Finite Group Databasebeta

← #29680898

Subgroups — #29680898

Order 15616.

1394 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459020C211yesmaximal, characteristic
2C2×#16839778C211yesmaximal, characteristic
3C2×#16839761C211yesmaximal, characteristic
4#21448443C211yesmaximal, characteristic, complemented
5#21446049C214yesmaximal, complemented
6#21446042C214yesmaximal, complemented
7#21445557C211yesmaximal, characteristic, complemented
8#21445528C211yesmaximal, characteristic, complemented
9C2×C61×#428C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12#16840243—21no—
13#16840243—21no—
14#16840242C2212yes—
15#16840242C2212yes—
16#16840239—21no—
17#16840236—21no—
18#16840235—22no—
19C4×#11758222—22no—
20#16840230—22no—
21C2×#14519736C2211yescharacteristic
22C2×#14519736C2211yescharacteristic
23#117×D61—22no—
24#16839778C418yescomplemented
25C2×#14519703C2214yes—
26#16839761—22no—
27#16839761C2214yes—
28C2×#14519700—21no—
29C2×#14519697—21no—
30#16839658—22no—
31#16839635—22no—
32#16839631C2212yescomplemented
33#16839627C2212yescomplemented
34#16839530C2212yescomplemented
35#16839530C2212yescomplemented
36#16839521C2214yescomplemented
37C22×M16×C61C2211yescharacteristic
38C2×#117×C61C2211yescharacteristic
39C22×C4⋊C4×C61C2211yescharacteristic
40C61×#428—22no—
41C61×#428C2214yescomplemented
42C2×#107×C61—21no—
43C2×#106×C61—21no—
44C2×#11758222—21no—
45C2×#11758222—21no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222C2×C414yes—
49C2×#11758222D412yes—
50C2×#11758222Q814yes—
51C2×#11758222C2312yes—
52C22×Dic122D411yescharacteristic
53C22×D244D411yescharacteristic
54C22×C22⋊D61—21no—
55C22×C4×D61—21no—
56C2×#11758194—21no—
57C2×#11758194—22no—
58C2×#11758194—22no—
59C2×#11758194C2312yes—
60C22×#92418—21no—
61C4×Dic122—22no—
62C4×Dic122—22no—
63#14519742—22no—
64#14519742—22no—
65C4×D244—22no—
66C4×D244—22no—
67#14519739—22no—
68#14519739—22no—
69C4×C22⋊D61—24no—
70C4×C22⋊D61—24no—
71#14519737—24no—
72#14519737—24no—
73#14519736—21no—
74#14519736—21no—
75#14519736—21no—
76#14519736—21no—
77#14519736—22no—
78#14519736C2×C414yescomplemented
79#14519735—22no—
80#14519735—22no—
81#14519734—24no—
82#14519733—24no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×#11758196—24no—
87C4⋊C4×D61—22no—
88C4⋊C4×D61—22no—
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61D412yes—
91C22⋊C4×D61—24no—
92C2×C4×Dic61D412yes—
93C2×C4×Dic61C2312yes—
94C42×D61—24no—
95#14519703C2×C4116yescomplemented
96#14519700—24no—
97#14519697—24no—
98#14519684—24no—
99C23×C4×C61—21no—
100C23×C4×C61C2311yescharacteristic
101C2×M16×C61—21no—
102C2×M16×C61—22no—
103C2×M16×C61C2×C414yescomplemented
104C2×M16×C61C2312yes—
105C22×C8×C61—21no—
106#117×C61—21no—
107#117×C61—21no—
108#117×C61D412yes—
109#117×C61C2312yes—
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61D411yescharacteristic
116C2×C4⋊C4×C61D411yescharacteristic
117C2×C4⋊C4×C61C2312yes—
118C2×C22⋊C4×C61—21no—
119C2×C42×C61—21no—
120#107×C61—24no—
121#106×C61—24no—
122C24×C61C2×D411yescharacteristic
123C23×D61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—22no—
127C22×C4×C61—41no—
128C22×C4×C61—41no—
129C22×C4×C61—82no—
130C22×C4×C61C22×C412yes—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×Q812yes—
137C22×C4×C61#4112yes—
138C22×C4×C61#4112yes—
139C22×C4×C61C2411yescharacteristic
140C2×C22⋊D61—24no—
141C2×C22⋊D61—22no—
142C2×C22⋊D61—24no—
143C2×C22⋊D61—22no—
144C22×Dic61—21no—
