Finite Group Databasebeta

← #29647008

Subgroups — #29647008

Order 15616.

1074 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459445C211yesmaximal, characteristic
2#21454948C211yesmaximal, characteristic, complemented
3C4×#14519783C211yesmaximal, characteristic, complemented
4#21444271C211yesmaximal, characteristic, complemented
5#21444268C212yesmaximal, complemented
6#21444258C211yesmaximal, characteristic, complemented
7#21444257C212yesmaximal, complemented
8#21444235C211yesmaximal, characteristic, complemented
9#21444234C212yesmaximal, complemented
10#21444200C212yesmaximal, complemented
11C61×#1476C211yesmaximal, characteristic, complemented
12C2×#14519783—21no—
13Q8×Dic122C411yescharacteristic
14#16840477—21no—
15#16840477C411yescharacteristic
16#16840475—22no—
17#16840475C412yes—
18#16840434C411yescharacteristic
19#16840433—21no—
20#16840433C411yescharacteristic
21#16840431C412yes—
22#16840312—21no—
23#16840312C2211yescharacteristic
24#16840309—22no—
25#16840309C2212yes—
26C4×#11758226C2212yes—
27C4×#11758222—21no—
28C4×#11758222C2211yescharacteristic
29C8×Dic122—21no—
30#16839841—21no—
31#16839837—22no—
32#16839814C2212yes—
33#16839786C2211yescharacteristic
34#16839784C2212yes—
35#16839717—21no—
36#16839716—22no—
37C4×#11758194C2211yescharacteristic
38C4×#11758189C2212yes—
39C4×C8×D61C2212yes—
40Q8⋊#92418C2212yes—
41#16839491C2212yes—
42#16839491C2212yes—
43D4⋊#92418C2212yes—
44#16839489C2212yes—
45#16839489C2212yes—
46C4×#41×C61C2211yescharacteristic
47C61×#431C2211yescharacteristic
48C2×C4×C8×C61C2211yescharacteristic
49C61×Q8⋊C8C2211yescharacteristic
50C61×Q8⋊C8C2211yescharacteristic
51C61×D4⋊C8C2211yescharacteristic
52C61×D4⋊C8C2211yescharacteristic
53C2×#11758226C2×C412yes—
54C2×#11758222—21no—
55C2×#11758222—22no—
56C2×#11758222—21no—
57C2×#11758222D411yescharacteristic
58#14519783—28no—
59C2×#11758194—21no—
60C2×#11758194—21no—
61C2×#11758189—22no—
62C2×#11758189—22no—
63C2×C8×D61—22no—
64#14519777—21no—
65#14519777C2×C411yescharacteristic
66#14519776—22no—
67#14519776C2×C412yes—
68#14519774—21no—
69#14519774C2×C411yescharacteristic
70#14519773—22no—
71#14519773C2×C412yes—
72#14519765—21no—
73#14519765—22no—
74#14519763—22no—
75#14519763—22no—
76#14519763—22no—
77#14519752—22no—
78#14519752C2×C412yes—
79#14519751—21no—
80#14519751—21no—
81#14519751—22no—
82#14519751—21no—
83#14519751C2×C412yes—
84#14519750—22no—
85#14519750—22no—
86#14519750—22no—
87#14519748—22no—
88#14519748C2×C412yes—
89Q8×Dic61C2×C412yes—
90C4×Dic122—21no—
91C4×Dic122C2×C411yescharacteristic
92C4×Dic122D411yescharacteristic
93C4×D244—22no—
94C4×D244—21no—
95C4×D244—21no—
96C4×D244C2×C411yescharacteristic
97C4×D244D411yescharacteristic
98C4×D244C2312yes—
99#14519739—22no—
100#14519739C2×C412yes—
101C4×C22⋊D61—22no—
102C4×C22⋊D61—22no—
103C4×C22⋊D61C2×C412yes—
