Finite Group Databasebeta

← #30619813

Subgroups — #30619813

Order 15616.

736 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21461941C211yesmaximal, characteristic
2#21461918C211yesmaximal, characteristic, complemented
3#21461895C211yesmaximal, characteristic, complemented
4#21461890C211yesmaximal, characteristic, complemented
5#21461890C211yesmaximal, characteristic, complemented
6#21461890C211yesmaximal, characteristic, complemented
7#21461877C211yesmaximal, characteristic, complemented
8#21461873C211yesmaximal, characteristic, complemented
9#21461795C211yesmaximal, characteristic
10#21461764C211yesmaximal, characteristic, complemented
11#21461678C211yesmaximal, characteristic, complemented
12#21461611C211yesmaximal, characteristic, complemented
13#21461553C211yesmaximal, characteristic, complemented
14#21461548C211yesmaximal, characteristic, complemented
15#21449085C211yesmaximal, characteristic
16C22×D61⋊C8C2211yescharacteristic
17C2×#14519844C2211yescharacteristic
18C2×#14519844C2211yescharacteristic
19C2×#14519844C2211yescharacteristic
20C2×C4⋊#92420C2211yescharacteristic
21C2×C4⋊#92420C2211yescharacteristic
22C2×#14519834C2211yescharacteristic
23C2×#14519834C2211yescharacteristic
24C2×#14519834C2211yescharacteristic
25C2×#14519834C2211yescharacteristic
26C2×#14519834C2211yescharacteristic
27C2×#14519834C2211yescharacteristic
28C2×#14519834C2211yescharacteristic
29C2×#14519832C2211yescharacteristic
30C2×C4×#92420C2211yescharacteristic
31C2×C4×#92420C2211yescharacteristic
32C2×C4×#92420C2211yescharacteristic
33C2×C4×#92420C2211yescharacteristic
34#16840628C2211yescharacteristic
35#16840628C2211yescharacteristic
36#16840628C2211yescharacteristic
37#16840628C2211yescharacteristic
38#16840628C2211yescharacteristic
39#16840628C2211yescharacteristic
40#16840628C2211yescharacteristic
41#16840628C2211yescharacteristic
42#16840628C2211yescharacteristic
43#16840628C2211yescharacteristic
44C2×#14519739C2211yescharacteristic
45C2×#14519735C2211yescharacteristic
46#16839697C411yescharacteristic
47#16839692C411yescharacteristic
48#16839691C411yescharacteristic
49#16839691C411yescharacteristic
50#16839689C411yescharacteristic
51#16839680C411yescharacteristic
52#16839680C411yescharacteristic
53#16839613C2211yescharacteristic
54#16839613C2211yescharacteristic
55#16839609C2211yescharacteristic
56#16839609C2211yescharacteristic
57#16839609C2211yescharacteristic
58C61×#400C411yescharacteristic
59C2×D61⋊C8—22no—
60C2×D61⋊C8—41no—
61C2×D61⋊C8—41no—
62C22×#92420C2311yescharacteristic
63C22×#92420C2311yescharacteristic
64C22×#92420C2311yescharacteristic
65C22×#92420C2311yescharacteristic
66C22×#92420C2311yescharacteristic
67C22×#92420C2311yescharacteristic
68C22×#92420C2311yescharacteristic
69C22×#92420C2311yescharacteristic
70#14519844—22no—
71#14519844—41no—
72#14519844—41no—
73C4⋊#92420—41no—
74C4⋊#92420—41no—
75#14519834—22no—
76#14519834—22no—
77#14519834—22no—
78#14519834—41no—
79#14519834—41no—
80#14519834—41no—
81#14519834—41no—
82#14519832—41no—
83C4×#92420—22no—
84C4×#92420—41no—
85C4×#92420—41no—
86C4×#92420—41no—
87C22×D244C2×C411yescharacteristic
88C22×C4×D61C2311yescharacteristic
89C22×C4×D61C2311yescharacteristic
90#14519739—41no—
91#14519739—41no—
92#14519735—41no—
93#14519735—41no—
94C2×C4⋊Dic61C2×C411yescharacteristic
95C2×#11758196C2×C411yescharacteristic
96C2×#11758196C2×C411yescharacteristic
97C2×#11758196C2×C411yescharacteristic
98C2×#11758196C2×C411yescharacteristic
99C2×#11758198C2×C411yescharacteristic
100C2×#11758198C2×C411yescharacteristic
