Finite Group Databasebeta

← #29734310

Subgroups — #29734310

Order 15616.

1082 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×#16840293C211yesmaximal, characteristic
2C2×#16840163C211yesmaximal, characteristic, complemented
3C2×#16840115C211yesmaximal, characteristic, complemented
4#21454127C211yesmaximal, characteristic, complemented
5#21447023C211yesmaximal, characteristic, complemented
6#21447015C211yesmaximal, characteristic, complemented
7#21447005C211yesmaximal, characteristic
8#21446782C211yesmaximal, characteristic, complemented
9#21446629C211yesmaximal, characteristic, complemented
10#21446088C211yesmaximal, characteristic, complemented
11#21446078C211yesmaximal, characteristic, complemented
12#21446070C211yesmaximal, characteristic, complemented
13#21446068C211yesmaximal, characteristic, complemented
14C22⋊C8×Dic61C211yesmaximal, characteristic, complemented
15C61×#2033C211yesmaximal, characteristic
16#16840293—24no—
17#16840293—24no—
18C2×#14519770—21no—
19C2×#14519768C2211yescharacteristic
20C2×#14519768C2211yescharacteristic
21C2×#14519764C2211yescharacteristic
22C2×#14519758—21no—
23C2×#14519756C2211yescharacteristic
24C2×#14519742C2211yescharacteristic
25C2×D4×Dic61—21no—
26C2×D4×Dic61C2211yescharacteristic
27C2×#14519733—21no—
28C22×C4×Dic61C2211yescharacteristic
29#16840163—24no—
30#16840115—24no—
31C2×#14519711C2211yescharacteristic
32C2×#14519708C2211yescharacteristic
33C2×#14519708C2211yescharacteristic
34C2×#14519702C2211yescharacteristic
35C2×#14519702C2211yescharacteristic
36C2×#14519700C2211yescharacteristic
37C2×#14519700C2211yescharacteristic
38C2×#14519698C2211yescharacteristic
39C2×#14519698C2211yescharacteristic
40C2×C22⋊#92418C2211yescharacteristic
41C2×C8×Dic61C2211yescharacteristic
42C2×C8×Dic61C2211yescharacteristic
43#16839661C2211yescharacteristic
44#16839658C2211yescharacteristic
45#16839649—21no—
46#16839645—21no—
47#16839637—21no—
48#16839634—21no—
49#16839627C2211yescharacteristic
50#16839622C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839516C2211yescharacteristic
55#16839516C2211yescharacteristic
56C2×#121×C61C2211yescharacteristic
57C2×#107×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C2×#102×C61C2211yescharacteristic
60C2×C22⋊C8×C61C2211yescharacteristic
61C61×#380C2211yescharacteristic
62C61×#340C2211yescharacteristic
63C22×Dic122C2311yescharacteristic
64C23×Dic61—21no—
65C23×Dic61D411yescharacteristic
66C23×Dic61D411yescharacteristic
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519770—28no—
70#14519768—22no—
71#14519768—22no—
72#14519768—22no—
73#14519768D414yes—
74#14519764—22no—
75#14519764D414yes—
76#14519758—28no—
77#14519756—22no—
78#14519756—22no—
79#14519742—22no—
80#14519742—22no—
81D4×Dic61—28no—
82D4×Dic61—22no—
83D4×Dic61—22no—
84#14519733—28no—
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61—21no—
87C2×C4⋊Dic61—21no—
88C2×C4⋊Dic61C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—21no—
96C2×C22⋊Dic61—21no—
97C2×C22⋊Dic61—21no—
98C2×C22⋊Dic61—21no—
99C2×C22⋊Dic61—21no—
100C2×C22⋊Dic61—21no—
101C2×C4×Dic61—21no—
102C2×C4×Dic61—22no—
