Finite Group Databasebeta

← #30623987

Subgroups — #30623987

Order 15616.

1252 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462419C212yesmaximal, complemented
2#21462102C212yesmaximal, complemented
3#21462097C212yesmaximal, complemented
4#21462096C212yesmaximal, complemented
5#21462093C212yesmaximal, complemented
6D61⋊#354C212yesmaximal, complemented
7C2×#16840668C212yesmaximal, complemented
8#21459608C211yesmaximal, characteristic
9#16840829—22no—
10#16840743C2214yes—
11#16840741—22no—
12C2×#14519836—22no—
13C2×#14519836C2214yes—
14D61⋊#99—22no—
15D61⋊#99C2214yes—
16#16840669C2214yescomplemented
17#16840668C2214yescomplemented
18#16840668C2214yescomplemented
19#16840667C2214yescomplemented
20C2×#14519825C411yescharacteristic, complemented
21C2×#14519823C2211yescharacteristic
22#16840422—22no—
23#16840404—22no—
24#16840404—22no—
25#16840403—22no—
26#16840337C411yescharacteristic, complemented
27#16840337C2211yescharacteristic
28#16840331C2211yescharacteristic
29#16840330C411yescharacteristic, complemented
30#16840330C2211yescharacteristic
31#16840313C411yescharacteristic, complemented
32#16840313C2211yescharacteristic
33#16840308C411yescharacteristic, complemented
34C2×#14519766C2211yescharacteristic
35#124×D61C411yescharacteristic, complemented
36#124×D61C2211yescharacteristic
37#16840230C411yescharacteristic, complemented
38C61×#536C411yescharacteristic, complemented
39D4×#6923—48no—
40#14519856—24no—
41C2×C22⋊#6923—24no—
42C2×C22⋊#6923—24no—
43#14519836—24no—
44#14519836—24no—
45#14519836C2318yes—
46#14519835—24no—
47#14519825—44no—
48#14519823—44no—
49C2×#11758226—21no—
50C2×#11758226C2×C411yescharacteristic
51C2×#11758226C2×C411yescharacteristic, complemented
52C2×#11758226C2×C411yescharacteristic, complemented
53C2×#11758226C2311yescharacteristic
54C2×#11758222—21no—
55C2×#11758222—21no—
56C2×#11758222C2×C411yescharacteristic
57C2×#11758222C2×C411yescharacteristic, complemented
58C2×#11758222D412yes—
59C2×Q8×D61—21no—
60C2×Q8×D61C2×C411yescharacteristic, complemented
61C2×Q8×D61C2×C411yescharacteristic, complemented
62C22×D244D412yes—
63C22×D244C2311yescharacteristic
64C2×D4×D61C2×C411yescharacteristic
65C2×D4×D61D412yes—
66C2×D4×D61D412yes—
67C2×D4×D61C2311yescharacteristic
68#14519775C2×C411yescharacteristic, complemented
69#14519774C2×C411yescharacteristic, complemented
70#14519774C2311yescharacteristic
71#14519773C2×C411yescharacteristic, complemented
72#14519766—22no—
73#14519766C2×C411yescharacteristic, complemented
74#14519766C2×C411yescharacteristic, complemented
75#14519766C2×C411yescharacteristic, complemented
76#14519766C2311yescharacteristic
77#14519766C2311yescharacteristic
78#14519765C2×C411yescharacteristic, complemented
79#14519765C2311yescharacteristic
80#14519764C2×C411yescharacteristic, complemented
81#14519763—22no—
82#14519763—22no—
83#14519762—22no—
84#14519762—22no—
85#14519755—22no—
86#14519755—22no—
87#14519752—22no—
88#14519752—22no—
89#14519752—22no—
90#14519750—22no—
91#14519750—22no—
92#14519749—22no—
93#14519747—22no—
94#14519747—22no—
95#14519747—22no—
96#14519742—21no—
97#14519739—21no—
98#14519739—22no—
99#14519739—22no—
