Finite Group Databasebeta

← #30623971

Subgroups — #30623971

Order 15616.

1252 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840743C212yesmaximal, complemented
2#21462102C212yesmaximal, complemented
3#21462099C212yesmaximal, complemented
4#21462096C212yesmaximal, complemented
5#21462094C212yesmaximal, complemented
6#21462093C212yesmaximal, complemented
7#21462091C212yesmaximal, complemented
8#21459151C211yesmaximal, characteristic
9C2×#14519856—22no—
10#16840743—22no—
11#16840743—22no—
12#16840743C2214yes—
13#16840743C2214yescomplemented
14C2×#14519836—22no—
15C2×#14519836C2214yescomplemented
16#16840669C2214yescomplemented
17#16840668C2214yescomplemented
18#16840667C2214yescomplemented
19#16840666C2214yescomplemented
20C22×#11758226C2211yescharacteristic
21C2×#41×D61C411yescharacteristic, complemented
22#16840342—22no—
23#16840337C411yescharacteristic, complemented
24#16840337C2211yescharacteristic
25#16840336C411yescharacteristic, complemented
26#16840331C2211yescharacteristic
27#16840330C411yescharacteristic, complemented
28#16840330C2211yescharacteristic
29#16840326C2211yescharacteristic
30#16840323—22no—
31#16840313C411yescharacteristic, complemented
32#16840313C2211yescharacteristic
33#16840311C411yescharacteristic, complemented
34#16840311C2211yescharacteristic
35#16840275—22no—
36#16840270—22no—
37#117×D61C411yescharacteristic, complemented
38C61×#536C411yescharacteristic, complemented
39#14519856—24no—
40#14519856—24no—
41#14519856—42no—
42#14519856—42no—
43C2×C4⋊#6923—42no—
44C2×C22⋊#6923—24no—
45C2×C4×#6923—42no—
46#14519836—24no—
47#14519836—24no—
48#14519836—24no—
49#14519836C2318yes—
50C2×#11758226—42no—
51C2×#11758226—42no—
52C2×#11758226C2×C411yescharacteristic
53C2×#11758226C2×C411yescharacteristic
54C2×#11758226C2×C411yescharacteristic, complemented
55C2×#11758226D412yes—
56C2×#11758226D412yes—
57C2×#11758226C2311yescharacteristic
58C2×#11758226C2311yescharacteristic
59C2×#11758222—21no—
60C2×#11758222C2×C411yescharacteristic
61C2×#11758224—21no—
62C2×#11758224C2×C411yescharacteristic, complemented
63#41×D61—42no—
64#41×D61—42no—
65C2×Q8×D61C2×C411yescharacteristic, complemented
66C22×D244D412yes—
67C22×D244C2311yescharacteristic
68C2×D4×D61—21no—
69C2×D4×D61C2×C411yescharacteristic, complemented
70C22×C4×D61—21no—
71C22×C4×D61C2×C411yescharacteristic
72C22×C4×D61D412yes—
73#14519775C2×C411yescharacteristic, complemented
74#14519775C2×C411yescharacteristic, complemented
75#14519774C2×C411yescharacteristic
76#14519774C2311yescharacteristic
77#14519773C2×C411yescharacteristic
78#14519773C2311yescharacteristic
79#14519766C2×C411yescharacteristic
80#14519766C2×C411yescharacteristic, complemented
81#14519766C2×C411yescharacteristic, complemented
82#14519766C2311yescharacteristic
