Finite Group Databasebeta

← #29647519

Subgroups — #29647519

Order 15616.

1274 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#532×D61C211yesmaximal, characteristic
2#432×D61C211yesmaximal, characteristic, complemented
3#21448216C211yesmaximal, characteristic, complemented
4#21444385C214yesmaximal, complemented
5#21444367C211yesmaximal, characteristic, complemented
6#21444366C211yesmaximal, characteristic, complemented
7#21444365C211yesmaximal, characteristic, complemented
8#21444325C214yesmaximal, complemented
9C61×#1514C211yesmaximal, characteristic, complemented
10C2×#41×D61—21no—
11#16840313C411yescharacteristic, complemented
12#16840312C2211yescharacteristic
13#16840311—21no—
14#16840310C411yescharacteristic, complemented
15#16840309—21no—
16#16840309C412yescomplemented
17#16840308—21no—
18#16840307—21no—
19#16840307C2212yes—
20C4⋊Q8×D61C411yescharacteristic, complemented
21#127×D61C411yescharacteristic, complemented
22#124×D61—21no—
23#121×D61—21no—
24#121×D61C2212yes—
25C4×D4×D61—21no—
26C4×D4×D61C412yescomplemented
27C2×C42×D61C2211yescharacteristic
28#16839819C2211yescharacteristic, complemented
29#16839818—21no—
30#16839817—21no—
31#16839816C2212yescomplemented
32C4⋊C8×D61—21no—
33C22⋊C8×D61C2212yes—
34#97×D61—21no—
35#16839630C2211yescharacteristic, complemented
36#16839626—21no—
37#16839624—21no—
38#16839621C2212yes—
39#16839613—21no—
40#16839609—21no—
41#16839607C2212yescomplemented
42#16839497C2214yescomplemented
43#16839496C2214yescomplemented
44#16839495C2214yescomplemented
45C61×#532C2211yescharacteristic
46C61×#432C2211yescharacteristic, complemented
47C61×#383C2211yescharacteristic, complemented
48C61×#327C2214yescomplemented
49C2×#11758226—21no—
50C2×#11758222—21no—
51C2×#11758222—22no—
52C2×#11758224—22no—
53C2×#11758224—21no—
54#41×D61—48no—
55C2×Q8×D61—21no—
56C2×D4×D61—22no—
57C2×D4×D61—21no—
58C2×D4×D61—21no—
59C2×D4×D61C2×C412yes—
60C22×C4×D61—22no—
61C22×C4×D61D411yescharacteristic
62C22×C4×D61D411yescharacteristic
63C22×C4×D61C2311yescharacteristic
64C2×C8×D61—22no—
65#14519777C2×C411yescharacteristic
66#14519776—21no—
67#14519776C2×C412yes—
68#14519775—21no—
69#14519774C2×C411yescharacteristic
70#14519773—21no—
71#14519773—21no—
72#14519773C2×C412yes—
73#14519766—21no—
74#14519765—21no—
75#14519764—21no—
76#14519763—22no—
77#14519763—22no—
78#14519752—21no—
79#14519752C2×C412yes—
80#14519751—21no—
81#14519751C2312yes—
82#14519750—22no—
83#14519750—22no—
84#14519749—22no—
85#14519749—22no—
86#14519748—21no—
87#14519748C2×C412yes—
88C4×D244—21no—
89C4×D244C2×C412yes—
90D4×Dic61—21no—
91D4×Dic61C2×C412yes—
92#14519739—21no—
93#14519739C2×C412yes—
