Finite Group Databasebeta

← #29696523

Subgroups — #29696523

Order 15616.

1126 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459383C211yesmaximal, characteristic
2M16×D244C211yesmaximal, characteristic, complemented
3#21448283C211yesmaximal, characteristic, complemented
4#21446796C211yesmaximal, characteristic, complemented
5#21446763C211yesmaximal, characteristic, complemented
6#21446264C212yesmaximal, complemented
7#21446118C212yesmaximal, complemented
8#21445917C212yesmaximal, complemented
9#21445839C211yesmaximal, characteristic, complemented
10#21445527C212yesmaximal, complemented
11C4×#104×C61C211yesmaximal, characteristic, complemented
12C2×M16×D61C2212yes—
13#16840485C411yescharacteristic, complemented
14#16840477C411yescharacteristic, complemented
15#16840472C411yescharacteristic, complemented
16#16840462C412yescomplemented
17#16840453C412yescomplemented
18#16840452—21no—
19#16840441—22no—
20#16840417C411yescharacteristic, complemented
21#16840408—21no—
22#16840319C2211yescharacteristic
23#16840296—21no—
24#16840290—22no—
25#16840269C2212yes—
26C4×#11758222—21no—
27C2×C4×D244C2211yescharacteristic
28C2×#14519734C2212yes—
29#117×D61C2212yes—
30#16839841—21no—
31#16839835—22no—
32C8×D244—21no—
33#16839786C2211yescharacteristic
34C2×#14519704C2212yes—
35#16839725C2212yes—
36#16839664—21no—
37#16839660—21no—
38#16839653—22no—
39C4×C4⋊Dic61C2211yescharacteristic
40C4×#11758198C2212yes—
41#16839523C2212yes—
42#16839523C2212yes—
43#16839522C2212yes—
44#16839522C2212yes—
45#16839521C2212yes—
46C4×#41×C61C2211yescharacteristic
47C2×#104×C61C2211yescharacteristic
48C2×#104×C61C2211yescharacteristic
49C4×M16×C61C2211yescharacteristic
50C43×C61C2211yescharacteristic
51C61×#339C2211yescharacteristic
52C61×#339C2211yescharacteristic
53C2×#11758222—21no—
54C22×D244D411yescharacteristic
55C22×C4×D61C2312yes—
56C2×#11758194C2312yes—
57C2×#11758189—22no—
58M16×D61—24no—
59M16×D61—42no—
60C2×C8×D61—22no—
61#14519777C2×C411yescharacteristic
62#14519777C2×C411yescharacteristic
63#14519772—22no—
64#14519771—22no—
65#14519770—22no—
66#14519769—21no—
67#14519769C2×C412yes—
68#14519767—22no—
69#14519765—21no—
70#14519763—22no—
71#14519762—22no—
72#14519760—22no—
73#14519759—22no—
74#14519758—22no—
75#14519756—21no—
76#14519756—21no—
77#14519756—21no—
78#14519755—22no—
79#14519753C2×C412yes—
80#14519753C2×C412yes—
81#14519751—21no—
82#14519750—22no—
83#14519748C2×C412yes—
84#14519748C2×C412yes—
85#14519745—22no—
86C4×Dic122—21no—
87C4×Dic122—21no—
88C4×Dic122—21no—
89C4×D244—21no—
90C4×D244C2×C412yes—
91C4×D244C2×C411yescharacteristic
92C4×D244C2×C411yescharacteristic
93C4×D244D411yescharacteristic
94C4×D244D412yes—
95C4×D244D411yescharacteristic