100C4×C22⋊D61—22no—
101#14519737—21no—
102#14519737—21no—
103#14519734C2×C411yescharacteristic, complemented
104#14519734C2×C411yescharacteristic, complemented
105C2×#11758198—22no—
106C22⋊C4×D61—22no—
107C22⋊C4×D61—22no—
108C2×C4×Dic61C2×C411yescharacteristic, complemented
109C42×D61C2×C411yescharacteristic, complemented
110C42×D61C2×C411yescharacteristic, complemented
111C2×#41×C61C2×C411yescharacteristic
112C22×Q8×C61C2×C411yescharacteristic, complemented
113C4⋊Q8×C61C2×C411yescharacteristic, complemented
114C4⋊Q8×C61C2×C411yescharacteristic, complemented
115#124×C61C2×C411yescharacteristic, complemented
116#124×C61C2×C411yescharacteristic, complemented
117#123×C61—22no—
118C22⋊Q8×C61—22no—
119#117×C61C2×C411yescharacteristic, complemented
120C23×D61C22⋊C412yes—
121C23×D61C2×D412yes—
122C23×D61C2×D412yes—
123C22×#6923—44no—
124C22×#6923—44no—
125#41×C61—42no—
126C2×Q8×C61—21no—
127C2×Q8×C61—42no—
128C2×Q8×C61C22×C411yescharacteristic
129C2×Q8×C61C22×C411yescharacteristic
130C2×Q8×C61C22×C411yescharacteristic
131C2×D4×C61C22⋊C412yes—
132C2×D4×C61C22×C411yescharacteristic
133C22×C4×C61—21no—
134C22×C4×C61C22⋊C412yes—
135C22×C4×C61C22×C411yescharacteristic
136C2×C22⋊D61—21no—
137C2×C22⋊D61—22no—
138C2×C22⋊D61C22⋊C412yes—
139C2×C22⋊D61C22⋊C411yescharacteristic
140C2×C22⋊D61C22⋊C411yescharacteristic
141C2×C22⋊D61C2×D412yes—
142C22×Dic61—21no—
143#11758226—42no—
144#11758226—42no—
145#11758226—42no—
146#11758226—42no—
147#11758226—44no—
148Q8×D61—44no—
149Q8×D61—42no—
150Q8×D61—42no—
151D4×D61—28no—
152D4×D61—42no—
153D4×D61—42no—
154D4×D61—44no—
155#11758222—24no—
156#11758222—44no—
157#11758222—44no—
158#11758222—44no—
159#11758222—42no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—22no—
163C2×D244—22no—
164C2×D244—44no—
165C2×D244—42no—
166C2×D244C22⋊C412yes—
167C2×D244C22⋊C411yescharacteristic
168C2×D244C22×C411yescharacteristic
169C2×D244C22×C411yescharacteristic
170C2×D244C2×D412yes—
171C2×D244C2×D412yes—
172C2×D244C2411yescharacteristic
173C2×C4×D61—21no—
174C2×C4×D61—21no—
175C2×C4×D61—21no—
176C2×C4×D61—21no—
177C2×C4×D61—21no—
178C2×C4×D61—21no—
179C2×C4×D61—21no—
180C2×C4×D61C22⋊C412yes—
181C2×C4×D61C22×C411yescharacteristic
182C2×C4×D61C22×C411yescharacteristic
183C2×C4×D61C2×D412yes—
184C2×Dic122—21no—
185C2×Dic122—21no—
186C2×Dic122C22⋊C411yescharacteristic
187C2×Dic122C22×C411yescharacteristic
188C22⋊#6923—28no—
189C22⋊#6923—44no—
190C22⋊#6923—44no—
191C4⋊#6923—44no—
192C4×#6923—44no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—22no—
195C4⋊C4×C61—22no—
196C22⋊C4×C61—21no—
197C22⋊C4×C61—22no—
198C22⋊C4×C61—22no—
199C42×C61C22×C411yescharacteristic
200C42×C61C22×C411yescharacteristic
201C22⋊Dic61—22no—
202#11758198—22no—
203#11758198—22no—
204#11758198—22no—
205#11758198—21no—
206#11758198—21no—
207#11758198—22no—
208#11758198—22no—
209#11758198—22no—
210#11758198—22no—
211#11758196—22no—
212#11758196—21no—
213#11758196—21no—
214C4×Dic61—21no—
215C4×Dic61—21no—
216C4×Dic61—21no—
217C4×Dic61C22×C411yescharacteristic
218C4×Dic61C22×C411yescharacteristic
219C23×C61C2×C22⋊C412yes—
220C23×C61C2×C22⋊C411yescharacteristic
221C22×D61—24no—
222C22×D61—24no—