94C4×C22⋊D61—22no—
95C4×C22⋊D61—22no—
96#14519737—22no—
97#14519737—22no—
98C2×#11758196—22no—
99C4⋊C4×D61—21no—
100C4⋊C4×D61C2×C412yes—
101C2×#11758198—22no—
102C22⋊C4×D61—22no—
103C22⋊C4×D61—22no—
104C2×C4×Dic61—22no—
105C2×C4×Dic61D411yescharacteristic
106C2×C4×Dic61D411yescharacteristic
107C2×C4×Dic61C2311yescharacteristic
108C42×D61C2×C411yescharacteristic
109C42×D61C2×C411yescharacteristic
110C42×D61C2×C411yescharacteristic
111C42×D61C2×C411yescharacteristic
112C42×D61D411yescharacteristic
113C42×D61D411yescharacteristic
114C42×D61D411yescharacteristic
115C42×D61D411yescharacteristic
116C4⋊#92418—21no—
117#14519706—22no—
118C22⋊#92418C2312yes—
119#14519689—22no—
120#14519688—21no—
121#14519687—22no—
122#14519684—21no—
123#14519684—21no—
124#14519684—22no—
125#14519684C2312yes—
126C2×#41×C61—21no—
127C4⋊Q8×C61C2×C411yescharacteristic
128#127×C61C2×C411yescharacteristic
129#124×C61—21no—
130#121×C61—21no—
131#121×C61C2312yes—
132C4×D4×C61—21no—
133C4×D4×C61C2×C412yes—
134C2×C42×C61C2311yescharacteristic
135C4⋊C8×C61—21no—
136C22⋊C8×C61C2312yes—
137#97×C61—21no—
138#95×C61—21no—
139#95×C61—21no—
140#95×C61C2312yes—
141C23×D61—22no—
142C23×D61C2×D411yescharacteristic
143#41×C61—44no—
144C2×Q8×C61—21no—
145C2×D4×C61—22no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61C22×C412yes—
149C22×C4×C61—22no—
150C22×C4×C61—22no—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2411yescharacteristic
154C2×C22⋊D61—22no—
155C2×C22⋊D61—22no—
156C2×C22⋊D61—22no—
157C2×C22⋊D61—22no—
158C2×C22⋊D61—22no—
159C22×Dic61—22no—
160C22×Dic61—22no—
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164#11758226—44no—
165Q8×D61—44no—
166#11758224—44no—
167#11758224—48no—
168D4×D61—44no—
169D4×D61—44no—
170D4×D61—44no—
171D4×D61—48no—
172#11758222—48no—
173#11758222—44no—
174C2×D244—22no—
175C2×D244—21no—
176C2×D244—21no—
177C2×D244C22×C412yes—
178C2×C4×D61—24no—
179C2×C4×D61—22no—
180C2×C4×D61—22no—
181C2×C4×D61—22no—
182C2×C4×D61—42no—
183C2×C4×D61—42no—
184C2×C4×D61—48no—
185C2×C4×D61C22⋊C411yescharacteristic
186C2×C4×D61C22⋊C411yescharacteristic
187C2×C4×D61C22⋊C411yescharacteristic
188C2×C4×D61C22⋊C411yescharacteristic
189C2×C4×D61C22⋊C411yescharacteristic
190C2×C4×D61C22⋊C411yescharacteristic
191C2×C4×D61C22⋊C411yescharacteristic
192C2×C4×D61C22⋊C411yescharacteristic
193C2×C4×D61C22×C411yescharacteristic
194C2×C4×D61C22×C411yescharacteristic
195C2×C4×D61C2×D411yescharacteristic
196C2×C4×D61C2×D411yescharacteristic
197C2×Dic122—21no—
198C2×Dic122—22no—
199C2×C8×C61—22no—
200C4⋊C4×C61—21no—
201C4⋊C4×C61C22×C412yes—
202C22⋊C4×C61—22no—
203C22⋊C4×C61—22no—
204C42×C61C22×C411yescharacteristic