96C4×C22⋊D61—22no—
97C4×C22⋊D61—22no—
98C4×C22⋊D61—22no—
99#14519736—21no—
100#14519736C2312yes—
101#14519735C2×C412yes—
102#14519735C2×C412yes—
103#14519734—22no—
104#14519734—22no—
105#14519734—24no—
106#14519734—42no—
107#14519733—22no—
108C2×C4⋊Dic61—21no—
109C2×C4⋊Dic61D411yescharacteristic
110C2×#11758196—22no—
111C4⋊C4×D61C2×C412yes—
112C4⋊C4×D61C2×C412yes—
113C2×#11758198—22no—
114C2×#11758198—22no—
115C2×#11758198D412yes—
116C22⋊C4×D61—22no—
117C2×C4×Dic61C2312yes—
118C42×D61—22no—
119C4⋊#92418—21no—
120C4⋊#92418—21no—
121#14519704—42no—
122#14519689—22no—
123#14519689—22no—
124#14519684—21no—
125#14519684—22no—
126C2×#41×C61C2311yescharacteristic
127C2×M16×C61C2311yescharacteristic
128C2×M16×C61C2311yescharacteristic
129C4×Q8×C61C2×C411yescharacteristic
130C4×Q8×C61C2×C411yescharacteristic
131C4×D4×C61—21no—
132C4×D4×C61—21no—
133C4×D4×C61C2×C411yescharacteristic
134C4×D4×C61C2×C411yescharacteristic
135#117×C61—21no—
136#117×C61C2311yescharacteristic
137C2×C42×C61—21no—
138C2×C42×C61—21no—
139C2×C42×C61—21no—
140C2×C42×C61C2311yescharacteristic
141C2×C42×C61C2311yescharacteristic
142C2×C42×C61C2311yescharacteristic
143#104×C61—22no—
144#104×C61—22no—
145#97×C61—21no—
146C4×C8×C61—21no—
147C23×D61C2×D412yes—
148#41×C61—21no—
149#41×C61—21no—
150#41×C61—21no—
151#41×C61—21no—
152C2×Q8×C61C22×C411yescharacteristic
153C2×D4×C61—21no—
154C2×D4×C61C22×C411yescharacteristic
155C22×C4×C61—21no—
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61C2411yescharacteristic
163C2×C22⋊D61—22no—
164C2×C22⋊D61—22no—
165C22×Dic61—22no—
166C22×Dic61C2×D412yes—
167#11758222—44no—
168C2×D244—42no—
169C2×D244C22⋊C411yescharacteristic
170C2×D244C22⋊C411yescharacteristic
171C2×D244C22⋊C412yes—
172C2×C4×D61—22no—
173C2×C4×D61—22no—
174C2×C4×D61—22no—
175C2×C4×D61—22no—
176C2×C4×D61—22no—
177C2×C4×D61—22no—
178C2×C4×D61C22×C412yes—
179C2×C4×D61C22×C412yes—
180C2×C4×D61C2×D412yes—
181C2×C4×D61C2×D412yes—
182C2×Dic122—21no—
183C2×Dic122—21no—
184M16×C61—21no—
185M16×C61—21no—
186M16×C61—21no—
187M16×C61#4112yescomplemented
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61C22×C411yescharacteristic
193C4⋊C4×C61C22×C411yescharacteristic
194C22⋊C4×C61—21no—
195C22⋊C4×C61—21no—
196C22⋊C4×C61—21no—
197C42×C61—21no—
198C42×C61—21no—
199C42×C61—21no—
200C42×C61—21no—
201C42×C61—21no—
202C42×C61—21no—
203C42×C61—42no—
204C42×C61—42no—
205C42×C61C22×C411yescharacteristic
206C42×C61C22×C411yescharacteristic
207C42×C61C2×D411yescharacteristic
208C42×C61C2×D411yescharacteristic
209C42×C61#4112yescomplemented
210C22⋊Dic61—22no—
211C22⋊Dic61—22no—
212C22⋊Dic61—22no—
213#11758198—22no—
214#11758198—22no—
215#11758198—22no—
216#11758198—22no—
