Finite Group Databasebeta

← #29748517

Subgroups — #29748517

Order 15616.

832 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450333C211yesmaximal, characteristic, complemented
2#21448884C211yesmaximal, characteristic
3#21447225C211yesmaximal, characteristic, complemented
4#21447205C211yesmaximal, characteristic, complemented
5#21447050C211yesmaximal, characteristic
6#21446499C211yesmaximal, characteristic, complemented
7#21446441C211yesmaximal, characteristic, complemented
8#21446391C211yesmaximal, characteristic, complemented
9#21446387C211yesmaximal, characteristic, complemented
10#21446375C211yesmaximal, characteristic, complemented
11#21446064C211yesmaximal, characteristic, complemented
12#21446064C211yesmaximal, characteristic, complemented
13#21445817C211yesmaximal, characteristic, complemented
14#102×Dic61C211yesmaximal, characteristic, complemented
15C61×#2041C211yesmaximal, characteristic
16C2×#14519767—21no—
17C2×D4×Dic61—21no—
18C2×D4×Dic61C2211yescharacteristic
19C2×D4×Dic61C2211yescharacteristic
20C2×#14519711C2211yescharacteristic
21C2×#14519710C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×C8×Dic61C2211yescharacteristic
28#16839703—21no—
29#16839695—21no—
30#16839665—21no—
31#16839665C2211yescharacteristic
32#16839665C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839658C2211yescharacteristic
35#16839658C2211yescharacteristic
36#16839644—21no—
37#16839642—21no—
38#16839642—21no—
39#16839631C2211yescharacteristic
40C4⋊C4×Dic61C2211yescharacteristic
41C4×C4⋊Dic61C2211yescharacteristic
42C4×C4⋊Dic61C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839517C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
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
63#14519767—44no—
64D4×Dic61—22no—
65D4×Dic61—22no—
66D4×Dic61—41no—
67D4×Dic61—41no—
68D4×Dic61—44no—
69C2×C4⋊Dic61D411yescharacteristic
70C2×C4⋊Dic61D411yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61C2311yescharacteristic
89C2×C4×Dic61C2311yescharacteristic
90#14519711—22no—
91#14519710—22no—
92#14519706—41no—
93#14519698—22no—
94#14519698—22no—
95#14519698—22no—
96#14519698—22no—
97C8×Dic61—41no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684D411yescharacteristic
104#14519684D411yescharacteristic
105C22×D4×C61—21no—
106C22×D4×C61—21no—
107C22×D4×C61C2311yescharacteristic
108C22×D4×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110C22×C8×C61C2311yescharacteristic
111#127×C61—22no—
112#127×C61—22no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C42×C61C2311yescharacteristic
117#102×C61—22no—
118#102×C61—22no—
119#102×C61—41no—
120#102×C61—41no—
121#95×C61—21no—
122C24×C61—21no—
123C24×C61—21no—
124C2×D4×C61—24no—
125C2×D4×C61—24no—
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—21no—
135C2×D4×C61—21no—
136C2×D4×C61—42no—
137C2×D4×C61—42no—
138C2×D4×C61—81no—
139C2×D4×C61—81no—
140C2×D4×C61#4112yes—
141C2×D4×C61#4112yes—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2411yescharacteristic
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—42no—
152C22×Dic61—42no—
153C22×Dic61—42no—
154C22×Dic61—42no—
155C22×Dic61—81no—
156C22×Dic61—81no—
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—42no—
172C42×C61—21no—
173C42×C61—21no—
174C22⋊Dic61—42no—
175C22⋊Dic61—42no—
176C22⋊Dic61—42no—
177C22⋊Dic61—42no—
178C22⋊Dic61—81no—
179C22⋊Dic61—81no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—41no—
187C4⋊Dic61D814yes—
188C4⋊Dic61#4112yes—
189C4⋊Dic61#4112yes—
190#11758196—42no—
191#11758196—42no—
192#11758196—42no—
193#11758196—42no—
194#11758196—42no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C4×Dic61—42no—
200C4×Dic61—42no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C23×C61—22no—