205C42×C61C22×C411yescharacteristic
206C42×C61C2×D411yescharacteristic
207C42×C61C2×D411yescharacteristic
208C22⋊Dic61—22no—
209C22⋊Dic61—22no—
210#11758198—22no—
211#11758198—22no—
212#11758198—22no—
213#11758198—22no—
214#11758198—44no—
215C4⋊Dic61—21no—
216C4⋊Dic61C22×C412yes—
217#11758196—22no—
218#11758196—22no—
219#11758196—44no—
220C4×Dic61—44no—
221C4×Dic61C22⋊C411yescharacteristic
222C4×Dic61C22⋊C411yescharacteristic
223C4×Dic61C22⋊C411yescharacteristic
224C4×Dic61C22⋊C411yescharacteristic
225C4×Dic61C22⋊C411yescharacteristic
226C4×Dic61C22⋊C411yescharacteristic
227C4×Dic61C22⋊C411yescharacteristic
228C4×Dic61C22⋊C411yescharacteristic
229C4×Dic61C22×C411yescharacteristic
230C4×Dic61C22×C411yescharacteristic
231C4×Dic61C2×D411yescharacteristic
232C4×Dic61C2×D411yescharacteristic
233C2×#92418—22no—
234C8×D61—44no—
235C23×C61—22no—
236C23×C61C22×D411yescharacteristic
237C22×D61—22no—
238C22×D61—41no—
239C22×D61—41no—
240C22×D61—44no—
241C22×D61—44no—
242C22×D61—44no—
243C22×D61C2×C22⋊C411yescharacteristic
244C22×D61C2×C22⋊C411yescharacteristic
245C22×D61AGL(1,ℤ/8)14yes—
246Q8×C61—42no—
247D4×C61—42no—
248D4×C61—42no—
249D4×C61—42no—
250D4×C61—44no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—44no—
256C2×C4×C61—42no—
257C2×C4×C61—42no—
258C2×C4×C61—42no—
259C2×C4×C61C2×C22⋊C411yescharacteristic
260C2×C4×C61C2×C22⋊C411yescharacteristic
261C2×C4×C61C2×C22⋊C411yescharacteristic
262C2×C4×C61C2×C22⋊C411yescharacteristic
263C2×C4×C61C23×C411yescharacteristic
264C2×C4×C61C22×D411yescharacteristic
265C22⋊D61—44no—
266C22⋊D61—44no—
267C22⋊D61—44no—
268C22⋊D61—44no—
269C22⋊D61—44no—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—22no—
274C2×Dic61—44no—
275C2×Dic61—42no—
276C2×Dic61—42no—
277C2×Dic61—42no—
278C2×Dic61C2×C22⋊C411yescharacteristic
279C2×Dic61C2×C22⋊C411yescharacteristic
280C2×Dic61C2×C22⋊C411yescharacteristic
281C2×Dic61C2×C22⋊C411yescharacteristic
282C2×Dic61C2×C22⋊C411yescharacteristic
283C2×Dic61C2×C22⋊C411yescharacteristic
284D244—42no—
285D244—42no—
286D244—42no—
287D244—44no—
288C4×D61—22no—
289C4×D61—22no—
290C4×D61—22no—
291C4×D61—22no—
292C4×D61—22no—
293C4×D61—22no—
294C4×D61—22no—
295C4×D61—22no—
296C4×D61—41no—
297C4×D61—41no—
298C4×D61—41no—
299C4×D61—41no—
300C4×D61—44no—
301C4×D61—44no—
302C4×D61—44no—
303Dic122—44no—
304Dic122—42no—
305C8×C61—42no—
306#92418—42no—
307#17595—611nomaximal, Sylow
308C22×C61—21no—
309C22×C61—21no—
310C22×C61—42no—
311C22×C61—42no—
312C22×C61—42no—
313C22×C61C22×C22⋊C411yescharacteristic
314C22×C61C2×AGL(1,ℤ/8)12yes—
315C2×D61—21no—
316C2×D61—21no—
317C2×D61—21no—
318C2×D61—21no—
319C2×D61—44no—
