Finite Group Databasebeta

← #30623980

Subgroups — #30623980

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#21462097C212yesmaximal, complemented
4#21462094C212yesmaximal, complemented
5#21462093C212yesmaximal, complemented
6C2×#16840667C212yesmaximal, complemented
7D61⋊#353C212yesmaximal, complemented
8#21459592C211yesmaximal, characteristic
9#16840829—22no—
10#16840743C2214yescomplemented
11#16840741—22no—
12C2×#14519836—22no—
13C2×#14519836C2214yescomplemented
14D61⋊#99—22no—
15D61⋊#99C2214yescomplemented
16#16840668C2214yescomplemented
17#16840667C2214yescomplemented
18#16840667C2214yescomplemented
19#16840666C2214yescomplemented
20C2×#14519823C411yescharacteristic, complemented
21C2×#14519823C2211yescharacteristic
22#16840433—22no—
23D4×D244—22no—
24#16840396—22no—
25#16840396—22no—
26#16840330C411yescharacteristic, complemented
27#16840330C411yescharacteristic, complemented
28#16840330C2211yescharacteristic
29#16840330C2211yescharacteristic
30#16840326C411yescharacteristic, complemented
31#16840326C2211yescharacteristic
32#16840313C411yescharacteristic, complemented
33#16840313C2211yescharacteristic
34C2×#14519774C2211yescharacteristic
35#127×D61C411yescharacteristic, complemented
36#127×D61C2211yescharacteristic
37C4×#11758222C411yescharacteristic, complemented
38C61×#532C411yescharacteristic, 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#14519823—44no—
49C2×#11758226C2×C411yescharacteristic
50C2×#11758226C2×C411yescharacteristic, complemented
51C2×#11758226C2×C411yescharacteristic, complemented
52C2×#11758226C2311yescharacteristic
53C2×#11758222—21no—
54C2×#11758222C2×C411yescharacteristic
55C2×#11758222D412yes—
56C22×D244—22no—
57C22×D244—21no—
58C22×D244C2×C411yescharacteristic, complemented
59C22×D244D412yes—
60C22×D244C2311yescharacteristic
61C2×D4×D61—21no—
62C2×D4×D61—21no—
63C2×D4×D61—22no—
64C2×D4×D61—22no—
65C2×D4×D61C2×C411yescharacteristic
66C2×D4×D61C2×C411yescharacteristic, complemented
67C2×D4×D61C2×C411yescharacteristic, complemented
68C2×D4×D61D412yes—
69C2×D4×D61D412yes—
70C2×D4×D61C2311yescharacteristic
71#14519774—22no—
72#14519774C2×C411yescharacteristic, complemented
73#14519774C2×C411yescharacteristic, complemented
74#14519774C2311yescharacteristic
75#14519774C2311yescharacteristic
76#14519773—22no—
77#14519773—22no—
78#14519773C2×C411yescharacteristic, complemented
79#14519773C2×C411yescharacteristic, complemented
80#14519773C2×C411yescharacteristic, complemented
81#14519773C2311yescharacteristic
82#14519766C2×C411yescharacteristic, complemented
83#14519766C2×C411yescharacteristic, complemented
84#14519766C2×C411yescharacteristic, complemented
85#14519766C2311yescharacteristic
86#14519752—22no—
87#14519752—22no—
88#14519751—22no—
89#14519751—22no—
90#14519750—22no—
91#14519750—22no—
92#14519750—22no—
93#14519750—22no—
94#14519747—22no—