145C22×Dic61—22no—
146C22×Dic61C2×D412yes—
147C22×Dic61C2×D412yes—
148#11758222—24no—
149#11758222—24no—
150#11758222—24no—
151#11758222—24no—
152#11758222—24no—
153#11758222—22no—
154#11758222—22no—
155#11758222C4⋊C418yescomplemented
156#11758222C22×C414yes—
157#11758222C2×Q814yes—
158C2×D244—21no—
159C2×D244—22no—
160C2×D244C4⋊C414yescomplemented
161C2×D244C2×D412yes—
162C2×C4×D61—24no—
163C2×C4×D61—22no—
164C2×C4×D61—24no—
165C2×C4×D61—22no—
166C2×C4×D61—24no—
167C2×C4×D61—24no—
168C2×Dic122—21no—
169C2×Dic122—22no—
170C2×Dic122C4⋊C414yescomplemented
171C2×Dic122C2×D412yes—
172M16×C61—24no—
173M16×C61—22no—
174M16×C61C22×C414yes—
175C2×C8×C61—24no—
176C2×C8×C61—22no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—41no—
182C4⋊C4×C61—41no—
183C4⋊C4×C61C2×D412yes—
184C4⋊C4×C61C2×D412yes—
185C22⋊C4×C61—22no—
186C22⋊C4×C61—22no—
187C42×C61—22no—
188C42×C61—22no—
189C22⋊Dic61—24no—
190C22⋊Dic61—22no—
191C22⋊Dic61—22no—
192#11758198—24no—
193#11758198—24no—
194C4⋊Dic61—24no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199#11758196—24no—
200#11758196—24no—
201#11758196—84no—
202C4×Dic61—24no—
203C4×Dic61—24no—
204C4×Dic61—22no—
205C4×Dic61—22no—
206C4×Dic61—22no—
207C4×Dic61C22×C414yes—
208#11758194—22no—
209#11758194—22no—
210#11758194—42no—
211C2×#92418—22no—
212C2×#92418—42no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61C4×D412yes—
216C23×C61#12011yescharacteristic
217C23×C61#12011yescharacteristic
218C23×C61C22⋊Q812yes—
219C23×C61C22×D411yescharacteristic
220C22×D61—22no—
221C22×D61—24no—
222C22×D61—81no—
223C22×D61—81no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—44no—
227C2×C4×C61—42no—
228C2×C4×C61—81no—
229C2×C4×C61—81no—
230C2×C4×C61—82no—
231C2×C4×C61—82no—
232C2×C4×C61C2×C4⋊C412yes—
233C2×C4×C61C2×C4⋊C412yes—
234C2×C4×C61C4×D412yes—
235C2×C4×C61C4×D412yes—
236C2×C4×C61C4×D412yes—
237C2×C4×C61#12011yescharacteristic
238C2×C4×C61#12011yescharacteristic
239C2×C4×C61C22⋊Q812yes—
240C2×C4×C61C22⋊Q812yes—
241C2×C4×C61C22⋊Q812yes—
242C2×C4×C61#12312yes—
243C2×C4×C61#12312yes—
244C2×C4×C61C23×C411yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×Q811yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C22⋊D61—24no—
251C22⋊D61—24no—
252C22⋊D61—28no—
253C2×Dic61—22no—
254C2×Dic61—24no—
255C2×Dic61—24no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—42no—
259C2×Dic61—42no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262D244—22no—
263D244—24no—
264D244C2×C4⋊C414yes—
265C4×D61—24no—
266C4×D61—24no—
267C4×D61—28no—
268C4×D61—84no—
269C4×D61—84no—
270Dic122—22no—
271Dic122—24no—
272Dic122C2×C4⋊C414yes—
273C8×C61—24no—
274#92418—42no—
275#20865—611nomaximal, Sylow
276C22×C61—21no—
277C22×C61—21no—
278C22×C61—21no—
279C22×C61—21no—
280C22×C61—21no—
281C22×C61—21no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61#38512yes—
285C22×C61#38512yes—
286C22×C61C2×C4×D411yescharacteristic
287C22×C61C2×#12011yescharacteristic
288C22×C61C2×C22⋊Q811yescharacteristic
289C2×D61—24no—
290C2×D61—42no—
291C2×D61—42no—
292C2×D61—84no—
293C4×C61—42no—
294C4×C61—42no—
295C4×C61—82no—
296C4×C61#38512yes—
297C4×C61#38512yes—
298C4×C61C22×C4⋊C411yescharacteristic
299C4×C61C2×C4×D411yescharacteristic
300C4×C61C2×C22⋊Q811yescharacteristic
301C4×C61C2×#12311yescharacteristic
302Dic61—24no—
303Dic61—44no—
304Dic61—42no—
305Dic61—42no—
306#3428—611no—
307C2×#428D6111yescharacteristic, complemented
308C2×#421—611no—