320C2×D61—42no—
321C2×D61—42no—
322C2×D61—42no—
323C2×D61—88no—
324C2×D61#41812yes—
325C2×D61#41812yes—
326C4×C61—21no—
327C4×C61—21no—
328C4×C61—21no—
329C4×C61—21no—
330C4×C61—21no—
331C4×C61—21no—
332C4×C61—42no—
333C4×C61—82no—
334Dic61—21no—
335Dic61—21no—
336Dic61—21no—
337Dic61—21no—
338Dic61—42no—
339Dic61—82no—
340Dic61#41012yes—
341Dic61#42112yes—
342C2×#532—611no—
343C2×#432—611no—
344#2283—611no—
345#1519—614no—
346#1514—611no—
347#1514—611no—
348#1514—611no—
349#1514D6111yescharacteristic, complemented
350#1505—614no—
351C2×C61—22no—
352C2×C61—82no—
353C2×C61C2×#41011yescharacteristic
354C2×C61C2×#41811yescharacteristic
355C2×C61C2×#42111yescharacteristic
356D61—22no—
357D61—22no—
358D61—82no—
359C22×#41—1221no—
360#532—611no—
361#532—611no—
362#532—612no—
363#532—612no—
364#532—611no—
365#532—1221no—
366#532—1221no—
367#532—1221no—
368#532—1221no—
369#532C2×D6111yescharacteristic
370C2×C4⋊Q8—611no—
371C2×#127—611no—
372C2×#124—1221no—
373C2×#121—612no—
374C2×#121—1221no—
375C2×C4×D4—612no—
376C2×C4×D4—1221no—
377C22×C42—611no—
378#432—611no—
379#432—612no—
380#432—1221no—
381#432—1221no—
382#432C2×D6111yescharacteristic, complemented
383C2×C4⋊C8—1221no—
384C2×C22⋊C8—612no—
385C2×#97—1221no—
386#383—611no—
387#383C2×D6111yescharacteristic, complemented
388#382—1221no—
389#381—1221no—
390#380—612no—
391C4×C4⋊C4—1221no—
392C4×C22⋊C4—1221no—
393#376—612no—
394#327—614no—
395#327—614no—
396#327—614no—
397#327C2×D6114yescomplemented
398C61#1759511yescharacteristic, complemented, Sylow
399C2×#41—21no—
400C2×#41—1222no—
401C2×#41—1222no—
402C2×#41—1221no—
403C2×#41—1221no—
404C2×#41—1221no—
405C2×#41—2448no—
406C22×Q8—1221no—
407C22×D4—612no—
408C22×D4—1221no—
409C22×D4—1221no—
410C22×D4—1222no—
411C23×C4—611no—
412C23×C4—611no—
413C23×C4—611no—
414C23×C4—1222no—
415C22×C8—1222no—
416C4⋊Q8—612no—
417C4⋊Q8—611no—
418C4⋊Q8—1221no—
419C4⋊Q8—1221no—
420C4⋊Q8C4×D6111yescharacteristic
421#127—612no—
422#127—611no—
423#127—1221no—
424#127—1221no—
425#127C4×D6111yescharacteristic
426#124—21no—
427#124—1222no—
428#124—1222no—
429#124—1221no—
430#124—1221no—
431#124—1221no—
432#121—21no—
433#121—612no—
434#121—612no—
435#121—612no—
436#121—1222no—
437#121—1222no—
438#121—1221no—
439#121—1222no—
440#121—1221no—
441#121—1222no—
442#121—1221no—
443#121C22×D6112yes—
444C4×D4—21no—
445C4×D4—612no—
446C4×D4—612no—
447C4×D4—612no—
448C4×D4—1222no—
449C4×D4—1222no—
450C4×D4—1221no—
451C4×D4—1222no—
452C4×D4—1222no—
453C4×D4—1221no—
454C4×D4—1221no—
455C4×D4C4×D6112yes—
456C2×C4⋊C4—612no—
457C2×C4⋊C4—1221no—
458C2×C4⋊C4—1222no—
459C2×C22⋊C4—1222no—
460C2×C22⋊C4—1222no—
461C2×C22⋊C4—1222no—