83#14519765C2×C411yescharacteristic
84#14519765C2311yescharacteristic
85#14519762—22no—
86#14519762—22no—
87#14519760—22no—
88#14519755—22no—
89#14519755—22no—
90#14519753—22no—
91#14519752—22no—
92#14519752—22no—
93#14519752—22no—
94#14519751—22no—
95#14519750—22no—
96#14519750—22no—
97#14519750—22no—
98#14519750—22no—
99#14519747—22no—
100#14519747—22no—
101#14519747—22no—
102#14519745—22no—
103#14519739—22no—
104#14519739—22no—
105#14519737—22no—
106#14519737—22no—
107#14519736C2×C411yescharacteristic
108#14519735—21no—
109#14519734C2×C411yescharacteristic, complemented
110#14519734C2×C411yescharacteristic, complemented
111#14519733—21no—
112C4⋊C4×D61—21no—
113C22⋊C4×D61—21no—
114C42×D61C2×C411yescharacteristic, complemented
115C42×D61C2×C411yescharacteristic, complemented
116C2×#41×C61C2×C411yescharacteristic
117C22×Q8×C61C2×C411yescharacteristic
118C4⋊Q8×C61C2×C411yescharacteristic, complemented
119C4⋊Q8×C61C2×C411yescharacteristic, complemented
120#124×C61C2×C411yescharacteristic, complemented
121#124×C61C2×C411yescharacteristic, complemented
122#123×C61—22no—
123C22⋊Q8×C61—22no—
124#117×C61C2×C411yescharacteristic
125C23×D61—21no—
126C23×D61C22⋊C411yescharacteristic
127C23×D61C2×D412yes—
128C22×#6923—44no—
129#41×C61—42no—
130C2×Q8×C61—42no—
131C2×Q8×C61C22⋊C412yes—
132C2×Q8×C61C22×C411yescharacteristic
133C2×Q8×C61C22×C411yescharacteristic
134C2×Q8×C61C22×C411yescharacteristic
135C2×D4×C61—21no—
136C2×D4×C61C22×C411yescharacteristic
137C22×C4×C61—21no—
138C22×C4×C61C22⋊C412yes—
139C22×C4×C61C22×C411yescharacteristic
140C2×C22⋊D61—21no—
141C2×C22⋊D61—21no—
142C2×C22⋊D61—22no—
143C2×C22⋊D61—22no—
144C2×C22⋊D61—21no—
145C22×Dic61—21no—
146C22×Dic61C22⋊C411yescharacteristic
147#11758226—42no—
148#11758226—41no—
149#11758226—41no—
150#11758226—42no—
151#11758226—42no—
152#11758226—42no—
153#11758226—42no—
154#11758226—41no—
155#11758226—42no—
156#11758226—41no—
157#11758226—41no—
158#11758226—41no—
159Q8×D61—42no—
160#11758224—41no—
161#11758224—41no—
162#11758224—42no—
163#11758224—42no—
164D4×D61—42no—
165D4×D61—44no—
166#11758222—44no—
167#11758222—42no—
168C2×D244—21no—
169C2×D244—22no—
170C2×D244—22no—
171C2×D244—42no—
172C2×D244—42no—
173C2×D244—42no—
174C2×D244C22⋊C412yes—
175C2×D244C22⋊C412yes—
176C2×D244C22×C411yescharacteristic
177C2×D244C22×C411yescharacteristic
178C2×D244C2×D412yes—
179C2×D244C2×D412yes—
180C2×D244C2×D412yes—
181C2×D244C2411yescharacteristic
182C2×C4×D61—22no—
183C2×C4×D61—22no—
184C2×C4×D61—22no—
185C2×C4×D61—22no—
186C2×C4×D61—22no—
187C2×C4×D61—22no—
188C2×C4×D61—21no—
189C2×C4×D61—21no—
190C2×C4×D61—21no—
191C2×C4×D61—21no—
192C2×C4×D61—21no—
193C2×C4×D61—42no—
194C2×C4×D61—42no—
195C2×C4×D61C22⋊C411yescharacteristic
196C2×C4×D61C22⋊C411yescharacteristic
