Finite Group Databasebeta

← #29646740

Subgroups — #29646740

Order 15616.

1238 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#532×D61C211yesmaximal, characteristic
2#21454450C211yesmaximal, characteristic, complemented
3#21454447C211yesmaximal, characteristic, complemented
4#21444286C214yesmaximal, complemented
5#21444242C214yesmaximal, complemented
6#21444237C211yesmaximal, characteristic, complemented
7#21444236C211yesmaximal, characteristic, complemented
8#21444235C211yesmaximal, characteristic, complemented
9C61×#1484C211yesmaximal, characteristic, complemented
10C2×#41×D61—21no—
11#16840313C411yescharacteristic
12#16840312C2211yescharacteristic
13#16840311—21no—
14#16840310C411yescharacteristic
15#16840309—21no—
16#16840309C2212yes—
17#16840308—21no—
18#16840307—21no—
19#16840307C412yes—
20C4⋊Q8×D61C411yescharacteristic
21#127×D61C411yescharacteristic
22#124×D61—21no—
23#121×D61—21no—
24#121×D61C412yes—
25C4×D4×D61—21no—
26C4×D4×D61C2212yes—
27C2×C42×D61C2211yescharacteristic
28#16839818C2212yescomplemented
29#16839817—21no—
30#16839817—21no—
31#16839816—21no—
32#16839815C2211yescharacteristic, complemented
33#16839813—21no—
34#16839813C2212yes—
35#16839813C2212yescomplemented
36#16839785—21no—
37#16839784C2212yes—
38#16839725—21no—
39#16839724—21no—
40C4×#11758194C2211yescharacteristic, complemented
41C4×#11758189—21no—
42#16839491C2214yescomplemented
43D4⋊#92418C2214yescomplemented
44#16839489C2214yescomplemented
45C61×#532C2211yescharacteristic
46C61×#431C2211yescharacteristic, complemented
47C4×M16×C61C2211yescharacteristic, complemented
48C61×D4⋊C8C2214yescomplemented
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×D61C2×C411yescharacteristic
62C22×C4×D61D411yescharacteristic
63C22×C4×D61D411yescharacteristic
64C2×#11758194—21no—
65C2×#11758189—22no—
66#14519777C2×C411yescharacteristic
67#14519776—21no—
68#14519776C2×C412yes—
69#14519775—21no—
70#14519774C2×C411yescharacteristic
71#14519773—21no—
72#14519773—21no—
73#14519773C2×C412yes—
74#14519766—21no—
75#14519765—21no—
76#14519764—21no—
77#14519763—22no—
78#14519763—22no—
79#14519752—21no—
80#14519752C2×C412yes—
81#14519751—21no—
82#14519751C2×C412yes—
83#14519750—22no—
84#14519750—22no—
85#14519749—22no—
86#14519749—22no—
87#14519748—21no—
88#14519748C2×C412yes—
89C4×D244—21no—
90C4×D244C2312yes—
91D4×Dic61—21no—
92D4×Dic61C2×C412yes—
93#14519739—21no—
94#14519739C2×C412yes—
95C4×C22⋊D61—22no—
96C4×C22⋊D61—22no—
97#14519737—22no—
98#14519737—22no—
99C4⋊C4×D61—21no—
100C4⋊C4×D61C2×C412yes—
101C22⋊C4×D61—22no—
102C22⋊C4×D61—22no—
103C2×C4×Dic61C2×C411yescharacteristic
104C2×C4×Dic61D411yescharacteristic
105C2×C4×Dic61D411yescharacteristic
106C42×D61C2×C411yescharacteristic
107C42×D61C2×C411yescharacteristic
108C42×D61D411yescharacteristic
109C42×D61D411yescharacteristic
110C42×D61D411yescharacteristic
111C42×D61D411yescharacteristic
112C42×D61C2311yescharacteristic
113C42×D61C2311yescharacteristic
