Finite Group Databasebeta

← #29651317

Subgroups — #29651317

Order 15616.

656 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455819C211yesmaximal, characteristic
2#21454885C211yesmaximal, characteristic, complemented
3#21444731C211yesmaximal, characteristic, complemented
4#21444702C211yesmaximal, characteristic, complemented
5#21444683C211yesmaximal, characteristic, complemented
6#21444681C211yesmaximal, characteristic, complemented
7#21444648C211yesmaximal, characteristic, complemented
8#21444646C211yesmaximal, characteristic, complemented
9#21444641C211yesmaximal, characteristic, complemented
10C8×#11758199C211yesmaximal, characteristic, complemented
11#21444600C211yesmaximal, characteristic, complemented
12#21444287C211yesmaximal, characteristic, complemented
13#21444210C211yesmaximal, characteristic, complemented
14#21444119C211yesmaximal, characteristic, complemented
15C61×#2974C211yesmaximal, characteristic, complemented
16#16840312C2211yescharacteristic
17#16840075C411yescharacteristic
18#16840075C411yescharacteristic
19#16840042C411yescharacteristic
20#16840042C411yescharacteristic
21#16839957C411yescharacteristic
22#16839957C411yescharacteristic
23#16839896C411yescharacteristic
24#16839896C411yescharacteristic
25C4×#11758199C2211yescharacteristic
26C4×#11758199C2211yescharacteristic
27C4×#11758199C2211yescharacteristic
28C4×#11758199C2211yescharacteristic
29#16839841C2211yescharacteristic, complemented
30#16839841C2211yescharacteristic, complemented
31#16839839—21no—
32C8×D244C2211yescharacteristic, complemented
33C8×D244C2211yescharacteristic, complemented
34#16839827—21no—
35#16839815C2211yescharacteristic
36#16839815C2211yescharacteristic
37#16839815C2211yescharacteristic
38#16839727—21no—
39#16839727—21no—
40#16839514C2211yescharacteristic, complemented
41#16839514C2211yescharacteristic, complemented
42#16839512C2211yescharacteristic, complemented
43#16839512C2211yescharacteristic, complemented
44D4⋊#92418C2211yescharacteristic
45D4⋊#92418C2211yescharacteristic
46D4⋊#92418C2211yescharacteristic
47D4⋊#92418C2211yescharacteristic
48#16839489C2211yescharacteristic, complemented
49#16839489C2211yescharacteristic, complemented
50#16839489C2211yescharacteristic, complemented
51#16839489C2211yescharacteristic, complemented
52#16839484C2211yescharacteristic, complemented
53#16839484C2211yescharacteristic, complemented
54#16839483C2211yescharacteristic, complemented
55C8×#92418C2211yescharacteristic, complemented
56C2×C4×D4×C61C2211yescharacteristic
57C61×#436C2211yescharacteristic
58C61×#436C2211yescharacteristic
59C61×#435—21no—
60C61×#435—21no—
61C8×D4×C61C2211yescharacteristic
62C8×D4×C61C2211yescharacteristic
63C61×#432C2211yescharacteristic
64C61×#431C2211yescharacteristic
65C61×#407—21no—
66C2×C22⋊C8×C61—21no—
67C2×#11758222—21no—
68C2×#11758199C2×C411yescharacteristic
69C2×#11758199C2×C411yescharacteristic
70C2×#11758199C2×C411yescharacteristic
71C2×#11758199C2×C411yescharacteristic
72C2×#11758189—21no—
73C2×#11758189—21no—
74C2×C8×D61—21no—
75C2×C8×D61—21no—
76#14519777C2×C411yescharacteristic
77#14519774C2×C411yescharacteristic
78#14519765—21no—
79#14519751—21no—
80#14519751C2×C411yescharacteristic
81#14519751C2×C411yescharacteristic
82C4×D244—21no—
83C4×D244C2311yescharacteristic
84C4×D244C2311yescharacteristic
85#14519710C2×C411yescharacteristic
86#14519710C2×C411yescharacteristic
87#14519710C2×C411yescharacteristic
88#14519710C2×C411yescharacteristic
89C4⋊#92418—21no—
90C4⋊#92418C2×C411yescharacteristic