462C2×C42—611no—
463C2×C42—611no—
464C2×C42—611no—
465C2×C42—611no—
466C2×C42—611no—
467C2×C42—611no—
468C2×C42—611no—
469C2×C42—611no—
470C2×C42—611no—
471C2×C42—611no—
472C2×C42—611no—
473C2×C42—1222no—
474C2×C42C22×D6111yescharacteristic
475C4⋊C8—21no—
476C4⋊C8—1222no—
477C4⋊C8—1221no—
478C22⋊C8—612no—
479C22⋊C8—1222no—
480C22⋊C8C22×D6112yes—
481#97—21no—
482#97—1222no—
483#97—1221no—
484#95—21no—
485#95—21no—
486#95—612no—
487#95—1222no—
488#95—1221no—
489#95—1221no—
490#95C22×D6112yes—
491C24—611no—
492C24—1222no—
493#41—44no—
494#41—2444no—
495#41—2448no—
496#41—2444no—
497#41—2448no—
498#41—2444no—
499C2×Q8—21no—
500C2×Q8—1222no—
501C2×Q8—1221no—
502C2×Q8—2444no—
503C2×D4—22no—
504C2×D4—21no—
505C2×D4—21no—
506C2×D4—612no—
507C2×D4—1222no—
508C2×D4—1222no—
509C2×D4—1222no—
510C2×D4—1222no—
511C2×D4—1222no—
512C2×D4—1222no—
513C2×D4—1221no—
514C2×D4—1221no—
515C2×D4—2444no—
516C2×D4—2444no—
517C2×D4—2444no—
518C2×D4—2448no—
519C2×D4C2×C4×D6112yes—
520C22×C4—22no—
521C22×C4—22no—
522C22×C4—611no—
523C22×C4—611no—
524C22×C4—611no—
525C22×C4—611no—
526C22×C4—611no—
527C22×C4—611no—
528C22×C4—611no—
529C22×C4—611no—
530C22×C4—611no—
531C22×C4—611no—
532C22×C4—611no—
533C22×C4—611no—
534C22×C4—611no—
535C22×C4—611no—
536C22×C4—611no—
537C22×C4—1222no—
538C22×C4—1224no—
539C22×C4—1222no—
540C22×C4—1222no—
541C22×C4—1222no—
542C22×C4—1222no—
543C22×C4—2442no—
544C22×C4—2442no—
545C22×C4—2448no—
546C22×C4D4×D6111yescharacteristic
547C22×C4D4×D6111yescharacteristic
548C22×C4C23×D6111yescharacteristic
549C2×C8—22no—
550C2×C8—1222no—
551C2×C8—2444no—
552C4⋊C4—21no—
553C4⋊C4—612no—
554C4⋊C4—1222no—
555C4⋊C4—1222no—
556C4⋊C4—1221no—
557C4⋊C4—2444no—
558C4⋊C4C2×C4×D6112yes—
559C22⋊C4—22no—
560C22⋊C4—22no—
561C22⋊C4—1222no—
562C22⋊C4—1222no—
563C22⋊C4—1222no—
564C22⋊C4—1222no—
565C22⋊C4—1222no—
566C22⋊C4—1222no—
567C22⋊C4—2444no—
568C42—611no—
569C42—611no—
570C42—611no—
571C42—611no—
572C42—611no—
573C42—611no—
574C42—611no—
575C42—611no—
576C42—611no—
577C42—611no—
578C42—611no—
579C42—611no—
580C42—2444no—
581C42C2×C4×D6111yescharacteristic
582C42C2×C4×D6111yescharacteristic
583C42D4×D6111yescharacteristic
584C42D4×D6111yescharacteristic
585C23—22no—
586C23—611no—
587C23—611no—
588C23—614no—
589C23—1222no—
590C23—2441no—
591C23—2441no—
592C23—2444no—
593C23—2444no—
594C23—2444no—
595C23C2×D4×D6111yescharacteristic
596Q8—42no—
597Q8—2442no—
598Q8—2444no—
599D4—42no—
600D4—42no—
601D4—42no—
602D4—44no—
603D4—2442no—
604D4—2442no—
605D4—2442no—
606D4—2444no—
607D4—2444no—
608D4—2444no—
609D4—2444no—
610D4—2444no—
611D4—2444no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—611no—