309C2×#417—611no—
310#2317—611no—
311#1860—614no—
312#1859—614no—
313#1742—611no—
314#1733—611no—
315C2×C61—21no—
316C2×C61—22no—
317C2×C61—21no—
318C2×C61—22no—
319C2×C61C2×#38511yescharacteristic
320C2×C61#302111yescharacteristic
321C2×C61#302211yescharacteristic
322D61—44no—
323C22×#41—611no—
324C22×M16—611no—
325C22×M16C2×D6111yescharacteristic
326#520—612no—
327#520—612no—
328#520—1221no—
329#520—1221no—
330#519—1221no—
331#518—1222no—
332#518—1221no—
333C4×#41—1222no—
334C4×#41—1222no—
335C2×#117—611no—
336C2×#117—611no—
337C2×#117—1222no—
338C2×#117C2×D6111yescharacteristic
339C22×C4⋊C4C2×D6111yescharacteristic
340#428—22no—
341#428C2×D6114yescomplemented
342C2×#107—21no—
343C2×#106—21no—
344#421—618no—
345C2×#104—614no—
346#417—614no—
347#417—1222no—
348C2×#103—1221no—
349C2×#102—1221no—
350#389—1222no—
351#385—1222no—
352#384—612no—
353#382—612no—
354#343—612no—
355#343—612no—
356#339—614no—
357C61#2086511yescharacteristic, complemented, Sylow
358C2×#41—614no—
359C2×#41—612no—
360C2×#41—614no—
361C2×#41—612no—
362C2×#41—1222no—
363C2×#41—1221no—
364C2×#41—1222no—
365C2×#41—1221no—
366C22×Q8—611no—
367C22×D4—611no—
368C22×D4—1221no—
369C23×C4—21no—
370C23×C4—1221no—
371C23×C4C22×D6111yescharacteristic
372C2×M16—22no—
373C2×M16—21no—
374C2×M16—612no—
375C2×M16—1221no—
376C2×M16—1222no—
377C2×M16—1222no—
378C2×M16C4×D6114yescomplemented
379C2×M16C22×D6112yes—
380C22×C8—21no—
381C22×C8—1221no—
382C4×Q8—1222no—
383C4×Q8—1222no—
384C4×Q8—1222no—
385C4×Q8—1222no—
386C4×D4—1222no—
387C4×D4—1222no—
388C4×D4—1224no—
389C4×D4—1224no—
390C4×D4—1222no—
391C4×D4—1222no—
392C4×D4—1224no—
393C4×D4—1224no—
394#117—21no—
395#117—21no—
396#117—614no—
397#117—1224no—
398#117—1224no—
399#117—1221no—
400#117—1221no—
401#117—1222no—
402#117—1221no—
403#117—1221no—
404#117—1222no—
405#117—1222no—
406#117C22⋊D6112yes—
407#117C22×D6112yes—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1222no—
417C2×C4⋊C4—1222no—
418C2×C4⋊C4—1224no—
419C2×C4⋊C4C22⋊D6111yescharacteristic
420C2×C4⋊C4C22⋊D6111yescharacteristic
421C2×C4⋊C4C22×D6112yes—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—612no—
424C2×C22⋊C4—1224no—
425C2×C22⋊C4—1221no—
426C2×C42—21no—
427C2×C42—612no—
428C2×C42—612no—
429C2×C42—1224no—
430#107—24no—
431#106—24no—
432#104—6116no—
433#103—1224no—
434#102—1224no—
435#95—1224no—
436C24—1221no—
437C24D4×D6111yescharacteristic
438#41—614no—
439#41—614no—
440#41—618no—
441#41—1222no—
442#41—1222no—
443#41—1224no—
444#41—1224no—
445#41—1224no—
446#41—1224no—
447#41—1224no—
448C2×Q8—612no—
449C2×Q8—614no—
450C2×Q8—1221no—
451C2×Q8—1222no—
452C2×D4—612no—
453C2×D4—614no—
454C2×D4—1222no—
455C2×D4—1224no—
456C2×D4—1222no—
457C2×D4—1224no—
458C2×D4—1221no—
459C2×D4—1222no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—82no—
466C22×C4—612no—
467C22×C4—612no—
468C22×C4—1224no—
469C22×C4—1224no—
470C22×C4—1221no—
471C22×C4—1222no—
472C22×C4—1224no—
473C22×C4—1222no—
474C22×C4—1224no—
475C22×C4—1222no—
476C22×C4C2×C4×D6112yes—
477C22×C4#1175822212yes—
478C22×C4D4×D6111yescharacteristic
479C22×C4D4×D6111yescharacteristic
480C22×C4D4×D6111yescharacteristic
481C22×C4Q8×D6112yes—
482C22×C4#1175822612yes—
483C22×C4C2×C22⋊D6111yescharacteristic
484C22×C4C2×C22⋊D6111yescharacteristic
485C22×C4C23×D6111yescharacteristic
486M16—22no—
487M16—24no—
488M16—1222no—
489M16—1222no—
490M16—2442no—
491M16C2×C4×D6114yes—
492C2×C8—22no—
493C2×C8—24no—
494C2×C8—1222no—
