Finite Group Databasebeta

← #29647959

Subgroups — #29647959

Order 15616.

1316 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#532×D61C211yesmaximal, characteristic
2#21454450C211yesmaximal, characteristic, complemented
3C4×M16×D61C211yesmaximal, characteristic, complemented
4#21444428C214yesmaximal, complemented
5#21444415C214yesmaximal, complemented
6#21444411C211yesmaximal, characteristic, complemented
7#21444410C211yesmaximal, characteristic, complemented
8#21444410C211yesmaximal, characteristic, complemented
9C61×#1525C211yesmaximal, characteristic, complemented
10C2×#41×D61C412yes—
11C2×M16×D61—21no—
12#16840313C411yescharacteristic
13#16840312C2211yescharacteristic
14#16840311C2212yes—
15#16840310C411yescharacteristic
16#16840309—21no—
17#16840309—21no—
18#16840308C412yes—
19#16840307—21no—
20#16840307—21no—
21C4⋊Q8×D61C411yescharacteristic
22#127×D61C411yescharacteristic
23#124×D61C2212yes—
24#121×D61—21no—
25#121×D61—21no—
26C4×D4×D61—21no—
27C4×D4×D61—21no—
28C2×C42×D61C2211yescharacteristic
29#16839817C2212yes—
30#16839817C2212yescomplemented
31#16839813—21no—
32#16839813—21no—
33#16839785—21no—
34#16839725—21no—
35C4×#11758194C2211yescharacteristic, complemented
36C4×#11758194C2211yescharacteristic, complemented
37#16839714C2212yes—
38M16×Dic61—21no—
39C4×#11758189—21no—
40#97×D61C2212yescomplemented
41C4×C8×D61—21no—
42#16839502C2214yescomplemented
43#16839501C2214yescomplemented
44#16839501C2214yescomplemented
45C61×#532C2211yescharacteristic
46C4×M16×C61C2211yescharacteristic, complemented
47C4×M16×C61C2211yescharacteristic, complemented
48C61×#329C2214yescomplemented
49C2×#11758226C2×C412yes—
50C2×#11758222—22no—
51C2×#11758222C2×C412yes—
52C2×#11758224—22no—
53C2×#11758224C2×C412yes—
54#41×D61—48no—
55C2×Q8×D61C2×C412yes—
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×#11758194—21no—
66C2×#11758189—22no—
67M16×D61—28no—
68C2×C8×D61—22no—
69#14519777C2×C411yescharacteristic
70#14519776—21no—
71#14519776—21no—
72#14519775C2×C412yes—
73#14519774C2×C411yescharacteristic
74#14519773—21no—
75#14519773—21no—
76#14519773C2×C412yes—
77#14519766C2×C412yes—
78#14519765C2312yes—
79#14519764C2×C412yes—
80#14519763—22no—
81#14519763—22no—
82#14519752—21no—
83#14519752—21no—
84#14519751—21no—
85#14519751—21no—
86#14519750—22no—
87#14519750—22no—
88#14519749—22no—
89#14519749—22no—
90#14519748—21no—
91#14519748—21no—
92C4×D244—21no—
93C4×D244—21no—
94D4×Dic61—21no—
95D4×Dic61—21no—
96#14519739—21no—
97#14519739—21no—
98C4×C22⋊D61—22no—
99C4×C22⋊D61—22no—
100#14519737—22no—
101#14519737—22no—
102C4⋊C4×D61—21no—
103C4⋊C4×D61—21no—
104C22⋊C4×D61—22no—
105C22⋊C4×D61—22no—
106C2×C4×Dic61C2×C411yescharacteristic
107C2×C4×Dic61D411yescharacteristic
108C2×C4×Dic61D411yescharacteristic
109C42×D61C2×C411yescharacteristic
110C42×D61C2×C411yescharacteristic
111C42×D61D411yescharacteristic
112C42×D61D411yescharacteristic
113C42×D61D411yescharacteristic
114C42×D61D411yescharacteristic