197C2×C4×D61C22⋊C412yes—
198C2×C4×D61C22⋊C412yes—
199C2×C4×D61C22×C411yescharacteristic
200C2×C4×D61C22×C411yescharacteristic
201C2×C4×D61C2×D412yes—
202C2×C4×D61C2×D412yes—
203C2×Dic122—21no—
204C2×Dic122C22×C411yescharacteristic
205C22⋊#6923—28no—
206C22⋊#6923—44no—
207C4⋊#6923—44no—
208C4⋊#6923—44no—
209C4×#6923—44no—
210C4×#6923—44no—
211C4⋊C4×C61—21no—
212C4⋊C4×C61—22no—
213C4⋊C4×C61—22no—
214C22⋊C4×C61—21no—
215C22⋊C4×C61—22no—
216C22⋊C4×C61—22no—
217C42×C61C22×C411yescharacteristic
218C42×C61C22×C411yescharacteristic
219C22⋊Dic61—21no—
220#11758198—21no—
221#11758198—21no—
222#11758198—22no—
223#11758198—22no—
224#11758198—22no—
225#11758198—22no—
226#11758198—22no—
227#11758198—22no—
228C4⋊Dic61—21no—
229#11758196—22no—
230#11758196—22no—
231#11758196—21no—
232#11758196—21no—
233C4×Dic61—21no—
234C4×Dic61—21no—
235C4×Dic61C22×C411yescharacteristic
236C4×Dic61C22×C411yescharacteristic
237C23×C61—21no—
238C23×C61C2×C22⋊C411yescharacteristic
239C22×D61—22no—
240C22×D61—22no—
241C22×D61—22no—
242C22×D61—21no—
243C22×D61—42no—
244C22×D61—82no—
245C22×D61—82no—
246C22×D61—81no—
247C22×D61C2×C22⋊C411yescharacteristic
248C22×D61C2×C22⋊C412yes—
249C22×D61C2×C22⋊C411yescharacteristic
250C22×D61C22×D412yes—
251C2×#6923—48no—
252C2×#6923—88no—
253Q8×C61—42no—
254Q8×C61—41no—
255Q8×C61—41no—
256Q8×C61—41no—
257D4×C61—41no—
258D4×C61—42no—
259C2×C4×C61—22no—
260C2×C4×C61—22no—
261C2×C4×C61—22no—
262C2×C4×C61—21no—
263C2×C4×C61—21no—
264C2×C4×C61—21no—
265C2×C4×C61—42no—
266C2×C4×C61C2×C22⋊C412yes—
267C2×C4×C61C2×C22⋊C412yes—
268C2×C4×C61C2×C22⋊C411yescharacteristic
269C2×C4×C61C23×C411yescharacteristic
270C22⋊D61—42no—
271C22⋊D61—42no—
272C22⋊D61—42no—
273C22⋊D61—82no—
274C22⋊D61—82no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61#9912yes—
281C2×Dic61#9912yes—
282C2×Dic61C2×C22⋊C411yescharacteristic
283C2×Dic61C2×C22⋊C411yescharacteristic
284D244—42no—
285D244—42no—
286D244—41no—
287D244—42no—
288D244—42no—
289D244—42no—
290D244—41no—
291D244—41no—
292D244—42no—
293D244—82no—
294D244—82no—
295C4×D61—22no—
296C4×D61—22no—
297C4×D61—22no—
298C4×D61—22no—
299C4×D61—21no—
300C4×D61—21no—
301C4×D61—22no—
302C4×D61—22no—
303C4×D61—22no—
304C4×D61—22no—
305C4×D61—22no—
306C4×D61—22no—
307C4×D61—21no—
308C4×D61—21no—
309C4×D61—42no—
310C4×D61—42no—
311C4×D61—41no—
312C4×D61—41no—
313C4×D61—81no—
314C4×D61—81no—
315Dic122—42no—
316Dic122—41no—
317#33358—611nomaximal, Sylow
318C22×C61—41no—
319C22×C61—42no—
320C22×C61—41no—
321C22×C61C2×#9912yes—
322C22×C61C22×C22⋊C411yescharacteristic
323C2×D61—21no—
324C2×D61—21no—
325C2×D61—22no—
326C2×D61—22no—
327C2×D61—44no—
328C2×D61—41no—
329C2×D61—42no—
330C2×D61—42no—
331C2×D61—41no—
332C2×D61—42no—
333C2×D61—41no—
334C2×D61—82no—
335C2×D61—82no—
336C2×D61—84no—