223C22×D61—21no—
224C22×D61—42no—
225C22×D61—82no—
226C22×D61—82no—
227C22×D61—82no—
228C22×D61C2×C22⋊C412yes—
229C22×D61C2×C22⋊C411yescharacteristic
230C22×D61C2×C22⋊C411yescharacteristic
231C22×D61C22×D412yes—
232C2×#6923—48no—
233C2×#6923—164no—
234C2×#6923—164no—
235Q8×C61—41no—
236Q8×C61—42no—
237Q8×C61—41no—
238Q8×C61—41no—
239D4×C61—24no—
240D4×C61—41no—
241C2×C4×C61—22no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C412yes—
250C2×C4×C61C23×C411yescharacteristic
251C22⋊D61—22no—
252C22⋊D61—24no—
253C22⋊D61—42no—
254C22⋊D61—42no—
255C22⋊D61—82no—
256C22⋊D61#9914yes—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—81no—
261C2×Dic61C2×C22⋊C411yescharacteristic
262C2×Dic61C2×C22⋊C411yescharacteristic
263D244—22no—
264D244—24no—
265D244—42no—
266D244—41no—
267D244—42no—
268D244—41no—
269D244—41no—
270D244—42no—
271D244—42no—
272D244—82no—
273D244—82no—
274C4×D61—22no—
275C4×D61—24no—
276C4×D61—22no—
277C4×D61—42no—
278C4×D61—42no—
279C4×D61—42no—
280C4×D61—42no—
281C4×D61—42no—
282C4×D61—42no—
283C4×D61—81no—
284C4×D61—81no—
285Dic122—22no—
286Dic122—42no—
287Dic122—42no—
288Dic122—41no—
289#33361—611nomaximal, Sylow
290C22×C61—22no—
291C22×C61—41no—
292C22×C61—42no—
293C22×C61C2×#9912yes—
294C22×C61C22×C22⋊C411yescharacteristic
295C2×D61—21no—
296C2×D61—22no—
297C2×D61—22no—
298C2×D61—41no—
299C2×D61—41no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—42no—
303C2×D61—41no—
304C2×D61—84no—
305C2×D61—84no—
306C2×D61—162no—
307C2×D61C2×#9912yes—
308#6923—88no—
309C4×C61—21no—
310C4×C61—21no—
311C4×C61—22no—
312C4×C61—41no—
313C4×C61—41no—
314C4×C61—41no—
315C4×C61—41no—
316Dic61—21no—
317Dic61—41no—
318Dic61—41no—
319Dic61—41no—
320Dic61C2×#9912yes—
321#3500—611no—
322#2883—612no—
323#2129—612no—
324#2127—612no—
325#2127—612no—
326#2126—612no—
327C2×#354—612no—
328C2×#354—612no—
329C2×C61—21no—
330C2×C61—22no—
331C2×C61—42no—
332C2×C61C22×#9911yescharacteristic
333D61—22no—
334D61—42no—
335D61—42no—
336D61—41no—
337D61—81no—
338C2×2−1+4—611no—
339C2×H(5,𝔽2)—611no—
340#548—1222no—
341#546—1222no—
342#546—1222no—
343#546—1222no—
344#536—611no—
345#536—611no—
346#536#692311yescharacteristic, complemented
347#535—611no—
348#535—611no—
349#535—611no—
350#532—611no—
351#532—611no—
352#532—611no—
353C2×#124—611no—
354C2×#124—611no—
355C2×#124—611no—
356#518—1222no—
357C4×#41—611no—
358#410—614no—
359#410—1222no—
360C2×#99—614no—
361C2×#99—614no—
362C2×#99—1222no—
363C2×#99—1222no—
364#354—614no—
365#354—614no—
366#354—614no—
367#353—614no—
368C61#3336111yescharacteristic, complemented, Sylow
3692−1+4—2444no—
370H(5,𝔽2)—2444no—
371C2×#41—611no—
372C2×#41—611no—
373C2×#41—612no—
374C2×#41—611no—
375C2×#41—611no—
376C2×#41—611no—
377C2×#41—611no—
378C2×#41—1221no—
379C2×#41—1221no—
380C2×#41—1221no—
381C2×#41C2×#692311yescharacteristic
382C22×Q8—611no—
383C22×Q8—611no—
384C22×Q8—1221no—
385C22×Q8C2×#692311yescharacteristic, complemented
386C22×D4—612no—
387C22×D4—612no—
388C22×D4—611no—
389C22×D4—611no—
390C22×D4—612no—
391C22×D4—611no—
392C4⋊Q8—611no—
393C4⋊Q8C2×#692311yescharacteristic, complemented
394C4⋊Q8C2×#692311yescharacteristic, complemented
395#127—611no—