115C42×D61C2311yescharacteristic
116C42×D61C2311yescharacteristic
117#14519706—22no—
118#14519706—22no—
119#14519689—22no—
120#14519689—22no—
121#14519688C2312yes—
122#14519688C2312yes—
123#14519687—22no—
124C4×#92418—21no—
125C4×#92418—21no—
126C8×Dic61—22no—
127C2×#41×C61C2×C412yes—
128C2×M16×C61—21no—
129C2×M16×C61—21no—
130C4⋊Q8×C61C2×C411yescharacteristic
131#127×C61C2×C411yescharacteristic
132#124×C61C2312yes—
133#121×C61—21no—
134#121×C61—21no—
135C4×D4×C61—21no—
136C4×D4×C61—21no—
137C2×C42×C61C2311yescharacteristic
138#97×C61C2312yes—
139#97×C61C2312yes—
140C4×C8×C61—21no—
141C4×C8×C61—21no—
142C23×D61—22no—
143C23×D61C22⋊C411yescharacteristic
144#41×C61—44no—
145C2×Q8×C61C22×C412yes—
146C2×D4×C61—22no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61C22×C412yes—
150C22×C4×C61—22no—
151C22×C4×C61C22×C411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C2×C22⋊D61—22no—
155C2×C22⋊D61—22no—
156C2×C22⋊D61—22no—
157C2×C22⋊D61—22no—
158C2×C22⋊D61—22no—
159C22×Dic61—22no—
160C22×Dic61C22⋊C411yescharacteristic
161C22×Dic61C22⋊C411yescharacteristic
162C22×Dic61C22⋊C411yescharacteristic
163#11758226—44no—
164Q8×D61—44no—
165#11758224—44no—
166#11758224—48no—
167D4×D61—44no—
168D4×D61—44no—
169D4×D61—48no—
170D4×D61—44no—
171#11758222—48no—
172#11758222—44no—
173C2×D244—22no—
174C2×D244—21no—
175C2×D244—21no—
176C2×D244C22×C412yes—
177C2×C4×D61—24no—
178C2×C4×D61—24no—
179C2×C4×D61—22no—
180C2×C4×D61—22no—
181C2×C4×D61—22no—
182C2×C4×D61—42no—
183C2×C4×D61—48no—
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×D61C22×C411yescharacteristic
192C2×C4×D61C2×D411yescharacteristic
193C2×C4×D61C2×D411yescharacteristic
194C2×C4×D61C2×D411yescharacteristic
195C2×C4×D61C2×D411yescharacteristic
196C2×Dic122—22no—
197C2×Dic122C22×C412yes—
198M16×C61—24no—
199M16×C61—42no—
200C2×C8×C61—22no—
201C2×C8×C61—22no—
202C4⋊C4×C61—21no—
203C4⋊C4×C61—21no—
204C22⋊C4×C61—22no—
205C22⋊C4×C61—22no—
206C42×C61C22×C411yescharacteristic
207C42×C61C2×D411yescharacteristic
208C42×C61C2×D411yescharacteristic
209C42×C61C2411yescharacteristic
210C22⋊Dic61—22no—
211C22⋊Dic61—22no—
212#11758198—22no—
213#11758198—22no—
214#11758198—22no—
215#11758198—22no—
216C4⋊Dic61—21no—
217C4⋊Dic61—21no—
218#11758196—22no—
219#11758196—22no—
220C4×Dic61C22⋊C411yescharacteristic
221C4×Dic61C22⋊C411yescharacteristic
222C4×Dic61C22⋊C411yescharacteristic
223C4×Dic61C22⋊C411yescharacteristic
224C4×Dic61C22⋊C411yescharacteristic
225C4×Dic61C22⋊C411yescharacteristic
226C4×Dic61C22×C411yescharacteristic
227C4×Dic61C22×C411yescharacteristic
228C4×Dic61C2×D411yescharacteristic
229C4×Dic61C2×D411yescharacteristic
230C4×Dic61C2×D411yescharacteristic
231C4×Dic61C2×D411yescharacteristic
232#11758194—24no—
233#11758194—42no—
234C2×#92418—22no—
235C2×#92418—22no—
236#11758189—28no—
237C8×D61—28no—
238C23×C61—22no—
239C23×C61C2×C22⋊C411yescharacteristic
240C22×D61—22no—
241C22×D61—22no—
242C22×D61—22no—
243C22×D61—41no—
244C22×D61—44no—