95#14519747—22no—
96#14519747—22no—
97#14519747—22no—
98#14519745—22no—
99#14519745—22no—
100C4×Dic122—21no—
101C4×D244—21no—
102#14519739—22no—
103#14519739—22no—
104C4×C22⋊D61—21no—
105C4×C22⋊D61—21no—
106#14519736C2×C411yescharacteristic, complemented
107#14519734C2×C411yescharacteristic, complemented
108#14519734C2×C411yescharacteristic, complemented
109C42×D61C2×C411yescharacteristic, complemented
110C42×D61C2×C411yescharacteristic, complemented
111C2×#41×C61C2×C411yescharacteristic
112C2×#41×C61C2×C411yescharacteristic, complemented
113C4⋊Q8×C61C2×C411yescharacteristic, complemented
114#127×C61C2×C411yescharacteristic, complemented
115#124×C61C2×C411yescharacteristic, complemented
116#124×C61C2×C411yescharacteristic, complemented
117#121×C61—22no—
118C4×D4×C61—22no—
119C2×C42×C61C2×C411yescharacteristic, complemented
120C23×D61—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—
129#41×C61—42no—
130C2×Q8×C61C22×C411yescharacteristic
131C2×Q8×C61C22×C411yescharacteristic
132C2×D4×C61—22no—
133C2×D4×C61—21no—
134C2×D4×C61C22⋊C412yes—
135C2×D4×C61C22×C411yescharacteristic
136C2×D4×C61C22×C411yescharacteristic
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61C22⋊C412yes—
140C22×C4×C61C22×C411yescharacteristic
141C2×C22⋊D61—22no—
142C2×C22⋊D61—22no—
143C2×C22⋊D61—22no—
144C2×C22⋊D61—22no—
145C2×C22⋊D61—21no—
146C2×C22⋊D61—21no—
147C2×C22⋊D61C22⋊C411yescharacteristic
148C2×C22⋊D61C22⋊C412yes—
149C2×C22⋊D61C22⋊C411yescharacteristic
150C2×C22⋊D61C2×D412yes—
151#11758226—42no—
152#11758226—42no—
153#11758226—42no—
154#11758226—42no—
155D4×D61—28no—
156D4×D61—42no—
157D4×D61—42no—
158D4×D61—42no—
159D4×D61—42no—
160D4×D61—44no—
161D4×D61—44no—
162D4×D61—44no—
163D4×D61—84no—
164D4×D61—84no—
165#11758222—24no—
166#11758222—44no—
167#11758222—44no—
168C2×D244—21no—
169C2×D244—21no—
170C2×D244—22no—
171C2×D244—22no—
172C2×D244—21no—
173C2×D244—22no—
174C2×D244—22no—
175C2×D244—44no—
176C2×D244—44no—
177C2×D244—42no—
178C2×D244—42no—
179C2×D244—84no—
180C2×D244C22⋊C412yes—
181C2×D244C22⋊C411yescharacteristic
182C2×D244C22×C411yescharacteristic
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×D61—21no—
192C2×C4×D61C22⋊C412yes—
193C2×C4×D61C22×C411yescharacteristic
194C2×C4×D61C22×C411yescharacteristic
195C2×C4×D61C2×D412yes—
196C2×Dic122C22⋊C411yescharacteristic
197C22⋊#6923—28no—
198C22⋊#6923—44no—
199C22⋊#6923—44no—
200C4⋊#6923—44no—
201C4×#6923—44no—
202C4⋊C4×C61—22no—
203C22⋊C4×C61—22no—
204C22⋊C4×C61—22no—
205C42×C61—21no—
206C42×C61C22×C411yescharacteristic
207C42×C61C22×C411yescharacteristic
208C22⋊Dic61—21no—
209#11758198—21no—
210#11758198—21no—
211#11758198—22no—
212#11758198—22no—
213#11758198—22no—
214#11758198—22no—
215C4⋊Dic61—21no—
216#11758196—21no—
217#11758196—21no—
218C4×Dic61—21no—
219C4×Dic61—21no—
220C4×Dic61C22×C411yescharacteristic
