Finite Group Databasebeta

← #29689161

Subgroups — #29689161

Order 15616.

1520 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459450C211yesmaximal, characteristic
2#21454280C211yesmaximal, characteristic, complemented
3#21446361C212yesmaximal, complemented
4#21446359C212yesmaximal, complemented
5#21446227C212yesmaximal, complemented
6#21446225C212yesmaximal, complemented
7#21445925C212yesmaximal, complemented
8#21445732C212yesmaximal, complemented
9C61×#2883C211yesmaximal, characteristic, complemented
10C22×D4×D61C2211yescharacteristic
11#16840400—22no—
12#16840400—22no—
13D4×C22⋊D61—22no—
14D4×C22⋊D61—22no—
15#16840326C412yescomplemented
16#16840321C412yescomplemented
17#16840321C2211yescharacteristic
18#16840321C2211yescharacteristic
19#16840269C412yescomplemented
20C2×#14519745C412yescomplemented
21C2×#14519745C2211yescharacteristic
22C2×#14519745C2211yescharacteristic
23#16840237C2211yescharacteristic
24#16840237C2212yescomplemented
25#16839736—22no—
26C2×#14519692—22no—
27C2×#14519692C2212yes—
28C2×#14519692C2212yescomplemented
29C2×#14519691C2214yescomplemented
30#16839617C2214yescomplemented
31#16839529C2214yescomplemented
32#16839528C2214yescomplemented
33C2×H(5,𝔽2)×C61C2211yescharacteristic
34C61×#518C2212yescomplemented
35C61×#410—22no—
36C2×#99×C61—22no—
37C2×#99×C61C2212yes—
38C2×#99×C61C2212yescomplemented
39C24×D61C2×C411yescharacteristic, complemented
40C24×D61C2311yescharacteristic
41C22×D244C2×C411yescharacteristic, complemented
42C22×C22⋊D61—22no—
43C22×C22⋊D61—21no—
44C22×C22⋊D61—21no—
45C22×C22⋊D61C2×C411yescharacteristic, complemented
46C22×C22⋊D61C2311yescharacteristic
47C2×D4×D61—22no—
48C2×D4×D61—22no—
49C2×D4×D61—44no—
50C22×C4×D61C2×C411yescharacteristic, complemented
51#14519773—22no—
52#14519766—22no—
53#14519760—22no—
54#14519760—21no—
55#14519760—21no—
56#14519758C2×C412yescomplemented
57#14519758C2×C412yescomplemented
58#14519753—22no—
59#14519751—22no—
60#14519751—22no—
61#14519749—22no—
62#14519749—22no—
63#14519749—22no—
64#14519749—22no—
65#14519748—22no—
66#14519746—22no—
67#14519746—21no—
68#14519746—21no—
69#14519745—22no—
70#14519745—22no—
71#14519745—22no—
72#14519745—22no—
73#14519745C2×C412yescomplemented
74#14519745C2×C412yescomplemented
75#14519745C2×C412yescomplemented
76#14519745C2×C412yescomplemented
77#14519745C2×C412yescomplemented
78#14519745C2×C412yescomplemented
79D4×Dic61—22no—
80D4×Dic61—22no—
81D4×Dic61—44no—
82C4×C22⋊D61—22no—
83C4×C22⋊D61—22no—
84#14519736—22no—
85#14519736C2×C412yescomplemented
86C2×C22⋊Dic61—22no—
87C2×C22⋊Dic61C2×C412yescomplemented
88C2×C22⋊Dic61C2314yes—
89C2×C22⋊Dic61C2311yescharacteristic
