Finite Group Databasebeta

← #29753294

Subgroups — #29753294

Order 15616.

774 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450416C211yesmaximal, characteristic, complemented
2#21447340C211yesmaximal, characteristic, complemented
3#21447129C211yesmaximal, characteristic
4#21447055C211yesmaximal, characteristic, complemented
5#21446896C211yesmaximal, characteristic
6#21446676C211yesmaximal, characteristic, complemented
7#21446457C211yesmaximal, characteristic, complemented
8#21446439C211yesmaximal, characteristic, complemented
9#21446397C211yesmaximal, characteristic, complemented
10#21446378C211yesmaximal, characteristic, complemented
11#21446174C211yesmaximal, characteristic, complemented
12#21446167C211yesmaximal, characteristic
13#21445834C211yesmaximal, characteristic, complemented
14C4×#14519698C211yesmaximal, characteristic, complemented
15C61×#2367C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C22×#11758199C2211yescharacteristic
18C2×#14519710C2211yescharacteristic
19C2×#14519708C2211yescharacteristic
20C2×C4⋊#92418C2211yescharacteristic
21C2×#14519701C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×C4×#92418C2211yescharacteristic
30#16839691—21no—
31#16839688—21no—
32#16839664C2211yescharacteristic
33#16839664C2211yescharacteristic
34#16839664C2211yescharacteristic
35#16839660C2211yescharacteristic
36#16839652—21no—
37#16839651—21no—
38#16839628C2211yescharacteristic
39C4×C4⋊Dic61C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839517C2211yescharacteristic
45#16839517C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48C2×#124×C61C2211yescharacteristic
49C2×C4×Q8×C61C2211yescharacteristic
50C2×C4×D4×C61C2211yescharacteristic
51C61×#391C2211yescharacteristic
52C61×#390C2211yescharacteristic
53C61×#388—21no—
54C61×#388—21no—
55C61×#385—21no—
56C4×C4⋊C4×C61C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C4×C22⋊C4×C61—21no—
59C22×D244C2311yescharacteristic
60C2×#11758201—22no—
61C2×#11758201—22no—
62C2×#11758201—41no—
63C2×#11758199—22no—
64C2×#11758199—41no—
65C2×#11758199—41no—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758198—21no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×C4×Dic61—21no—
81#14519710—41no—
82#14519708—22no—
83C4⋊#92418—41no—
84#14519701—22no—
85#14519701—41no—
86#14519701—41no—
87#14519698—22no—
88#14519698—22no—
89#14519698—41no—
90#14519697—22no—
91#14519697—41no—
92C4×#92418—41no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97C22×Q8×C61C2311yescharacteristic
98C22×D4×C61C2311yescharacteristic
99C23×C4×C61—21no—
100#124×C61—22no—
101#124×C61—22no—
102C4×Q8×C61—22no—
103C4×Q8×C61—41no—
104C4×D4×C61—22no—
105C4×D4×C61—41no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61D411yescharacteristic
108C2×C4⋊C4×C61D411yescharacteristic
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61—21no—
115C2×C22⋊C4×C61—21no—
116C2×C42×C61—21no—
117C2×C42×C61—21no—
118C2×C42×C61C2311yescharacteristic
119C2×C42×C61C2311yescharacteristic
120C2×C42×C61C2311yescharacteristic
121#95×C61—21no—
122#95×C61—21no—
123#95×C61—21no—
124#95×C61D411yescharacteristic
125#95×C61D411yescharacteristic
126C24×C61—21no—
127C23×D61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61#4112yes—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61#4112yes—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—42no—
144C22×C4×C61—42no—
145C22×C4×C61—81no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2411yescharacteristic
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—41no—
167C4⋊C4×C61—42no—
168C4⋊C4×C61#4112yes—
169C22⋊C4×C61—24no—
170C22⋊C4×C61—24no—
171C22⋊C4×C61—42no—
172C22⋊C4×C61—42no—
173C22⋊C4×C61—81no—
174C42×C61—21no—
175C42×C61—21no—
176C42×C61—21no—
177C42×C61—21no—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61—42no—
181C42×C61—81no—
182#11758201—44no—
183#11758199—44no—
184#11758198—42no—
185#11758198—42no—
186#11758198—81no—
187#11758198—81no—
188#11758198—81no—
189#11758198—81no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196#11758196—81no—
197C4×Dic61—81no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418#4112yes—