101C2×#11758198C2×C411yescharacteristic
102C2×#11758198C2×C411yescharacteristic
103C2×#11758198C2×C411yescharacteristic
104C2×#11758198C2×C411yescharacteristic
105C2×#11758198C2×C411yescharacteristic
106C2×#11758198C2×C411yescharacteristic
107C2×#11758198C2×C411yescharacteristic
108C2×#11758198C2×C411yescharacteristic
109C2×C4×Dic61C2311yescharacteristic
110C2×C4×Dic61C2311yescharacteristic
111C2×C4×Dic61C2311yescharacteristic
112C2×C4×Dic61C2311yescharacteristic
113C2×C4×Dic61C2311yescharacteristic
114#14519684C2×C411yescharacteristic
115#14519684C2×C411yescharacteristic
116#14519684C2×C411yescharacteristic
117#14519684C2×C411yescharacteristic
118#14519684C2×C411yescharacteristic
119C2×C4⋊C4×C61C2×C411yescharacteristic
120C2×C4⋊C4×C61C2×C411yescharacteristic
121#95×C61C2×C411yescharacteristic
122#95×C61C2×C411yescharacteristic
123#95×C61C2×C411yescharacteristic
124#95×C61C2×C411yescharacteristic
125#95×C61C2×C411yescharacteristic
126C23×D61C22×C411yescharacteristic
127C23×D61C22×C411yescharacteristic
128C22×C4×C61C22×C411yescharacteristic
129C22×C4×C61C22×C411yescharacteristic
130C22×C4×C61C22×C411yescharacteristic
131C22×C4×C61C22×C411yescharacteristic
132C22×C4×C61C22×C411yescharacteristic
133C22×C4×C61C22×C411yescharacteristic
134C22×C4×C61C22×C411yescharacteristic
135C22×Dic61C22×C411yescharacteristic
136C22×Dic61C22×C411yescharacteristic
137C22×Dic61C22×C411yescharacteristic
138C22×Dic61C22×C411yescharacteristic
139C22×Dic61C22×C411yescharacteristic
140C22×Dic61C2411yescharacteristic
141C2×D244—22no—
142C2×D244—22no—
143C2×D244—41no—
144C2×C4×D61—21no—
145C2×C4×D61—21no—
146C2×C4×D61—21no—
147C2×C4×D61—21no—
148C2×C4×D61—21no—
149C2×C4×D61—21no—
150C2×C4×D61—21no—
151C2×C4×D61—21no—
152C2×C4×D61—21no—
153C2×C4×D61—21no—
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×#92420—21no—
157C2×#92420—21no—
158C2×#92420—21no—
159C2×#92420—21no—
160C2×#92420—21no—
161C2×#92420—21no—
162C2×#92420—21no—
163C2×#92420—21no—
164C2×#92420—21no—
165C2×#92420—21no—
166C2×#92420—21no—
167C2×#92420—21no—
168C2×#92420—21no—
169C2×#92420—21no—
170C2×#92420—21no—
171C2×#92420—21no—
172C2×#92420—21no—
173C2×#92420—21no—
174C2×#92420—21no—
175C2×#92420—21no—
176C2×#92420—21no—
177C2×#92420—21no—
178C2×#92420#4112yes—
179C2×#92420#4112yes—
180D61⋊C8—44no—
181C4⋊C4×C61—41no—
182C4⋊C4×C61—41no—
183#11758198—22no—
184#11758198—22no—
185#11758198—22no—
186#11758198—22no—
187#11758198—22no—
188#11758198—22no—
189#11758198—41no—
190#11758198—41no—
191#11758198—41no—
192#11758198—41no—
193C4⋊Dic61—41no—
194#11758196—22no—
195#11758196—22no—
196#11758196—22no—
197#11758196—22no—
198C4×Dic61—21no—
199C4×Dic61—21no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—21no—
208C23×C61C23×C411yescharacteristic
209C22×D61—21no—
210C22×D61—21no—
211C22×D61—21no—
212C22×D61—21no—
213C22×D61—21no—
214C22×D61—21no—
215C22×D61—21no—
216C22×D61—21no—
217C22×D61—21no—
218C22×D61—21no—
219C22×D61—21no—
220C22×D61—21no—
221C22×D61—21no—
222C22×D61#13112yes—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
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—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61#13112yes—
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—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61#13112yes—
258C2×Dic61#13112yes—
259C2×Dic61C2×#4111yescharacteristic
260C2×Dic61C2×#4111yescharacteristic
261C2×Dic61H(5,𝔽2)11yescharacteristic
262C2×Dic61H(5,𝔽2)11yescharacteristic
263C2×Dic61H(5,𝔽2)11yescharacteristic
264C2×Dic612−1+411yescharacteristic