245C22×D61—44no—
246C22×D61—44no—
247C22×D61C2×C22⋊C411yescharacteristic
248C22×D61C2×C22⋊C411yescharacteristic
249Q8×C61—42no—
250D4×C61—42no—
251D4×C61—44no—
252D4×C61—42no—
253D4×C61—42no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—22no—
257C2×C4×C61—22no—
258C2×C4×C61—44no—
259C2×C4×C61C2×C22⋊C411yescharacteristic
260C2×C4×C61C2×C22⋊C411yescharacteristic
261C2×C4×C61C2×C22⋊C411yescharacteristic
262C2×C4×C61C23×C411yescharacteristic
263C2×C4×C61C22×D411yescharacteristic
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—41no—
274C2×Dic61—44no—
275C2×Dic61C2×C22⋊C411yescharacteristic
276C2×Dic61C2×C22⋊C411yescharacteristic
277C2×Dic61C2×C22⋊C411yescharacteristic
278C2×Dic61C2×C22⋊C411yescharacteristic
279C2×Dic61C2×C22⋊C411yescharacteristic
280C2×Dic61C2×C22⋊C411yescharacteristic
281D244—42no—
282D244—44no—
283D244—42no—
284D244—42no—
285C4×D61—22no—
286C4×D61—22no—
287C4×D61—22no—
288C4×D61—22no—
289C4×D61—22no—
290C4×D61—22no—
291C4×D61—22no—
292C4×D61—22no—
293C4×D61—41no—
294C4×D61—41no—
295C4×D61—44no—
296C4×D61—44no—
297C4×D61—44no—
298C4×D61#10014yes—
299C4×D61#10414yes—
300Dic122—44no—
301Dic122—42no—
302C8×C61—24no—
303C8×C61—42no—
304#92418—24no—
305#92418—42no—
306#17647—611nomaximal, Sylow
307C22×C61—21no—
308C22×C61—21no—
309C22×C61—21no—
310C22×C61—42no—
311C22×C61—42no—
312C22×C61—42no—
313C22×C61C22×C22⋊C411yescharacteristic
314C2×D61—21no—
315C2×D61—21no—
316C2×D61—21no—
317C2×D61—21no—
318C2×D61—44no—
319C2×D61—42no—
320C2×D61—42no—
321C2×D61—42no—
322C2×D61—88no—
323C2×D61C2×#10012yes—
324C2×D61C2×#10412yes—
325C4×C61—21no—
326C4×C61—21no—
327C4×C61—21no—
328C4×C61—42no—
329C4×C61C2×#10012yes—
330C4×C61C2×#10412yes—
331C4×C61#42112yes—
332Dic61—21no—
333Dic61—21no—
334Dic61—21no—
335Dic61—21no—
336Dic61—42no—
337Dic61C2×#10012yes—
338Dic61C2×#10412yes—
339C2×#532—611no—
340#2920—611no—
341C2×C4×M16—611no—
342#1530—614no—
343#1527—614no—
344#1525—611no—
345#1525—611no—
346#1525—611no—
347#1525D6111yescharacteristic, complemented
348C2×C61—41no—
349C2×C61—82no—
350C2×C61C22×#10011yescharacteristic
351C2×C61C22×#10411yescharacteristic
352C2×C61C2×#42111yescharacteristic
353D61—22no—
354D61—22no—
355D61—82no—
356C22×#41—612no—
357C22×M16—1221no—
358#532—612no—
359#532—611no—
360#532—611no—
361#532—612no—
362#532—611no—
363#532—1221no—
364#532—1221no—
365#532—1221no—
366#532—1221no—
367#532C2×D6111yescharacteristic
368C2×C4⋊Q8—611no—
369C2×#127—611no—
370C2×#124—612no—
371C2×#121—1221no—
372C2×#121—1221no—
373C2×C4×D4—1221no—
374C2×C4×D4—1221no—
375C22×C42—611no—
376#432—612no—
377#432—612no—
378#431—1221no—
379#431—1221no—
380#423—1221no—
381#407—1221no—
382C4×M16—611no—
383C4×M16—611no—
384C4×M16—612no—
385C4×M16—1221no—
386C4×M16—1221no—
387C4×M16C2×D6111yescharacteristic, complemented
388C4×M16C2×D6111yescharacteristic, complemented
389C2×#97—612no—
390C2×C4×C8—1221no—
391#329—614no—
392#329—614no—
393#329—614no—
394#329C2×D6114yescomplemented
