Finite Group Databasebeta

← #29752678

Subgroups — #29752678

Order 15616.

784 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450232C211yesmaximal, characteristic, complemented
2#21448839C211yesmaximal, characteristic
3#21447235C211yesmaximal, characteristic, complemented
4#21447189C211yesmaximal, characteristic, complemented
5#21447005C211yesmaximal, characteristic
6#21446584C211yesmaximal, characteristic, complemented
7#21446546C211yesmaximal, characteristic, complemented
8#21446422C211yesmaximal, characteristic, complemented
9#21446400C211yesmaximal, characteristic, complemented
10#21446392C211yesmaximal, characteristic, complemented
11#21446078C211yesmaximal, characteristic, complemented
12#21446011C211yesmaximal, characteristic, complemented
13#21445910C211yesmaximal, characteristic, complemented
14#102×Dic61C211yesmaximal, characteristic, complemented
15C61×#2033C211yesmaximal, characteristic
16C2×D4×Dic61—21no—
17C2×D4×Dic61C2211yescharacteristic
18C22×C4⋊Dic61C2211yescharacteristic
19C2×#14519708C2211yescharacteristic
20C2×#14519708C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×C22⋊#92418C2211yescharacteristic
26C2×C8×Dic61C2211yescharacteristic
27#16839702—21no—
28#16839690—21no—
29#16839665C2211yescharacteristic
30#16839661C2211yescharacteristic
31#16839661C2211yescharacteristic
32#16839658C2211yescharacteristic
33#16839655—21no—
34#16839655—21no—
35#16839642—21no—
36#16839637—21no—
37#16839629—21no—
38#16839627C2211yescharacteristic
39#16839622C2211yescharacteristic
40C4⋊C4×Dic61C2211yescharacteristic
41C4⋊C4×Dic61C2211yescharacteristic
42C4×C22⋊Dic61C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#121×C61C2211yescharacteristic
53C2×#107×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×C22⋊C8×C61C2211yescharacteristic
57C61×#380C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C23×Dic61—21no—
60C23×Dic61—21no—
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63D4×Dic61—22no—
64D4×Dic61—41no—
65D4×Dic61—44no—
66C2×C4⋊Dic61—22no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—41no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61D411yescharacteristic
81C2×C22⋊Dic61D411yescharacteristic
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61C2311yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86#14519708—22no—
87#14519708—22no—
88#14519708—41no—
89#14519706—22no—
90#14519698—22no—
91#14519698—22no—
92C22⋊#92418—22no—
93C8×Dic61—41no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684D411yescharacteristic
103#14519684D411yescharacteristic
104C22×D4×C61—21no—
105C22×D4×C61C2311yescharacteristic
106C23×C4×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109#121×C61—22no—
110#121×C61—22no—
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C22⋊C4×C61—21no—
115C2×C22⋊C4×C61—21no—
116#107×C61—22no—
117#102×C61—22no—
118#102×C61—41no—
119C22⋊C8×C61—41no—
120#95×C61—21no—
121C24×C61—21no—
122C24×C61C2×D411yescharacteristic
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—42no—
130C2×D4×C61—42no—
131C2×D4×C61#4112yes—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2411yescharacteristic
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—42no—
148C22×Dic61—42no—
149C22×Dic61—42no—
150C22×Dic61—42no—
151C22×Dic61—81no—
152C22×Dic61—81no—
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61#4112yes—
169C22⋊C4×C61—24no—
170C22⋊C4×C61—42no—
171C22⋊Dic61—22no—
172C22⋊Dic61—22no—
173C22⋊Dic61—42no—
174C22⋊Dic61—42no—
175C22⋊Dic61—81no—
176C22⋊Dic61—81no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—42no—
185C4⋊Dic61—42no—
186C4⋊Dic61—81no—
187C4⋊Dic61#4112yes—
188#11758196—81no—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61—21no—
193C4×Dic61—42no—
194C4×Dic61—42no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418#4112yes—