104C4×C22⋊D61D412yes—
105#14519736—21no—
106#14519736D411yescharacteristic
107#14519735—22no—
108#14519735C2×C412yes—
109#14519734—22no—
110#14519734D412yes—
111C42×D61—22no—
112C42×D61C2312yes—
113C4⋊#92418—21no—
114C4⋊#92418C2312yes—
115#14519706—22no—
116#14519689—22no—
117#14519688—21no—
118#14519687—22no—
119C4×#92418C2312yes—
120C8×Dic61—22no—
121C8×Dic61—22no—
122C2×#41×C61C2×C411yescharacteristic
123C22×C8×C61—21no—
124C4×Q8×C61C2×C411yescharacteristic
125C4×Q8×C61C2311yescharacteristic
126C4×D4×C61—21no—
127C4×D4×C61—21no—
128C4×D4×C61C2×C411yescharacteristic
129C4×D4×C61C2311yescharacteristic
130#117×C61—21no—
131#117×C61C2×C411yescharacteristic
132C2×C42×C61—21no—
133C2×C42×C61C2311yescharacteristic
134C4⋊C8×C61C2311yescharacteristic
135C4⋊C8×C61C2311yescharacteristic
136C22⋊C8×C61—21no—
137C4×C8×C61—21no—
138C4×C8×C61—21no—
139C4×C8×C61C2311yescharacteristic
140C4×C8×C61C2311yescharacteristic
141#41×C61—22no—
142#41×C61—41no—
143C2×Q8×C61C22×C411yescharacteristic
144C2×D4×C61—21no—
145C2×D4×C61C22×C411yescharacteristic
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61C22×C411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C2×C22⋊D61—22no—
153C2×C22⋊D61—22no—
154C2×C22⋊D61—22no—
155C2×C22⋊D61—22no—
156C2×C22⋊D61C22⋊C412yes—
157#11758226—24no—
158#11758226—42no—
159#11758222—24no—
160#11758222—44no—
161#11758222—82no—
162#11758222—82no—
163#11758222—81no—
164#11758222—81no—
165C2×D244—22no—
166C2×D244—21no—
167C2×D244—22no—
168C2×D244—21no—
169C2×D244C22⋊C411yescharacteristic
170C2×D244C22×C412yes—
171C2×C4×D61—22no—
172C2×C4×D61—22no—
173C2×C4×D61—22no—
174C2×C4×D61C22×C412yes—
175C2×C4×D61C2×D412yes—
176C2×C4×D61C2×D412yes—
177C2×Dic122—21no—
178C2×Dic122—22no—
179C2×Dic122C22⋊C411yescharacteristic
180C2×C8×C61—24no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61C22×C411yescharacteristic
188C4⋊C4×C61C22×C411yescharacteristic
189C22⋊C4×C61—21no—
190C22⋊C4×C61—21no—
191C22⋊C4×C61—21no—
192C42×C61—21no—
193C42×C61—21no—
194C42×C61—21no—
195C42×C61C22×C411yescharacteristic
196C42×C61C2×D411yescharacteristic
197C42×C61C2×D411yescharacteristic
198C42×C61C2411yescharacteristic
199C22⋊Dic61—22no—
200C22⋊Dic61C22⋊C412yes—
201#11758198—22no—
202#11758198—22no—
203#11758198—22no—
204#11758198—22no—
205#11758198—22no—
206#11758198C22⋊C412yes—
207#11758198C22⋊C412yes—
208C4⋊Dic61—21no—
209C4⋊Dic61—22no—
210C4⋊Dic61C22⋊C411yescharacteristic
211C4⋊Dic61C22⋊C411yescharacteristic
212C4⋊Dic61C22×C412yes—
213#11758196—22no—
214#11758196C22⋊C412yes—
215#11758196C22⋊C412yes—
216C4×Dic61—22no—
217C4×Dic61—22no—
218C4×Dic61C22×C412yes—
219C4×Dic61C2×D412yes—
220C4×Dic61C2×D412yes—
221#11758194—24no—
222#11758194—81no—
223C2×#92418—22no—
224C2×#92418—22no—
225#11758189—28no—
226#11758189—82no—
227C8×D61—28no—
228C23×C61—21no—
229C23×C61C2×C22⋊C411yescharacteristic