114C4⋊#92418C2312yes—
115#14519706—22no—
116#14519706—22no—
117#14519706—22no—
118C22⋊#92418—21no—
119#14519689—22no—
120#14519689—22no—
121#14519689—22no—
122#14519688—21no—
123#14519687—22no—
124C4×#92418—21no—
125C4×#92418C2312yes—
126C2×#41×C61—21no—
127C2×M16×C61—21no—
128C4⋊Q8×C61C2×C411yescharacteristic
129#127×C61C2×C411yescharacteristic
130#124×C61—21no—
131#121×C61—21no—
132#121×C61C2×C412yes—
133C4×D4×C61—21no—
134C4×D4×C61C2312yes—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61C2312yes—
137C22⋊C8×C61—21no—
138#97×C61—21no—
139C4×C8×C61—21no—
140C4×C8×C61C2312yes—
141C23×D61—22no—
142C23×D61C22⋊C411yescharacteristic
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×C61C22×C411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C2×C22⋊D61—22no—
154C2×C22⋊D61—22no—
155C2×C22⋊D61—22no—
156C2×C22⋊D61—22no—
157C2×C22⋊D61—22no—
158C22×Dic61—22no—
159C22×Dic61C22⋊C411yescharacteristic
160C22×Dic61C22⋊C411yescharacteristic
161C22×Dic61C22⋊C411yescharacteristic
162#11758226—44no—
163Q8×D61—44no—
164#11758224—44no—
165#11758224—48no—
166D4×D61—44no—
167D4×D61—44no—
168D4×D61—44no—
169D4×D61—48no—
170#11758222—48no—
171#11758222—44no—
172C2×D244—22no—
173C2×D244—21no—
174C2×D244—21no—
175C2×D244C22×C412yes—
176C2×C4×D61—24no—
177C2×C4×D61—22no—
178C2×C4×D61—22no—
179C2×C4×D61—22no—
180C2×C4×D61—42no—
181C2×C4×D61—42no—
182C2×C4×D61—48no—
183C2×C4×D61C22⋊C411yescharacteristic
184C2×C4×D61C22⋊C411yescharacteristic
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×D61C2×D411yescharacteristic
192C2×C4×D61C2×D411yescharacteristic
193C2×C4×D61C2×D411yescharacteristic
194C2×C4×D61C2×D411yescharacteristic
195C2×Dic122—21no—
196C2×Dic122—22no—
197M16×C61—42no—
198C2×C8×C61—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×C61C2×D411yescharacteristic
206C42×C61C2×D411yescharacteristic
207C42×C61C2411yescharacteristic
208C22⋊Dic61—22no—
209C22⋊Dic61—22no—
210#11758198—22no—
211#11758198—22no—
212#11758198—22no—
213#11758198—22no—
214C4⋊Dic61—21no—
215C4⋊Dic61C22×C412yes—
216#11758196—22no—
217#11758196—22no—
218C4×Dic61C22⋊C411yescharacteristic
219C4×Dic61C22⋊C411yescharacteristic
220C4×Dic61C22⋊C411yescharacteristic
221C4×Dic61C22⋊C411yescharacteristic
222C4×Dic61C22⋊C411yescharacteristic
223C4×Dic61C22⋊C411yescharacteristic
224C4×Dic61C22×C411yescharacteristic
225C4×Dic61C22×C411yescharacteristic
226C4×Dic61C2×D411yescharacteristic
227C4×Dic61C2×D411yescharacteristic
228C4×Dic61C2×D411yescharacteristic
229C4×Dic61C2×D411yescharacteristic
230#11758194—42no—
231C2×#92418—22no—
232C2×#92418—22no—
233#11758189—44no—
234C23×C61—22no—
235C23×C61C2×C22⋊C411yescharacteristic
236C22×D61—22no—
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
245Q8×C61—42no—
246D4×C61—24no—
247D4×C61—42no—
248D4×C61—44no—
249D4×C61—42no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—22no—
254C2×C4×C61—44no—
255C2×C4×C61C2×C22⋊C411yescharacteristic
256C2×C4×C61C2×C22⋊C411yescharacteristic
