Finite Group Databasebeta

← #29734319

Subgroups — #29734319

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×#16840291C211yesmaximal, characteristic
2C2×#16840164C211yesmaximal, characteristic, complemented
3C2×#16840116C211yesmaximal, characteristic, complemented
4#21454130C211yesmaximal, characteristic, complemented
5#21447021C211yesmaximal, characteristic, complemented
6#21447014C211yesmaximal, characteristic, complemented
7#21447007C211yesmaximal, characteristic
8#21447000C211yesmaximal, characteristic
9#21446783C211yesmaximal, characteristic, complemented
10#21446089C211yesmaximal, characteristic, complemented
11#21446082C211yesmaximal, characteristic, complemented
12#21446075C211yesmaximal, characteristic, complemented
13#21446068C211yesmaximal, characteristic, complemented
14C4×#14519689C211yesmaximal, characteristic, complemented
15C61×#1958C211yesmaximal, characteristic, complemented
16#16840291—24no—
17#16840291—24no—
18C2×#14519770—21no—
19C2×#14519769C2211yescharacteristic
20C2×#14519769C2211yescharacteristic
21C2×#14519759—21no—
22C2×#14519753C2211yescharacteristic
23C2×#14519748C2211yescharacteristic
24C2×Q8×Dic61C2211yescharacteristic
25C2×D4×Dic61C2211yescharacteristic
26C2×#14519737—21no—
27C2×#14519733—21no—
28C2×C42×D61C2211yescharacteristic
29#16840164—24no—
30#16840116—24no—
31C2×#14519711C2211yescharacteristic
32C2×#14519711C2211yescharacteristic
33C2×#14519709C2211yescharacteristic
34C2×#14519709C2211yescharacteristic
35C2×#14519701C2211yescharacteristic
36C2×#14519701C2211yescharacteristic
37C2×#14519698C2211yescharacteristic
38C2×#14519698C2211yescharacteristic
39C2×#14519689C2211yescharacteristic
40C2×#14519689C2211yescharacteristic
41C2×C4×#92418C2211yescharacteristic
42#16839661C2211yescharacteristic
43#16839661C2211yescharacteristic
44#16839647—21no—
45#16839647—21no—
46#16839636—21no—
47#16839636—21no—
48#16839619C2211yescharacteristic
49#16839619C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839516C2211yescharacteristic
55#16839516C2211yescharacteristic
56C2×#124×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C2×#102×C61C2211yescharacteristic
60C2×#102×C61C2211yescharacteristic
61C2×C4×C8×C61C2211yescharacteristic
62C61×#382C2211yescharacteristic
63C22×C22⋊D61—21no—
64C23×Dic61—21no—
65C22×C4×D61D411yescharacteristic
66C22×C4×D61D411yescharacteristic
67C22×C4×D61C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70#14519770—28no—
71#14519769—22no—
72#14519769—22no—
73#14519769—22no—
74#14519769D414yes—
75#14519759—28no—
76#14519753—22no—
77#14519753—22no—
78#14519748—22no—
79#14519748—22no—
80Q8×Dic61—22no—
81Q8×Dic61—22no—
82D4×Dic61—22no—
83D4×Dic61—22no—
84#14519737—28no—
85#14519733—28no—
86C2×C4⋊Dic61—21no—
87C2×C4⋊Dic61—21no—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×C4⋊Dic61C2311yescharacteristic
92C2×C4⋊Dic61C2311yescharacteristic
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—21no—