210C23×C61—22no—
211C23×C61—41no—
212C23×C61—41no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61C22×D411yescharacteristic
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—41no—
223C22×D61—41no—
224C22×D61—41no—
225Q8×C61—22no—
226Q8×C61—22no—
227Q8×C61—41no—
228Q8×C61—41no—
229D4×C61—22no—
230D4×C61—22no—
231D4×C61—41no—
232D4×C61—41no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—22no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—22no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—84no—
260C2×C4×C61#13512yes—
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61H(5,𝔽2)11yescharacteristic
266C2×C4×C61H(5,𝔽2)11yescharacteristic
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276D244—41no—
277D244—41no—
278D244—41no—
279D244—41no—
280#92418—22no—
281#92418—41no—
282#92418—41no—
283#92418—41no—
284#27072—611nomaximal, Sylow
285C22×C61—42no—
286C22×C61—42no—
287C22×C61—42no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61#52511yescharacteristic
293C22×C61#54611yescharacteristic
294C22×C61#55011yescharacteristic
295C22×C61C2×#13511yescharacteristic
296C22×C61#57511yescharacteristic
297C22×C61#57611yescharacteristic
298C22×C61#57711yescharacteristic
299C2×D61—81no—
300C2×D61—81no—
301C2×D61—81no—
302C2×D61—81no—
303C2×D61—81no—
304C2×D61—81no—
305C2×D61—81no—
306C4×C61—22no—
307C4×C61—41no—
308C4×C61—41no—
309C4×C61—42no—
310C4×C61—41no—
311C4×C61—42no—
312C4×C61—41no—
313C4×C61—41no—
314C4×C61—81no—
315C4×C61#52911yescharacteristic
316C4×C61#53811yescharacteristic
317C4×C61#55111yescharacteristic
318C4×C61#55311yescharacteristic
319Dic61—81no—
320Dic61—81no—
321Dic61—81no—
322#2513—611no—
323#2367D6111yescharacteristic, complemented
324#2092—611no—
325#2054—611no—
326#2042—611no—
327#2012—611no—
328#1968—611no—
329#1928—611no—
330#1925—611no—
331#1919—611no—
332#1916—611no—
333#1880—611no—
334#1879—611no—
335#1805—611no—
336C4×#102—611no—
337C2×C61—82no—
338C2×C61#246711yescharacteristic
339C2×C61#310211yescharacteristic
340C2×C61#311911yescharacteristic
341C2×C61#329711yescharacteristic
342C2×C61#332111yescharacteristic
343C2×C61#339411yescharacteristic
344C2×C61#340311yescharacteristic
345D61—161no—
346C22×SD16—611no—
347C22×D8—611no—
348C2×#124C2×D6111yescharacteristic
349C2×C4×Q8C2×D6111yescharacteristic
350C2×C4×D4C2×D6111yescharacteristic
351C2×#107—611no—
352C2×#106—611no—
353C2×C4⋊C8—611no—
354C2×#103—611no—
355C2×#103—611no—
356C2×#103—611no—
357C2×#102—611no—
358C2×#102—611no—
359C2×#102—611no—
360C2×#102—611no—
361C2×#102—611no—
362C2×C4×C8—611no—
363#397—1221no—
364#397—1221no—
365#391C2×D6111yescharacteristic
366#390—611no—
367#390—611no—
368#390—611no—
369#390C2×D6111yescharacteristic
370#389—611no—
371#388—21no—
372#388—21no—
373#388—1221no—
374#388—1221no—
375#385—21no—
376#382—611no—
377C4×C4⋊C4—611no—
378C4×C4⋊C4C2×D6111yescharacteristic
379C4×C4⋊C4C2×D6111yescharacteristic
380C4×C22⋊C4—21no—
381#340—611no—
382#340—611no—
383#340—611no—
384#340—611no—
385#336—611no—
386#336—611no—
387#336—611no—
388#336—611no—
389C61#2707211yescharacteristic, complemented, Sylow
390C22×Q8C22×D6111yescharacteristic
391C22×D4—611no—
392C22×D4C22×D6111yescharacteristic
393C23×C4—21no—
394C2×SD16—1222no—
395C2×SD16—1222no—
396C2×SD16—2441no—
397C2×D8—1222no—
398C2×D8—2441no—
399C2×D8—2441no—
400C22×C8—611no—
401C22×C8—611no—
402C22×C8—611no—
403C22×C8—611no—
404#124—22no—
405#124—22no—
406C4×Q8—22no—
407C4×Q8—41no—
408C4×D4—22no—
409C4×D4—41no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4C22⋊D6111yescharacteristic
416C2×C4⋊C4C22⋊D6111yescharacteristic
417C2×C4⋊C4C22×D6111yescharacteristic
418C2×C4⋊C4C22×D6111yescharacteristic
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1221no—
426C2×C22⋊C4—1221no—
427C2×C22⋊C4—1221no—
428C2×C22⋊C4—1221no—
429C2×C22⋊C4—1221no—
430C2×C42—21no—
431C2×C42—21no—
432C2×C42—1221no—
433C2×C42C22×D6111yescharacteristic
434C2×C42C22×D6111yescharacteristic
435C2×C42C22×D6111yescharacteristic
436#107—2441no—
437#106—1222no—
438C4⋊C8—2441no—
439#103—1222no—
440#103—2441no—