103C2×C4×Dic61—22no—
104C2×C4×Dic61D411yescharacteristic
105C2×C4×Dic61D411yescharacteristic
106C2×C4×Dic61C2311yescharacteristic
107C2×C4×Dic61C2311yescharacteristic
108#14519711—22no—
109#14519708—22no—
110#14519708—22no—
111#14519702—22no—
112#14519702—22no—
113#14519700—22no—
114#14519700—22no—
115#14519698—22no—
116#14519698—22no—
117C22⋊#92418—22no—
118C8×Dic61—22no—
119C8×Dic61—22no—
120#14519684—21no—
121#14519684—21no—
122#14519684—21no—
123#14519684—21no—
124#14519684—21no—
125C22×D4×C61—21no—
126C22×D4×C61C2311yescharacteristic
127C23×C4×C61C2311yescharacteristic
128C22×C8×C61C2311yescharacteristic
129C22×C8×C61C2311yescharacteristic
130#121×C61—22no—
131#121×C61—22no—
132C2×C4⋊C4×C61C2311yescharacteristic
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C4⋊C4×C61C2311yescharacteristic
135C2×C22⋊C4×C61—21no—
136C2×C22⋊C4×C61—21no—
137#107×C61—22no—
138#102×C61—22no—
139#102×C61—22no—
140C22⋊C8×C61—22no—
141#95×C61—21no—
142C24×C61—21no—
143C24×C61C2×D411yescharacteristic
144C2×D4×C61—24no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—42no—
151C2×D4×C61—42no—
152C2×D4×C61C2×D412yes—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61—21no—
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C22×C4×C61—21no—
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2411yescharacteristic
164C22×Dic61—24no—
165C22×Dic61—22no—
166C22×Dic61—21no—
167C22×Dic61—21no—
168C22×Dic61—22no—
169C22×Dic61—22no—
170C22×Dic61—22no—
171C22×Dic61—21no—
172C22×Dic61—21no—
173C22×Dic61—41no—
174C22×Dic61—41no—
175C22×Dic61—42no—
176C22×Dic61—42no—
177C22×Dic61C2×D411yescharacteristic
178C22×Dic61C2×D411yescharacteristic
179C22×Dic61C2×D411yescharacteristic
180C22×Dic61C2×D411yescharacteristic
181C2×Dic122—21no—
182C2×Dic122—21no—
183C2×Dic122—21no—
184C2×Dic122—21no—
185C2×Dic122—21no—
186C2×Dic122C2×D412yes—
187C2×C8×C61—21no—
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C2×C8×C61—21no—
191C2×C8×C61—21no—
192C2×C8×C61#4112yes—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61—21no—
197C4⋊C4×C61—21no—
198C4⋊C4×C61C2×D412yes—
199C22⋊C4×C61—24no—
200C22⋊C4×C61—42no—
201C22⋊Dic61—24no—
202C22⋊Dic61—24no—
203C22⋊Dic61—24no—
204C22⋊Dic61—24no—
205C22⋊Dic61—42no—
206C4⋊Dic61—21no—
207C4⋊Dic61—24no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—42no—
213C4⋊Dic61—42no—
214C4⋊Dic61C2×D412yes—
215#11758196—24no—
216#11758196—24no—
217#11758196—24no—
218#11758196—42no—
219#11758196—42no—
220C4×Dic61—24no—
221C4×Dic61—21no—
222C4×Dic61—21no—
223C4×Dic61—22no—
224C4×Dic61—22no—
225C4×Dic61C2×D412yes—
226C4×Dic61C2×D412yes—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418—21no—
231C2×#92418—21no—
232C2×#92418#4112yes—
233C23×C61—21no—
234C23×C61—21no—
235C23×C61—21no—
236C23×C61—21no—
237C23×C61—21no—
238C23×C61—21no—
239C23×C61—22no—
240C23×C61—41no—
241C23×C61—41no—
242C23×C61—41no—
243C23×C61—41no—
244C23×C61—41no—
245C23×C61—41no—
246C23×C61#13512yes—
247C23×C61C22×D411yescharacteristic
248D4×C61—22no—
249D4×C61—22no—
250D4×C61—48no—
251D4×C61—41no—
252D4×C61—41no—
253C2×C4×C61—21no—