265D244—44no—
266C4×D61—41no—
267C4×D61—41no—
268C4×D61—41no—
269C4×D61—41no—
270C4×D61—41no—
271C4×D61—41no—
272C4×D61—41no—
273C4×D61—41no—
274#92420—22no—
275#92420—22no—
276#92420—41no—
277#92420—41no—
278#92420—41no—
279#92420—41no—
280#92420—41no—
281#92420—41no—
282#22358—611nomaximal, Sylow
283C22×C61C4×#4111yescharacteristic
284C22×C61#51811yescharacteristic
285C22×C61#52011yescharacteristic
286C22×C61C2×#13111yescharacteristic
287C22×C61C2×#13111yescharacteristic
288C22×C61C2×#13111yescharacteristic
289C22×C61C2×#13111yescharacteristic
290C2×D61—22no—
291C2×D61—22no—
292C2×D61—22no—
293C2×D61—41no—
294C2×D61—41no—
295C2×D61—41no—
296C2×D61—41no—
297C2×D61—41no—
298C2×D61—41no—
299C2×D61—41no—
300C2×D61—41no—
301C2×D61—41no—
302C2×D61—41no—
303C2×D61—41no—
304C4×C61—22no—
305C4×C61—41no—
306C4×C61—41no—
307C4×C61—41no—
308C4×C61—41no—
309C4×C61—41no—
310C4×C61—41no—
311Dic61—22no—
312Dic61—22no—
313Dic61—41no—
314Dic61—41no—
315Dic61—41no—
316Dic61#53811yescharacteristic
317Dic61#53911yescharacteristic
318Dic61#54011yescharacteristic
319Dic61#54211yescharacteristic
320#2395—611no—
321#1954—611no—
322#1939—611no—
323#1924—611no—
324#1923—611no—
325#1923—611no—
326#1923—611no—
327#1917—611no—
328#1916—611no—
329#1869—611no—
330#1850—611no—
331#1798—611no—
332C8×C4⋊C4—611no—
333C8×C22⋊C4—611no—
334#1750—611no—
335C2×C61#233511yescharacteristic
336C2×C61#296611yescharacteristic
337C2×C61#296611yescharacteristic
338C2×C61#297711yescharacteristic
339C2×C61#297911yescharacteristic
340C2×C61#299411yescharacteristic
341C2×C61#299411yescharacteristic
342D61—42no—
343D61—81no—
344C23×C8—611no—
345C2×C4×D4—611no—
346C2×#117—611no—
347C2×C4⋊C8—611no—
348C2×C4⋊C8—611no—
349C2×C4⋊C8—611no—
350C2×C4⋊C8—611no—
351C2×C4⋊C8—611no—
352C2×C22⋊C8—611no—
353C2×C22⋊C8—611no—
354C2×C22⋊C8—611no—
355C2×C22⋊C8—611no—
356C2×C22⋊C8—611no—
357C2×C22⋊C8—611no—
358C2×C22⋊C8—611no—
359C2×#97—611no—
360C2×C4×C8—611no—
361C2×C4×C8—611no—
362C2×C4×C8—611no—
363C2×C4×C8—611no—
364#400#692311yescharacteristic
365#399—611no—
366#398—611no—
367#397—611no—
368#397—611no—
369#397—611no—
370#394—611no—
371#394—611no—
372C4×C4⋊C4—611no—
373C4×C4⋊C4—611no—
374C4×C22⋊C4—611no—
375C4×C22⋊C4—611no—
376C4×C22⋊C4—611no—
377#336—611no—
378#336—611no—
379#336—611no—
380#336—611no—
381#336—611no—
382#336—611no—
383#336—611no—
384#336—611no—
385#336—611no—
386#336—611no—
387C61#2235811yescharacteristic, complemented, Sylow
388C22×D4—611no—
389C23×C4—611no—
390C23×C4—611no—
391C22×C8—611no—
392C22×C8—611no—
393C22×C8—611no—
394C22×C8—611no—
395C22×C8—611no—
396C22×C8—611no—
397C22×C8—611no—
398C22×C8—611no—
399C22×C8—1222no—
400C22×C8—2441no—
401C22×C8—2441no—
402C4×D4—2441no—
403C4×D4—2441no—
404#117—2441no—
405#117—2441no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4C2×#692311yescharacteristic
412C2×C4⋊C4C2×#692311yescharacteristic
413C2×C22⋊C4—611no—
414C2×C22⋊C4—611no—
415C2×C22⋊C4—611no—
416C2×C22⋊C4—611no—
417C2×C22⋊C4—611no—
418C2×C22⋊C4—611no—
419C2×C22⋊C4—611no—
420C2×C22⋊C4—611no—
421C2×C22⋊C4—611no—
422C2×C22⋊C4—611no—
423C2×C42—611no—
424C2×C42—611no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—611no—
428C4⋊C8—1222no—
429C4⋊C8—2441no—
430C4⋊C8—2441no—
431C4⋊C8—2441no—
432C4⋊C8—2441no—
433C22⋊C8—1222no—
434C22⋊C8—1222no—
435C22⋊C8—1222no—
436C22⋊C8—2441no—
437C22⋊C8—2441no—
438C22⋊C8—2441no—
439C22⋊C8—2441no—
440#97—2441no—
441C4×C8—1222no—
442C4×C8—2441no—
443C4×C8—2441no—
444C4×C8—2441no—
445#95—611no—
446#95—611no—
447#95—611no—
448#95—611no—
449#95—611no—