208C23×C61—22no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—41no—
212C23×C61—41no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61C22×D411yescharacteristic
222D4×C61—22no—
223D4×C61—22no—
224D4×C61—22no—
225D4×C61—22no—
226D4×C61—48no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—41no—
231D4×C61—84no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61C2×D812yes—
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×C4×C61C2×#4111yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C2×C4×C61H(5,𝔽2)11yescharacteristic
249C2×C4×C612−1+411yescharacteristic
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—84no—
273C2×Dic61—84no—
274C2×Dic61C2×D812yes—
275C2×Dic61C2×D812yes—
276C8×C61—41no—
277C8×C61—41no—
278#92418—41no—
279#92418—41no—
280#26752—611nomaximal, Sylow
281C22×C61—42no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61—42no—
285C22×C61—42no—
286C22×C61—42no—
287C22×C61—81no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61—81no—
293C22×C61—81no—
294C22×C61—81no—
295C22×C61#52511yescharacteristic
296C22×C61#54611yescharacteristic
297C22×C61#54811yescharacteristic
298C22×C61C22×D811yescharacteristic
299C22×C61#57511yescharacteristic
300C22×C61#57711yescharacteristic
301C22×C61#57711yescharacteristic
302C4×C61—22no—
303C4×C61—41no—
304C4×C61—81no—
305C4×C61—81no—
306C4×C61#52911yescharacteristic
307C4×C61#53911yescharacteristic
308C4×C61#55211yescharacteristic
309C4×C61#55311yescharacteristic
310Dic61—22no—
311Dic61—22no—
312Dic61—42no—
313Dic61—42no—
314Dic61—41no—
315Dic61—41no—
316Dic61—81no—
317Dic61—81no—
318#2506—611no—
319#2366—611no—
320#2071—611no—
321#2067—611no—
322#2041—611no—
323#2041D6111yescharacteristic
324#1934—611no—
325#1925—611no—
326#1918—611no—
327#1918—611no—
328#1916—611no—
329#1862—611no—
330#1862—611no—
331#1802—611no—
332C4×#102—611no—
333C2×C61—82no—
334C2×C61—82no—
335C2×C61#246011yescharacteristic
336C2×C61#310011yescharacteristic
337C2×C61#312311yescharacteristic
338C2×C61#329411yescharacteristic
339C2×C61#331711yescharacteristic
340C2×C61#335811yescharacteristic
341C2×C61#337111yescharacteristic
342C2×#127C2×D6111yescharacteristic
343C2×#125—1221no—
344C2×C4×D4—611no—
345C2×C4×D4—611no—
346C2×C4×D4—1221no—
347C2×#107—611no—
348C2×#107—611no—
349C2×C4⋊C8—611no—
350C2×#102—611no—
351C2×#102—611no—
352C2×#102—611no—
353C2×#102—611no—
354C2×#102C2×D6111yescharacteristic
355C2×#102C2×D6111yescharacteristic
356C2×#102C2×D6111yescharacteristic
357C2×#102C2×D6111yescharacteristic
358C2×C4×C8—611no—
359#400—1221no—
360#398—1221no—
361#390—611no—
362#390—611no—
363#390—1221no—
364#389—611no—
365#389—611no—
366#389—611no—
367#389C2×D6111yescharacteristic
368#387—1221no—
369#386—1221no—
370#386—1221no—
371#384—611no—
372C4×C4⋊C4—611no—
373C4×C4⋊C4—611no—
374C4×C4⋊C4—611no—
375#340—611no—
376#340—611no—
377#340—611no—
378#340—611no—
379#340—611no—
380#340—611no—
381#336—611no—
382#336—611no—
383#336—611no—
384#336C2×D6111yescharacteristic
385C61#2675211yescharacteristic, complemented, Sylow
386C22×D4—21no—
387C22×D4—21no—
388C22×D4C22×D6111yescharacteristic
389C22×D4C22×D6111yescharacteristic
390C23×C4—1221no—
391C23×C4—1221no—
392C22×C8—611no—
393C22×C8—611no—
394C22×C8C22×D6111yescharacteristic
395C22×C8C22×D6111yescharacteristic
396#127—22no—
397#127—22no—
398#125—2444no—
399C4×D4—1222no—
400C4×D4—1222no—
401C4×D4—2441no—
402C4×D4—2441no—
403C4×D4—2444no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4C22×D6111yescharacteristic
417C2×C4⋊C4C22×D6111yescharacteristic
418C2×C22⋊C4—1221no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C42—611no—
425C2×C42—611no—
426C2×C42—1221no—
427C2×C42—1221no—
428C2×C42C22×D6111yescharacteristic
429#107—1222no—
430#107—1222no—
431C4⋊C8—2441no—
432#102—22no—
433#102—22no—
434#102—41no—
435#102—41no—