90C2×C22⋊Dic61C2311yescharacteristic
91C2×#11758198—22no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×#11758198—44no—
95#14519692—24no—
96#14519692—24no—
97#14519691—44no—
98H(5,𝔽2)×C61—44no—
99C2×#41×C61C2×C412yes—
100C2×#41×C61D412yes—
101C2×#41×C61D412yes—
102C22×D4×C61C2×C412yes—
103C22×D4×C61D412yes—
104C22×D4×C61D412yes—
105C22×D4×C61C2311yescharacteristic
106C22×D4×C61C2311yescharacteristic
107C22×D4×C61C2311yescharacteristic
108C4×D4×C61—44no—
109#117×C61—22no—
110C2×C22⋊C4×C61—22no—
111C2×C22⋊C4×C61C2314yes—
112#99×C61—24no—
113#99×C61—24no—
114C24×C61C22⋊C412yescomplemented
115C24×C61C22×C411yescharacteristic
116C24×C61C2×D411yescharacteristic
117C24×C61C2×D411yescharacteristic
118C24×C61C2411yescharacteristic
119C23×D61—22no—
120C23×D61—21no—
121C23×D61—21no—
122C23×D61—24no—
123C23×D61—42no—
124C23×D61—82no—
125C23×D61—82no—
126C23×D61C22×C411yescharacteristic
127C23×D61C22×C411yescharacteristic
128C23×D61C22×C411yescharacteristic
129#41×C61—44no—
130#41×C61—44no—
131#41×C61—44no—
132C2×Q8×C61C22⋊C412yescomplemented
133C2×D4×C61—24no—
134C2×D4×C61—24no—
135C2×D4×C61—24no—
136C2×D4×C61—44no—
137C2×D4×C61—44no—
138C2×D4×C61—44no—
139C2×D4×C61C22⋊C412yescomplemented
140C2×D4×C61C22⋊C412yescomplemented
141C2×D4×C61C22⋊C412yescomplemented
142C2×D4×C61C22×C412yes—
143C2×D4×C61C22×C412yes—
144C2×D4×C61C2×D412yes—
145C2×D4×C61C2×D412yes—
146C2×D4×C61C2×D412yes—
147C2×D4×C61C2×D412yes—
148C22×C4×C61—24no—
149C22×C4×C61—42no—
150C22×C4×C61C22⋊C412yescomplemented
151C22×C4×C61C22⋊C412yescomplemented
152C22×C4×C61C22⋊C412yescomplemented
153C22×C4×C61C22×C411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C2×C22⋊D61—22no—
157C2×C22⋊D61—22no—
158C2×C22⋊D61—22no—
159C2×C22⋊D61—21no—
160C2×C22⋊D61—21no—
161C2×C22⋊D61—24no—
162C2×C22⋊D61—22no—
163C2×C22⋊D61—22no—
164C2×C22⋊D61—22no—
165C2×C22⋊D61—22no—
166C2×C22⋊D61—42no—
167C2×C22⋊D61—82no—
168C2×C22⋊D61—82no—
169C2×C22⋊D61—84no—
170C22×Dic61—22no—
171C22×Dic61—21no—
172C22×Dic61—21no—
173C22×Dic61—24no—
174C22×Dic61—42no—
175C22×Dic61C22×C411yescharacteristic
176D4×D61—48no—
177D4×D61—48no—
178C2×D244—22no—
179C2×D244—42no—
180C2×C4×D61—22no—
181C2×C4×D61—42no—
182C4⋊C4×C61—42no—
183C22⋊C4×C61—24no—
184C22⋊C4×C61—24no—
185C22⋊C4×C61—42no—
186C42×C61—42no—
187C22⋊Dic61—22no—
188C22⋊Dic61—22no—
189C22⋊Dic61—22no—
190C22⋊Dic61—21no—
191C22⋊Dic61—21no—
192C22⋊Dic61—24no—
193C22⋊Dic61—24no—
194C22⋊Dic61—42no—
195C22⋊Dic61C22×C412yes—
196C22⋊Dic61C22×C412yes—
197#11758198—22no—
198#11758198—22no—
199#11758198—22no—
200#11758198—22no—
201#11758198—22no—
202#11758198—22no—
203#11758198—88no—
204C4⋊Dic61—22no—
205C4⋊Dic61—42no—
206#11758196—22no—