230C22×D61—22no—
231C22×D61C2×C22⋊C412yes—
232Q8×C61—21no—
233Q8×C61#13112yescomplemented
234D4×C61—21no—
235D4×C61—21no—
236D4×C61—42no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—22no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—42no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—42no—
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61C23×C411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C22⋊D61—24no—
256C22⋊D61—44no—
257C22⋊D61—82no—
258C22⋊D61—82no—
259C22⋊D61—82no—
260C2×Dic61—22no—
261C2×Dic61C2×C22⋊C412yes—
262C2×Dic61C2×C22⋊C412yes—
263C2×Dic61C2×C22⋊C412yes—
264D244—22no—
265D244—22no—
266D244—22no—
267D244—44no—
268D244—42no—
269D244—81no—
270D244—82no—
271C4×D61—22no—
272C4×D61—22no—
273C4×D61—22no—
274C4×D61—22no—
275C4×D61—24no—
276C4×D61—44no—
277C4×D61—42no—
278C4×D61—42no—
279C4×D61—44no—
280Dic122—82no—
281Dic122—81no—
282Dic122#10414yes—
283C8×C61—22no—
284C8×C61—22no—
285C8×C61—41no—
286C8×C61—41no—
287#92418—24no—
288#92418—42no—
289#17520—611nomaximal, Sylow
290C22×C61—21no—
291C22×C61—21no—
292C22×C61—21no—
293C22×C61—41no—
294C22×C61—41no—
295C22×C61—41no—
296C22×C61C22×C22⋊C411yescharacteristic
297C2×D61—22no—
298C2×D61—22no—
299C2×D61—22no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—42no—
303C4×C61—22no—
304C4×C61—21no—
305C4×C61—21no—
306C4×C61—21no—
307C4×C61—41no—
308C4×C61—41no—
309C4×C61C2×#10412yes—
310C4×C61C2×#13111yescharacteristic
311C4×C61#56811yescharacteristic
312C4×C61#57611yescharacteristic
313C4×C61#57711yescharacteristic
314Dic61—22no—
315Dic61—22no—
316Dic61—42no—
317Dic61C2×#10414yes—
318#3483—611no—
319#2991—611no—
320C4×#131—611no—
321#1492—611no—
322#1492—612no—
323#1489—612no—
324#1489—611no—
325#1484—611no—
326#1484—612no—
327#1476—612no—
328#1476D6111yescharacteristic, complemented
329C2×C61—41no—
330C2×C61—81no—
331C2×C61#288011yescharacteristic
332C2×C61#289711yescharacteristic
333C2×C61C22×#10411yescharacteristic
334D61—42no—
335D61—82no—
336C2×#131—1221no—
337Q82—611no—
338#555—611no—
339#555—612no—
340#555—1222no—
341#555—1221no—
342#550—611no—
343#550—612no—
344#550—611no—
345#550—1221no—
346#532—612no—
347#532—611no—
348#532—1221no—
349#532—1222no—
350C4×#41—611no—
351C4×#41—612no—
352C4×#41—1221no—
353C4×#41C2×D6111yescharacteristic
354C8×Q8—1221no—
355#436—1221no—
356#436—1222no—
357#431—612no—
358#431C2×D6111yescharacteristic
359#423—611no—
360#423—612no—
361#405—1222no—
362#405—1221no—
363C4×M16—611no—
364C4×M16—612no—
365C2×C4×C8—612no—
366C2×C4×C8C2×D6111yescharacteristic
367Q8⋊C8—612no—
368Q8⋊C8C2×D6111yescharacteristic
369Q8⋊C8C2×D6111yescharacteristic
370D4⋊C8—612no—
371D4⋊C8—612no—
372D4⋊C8—612no—
373D4⋊C8—612no—
374D4⋊C8—612no—
375D4⋊C8C2×D6111yescharacteristic
376D4⋊C8C2×D6111yescharacteristic
377C61#1752011yescharacteristic, complemented, Sylow