91C4⋊#92418C2×C411yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94#14519698C2×C411yescharacteristic
95#14519698C2×C411yescharacteristic
96#14519698C2×C411yescharacteristic
97#14519698C2×C411yescharacteristic
98#14519697C2×C411yescharacteristic
99#14519697C2×C411yescharacteristic
100#14519697C2×C411yescharacteristic
101#14519697C2×C411yescharacteristic
102C22⋊#92418—21no—
103C22⋊#92418—21no—
104C22⋊#92418C2×C411yescharacteristic
105C22⋊#92418C2×C411yescharacteristic
106#14519689—21no—
107#14519689—21no—
108#14519689—21no—
109#14519689—21no—
110C4×#92418—21no—
111C4×#92418C2311yescharacteristic
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C22×D4×C61C2×C411yescharacteristic
116C23×C4×C61—21no—
117C2×M16×C61—21no—
118C2×M16×C61—21no—
119C22×C8×C61—21no—
120C22×C8×C61—21no—
121C4×D4×C61—21no—
122C4×D4×C61—21no—
123C4×D4×C61C2×C411yescharacteristic
124C4×D4×C61C2×C411yescharacteristic
125C4×D4×C61C2311yescharacteristic
126C4×D4×C61C2311yescharacteristic
127C2×C4⋊C4×C61C2×C411yescharacteristic
128C2×C22⋊C4×C61—21no—
129C2×C42×C61C2311yescharacteristic
130C4⋊C8×C61D411yescharacteristic
131C4⋊C8×C61D411yescharacteristic
132C4⋊C8×C61C2311yescharacteristic
133C4⋊C8×C61C2311yescharacteristic
134C22⋊C8×C61—21no—
135C22⋊C8×C61—21no—
136C22⋊C8×C61—21no—
137C22⋊C8×C61—21no—
138C22⋊C8×C61—21no—
139C22⋊C8×C61D411yescharacteristic
140C22⋊C8×C61D411yescharacteristic
141#97×C61—21no—
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144C24×C61—21no—
145C2×D4×C61—22no—
146C2×D4×C61—21no—
147C2×D4×C61—41no—
148C2×D4×C61C22×C411yescharacteristic
149C2×D4×C61C22×C411yescharacteristic
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61—81no—
156C22×C4×C61—81no—
157C22×C4×C61C22×C411yescharacteristic
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61#4111yescharacteristic
160C2×C22⋊D61—21no—
161C2×C22⋊D61—21no—
162#11758222—81no—
163#11758222—81no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244C22×C411yescharacteristic
167C2×D244C22×C411yescharacteristic
168C2×C4×D61—21no—
169C2×C4×D61—21no—
170C2×Dic122—21no—
171M16×C61—81no—
172M16×C61—81no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—81no—
176C2×C8×C61—81no—
177C2×C8×C61C2×D411yescharacteristic
178C2×C8×C61C2×D411yescharacteristic
179C2×C8×C61C2×D411yescharacteristic
180C2×C8×C61C2×D411yescharacteristic
181C4⋊C4×C61—21no—
182C4⋊C4×C61C22×C411yescharacteristic
183C4⋊C4×C61C22×C411yescharacteristic
184C22⋊C4×C61—21no—
185C22⋊C4×C61—21no—
186C22⋊C4×C61—21no—
187C22⋊C4×C61—21no—
188C42×C61C22×C411yescharacteristic
189C42×C61C2×D411yescharacteristic
190C42×C61#4111yescharacteristic
191C42×C61C2411yescharacteristic
192#11758199—22no—
193#11758199—22no—
194#11758199—41no—
195#11758199—41no—
196#11758198—21no—
197#11758198—21no—
198#11758198—21no—
199#11758198—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61C22×C411yescharacteristic
202C4⋊Dic61C22×C411yescharacteristic
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418C22×C411yescharacteristic
208C2×#92418C22×C411yescharacteristic
209C2×#92418C22×C411yescharacteristic
210C2×#92418C22×C411yescharacteristic
211#11758189—81no—
212#11758189—81no—
213C8×D61—81no—
214C8×D61—81no—
215C23×C61—21no—
216C23×C61—21no—
217C23×C61—42no—
218C23×C61—42no—
219C23×C61—81no—
220C23×C61C4×D411yescharacteristic
221C22×D61—21no—
222C22×D61—21no—
223D4×C61—21no—
224D4×C61—21no—
225D4×C61—21no—
226D4×C61—42no—
227D4×C61#13112yes—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—81no—
240C2×C4×C61—81no—
241C2×C4×C61—81no—
242C2×C4×C61C4×D411yescharacteristic
243C2×C4×C61C4×D411yescharacteristic
244C2×C4×C61C4×D411yescharacteristic