96C2×#11758196—21no—
97C2×#11758196—21no—
98C2×C22⋊Dic61—21no—
99C2×C22⋊Dic61—21no—
100C2×#11758198—21no—
101C2×#11758198—21no—
102C2×#11758198—21no—
103C2×C4×Dic61—21no—
104C2×C4×Dic61D411yescharacteristic
105C2×C4×Dic61D411yescharacteristic
106C2×C4×Dic61C2311yescharacteristic
107C42×D61—22no—
108C42×D61—22no—
109#14519711—22no—
110#14519711—22no—
111#14519709—22no—
112#14519709—22no—
113#14519701—22no—
114#14519701—22no—
115#14519698—22no—
116#14519698—22no—
117#14519689—22no—
118#14519689—22no—
119C4×#92418—22no—
120#14519684—21no—
121#14519684—21no—
122#14519684—21no—
123#14519684—21no—
124C22×Q8×C61C2311yescharacteristic
125C22×D4×C61C2311yescharacteristic
126C22×C8×C61C2311yescharacteristic
127C22×C8×C61C2311yescharacteristic
128#124×C61—22no—
129#124×C61D414yes—
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C4⋊C4×C61C2311yescharacteristic
132C2×C22⋊C4×C61—21no—
133C2×C22⋊C4×C61—21no—
134C2×C42×C61C2311yescharacteristic
135#103×C61—22no—
136#103×C61—22no—
137#102×C61—22no—
138#102×C61—22no—
139C4×C8×C61—22no—
140#95×C61—21no—
141#95×C61—21no—
142C24×C61—21no—
143C23×D61C2×D411yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×Q8×C61C2×D412yes—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—21no—
155C2×D4×C61C2×D412yes—
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61—21no—
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2411yescharacteristic
162C2×C22⋊D61—24no—
163C2×C22⋊D61—42no—
164C2×C22⋊D61—42no—
165C22×Dic61—24no—
166C22×Dic61—21no—
167C22×Dic61—21no—
168C22×Dic61—21no—
169C22×Dic61—21no—
170C22×Dic61—42no—
171C22×Dic61—42no—
172C22×Dic61C2×D411yescharacteristic
173C22×Dic61C2×D411yescharacteristic
174C22×Dic61C2×D411yescharacteristic
175C2×C4×D61—22no—
176C2×C4×D61—22no—
177C2×C4×D61—22no—
178C2×C4×D61—21no—
179C2×C4×D61—21no—
180C2×C4×D61—21no—
181C2×C4×D61—21no—
182C2×C4×D61—21no—
183C2×C4×D61—21no—
184C2×C4×D61—22no—
185C2×C4×D61—41no—
186C2×C4×D61—41no—
187C2×C8×C61—21no—
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C2×C8×C61—21no—
191C2×C8×C61#4112yes—
192C2×C8×C61#4112yes—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61—21no—
197C22⋊C4×C61—24no—
198C22⋊C4×C61—24no—
199C42×C61C2×D412yes—
200C42×C61C2×D412yes—
201C22⋊Dic61—24no—
202C22⋊Dic61—24no—
203#11758198—24no—
204#11758198—42no—
205#11758198—42no—
206C4⋊Dic61—24no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—42no—
214C4⋊Dic61—42no—
215C4⋊Dic61C2×D412yes—
216C4⋊Dic61C2×D412yes—
217#11758196—24no—
218#11758196—24no—
219#11758196—24no—
220#11758196—42no—
221#11758196—42no—
222C4×Dic61—24no—
223C4×Dic61—22no—
224C4×Dic61—22no—
225C4×Dic61—21no—
226C4×Dic61—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418—21no—
231C2×#92418—21no—
232C2×#92418—21no—
233C23×C61—22no—
234C23×C61—41no—
235C23×C61—41no—
236C23×C61—41no—
237C23×C61—41no—
238C23×C61—41no—
239C23×C61—41no—