436#102—1222no—
437#102—1222no—
438#102—1222no—
439#102—1222no—
440C4×C8—2441no—
441#95—21no—
442#95—611no—
443#95—611no—
444#95—1221no—
445#95—1221no—
446#95—1221no—
447#95—1221no—
448#95—1221no—
449C24—21no—
450C24—21no—
451C2×D4—24no—
452C2×D4—24no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—81no—
466C2×D4—81no—
467C2×D4#1175822412yes—
468C2×D4#1175822412yes—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—611no—
472C22×C4—611no—
473C22×C4—611no—
474C22×C4—611no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—1221no—
479C22×C4—4881no—
480C22×C4—4881no—
481C22×C4—2442no—
482C22×C4—2442no—
483C22×C4—2442no—
484C22×C4—2442no—
485C22×C4D4×D6111yescharacteristic
486C22×C4D4×D6111yescharacteristic
487C22×C4C23×D6111yescharacteristic
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—1221no—
495C2×C8—1221no—
496C2×C8—1221no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—42no—
505C4⋊C4—614no—
506C4⋊C4—612no—
507C4⋊C4—612no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—2441no—
515C4⋊C4—2442no—
516C4⋊C4—2442no—
517C4⋊C4—2442no—
518C4⋊C4—2442no—
519C4⋊C4—2442no—
520C22⋊C4—4881no—
521C22⋊C4—4881no—
522C22⋊C4—2442no—
523C22⋊C4—2442no—
524C22⋊C4—2442no—
525C22⋊C4—2442no—
526C42—21no—
527C42—21no—
528C42—1221no—
529C42—1221no—
530C42—1221no—
531C42—1221no—
532C42—2442no—
533C42—2442no—
534C23—22no—
535C23—22no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23C2×D4×D6111yescharacteristic
549D4—22no—
550D4—22no—
551D4—22no—
552D4—22no—
553D4—48no—
554D4—41no—
555D4—41no—
556D4—41no—
557D4—41no—
558D4—84no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—612no—
571C2×C4—612no—
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—1222no—
583C2×C4—1222no—
584C2×C4—4884no—
585C2×C4—4884no—
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—2441no—
595C2×C4—2441no—
596C2×C4D8×D6112yes—
597C2×C4#41×D6111yescharacteristic
598C2×C4#41×D6111yescharacteristic
599C2×C4C2×#1175822411yescharacteristic
600C2×C4C2×#1175822411yescharacteristic
601C2×C4#1451982211yescharacteristic
602C2×C4#1451982411yescharacteristic
603C8—41no—
604C8—41no—
605C8—2441no—
606C8—2441no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22#1684026811yescharacteristic
622C22#1684040911yescharacteristic
623C22#1684042011yescharacteristic
624C22C2×D8×D6111yescharacteristic
625C22#1684055811yescharacteristic
626C22#1684056811yescharacteristic
627C22#1684057211yescharacteristic
628C4—22no—
629C4—41no—
630C4—81no—
631C4—81no—
632C4—1222no—
633C4—1222no—
634C4—4881no—
635C4—4881no—
636C4—2442no—
637C4—2441no—
638C4—2441no—
639C4—2442no—
640C4#1684028911yescharacteristic
641C4#1684035711yescharacteristic
642C4#1684044911yescharacteristic
643C4#1684045511yescharacteristic
644C2—82no—
645C2—82no—
646C2#2144976411yescharacteristic
647C2#2145581411yescharacteristic
648C2#2145606511yescharacteristic
649C2#2145763511yescharacteristic
650C2#2145795211yescharacteristic
651C2#2145845111yescharacteristic
652C2#2145858211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2041C2
socleC23×C61C22×D4
prime radicalC2×#127×C61C22
hypercenter#2041D61
hypocenterC61#26752
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. #21450333C2semidirect
3. #21447225C2semidirect
4. #21447205C2semidirect
6. #21446499C2semidirect
7. #21446441C2semidirect
8. #21446391C2semidirect
9. #21446387C2semidirect
10. #21446375C2semidirect
11. #21446064C2semidirect
12. #21446064C2semidirect
13. #21445817C2semidirect
14. #102×Dic61C2semidirect
385. C61#26752semidirect

Maximal subgroups

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

subgroupindex
#214503332
#214488842
#214472252
#214472052
#214470502
#214464992
#214464412
#214463912
#214463872
#214463752
#214460642
#214458172
#102×Dic612
C61×#20412
#2675261