257C2×C4×C61C2×C22⋊C411yescharacteristic
258C2×C4×C61C23×C411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C22×D411yescharacteristic
261C22⋊D61—44no—
262C22⋊D61—44no—
263C22⋊D61—44no—
264C22⋊D61—44no—
265C22⋊D61—44no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—44no—
271C2×Dic61C2×C22⋊C411yescharacteristic
272C2×Dic61C2×C22⋊C411yescharacteristic
273C2×Dic61C2×C22⋊C411yescharacteristic
274C2×Dic61C2×C22⋊C411yescharacteristic
275C2×Dic61C2×C22⋊C411yescharacteristic
276C2×Dic61C2×C22⋊C411yescharacteristic
277D244—24no—
278D244—42no—
279D244—44no—
280D244—42no—
281C4×D61—22no—
282C4×D61—22no—
283C4×D61—22no—
284C4×D61—22no—
285C4×D61—22no—
286C4×D61—22no—
287C4×D61—22no—
288C4×D61—41no—
289C4×D61—41no—
290C4×D61—41no—
291C4×D61—41no—
292C4×D61—44no—
293C4×D61—44no—
294C4×D61—44no—
295C4×D61AGL(1,ℤ/8)14yes—
296Dic122—44no—
297Dic122—42no—
298C8×C61—42no—
299C8×C61—42no—
300#92418—42no—
301#92418—42no—
302#17490—611nomaximal, Sylow
303C22×C61—21no—
304C22×C61—21no—
305C22×C61—21no—
306C22×C61—42no—
307C22×C61—42no—
308C22×C61—42no—
309C22×C61C22×C22⋊C411yescharacteristic
310C2×D61—21no—
311C2×D61—21no—
312C2×D61—21no—
313C2×D61—21no—
314C2×D61—21no—
315C2×D61—44no—
316C2×D61—42no—
317C2×D61—42no—
318C2×D61—42no—
319C2×D61—88no—
320C2×D61#41812yes—
321C4×C61—21no—
322C4×C61—21no—
323C4×C61—21no—
324C4×C61—42no—
325C4×C61#42112yes—
326C4×C61#56812yes—
327C4×C61C2×AGL(1,ℤ/8)12yes—
328Dic61—21no—
329Dic61—21no—
330Dic61—21no—
331Dic61—21no—
332Dic61—21no—
333Dic61—42no—
334Dic61#41812yes—
335C2×#532—611no—
336#2920—611no—
337#2920—611no—
338#1496—614no—
339#1486—614no—
340#1484—611no—
341#1484—611no—
342#1484—611no—
343#1484D6111yescharacteristic, complemented
344C2×C61—41no—
345C2×C61—82no—
346C2×C61#287911yescharacteristic
347C2×C61C2×#41811yescharacteristic
348C2×C61C2×#42111yescharacteristic
349D61—22no—
350D61—41no—
351D61—82no—
352C22×#41—1221no—
353#532—611no—
354#532—611no—
355#532—612no—
356#532—612no—
357#532—611no—
358#532—1221no—
359#532—1221no—
360#532—1221no—
361#532—1221no—
362#532C2×D6111yescharacteristic
363C2×C4⋊Q8—611no—
364C2×#127—611no—
365C2×#124—1221no—
366C2×#121—612no—
367C2×#121—1221no—
368C2×C4×D4—612no—
369C2×C4×D4—1221no—
370C22×C42—611no—
371#432—612no—
372#432—1221no—
373#432—1221no—
374#432—1221no—
375#431—612no—
376#431—612no—
377#431—611no—
378#431—1221no—
379#431C2×D6111yescharacteristic, complemented
380#423—612no—
381#423—1221no—
382#407—1221no—
383#407—1221no—
384C4×M16—611no—
385C4×M16—1221no—
386C4×M16C2×D6111yescharacteristic, complemented
387D4⋊C8—614no—
388D4⋊C8—614no—
389D4⋊C8—614no—
390D4⋊C8C2×D6114yescomplemented
391C61#1749011yescharacteristic, complemented, Sylow
392C2×#41—21no—
393C2×#41—1222no—
394C2×#41—1222no—
395C2×#41—1221no—
396C2×#41—1221no—
397C2×#41—1221no—
398C2×#41—2448no—
399C22×Q8—1221no—
400C22×D4—612no—
401C22×D4—1221no—
402C22×D4—1221no—
403C22×D4—1222no—
404C23×C4—611no—
405C23×C4—611no—
406C23×C4—611no—