240C23×C61C22×D411yescharacteristic
241C22×D61—21no—
242C22×D61—21no—
243C22×D61—21no—
244C22×D61—21no—
245C22×D61—21no—
246C22×D61—21no—
247C22×D61#13512yes—
248Q8×C61—22no—
249Q8×C61—22no—
250Q8×C61—41no—
251Q8×C61—41no—
252D4×C61—22no—
253D4×C61—22no—
254D4×C61—41no—
255D4×C61—41no—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—21no—
259C2×C4×C61—21no—
260C2×C4×C61—22no—
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×C61#12012yes—
270C2×C4×C61#12012yes—
271C2×C4×C61C22×D411yescharacteristic
272C2×C4×C61C22×D411yescharacteristic
273C2×C4×C61C2×#4111yescharacteristic
274C2×C4×C61C2×#4111yescharacteristic
275C2×C4×C612−1+411yescharacteristic
276C2×C4×C612−1+411yescharacteristic
277C22⋊D61—48no—
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—21no—
287C2×Dic61—21no—
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#13512yes—
300C4×D61—44no—
301C4×D61—44no—
302C4×D61—41no—
303C4×D61—41no—
304C4×D61—41no—
305C4×D61—41no—
306C8×C61—22no—
307C8×C61—22no—
308#92418—41no—
309#92418—41no—
310#25675—611nomaximal, Sylow
311C22×C61—42no—
312C22×C61—42no—
313C22×C61—42no—
314C22×C61—81no—
315C22×C61—81no—
316C22×C61—81no—
317C22×C61—81no—
318C22×C61C2×#12011yescharacteristic
319C22×C61#53211yescharacteristic
320C22×C61#53611yescharacteristic
321C22×C61C2×#13511yescharacteristic
322C22×C61C2×#13511yescharacteristic
323C22×C61#57511yescharacteristic
324C22×C61#57511yescharacteristic
325C2×D61—22no—
326C2×D61—22no—
327C2×D61—22no—
328C2×D61—41no—
329C2×D61—41no—
330C2×D61—41no—
331C2×D61—41no—
332C4×C61—22no—
333C4×C61—22no—
334C4×C61—42no—
335C4×C61—81no—
336C4×C61—81no—
337C4×C61#54711yescharacteristic
338C4×C61#54711yescharacteristic
339C4×C61#54811yescharacteristic
340C4×C61#54811yescharacteristic
341Dic61—22no—
342Dic61—41no—
343Dic61—41no—
344Dic61—42no—
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#2033—611no—
356#1991—611no—
357#1958D6111yescharacteristic, complemented
358#1865—611no—
359#1864—611no—
360#1863—611no—
361#1862—611no—
362C4×C22⋊C8—611no—
363C2×C61—82no—
364C2×C61#243211yescharacteristic
365C2×C61#300211yescharacteristic
366C2×C61#300211yescharacteristic
367C2×C61#316811yescharacteristic
368C2×C61#318311yescharacteristic
369C2×C61#323211yescharacteristic
370C2×C61#323311yescharacteristic
371D61—42no—
372#529—1224no—
373#529—1224no—
374C2×#126—1221no—
375C2×#125—611no—
376C2×#125—611no—
377C2×#124C2×D6111yescharacteristic
378C2×#123—1221no—
379C2×C22⋊Q8—611no—
380C2×#121—611no—
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×#107—611no—
390C2×#106—611no—
391C2×#106—611no—
392C2×#103—611no—
393C2×#103—611no—
394C2×#103C2×D6111yescharacteristic
395C2×#103C2×D6111yescharacteristic
396C2×#102—611no—
397C2×#102—611no—
398C2×#102C2×D6111yescharacteristic
399C2×#102C2×D6111yescharacteristic
400C2×C22⋊C8—611no—
401C2×C22⋊C8—611no—
402C2×C4×C8—611no—
403C2×C4×C8C2×D6111yescharacteristic
404#389—611no—
405#389—611no—