254C2×C4×C61—22no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
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×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61C22×D411yescharacteristic
270C2×C4×C61C22×D411yescharacteristic
271C2×C4×C61C2×#4111yescharacteristic
272C2×C4×C61C2×#4111yescharacteristic
273C2×C4×C612−1+411yescharacteristic
274C2×C4×C612−1+411yescharacteristic
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—21no—
281C2×Dic61—21no—
282C2×Dic61—22no—
283C2×Dic61—22no—
284C2×Dic61—21no—
285C2×Dic61—21no—
286C2×Dic61—44no—
287C2×Dic61—44no—
288C2×Dic61—48no—
289C2×Dic61—41no—
290C2×Dic61—41no—
291C2×Dic61—41no—
292C2×Dic61—41no—
293C2×Dic61—41no—
294C2×Dic61—41no—
295C2×Dic61—41no—
296C2×Dic61—41no—
297C2×Dic61—41no—
298C2×Dic61—41no—
299C2×Dic61#12012yes—
300C2×Dic61#12012yes—
301C2×Dic61#13512yes—
302Dic122—22no—
303Dic122—22no—
304Dic122—41no—
305Dic122—41no—
306C8×C61—22no—
307C8×C61—41no—
308#92418—22no—
309#92418—41no—
310#25675—611nomaximal, Sylow
311C22×C61—22no—
312C22×C61—22no—
313C22×C61—22no—
314C22×C61—41no—
315C22×C61—41no—
316C22×C61—41no—
317C22×C61—41no—
318C22×C61—42no—
319C22×C61—42no—
320C22×C61—42no—
321C22×C61—81no—
322C22×C61—81no—
323C22×C61—81no—
324C22×C61—81no—
325C22×C61C2×#12011yescharacteristic
326C22×C61#53211yescharacteristic
327C22×C61#53611yescharacteristic
328C22×C61C2×#13511yescharacteristic
329C22×C61C2×#13511yescharacteristic
330C22×C61#57511yescharacteristic
331C22×C61#57511yescharacteristic
332C4×C61—41no—
333C4×C61—42no—
334C4×C61—81no—
335C4×C61—81no—
336C4×C61#54711yescharacteristic
337C4×C61#54711yescharacteristic
338C4×C61#54811yescharacteristic
339C4×C61#54811yescharacteristic
340Dic61—22no—
341Dic61—22no—
342Dic61—22no—
343Dic61—42no—
344Dic61—41no—
345Dic61—42no—
346Dic61—81no—
347Dic61—81no—
348C2×#529—611no—
349C2×#499—611no—
350C2×#488—611no—
351#2851—611no—
352#2036—611no—
353#2035—611no—
354#2034—611no—
355#2033D6111yescharacteristic
356#1991—611no—
357#1958—611no—
358#1865—611no—
359#1864—611no—
360#1863—611no—
361#1862—611no—
362C4×C22⋊C8—611no—
363C2×C61—42no—
364C2×C61—82no—
365C2×C61#243211yescharacteristic
366C2×C61#300211yescharacteristic
367C2×C61#300211yescharacteristic
368C2×C61#316811yescharacteristic
369C2×C61#318311yescharacteristic
370C2×C61#323211yescharacteristic
371C2×C61#323311yescharacteristic
372#529—1224no—
373#529—1224no—
374C2×#126—1221no—
375C2×#125—611no—
376C2×#125—611no—
377C2×#124—611no—
378C2×#123—1221no—
379C2×C22⋊Q8—611no—
380C2×#121C2×D6111yescharacteristic
381C2×C4×Q8—611no—
382C2×C4×D4—611no—
383C2×C4×D4—1221no—
384C2×#117—1221no—
385C22×C42—611no—
386#499—1224no—
387#488—1224no—
388C2×#107—611no—
389C2×#107C2×D6111yescharacteristic
390C2×#106—611no—
391C2×#106—611no—
392C2×#103—611no—
393C2×#103—611no—
394C2×#103—611no—
395C2×#103—611no—
396C2×#102—611no—
397C2×#102—611no—
398C2×#102C2×D6111yescharacteristic
399C2×#102C2×D6111yescharacteristic
400C2×C22⋊C8—611no—
401C2×C22⋊C8C2×D6111yescharacteristic
402C2×C4×C8—611no—
403C2×C4×C8—611no—
404#389—611no—
405#389—611no—
406#387—1221no—
407#387—1221no—
408#385—1221no—
409#385—1221no—
410#382—611no—
