Finite Group Databasebeta

← #29730169

Subgroups — #29730169

Order 15616.

914 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840064C211yesmaximal, characteristic, complemented
2C2×#16839825C211yesmaximal, characteristic, complemented
3#21450984C211yesmaximal, characteristic, complemented
4#21450718C211yesmaximal, characteristic
5#21447252C211yesmaximal, characteristic, complemented
6#21447251C211yesmaximal, characteristic, complemented
7#21446999C211yesmaximal, characteristic, complemented
8#21446782C211yesmaximal, characteristic, complemented
9#21446588C211yesmaximal, characteristic, complemented
10#21446248C211yesmaximal, characteristic, complemented
11#21446244C211yesmaximal, characteristic, complemented
12#21445846C211yesmaximal, characteristic, complemented
13#21445841C211yesmaximal, characteristic, complemented
14#102×Dic61C211yesmaximal, characteristic, complemented
15C61×#2034C211yesmaximal, characteristic, complemented
16C23×#92418C2211yescharacteristic
17C22×C8×D61C2211yescharacteristic
18C2×#14519768C2211yescharacteristic
19C2×#14519748C2211yescharacteristic
20C2×D4×Dic61C2211yescharacteristic
21C2×C4×C22⋊D61C2211yescharacteristic
22#16840064—24no—
23#16839825—24no—
24C2×#14519710C2211yescharacteristic
25C2×#14519710C2211yescharacteristic
26C2×#14519709C2211yescharacteristic
27C2×#14519706C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519698C2211yescharacteristic
30C2×#14519689C2211yescharacteristic
31C2×C8×Dic61C2211yescharacteristic
32#16839704—21no—
33#16839690—21no—
34#16839678—21no—
35#16839670—21no—
36#16839651—21no—
37#16839650—21no—
38#16839642—21no—
39#16839639—21no—
40#16839627C2211yescharacteristic
41#16839622C2211yescharacteristic
42#16839622C2211yescharacteristic
43#16839619C2211yescharacteristic
44#16839619C2211yescharacteristic
45#16839619C2211yescharacteristic
46C4⋊C4×Dic61C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54C2×#121×C61C2211yescharacteristic
55C2×#106×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C2×C22⋊C8×C61C2211yescharacteristic
59C61×#380C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×C22⋊D61—21no—
62C22×C22⋊D61D411yescharacteristic
63C23×Dic61—21no—
64C22×C4×D61C2311yescharacteristic
65C22×#92418—22no—
66C22×#92418—22no—
67C22×#92418—41no—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C2×C8×D61—22no—
71C2×C8×D61—22no—
72C2×C8×D61—41no—
73#14519768—22no—
74#14519768—22no—
75#14519748—22no—
76#14519748—41no—
77D4×Dic61—22no—
78D4×Dic61—22no—
79C4×C22⋊D61—22no—
80C4×C22⋊D61—22no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×#11758196—21no—
87C2×#11758196D411yescharacteristic
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×#11758198—21no—
96C2×#11758198D411yescharacteristic
97C2×C4×Dic61—21no—
98C2×C4×Dic61C2311yescharacteristic
99#14519710—22no—
100#14519710—41no—
101#14519709—22no—
102#14519706D414yescomplemented
103#14519698—22no—
104#14519698—22no—
105#14519689—22no—
106C8×Dic61—22no—
107#14519684—21no—
108#14519684—21no—
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#106×C61—41no—
128#102×C61—22no—
129#102×C61—22no—
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—42no—
141C2×D4×C61—42no—
142C2×D4×C61—81no—
143C2×D4×C61#4112yes—
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—22no—
