Finite Group Databasebeta

← #30623977

Subgroups — #30623977

Order 15616.

1356 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462419C212yesmaximal, complemented
2#21462099C212yesmaximal, complemented
3#21462096C212yesmaximal, complemented
4#21462095C212yesmaximal, complemented
5#21462091C212yesmaximal, complemented
6D61⋊#353C212yesmaximal, complemented
7C2×#16840666C212yesmaximal, complemented
8#21459594C211yesmaximal, characteristic
9#16840829—22no—
10#16840743C2214yescomplemented
11#16840741—22no—
12C2×#14519836—22no—
13C2×#14519836C2214yescomplemented
14D61⋊#99—22no—
15D61⋊#99C2214yescomplemented
16#16840669C2214yescomplemented
17#16840667C2214yescomplemented
18#16840666C2214yescomplemented
19#16840666C2214yescomplemented
20C2×#14519823C2211yescharacteristic
21C2×#14519822C411yescharacteristic, complemented
22#16840430—22no—
23#16840396—22no—
24#16840394—22no—
25#16840394—22no—
26#16840331C2211yescharacteristic
27#16840330C411yescharacteristic, complemented
28#16840330C2211yescharacteristic
29#16840325C411yescharacteristic, complemented
30#16840325C2211yescharacteristic
31#16840311C411yescharacteristic, complemented
32#16840311C2211yescharacteristic
33#16840308C411yescharacteristic, complemented
34C2×#14519773C2211yescharacteristic
35#127×D61C411yescharacteristic, complemented
36#127×D61C2211yescharacteristic
37#16840230C411yescharacteristic, complemented
38C61×#535C411yescharacteristic, 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#14519823—44no—
48#14519822—44no—
49C2×#11758226C2×C411yescharacteristic
50C2×#11758226C2311yescharacteristic
51C2×#11758222C2×C411yescharacteristic
52C2×#11758222C2×C411yescharacteristic, complemented
53C2×#11758222D412yes—
54C2×#11758224—21no—
55C2×#11758224C2×C411yescharacteristic, complemented
56C2×#11758224C2×C411yescharacteristic, complemented
57C22×D244D412yes—
58C22×D244C2311yescharacteristic
59C22×C22⋊D61—22no—
60C22×C22⋊D61—21no—
61C22×C22⋊D61—21no—
62C2×D4×D61—22no—
63C2×D4×D61—22no—
64C2×D4×D61—21no—
65C2×D4×D61C2×C411yescharacteristic
66C2×D4×D61C2×C411yescharacteristic, complemented
67C2×D4×D61C2×C411yescharacteristic, complemented
68C2×D4×D61D412yes—
69C2×D4×D61D412yes—
70C2×D4×D61C2311yescharacteristic
71#14519775C2×C411yescharacteristic, complemented
72#14519774C2×C411yescharacteristic, complemented
73#14519774C2311yescharacteristic
74#14519773—22no—
75#14519773—22no—
76#14519773—22no—
77#14519773C2×C411yescharacteristic, complemented
78#14519773C2×C411yescharacteristic, complemented
79#14519773C2×C411yescharacteristic, complemented
80#14519773C2311yescharacteristic
81#14519773C2311yescharacteristic
82#14519766C2×C411yescharacteristic, complemented
83#14519765C2×C411yescharacteristic, complemented
84#14519765C2311yescharacteristic