450#95C2×#692311yescharacteristic
451#95C2×#692311yescharacteristic
452#95C2×#692311yescharacteristic
453#95C2×#692311yescharacteristic
454#95C2×#692311yescharacteristic
455C24—611no—
456C24—611no—
457C2×D4—1222no—
458C2×D4—1222no—
459C2×D4—2441no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4C22×#692311yescharacteristic
479C22×C4C22×#692311yescharacteristic
480C22×C4C22×#692311yescharacteristic
481C22×C4C22×#692311yescharacteristic
482C22×C4C22×#692311yescharacteristic
483C22×C4C22×#692311yescharacteristic
484C22×C4C22×#692311yescharacteristic
485C2×C8—612no—
486C2×C8—612no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8—1221no—
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—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—2444no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4—1222no—
513C4⋊C4—1222no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—2441no—
517C22⋊C4—1222no—
518C22⋊C4—1222no—
519C22⋊C4—1222no—
520C22⋊C4—1222no—
521C22⋊C4—1222no—
522C22⋊C4—1222no—
523C22⋊C4—2441no—
524C22⋊C4—2441no—
525C22⋊C4—2441no—
526C22⋊C4—2441no—
527C42—1221no—
528C42—1221no—
529C42—1221no—
530C42—1221no—
531C42—1221no—
532C42—1221no—
533C42—1221no—
534C42—1221no—
535C42—1221no—
536C42—1221no—
537C23—612no—
538C23—1221no—
539C23—1221no—
540C23—1221no—
541C23—1221no—
542C23—1221no—
543C23—1221no—
544C23—1221no—
545C23—1221no—
546C23—1221no—
547C23—1221no—
548C23—1221no—
549C23—1221no—
550C23—1221no—
551C23C23×#692311yescharacteristic
552D4—2444no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—611no—
574C2×C4—611no—
575C2×C4—611no—
576C2×C4—611no—
577C2×C4—611no—
578C2×C4—611no—
579C2×C4—612no—
580C2×C4—612no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4#1451986512yes—
603C8—1222no—
604C8—1222no—
605C8—2441no—
606C8—2441no—
607C8—2441no—
608C8—2441no—
609C8—2441no—
610C8—2441no—
611C22—1222no—
612C22—1222no—
613C22—1222no—
614C22—2441no—
615C22—2441no—
616C22—2441no—
617C22—2441no—
618C22—2441no—
619C22—2441no—
620C22—2441no—
621C22—2441no—
622C22—2441no—
623C22—2441no—
624C22—2441no—
625C22#41×#692311yescharacteristic
626C22#1684082911yescharacteristic
627C22#1684083111yescharacteristic
628C22D61⋊#13111yescharacteristic
629C22D61⋊#13111yescharacteristic
630C22D61⋊#13111yescharacteristic
631C22C2×#1451986511yescharacteristic
632C4—22no—
633C4—41no—
634C4—41no—
635C4—41no—
636C4—41no—
637C4—41no—
638C4—41no—
639C4—611no—
640C4—611no—
641C4—611no—
642C4—611no—
643C4—1222no—
644C4—1222no—
645C4—2441no—
646C4—2441no—
647C4—2441no—
648C2—4881no—
649C2—2442no—
650C2#2146231811yescharacteristic
651C2#2146263611yescharacteristic
652C2#2146263711yescharacteristic
653C2#2146269411yescharacteristic
654C2#2146270111yescharacteristic
655C2#2146276811yescharacteristic
656C2#2146277111yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC61×#400C4
prime radicalC22×D244C2×C4
cosocleC22×Dic61C24
hypercenter#400#6923
hypocenterC61#22358
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. #21461918C2semidirect
3. #21461895C2semidirect
4. #21461890C2semidirect
5. #21461890C2semidirect
6. #21461890C2semidirect
7. #21461877C2semidirect
8. #21461873C2semidirect
10. #21461764C2semidirect
11. #21461678C2semidirect
12. #21461611C2semidirect
13. #21461553C2semidirect
14. #21461548C2semidirect
387. C61#22358semidirect

Maximal subgroups

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

subgroupindex
#214619412
#214619182
#214618952
#214618902
#214618772
#214618732
#214617952
#214617642
#214616782
#214616112
#214615532
#214615482
#214490852
#2235861