221C4×Dic61C22×C411yescharacteristic
222C23×C61—21no—
223C23×C61C2×C22⋊C412yes—
224C23×C61C2×C22⋊C411yescharacteristic
225C22×D61—24no—
226C22×D61—24no—
227C22×D61—21no—
228C22×D61—21no—
229C22×D61—42no—
230C22×D61—42no—
231C22×D61—42no—
232C22×D61—42no—
233C22×D61—82no—
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—
247Q8×C61—41no—
248D4×C61—24no—
249D4×C61—41no—
250D4×C61—41no—
251D4×C61—44no—
252D4×C61—42no—
253C2×C4×C61—21no—
254C2×C4×C61—22no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—42no—
258C2×C4×C61—42no—
259C2×C4×C61—42no—
260C2×C4×C61C2×C22⋊C411yescharacteristic
261C2×C4×C61C2×C22⋊C412yes—
262C2×C4×C61C23×C411yescharacteristic
263C22⋊D61—22no—
264C22⋊D61—24no—
265C22⋊D61—42no—
266C22⋊D61—42no—
267C22⋊D61—42no—
268C22⋊D61—82no—
269C22⋊D61—82no—
270C22⋊D61—82no—
271C22⋊D61—82no—
272C22⋊D61#9914yes—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61C2×C22⋊C411yescharacteristic
276C2×Dic61C2×C22⋊C411yescharacteristic
277D244—22no—
278D244—24no—
279D244—42no—
280D244—41no—
281D244—42no—
282D244—41no—
283D244—41no—
284D244—41no—
285D244—42no—
286D244—44no—
287D244—44no—
288D244—42no—
289D244—42no—
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—
299C4×D61—42no—
300Dic122—22no—
301#33359—611nomaximal, Sylow
302C22×C61—22no—
303C22×C61—41no—
304C22×C61—42no—
305C22×C61—42no—
306C22×C61—41no—
307C22×C61C2×#9912yes—
308C22×C61C22×C22⋊C411yescharacteristic
309C2×D61—21no—
310C2×D61—22no—
311C2×D61—22no—
312C2×D61—41no—
313C2×D61—41no—
314C2×D61—42no—
315C2×D61—42no—
316C2×D61—42no—
317C2×D61—42no—
318C2×D61—41no—
319C2×D61—41no—
320C2×D61—84no—
321C2×D61—84no—
322C2×D61—162no—
323C2×D61—162no—
324C2×D61—162no—
325C2×D61—162no—
326C2×D61C2×#9912yes—
327#6923—88no—
328C4×C61—22no—
329C4×C61—21no—
330C4×C61—21no—
331C4×C61—22no—
332C4×C61—41no—
333C4×C61—41no—
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×C61C22×#9911yescharacteristic
351D61—22no—
352D61—42no—
353D61—42no—
354D61—41no—
355D61—81no—
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—611no—
369#532—611no—
370#532—611no—
371#532#692311yescharacteristic, complemented
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—1221no—
397C2×#41C2×#692311yescharacteristic
398C2×#41C2×#692311yescharacteristic, complemented
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—611no—
408C22×D4—1221no—
409C22×D4—1222no—
410C22×D4—1221no—
411C22×D4—1221no—
412C22×D4—1222no—
413C22×D4—1222no—
414C4⋊Q8C2×#692311yescharacteristic, complemented
415#127—611no—
416#127—611no—
417#127—611no—
418#127—611no—
419#127—611no—
420#127—611no—
421#127—611no—
422#127—611no—
423#127—1222no—
424#127—1222no—
425#127—1222no—
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—1222no—
443#120—1222no—
444#120—1222no—
445#120—1222no—
446#120—1222no—
447#120—1222no—
448C4×Q8—1221no—
449C4×D4—22no—
450C4×D4—1221no—
451C4×D4—1222no—