157C2×C22⋊D61—22no—
158C2×C22⋊D61—42no—
159C2×C22⋊D61—42no—
160C2×C22⋊D61—81no—
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61—21no—
165C22×Dic61—42no—
166C22×Dic61—81no—
167C22×Dic61—81no—
168C22×Dic61C2×D411yescharacteristic
169C22×Dic61C2×D411yescharacteristic
170C22×Dic61C2×D411yescharacteristic
171C2×C4×D61—21no—
172C2×C4×D61—21no—
173C2×C4×D61—21no—
174C2×C4×D61—21no—
175C2×C4×D61—21no—
176C2×C4×D61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61C2×D412yes—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C22⋊C4×C61—42no—
190C22⋊C4×C61—81no—
191C22⋊Dic61—22no—
192C22⋊Dic61—42no—
193C22⋊Dic61—42no—
194C22⋊Dic61—42no—
195C22⋊Dic61—81no—
196#11758198—22no—
197#11758198—42no—
198#11758198—81no—
199#11758198—81no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—42no—
209#11758196—22no—
210#11758196—42no—
211C4×Dic61—81no—
212C4×Dic61C2×D412yes—
213C4×Dic61#4112yes—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—44no—
219C2×#92418C2×D412yes—
220C2×#92418#4112yes—
221C8×D61—44no—
222C23×C61—21no—
223C23×C61—21no—
224C23×C61—21no—
225C23×C61—21no—
226C23×C61—21no—
227C23×C61—21no—
228C23×C61—22no—
229C23×C61—41no—
230C23×C61—41no—
231C23×C61—41no—
232C23×C61—41no—
233C23×C61—41no—
234C23×C61—41no—
235C23×C61#13512yes—
236C23×C61C22×D411yescharacteristic
237C22×D61—21no—
238C22×D61—21no—
239C22×D61—21no—
240C22×D61—21no—
241C22×D61—21no—
242C22×D61—21no—
243C22×D61#13512yes—
244D4×C61—22no—
245D4×C61—22no—
246D4×C61—41no—
247D4×C61—41no—
248D4×C61—84no—
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—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C2×C4×C61C2×#4111yescharacteristic
269C2×C4×C61H(5,𝔽2)11yescharacteristic
270C2×C4×C612−1+411yescharacteristic
271C22⋊D61—28no—
272C22⋊D61—84no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
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—84no—
293C2×Dic61#12112yes—
294C2×Dic61#12112yes—
295C4×D61—41no—
296C4×D61—41no—
297C4×D61—41no—
298C4×D61—41no—
299C8×C61—22no—
300C8×C61—41no—
301#92418—22no—
302#92418—22no—
303#25342—611nomaximal, Sylow
304C22×C61—22no—
305C22×C61—22no—
306C22×C61—22no—
307C22×C61—41no—
308C22×C61—41no—
309C22×C61—41no—
310C22×C61—41no—
311C22×C61—42no—
312C22×C61—42no—
313C22×C61—42no—
314C22×C61—81no—
315C22×C61—81no—
316C22×C61—81no—
317C22×C61—81no—
318C22×C61C2×#12111yescharacteristic
319C22×C61#54611yescharacteristic
320C22×C61#54711yescharacteristic
321C22×C61C2×#13511yescharacteristic
322C22×C61C2×#13511yescharacteristic
323C22×C61#57611yescharacteristic
324C22×C61#57711yescharacteristic
325C2×D61—22no—
326C2×D61—22no—
327C2×D61—22no—
328C2×D61—41no—
329C2×D61—41no—
330C2×D61—41no—
331C2×D61—41no—
332C4×C61—41no—
333C4×C61—81no—
334C4×C61—81no—
335C4×C61—81no—
336C4×C61#52911yescharacteristic
337C4×C61#54211yescharacteristic
338C4×C61#54611yescharacteristic
339C4×C61#54811yescharacteristic
340Dic61—22no—
341Dic61—22no—
342Dic61—41no—
343Dic61—81no—
344Dic61—81no—
345Dic61—81no—
346Dic61—81no—
347C2×#478—611no—
348C2×C8×D4—611no—
349#2562—611no—
350#2535—611no—
351#2077—611no—
352#2076—611no—
353#2034D6111yescharacteristic, complemented
354#2033—611no—
355#1991—611no—
356#1949—611no—
357#1893—611no—
358#1893—611no—
359#1808—611no—
360#1807—611no—
361C4×#102—611no—
362C2×C61—42no—