337#6923—88no—
338C4×C61—21no—
339C4×C61—21no—
340C4×C61—22no—
341C4×C61—22no—
342C4×C61—41no—
343C4×C61—41no—
344C4×C61—41no—
345Dic61—41no—
346Dic61—41no—
347Dic61—41no—
348Dic61C2×#9912yes—
349Dic61C2×#9911yescharacteristic
350Dic61C2×#9911yescharacteristic
351#3449—611no—
352C2×#410—612no—
353#2129—612no—
354#2128—612no—
355#2127—612no—
356#2127—612no—
357#2126—612no—
358#2126—612no—
359C2×C61—21no—
360C2×C61—22no—
361C2×C61—81no—
362C2×C61C22×#9911yescharacteristic
363D61—42no—
364D61—42no—
365D61—41no—
366D61—41no—
367D61—81no—
368C22×#41—611no—
369C22×#41—611no—
370#537—1222no—
371#536—611no—
372#536—611no—
373#536—611no—
374#536#692311yescharacteristic, complemented
375#535—611no—
376#535—611no—
377#535—611no—
378#535—611no—
379#534—1222no—
380#532—611no—
381#532—611no—
382#532—611no—
383#532—611no—
384#525—1222no—
385#525—1222no—
386C2×#117—611no—
387C2×#117—1222no—
388#410—614no—
389#410—614no—
390#410—1222no—
391#410—1222no—
392C2×#99—614no—
393C2×#99—1222no—
394#354—614no—
395#354—614no—
396#353—614no—
397#353—614no—
398C61#3335811yescharacteristic, complemented, Sylow
399C2×#41—612no—
400C2×#41—612no—
401C2×#41—611no—
402C2×#41—611no—
403C2×#41—611no—
404C2×#41—611no—
405C2×#41—611no—
406C2×#41—611no—
407C2×#41—611no—
408C2×#41—1221no—
409C2×#41—1221no—
410C2×#41—2442no—
411C2×#41—2442no—
412C2×#41—2442no—
413C2×#41—2442no—
414C2×#41C2×#692311yescharacteristic
415C22×Q8—611no—
416C22×Q8C2×#692311yescharacteristic
417C22×D4—611no—
418C22×D4—612no—
419C22×D4—611no—
420C22×D4—1221no—
421C23×C4—612no—
422C23×C4—611no—
423C23×C4—1221no—
424C4⋊Q8—611no—
425C4⋊Q8—611no—
426C4⋊Q8C2×#692311yescharacteristic, complemented
427C4⋊Q8C2×#692311yescharacteristic, complemented
428#127—611no—
429#127—611no—
430#127—611no—
431#127—611no—
432#124—611no—
433#124—611no—
434#124—611no—
435#124—611no—
436#124—611no—
437#124—611no—
438#124C2×#692311yescharacteristic, complemented
439#124C2×#692311yescharacteristic, complemented
440#123—22no—
441#123—1222no—
442#123—1222no—
443#123—1222no—
444C22⋊Q8—22no—
445C22⋊Q8—1222no—
446C22⋊Q8—1222no—
447C22⋊Q8—1222no—
448#121—1222no—
449#121—1222no—
450#121—1222no—
451#121—1222no—
452#121—1222no—
453#121—1222no—
454#121—1222no—
455#121—1222no—
456#120—1222no—
457#120—1222no—
458#120—1222no—
459#120—1222no—
460C4×D4—1222no—
461C4×D4—1222no—
462C4×D4—1222no—
463C4×D4—1222no—
464#117—611no—
465#117—611no—
466#117—611no—
467#117—1221no—
468#117—1221no—
469#117—1224no—
470#117—1224no—
471#117—2442no—
472#117—2442no—
473#117C2×#692311yescharacteristic
474C2×C4⋊C4—1221no—
475C2×C4⋊C4—2442no—
476C2×C22⋊C4—1221no—
477C2×C22⋊C4—1224no—
478C2×C42—611no—
479C2×C42—611no—
480C2×C42—2442no—
481#99—618no—
482#99—1224no—
483#99—1224no—
484#99—1224no—
485C24—612no—
486C24—611no—