85#14519764C2×C411yescharacteristic, complemented
86#14519752—22no—
87#14519750—22no—
88#14519750—22no—
89#14519750—22no—
90#14519750—22no—
91#14519749—22no—
92#14519749—22no—
93#14519749—22no—
94#14519747—22no—
95#14519747—22no—
96#14519747—22no—
97#14519745—22no—
98#14519745—22no—
99#14519742—21no—
100#14519739—21no—
101C4×C22⋊D61—22no—
102#14519737—21no—
103#14519737—21no—
104#14519737—22no—
105#14519737—22no—
106#14519734C2×C411yescharacteristic, complemented
107#14519734C2×C411yescharacteristic, complemented
108C2×C4×Dic61C2×C411yescharacteristic, complemented
109C42×D61C2×C411yescharacteristic, complemented
110C42×D61C2×C411yescharacteristic, complemented
111C2×#41×C61C2×C411yescharacteristic
112C22×D4×C61C2×C411yescharacteristic, complemented
113#127×C61C2×C411yescharacteristic, complemented
114#127×C61C2×C411yescharacteristic, complemented
115#124×C61C2×C411yescharacteristic, complemented
116#124×C61C2×C411yescharacteristic, complemented
117#121×C61—22no—
118#120×C61—22no—
119#117×C61C2×C411yescharacteristic, complemented
120C24×C61—21no—
121C23×D61—21no—
122C23×D61—21no—
123C23×D61C22⋊C412yes—
124C23×D61C2×D412yes—
125C23×D61C2×D412yes—
126C22×#6923—44no—
127C22×#6923—44no—
128#41×C61—42no—
129C2×Q8×C61C22×C411yescharacteristic
130C2×D4×C61—22no—
131C2×D4×C61—22no—
132C2×D4×C61—21no—
133C2×D4×C61—42no—
134C2×D4×C61C22⋊C412yes—
135C2×D4×C61C22×C411yescharacteristic
136C2×D4×C61C22×C411yescharacteristic
137C2×D4×C61C22×C411yescharacteristic
138C22×C4×C61C22⋊C412yes—
139C22×C4×C61C22×C411yescharacteristic
140C2×C22⋊D61—22no—
141C2×C22⋊D61—21no—
142C2×C22⋊D61—21no—
143C2×C22⋊D61—21no—
144C2×C22⋊D61—22no—
145C2×C22⋊D61—22no—
146C2×C22⋊D61—22no—
147C2×C22⋊D61—22no—
148C2×C22⋊D61—21no—
149C2×C22⋊D61—21no—
150C2×C22⋊D61—44no—
151C2×C22⋊D61—44no—
152C2×C22⋊D61—84no—
153C2×C22⋊D61C22⋊C412yes—
154C2×C22⋊D61C22⋊C411yescharacteristic
155C2×C22⋊D61C22⋊C411yescharacteristic
156C2×C22⋊D61C2×D412yes—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160#11758226—42no—
161#11758226—42no—
162#11758224—44no—
163#11758224—42no—
164#11758224—42no—
165D4×D61—28no—
166D4×D61—42no—
167D4×D61—42no—
168D4×D61—42no—
169D4×D61—42no—
170D4×D61—44no—
171D4×D61—44no—
172D4×D61—84no—
173D4×D61—84no—
174#11758222—24no—
175#11758222—44no—
176#11758222—42no—
177C2×D244—22no—
178C2×D244—22no—
179C2×D244—44no—
180C2×D244—42no—
181C2×D244C22⋊C411yescharacteristic
182C2×D244C22⋊C412yes—
183C2×D244C22×C411yescharacteristic
184C2×D244C22×C411yescharacteristic
185C2×D244C2×D412yes—
186C2×D244C2×D412yes—
187C2×D244C2411yescharacteristic
188C2×C4×D61—21no—
189C2×C4×D61—21no—
190C2×C4×D61—21no—
191C2×C4×D61C22⋊C412yes—
192C2×C4×D61C22×C411yescharacteristic
193C2×C4×D61C22×C411yescharacteristic
194C2×C4×D61C2×D412yes—