406#387—1221no—
407#387—1221no—
408#385—1221no—
409#385—1221no—
410#382C2×D6111yescharacteristic
411#380—611no—
412#380—611no—
413#340—611no—
414#340—611no—
415#340—611no—
416#340—611no—
417#336—611no—
418#336—611no—
419C61#2567511yescharacteristic, complemented, Sylow
420C22×Q8C22×D6111yescharacteristic
421C22×D4—1221no—
422C22×D4C22×D6111yescharacteristic
423C23×C4—611no—
424C23×C4—611no—
425C23×C4—611no—
426C23×C4—1221no—
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—22no—
437#124D24414yes—
438#123—1228no—
439C22⋊Q8—1222no—
440C22⋊Q8—1222no—
441#121—1222no—
442#121—1222no—
443C4×Q8—1222no—
444C4×Q8—1222no—
445C4×D4—1228no—
446C4×D4—1222no—
447C4×D4—1222no—
448#117—1228no—
449C2×C4⋊C4—611no—
450C2×C4⋊C4—611no—
451C2×C4⋊C4—611no—
452C2×C4⋊C4—611no—
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⋊C4—1221no—
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—1221no—
474C2×C42—1222no—
475C2×C42—1222no—
476C2×C42C22×D6111yescharacteristic
477#107—1222no—
478#107—1222no—
479#106—1222no—
480#106—1222no—
481#103—22no—
482#103—22no—
483#103—1222no—
484#103—1222no—
485#102—22no—
486#102—22no—
487#102—1222no—
488#102—1222no—
489C22⋊C8—1222no—
490C22⋊C8—1222no—
491C4×C8—22no—
492C4×C8—1222no—
493#95—21no—
494#95—21no—
495#95—1221no—
496#95—1221no—
497#95—1221no—
498#95—1221no—
499C24—21no—
500C24—611no—
501C2×Q8—21no—
502C2×Q8—21no—
503C2×Q8—21no—
504C2×Q8—21no—
505C2×Q8—21no—
506C2×Q8C2×D24412yes—
507C2×D4—21no—
508C2×D4—21no—
509C2×D4—21no—
510C2×D4—21no—
511C2×D4—21no—
512C2×D4—1224no—
513C2×D4—2442no—
514C2×D4—2442no—
515C2×D4C2×D24412yes—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—611no—
520C22×C4—611no—
521C22×C4—611no—
522C22×C4—1222no—
523C22×C4—1222no—
524C22×C4—1224no—
525C22×C4—1221no—
526C22×C4—1221no—
527C22×C4—1221no—
528C22×C4—1221no—
529C22×C4—1221no—
530C22×C4—1221no—
531C22×C4—1221no—
532C22×C4—1221no—
533C22×C4—1222no—
534C22×C4—1222no—
535C22×C4—1221no—
536C22×C4—1221no—
537C22×C4—2441no—
538C22×C4—2441no—
539C22×C4—2442no—
540C22×C4—2442no—
541C22×C4D4×D6111yescharacteristic
542C22×C4D4×D6111yescharacteristic
543C22×C4C23×D6111yescharacteristic
544C2×C8—21no—
545C2×C8—21no—
546C2×C8—21no—
547C2×C8—21no—
548C2×C8—1221no—
549C2×C8—1221no—
550C2×C8—1221no—
551C2×C8—1221no—
552C2×C8—1221no—
553C2×C8—1221no—
554C2×C8#1175822412yes—
555C2×C8#1175822412yes—
556C4⋊C4—21no—
557C4⋊C4—21no—
558C4⋊C4—21no—
559C4⋊C4—21no—
560C4⋊C4—612no—
561C4⋊C4—612no—
562C4⋊C4—1224no—
563C4⋊C4—1224no—
564C4⋊C4—1221no—
565C4⋊C4—1221no—
566C4⋊C4—1224no—
567C4⋊C4—1224no—
568C4⋊C4—1221no—
569C4⋊C4—1221no—
570C4⋊C4—1221no—
571C4⋊C4—1221no—
572C4⋊C4—2442no—
573C4⋊C4—2442no—
574C4⋊C4—2442no—
575C4⋊C4—2442no—
576C22⋊C4—24no—
577C22⋊C4—24no—
578C22⋊C4—1224no—
579C22⋊C4—1224no—
580C22⋊C4—1224no—
581C22⋊C4—2442no—
582C22⋊C4—2442no—
583C42—1224no—
584C42—1221no—
585C42—1221no—
586C42—1222no—
587C42—1222no—
588C42C2×D24412yes—