396#127—611no—
397#127—611no—
398#124—611no—
399#124—611no—
400#124—611no—
401#124—611no—
402#124—611no—
403#124—611no—
404#124—611no—
405#124—611no—
406#124—1222no—
407#124—1222no—
408#124—1222no—
409#124C2×#692311yescharacteristic, complemented
410#124C2×#692311yescharacteristic, complemented
411#123—22no—
412#123—1222no—
413#123—1222no—
414C22⋊Q8—22no—
415C22⋊Q8—1222no—
416C22⋊Q8—1222no—
417#121—1222no—
418#121—1222no—
419#121—1222no—
420#121—1222no—
421#121—1222no—
422#121—1222no—
423#120—1222no—
424#120—1222no—
425#120—1222no—
426C4×Q8—1221no—
427C4×D4—1221no—
428C4×D4—1221no—
429C4×D4—1222no—
430C4×D4—1222no—
431C4×D4—1221no—
432C4×D4—1222no—
433C4×D4—2448no—
434#117—611no—
435#117—611no—
436#117—1224no—
437#117C2×#692311yescharacteristic, complemented
438C2×C22⋊C4—1222no—
439C2×C22⋊C4—1222no—
440C2×C22⋊C4—1222no—
441C2×C22⋊C4—1224no—
442C2×C22⋊C4—1224no—
443C2×C42—611no—
444C2×C42—611no—
445C2×C42—611no—
446#99—618no—
447#99—1224no—
448#99—1224no—
449#99—1224no—
450C24—612no—
451C24—612no—
452C24—612no—
453#41—42no—
454#41—1224no—
455#41—2442no—
456#41—2444no—
457#41—2444no—
458#41—2444no—
459#41—2442no—
460#41—2442no—
461#41—2442no—
462#41—2442no—
463#41—2444no—
464C2×Q8—21no—
465C2×Q8—42no—
466C2×Q8—611no—
467C2×Q8—611no—
468C2×Q8—1221no—
469C2×Q8—1221no—
470C2×Q8—2442no—
471C2×Q8—2442no—
472C2×Q8—2444no—
473C2×Q8C22×#692311yescharacteristic
474C2×Q8C22×#692311yescharacteristic
475C2×Q8C22×#692311yescharacteristic
476C2×D4—611no—
477C2×D4—612no—
478C2×D4—611no—
479C2×D4—612no—
480C2×D4—611no—
481C2×D4—612no—
482C2×D4—612no—
483C2×D4—612no—
484C2×D4—611no—
485C2×D4—611no—
486C2×D4—611no—
487C2×D4—1222no—
488C2×D4—1222no—
489C2×D4—1221no—
490C2×D4—1221no—
491C2×D4—1228no—
492C2×D4—1222no—
493C2×D4—1221no—
494C2×D4—2442no—
495C2×D4—2444no—
496C2×D4—2442no—
497C2×D4—2442no—
498C2×D4—2444no—
499C2×D4C22⋊#692312yes—
500C2×D4C22×#692311yescharacteristic
501C22×C4—21no—
502C22×C4—612no—
503C22×C4—611no—
504C22×C4—611no—
505C22×C4—612no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4—1221no—
509C22×C4—1221no—
510C22×C4—1221no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—2444no—
515C22×C4—2444no—
516C22×C4C22⋊#692312yes—
517C22×C4C22×#692311yescharacteristic
518C4⋊C4—21no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—1222no—
524C4⋊C4—2444no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—21no—
528C22⋊C4—1222no—
529C22⋊C4—1222no—
530C22⋊C4—1222no—
531C22⋊C4—1222no—
532C22⋊C4—1222no—
533C22⋊C4—1222no—
534C22⋊C4—1221no—
535C22⋊C4—1221no—
536C22⋊C4—1222no—
537C22⋊C4—1222no—
538C22⋊C4—1228no—
539C22⋊C4—2444no—
540C22⋊C4—2444no—
541C42—611no—
542C42—611no—
543C42—1221no—
544C42—1221no—
545C42—1221no—
546C42—2444no—
547C42C22×#692311yescharacteristic
548C42C22×#692311yescharacteristic
549C23—611no—
550C23—612no—
551C23—611no—
552C23—612no—
553C23—1224no—
554C23—1224no—
555C23—1221no—
556C23—4882no—
557C23—4882no—
558C23—4882no—
559C23—2442no—
560C23C2×C22⋊#692312yes—
561C23C2×C22⋊#692311yescharacteristic
562Q8—41no—
563Q8—41no—
564Q8—41no—
565Q8—42no—
566Q8—1222no—
567Q8—2441no—
568Q8—2442no—
569Q8—2442no—
570D4—24no—
571D4—41no—
572D4—614no—
573D4—1224no—