411#380—611no—
412#380C2×D6111yescharacteristic
413#340—611no—
414#340—611no—
415#340—611no—
416#340C2×D6111yescharacteristic
417#336—611no—
418#336—611no—
419C61#2567511yescharacteristic, complemented, Sylow
420C22×Q8—611no—
421C22×D4—21no—
422C22×D4C22×D6111yescharacteristic
423C23×C4—611no—
424C23×C4—611no—
425C23×C4—1221no—
426C23×C4C22×D6111yescharacteristic
427C22×C8—611no—
428C22×C8—611no—
429C22×C8C22×D6111yescharacteristic
430C22×C8C22×D6111yescharacteristic
431#126—1228no—
432#125—614no—
433#125—1222no—
434#125—1222no—
435#125—1222no—
436#124—614no—
437#124—1222no—
438#123—1228no—
439C22⋊Q8—1222no—
440C22⋊Q8—1222no—
441#121—22no—
442#121—22no—
443C4×Q8—1222no—
444C4×Q8—1222no—
445C4×D4—1222no—
446C4×D4—1228no—
447C4×D4—1222no—
448#117—1228no—
449C2×C4⋊C4—611no—
450C2×C4⋊C4—611no—
451C2×C4⋊C4—611no—
452C2×C4⋊C4—1221no—
453C2×C4⋊C4—1221no—
454C2×C4⋊C4—1221no—
455C2×C4⋊C4—1221no—
456C2×C4⋊C4—1221no—
457C2×C4⋊C4—1221no—
458C2×C4⋊C4—1221no—
459C2×C4⋊C4—1221no—
460C2×C4⋊C4C22×D6111yescharacteristic
461C2×C4⋊C4C22×D6111yescharacteristic
462C2×C4⋊C4C22×D6111yescharacteristic
463C2×C22⋊C4—21no—
464C2×C22⋊C4—21no—
465C2×C22⋊C4—1221no—
466C2×C22⋊C4—1221no—
467C2×C22⋊C4—1221no—
468C2×C22⋊C4—1221no—
469C2×C22⋊C4—1221no—
470C2×C42—611no—
471C2×C42—611no—
472C2×C42—611no—
473C2×C42—611no—
474C2×C42—1222no—
475C2×C42—1222no—
476C2×C42—1221no—
477#107—22no—
478#107—1222no—
479#106—1222no—
480#106—1222no—
481#103—1222no—
482#103—1222no—
483#103—1222no—
484#103—1222no—
485#102—22no—
486#102—22no—
487#102—1222no—
488#102—1222no—
489C22⋊C8—22no—
490C22⋊C8—1222no—
491C4×C8—1222no—
492C4×C8—1222no—
493#95—21no—
494#95—1221no—
495#95—1221no—
496#95—1221no—
497#95—1221no—
498#95—1221no—
499C24—21no—
500C24D4×D6111yescharacteristic
501C2×Q8—612no—
502C2×Q8—1221no—
503C2×Q8—1221no—
504C2×Q8—1221no—
505C2×Q8—1221no—
506C2×Q8—1221no—
507C2×D4—24no—
508C2×D4—21no—
509C2×D4—21no—
510C2×D4—21no—
511C2×D4—21no—
512C2×D4—21no—
513C2×D4—42no—
514C2×D4—42no—
515C2×D4D4×D6112yes—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—21no—
520C22×C4—21no—
521C22×C4—21no—
522C22×C4—21no—
523C22×C4—21no—
524C22×C4—21no—
525C22×C4—611no—
526C22×C4—611no—
527C22×C4—611no—
528C22×C4—611no—
529C22×C4—1222no—
530C22×C4—1222no—
531C22×C4—1222no—
532C22×C4—1224no—
533C22×C4—1221no—
534C22×C4—1221no—
535C22×C4—1222no—
536C22×C4—1221no—
537C22×C4—1221no—
538C22×C4—2442no—
539C22×C4—2441no—
540C22×C4—2441no—
541C22×C4—2442no—
542C22×C4D4×D6111yescharacteristic
543C22×C4C23×D6111yescharacteristic
544C2×C8—21no—
545C2×C8—21no—
546C2×C8—21no—
547C2×C8—21no—
548C2×C8—21no—
549C2×C8—612no—
550C2×C8—1221no—
551C2×C8—1221no—
552C2×C8—1221no—
553C2×C8—1221no—
554C2×C8—1221no—
555C2×C8#1175822412yes—
556C4⋊C4—21no—
557C4⋊C4—21no—
558C4⋊C4—21no—
559C4⋊C4—21no—
560C4⋊C4—21no—
561C4⋊C4—612no—
562C4⋊C4—1224no—
563C4⋊C4—1224no—
564C4⋊C4—1224no—
565C4⋊C4—1224no—
566C4⋊C4—1221no—
567C4⋊C4—1221no—
568C4⋊C4—1221no—
569C4⋊C4—1221no—
570C4⋊C4—1221no—
571C4⋊C4—2442no—
572C4⋊C4—2442no—
573C4⋊C4—2442no—
574C4⋊C4—2442no—
575C4⋊C4D4×D6112yes—
576C22⋊C4—24no—
577C22⋊C4—42no—
578C22⋊C4—1224no—