441#103—2441no—
442#102—1222no—
443#102—1222no—
444#102—1222no—
445#102—2441no—
446#102—2441no—
447C4×C8—2441no—
448#95—21no—
449#95—21no—
450#95—21no—
451#95—1221no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455#95C22⋊D6111yescharacteristic
456#95C22⋊D6111yescharacteristic
457C24—21no—
458C24—1221no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8#1175822212yes—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—1221no—
471C2×D4—1221no—
472C2×D4—1221no—
473C2×D4—1221no—
474C2×D4—1221no—
475C2×D4—1221no—
476C2×D4#1175822212yes—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—81no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4C2×C22⋊D6111yescharacteristic
487C22×C4C2×C22⋊D6111yescharacteristic
488C22×C4C2×C22⋊D6111yescharacteristic
489C22×C4C2×C22⋊D6111yescharacteristic
490C22×C4C2×C22⋊D6111yescharacteristic
491C22×C4C2×C22⋊D6111yescharacteristic
492C22×C4C23×D6111yescharacteristic
493SD16—2444no—
494D8—2444no—
495C2×C8—612no—
496C2×C8—1221no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1221no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—41no—
512C4⋊C4—42no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—4881no—
520C4⋊C4#1175822412yes—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—81no—
526C22⋊C4—4881no—
527C22⋊C4—4881no—
528C22⋊C4—4881no—
529C22⋊C4—4881no—
530C22⋊C4—2442no—
531C22⋊C4—2442no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—42no—
539C42—81no—
540C42—4881no—
541C23—22no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—2441no—
549C23—2441no—
550C23—2441no—
551C23—2441no—
552C23—2441no—
553C23—2441no—
554C23—2441no—
555C23C22×C22⋊D6111yescharacteristic
556Q8—22no—
557Q8—22no—
558Q8—41no—
559Q8—41no—
560D4—22no—
561D4—22no—
562D4—41no—
563D4—41no—
564D4—2441no—
565D4—2441no—
566D4—2441no—
567D4—2441no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—84no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4#1451979712yes—
605C2×C4#41×D6111yescharacteristic
606C2×C4C2×#1175822411yescharacteristic
607C2×C4C2×#1175822211yescharacteristic
608C2×C4C2×#1175822211yescharacteristic
609C2×C4#1451982211yescharacteristic
610C2×C4#1451982311yescharacteristic
611C8—1222no—
612C8—2441no—
613C8—2441no—
614C8—2441no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—4881no—
623C22—4881no—
624C22—4881no—
625C22—4881no—
626C22—4881no—
627C22—4881no—
628C22—4881no—
629C22#1684027311yescharacteristic
630C22#1684040611yescharacteristic
631C22#1684043111yescharacteristic
632C22C2×#1451979711yescharacteristic
633C22#1684056011yescharacteristic
634C22#1684056511yescharacteristic
635C22#1684057011yescharacteristic
636C4—22no—
637C4—41no—
638C4—42no—
639C4—41no—
640C4—41no—
641C4—42no—
642C4—41no—
643C4—41no—
644C4—81no—
645C4—4881no—
646C4—4881no—
647C4—4881no—
648C4#1684029211yescharacteristic
649C4#1684034511yescharacteristic
650C4#1684044111yescharacteristic
651C4#1684045811yescharacteristic
652C2—82no—
653C2—9761no—
654C2#2144986411yescharacteristic
655C2#2145583711yescharacteristic
656C2#2145601811yescharacteristic
657C2#2145766911yescharacteristic
658C2#2145800511yescharacteristic
659C2#2145881511yescharacteristic
660C2#2145890311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2367C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
hypercenter#2367D61
hypocenterC61#27072
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2367

Complements

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

NH ≅ G/Ntype
1. #21450416C2semidirect
2. #21447340C2semidirect
4. #21447055C2semidirect
6. #21446676C2semidirect
7. #21446457C2semidirect
8. #21446439C2semidirect
9. #21446397C2semidirect
10. #21446378C2semidirect
11. #21446174C2semidirect
13. #21445834C2semidirect
14. C4×#14519698C2semidirect
15. C61×#2367C2semidirect
323. #2367D61semidirect
389. C61#27072semidirect

Maximal subgroups

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

subgroupindex
#214504162
#214473402
#214471292
#214470552
#214468962
#214466762
#214464572
#214464392
#214463972
#214463782
#214461742
#214461672
#214458342
C4×#145196982
C61×#23672
#2707261