395C61#1764711yescharacteristic, complemented, Sylow
396C2×#41—612no—
397C2×#41—612no—
398C2×#41—612no—
399C2×#41—1222no—
400C2×#41—1222no—
401C2×#41—2448no—
402C2×#41C4×D6112yes—
403C22×Q8—612no—
404C22×D4—612no—
405C22×D4—1221no—
406C22×D4—1221no—
407C22×D4—1222no—
408C23×C4—611no—
409C23×C4—611no—
410C23×C4—611no—
411C23×C4—1222no—
412C2×M16—21no—
413C2×M16—21no—
414C2×M16—1222no—
415C2×M16—1228no—
416C2×M16—1221no—
417C2×M16—1221no—
418C22×C8—1222no—
419C4⋊Q8—612no—
420C4⋊Q8—611no—
421C4⋊Q8—1221no—
422C4⋊Q8—1221no—
423C4⋊Q8C4×D6111yescharacteristic
424#127—612no—
425#127—611no—
426#127—1221no—
427#127—1221no—
428#127C4×D6111yescharacteristic
429#124—612no—
430#124—612no—
431#124—612no—
432#124—1222no—
433#124—1222no—
434#124C22×D6112yes—
435#121—21no—
436#121—21no—
437#121—1222no—
438#121—1222no—
439#121—1221no—
440#121—1221no—
441#121—1221no—
442#121—1221no—
443#121—1222no—
444#121—1222no—
445#121—1221no—
446#121—1221no—
447C4×D4—21no—
448C4×D4—21no—
449C4×D4—1222no—
450C4×D4—1222no—
451C4×D4—1221no—
452C4×D4—1221no—
453C4×D4—1222no—
454C4×D4—1222no—
455C4×D4—1221no—
456C4×D4—1221no—
457C4×D4—1221no—
458C4×D4—1221no—
459C2×C4⋊C4—1221no—
460C2×C4⋊C4—1221no—
461C2×C22⋊C4—1222no—
462C2×C22⋊C4—1222no—
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—611no—
474C2×C42C22×D6111yescharacteristic
475C4⋊C8—1222no—
476C4⋊C8—1222no—
477C22⋊C8—1222no—
478C22⋊C8—1222no—
479#97—612no—
480#97—612no—
481#97—1222no—
482#97C22×D6112yes—
483#97C22×D6112yes—
484C4×C8—21no—
485C4×C8—21no—
486C4×C8—1222no—
487C4×C8—1221no—
488C4×C8—1221no—
489C24—611no—
490C24—1222no—
491#41—44no—
492#41—2444no—
493#41—2448no—
494#41—2448no—
495#41—2444no—
496#41—2444no—
497C2×Q8—612no—
498C2×Q8—1222no—
499C2×Q8—2444no—
500C2×Q8C2×C4×D6112yes—
501C2×D4—22no—
502C2×D4—21no—
503C2×D4—21no—
504C2×D4—612no—
505C2×D4—1222no—
506C2×D4—1222no—
507C2×D4—1222no—
508C2×D4—1222no—
509C2×D4—1222no—
510C2×D4—1222no—
511C2×D4—1221no—
512C2×D4—1221no—
513C2×D4—2444no—
514C2×D4—2444no—
515C2×D4—2444no—
516C2×D4—2448no—
517C2×D4C2×C4×D6112yes—
518C22×C4—22no—
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—611no—
530C22×C4—611no—
531C22×C4—611no—
532C22×C4—611no—
533C22×C4—611no—
534C22×C4—1224no—
535C22×C4—1224no—
536C22×C4—1222no—
537C22×C4—1222no—
538C22×C4—1222no—
539C22×C4—1222no—
540C22×C4—2442no—
541C22×C4—2448no—
542C22×C4C2×C4×D6111yescharacteristic
543C22×C4D4×D6111yescharacteristic
544C22×C4D4×D6111yescharacteristic
545M16—24no—
546M16—42no—
547M16—1228no—
548M16—1224no—
549M16—2442no—
550C2×C8—22no—
551C2×C8—22no—
552C2×C8—1228no—
553C2×C8—1222no—
554C2×C8—1222no—
555C4⋊C4—21no—
556C4⋊C4—21no—
557C4⋊C4—1222no—
558C4⋊C4—1222no—
559C4⋊C4—1221no—
560C4⋊C4—1221no—
561C22⋊C4—22no—
562C22⋊C4—22no—
563C22⋊C4—1222no—
564C22⋊C4—1222no—
565C22⋊C4—1222no—
566C22⋊C4—1222no—
567C22⋊C4—1222no—
568C22⋊C4—1222no—
569C42—611no—
570C42—611no—
571C42—611no—
572C42—611no—
573C42—611no—
574C42—611no—
575C42—611no—
576C42—611no—
577C42—611no—
578C42—611no—