245C2×C4×C61C23×C411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C22⋊D61—81no—
249C22⋊D61—81no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252D244—21no—
253D244—21no—
254D244—21no—
255D244—21no—
256D244—42no—
257D244—81no—
258C4×D61—41no—
259C4×D61—41no—
260C4×D61—41no—
261C4×D61—41no—
262Dic122—81no—
263C8×C61—21no—
264C8×C61—21no—
265C8×C61—21no—
266C8×C61—21no—
267C8×C61—41no—
268C8×C61—41no—
269#92418—21no—
270#92418—21no—
271#92418—21no—
272#92418—21no—
273#92418—41no—
274#92418—41no—
275#92418—41no—
276#92418—41no—
277#18026—611nomaximal, Sylow
278C22×C61—21no—
279C22×C61—22no—
280C22×C61—21no—
281C22×C61—21no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61—84no—
288C22×C61C2×C4×D411yescharacteristic
289C2×D61—41no—
290C2×D61—41no—
291C2×D61—41no—
292C2×D61—41no—
293C2×D61—41no—
294C2×D61—41no—
295C4×C61—21no—
296C4×C61—21no—
297C4×C61—21no—
298C4×C61—21no—
299C4×C61—41no—
300C4×C61—41no—
301C4×C61C2×#13111yescharacteristic
302C4×C61#56811yescharacteristic
303C4×C61#57611yescharacteristic
304C4×C61#57611yescharacteristic
305Dic61—41no—
306Dic61—41no—
307#3100—611no—
308#2979—611no—
309#2974D6111yescharacteristic, complemented
310#1594—611no—
311#1589—611no—
312#1586—611no—
313#1586—611no—
314#1581—611no—
315#1581—611no—
316#1580—611no—
317C8×D8—611no—
318#1572—611no—
319#1496—611no—
320#1478—611no—
321#1454—611no—
322C2×C61—41no—
323C2×C61—42no—
324C2×C61—81no—
325C2×C61#293311yescharacteristic
326C2×C61#294811yescharacteristic
327C2×C61#295511yescharacteristic
328D61—81no—
329D61—81no—
330#532—611no—
331C2×C4×D4C2×D6111yescharacteristic
332#480—611no—
333#480—611no—
334#474—611no—
335#474—611no—
336#459—611no—
337#459—611no—
338#447—611no—
339#447—611no—
340C4×D8—611no—
341C4×D8—611no—
342C4×D8—611no—
343C4×D8—611no—
344#436—611no—
345#436—611no—
346#436—1221no—
347#436C2×D6111yescharacteristic
348#436C2×D6111yescharacteristic
349#435—21no—
350#435—21no—
351C8×D4—611no—
352C8×D4—611no—
353C8×D4—1221no—
354C8×D4C2×D6111yescharacteristic
355C8×D4C2×D6111yescharacteristic
356#432C2×D6111yescharacteristic
357#431—611no—
358#431—611no—
359#431—611no—
360#431C2×D6111yescharacteristic
361#408—1221no—
362#408—1221no—
363#407—21no—
364C2×C22⋊C8—21no—
365#335—611no—
366#335—611no—
367#334—611no—
368#334—611no—
369D4⋊C8—611no—
370D4⋊C8—611no—
371D4⋊C8—611no—
372D4⋊C8—611no—
373D4⋊C8—611no—
374D4⋊C8—611no—
375D4⋊C8—611no—
376D4⋊C8—611no—
377#322—611no—
378#322—611no—
379#322—611no—
380C82—611no—
381C61#1802611yescharacteristic, complemented, Sylow
382C2×#41—1221no—
383C22×D4C4×D6111yescharacteristic
384C23×C4—21no—
385C2×D8—611no—
386C2×D8—611no—
387C2×D8—611no—
388C2×D8—611no—
389C2×M16—21no—
390C2×M16—21no—
391C2×M16—1221no—
392C2×M16—1221no—
393C22×C8—21no—
394C22×C8—21no—
395C22×C8—1221no—
396C22×C8—1221no—
397C4⋊Q8—611no—
398#127—611no—
399#124—1221no—
400#121—611no—
401#121—611no—
402#121—1221no—
403C4×D4—21no—
404C4×D4—21no—
405C4×D4—611no—
406C4×D4—611no—
407C4×D4—1221no—
408C4×D4C4×D6111yescharacteristic
409C4×D4C4×D6111yescharacteristic
410C4×D4C22×D6111yescharacteristic
411C4×D4C22×D6111yescharacteristic
412C2×C4⋊C4C4×D6111yescharacteristic
413C2×C22⋊C4—21no—
414C2×C42C22×D6111yescharacteristic
415#107—611no—
416#107—611no—
417#107—611no—
418#107—611no—
419C4⋊C8—611no—
420C4⋊C8—611no—
421C4⋊C8—611no—
422C4⋊C8—611no—
423C4⋊C8—1221no—
424C4⋊C8C22⋊D6111yescharacteristic
425C4⋊C8C22⋊D6111yescharacteristic
426C4⋊C8C22×D6111yescharacteristic
427C4⋊C8C22×D6111yescharacteristic
428#102—611no—
429#102—611no—
430#102—611no—
431#102—611no—
432#102—611no—