407C23×C4—1222no—
408C2×M16—21no—
409C2×M16—1222no—
410C2×M16—1221no—
411C4⋊Q8—612no—
412C4⋊Q8—611no—
413C4⋊Q8—1221no—
414C4⋊Q8—1221no—
415C4⋊Q8C4×D6111yescharacteristic
416#127—612no—
417#127—611no—
418#127—1221no—
419#127—1221no—
420#127C4×D6111yescharacteristic
421#124—21no—
422#124—1222no—
423#124—1222no—
424#124—1221no—
425#124—1221no—
426#124—1221no—
427#121—21no—
428#121—612no—
429#121—612no—
430#121—612no—
431#121—1222no—
432#121—1222no—
433#121—1221no—
434#121—1222no—
435#121—1221no—
436#121—1222no—
437#121—1221no—
438#121C4×D6112yes—
439C4×D4—21no—
440C4×D4—612no—
441C4×D4—612no—
442C4×D4—612no—
443C4×D4—1222no—
444C4×D4—1222no—
445C4×D4—1221no—
446C4×D4—1222no—
447C4×D4—1222no—
448C4×D4—1221no—
449C4×D4—1221no—
450C4×D4C22×D6112yes—
451C2×C4⋊C4—612no—
452C2×C4⋊C4—1221no—
453C2×C22⋊C4—1222no—
454C2×C22⋊C4—1222no—
455C2×C42—611no—
456C2×C42—611no—
457C2×C42—611no—
458C2×C42—611no—
459C2×C42—611no—
460C2×C42—611no—
461C2×C42—611no—
462C2×C42—611no—
463C2×C42—611no—
464C2×C42—611no—
465C2×C42—611no—
466C2×C42C22×D6111yescharacteristic
467C4⋊C8—612no—
468C4⋊C8—1222no—
469C4⋊C8—1222no—
470C4⋊C8—1222no—
471C4⋊C8C22×D6112yes—
472C22⋊C8—21no—
473C22⋊C8—1222no—
474C22⋊C8—1222no—
475C22⋊C8—1222no—
476C22⋊C8—1221no—
477#97—21no—
478#97—1222no—
479#97—1221no—
480C4×C8—21no—
481C4×C8—612no—
482C4×C8—1221no—
483C4×C8C22×D6112yes—
484C24—611no—
485C24—1222no—
486#41—44no—
487#41—2444no—
488#41—2448no—
489#41—2444no—
490#41—2448no—
491#41—2444no—
492C2×Q8—21no—
493C2×Q8—1222no—
494C2×Q8—1221no—
495C2×Q8—2444no—
496C2×D4—22no—
497C2×D4—21no—
498C2×D4—21no—
499C2×D4—612no—
500C2×D4—1222no—
501C2×D4—1222no—
502C2×D4—1222no—
503C2×D4—1222no—
504C2×D4—1222no—
505C2×D4—1222no—
506C2×D4—1221no—
507C2×D4—1221no—
508C2×D4—2444no—
509C2×D4—2444no—
510C2×D4—2444no—
511C2×D4—2448no—
512C2×D4C2×C4×D6112yes—
513C22×C4—22no—
514C22×C4—611no—
515C22×C4—611no—
516C22×C4—611no—
517C22×C4—611no—
518C22×C4—611no—
519C22×C4—611no—
520C22×C4—611no—
521C22×C4—611no—
522C22×C4—611no—
523C22×C4—611no—
524C22×C4—611no—
525C22×C4—611no—
526C22×C4—611no—
527C22×C4—611no—
528C22×C4—611no—
529C22×C4—1224no—
530C22×C4—1222no—
531C22×C4—1222no—
532C22×C4—1222no—
533C22×C4—1222no—
534C22×C4—2442no—
535C22×C4—2442no—
536C22×C4—2448no—
537C22×C4C2×C4×D6111yescharacteristic
538C22×C4D4×D6111yescharacteristic
539C22×C4D4×D6111yescharacteristic
540M16—42no—
541M16—2444no—
542M16—2442no—
543C2×C8—22no—
544C2×C8—22no—
545C2×C8—1222no—
546C2×C8—1222no—
547C4⋊C4—21no—
548C4⋊C4—612no—
549C4⋊C4—1222no—
550C4⋊C4—1222no—
551C4⋊C4—1221no—
552C4⋊C4C2×C4×D6112yes—
553C22⋊C4—22no—
554C22⋊C4—22no—
555C22⋊C4—1222no—
556C22⋊C4—1222no—
557C22⋊C4—1222no—
558C22⋊C4—1222no—
559C22⋊C4—1222no—
560C22⋊C4—1222no—
561C42—611no—
562C42—611no—
563C42—611no—
564C42—611no—
565C42—611no—
566C42—611no—
567C42—611no—
568C42—611no—
569C42—611no—
570C42—611no—
571C42—611no—
572C42—611no—
573C42C2×C4×D6111yescharacteristic
574C42D4×D6111yescharacteristic
