H(5,𝔽2)11yescharacteristic
243C2×C4×C612−1+411yescharacteristic
244C2×Dic61—21no—
245C2×Dic61—21no—
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—41no—
255C2×Dic61—41no—
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—84no—
267C2×Dic61—84no—
268C2×Dic61#12412yes—
269C2×Dic61#12412yes—
270C8×C61—41no—
271C8×C61—41no—
272#92418—41no—
273#92418—41no—
274#29133—611nomaximal, Sylow
275C22×C61—42no—
276C22×C61—42no—
277C22×C61—42no—
278C22×C61—42no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61—81no—
282C22×C61—81no—
283C22×C61—81no—
284C22×C61—81no—
285C22×C61—81no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61—81no—
289C22×C61#52511yescharacteristic
290C22×C61C2×#12411yescharacteristic
291C22×C61#53711yescharacteristic
292C22×C61#57611yescharacteristic
293C22×C61#57611yescharacteristic
294C22×C61#57711yescharacteristic
295C22×C61#57711yescharacteristic
296C4×C61—41no—
297C4×C61—41no—
298C4×C61—81no—
299C4×C61—81no—
300C4×C61#55211yescharacteristic
301C4×C61#55211yescharacteristic
302C4×C61#55311yescharacteristic
303C4×C61#55311yescharacteristic
304Dic61—22no—
305Dic61—22no—
306Dic61—41no—
307Dic61—41no—
308Dic61—81no—
309Dic61—81no—
310Dic61—81no—
311Dic61—81no—
312#2647—611no—
313#2328—611no—
314#2101—611no—
315#2101—611no—
316#2041—611no—
317#2041D6111yescharacteristic
318#1936—611no—
319#1936—611no—
320#1934—611no—
321#1934—611no—
322#1804—611no—
323#1804—611no—
324#1802—611no—
325#1802—611no—
326#1768—611no—
327C2×C61—82no—
328C2×C61—82no—
329C2×C61#256211yescharacteristic
330C2×C61#312211yescharacteristic
331C2×C61#312311yescharacteristic
332C2×C61#313611yescharacteristic
333C2×C61#313611yescharacteristic
334C2×C61#325711yescharacteristic
335C2×C61#325711yescharacteristic
336C2×#127C2×D6111yescharacteristic
337C2×C4×D4—611no—
338C2×C4×D4—611no—
339C2×C4⋊C8—611no—
340C2×C4⋊C8—611no—
341C2×#102—611no—
342C2×#102—611no—
343C2×#102—611no—
344C2×#102—611no—
345C2×#102C2×D6111yescharacteristic
346C2×#102C2×D6111yescharacteristic
347C2×#102C2×D6111yescharacteristic
348C2×#102C2×D6111yescharacteristic
349C2×#97—611no—
350C2×C4×C8—611no—
351#400—1221no—
352#400—1221no—
353#398—1221no—
354#398—1221no—
355#390—611no—
356#390—611no—
357#389—611no—
358#389—611no—
359#389—611no—
360#389C2×D6111yescharacteristic
361#386—1221no—
362#386—1221no—
363#386—1221no—
364#386—1221no—
365#384—611no—
366#382—611no—
367#382—611no—
368#376—611no—
369#340—611no—
370#340—611no—
371#340—611no—
372#340—611no—
373#340—611no—
374#340—611no—
375#340—611no—
376#340—611no—
377#336—611no—
378#336C2×D6111yescharacteristic
379C61#2913311yescharacteristic, complemented, Sylow
380C22×D4—21no—
381C22×D4—21no—
382C22×D4C22×D6111yescharacteristic
383C22×D4C22×D6111yescharacteristic
384C23×C4—1221no—
385C23×C4—1221no—
386C22×C8—611no—
387C22×C8—611no—
388C22×C8C22×D6111yescharacteristic
389C22×C8C22×D6111yescharacteristic
390#127—22no—
391#127—22no—
392C4×D4—1222no—
393C4×D4—1222no—
394C4×D4—1222no—
395C4×D4—1222no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4C22×D6111yescharacteristic
407C2×C4⋊C4C22×D6111yescharacteristic
408C2×C22⋊C4—1221no—
409C2×C22⋊C4—1221no—
410C2×C22⋊C4—1221no—
411C2×C22⋊C4—1221no—
412C2×C22⋊C4—1221no—
413C2×C22⋊C4—1221no—
414C2×C42—611no—
415C2×C42—611no—
416C2×C42C22×D6111yescharacteristic
417C4⋊C8—1222no—
418C4⋊C8—1222no—
419#102—22no—
420#102—22no—
421#102—41no—
422#102—41no—
423#102—1222no—
424#102—1222no—
425#102—2441no—
426#102—2441no—
427#97—2441no—
428C4×C8—2441no—
429#95—21no—
