Finite Group Databasebeta

← #29739953

Subgroups — #29739953

Order 15616.

896 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840083C211yesmaximal, characteristic, complemented
2C2×#16839803C211yesmaximal, characteristic, complemented
3#21449233C211yesmaximal, characteristic
4#21448741C211yesmaximal, characteristic, complemented
5#21447338C211yesmaximal, characteristic, complemented
6#21447303C211yesmaximal, characteristic, complemented
7#21447256C211yesmaximal, characteristic, complemented
8#21446448C211yesmaximal, characteristic, complemented
9#21446393C211yesmaximal, characteristic, complemented
10#21446254C211yesmaximal, characteristic, complemented
11#21446133C211yesmaximal, characteristic, complemented
12#21446113C211yesmaximal, characteristic, complemented
13#21446021C211yesmaximal, characteristic, complemented
14#21446020C211yesmaximal, characteristic, complemented
15C61×#2033C211yesmaximal, characteristic, complemented
16C22×#11758189C2211yescharacteristic
17C2×#14519748C2211yescharacteristic
18C2×C4×C22⋊D61C2211yescharacteristic
19C2×#14519736C2211yescharacteristic
20C2×#14519735C2211yescharacteristic
21C2×C4⋊C4×D61C2211yescharacteristic
22#16840083—24no—
23#16839803—24no—
24C2×#14519711C2211yescharacteristic
25C2×#14519710C2211yescharacteristic
26C2×#14519709C2211yescharacteristic
27C2×#14519708C2211yescharacteristic
28C2×#14519708C2211yescharacteristic
29C2×#14519698C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×C22⋊#92418C2211yescharacteristic
32C2×#14519689C2211yescharacteristic
33#16839678—21no—
34#16839670—21no—
35#16839665—21no—
36#16839665C2211yescharacteristic
37#16839661C2211yescharacteristic
38#16839650—21no—
39#16839650—21no—
40#16839644—21no—
41#16839638—21no—
42#16839627C2211yescharacteristic
43#16839622C2211yescharacteristic
44#16839619C2211yescharacteristic
45C4⋊C4×Dic61C2211yescharacteristic
46#16839609—21no—
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×#121×C61C2211yescharacteristic
55C2×#107×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C2×C22⋊C8×C61C2211yescharacteristic
59C61×#380C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×C22⋊D61—21no—
62C22×C22⋊D61D411yescharacteristic
63C22×C4×D61—21no—
64C22×C4×D61C2311yescharacteristic
65C2×#11758189—22no—
66C2×#11758189—22no—
67C2×#11758189—22no—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70#14519748—22no—
71#14519748—22no—
72C4×C22⋊D61—22no—
73C4×C22⋊D61—22no—
74#14519736—22no—
75#14519736—41no—
76#14519735—22no—
77#14519735—41no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×#11758196—21no—
84C2×#11758196D411yescharacteristic
85C4⋊C4×D61—22no—
86C4⋊C4×D61—41no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61—21no—
91C2×C22⋊Dic61—21no—
92C2×C22⋊Dic61D411yescharacteristic
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×#11758198D411yescharacteristic
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61C2311yescharacteristic
100#14519711—41no—
101#14519710D414yescomplemented
102#14519709—22no—
103#14519708—22no—
104#14519708—22no—
105#14519698—22no—
106#14519698—22no—
107C22⋊#92418—22no—
108#14519689—22no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115C22×D4×C61—21no—
116C22×D4×C61C2311yescharacteristic
117C23×C4×C61C2311yescharacteristic
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120#121×C61—22no—
121#121×C61—41no—
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C22⋊C4×C61—21no—
126C2×C22⋊C4×C61—21no—
127#107×C61—22no—
128#102×C61—22no—
129#102×C61—41no—
130C22⋊C8×C61—22no—
131#95×C61—21no—
132C24×C61—21no—
133C24×C61C2×D411yescharacteristic