195C2×Dic122C22⋊C411yescharacteristic
196C2×Dic122C22×C411yescharacteristic
197C22⋊#6923—28no—
198C22⋊#6923—44no—
199C22⋊#6923—44no—
200C4⋊#6923—44no—
201C4×#6923—44no—
202C4⋊C4×C61—21no—
203C22⋊C4×C61—21no—
204C22⋊C4×C61—22no—
205C22⋊C4×C61—22no—
206C42×C61C22×C411yescharacteristic
207C42×C61C22×C411yescharacteristic
208C22⋊Dic61—22no—
209#11758198—21no—
210#11758198—21no—
211#11758198—22no—
212#11758198—22no—
213#11758198—22no—
214#11758196—22no—
215#11758196—21no—
216#11758196—21no—
217C4×Dic61—21no—
218C4×Dic61—21no—
219C4×Dic61—21no—
220C4×Dic61C22×C411yescharacteristic
221C4×Dic61C22×C411yescharacteristic
222C23×C61—21no—
223C23×C61—21no—
224C23×C61—42no—
225C23×C61—82no—
226C23×C61C2×C22⋊C412yes—
227C23×C61C2×C22⋊C411yescharacteristic
228C22×D61—24no—
229C22×D61—24no—
230C22×D61—21no—
231C22×D61—42no—
232C22×D61—42no—
233C22×D61—42no—
234C22×D61—82no—
235C22×D61—82no—
236C22×D61—82no—
237C22×D61—82no—
238C22×D61—82no—
239C22×D61C2×C22⋊C411yescharacteristic
240C22×D61C2×C22⋊C412yes—
241C22×D61C2×C22⋊C411yescharacteristic
242C22×D61C22×D412yes—
243C2×#6923—48no—
244C2×#6923—164no—
245C2×#6923—164no—
246Q8×C61—41no—
247D4×C61—24no—
248D4×C61—41no—
249D4×C61—41no—
250D4×C61—41no—
251D4×C61—42no—
252D4×C61—82no—
253D4×C61—82no—
254C2×C4×C61—22no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—42no—
258C2×C4×C61C2×C22⋊C411yescharacteristic
259C2×C4×C61C2×C22⋊C412yes—
260C2×C4×C61C23×C411yescharacteristic
261C22⋊D61—22no—
262C22⋊D61—24no—
263C22⋊D61—42no—
264C22⋊D61—42no—
265C22⋊D61—44no—
266C22⋊D61—44no—
267C22⋊D61—44no—
268C22⋊D61—42no—
269C22⋊D61—42no—
270C22⋊D61—42no—
271C22⋊D61—42no—
272C22⋊D61—82no—
273C22⋊D61—82no—
274C22⋊D61#9914yes—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—42no—
279C2×Dic61—42no—
280C2×Dic61—42no—
281C2×Dic61C2×C22⋊C411yescharacteristic
282C2×Dic61C2×C22⋊C411yescharacteristic
283D244—22no—
284D244—24no—
285D244—42no—
286D244—41no—
287D244—42no—
288D244—41no—
289D244—41no—
290D244—82no—
291D244—82no—
292C4×D61—22no—
293C4×D61—24no—
294C4×D61—22no—
295C4×D61—42no—
296C4×D61—42no—
297C4×D61—42no—
298C4×D61—42no—
299Dic122—22no—
300Dic122—41no—
301#33359—611nomaximal, Sylow
302C22×C61—22no—
303C22×C61—41no—
304C22×C61—42no—
305C22×C61—42no—
306C22×C61—42no—
307C22×C61—41no—
308C22×C61—41no—
309C22×C61—162no—
310C22×C61C2×#9912yes—
311C22×C61C22×C22⋊C411yescharacteristic
312C2×D61—21no—
313C2×D61—22no—
314C2×D61—22no—
315C2×D61—41no—
316C2×D61—41no—
317C2×D61—42no—
318C2×D61—42no—
319C2×D61—42no—
320C2×D61—41no—
321C2×D61—84no—
322C2×D61—84no—
323C2×D61—162no—
324C2×D61—162no—
325C2×D61—162no—
326C2×D61C2×#9912yes—
327#6923—88no—
328C4×C61—21no—
329C4×C61—21no—
330C4×C61—22no—
331C4×C61—41no—
332C4×C61—41no—
333Dic61—22no—
334Dic61—21no—
335Dic61—41no—