363C2×C61—82no—
364C2×C61#242711yescharacteristic
365C2×C61#305011yescharacteristic
366C2×C61#306211yescharacteristic
367C2×C61#329611yescharacteristic
368C2×C61#329711yescharacteristic
369C2×C61#333711yescharacteristic
370C2×C61#333911yescharacteristic
371D61—42no—
372C23×C8—611no—
373C23×C8—611no—
374C2×#125—611no—
375C2×#121—611no—
376C2×#121C2×D6111yescharacteristic
377C2×C4×D4—611no—
378C2×C4×D4—611no—
379#478—1224no—
380C8×D4—1224no—
381C2×#107—611no—
382C2×#107—611no—
383C2×#106—611no—
384C2×#106C2×D6111yescharacteristic
385C2×C4⋊C8—611no—
386C2×#102—611no—
387C2×#102—611no—
388C2×#102C2×D6111yescharacteristic
389C2×#102C2×D6111yescharacteristic
390C2×C22⋊C8—611no—
391C2×C22⋊C8C2×D6111yescharacteristic
392C2×C4×C8—611no—
393#400—1221no—
394#397—1221no—
395#394—1221no—
396#392—1221no—
397#388—1221no—
398#388—1221no—
399#386—1221no—
400#386—1221no—
401#382—611no—
402#380—611no—
403#380—611no—
404#380—611no—
405#380—611no—
406#380—611no—
407#380C2×D6111yescharacteristic
408C4×C4⋊C4—611no—
409#340—611no—
410#340—611no—
411#340—611no—
412#340—611no—
413#340—611no—
414#340—611no—
415#340—611no—
416#340C2×D6111yescharacteristic
417C61#2534211yescharacteristic, complemented, Sylow
418C22×D4—21no—
419C22×D4—611no—
420C22×D4—1221no—
421C22×D4C22×D6111yescharacteristic
422C23×C4—611no—
423C23×C4—1221no—
424C23×C4C22×D6111yescharacteristic
425C22×C8—611no—
426C22×C8—611no—
427C22×C8—1222no—
428C22×C8—1222no—
429C22×C8—1222no—
430C22×C8—1222no—
431C22×C8—2441no—
432C22×C8—2441no—
433C22×C8C22×D6111yescharacteristic
434C22×C8C22×D6111yescharacteristic
435#125—1222no—
436#125—1222no—
437#121—22no—
438#121—41no—
439#121—1222no—
440#121—2441no—
441C4×D4—1222no—
442C4×D4—1222no—
443C4×D4—1222no—
444C4×D4—1222no—
445C2×C4⋊C4—611no—
446C2×C4⋊C4—611no—
447C2×C4⋊C4—611no—
448C2×C4⋊C4—611no—
449C2×C4⋊C4—611no—
450C2×C4⋊C4—1221no—
451C2×C4⋊C4—1221no—
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—
468#107—1222no—
469#107—2441no—
470#106—41no—
471#106—1222no—
472C4⋊C8—614no—
473#102—22no—
474#102—22no—
475#102—1222no—
476#102—1222no—
477C22⋊C8—22no—
478C22⋊C8—1222no—
479C4×C8—1222no—
480#95—21no—
481#95—1221no—
482#95—1221no—
483#95—1221no—
484#95—1221no—
485#95—1221no—
486#95—1221no—
487#95—1221no—
488#95—1221no—
489C24—21no—
490C24—611no—
491C24D4×D6111yescharacteristic
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—42no—
498C2×D4—42no—
499C2×D4—81no—
500C2×D4—1222no—
501C2×D4—1222no—
502C2×D4—1222no—
503C2×D4—4881no—
504C2×D4—2442no—
505C2×D4—2442no—
506C2×D4#1175822412yes—
507C22×C4—21no—
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—611no—
517C22×C4—611no—
518C22×C4—611no—
519C22×C4—1221no—
520C22×C4—1221no—
521C22×C4—1221no—
522C22×C4—1221no—
523C22×C4—1221no—
524C22×C4—1221no—
525C22×C4—1221no—
526C22×C4—1221no—
527C22×C4—1221no—
528C22×C4—1221no—
529C22×C4—4881no—
530C22×C4—4881no—
531C22×C4—2442no—
532C22×C4D4×D6111yescharacteristic
533C22×C4C23×D6111yescharacteristic
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—612no—
540C2×C8—612no—
541C2×C8—1221no—
542C2×C8—1221no—
543C2×C8—1221no—
544C2×C8—1221no—
545C2×C8—2444no—
546C2×C8—2444no—
547C2×C8D4×D6112yes—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—21no—
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—1221no—