134C23×D61C2×D411yescharacteristic
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—42no—
142C2×D4×C61—42no—
143C2×D4×C61—81no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2411yescharacteristic
155C2×C22⋊D61—22no—
156C2×C22⋊D61—41no—
157C2×C22⋊D61—41no—
158C2×C22⋊D61—42no—
159C2×C22⋊D61—81no—
160C2×C22⋊D61—81no—
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61—21no—
165C22×Dic61C2×D411yescharacteristic
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61C2×D411yescharacteristic
168C2×C4×D61—21no—
169C2×C4×D61—21no—
170C2×C4×D61—21no—
171C2×C4×D61—21no—
172C2×C4×D61—42no—
173C2×C4×D61—42no—
174C2×C4×D61—81no—
175C2×C4×D61#4112yes—
176C2×C4×D61#4112yes—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61#4112yes—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61C2×D412yes—
189C22⋊C4×C61—42no—
190C22⋊C4×C61—81no—
191C22⋊Dic61—22no—
192C22⋊Dic61—42no—
193C22⋊Dic61—42no—
194C22⋊Dic61—42no—
195C22⋊Dic61—81no—
196C22⋊Dic61—81no—
197#11758198—22no—
198#11758198—42no—
199#11758198—42no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—81no—
208C4⋊Dic61C2×D412yes—
209#11758196—22no—
210#11758196—42no—
211C4×Dic61—21no—
212C4×Dic61—21no—
213C4×Dic61—42no—
214C4×Dic61—42no—
215C4×Dic61—81no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418C2×D412yes—
222#11758189—28no—
223C23×C61—21no—
224C23×C61—21no—
225C23×C61—21no—
226C23×C61—21no—
227C23×C61—21no—
228C23×C61—21no—
229C23×C61—21no—
230C23×C61—41no—
231C23×C61—41no—
232C23×C61—41no—
233C23×C61—41no—
234C23×C61—41no—
235C23×C61—41no—
236C23×C61—41no—
237C23×C61C22×D411yescharacteristic
238C22×D61—21no—
239C22×D61—21no—
240C22×D61—21no—
241C22×D61—21no—
242C22×D61—21no—
243C22×D61—21no—
244C22×D61#12312yes—
245D4×C61—41no—
246D4×C61—41no—
247D4×C61—41no—
248D4×C61—41no—
249D4×C61—84no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—22no—
253C2×C4×C61—21no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61—41no—
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C2×C4×C61C2×#4111yescharacteristic
269C2×C4×C61C2×#4111yescharacteristic
270C2×C4×C61H(5,𝔽2)11yescharacteristic
271C2×C4×C612−1+411yescharacteristic
272C22⋊D61—44no—
273C22⋊D61—84no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—22no—
279C2×Dic61—21no—
280C2×Dic61—21no—
281C2×Dic61—21no—
282C2×Dic61—21no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—41no—
291C2×Dic61—41no—
292C2×Dic61—41no—
293C2×Dic61—41no—
294C2×Dic61#12312yes—
295C4×D61—22no—
296C4×D61—22no—
297C4×D61—22no—
298C4×D61—22no—
299C4×D61—84no—
300C8×C61—22no—
301C8×C61—41no—
302#92418—22no—
303#92418—41no—
304#26156—611nomaximal, Sylow
305C22×C61—41no—
306C22×C61—41no—
307C22×C61—41no—
308C22×C61—41no—
309C22×C61—41no—
310C22×C61—41no—
311C22×C61—41no—
312C22×C61—81no—
313C22×C61—81no—
314C22×C61—81no—
315C22×C61—81no—
316C22×C61—81no—
317C22×C61—81no—
318C22×C61—81no—
319C22×C61C2×#12311yescharacteristic
320C22×C61D4211yescharacteristic
321C22×C61#54711yescharacteristic
322C22×C61#57511yescharacteristic
323C22×C61#57511yescharacteristic
324C22×C61#57611yescharacteristic
325C22×C61#57711yescharacteristic
326C2×D61—22no—
327C2×D61—22no—
328C2×D61—22no—
329C2×D61—41no—
330C2×D61—41no—
331C2×D61—41no—
332C2×D61—41no—
333C4×C61—41no—
334C4×C61—42no—
335C4×C61—81no—
336C4×C61—81no—
337C4×C61#52511yescharacteristic
338C4×C61#53911yescharacteristic
339C4×C61#54811yescharacteristic
340C4×C61#55311yescharacteristic
341Dic61—22no—
342Dic61—41no—
343Dic61—41no—
344Dic61—42no—
345Dic61—81no—
346Dic61—81no—
347Dic61—81no—
348C2×#482—611no—
349C2×#428—611no—
350#2414—611no—