336Dic61—41no—
337Dic61C2×#9912yes—
338#3498—611no—
339#2883—612no—
340#2128—612no—
341#2127—612no—
342#2127—612no—
343#2126—612no—
344C2×#353—612no—
345C2×#353—612no—
346C2×C61—21no—
347C2×C61—22no—
348C2×C61—42no—
349C2×C61—81no—
350C2×C61—81no—
351C2×C61C22×#9911yescharacteristic
352D61—22no—
353D61—42no—
354D61—42no—
355D61—41no—
356D61—81no—
357C2×H(5,𝔽2)—611no—
358C2×H(5,𝔽2)—611no—
359#550—1222no—
360D42—1222no—
361D42—1222no—
362D42—1222no—
363#535—611no—
364#535—611no—
365#535—611no—
366#535—611no—
367#535—611no—
368#535#692311yescharacteristic, complemented
369#532—611no—
370#532—611no—
371#532—611no—
372C2×#127—611no—
373C2×#127—611no—
374C2×#127—611no—
375#518—1222no—
376C4×#41—611no—
377#410—614no—
378#410—1222no—
379C2×#99—614no—
380C2×#99—614no—
381C2×#99—1222no—
382C2×#99—1222no—
383#354—614no—
384#353—614no—
385#353—614no—
386#353—614no—
387C61#3335911yescharacteristic, complemented, Sylow
388H(5,𝔽2)—2444no—
389H(5,𝔽2)—2444no—
390C2×#41—611no—
391C2×#41—611no—
392C2×#41—612no—
393C2×#41—611no—
394C2×#41—611no—
395C2×#41—611no—
396C2×#41—611no—
397C2×#41—1221no—
398C2×#41C2×#692311yescharacteristic
399C22×D4—612no—
400C22×D4—612no—
401C22×D4—611no—
402C22×D4—611no—
403C22×D4—611no—
404C22×D4—611no—
405C22×D4—612no—
406C22×D4—611no—
407C22×D4—1221no—
408C22×D4—1222no—
409C22×D4—1222no—
410C22×D4—1221no—
411C22×D4—1221no—
412C22×D4—1222no—
413C22×D4C2×#692311yescharacteristic, complemented
414C4⋊Q8—611no—
415#127—611no—
416#127—611no—
417#127—611no—
418#127—611no—
419#127—611no—
420#127—611no—
421#127—611no—
422#127—1222no—
423#127—1222no—
424#127—1222no—
425#127C2×#692311yescharacteristic, complemented
426#127C2×#692311yescharacteristic, complemented
427#124—611no—
428#124—611no—
429#124—611no—
430#124—611no—
431#124C2×#692311yescharacteristic, complemented
432#124C2×#692311yescharacteristic, complemented
433#121—22no—
434#121—1222no—
435#121—1222no—
436#121—1222no—
437#121—1222no—
438#121—1222no—
439#121—1222no—
440#121—1222no—
441#121—1222no—
442#120—22no—
443#120—1222no—
444#120—1222no—
445#120—1222no—
446#120—1222no—
447#120—1222no—
448C4×Q8—1221no—
449C4×D4—1222no—
450C4×D4—1222no—
451C4×D4—1221no—
452C4×D4—1221no—
453C4×D4—1221no—
454C4×D4—1222no—
455C4×D4—2448no—
456#117—611no—
457#117—611no—
458#117—1224no—
459#117C2×#692311yescharacteristic, complemented
460C2×C22⋊C4—1224no—
461C2×C22⋊C4—1224no—
462C2×C42—611no—
463C2×C42—611no—
464C2×C42—611no—
465#99—618no—
466#99—1224no—
467#99—1224no—
468#99—1224no—
469C24—21no—
470C24—612no—
471C24—612no—
472C24—612no—
473C24—1221no—
474C24—1221no—
475#41—42no—
476#41—1224no—
477#41—2442no—
478#41—2444no—
479#41—2442no—
480#41—2442no—
481#41—2442no—
482#41—2442no—
483#41—2444no—
484C2×Q8—611no—
485C2×Q8—611no—
486C2×Q8C22×#692311yescharacteristic