378C2×#41—611no—
379C2×#41—612no—
380C2×#41—1222no—
381C2×#41—1221no—
382C2×#41—1221no—
383C2×#41C4×D6111yescharacteristic
384#131—1228no—
385C2×M16—1222no—
386C2×M16—1222no—
387C2×M16—1221no—
388C2×M16—1221no—
389C22×C8—21no—
390C22×C8—1222no—
391C4⋊Q8—612no—
392C4⋊Q8—611no—
393C4⋊Q8—1221no—
394C4⋊Q8—1222no—
395#127—612no—
396#127—611no—
397#127—1221no—
398#127—1222no—
399#124—1222no—
400#124—1222no—
401#124—1222no—
402#124—1222no—
403#124—1221no—
404#121—612no—
405#121—612no—
406#121—612no—
407#121—1222no—
408#121—1222no—
409#121—1222no—
410#121—1222no—
411#121—1222no—
412#121—1222no—
413#121—1221no—
414#121—1221no—
415#121—1221no—
416C4×Q8—611no—
417C4×Q8—611no—
418C4×Q8—612no—
419C4×Q8—1221no—
420C4×Q8C4×D6111yescharacteristic
421C4×Q8C22×D6111yescharacteristic
422C4×D4—21no—
423C4×D4—21no—
424C4×D4—612no—
425C4×D4—612no—
426C4×D4—612no—
427C4×D4—611no—
428C4×D4—611no—
429C4×D4—612no—
430C4×D4—1221no—
431C4×D4—1222no—
432C4×D4—1222no—
433C4×D4—1222no—
434C4×D4—1222no—
435C4×D4—1221no—
436C4×D4C4×D6111yescharacteristic
437C4×D4C22×D6111yescharacteristic
438#117—21no—
439#117—611no—
440#117—612no—
441#117—612no—
442#117—1222no—
443#117—1222no—
444#117—1221no—
445#117C4×D6111yescharacteristic
446C2×C42—21no—
447C2×C42—612no—
448C2×C42—1222no—
449C2×C42C22×D6111yescharacteristic
450C4⋊C8—612no—
451C4⋊C8—1222no—
452C4⋊C8—1221no—
453C4⋊C8C22×D6111yescharacteristic
454C4⋊C8C22×D6111yescharacteristic
455C22⋊C8—21no—
456C22⋊C8—1222no—
457#97—1222no—
458#97—1221no—
459C4×C8—21no—
460C4×C8—21no—
461C4×C8—612no—
462C4×C8—1222no—
463C4×C8—1222no—
464C4×C8C22×D6111yescharacteristic
465C4×C8C22×D6111yescharacteristic
466#41—22no—
467#41—41no—
468#41—1224no—
469#41—1224no—
470#41—4881no—
471#41—4881no—
472#41—4882no—
473#41—4882no—
474#41—2444no—
475#41—2442no—
476C2×Q8—611no—
477C2×Q8—1222no—
478C2×Q8—1221no—
479C2×Q8C2×C4×D6111yescharacteristic
480C2×D4—21no—
481C2×D4—612no—
482C2×D4—612no—
483C2×D4—611no—
484C2×D4—1222no—
485C2×D4—1222no—
486C2×D4—1222no—
487C2×D4—1222no—
488C2×D4—1222no—
489C2×D4—1222no—
490C2×D4—1221no—
491C2×D4—1221no—
492C2×D4C2×C4×D6111yescharacteristic
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—612no—
497C22×C4—612no—
498C22×C4—612no—
499C22×C4—1222no—
500C22×C4—1222no—
501C22×C4—1222no—
502C22×C4C2×C4×D6111yescharacteristic
503C22×C4D4×D6111yescharacteristic
504C22×C4D4×D6111yescharacteristic
505M16—1228no—
506M16—1224no—
507M16—4882no—
508M16—4881no—
509C2×C8—24no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—1228no—
515C2×C8—1222no—
516C2×C8—1222no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—612no—
520C4⋊C4—612no—
521C4⋊C4—611no—
522C4⋊C4—611no—
523C4⋊C4—612no—
524C4⋊C4—1222no—
525C4⋊C4—1222no—
526C4⋊C4—1221no—
527C4⋊C4C2×C4×D6111yescharacteristic
528C4⋊C4C2×C4×D6111yescharacteristic
529C22⋊C4—21no—