207#11758196—22no—
208C4×Dic61—22no—
209C4×Dic61—42no—
210C23×C61—22no—
211C23×C61—22no—
212C23×C61—22no—
213C23×C61—42no—
214C23×C61—81no—
215C23×C61—81no—
216C23×C61—81no—
217C23×C61—81no—
218C23×C61—82no—
219C23×C61C2×C22⋊C412yes—
220C23×C61C2×C22⋊C412yes—
221C23×C61C2×C22⋊C412yes—
222C23×C61C23×C411yescharacteristic
223C23×C61C22×D411yescharacteristic
224C23×C61C22×D411yescharacteristic
225C23×C61H(5,𝔽2)14yes—
226C22×D61—21no—
227C22×D61—22no—
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—22no—
231C22×D61—22no—
232C22×D61—42no—
233C22×D61—48no—
234C22×D61—42no—
235C22×D61—42no—
236C22×D61—44no—
237C22×D61—44no—
238C22×D61—42no—
239C22×D61—42no—
240C22×D61—84no—
241C22×D61—81no—
242C22×D61—81no—
243C22×D61—81no—
244C22×D61—82no—
245C22×D61—84no—
246C22×D61—81no—
247C22×D61—84no—
248Q8×C61—42no—
249D4×C61—24no—
250D4×C61—24no—
251D4×C61—24no—
252D4×C61—24no—
253D4×C61—24no—
254D4×C61—24no—
255D4×C61—42no—
256D4×C61—42no—
257D4×C61—42no—
258C2×C4×C61—22no—
259C2×C4×C61—22no—
260C2×C4×C61—22no—
261C2×C4×C61—42no—
262C2×C4×C61—42no—
263C2×C4×C61—42no—
264C2×C4×C61—44no—
265C2×C4×C61—84no—
266C2×C4×C61—162no—
267C2×C4×C61C2×C22⋊C412yes—
268C2×C4×C61C2×C22⋊C412yes—
269C2×C4×C61C2×C22⋊C412yes—
270C22⋊D61—44no—
271C22⋊D61—44no—
272C22⋊D61—44no—
273C22⋊D61—42no—
274C22⋊D61—42no—
275C22⋊D61—82no—
276C22⋊D61—82no—
277C22⋊D61—82no—
278C22⋊D61—82no—
279C2×Dic61—22no—
280C2×Dic61—22no—
281C2×Dic61—21no—
282C2×Dic61—22no—
283C2×Dic61—44no—
284C2×Dic61—81no—
285C2×Dic61—81no—
286C2×Dic61—82no—
287C2×Dic61—84no—
288C2×Dic61—162no—
289D244—44no—
290C4×D61—44no—
291#21776—611nomaximal, Sylow
292C22×C61—22no—
293C22×C61—22no—
294C22×C61—22no—
295C22×C61—44no—
296C22×C61—41no—
297C22×C61—41no—
298C22×C61—41no—
299C22×C61—42no—
300C22×C61—42no—
301C22×C61—42no—
302C22×C61—82no—
303C22×C61—82no—
304C22×C61—82no—
305C22×C61—162no—
306C22×C61C22×C22⋊C411yescharacteristic
307C22×C61#51812yes—
308C22×C61#53412yes—
309C22×C61#53412yes—
310C2×D61—22no—
311C2×D61—24no—
312C2×D61—41no—
313C2×D61—42no—
314C2×D61—41no—
315C2×D61—41no—
316C2×D61—41no—
317C2×D61—42no—
318C2×D61—44no—
319C2×D61—41no—
320C2×D61—41no—
321C2×D61—84no—
322C2×D61—84no—
323C2×D61—88no—
324C2×D61—82no—
325C4×C61—22no—
326C4×C61—22no—
327C4×C61—22no—
328C4×C61—84no—
329Dic61—42no—
330Dic61—42no—
331Dic61—41no—
332Dic61—84no—
333#3484—611no—
334#2883—611no—
335#2883D6111yescharacteristic, complemented
336#1913—612no—
337#1913—612no—
338#1889—612no—
339#1889—612no—
340#1828—612no—
341#1781—612no—
342C2×C61—21no—
343C2×C61—22no—
344C2×C61—22no—
345C2×C61—22no—
346C2×C61—42no—
347C2×C61—42no—
348C2×C61—42no—
349C2×C61#230811yescharacteristic