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—21no—
490C2×D4—42no—
491C2×D4—611no—
492C2×D4—612no—
493C2×D4—611no—
494C2×D4—612no—
495C2×D4—611no—
496C2×D4—612no—
497C2×D4—612no—
498C2×D4—612no—
499C2×D4—611no—
500C2×D4—611no—
501C2×D4—611no—
502C2×D4—1222no—
503C2×D4—1222no—
504C2×D4—1221no—
505C2×D4—1221no—
506C2×D4—1228no—
507C2×D4—1222no—
508C2×D4—1222no—
509C2×D4—1222no—
510C2×D4—1222no—
511C2×D4—1221no—
512C2×D4—1221no—
513C2×D4—1222no—
514C2×D4—1221no—
515C2×D4—4884no—
516C2×D4—4884no—
517C2×D4—4884no—
518C2×D4—2442no—
519C2×D4—2444no—
520C2×D4—2442no—
521C2×D4—2442no—
522C2×D4—2442no—
523C2×D4—2442no—
524C2×D4—2444no—
525C2×D4—2444no—
526C2×D4—2444no—
527C2×D4—2444no—
528C2×D4C22⋊#692312yes—
529C2×D4C22×#692311yescharacteristic
530C2×D4C22×#692311yescharacteristic
531C2×D4C22×#692311yescharacteristic
532C22×C4—612no—
533C22×C4—611no—
534C22×C4—611no—
535C22×C4—612no—
536C22×C4—1221no—
537C22×C4—1221no—
538C22×C4—1221no—
539C22×C4—1221no—
540C22×C4—1221no—
541C22×C4—1221no—
542C22×C4—2444no—
543C22×C4—2444no—
544C22×C4C22⋊#692312yes—
545C22×C4C22×#692311yescharacteristic
546C4⋊C4—21no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1222no—
550C4⋊C4—2444no—
551C22⋊C4—21no—
552C22⋊C4—22no—
553C22⋊C4—22no—
554C22⋊C4—1221no—
555C22⋊C4—1221no—
556C22⋊C4—1222no—
557C22⋊C4—1222no—
558C22⋊C4—1222no—
559C22⋊C4—1222no—
560C22⋊C4—1228no—
561C22⋊C4—2444no—
562C22⋊C4—2444no—
563C42—611no—
564C42—611no—
565C42—1221no—
566C42—1221no—
567C42—1221no—
568C42—2444no—
569C42C22×#692311yescharacteristic
570C42C22×#692311yescharacteristic
571C23—21no—
572C23—21no—
573C23—42no—
574C23—82no—
575C23—611no—
576C23—611no—
577C23—612no—
578C23—612no—
579C23—1224no—
580C23—1224no—
581C23—1221no—
582C23—4882no—
583C23—4882no—
584C23—4882no—
585C23—4882no—
586C23—4882no—
587C23—2442no—
588C23—2442no—
589C23—2442no—
590C23C2×C22⋊#692312yes—
591C23C2×C22⋊#692311yescharacteristic
592Q8—41no—
593Q8—1222no—
594Q8—2441no—
595D4—24no—
596D4—41no—
597D4—41no—
598D4—41no—
599D4—42no—
600D4—82no—
601D4—82no—
602D4—614no—
603D4—1224no—
604D4—1222no—
605D4—1224no—
606D4—1222no—
607D4—4882no—
608D4—4882no—
609D4—4882no—
610D4—4882no—
611D4—2442no—
612D4—2441no—
613D4—2441no—
614D4—2442no—
615D4—2441no—
616D4—2442no—
617D4—2442no—
618D4—2444no—
619D4—2444no—
620D4—2444no—
621D4—2442no—
622D4—2442no—
623D4—2442no—
624D4—2442no—
625C2×C4—22no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—42no—
629C2×C4—611no—
630C2×C4—611no—
631C2×C4—1224no—
632C2×C4—1222no—
633C2×C4—1222no—
634C2×C4—1221no—
635C2×C4—1221no—
636C2×C4—1221no—
637C2×C4—9764no—
638C2×C4—9764no—
639C2×C4—2442no—
640C2×C4—2442no—
641C2×C4—2442no—
642C2×C4—2442no—
643C2×C4—2442no—
644C2×C4—2442no—
645C2×C4—2442no—
646C2×C4—2448no—