575C42D4×D6111yescharacteristic
576C42C23×D6111yescharacteristic
577C23—22no—
578C23—611no—
579C23—611no—
580C23—1222no—
581C23—1222no—
582C23—2441no—
583C23—2441no—
584C23—2444no—
585C23—2444no—
586C23—2444no—
587C23C22⋊C4×D6111yescharacteristic
588Q8—42no—
589Q8—2442no—
590Q8—2444no—
591D4—24no—
592D4—42no—
593D4—44no—
594D4—42no—
595D4—1224no—
596D4—2442no—
597D4—2444no—
598D4—2444no—
599D4—2444no—
600D4—2444no—
601D4—2444no—
602D4—2444no—
603D4—2442no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—44no—
609C2×C4—611no—
610C2×C4—614no—
611C2×C4—611no—
612C2×C4—611no—
613C2×C4—611no—
614C2×C4—611no—
615C2×C4—611no—
616C2×C4—1222no—
617C2×C4—1222no—
618C2×C4—1222no—
619C2×C4—1222no—
620C2×C4—1222no—
621C2×C4—1222no—
622C2×C4—1222no—
623C2×C4—1222no—
624C2×C4—1222no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2444no—
630C2×C4—2444no—
631C2×C4—2444no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4—2444no—
635C2×C4C22⋊C4×D6111yescharacteristic
636C2×C4C22⋊C4×D6111yescharacteristic
637C2×C4C22⋊C4×D6111yescharacteristic
638C2×C4C22×C4×D6111yescharacteristic
639C2×C4C2×D4×D6111yescharacteristic
640C2×C4C2×D4×D6111yescharacteristic
641C8—42no—
642C8—42no—
643C8—2442no—
644C8—2442no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—612no—
652C22—1221no—
653C22—1221no—
654C22—1221no—
655C22—1221no—
656C22—1221no—
657C22—4888no—
658C22—2444no—
659C22—2442no—
660C22—2442no—
661C22—2442no—
662C22C2×C22⋊C4×D6111yescharacteristic
663C4—21no—
664C4—21no—
665C4—21no—
666C4—42no—
667C4—612no—
668C4—1221no—
669C4—1221no—
670C4—1221no—
671C4—1221no—
672C4—1221no—
673C4—2442no—
674C4#1683977712yes—
675C4#1684052712yes—
676C4AGL(1,ℤ/8)×D6112yes—
677C2—41no—
678C2—82no—
679C2—1222no—
680C2—4882no—
681C2—2441no—
682C2#2145426111yescharacteristic
683C2#418×D6111yescharacteristic
684C2C2×#1683977711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1484C2
socleC22×C61C22×C22⋊C4
prime radical#532×D61C2
cosocleC42×C61C24
hypercenter#1484D61
hypocenterC61#17490
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1484

Complements

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

NH ≅ G/Ntype
2. #21454450C2semidirect
3. #21454447C2semidirect
4. #21444286C2semidirect
5. #21444242C2semidirect
6. #21444237C2semidirect
7. #21444236C2semidirect
8. #21444235C2semidirect
9. C61×#1484C2semidirect
28. #16839818C22semidirect
32. #16839815C22semidirect
35. #16839813C22semidirect
40. C4×#11758194C22semidirect
42. #16839491C22semidirect
43. D4⋊#92418C22semidirect
44. #16839489C22semidirect
46. C61×#431C22semidirect
47. C4×M16×C61C22semidirect
48. C61×D4⋊C8C22semidirect
343. #1484D61semidirect
379. #431C2×D61semidirect
386. C4×M16C2×D61semidirect
390. D4⋊C8C2×D61semidirect
391. C61#17490semidirect

Maximal subgroups

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

subgroupindex
#532×D612
#214544502
#214544472
#214442862
#214442422
#214442372
#214442362
#214442352
C61×#14842
#1749061