452C4×D4—1222no—
453C4×D4—1221no—
454C4×D4—1221no—
455C4×D4—2448no—
456#117—611no—
457#117—611no—
458#117—611no—
459#117—1224no—
460C2×C22⋊C4—1224no—
461C2×C22⋊C4—1224no—
462C2×C42—611no—
463C2×C42—611no—
464C2×C42C2×#692311yescharacteristic, complemented
465#99—618no—
466#99—1224no—
467#99—1224no—
468#99—1224no—
469C24—612no—
470C24—612no—
471C24—612no—
472C24—1221no—
473C24—1221no—
474C24—1221no—
475#41—42no—
476#41—42no—
477#41—1224no—
478#41—2444no—
479#41—2444no—
480#41—2442no—
481#41—2442no—
482#41—2442no—
483#41—2442no—
484C2×Q8—611no—
485C2×Q8C22×#692311yescharacteristic
486C2×Q8C22×#692311yescharacteristic
487C2×D4—22no—
488C2×D4—21no—
489C2×D4—611no—
490C2×D4—612no—
491C2×D4—611no—
492C2×D4—612no—
493C2×D4—611no—
494C2×D4—612no—
495C2×D4—612no—
496C2×D4—612no—
497C2×D4—611no—
498C2×D4—611no—
499C2×D4—611no—
500C2×D4—611no—
501C2×D4—1222no—
502C2×D4—1222no—
503C2×D4—1222no—
504C2×D4—1222no—
505C2×D4—1221no—
506C2×D4—1221no—
507C2×D4—1221no—
508C2×D4—1221no—
509C2×D4—1228no—
510C2×D4—1222no—
511C2×D4—1222no—
512C2×D4—1222no—
513C2×D4—1222no—
514C2×D4—1221no—
515C2×D4—4884no—
516C2×D4—4884no—
517C2×D4—4884no—
518C2×D4—2442no—
519C2×D4—2442no—
520C2×D4—2444no—
521C2×D4—2444no—
522C2×D4—2442no—
523C2×D4—2442no—
524C2×D4—2442no—
525C2×D4—2442no—
526C2×D4—2444no—
527C2×D4—2444no—
528C2×D4—2444no—
529C2×D4C22⋊#692312yes—
530C2×D4C22×#692311yescharacteristic
531C2×D4C22×#692311yescharacteristic
532C22×C4—21no—
533C22×C4—21no—
534C22×C4—612no—
535C22×C4—611no—
536C22×C4—611no—
537C22×C4—612no—
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—22no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—2444no—
551C22⋊C4—22no—
552C22⋊C4—22no—
553C22⋊C4—1221no—
554C22⋊C4—1221no—
555C22⋊C4—1222no—
556C22⋊C4—1222no—
557C22⋊C4—1221no—
558C22⋊C4—1222no—
559C22⋊C4—1222no—
560C22⋊C4—1228no—
561C22⋊C4—2444no—
562C22⋊C4—2444no—
563C42—21no—
564C42—611no—
565C42—611no—
566C42—1221no—
567C42—1221no—
568C42—2444no—
569C42C22×#692311yescharacteristic
570C42C22×#692311yescharacteristic
571C23—21no—
572C23—611no—
573C23—611no—
574C23—612no—
575C23—612no—
576C23—1224no—
577C23—1224no—
578C23—1221no—
579C23—1221no—
580C23—4882no—
581C23—4882no—
582C23—4882no—
583C23—4882no—
584C23—4882no—
585C23—4882no—
586C23—2442no—
587C23—2442no—
588C23—2442no—
589C23—2442no—
590C23C2×C22⋊#692312yes—
591C23C2×C22⋊#692311yescharacteristic
592Q8—41no—
593Q8—41no—
594Q8—1222no—
595D4—24no—
596D4—41no—
597D4—41no—
598D4—44no—
599D4—42no—
600D4—614no—
601D4—1224no—
602D4—1222no—
603D4—1224no—
604D4—1222no—
605D4—4882no—
606D4—4882no—
607D4—4882no—
608D4—4882no—
609D4—4882no—
610D4—4882no—
611D4—2442no—
612D4—2441no—
613D4—2441no—
614D4—2442no—
615D4—2441no—
616D4—2441no—
617D4—2444no—
618D4—2444no—
619D4—2442no—
620D4—2442no—
621D4—2442no—
622D4—2442no—
623D4—2442no—
624D4—2442no—