350D61—42no—
351D61—81no—
352D61—81no—
353C2×H(5,𝔽2)C2×D6111yescharacteristic
354C23×D4—611no—
355#546—1222no—
356#546—1222no—
357D42—1222no—
358D42—1222no—
359#535—612no—
360#534—612no—
361#534—611no—
362#534—611no—
363#525—612no—
364C2×#120—612no—
365C2×#120—611no—
366C2×#120—611no—
367#518—611no—
368#518—612no—
369#518C2×D6112yescomplemented
370#410—22no—
371#410—1222no—
372C2×#99—22no—
373C2×#99—612no—
374C2×#99—612no—
375C2×#99—614no—
376C2×#99—1222no—
377C2×#99C2×D6112yes—
378C2×#99C2×D6112yescomplemented
379#379—614no—
380#342—614no—
381#342—614no—
382C61#2177611yescharacteristic, complemented, Sylow
383C25—611no—
384C25—611no—
385H(5,𝔽2)—44no—
386C2×#41C4×D6112yes—
387C2×#41D24412yes—
388C2×#41C22⋊D6112yes—
389C22×D4—611no—
390C22×D4—611no—
391C22×D4—611no—
392C22×D4—1222no—
393C22×D4—1221no—
394C22×D4—1221no—
395C22×D4—1222no—
396C22×D4—1222no—
397C22×D4—2444no—
398C22×D4C4×D6112yes—
399C22×D4D24412yes—
400C22×D4C22⋊D6112yes—
401C22×D4C22×D6111yescharacteristic
402C22×D4C22×D6111yescharacteristic
403C22×D4C22×D6111yescharacteristic
404C23×C4—611no—
405#127—1222no—
406#124—1222no—
407#123—612no—
408#123—612no—
409#123—1222no—
410#123—1221no—
411#123—1221no—
412C22⋊Q8—1222no—
413#121—1222no—
414#121—1222no—
415#121—1222no—
416#121—1222no—
417#121—1222no—
418#121—1222no—
419#121—1222no—
420#120—612no—
421#120—612no—
422#120—612no—
423#120—612no—
424#120—612no—
425#120—612no—
426#120—1222no—
427#120—1221no—
428#120—1221no—
429#120—1222no—
430#120—1222no—
431#120—1222no—
432#120—1222no—
433C4×D4—44no—
434C4×D4—1222no—
435C4×D4—1222no—
436C4×D4—1222no—
437C4×D4—1222no—
438C4×D4—2444no—
439#117—22no—
440#117—612no—
441#117—1222no—
442C2×C22⋊C4—22no—
443C2×C22⋊C4—612no—
444C2×C22⋊C4—611no—
445C2×C22⋊C4—611no—
446C2×C22⋊C4—614no—
447C2×C22⋊C4—1222no—
448C2×C22⋊C4—1221no—
449C2×C22⋊C4—1221no—
450C2×C22⋊C4—1222no—
451C2×C22⋊C4—2444no—
452C2×C22⋊C4C22×D6114yes—
453#99—24no—
454#99—24no—
455#99—1224no—
456#99—1224no—
457#99—2444no—
458C24—611no—
459C24—611no—
460C24—611no—
461C24—1221no—
462C24—1221no—
463C24—1224no—
464C24—1222no—
465C24—4882no—
466C24—4882no—
467C24—2442no—
468C24#1175819812yescomplemented
469C24C2×C4×D6111yescharacteristic
470C24C2×D24411yescharacteristic
471C24C2×C22⋊D6111yescharacteristic
472C24C23×D6111yescharacteristic
473#41—44no—
474#41—44no—
475#41—44no—
476C2×Q8#1175819812yescomplemented
477C2×D4—24no—
478C2×D4—24no—
479C2×D4—24no—
480C2×D4—44no—
481C2×D4—44no—
482C2×D4—44no—
483C2×D4—1222no—
484C2×D4—1222no—
485C2×D4—1222no—
486C2×D4—1222no—
487C2×D4—1221no—
488C2×D4—1221no—
489C2×D4—1224no—
490C2×D4—1222no—
491C2×D4—1222no—
492C2×D4—1222no—
493C2×D4—1222no—
494C2×D4—4882no—