579C42—611no—
580C42—611no—
581C42C2×C4×D6111yescharacteristic
582C42D4×D6111yescharacteristic
583C42D4×D6111yescharacteristic
584C42C23×D6111yescharacteristic
585C23—22no—
586C23—611no—
587C23—611no—
588C23—1222no—
589C23—1222no—
590C23—1222no—
591C23—2441no—
592C23—2444no—
593C23—2444no—
594C23—2444no—
595C23C22⋊C4×D6111yescharacteristic
596Q8—42no—
597Q8—2442no—
598Q8—2444no—
599D4—42no—
600D4—44no—
601D4—42no—
602D4—42no—
603D4—2442no—
604D4—2444no—
605D4—2444no—
606D4—2444no—
607D4—2444no—
608D4—2444no—
609D4—2444no—
610D4—2442no—
611D4—2442no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—44no—
617C2×C4—611no—
618C2×C4—611no—
619C2×C4—614no—
620C2×C4—614no—
621C2×C4—611no—
622C2×C4—611no—
623C2×C4—611no—
624C2×C4—611no—
625C2×C4—1222no—
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—2441no—
637C2×C4—2441no—
638C2×C4—2444no—
639C2×C4—2444no—
640C2×C4—2441no—
641C2×C4—2444no—
642C2×C4—2444no—
643C2×C4C22⋊C4×D6111yescharacteristic
644C2×C4C22⋊C4×D6111yescharacteristic
645C2×C4C22⋊C4×D6111yescharacteristic
646C2×C4C22×C4×D6111yescharacteristic
647C2×C4C2×D4×D6111yescharacteristic
648C2×C4C2×D4×D6111yescharacteristic
649C8—24no—
650C8—42no—
651C8—1224no—
652C8—2442no—
653C22—21no—
654C22—21no—
655C22—21no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—612no—
660C22—612no—
661C22—1221no—
662C22—1221no—
663C22—1221no—
664C22—1221no—
665C22—4888no—
666C22—2444no—
667C22—2442no—
668C22—2442no—
669C22—2442no—
670C22C2×C22⋊C4×D6111yescharacteristic
671C4—21no—
672C4—21no—
673C4—21no—
674C4—42no—
675C4—612no—
676C4—612no—
677C4—1221no—
678C4—1221no—
679C4—1221no—
680C4—1221no—
681C4—2442no—
682C4#100×D6112yes—
683C4#104×D6112yes—
684C4#1683977712yes—
685C2—41no—
686C2—82no—
687C2—1222no—
688C2—1222no—
689C2—4882no—
690C2C2×#100×D6111yescharacteristic
691C2C2×#104×D6111yescharacteristic
692C2C2×#1683977711yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1525C2
socleC22×C61C22×C22⋊C4
prime radical#532×D61C2
cosocleC42×C61C24
hypercenter#1525D61
hypocenterC61#17647
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1525

Complements

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

NH ≅ G/Ntype
2. #21454450C2semidirect
3. C4×M16×D61C2semidirect
4. #21444428C2semidirect
5. #21444415C2semidirect
6. #21444411C2semidirect
7. #21444410C2semidirect
8. #21444410C2semidirect
9. C61×#1525C2semidirect
30. #16839817C22semidirect
35. C4×#11758194C22semidirect
36. C4×#11758194C22semidirect
40. #97×D61C22semidirect
42. #16839502C22semidirect
43. #16839501C22semidirect
44. #16839501C22semidirect
46. C4×M16×C61C22semidirect
47. C4×M16×C61C22semidirect
48. C61×#329C22semidirect
347. #1525D61semidirect
387. C4×M16C2×D61semidirect
388. C4×M16C2×D61semidirect
394. #329C2×D61semidirect
395. C61#17647semidirect

Maximal subgroups

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

subgroupindex
#532×D612
#214544502
C4×M16×D612
#214444282
#214444152
#214444112
#214444102
C61×#15252
#1764761