430#95—611no—
431#95—611no—
432#95—611no—
433#95—611no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437#95—1221no—
438#95—1221no—
439#95—1221no—
440#95—1221no—
441C24—21no—
442C24—21no—
443C2×D4—24no—
444C2×D4—24no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—81no—
456C2×D4—81no—
457C2×D4—81no—
458C2×D4—81no—
459C2×D4#1175822412yes—
460C2×D4#1175822412yes—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—4881no—
472C22×C4—4881no—
473C22×C4—4881no—
474C22×C4—4881no—
475C22×C4—2442no—
476C22×C4—2442no—
477C22×C4D4×D6111yescharacteristic
478C22×C4D4×D6111yescharacteristic
479C22×C4C23×D6111yescharacteristic
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8—1221no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—81no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—4881no—
506C4⋊C4—4881no—
507C4⋊C4—4881no—
508C4⋊C4—4881no—
509C4⋊C4—4881no—
510C22⋊C4—4881no—
511C22⋊C4—4881no—
512C22⋊C4—4881no—
513C22⋊C4—4881no—
514C22⋊C4—2442no—
515C22⋊C4—2442no—
516C42—21no—
517C42—21no—
518C42—612no—
519C42—612no—
520C42—1221no—
521C42—1221no—
522C23—22no—
523C23—22no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23C2×D4×D6111yescharacteristic
537D4—22no—
538D4—22no—
539D4—22no—
540D4—22no—
541D4—41no—
542D4—41no—
543D4—41no—
544D4—41no—
545D4—84no—
546D4—84no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—612no—
560C2×C4—612no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—4884no—
572C2×C4—4884no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4#41×D6111yescharacteristic
586C2×C4#41×D6111yescharacteristic
587C2×C4C2×#1175822411yescharacteristic
588C2×C4C2×#1175822411yescharacteristic
589C2×C4#1451982211yescharacteristic
590C2×C4#1451982411yescharacteristic
591C8—41no—
592C8—41no—
593C8—2441no—
594C8—2441no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22#1684026811yescharacteristic
610C22#124×D6111yescharacteristic
611C22#1684034111yescharacteristic
612C22#1684056311yescharacteristic
613C22#1684056611yescharacteristic
614C22#1684056811yescharacteristic
615C22#1684057211yescharacteristic
616C4—41no—
617C4—41no—
618C4—81no—
619C4—81no—
620C4—1222no—
621C4—1222no—
622C4—4881no—
623C4—4881no—
624C4—4881no—
625C4—4881no—
626C4—2441no—
627C4—2441no—
628C4#1684044911yescharacteristic
629C4#1684044911yescharacteristic
630C4#1684045511yescharacteristic
631C4#1684045511yescharacteristic
632C2—82no—
633C2—82no—
634C2#2145097911yescharacteristic
635C2#2145605611yescharacteristic
636C2#2145606511yescharacteristic
637C2#2145616811yescharacteristic
638C2#2145617411yescharacteristic
639C2#2145722611yescharacteristic
640C2#2145722811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2041C2
socleC23×C61C22×D4
prime radicalC2×#127×C61C22
hypercenter#2041D61
hypocenterC61#29133
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2041

Complements

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

NH ≅ G/Ntype
1. #21451795C2semidirect
3. #21447388C2semidirect
4. #21447387C2semidirect
6. #21446518C2semidirect
7. #21446513C2semidirect
8. #21446504C2semidirect
9. #21446499C2semidirect
10. #21445827C2semidirect
11. #21445827C2semidirect
12. #21445817C2semidirect
13. #21445817C2semidirect
14. #21445675C2semidirect
379. C61#29133semidirect

Maximal subgroups

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

subgroupindex
#214517952
#214485452
#214473882
#214473872
#214470502
#214465182
#214465132
#214465042
#214464992
#214458272
#214458172
#214456752
C61×#20412
#2913361