433#102—611no—
434#102—611no—
435#102—611no—
436C22⋊C8—21no—
437C22⋊C8—21no—
438C22⋊C8—21no—
439C22⋊C8—21no—
440C22⋊C8—21no—
441C22⋊C8—611no—
442C22⋊C8—611no—
443C22⋊C8—1221no—
444C22⋊C8—1221no—
445C22⋊C8—1221no—
446C22⋊C8—1221no—
447C22⋊C8—1221no—
448C22⋊C8—1221no—
449C22⋊C8C22⋊D6111yescharacteristic
450C22⋊C8C22⋊D6111yescharacteristic
451#97—21no—
452C4×C8—611no—
453C4×C8—611no—
454C4×C8—611no—
455C4×C8—611no—
456C4×C8—1221no—
457C4×C8C22×D6111yescharacteristic
458C4×C8C22×D6111yescharacteristic
459C24—21no—
460#41—4881no—
461#41—4881no—
462C2×Q8—1221no—
463C2×D4—22no—
464C2×D4—21no—
465C2×D4—41no—
466C2×D4—611no—
467C2×D4—611no—
468C2×D4—1221no—
469C2×D4—1221no—
470C2×D4—1221no—
471C2×D4—1221no—
472C2×D4C2×C4×D6111yescharacteristic
473C2×D4C2×C4×D6111yescharacteristic
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—81no—
480C22×C4—81no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4C2×C4×D6111yescharacteristic
484C22×C4#1175822211yescharacteristic
485C22×C4C2×C22⋊D6111yescharacteristic
486D8—1222no—
487D8—1222no—
488D8—2441no—
489D8—2441no—
490M16—81no—
491M16—81no—
492M16—4881no—
493M16—4881no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—81no—
497C2×C8—81no—
498C2×C8—611no—
499C2×C8—611no—
500C2×C8—611no—
501C2×C8—611no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—4881no—
507C2×C8—4881no—
508C2×C8C2×C22⋊D6111yescharacteristic
509C2×C8C2×C22⋊D6111yescharacteristic
510C2×C8C2×C22⋊D6111yescharacteristic
511C2×C8C2×C22⋊D6111yescharacteristic
512C4⋊C4—21no—
513C4⋊C4—611no—
514C4⋊C4—611no—
515C4⋊C4—1221no—
516C4⋊C4C2×C4×D6111yescharacteristic
517C4⋊C4C2×C4×D6111yescharacteristic
518C22⋊C4—21no—
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—1221no—
523C22⋊C4—1221no—
524C22⋊C4—1221no—
525C22⋊C4—1221no—
526C42C2×C4×D6111yescharacteristic
527C42#1175822211yescharacteristic
528C42C2×C22⋊D6111yescharacteristic
529C42C23×D6111yescharacteristic
530C23—21no—
531C23—21no—
532C23—42no—
533C23—42no—
534C23—81no—
535C23—1221no—
536C23—1221no—
537C23C4×C22⋊D6111yescharacteristic
538Q8—4881no—
539D4—21no—
540D4—21no—
541D4—21no—
542D4—42no—
543D4—1221no—
544D4—1221no—
545D4—1221no—
546D4—1221no—
547D4—4881no—
548D4—4881no—
549D4—4881no—
550D4—2442no—
551D4#1451978512yes—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4C4×C22⋊D6111yescharacteristic
573C2×C4C4×C22⋊D6111yescharacteristic
574C2×C4C4×C22⋊D6111yescharacteristic
575C2×C4C22×C4×D6111yescharacteristic
576C2×C4C22×C22⋊D6111yescharacteristic
577C2×C4C2×#1175822211yescharacteristic
578C8—21no—
579C8—21no—
580C8—21no—
581C8—21no—
582C8—41no—
583C8—41no—
584C8—1221no—
585C8—1221no—
586C8—1221no—
587C8—1221no—
588C8—2441no—
589C8—2441no—
590C8—2441no—
591C8—2441no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—22no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—84no—
602C22—2441no—
603C22—2441no—
604C22—2441no—
605C22—2441no—
606C22—2441no—
607C22—2441no—
608C22C2×C4×C22⋊D6111yescharacteristic
609C4—21no—
610C4—21no—
611C4—21no—
612C4—21no—
613C4—41no—
614C4—41no—
615C4—2441no—
616C4—2441no—
617C4C2×#1451978511yescharacteristic
618C4#1684052811yescharacteristic
619C4#1684056511yescharacteristic
620C4#1684056511yescharacteristic
621C2—41no—
622C2—42no—
623C2—81no—
624C2—4881no—
625C2—4881no—
626C2#2145454411yescharacteristic
627C2#2145466911yescharacteristic
628C2#2145472011yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#2974C2
socleC22×C61C2×C4×D4
prime radical#21455819C2
cosocleC42×C61C24
hypercenter#2974D61
hypocenterC61#18026
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2974