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—612no—
533C22⋊C4—612no—
534C22⋊C4—612no—
535C22⋊C4—1222no—
536C22⋊C4—1222no—
537C22⋊C4—1222no—
538C22⋊C4—1222no—
539C22⋊C4—1222no—
540C22⋊C4—1222no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—612no—
545C42—612no—
546C42—612no—
547C42—1222no—
548C42—1222no—
549C42C2×C4×D6111yescharacteristic
550C42D4×D6111yescharacteristic
551C42D4×D6111yescharacteristic
552C42C23×D6111yescharacteristic
553C23—21no—
554C23—612no—
555C23—1222no—
556C23C22⋊C4×D6111yescharacteristic
557Q8—21no—
558Q8—614no—
559Q8—4881no—
560Q8—4882no—
561Q8#1451978312yescomplemented
562D4—21no—
563D4—21no—
564D4—42no—
565D4—1224no—
566D4—1222no—
567D4—1222no—
568D4—1222no—
569D4—4881no—
570D4—4882no—
571D4—4882no—
572D4—4882no—
573D4—4882no—
574D4—2444no—
575D4—2444no—
576D4—2442no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—612no—
590C2×C4—612no—
591C2×C4—612no—
592C2×C4—1224no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—2444no—
599C2×C4—2444no—
600C2×C4—2442no—
601C2×C4—2442no—
602C2×C4C22⋊C4×D6111yescharacteristic
603C2×C4C22⋊C4×D6111yescharacteristic
604C2×C4C22⋊C4×D6111yescharacteristic
605C2×C4C22×C4×D6111yescharacteristic
606C2×C4C2×D4×D6111yescharacteristic
607C2×C4C2×D4×D6111yescharacteristic
608C8—22no—
609C8—22no—
610C8—41no—
611C8—41no—
612C8—1224no—
613C8—2442no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—1222no—
621C22—1222no—
622C22—1222no—
623C22—2442no—
624C22—2442no—
625C22—2442no—
626C22C2×C22⋊C4×D6111yescharacteristic
627C4—21no—
628C4—21no—
629C4—21no—
630C4—22no—
631C4—41no—
632C4—41no—
633C4—614no—
634C4—1222no—
635C4—1222no—
636C4—2442no—
637C4#104×D6112yes—
638C4C2×#1451978311yescharacteristic
639C4#1684052811yescharacteristic
640C4#1684056411yescharacteristic
641C4#1684056911yescharacteristic
642C2—41no—
643C2—81no—
644C2—4882no—
645C2—2442no—
646C2#2145426511yescharacteristic
647C2#2145434111yescharacteristic
648C2C2×#104×D6111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1476C2
socleC22×C61C22×C22⋊C4
prime radical#21459445C2
cosocleC42×C61C24
hypercenter#1476D61
hypocenterC61#17520
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1476

Complements

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

NH ≅ G/Ntype
2. #21454948C2semidirect
3. C4×#14519783C2semidirect
4. #21444271C2semidirect
5. #21444268C2semidirect
6. #21444258C2semidirect
7. #21444257C2semidirect
8. #21444235C2semidirect
9. #21444234C2semidirect
10. #21444200C2semidirect
11. C61×#1476C2semidirect
233. Q8×C61#131semidirect
328. #1476D61semidirect
377. C61#17520semidirect
561. Q8#14519783semidirect

Maximal subgroups

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

subgroupindex
#214594452
#214549482
C4×#145197832
#214442712
#214442682
#214442582
#214442572
#214442352
#214442342
#214442002
C61×#14762
#1752061