201C23×C61—21no—
202C23×C61—21no—
203C23×C61—21no—
204C23×C61—21no—
205C23×C61—21no—
206C23×C61—21no—
207C23×C61—22no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—41no—
212C23×C61—41no—
213C23×C61—41no—
214C23×C61#13512yes—
215C23×C61C22×D411yescharacteristic
216D4×C61—22no—
217D4×C61—22no—
218D4×C61—48no—
219D4×C61—41no—
220D4×C61—41no—
221C2×C4×C61—21no—
222C2×C4×C61—22no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
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×C61C2×#4111yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61H(5,𝔽2)11yescharacteristic
242C2×C4×C612−1+411yescharacteristic
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—22no—
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—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×Dic61AGL(1,ℤ/8)12yes—
269C8×C61—41no—
270C8×C61—41no—
271#92418—22no—
272#92418—41no—
273#27031—611nomaximal, Sylow
274C22×C61—22no—
275C22×C61—22no—
276C22×C61—22no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—42no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61—81no—
285C22×C61—81no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61#52511yescharacteristic
289C22×C61#54611yescharacteristic
290C22×C61#54711yescharacteristic
291C22×C61C2×#13511yescharacteristic
292C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
293C22×C61#57711yescharacteristic
294C22×C61#57711yescharacteristic
295C4×C61—42no—
296C4×C61—41no—
297C4×C61—81no—
298C4×C61—81no—
299C4×C61#52911yescharacteristic
300C4×C61#53911yescharacteristic
301C4×C61#55311yescharacteristic
302C4×C61#55611yescharacteristic
303Dic61—22no—
304Dic61—41no—
305Dic61—41no—
306Dic61—41no—
307Dic61—42no—
308Dic61—81no—
309Dic61—81no—
310Dic61—81no—
311#2495—611no—
312#2361—611no—
313#2073—611no—
314#2065—611no—
315#2034—611no—
316#2033D6111yescharacteristic
317#1949—611no—
318#1941—611no—
319#1923—611no—
320#1919—611no—
321#1918—611no—
322#1864—611no—
323#1853—611no—
324#1824—611no—
325C4×#102—611no—
326C2×C61—42no—
327C2×C61—82no—
328C2×C61#246611yescharacteristic
329C2×C61#310511yescharacteristic
330C2×C61#312511yescharacteristic
331C2×C61#330111yescharacteristic
332C2×C61#331011yescharacteristic
333C2×C61#333711yescharacteristic
334C2×C61#335411yescharacteristic
335C2×#121C2×D6111yescharacteristic
336C2×C4×D4—611no—
337C2×C4×D4—1221no—
338C22×C4⋊C4—611no—
339C2×#107C2×D6111yescharacteristic
340C2×#106—611no—
341C2×#106—611no—
342C2×#106—611no—
343C2×C4⋊C8—611no—
344C2×#102—611no—
345C2×#102—611no—
346C2×#102C2×D6111yescharacteristic
347C2×#102C2×D6111yescharacteristic
348C2×C22⋊C8—611no—
349C2×C22⋊C8C2×D6111yescharacteristic
350C2×C4×C8—611no—
351#400—1221no—
352#397—1221no—
353#390—611no—
354#389—611no—
355#389—611no—
356#389—611no—
357#388—1221no—
358#388—1221no—
359#386—1221no—
360#385—1221no—
361#383—1221no—
362#382—611no—
363#380—611no—
364#380C2×D6111yescharacteristic
365C4×C4⋊C4—611no—
366C4×C4⋊C4—611no—
367C4×C22⋊C4—611no—
368#340—611no—
369#340—611no—
370#340—611no—
371#340—611no—
372#340—611no—
373#340—611no—
374#340—611no—
375#340C2×D6111yescharacteristic
376#336—611no—
377#336—611no—
378C61#2703111yescharacteristic, complemented, Sylow
379C22×D4—21no—
380C22×D4C22×D6111yescharacteristic
381C23×C4—1221no—
382C23×C4—1221no—
383C23×C4C22×D6111yescharacteristic
384C22×C8—611no—
385C22×C8—611no—
386C22×C8C22×D6111yescharacteristic
387C22×C8C22×D6111yescharacteristic
388#121—22no—
389#121—22no—
390C4×D4—1222no—
391C4×D4—2441no—
392C4×D4—2444no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—1222no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—2441no—
403C2×C4⋊C4C22×D6111yescharacteristic
404C2×C4⋊C4C22×D6111yescharacteristic
405C2×C4⋊C4C22×D6111yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—611no—
409C2×C22⋊C4—611no—
410C2×C22⋊C4—1221no—