647C2×C4C2×C22⋊#692311yescharacteristic
648C2×C4C2×C22⋊#692312yes—
649C2×C4C23×#692311yescharacteristic
650C22—22no—
651C22—41no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—41no—
656C22—41no—
657C22—162no—
658C22—612no—
659C22—1222no—
660C22—1221no—
661C22—1222no—
662C22—9762no—
663C22—9762no—
664C22—9762no—
665C22—4884no—
666C22—4884no—
667C22—2441no—
668C22—2442no—
669C22—2441no—
670C22—2442no—
671C22—2442no—
672C22—2441no—
673C22D61⋊#9912yes—
674C22C22×C22⋊#692311yescharacteristic
675C4—21no—
676C4—21no—
677C4—22no—
678C4—41no—
679C4—41no—
680C4—612no—
681C4—1222no—
682C4—1221no—
683C4—4888no—
684C4—2441no—
685C4—2441no—
686C2—21no—
687C2—22no—
688C2—42no—
689C2—81no—
690C2—81no—
691C2—1222no—
692C2—4881no—
693C2—2442no—
694C2—2441no—
695C2—2442no—
696C2C2×D61⋊#9911yescharacteristic
subgroupG/H
centerC2C2×D61⋊#99
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#535C4
socleC2×C61C22×#99
prime radical#21459594C2
cosocleC2×D244C24
hypercenter#535#6923
hypocenterC61#33359
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C23 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #21462419C2semidirect
2. #21462099C2semidirect
3. #21462096C2semidirect
4. #21462095C2semidirect
5. #21462091C2semidirect
6. D61⋊#353C2semidirect
7. C2×#16840666C2semidirect
10. #16840743C22semidirect
13. C2×#14519836C22semidirect
15. D61⋊#99C22semidirect
16. #16840669C22semidirect
17. #16840667C22semidirect
18. #16840666C22semidirect
19. #16840666C22semidirect
21. C2×#14519822C4semidirect
27. #16840330C4semidirect
29. #16840325C4semidirect
31. #16840311C4semidirect
33. #16840308C4semidirect
35. #127×D61C4semidirect
37. #16840230C4semidirect
38. C61×#535C4semidirect
52. C2×#11758222C2×C4semidirect
55. C2×#11758224C2×C4semidirect
56. C2×#11758224C2×C4semidirect
66. C2×D4×D61C2×C4semidirect
67. C2×D4×D61C2×C4semidirect
71. #14519775C2×C4semidirect
72. #14519774C2×C4semidirect
77. #14519773C2×C4semidirect
78. #14519773C2×C4semidirect
79. #14519773C2×C4semidirect
82. #14519766C2×C4semidirect
83. #14519765C2×C4semidirect
85. #14519764C2×C4semidirect
106. #14519734C2×C4semidirect
107. #14519734C2×C4semidirect
108. C2×C4×Dic61C2×C4semidirect
109. C42×D61C2×C4semidirect
110. C42×D61C2×C4semidirect
112. C22×D4×C61C2×C4semidirect
113. #127×C61C2×C4semidirect
114. #127×C61C2×C4semidirect
115. #124×C61C2×C4semidirect
116. #124×C61C2×C4semidirect
119. #117×C61C2×C4semidirect
368. #535#6923semidirect
387. C61#33359semidirect
413. C22×D4C2×#6923semidirect
425. #127C2×#6923semidirect
426. #127C2×#6923semidirect
431. #124C2×#6923semidirect
432. #124C2×#6923semidirect
459. #117C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214624192
#214620992
#214620962
#214620952
#214620912
D61⋊#3532
C2×#168406662
#214595942
#3335961