495C2×C8—2442no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—1224no—
503C4⋊C4—1224no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—1224no—
509C4⋊C4—4884no—
510C4⋊C4C2×C22⋊D6112yes—
511C4⋊C4C2×C22⋊D6112yes—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—1224no—
515C22⋊C4—1224no—
516C22⋊C4—1224no—
517C22⋊C4—1222no—
518C22⋊C4—1222no—
519C42—22no—
520C42—22no—
521C42—614no—
522C42—1224no—
523C42—1224no—
524C42—1222no—
525C42—1222no—
526C42—1222no—
527C23—41no—
528C23—41no—
529C23—1224no—
530C23—1222no—
531C23—4881no—
532C23—4881no—
533C23#1451973912yes—
534C23#1451974511yescharacteristic
535C23#1451974511yescharacteristic
536C23#1451975512yes—
537C23C2×D4×D6111yescharacteristic
538Q8—614no—
539Q8—1222no—
540Q8—1224no—
541D4—614no—
542D4—1224no—
543D4—1224no—
544D4—1228no—
545D4—1222no—
546D4—1224no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—1222no—
556C2×C4—1224no—
557C2×C4—1224no—
558C2×C4—1224no—
559C2×C4—1228no—
560C2×C4—1224no—
561C2×C4—1222no—
562C2×C4—1222no—
563C2×C4—4884no—
564C2×C4—4884no—
565C2×C4—2442no—
566C2×C4—2442no—
567C2×C4—2442no—
568C2×C4—2442no—
569C2×C4C4⋊C4×D6112yes—
570C2×C4C4⋊C4×D6112yes—
571C2×C4C4×C22⋊D6112yes—
572C2×C4C4×C22⋊D6112yes—
573C2×C4#1451973912yes—
574C2×C4#1451974511yescharacteristic
575C2×C4#1451974511yescharacteristic
576C2×C4#1451975312yes—
577C2×C4#1451975312yes—
578C2×C4#1451975512yes—
579C2×C4#1451976212yes—
580C2×C4#1451976212yes—
581C2×C4C22×C4×D6111yescharacteristic
582C2×C4C2×D4×D6111yescharacteristic
583C2×C4C22×C22⋊D6111yescharacteristic
584C2×C4C2×Q8×D6111yescharacteristic
585C2×C4C2×#1175822211yescharacteristic
586C2×C4C2×#1175822611yescharacteristic
587C8—24no—
588C8—2442no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—42no—
596C22—42no—
597C22—1224no—
598C22—4884no—
599C22—2442no—
600C22—2442no—
601C22#1683963312yes—
602C22#1683963312yes—
603C22C2×#1451973911yescharacteristic
604C22C2×#1451974511yescharacteristic
605C22C2×#1451975511yescharacteristic
606C4—42no—
607C4—42no—
608C4—82no—
609C4—1224no—
610C4—2444no—
611C4—2442no—
612C4—2442no—
613C4#1683963312yes—
614C4#1683963312yes—
615C4C2×C4⋊C4×D6111yescharacteristic
616C4C2×C4×C22⋊D6111yescharacteristic
617C4C2×#1451975311yescharacteristic
618C4C2×#1451976211yescharacteristic
619C2—21no—
620C2—22no—
621C2—21no—
622C2—22no—
623C2—2444no—
624C2C2×#1683963311yescharacteristic
625C2#2145518811yescharacteristic
626C2#2145519511yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#428C2
socleC22×C61C2×#120
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenterC2×#428D61
hypocenterC61#20865
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
4. #21448443C2semidirect
5. #21446049C2semidirect
6. #21446042C2semidirect
7. #21445557C2semidirect
8. #21445528C2semidirect
9. C2×C61×#428C2semidirect
24. #16839778C4semidirect
32. #16839631C22semidirect
33. #16839627C22semidirect
34. #16839530C22semidirect
35. #16839530C22semidirect
36. #16839521C22semidirect
41. C61×#428C22semidirect
78. #14519736C2×C4semidirect
95. #14519703C2×C4semidirect
103. C2×M16×C61C2×C4semidirect
155. #11758222C4⋊C4semidirect
160. C2×D244C4⋊C4semidirect
170. C2×Dic122C4⋊C4semidirect
307. C2×#428D61semidirect
341. #428C2×D61semidirect
357. C61#20865semidirect
378. C2×M16C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214590202
C2×#168397782
C2×#168397612
#214484432
#214460492
#214460422
#214455572
#214455282
C2×C61×#4282
#2086561