217#11758198—44no—
218#11758198—44no—
219#11758198C22⋊C412yes—
220#11758198C22⋊C412yes—
221#11758198C22⋊C412yes—
222#11758198C22⋊C412yes—
223C4⋊Dic61—21no—
224C4⋊Dic61—21no—
225C4⋊Dic61—42no—
226C4⋊Dic61C22⋊C411yescharacteristic
227C4⋊Dic61C22⋊C411yescharacteristic
228C4⋊Dic61C22⋊C412yes—
229C4⋊Dic61C22×C412yes—
230C4⋊Dic61C22×C412yes—
231#11758196—22no—
232#11758196—22no—
233#11758196—22no—
234#11758196—22no—
235#11758196—22no—
236#11758196—22no—
237#11758196—44no—
238C4×Dic61—22no—
239C4×Dic61—22no—
240C4×Dic61—22no—
241C4×Dic61—22no—
242C4×Dic61—22no—
243#11758194—22no—
244#11758194—22no—
245C2×#92418—22no—
246#11758189—44no—
247C8×D61—44no—
248C23×C61—21no—
249C23×C61C22×D411yescharacteristic
250C22×D61—22no—
251C22×D61—22no—
252C22×D61—42no—
253C22×D61—42no—
254C22×D61C2×C22⋊C412yes—
255C22×D61C2×C22⋊C412yes—
256Q8×C61—21no—
257Q8×C61—21no—
258D4×C61—21no—
259D4×C61—21no—
260D4×C61—41no—
261D4×C61—41no—
262C2×C4×C61—22no—
263C2×C4×C61—22no—
264C2×C4×C61—21no—
265C2×C4×C61—21no—
266C2×C4×C61—21no—
267C2×C4×C61—22no—
268C2×C4×C61—41no—
269C2×C4×C61—41no—
270C2×C4×C61—41no—
271C2×C4×C61—41no—
272C2×C4×C61—41no—
273C2×C4×C61—41no—
274C2×C4×C61—41no—
275C2×C4×C61—42no—
276C2×C4×C61—81no—
277C2×C4×C61C2×C22⋊C411yescharacteristic
278C2×C4×C61C2×C22⋊C411yescharacteristic
279C2×C4×C61C2×C22⋊C411yescharacteristic
280C2×C4×C61C2×C22⋊C411yescharacteristic
281C2×C4×C61C23×C411yescharacteristic
282C2×C4×C61C22×D411yescharacteristic
283C2×C4×C61C2×#4111yescharacteristic
284C2×C4×C61C2×#4111yescharacteristic
285C2×C4×C61H(5,𝔽2)11yescharacteristic
286C2×C4×C612−1+411yescharacteristic
287C22⋊D61—44no—
288C22⋊D61—82no—
289C2×Dic61—22no—
290C2×Dic61—22no—
291C2×Dic61—22no—
292C2×Dic61—22no—
293C2×Dic61—42no—
294C2×Dic61—42no—
295C2×Dic61—42no—
296C2×Dic61C2×C22⋊C412yes—
297C2×Dic61C2×C22⋊C412yes—
298D244—22no—
299D244—22no—
300D244—42no—
301C4×D61—22no—
302C4×D61—22no—
303C4×D61—22no—
304C4×D61—22no—
305C4×D61—22no—
306C4×D61—22no—
307C4×D61—22no—
308C4×D61—22no—
309C4×D61—44no—
310C4×D61—82no—
311Dic122—42no—
312Dic122—81no—
313C8×C61—22no—
314C8×C61—41no—
315#92418—42no—
316#22507—611nomaximal, Sylow
317C22×C61—21no—
318C22×C61—21no—
319C22×C61—41no—
320C22×C61—41no—
321C22×C61—41no—
322C22×C61C22×C22⋊C411yescharacteristic
323C22×C61#54611yescharacteristic
324C22×C61#54711yescharacteristic
325C2×D61—22no—
326C2×D61—22no—
327C2×D61—22no—
328C2×D61—22no—
329C2×D61—22no—
330C2×D61—22no—
331C2×D61—82no—
332C2×D61—82no—
333C4×C61—21no—
334C4×C61—21no—
335C4×C61—21no—
336C4×C61—42no—
337C4×C61—42no—
338C4×C61—41no—
339C4×C61—41no—
340C4×C61—41no—
341C4×C61#42112yes—
342C4×C61C4×#4111yescharacteristic
343C4×C61#52011yescharacteristic