Complements

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

NH ≅ G/Ntype
2. #21454885C2semidirect
3. #21444731C2semidirect
4. #21444702C2semidirect
5. #21444683C2semidirect
6. #21444681C2semidirect
7. #21444648C2semidirect
8. #21444646C2semidirect
9. #21444641C2semidirect
10. C8×#11758199C2semidirect
11. #21444600C2semidirect
12. #21444287C2semidirect
13. #21444210C2semidirect
14. #21444119C2semidirect
15. C61×#2974C2semidirect
29. #16839841C22semidirect
30. #16839841C22semidirect
32. C8×D244C22semidirect
33. C8×D244C22semidirect
40. #16839514C22semidirect
41. #16839514C22semidirect
42. #16839512C22semidirect
43. #16839512C22semidirect
48. #16839489C22semidirect
49. #16839489C22semidirect
50. #16839489C22semidirect
51. #16839489C22semidirect
52. #16839484C22semidirect
53. #16839484C22semidirect
54. #16839483C22semidirect
55. C8×#92418C22semidirect
309. #2974D61semidirect
381. C61#18026semidirect

Maximal subgroups

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

subgroupindex
#214558192
#214548852
#214447312
#214447022
#214446832
#214446812
#214446482
#214446462
#214446412
C8×#117581992
#214446002
#214442872
#214442102
#214441192
C61×#29742
#1802661