574D4—1222no—
575D4—1224no—
576D4—1222no—
577D4—4882no—
578D4—4882no—
579D4—4882no—
580D4—2442no—
581D4—2441no—
582D4—2441no—
583D4—2442no—
584D4—2441no—
585D4—2442no—
586D4—2442no—
587D4—2442no—
588D4—2442no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—611no—
597C2×C4—611no—
598C2×C4—1224no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—9764no—
605C2×C4—9764no—
606C2×C4—4881no—
607C2×C4—4881no—
608C2×C4—4881no—
609C2×C4—2442no—
610C2×C4—2442no—
611C2×C4—2442no—
612C2×C4—2442no—
613C2×C4—2442no—
614C2×C4—2442no—
615C2×C4—2448no—
616C2×C4C2×C22⋊#692311yescharacteristic
617C2×C4C2×C22⋊#692312yes—
618C2×C4C23×#692311yescharacteristic
619C22—22no—
620C22—41no—
621C22—42no—
622C22—612no—
623C22—1222no—
624C22—1221no—
625C22—1222no—
626C22—9762no—
627C22—4884no—
628C22—4884no—
629C22—2441no—
630C22—2442no—
631C22—2441no—
632C22—2442no—
633C22—2442no—
634C22—2441no—
635C22D61⋊#9912yes—
636C22C22×C22⋊#692311yescharacteristic
637C4—21no—
638C4—21no—
639C4—22no—
640C4—41no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—612no—
645C4—1221no—
646C4—4888no—
647C4—2441no—
648C4—2441no—
649C4—2441no—
650C2—21no—
651C2—22no—
652C2—42no—
653C2—1222no—
654C2—4881no—
655C2—2442no—
656C2—2441no—
657C2—2442no—
658C2C2×D61⋊#9911yescharacteristic
subgroupG/H
centerC2C2×D61⋊#99
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#536C4
socleC2×C61C22×#99
prime radical#21459608C2
cosocleC2×D244C24
hypercenter#536#6923
hypocenterC61#33361
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C23 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
1. #21462419C2semidirect
2. #21462102C2semidirect
3. #21462097C2semidirect
4. #21462096C2semidirect
5. #21462093C2semidirect
6. D61⋊#354C2semidirect
7. C2×#16840668C2semidirect
16. #16840669C22semidirect
17. #16840668C22semidirect
18. #16840668C22semidirect
19. #16840667C22semidirect
20. C2×#14519825C4semidirect
26. #16840337C4semidirect
29. #16840330C4semidirect
31. #16840313C4semidirect
33. #16840308C4semidirect
35. #124×D61C4semidirect
37. #16840230C4semidirect
38. C61×#536C4semidirect
51. C2×#11758226C2×C4semidirect
52. C2×#11758226C2×C4semidirect
57. C2×#11758222C2×C4semidirect
60. C2×Q8×D61C2×C4semidirect
61. C2×Q8×D61C2×C4semidirect
68. #14519775C2×C4semidirect
69. #14519774C2×C4semidirect
71. #14519773C2×C4semidirect
73. #14519766C2×C4semidirect
74. #14519766C2×C4semidirect
75. #14519766C2×C4semidirect
78. #14519765C2×C4semidirect
80. #14519764C2×C4semidirect
103. #14519734C2×C4semidirect
104. #14519734C2×C4semidirect
108. C2×C4×Dic61C2×C4semidirect
109. C42×D61C2×C4semidirect
110. C42×D61C2×C4semidirect
112. C22×Q8×C61C2×C4semidirect
113. C4⋊Q8×C61C2×C4semidirect
114. C4⋊Q8×C61C2×C4semidirect
115. #124×C61C2×C4semidirect
116. #124×C61C2×C4semidirect
119. #117×C61C2×C4semidirect
346. #536#6923semidirect
368. C61#33361semidirect
385. C22×Q8C2×#6923semidirect
393. C4⋊Q8C2×#6923semidirect
394. C4⋊Q8C2×#6923semidirect
409. #124C2×#6923semidirect
410. #124C2×#6923semidirect
437. #117C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214624192
#214621022
#214620972
#214620962
#214620932
D61⋊#3542
C2×#168406682
#214596082
#3336161