621C2×C4—611no—
622C2×C4—611no—
623C2×C4—611no—
624C2×C4—611no—
625C2×C4—611no—
626C2×C4—1222no—
627C2×C4—1222no—
628C2×C4—1222no—
629C2×C4—1222no—
630C2×C4—1222no—
631C2×C4—1222no—
632C2×C4—1222no—
633C2×C4—1222no—
634C2×C4—1222no—
635C2×C4—1222no—
636C2×C4—1222no—
637C2×C4—1222no—
638C2×C4—2441no—
639C2×C4—2441no—
640C2×C4—2441no—
641C2×C4—2441no—
642C2×C4—2444no—
643C2×C4—2444no—
644C2×C4—2444no—
645C2×C4—2444no—
646C2×C4—2442no—
647C2×C4—2442no—
648C2×C4—2442no—
649C2×C4C22⋊C4×D6111yescharacteristic
650C2×C4C22⋊C4×D6111yescharacteristic
651C2×C4C22⋊C4×D6111yescharacteristic
652C2×C4C22⋊C4×D6111yescharacteristic
653C2×C4C22×C4×D6111yescharacteristic
654C2×C4C2×D4×D6111yescharacteristic
655C8—42no—
656C8—2442no—
657C22—21no—
658C22—21no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—612no—
663C22—612no—
664C22—1221no—
665C22—1221no—
666C22—1221no—
667C22—1221no—
668C22—4888no—
669C22—2444no—
670C22—2442no—
671C22—2442no—
672C22—2442no—
673C22C2×C22⋊C4×D6111yescharacteristic
674C22AGL(1,ℤ/8)×D6112yes—
675C4—21no—
676C4—21no—
677C4—21no—
678C4—21no—
679C4—21no—
680C4—21no—
681C4—42no—
682C4—82no—
683C4—612no—
684C4—612no—
685C4—1221no—
686C4—1221no—
687C4—1221no—
688C4—1221no—
689C4—4882no—
690C4—2442no—
691C2—22no—
692C2—82no—
693C2—1222no—
694C2—1222no—
695C2—4882no—
696C2#410×D6111yescharacteristic
697C2#418×D6111yescharacteristic
698C2#421×D6111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1514C2
socleC22×C61C22×C22⋊C4
prime radical#532×D61C2
cosocleC22×C4×C61C24
hypercenter#1514D61
hypocenterC61#17595
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1514

Complements

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

NH ≅ G/Ntype
2. #432×D61C2semidirect
3. #21448216C2semidirect
4. #21444385C2semidirect
5. #21444367C2semidirect
6. #21444366C2semidirect
7. #21444365C2semidirect
8. #21444325C2semidirect
9. C61×#1514C2semidirect
11. #16840313C4semidirect
14. #16840310C4semidirect
16. #16840309C4semidirect
20. C4⋊Q8×D61C4semidirect
21. #127×D61C4semidirect
26. C4×D4×D61C4semidirect
28. #16839819C22semidirect
31. #16839816C22semidirect
35. #16839630C22semidirect
41. #16839607C22semidirect
42. #16839497C22semidirect
43. #16839496C22semidirect
44. #16839495C22semidirect
46. C61×#432C22semidirect
47. C61×#383C22semidirect
48. C61×#327C22semidirect
349. #1514D61semidirect
382. #432C2×D61semidirect
387. #383C2×D61semidirect
397. #327C2×D61semidirect
398. C61#17595semidirect

Maximal subgroups

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

subgroupindex
#532×D612
#432×D612
#214482162
#214443852
#214443672
#214443662
#214443652
#214443252
C61×#15142
#1759561