411C2×C22⋊C4—1221no—
412C2×C22⋊C4—1221no—
413C2×C22⋊C4—1221no—
414C2×C42—611no—
415C2×C42—611no—
416C2×C42—1221no—
417C2×C42—1221no—
418#107—22no—
419#106—1222no—
420#106—1222no—
421#106—2441no—
422C4⋊C8—1222no—
423#102—22no—
424#102—41no—
425#102—1222no—
426#102—1222no—
427C22⋊C8—41no—
428C22⋊C8—1222no—
429C4×C8—2441no—
430#95—21no—
431#95—611no—
432#95—611no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437#95—1221no—
438#95—1221no—
439#95—1221no—
440#95—1221no—
441C24—21no—
442C24D4×D6111yescharacteristic
443C2×D4—24no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4#1175822412yes—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—4881no—
470C22×C4—4881no—
471C22×C4—2442no—
472C22×C4—2442no—
473C22×C4—2442no—
474C22×C4—2442no—
475C22×C4D4×D6111yescharacteristic
476C22×C4C23×D6111yescharacteristic
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—612no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—612no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—4881no—
503C4⋊C4—4881no—
504C4⋊C4—2442no—
505C4⋊C4—2442no—
506C4⋊C4#1175822412yes—
507C22⋊C4—24no—
508C22⋊C4—42no—
509C22⋊C4—1222no—
510C22⋊C4—1222no—
511C22⋊C4—4881no—
512C22⋊C4—4881no—
513C22⋊C4—2442no—
514C22⋊C4—2442no—
515C42—1221no—
516C42—1221no—
517C42—1221no—
518C42—1221no—
519C42—2442no—
520C42—2442no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—22no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23#1451979612yes—
535C23C2×D4×D6111yescharacteristic
536D4—22no—
537D4—22no—
538D4—48no—
539D4—41no—
540D4—41no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—612no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1222no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—4884no—
571C2×C4—4884no—
572C2×C4—2441no—
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#41×D6111yescharacteristic
584C2×C4#41×D6111yescharacteristic
585C2×C4C2×#1175822411yescharacteristic
586C2×C4C2×#1175822411yescharacteristic
587C2×C4#1451982211yescharacteristic
588C2×C4#1451982411yescharacteristic
589C8—41no—
590C8—41no—
591C8—1222no—
592C8—2441no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22#1684026811yescharacteristic
608C22#1684040111yescharacteristic
609C22#1684041011yescharacteristic
610C22C2×#1451979611yescharacteristic
611C22AGL(1,ℤ/8)×D6111yescharacteristic
612C22#1684056911yescharacteristic
613C22#1684057111yescharacteristic
614C4—41no—
615C4—42no—
616C4—81no—
617C4—81no—
618C4—1222no—
619C4—4881no—
620C4—4881no—
621C4—4881no—
622C4—2441no—
623C4—2441no—
624C4—2442no—
625C4—2441no—
626C4#1684028911yescharacteristic
627C4#1684035711yescharacteristic
628C4#1684045911yescharacteristic
629C4#1684048111yescharacteristic
630C2—42no—
631C2—82no—
632C2#2144984811yescharacteristic
633C2#2145586111yescharacteristic
634C2#2145608911yescharacteristic
635C2#2145772711yescharacteristic
636C2#2145783511yescharacteristic
637C2#2145820211yescharacteristic
638C2#2145841211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2033C2
socleC23×C61C22×D4
prime radicalC2×#121×C61C22
hypercenter#2033D61
hypocenterC61#27031
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2033

Complements

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

NH ≅ G/Ntype
1. #21450232C2semidirect
3. #21447235C2semidirect
4. #21447189C2semidirect
6. #21446584C2semidirect
7. #21446546C2semidirect
8. #21446422C2semidirect
9. #21446400C2semidirect
10. #21446392C2semidirect
11. #21446078C2semidirect
12. #21446011C2semidirect
13. #21445910C2semidirect
14. #102×Dic61C2semidirect
378. C61#27031semidirect

Maximal subgroups

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

subgroupindex
#214502322
#214488392
#214472352
#214471892
#214470052
#214465842
#214465462
#214464222
#214464002
#214463922
#214460782
#214460112
#214459102
#102×Dic612
C61×#20332
#2703161