344C4×C61#54611yescharacteristic
345C4×C61#54711yescharacteristic
346Dic61—22no—
347Dic61—22no—
348Dic61—42no—
349Dic61—42no—
350Dic61—82no—
351#3475—611no—
352D4×M16—611no—
353#2293—611no—
354#1994—611no—
355#1986—611no—
356#1897—612no—
357#1871—612no—
358#1826—612no—
359#1806—611no—
360C4×#104D6111yescharacteristic, complemented
361#1733—612no—
362C2×C61—21no—
363C2×C61—21no—
364C2×C61—81no—
365C2×C61#233911yescharacteristic
366C2×C61C2×#42111yescharacteristic
367C2×C61#290111yescharacteristic
368D61—42no—
369D61—42no—
370C22×M16—612no—
371#556—611no—
372#555—611no—
373#554—611no—
374#553—612no—
375#552—612no—
376#552—1221no—
377#551—1222no—
378#547—611no—
379#546—1221no—
380#533—611no—
381#529—1221no—
382#529—1222no—
383#525—612no—
384C4×#41—1221no—
385C4×#41C2×D6111yescharacteristic
386C2×C4×D4—611no—
387C2×#117—612no—
388C2×#117—612no—
389#436—1221no—
390#435—1222no—
391C8×D4—1221no—
392#423—611no—
393C2×#104—612no—
394C2×#104C2×D6111yescharacteristic
395C2×#104C2×D6111yescharacteristic
396#407—612no—
397C4×M16C2×D6111yescharacteristic
398#390—1221no—
399#389—1221no—
400#388—1222no—
401C4×C4⋊C4—611no—
402C4×C22⋊C4—612no—
403C43C2×D6111yescharacteristic
404#339—612no—
405#339—612no—
406#339—612no—
407#339—612no—
408#339—612no—
409#339C2×D6111yescharacteristic
410#339C2×D6111yescharacteristic
411C61#2250711yescharacteristic, complemented, Sylow
412C2×#41—1221no—
413C2×#41C22×D6111yescharacteristic
414C22×D4—611no—
415C23×C4—612no—
416C2×M16—612no—
417C2×M16—1222no—
418C2×M16—1224no—
419C2×M16—2442no—
420C2×M16C22×D6111yescharacteristic
421C2×M16C22×D6111yescharacteristic
422C22×C8—1222no—
423C4⋊Q8—611no—
424C4⋊Q8—611no—
425#126—1222no—
426#126—1222no—
427#126—1222no—
428#125—612no—
429#125—1221no—
430#125—1222no—
431#124—1221no—
432#124—1222no—
433#123—1222no—
434#123—1222no—
435#123—1222no—
436#123—1222no—
437C22⋊Q8—612no—
438C22⋊Q8—612no—
439C22⋊Q8—1222no—
440C22⋊Q8—1221no—
441C22⋊Q8—1221no—
442C22⋊Q8—1221no—
443#121—612no—
444#121—612no—
445#121—1222no—
446#121—1221no—
447#120—1222no—
448C4×Q8—1221no—
449C4×Q8—1221no—
450C4×Q8—1221no—
451C4×Q8C4×D6111yescharacteristic
452C4×Q8C4×D6111yescharacteristic
453C4×D4—21no—
454C4×D4—21no—
455C4×D4—611no—
456C4×D4—612no—
457C4×D4—611no—
458C4×D4—611no—
459C4×D4—611no—
460C4×D4—612no—
461C4×D4—1221no—
462C4×D4—1222no—
463C4×D4—1222no—
464C4×D4—1222no—
465C4×D4C4×D6111yescharacteristic
466C4×D4C4×D6111yescharacteristic
467#117—21no—
468#117—612no—
469#117—612no—
470#117—612no—
471#117—1222no—
472#117—1222no—
473#117—1222no—
474#117—1221no—
475#117—1224no—
476#117—2442no—
477#117C22×D6111yescharacteristic
478C2×C4⋊C4—611no—
479C2×C4⋊C4—612no—
480C2×C4⋊C4—612no—
481C2×C4⋊C4—1222no—
482C2×C4⋊C4—1221no—