625C2×C4—22no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—611no—
633C2×C4—611no—
634C2×C4—1224no—
635C2×C4—1222no—
636C2×C4—1222no—
637C2×C4—1221no—
638C2×C4—1221no—
639C2×C4—9764no—
640C2×C4—9764no—
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—41no—
655C22—612no—
656C22—1222no—
657C22—1221no—
658C22—1222no—
659C22—9762no—
660C22—9762no—
661C22—9762no—
662C22—9762no—
663C22—4884no—
664C22—4884no—
665C22—2441no—
666C22—2442no—
667C22—2442no—
668C22—2441no—
669C22—2442no—
670C22—2442no—
671C22—2441no—
672C22—2441no—
673C22D61⋊#9912yes—
674C22C22×C22⋊#692311yescharacteristic
675C4—22no—
676C4—21no—
677C4—21no—
678C4—22no—
679C4—41no—
680C4—41no—
681C4—612no—
682C4—1221no—
683C4—4888no—
684C4—2441no—
685C4—2441no—
686C2—21no—
687C2—22no—
688C2—42no—
689C2—81no—
690C2—1222no—
691C2—4881no—
692C2—4881no—
693C2—2442no—
694C2—2441no—
695C2—2442no—
696C2C2×D61⋊#9911yescharacteristic
subgroupG/H
centerC2C2×D61⋊#99
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#532C4
socleC2×C61C22×#99
prime radical#21459592C2
cosocleC2×D244C24
hypercenter#532#6923
hypocenterC61#33359
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C23 ◁ C2×#41 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #21462419C2semidirect
2. #21462099C2semidirect
3. #21462097C2semidirect
4. #21462094C2semidirect
5. #21462093C2semidirect
6. C2×#16840667C2semidirect
7. D61⋊#353C2semidirect
10. #16840743C22semidirect
13. C2×#14519836C22semidirect
15. D61⋊#99C22semidirect
16. #16840668C22semidirect
17. #16840667C22semidirect
18. #16840667C22semidirect
19. #16840666C22semidirect
20. C2×#14519823C4semidirect
26. #16840330C4semidirect
27. #16840330C4semidirect
30. #16840326C4semidirect
32. #16840313C4semidirect
35. #127×D61C4semidirect
37. C4×#11758222C4semidirect
38. C61×#532C4semidirect
50. C2×#11758226C2×C4semidirect
51. C2×#11758226C2×C4semidirect
58. C22×D244C2×C4semidirect
66. C2×D4×D61C2×C4semidirect
67. C2×D4×D61C2×C4semidirect
72. #14519774C2×C4semidirect
73. #14519774C2×C4semidirect
78. #14519773C2×C4semidirect
79. #14519773C2×C4semidirect
80. #14519773C2×C4semidirect
82. #14519766C2×C4semidirect
83. #14519766C2×C4semidirect
84. #14519766C2×C4semidirect
106. #14519736C2×C4semidirect
107. #14519734C2×C4semidirect
108. #14519734C2×C4semidirect
109. C42×D61C2×C4semidirect
110. C42×D61C2×C4semidirect
112. C2×#41×C61C2×C4semidirect
113. C4⋊Q8×C61C2×C4semidirect
114. #127×C61C2×C4semidirect
115. #124×C61C2×C4semidirect
116. #124×C61C2×C4semidirect
119. C2×C42×C61C2×C4semidirect
371. #532#6923semidirect
387. C61#33359semidirect
398. C2×#41C2×#6923semidirect
414. C4⋊Q8C2×#6923semidirect
426. #127C2×#6923semidirect
431. #124C2×#6923semidirect
432. #124C2×#6923semidirect
464. C2×C42C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214624192
#214620992
#214620972
#214620942
#214620932
C2×#168406672
D61⋊#3532
#214595922
#3335961