487C24—1221no—
488#41—42no—
489#41—2442no—
490#41—2444no—
491#41—2442no—
492#41—2442no—
493#41—2441no—
494#41—2442no—
495#41—2442no—
496#41—2442no—
497#41—2441no—
498#41—2442no—
499#41—2442no—
500#41—2442no—
501#41—2441no—
502#41—2441no—
503#41—2441no—
504#41—2441no—
505#41—2441no—
506#41—2441no—
507C2×Q8—42no—
508C2×Q8—611no—
509C2×Q8—1221no—
510C2×Q8—2442no—
511C2×Q8C22⋊#692312yes—
512C2×Q8C22×#692311yescharacteristic
513C2×Q8C22×#692311yescharacteristic
514C2×Q8C22×#692311yescharacteristic
515C2×D4—21no—
516C2×D4—612no—
517C2×D4—612no—
518C2×D4—612no—
519C2×D4—612no—
520C2×D4—612no—
521C2×D4—611no—
522C2×D4—611no—
523C2×D4—611no—
524C2×D4—1222no—
525C2×D4—1222no—
526C2×D4—1221no—
527C2×D4—1221no—
528C2×D4—1222no—
529C2×D4—1222no—
530C2×D4—1221no—
531C2×D4—1221no—
532C2×D4—2442no—
533C2×D4—2442no—
534C2×D4—2442no—
535C2×D4—2442no—
536C2×D4—2444no—
537C2×D4C22×#692311yescharacteristic
538C22×C4—21no—
539C22×C4—611no—
540C22×C4—611no—
541C22×C4—612no—
542C22×C4—612no—
543C22×C4—611no—
544C22×C4—612no—
545C22×C4—612no—
546C22×C4—611no—
547C22×C4—611no—
548C22×C4—1222no—
549C22×C4—1222no—
550C22×C4—1222no—
551C22×C4—1222no—
552C22×C4—1222no—
553C22×C4—1222no—
554C22×C4—1221no—
555C22×C4—1221no—
556C22×C4—1221no—
557C22×C4—1221no—
558C22×C4—1221no—
559C22×C4—1221no—
560C22×C4—2442no—
561C22×C4—2442no—
562C22×C4—2444no—
563C22×C4C22⋊#692312yes—
564C22×C4C22×#692311yescharacteristic
565C4⋊C4—21no—
566C4⋊C4—22no—
567C4⋊C4—22no—
568C4⋊C4—1221no—
569C4⋊C4—1221no—
570C4⋊C4—1221no—
571C4⋊C4—1222no—
572C4⋊C4—1222no—
573C4⋊C4—2444no—
574C4⋊C4—2444no—
575C22⋊C4—21no—
576C22⋊C4—22no—
577C22⋊C4—22no—
578C22⋊C4—1221no—
579C22⋊C4—1221no—
580C22⋊C4—1222no—
581C22⋊C4—1222no—
582C22⋊C4—1222no—
583C22⋊C4—1222no—
584C22⋊C4—1221no—
585C22⋊C4—1222no—
586C22⋊C4—1222no—
587C22⋊C4—1228no—
588C22⋊C4—2444no—
589C42—611no—
590C42—611no—
591C42—1221no—
592C42—1221no—
593C42—2444no—
594C42—2444no—
595C42C22×#692311yescharacteristic
596C42C22×#692311yescharacteristic
597C23—21no—
598C23—611no—
599C23—612no—
600C23—611no—
601C23—612no—
602C23—1222no—
603C23—1222no—
604C23—1222no—
605C23—1221no—
606C23—4882no—
607C23—4881no—
608C23—4882no—
609C23—2442no—
610C23C2×C22⋊#692311yescharacteristic
611Q8—42no—
612Q8—41no—
613Q8—41no—
614Q8—41no—
615Q8—2441no—
616Q8—2442no—
617D4—41no—
618D4—42no—
619D4—4882no—
620D4—4882no—
621D4—4882no—
622D4—4882no—
623D4—2442no—
624D4—2442no—
625D4—2442no—
626D4—2442no—
627D4—2442no—
628D4—2441no—
629D4—2441no—
630D4—2441no—
631D4—2442no—
632D4—2442no—
633D4—2442no—
634D4—2442no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—42no—
642C2×C4—612no—
643C2×C4—612no—
644C2×C4—611no—
645C2×C4—611no—
646C2×C4—1222no—
647C2×C4—1222no—
648C2×C4—1222no—
649C2×C4—1222no—
650C2×C4—1221no—
651C2×C4—1221no—