483C2×C22⋊C4—612no—
484C2×C22⋊C4—1222no—
485C2×C22⋊C4—1222no—
486C2×C22⋊C4—1222no—
487C2×C42—21no—
488C2×C42—21no—
489C2×C42—21no—
490C2×C42—612no—
491C2×C42—1222no—
492C2×C42C22×D6111yescharacteristic
493C2×C42C22×D6111yescharacteristic
494C2×C42C22×D6111yescharacteristic
495C4⋊C8—1221no—
496C4⋊C8—1221no—
497#104—22no—
498#104—22no—
499#104—2442no—
500C22⋊C8—1222no—
501C22⋊C8—1222no—
502#97—21no—
503C4×C8—21no—
504#95—1222no—
505#95—1221no—
506C24—612no—
507#41—21no—
508#41—21no—
509#41—21no—
510#41—21no—
511#41—2444no—
512C2×Q8—1221no—
513C2×Q8—1221no—
514C2×Q8C2×C4×D6111yescharacteristic
515C2×D4—21no—
516C2×D4—611no—
517C2×D4—611no—
518C2×D4—612no—
519C2×D4—1222no—
520C2×D4—1222no—
521C2×D4—2442no—
522C2×D4C2×C4×D6111yescharacteristic
523C22×C4—21no—
524C22×C4—21no—
525C22×C4—21no—
526C22×C4—21no—
527C22×C4—21no—
528C22×C4—612no—
529C22×C4—612no—
530C22×C4—612no—
531C22×C4—612no—
532C22×C4—612no—
533C22×C4—1222no—
534C22×C4—1222no—
535C22×C4—1222no—
536C22×C4—1222no—
537C22×C4—1222no—
538C22×C4—1222no—
539C22×C4—1222no—
540C22×C4D4×D6111yescharacteristic
541C22×C4D4×D6111yescharacteristic
542C22×C4C23×D6111yescharacteristic
543M16—21no—
544M16—21no—
545M16—21no—
546M16—1222no—
547M16—1222no—
548M16—2444no—
549M16#1175822212yescomplemented
550C2×C8—21no—
551C2×C8—21no—
552C2×C8—1222no—
553C2×C8—2444no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—611no—
557C4⋊C4—611no—
558C4⋊C4—612no—
559C4⋊C4—612no—
560C4⋊C4—612no—
561C4⋊C4—1222no—
562C4⋊C4—1222no—
563C4⋊C4—1222no—
564C4⋊C4—1222no—
565C4⋊C4—1222no—
566C4⋊C4—1222no—
567C4⋊C4—1221no—
568C4⋊C4—1221no—
569C4⋊C4—2444no—
570C4⋊C4—2442no—
571C4⋊C4C2×C4×D6111yescharacteristic
572C4⋊C4C2×C4×D6111yescharacteristic
573C22⋊C4—21no—
574C22⋊C4—21no—
575C22⋊C4—21no—
576C22⋊C4—612no—
577C22⋊C4—612no—
578C22⋊C4—612no—
579C22⋊C4—612no—
580C22⋊C4—1222no—
581C22⋊C4—1222no—
582C22⋊C4—1222no—
583C22⋊C4—1222no—
584C22⋊C4—1222no—
585C22⋊C4—1222no—
586C22⋊C4—1222no—
587C22⋊C4—2444no—
588C22⋊C4—2444no—
589C42—21no—
590C42—21no—
591C42—21no—
592C42—21no—
593C42—21no—
594C42—21no—
595C42—42no—
596C42—42no—
597C42—1222no—
598C42—1222no—
599C42—1222no—
600C42—1222no—
601C42—1222no—
602C42C2×C4×D6111yescharacteristic
603C42C2×C4×D6111yescharacteristic
604C42#1175822212yescomplemented
605C42D4×D6111yescharacteristic
606C42D4×D6111yescharacteristic
607C23—21no—
608C23—612no—
609C23—612no—
610C23—1222no—
611C23—1222no—
612C23—2442no—
613C23—2442no—
614C23C2×D4×D6111yescharacteristic
615Q8—21no—
616Q8—21no—
617Q8—4881no—
618Q8—2442no—
619D4—21no—
620D4—21no—
621D4—41no—
622D4—41no—
623D4—1222no—
624D4—1222no—
625D4—4882no—
626D4—2442no—
627D4—2444no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—22no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—81no—