351#2352—611no—
352#2092—611no—
353#2085—611no—
354#2077—611no—
355#2033D6111yescharacteristic, complemented
356#1926—611no—
357#1918—611no—
358#1895—611no—
359#1874—611no—
360#1870—611no—
361#1854—611no—
362#1854—611no—
363C2×C61—81no—
364C2×C61—161no—
365C2×C61#244311yescharacteristic
366C2×C61#309311yescharacteristic
367C2×C61#309611yescharacteristic
368C2×C61#327111yescharacteristic
369C2×C61#327311yescharacteristic
370C2×C61#334711yescharacteristic
371C2×C61#335111yescharacteristic
372D61—42no—
373C22×M16—611no—
374C2×#121—611no—
375C2×#121C2×D6111yescharacteristic
376C2×C4×D4—611no—
377C2×#117—611no—
378C2×#117—611no—
379C22×C4⋊C4—611no—
380#482—1224no—
381#428—1224no—
382C2×#107—611no—
383C2×#107—611no—
384C2×#107C2×D6111yescharacteristic
385C2×#106—611no—
386C2×#106—611no—
387C2×#106—611no—
388C2×#102—611no—
389C2×#102—611no—
390C2×#102C2×D6111yescharacteristic
391C2×#102C2×D6111yescharacteristic
392C2×C22⋊C8—611no—
393C2×C22⋊C8—611no—
394C2×C22⋊C8C2×D6111yescharacteristic
395#394—1221no—
396#392—1221no—
397#390—611no—
398#390—1221no—
399#389—611no—
400#388—1221no—
401#388—1221no—
402#387—1221no—
403#386—1221no—
404#382—611no—
405#380—611no—
406#380—611no—
407#380C2×D6111yescharacteristic
408C4×C4⋊C4—611no—
409C4×C22⋊C4—1221no—
410#340—611no—
411#340—611no—
412#340—611no—
413#340—611no—
414#340—611no—
415#340C2×D6111yescharacteristic
416#336—611no—
417#336—611no—
418C61#2615611yescharacteristic, complemented, Sylow
419C22×D4—21no—
420C22×D4—611no—
421C22×D4—1221no—
422C22×D4C22×D6111yescharacteristic
423C23×C4—611no—
424C23×C4—1221no—
425C23×C4C22×D6111yescharacteristic
426C2×M16—1222no—
427C2×M16—1222no—
428C2×M16—1222no—
429C22×C8—611no—
430C22×C8—611no—
431C22×C8C22×D6111yescharacteristic
432C22×C8C22×D6111yescharacteristic
433#121—22no—
434#121—41no—
435#121—1222no—
436#121—1222no—
437C4×D4—1222no—
438C4×D4—1222no—
439#117—1222no—
440#117—1222no—
441#117—2441no—
442#117—2441no—
443C2×C4⋊C4—611no—
444C2×C4⋊C4—611no—
445C2×C4⋊C4—611no—
446C2×C4⋊C4—611no—
447C2×C4⋊C4—611no—
448C2×C4⋊C4—1222no—
449C2×C4⋊C4—1221no—
450C2×C4⋊C4—1221no—
451C2×C4⋊C4—2441no—
452C2×C4⋊C4C22×D6111yescharacteristic
453C2×C4⋊C4C22×D6111yescharacteristic
454C2×C4⋊C4C22×D6111yescharacteristic
455C2×C22⋊C4—21no—
456C2×C22⋊C4—21no—
457C2×C22⋊C4—611no—
458C2×C22⋊C4—611no—
459C2×C22⋊C4—1221no—
460C2×C22⋊C4—1221no—
461C2×C22⋊C4—1221no—
462C2×C22⋊C4—1221no—
463C2×C22⋊C4—1221no—
464C2×C22⋊C4—1221no—
465C2×C22⋊C4—1221no—
466C2×C42—611no—
467C2×C42—1221no—
468C2×C42—1221no—
469C2×C42—1221no—
470#107—22no—
471#107—614no—
472#107—2441no—
473#106—1222no—
474#106—1222no—
475#106—1222no—
476#102—22no—
477#102—41no—
478#102—1222no—
479#102—1222no—
480C22⋊C8—22no—
481C22⋊C8—1222no—
482C22⋊C8—1222no—
483#95—21no—
484#95—1221no—
485#95—1221no—
486#95—1221no—
487#95—1221no—
488#95—1221no—
489#95—1221no—
490C24—21no—
491C24—611no—
492C24D4×D6111yescharacteristic
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—21no—
498C2×D4—21no—
499C2×D4—42no—
500C2×D4—42no—
501C2×D4—81no—
502C2×D4—1222no—
503C2×D4—4881no—
504C2×D4—4881no—
505C2×D4—2441no—
506C2×D4—2441no—
507C2×D4—2442no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—611no—
518C22×C4—612no—
519C22×C4—612no—
520C22×C4—611no—
521C22×C4—611no—
522C22×C4—1221no—
523C22×C4—1221no—
524C22×C4—1221no—
525C22×C4—1221no—
526C22×C4—1221no—
527C22×C4—1221no—
528C22×C4—1221no—
529C22×C4—1221no—
530C22×C4—4881no—
531C22×C4—2442no—
532C22×C4—2442no—
533C22×C4D4×D6111yescharacteristic
534C22×C4C23×D6111yescharacteristic
535M16—1228no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—612no—