563C4⋊C4—2442no—
564C4⋊C4—2442no—
565C22⋊C4—42no—
566C22⋊C4—81no—
567C22⋊C4—1222no—
568C22⋊C4—1222no—
569C22⋊C4—4881no—
570C22⋊C4—4881no—
571C22⋊C4—4881no—
572C22⋊C4—2442no—
573C22⋊C4—2442no—
574C22⋊C4—2442no—
575C22⋊C4—2442no—
576C42—612no—
577C42—612no—
578C42—4881no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—22no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23—612no—
593C23—1221no—
594C23—1221no—
595C23—1221no—
596C23—1221no—
597C23—1221no—
598C23—1221no—
599C23#1451979812yes—
600C23C2×D4×D6111yescharacteristic
601D4—22no—
602D4—22no—
603D4—41no—
604D4—41no—
605D4—84no—
606D4—1228no—
607D4—4884no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
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—612no—
625C2×C4—612no—
626C2×C4—1221no—
627C2×C4—1221no—
628C2×C4—1221no—
629C2×C4—1221no—
630C2×C4—1221no—
631C2×C4—1221no—
632C2×C4—1221no—
633C2×C4—4884no—
634C2×C4—2441no—
635C2×C4—2441no—
636C2×C4—2441no—
637C2×C4—2441no—
638C2×C4—2441no—
639C2×C4—2441no—
640C2×C4—2441no—
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×C4C2×D4×D6111yescharacteristic
651C2×C4#41×D6111yescharacteristic
652C2×C4#41×D6111yescharacteristic
653C2×C4C2×#1175822411yescharacteristic
654C2×C4#1451982211yescharacteristic
655C2×C4#1451982411yescharacteristic
656C8—22no—
657C8—41no—
658C8—1222no—
659C8—1222no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22—1222no—
675C22—1222no—
676C22—1222no—
677C22—2441no—
678C22—2441no—
679C22—2441no—
680C22—2441no—
681C22#121×D6111yescharacteristic
682C22#1684039911yescharacteristic
683C22#1684041011yescharacteristic
684C22#135×D6111yescharacteristic
685C22C2×#1451979811yescharacteristic
686C22#1684056611yescharacteristic
687C22#1684057211yescharacteristic
688C4—41no—
689C4—81no—
690C4—81no—
691C4—81no—
692C4—1222no—
693C4—1222no—
694C4—4881no—
695C4—4881no—
696C4—4881no—
697C4—4881no—
698C4—2441no—
699C4#1684028911yescharacteristic
700C4#1684037611yescharacteristic
701C4#1684040111yescharacteristic
702C4#1684041811yescharacteristic
703C2—42no—
704C2—82no—
705C2—2442no—
706C2#2144938911yescharacteristic
707C2#2145539511yescharacteristic
708C2#2145549611yescharacteristic
709C2#2145765811yescharacteristic
710C2#2145766211yescharacteristic
711C2#2145819111yescharacteristic
712C2#2145821911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2034C2
socleC23×C61C22×D4
prime radical#21450718C2
hypercenter#2034D61
hypocenterC61#25342
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2034

Complements

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

NH ≅ G/Ntype
1. C2×#16840064C2semidirect
2. C2×#16839825C2semidirect
3. #21450984C2semidirect
5. #21447252C2semidirect
6. #21447251C2semidirect
7. #21446999C2semidirect
8. #21446782C2semidirect
9. #21446588C2semidirect
10. #21446248C2semidirect
11. #21446244C2semidirect
12. #21445846C2semidirect
13. #21445841C2semidirect
14. #102×Dic61C2semidirect
15. C61×#2034C2semidirect
102. #14519706D4semidirect
353. #2034D61semidirect
417. C61#25342semidirect

Maximal subgroups

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

subgroupindex
C2×#168400642
C2×#168398252
#214509842
#214507182
#214472522
#214472512
#214469992
#214467822
#214465882
#214462482
#214462442
#214458462
#214458412
#102×Dic612
C61×#20342
#2534261