643C2×C4—612no—
644C2×C4—612no—
645C2×C4—1222no—
646C2×C4—1222no—
647C2×C4—1222no—
648C2×C4—1222no—
649C2×C4—1222no—
650C2×C4—1222no—
651C2×C4—1222no—
652C2×C4—1222no—
653C2×C4—1222no—
654C2×C4—1222no—
655C2×C4—1222no—
656C2×C4—1222no—
657C2×C4—4882no—
658C2×C4—2444no—
659C2×C4—2442no—
660C2×C4—2442no—
661C2×C4—2442no—
662C2×C4C22⋊C4×D6111yescharacteristic
663C2×C4C22⋊C4×D6111yescharacteristic
664C2×C4C22⋊C4×D6111yescharacteristic
665C2×C4C22⋊C4×D6111yescharacteristic
666C2×C4C22×C4×D6111yescharacteristic
667C2×C4C2×D4×D6111yescharacteristic
668C2×C4C2×#1175822211yescharacteristic
669C2×C4C2×#1175822211yescharacteristic
670C2×C4#1451982311yescharacteristic
671C2×C4#1451982411yescharacteristic
672C8—22no—
673C8—41no—
674C8—2442no—
675C22—21no—
676C22—21no—
677C22—41no—
678C22—41no—
679C22—41no—
680C22—1222no—
681C22—1222no—
682C22—1222no—
683C22—1222no—
684C22—1222no—
685C22—1222no—
686C22—4882no—
687C22—4882no—
688C22C2×C22⋊C4×D6111yescharacteristic
689C22#1684040811yescharacteristic
690C22#1684041711yescharacteristic
691C4—21no—
692C4—21no—
693C4—21no—
694C4—42no—
695C4—42no—
696C4—41no—
697C4—41no—
698C4—41no—
699C4—1222no—
700C4—1222no—
701C4—4882no—
702C4—2442no—
703C4—2442no—
704C4#1683977712yes—
705C4C4×#1175822211yescharacteristic
706C4#1684024211yescharacteristic
707C4#1684040811yescharacteristic
708C4#1684041711yescharacteristic
709C2—21no—
710C2—21no—
711C2—81no—
712C2—2442no—
713C2—2442no—
714C2#2144865011yescharacteristic
715C2C2×#1683977711yescharacteristic
716C2#2145436311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×#104×C61C2
socleC22×C61C22×C22⋊C4
prime radical#21459383C2
cosocleC22×C4×C61C24
hypercenterC4×#104D61
hypocenterC61#22507
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#104

Complements

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

NH ≅ G/Ntype
2. M16×D244C2semidirect
3. #21448283C2semidirect
4. #21446796C2semidirect
5. #21446763C2semidirect
6. #21446264C2semidirect
7. #21446118C2semidirect
8. #21445917C2semidirect
9. #21445839C2semidirect
10. #21445527C2semidirect
11. C4×#104×C61C2semidirect
13. #16840485C4semidirect
14. #16840477C4semidirect
15. #16840472C4semidirect
16. #16840462C4semidirect
17. #16840453C4semidirect
20. #16840417C4semidirect
187. M16×C61#41semidirect
209. C42×C61#41semidirect
360. C4×#104D61semidirect
411. C61#22507semidirect
549. M16#11758222semidirect
604. C42#11758222semidirect

Maximal subgroups

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

subgroupindex
#214593832
M16×D2442
#214482832
#214467962
#214467632
#214462642
#214461182
#214459172
#214458392
#214455272
C4×#104×C612
#2250761