652C2×C4—1222no—
653C2×C4—1222no—
654C2×C4—1222no—
655C2×C4—1222no—
656C2×C4—1222no—
657C2×C4—1222no—
658C2×C4—1221no—
659C2×C4—1221no—
660C2×C4—1221no—
661C2×C4—1221no—
662C2×C4—1221no—
663C2×C4—4881no—
664C2×C4—4881no—
665C2×C4—4888no—
666C2×C4—2442no—
667C2×C4—2441no—
668C2×C4—2441no—
669C2×C4—2442no—
670C2×C4—2441no—
671C2×C4—2441no—
672C2×C4—2448no—
673C2×C4C2×C22⋊#692312yes—
674C2×C4C2×C22⋊#692312yes—
675C2×C4C2×C22⋊#692311yescharacteristic
676C2×C4C23×#692311yescharacteristic
677C22—41no—
678C22—42no—
679C22—41no—
680C22—1222no—
681C22—1221no—
682C22—1221no—
683C22—1222no—
684C22—4882no—
685C22—4882no—
686C22—4884no—
687C22—2442no—
688C22—2441no—
689C22—2442no—
690C22—2442no—
691C22—2441no—
692C22—2444no—
693C22—2441no—
694C22D61⋊#9912yes—
695C22C22×C22⋊#692311yescharacteristic
696C4—21no—
697C4—22no—
698C4—22no—
699C4—21no—
700C4—41no—
701C4—41no—
702C4—41no—
703C4—611no—
704C4—611no—
705C4—612no—
706C4—4888no—
707C4—2441no—
708C4—2441no—
709C4—2441no—
710C2—21no—
711C2—22no—
712C2—81no—
713C2—4881no—
714C2—2442no—
715C2—2441no—
716C2—2441no—
717C2—2442no—
718C2C2×D61⋊#9911yescharacteristic
subgroupG/H
centerC2C2×D61⋊#99
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#536C4
socleC2×C61C22×#99
prime radical#21459151C2
cosocleC2×D244C24
hypercenter#536#6923
hypocenterC61#33358
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C23 ◁ C22×Q8 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C2×#16840743C2semidirect
2. #21462102C2semidirect
3. #21462099C2semidirect
4. #21462096C2semidirect
5. #21462094C2semidirect
6. #21462093C2semidirect
7. #21462091C2semidirect
13. #16840743C22semidirect
15. C2×#14519836C22semidirect
16. #16840669C22semidirect
17. #16840668C22semidirect
18. #16840667C22semidirect
19. #16840666C22semidirect
21. C2×#41×D61C4semidirect
23. #16840337C4semidirect
25. #16840336C4semidirect
27. #16840330C4semidirect
31. #16840313C4semidirect
33. #16840311C4semidirect
37. #117×D61C4semidirect
38. C61×#536C4semidirect
54. C2×#11758226C2×C4semidirect
62. C2×#11758224C2×C4semidirect
65. C2×Q8×D61C2×C4semidirect
69. C2×D4×D61C2×C4semidirect
73. #14519775C2×C4semidirect
74. #14519775C2×C4semidirect
80. #14519766C2×C4semidirect
81. #14519766C2×C4semidirect
109. #14519734C2×C4semidirect
110. #14519734C2×C4semidirect
114. C42×D61C2×C4semidirect
115. C42×D61C2×C4semidirect
118. C4⋊Q8×C61C2×C4semidirect
119. C4⋊Q8×C61C2×C4semidirect
120. #124×C61C2×C4semidirect
121. #124×C61C2×C4semidirect
374. #536#6923semidirect
398. C61#33358semidirect
426. C4⋊Q8C2×#6923semidirect
427. C4⋊Q8C2×#6923semidirect
438. #124C2×#6923semidirect
439. #124C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168407432
#214621022
#214620992
#214620962
#214620942
#214620932
#214620912
#214591512
#3335861