589C42D4×D6112yes—
590C23—22no—
591C23—41no—
592C23—41no—
593C23—41no—
594C23—41no—
595C23—41no—
596C23—41no—
597C23—612no—
598C23—1221no—
599C23—1221no—
600C23—1221no—
601C23—1221no—
602C23—1221no—
603C23—1221no—
604C23C2×D4×D6111yescharacteristic
605Q8—22no—
606Q8—22no—
607Q8—41no—
608Q8—41no—
609D4—22no—
610D4—22no—
611D4—41no—
612D4—41no—
613D4—2448no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—22no—
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—612no—
628C2×C4—1221no—
629C2×C4—1221no—
630C2×C4—1221no—
631C2×C4—1221no—
632C2×C4—1222no—
633C2×C4—1222no—
634C2×C4—1221no—
635C2×C4—1221no—
636C2×C4—1221no—
637C2×C4—1221no—
638C2×C4—2444no—
639C2×C4—2444no—
640C2×C4—2448no—
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×C4—2441no—
653C2×C4—2441no—
654C2×C4—2441no—
655C2×C4#1451974712yes—
656C2×C4#1451974712yes—
657C2×C4C2×D4×D6111yescharacteristic
658C2×C4C22×D24411yescharacteristic
659C2×C4C2×#1175822411yescharacteristic
660C2×C4C2×#1175822411yescharacteristic
661C2×C4#1451982411yescharacteristic
662C2×C4#1451982411yescharacteristic
663C8—22no—
664C8—22no—
665C8—2441no—
666C8—2441no—
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—
681C22C2×#1451974711yescharacteristic
682C22#1684031011yescharacteristic
683C22#1684033211yescharacteristic
684C22#135×D6111yescharacteristic
685C22#135×D6111yescharacteristic
686C22#1684055911yescharacteristic
687C22#1684055911yescharacteristic
688C4—22no—
689C4—22no—
690C4—42no—
691C4—81no—
692C4—81no—
693C4—1222no—
694C4—4881no—
695C4—4881no—
696C4—2441no—
697C4—2442no—
698C4—2442no—
699C4—2441no—
700C4#1684041611yescharacteristic
701C4#1684041611yescharacteristic
702C4#1684041811yescharacteristic
703C4#1684041811yescharacteristic
704C2—82no—
705C2—2442no—
706C2#2144944911yescharacteristic
707C2#2145503911yescharacteristic
708C2#2145503911yescharacteristic
709C2#2145638611yescharacteristic
710C2#2145654511yescharacteristic
711C2#2145698211yescharacteristic
712C2#2145699311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1958C2
socleC23×C61C22×D4
prime radicalC2×#14519748C22
hypercenter#1958D61
hypocenterC61#25675
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1958

Complements

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

NH ≅ G/Ntype
2. C2×#16840164C2semidirect
3. C2×#16840116C2semidirect
4. #21454130C2semidirect
5. #21447021C2semidirect
6. #21447014C2semidirect
9. #21446783C2semidirect
10. #21446089C2semidirect
11. #21446082C2semidirect
12. #21446075C2semidirect
13. #21446068C2semidirect
14. C4×#14519689C2semidirect
15. C61×#1958C2semidirect
357. #1958D61semidirect
419. C61#25675semidirect

Maximal subgroups

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

subgroupindex
C2×#168402912
C2×#168401642
C2×#168401162
#214541302
#214470212
#214470142
#214470072
#214470002
#214467832
#214460892
#214460822
#214460752
#214460682
C4×#145196892
C61×#19582
#2567561