495C2×D4—4882no—
496C2×D4—4884no—
497C2×D4—2442no—
498C2×D4—2442no—
499C2×D4—2448no—
500C2×D4—2448no—
501C2×D4#1175819812yescomplemented
502C2×D4#1175819812yescomplemented
503C2×D4#1175819812yescomplemented
504C2×D4C2×C4×D6112yes—
505C2×D4C2×C4×D6112yes—
506C2×D4C2×D24412yes—
507C2×D4C2×D24412yes—
508C2×D4C2×C22⋊D6112yes—
509C2×D4C2×C22⋊D6112yes—
510C22×C4—24no—
511C22×C4—42no—
512C22×C4—611no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—1222no—
516C22×C4—1222no—
517C22×C4—1224no—
518C22×C4—2442no—
519C22×C4—2442no—
520C22×C4#1175819812yescomplemented
521C22×C4#1175819812yescomplemented
522C22×C4#1175819812yescomplemented
523C22×C4C2×C4×D6111yescharacteristic
524C22×C4C2×D24411yescharacteristic
525C22×C4C2×C22⋊D6111yescharacteristic
526C4⋊C4—42no—
527C4⋊C4—1222no—
528C4⋊C4—1222no—
529C4⋊C4—1222no—
530C4⋊C4—2442no—
531C22⋊C4—24no—
532C22⋊C4—24no—
533C22⋊C4—42no—
534C22⋊C4—612no—
535C22⋊C4—612no—
536C22⋊C4—1221no—
537C22⋊C4—1221no—
538C22⋊C4—1222no—
539C22⋊C4—1222no—
540C22⋊C4—1222no—
541C22⋊C4—1222no—
542C22⋊C4—1222no—
543C22⋊C4—1222no—
544C22⋊C4—1222no—
545C22⋊C4—1222no—
546C22⋊C4—1222no—
547C22⋊C4—1224no—
548C22⋊C4—1224no—
549C22⋊C4—4888no—
550C22⋊C4—2442no—
551C42—42no—
552C42—1222no—
553C42—2442no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—42no—
558C23—81no—
559C23—81no—
560C23—81no—
561C23—81no—
562C23—82no—
563C23—1221no—
564C23—1222no—
565C23—1222no—
566C23—1221no—
567C23—1221no—
568C23—1222no—
569C23—4884no—
570C23—4881no—
571C23—4881no—
572C23—4882no—
573C23—4881no—
574C23—4881no—
575C23—4884no—
576C23—4884no—
577C23—2444no—
578C23—2442no—
579C23—2448no—
580C23—2442no—
581C23—2442no—
582C23—2444no—
583C23—2442no—
584C23—2442no—
585C23C2×#1175819812yes—
586C23C2×#1175819812yes—
587C23C2×#1175819812yes—
588C23C22×C4×D6111yescharacteristic
589C23C22×C22⋊D6111yescharacteristic
590C23C22×D24411yescharacteristic
591C23#1451982214yes—
592Q8—42no—
593D4—24no—
594D4—24no—
595D4—24no—
596D4—24no—
597D4—24no—
598D4—24no—
599D4—42no—
600D4—42no—
601D4—42no—
602D4—4882no—
603D4—4882no—
604D4—4882no—
605D4—4882no—
606D4—2444no—
607D4—2444no—
608D4—2444no—
609D4—2444no—
610D4—2442no—
611D4—2442no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—84no—
620C2×C4—162no—
621C2×C4—1222no—
622C2×C4—1222no—
623C2×C4—1221no—
624C2×C4—1222no—
625C2×C4—9762no—
626C2×C4—4884no—
627C2×C4—4881no—
628C2×C4—4881no—
629C2×C4—4882no—
630C2×C4—2444no—
631C2×C4—2444no—
632C2×C4C2×#1175819812yes—
633C2×C4C2×#1175819812yes—
634C2×C4C2×#1175819812yes—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—44no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—162no—
649C22—1222no—
650C22—1224no—