542C2×C8—1221no—
543C2×C8—1221no—
544C2×C8—1221no—
545C2×C8—1221no—
546C2×C8—1221no—
547C2×C8#1175822412yes—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—612no—
554C4⋊C4—1222no—
555C4⋊C4—1221no—
556C4⋊C4—1221no—
557C4⋊C4—1221no—
558C4⋊C4—1221no—
559C4⋊C4—1221no—
560C4⋊C4—1221no—
561C4⋊C4—1221no—
562C4⋊C4—4881no—
563C4⋊C4—2442no—
564C4⋊C4D4×D6112yes—
565C22⋊C4—42no—
566C22⋊C4—81no—
567C22⋊C4—1222no—
568C22⋊C4—1222no—
569C22⋊C4—4881no—
570C22⋊C4—4881no—
571C22⋊C4—2442no—
572C22⋊C4—2442no—
573C22⋊C4—2442no—
574C22⋊C4—2442no—
575C22⋊C4—2442no—
576C42—1221no—
577C42—1221no—
578C42—4881no—
579C42—2442no—
580C42—2442no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23—41no—
593C23—41no—
594C23—41no—
595C23—612no—
596C23—1221no—
597C23—1221no—
598C23—1221no—
599C23—1221no—
600C23—1221no—
601C23—1221no—
602C23C2×D4×D6111yescharacteristic
603D4—41no—
604D4—41no—
605D4—41no—
606D4—41no—
607D4—84no—
608D4—4884no—
609D4—2444no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—612no—
627C2×C4—1221no—
628C2×C4—1221no—
629C2×C4—1222no—
630C2×C4—1222no—
631C2×C4—1222no—
632C2×C4—1222no—
633C2×C4—1221no—
634C2×C4—1221no—
635C2×C4—1221no—
636C2×C4—1221no—
637C2×C4—1221no—
638C2×C4—1221no—
639C2×C4—1222no—
640C2×C4—4884no—
641C2×C4—2441no—
642C2×C4—2441no—
643C2×C4—2441no—
644C2×C4—2441no—
645C2×C4—2441no—
646C2×C4—2441no—
647C2×C4—2441no—
648C2×C4—2441no—
649C2×C4—2441no—
650C2×C4—2441no—
651C2×C4—2441no—
652C2×C4C2×D4×D6111yescharacteristic
653C2×C4#41×D6111yescharacteristic
654C2×C4#41×D6111yescharacteristic
655C2×C4C2×#1175822411yescharacteristic
656C2×C4#1451982211yescharacteristic
657C2×C4#1451982411yescharacteristic
658C8—22no—
659C8—41no—
660C8—1222no—
661C8—2441no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22—1222no—
677C22—1222no—
678C22—1222no—
679C22—2441no—
680C22—2441no—
681C22—2441no—
682C22—2441no—
683C22#123×D6111yescharacteristic
684C22#1684039411yescharacteristic
685C22#1684041211yescharacteristic
686C22#1684055811yescharacteristic
687C22#1684055911yescharacteristic
688C22#1684056311yescharacteristic
689C22#1684057111yescharacteristic
690C4—42no—
691C4—41no—
692C4—81no—
693C4—81no—
694C4—1222no—
695C4—4881no—
696C4—4881no—
697C4—4881no—
698C4—2442no—
699C4—2441no—
700C4—2441no—
701C4#1684027211yescharacteristic
702C4#1684035511yescharacteristic
703C4#1684042011yescharacteristic
704C4#1684045911yescharacteristic
705C2—81no—
706C2—161no—
707C2—2442no—
708C2#2144954711yescharacteristic
709C2#2145574711yescharacteristic
710C2#2145577011yescharacteristic
711C2#2145740811yescharacteristic
712C2#2145743011yescharacteristic
713C2#2145832511yescharacteristic
714C2#2145837411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2033C2
socleC23×C61C22×D4
prime radical#21449233C2
hypercenter#2033D61
hypocenterC61#26156
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. C2×#16840083C2semidirect
2. C2×#16839803C2semidirect
4. #21448741C2semidirect
5. #21447338C2semidirect
6. #21447303C2semidirect
7. #21447256C2semidirect
8. #21446448C2semidirect
9. #21446393C2semidirect
10. #21446254C2semidirect
11. #21446133C2semidirect
12. #21446113C2semidirect
13. #21446021C2semidirect
14. #21446020C2semidirect
15. C61×#2033C2semidirect
101. #14519710D4semidirect
355. #2033D61semidirect
418. C61#26156semidirect

Maximal subgroups

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

subgroupindex
C2×#168400832
C2×#168398032
#214492332
#214487412
#214473382
#214473032
#214472562
#214464482
#214463932
#214462542
#214461332
#214461132
#214460212
#214460202
C61×#20332
#2615661