579C22⋊C4—1224no—
580C22⋊C4—1224no—
581C22⋊C4—1224no—
582C22⋊C4—2442no—
583C42—612no—
584C42—612no—
585C42—1222no—
586C42—1222no—
587C42—1224no—
588C42—1221no—
589C42—1221no—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—21no—
596C23—22no—
597C23—41no—
598C23—41no—
599C23—41no—
600C23—41no—
601C23—41no—
602C23—41no—
603C23#1451979612yes—
604C23C2×D4×D6111yescharacteristic
605Q8—1222no—
606Q8—1222no—
607Q8—2441no—
608Q8—2441no—
609D4—22no—
610D4—22no—
611D4—48no—
612D4—41no—
613D4—41no—
614C2×C4—21no—
615C2×C4—22no—
616C2×C4—21no—
617C2×C4—21no—
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—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—612no—
631C2×C4—612no—
632C2×C4—612no—
633C2×C4—1221no—
634C2×C4—1221no—
635C2×C4—1221no—
636C2×C4—1221no—
637C2×C4—1221no—
638C2×C4—1221no—
639C2×C4—1221no—
640C2×C4—1222no—
641C2×C4—1222no—
642C2×C4—1221no—
643C2×C4—1221no—
644C2×C4—2444no—
645C2×C4—2444no—
646C2×C4—2448no—
647C2×C4—2441no—
648C2×C4—2441no—
649C2×C4—2441no—
650C2×C4—2441no—
651C2×C4—2441no—
652C2×C4—2441no—
653C2×C4—2441no—
654C2×C4—2441no—
655C2×C4—2441no—
656C2×C4—2441no—
657C2×C4C2×D4×D6111yescharacteristic
658C2×C4C2×D4×D6111yescharacteristic
659C2×C4#41×D6111yescharacteristic
660C2×C4C2×#1175822411yescharacteristic
661C2×C4#1451982411yescharacteristic
662C2×C4#1451982411yescharacteristic
663C8—22no—
664C8—41no—
665C8—1222no—
666C8—2441no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—81no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22#120×D6111yescharacteristic
682C22#1684030711yescharacteristic
683C22#1684033411yescharacteristic
684C22#135×D6111yescharacteristic
685C22C2×#1451979611yescharacteristic
686C22#1684055811yescharacteristic
687C22#1684056111yescharacteristic
688C4—42no—
689C4—41no—
690C4—81no—
691C4—81no—
692C4—1222no—
693C4—1222no—
694C4—1222no—
695C4—4881no—
696C4—4881no—
697C4—2441no—
698C4—2442no—
699C4—2442no—
700C4#1684041011yescharacteristic
701C4#1684041211yescharacteristic
702C4#1684041811yescharacteristic
703C4#1684042011yescharacteristic
704C2—42no—
705C2—82no—
706C2#2144944211yescharacteristic
707C2#2145503211yescharacteristic
708C2#2145504011yescharacteristic
709C2#2145638411yescharacteristic
710C2#2145654111yescharacteristic
711C2#2145698411yescharacteristic
712C2#2145699511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2033C2
socleC23×C61C22×D4
prime radicalC2×#121×C61C22
hypercenter#2033D61
hypocenterC61#25675
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
2. C2×#16840163C2semidirect
3. C2×#16840115C2semidirect
4. #21454127C2semidirect
5. #21447023C2semidirect
6. #21447015C2semidirect
8. #21446782C2semidirect
9. #21446629C2semidirect
10. #21446088C2semidirect
11. #21446078C2semidirect
12. #21446070C2semidirect
13. #21446068C2semidirect
14. C22⋊C8×Dic61C2semidirect
419. C61#25675semidirect

Maximal subgroups

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

subgroupindex
C2×#168402932
C2×#168401632
C2×#168401152
#214541272
#214470232
#214470152
#214470052
#214467822
#214466292
#214460882
#214460782
#214460702
#214460682
C22⋊C8×Dic612
C61×#20332
#2567561