651C22—4882no—
652C22—4884no—
653C22—4884no—
654C22—4888no—
655C22—2442no—
656C22—2441no—
657C22—2441no—
658C22—2444no—
659C22—2441no—
660C22—2442no—
661C22—2441no—
662C22—2441no—
663C22—2441no—
664C22C22×#1175819811yescharacteristic
665C22#1684023512yes—
666C22#1684032112yes—
667C22#1684032412yes—
668C4—22no—
669C4—22no—
670C4—22no—
671C4—84no—
672C4—4884no—
673C4—2442no—
674C4—2442no—
675C4—2441no—
676C2—22no—
677C2—22no—
678C2—21no—
679C2—22no—
680C2—42no—
681C2—42no—
682C2—42no—
683C2—4881no—
684C2—4881no—
685C2—2442no—
686C2#2144839411yescharacteristic
subgroupG/H
centerC2#21448394
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC61×#2883C2
socleC2×C61#2308
prime radical#21459450C2
cosocleC24×C61C24
hypercenter#2883D61
hypocenterC61#21776
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C24 ◁ #2883

Complements

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

NH ≅ G/Ntype
2. #21454280C2semidirect
3. #21446361C2semidirect
4. #21446359C2semidirect
5. #21446227C2semidirect
6. #21446225C2semidirect
7. #21445925C2semidirect
8. #21445732C2semidirect
9. C61×#2883C2semidirect
15. #16840326C4semidirect
16. #16840321C4semidirect
19. #16840269C4semidirect
20. C2×#14519745C4semidirect
24. #16840237C22semidirect
28. C2×#14519692C22semidirect
29. C2×#14519691C22semidirect
30. #16839617C22semidirect
31. #16839529C22semidirect
32. #16839528C22semidirect
34. C61×#518C22semidirect
38. C2×#99×C61C22semidirect
39. C24×D61C2×C4semidirect
41. C22×D244C2×C4semidirect
45. C22×C22⋊D61C2×C4semidirect
50. C22×C4×D61C2×C4semidirect
56. #14519758C2×C4semidirect
57. #14519758C2×C4semidirect
73. #14519745C2×C4semidirect
74. #14519745C2×C4semidirect
75. #14519745C2×C4semidirect
76. #14519745C2×C4semidirect
77. #14519745C2×C4semidirect
78. #14519745C2×C4semidirect
85. #14519736C2×C4semidirect
87. C2×C22⋊Dic61C2×C4semidirect
114. C24×C61C22⋊C4semidirect
132. C2×Q8×C61C22⋊C4semidirect
139. C2×D4×C61C22⋊C4semidirect
140. C2×D4×C61C22⋊C4semidirect
141. C2×D4×C61C22⋊C4semidirect
150. C22×C4×C61C22⋊C4semidirect
151. C22×C4×C61C22⋊C4semidirect
152. C22×C4×C61C22⋊C4semidirect
335. #2883D61semidirect
369. #518C2×D61semidirect
378. C2×#99C2×D61semidirect
382. C61#21776semidirect
468. C24#11758198semidirect
476. C2×Q8#11758198semidirect
501. C2×D4#11758198semidirect
502. C2×D4#11758198semidirect
503. C2×D4#11758198semidirect
520. C22×C4#11758198semidirect
521. C22×C4#11758198semidirect
522. C22×C4#11758198semidirect

Maximal subgroups

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

subgroupindex
#214594502
#214542802
#214463612
#214463592
#214462272
#214462252
#214459252
#214457322
C61×#28832
#2177661