Finite Group Databasebeta

← #29680905

Subgroups — #29680905

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×#16839804C211yesmaximal, characteristic, complemented
3C2×#16839778C211yesmaximal, characteristic
4C2×#16839764C211yesmaximal, characteristic
5#21448439C211yesmaximal, characteristic, complemented
6#21446055C214yesmaximal, complemented
7#21446048C214yesmaximal, complemented
8#21445530C211yesmaximal, characteristic, complemented
9C61×#1742C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11#16840243C2212yes—
12#16840243C2212yes—
13#16840242—21no—
14#16840242—21no—
15#16840239—21no—
16#16840236—21no—
17#16840235—22no—
18C4×#11758222—22no—
19#16840230—22no—
20C2×#14519736C2211yescharacteristic
21C2×#14519736C2211yescharacteristic
22#117×D61—22no—
23C22×C4⋊Dic61C2211yescharacteristic
24#16839804—22no—
25#16839804C2214yescomplemented
26C2×#14519711—21no—
27C2×#14519709—21no—
28#16839778C418yescomplemented
29C2×#14519703C2214yes—
30#16839764—22no—
31#16839764C2214yes—
32C2×#14519702—21no—
33C2×#14519699—21no—
34#16839660—22no—
35#16839634—22no—
36#16839631C2212yescomplemented
37#16839627C2212yescomplemented
38#16839523C2214yescomplemented
39C22×M16×C61C2211yescharacteristic
40C22×M16×C61C2211yescharacteristic
41C2×#117×C61C2211yescharacteristic
42C61×#343C2212yescomplemented
43C61×#343C2212yescomplemented
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—
55C23×Dic61—21no—
56C22×C4×D61—21no—
57C4×Dic122—22no—
58C4×Dic122—22no—
59#14519742—22no—
60#14519742—22no—
61C4×D244—22no—
62C4×D244—22no—
63#14519739—22no—
64#14519739—22no—
65C4×C22⋊D61—24no—
66C4×C22⋊D61—24no—
67#14519737—24no—
68#14519737—24no—
69#14519736—21no—
70#14519736—21no—
71#14519736—22no—
72#14519736C2×C414yescomplemented
73#14519736D412yes—
74#14519736C2312yes—
75#14519735—22no—
76#14519735—22no—
77#14519734—24no—
78#14519733—24no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61D411yescharacteristic
86C2×C4⋊Dic61D411yescharacteristic
87C2×C4⋊Dic61C2312yes—
88C2×#11758196—24no—
89C4⋊C4×D61—22no—
90C4⋊C4×D61—22no—
91C2×C22⋊Dic61—21no—
92C2×C22⋊Dic61—21no—
93C22⋊C4×D61—24no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61C2312yes—
96C42×D61—24no—
97#14519711—24no—
98#14519709—24no—
99#14519703C2×C4116yescomplemented
100#14519702—24no—
101#14519699—24no—
102#14519684—24no—
103C23×C4×C61C2311yescharacteristic
104C2×M16×C61—21no—
105C2×M16×C61—22no—
106C2×M16×C61—21no—
107C2×M16×C61—22no—
108C2×M16×C61—22no—
109C2×M16×C61C2×C414yescomplemented
110C2×M16×C61C2312yes—
111C2×M16×C61C2312yes—
112C22×C8×C61—21no—
113C22×C8×C61—21no—
114#117×C61—21no—
115#117×C61—21no—
116#117×C61—21no—
117#117×C61—21no—
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C22⋊C4×C61D412yes—
121C2×C42×C61D412yes—
122C24×C61C2×D411yescharacteristic
123C23×D61—21no—
124C22×C4×C61C22×C412yes—
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2×D412yes—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×Q812yes—
133C22×C4×C61#4112yes—
134C22×C4×C61#4112yes—
135C22×C4×C61C2411yescharacteristic
136C2×C22⋊D61—24no—
137C2×C22⋊D61—22no—
138C2×C22⋊D61—24no—
139C2×C22⋊D61—22no—
140C22×Dic61—21no—
141C22×Dic61—22no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—22no—
145C22×Dic61—41no—
146C22×Dic61—41no—
147C22×Dic61—82no—
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×C61—22no—
175M16×C61—22no—
176M16×C61—42no—
177M16×C61C22×C414yes—
178C2×C8×C61—24no—
179C2×C8×C61—22no—
180C2×C8×C61—22no—
181C2×C8×C61—42no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—22no—
188C42×C61—22no—
189C42×C61—22no—
190C22⋊Dic61—24no—
191C22⋊Dic61—22no—
192C22⋊Dic61—22no—
193#11758198—24no—
194#11758198—24no—
195C4⋊Dic61—24no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—22no—
199C4⋊Dic61—22no—
200C4⋊Dic61—41no—
201C4⋊Dic61—41no—
202C4⋊Dic61C2×D412yes—
203C4⋊Dic61C2×D412yes—
204#11758196—24no—
205#11758196—24no—
206#11758196—84no—
207C4×Dic61—24no—
208C4×Dic61—24no—
209C4×Dic61—22no—
210C4×Dic61—22no—
211C4×Dic61—22no—
212C4×Dic61C22×C414yes—
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—42no—
227C2×C4×C61—42no—
228C2×C4×C61C2×C4⋊C412yes—
229C2×C4×C61C2×C4⋊C412yes—
230C2×C4×C61C4×D412yes—
231C2×C4×C61C4×D412yes—
232C2×C4×C61C4×D412yes—
233C2×C4×C61#12011yescharacteristic
234C2×C4×C61#12011yescharacteristic
235C2×C4×C61C22⋊Q812yes—
236C2×C4×C61C22⋊Q812yes—
237C2×C4×C61C22⋊Q812yes—
238C2×C4×C61#12312yes—
239C2×C4×C61#12312yes—
240C2×C4×C61C23×C411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C22×Q811yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C22⋊D61—24no—
247C22⋊D61—24no—
248C22⋊D61—28no—
249C2×Dic61—22no—
250C2×Dic61—24no—
251C2×Dic61—24no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—44no—
257C2×Dic61—42no—
258C2×Dic61—81no—
259C2×Dic61—81no—
260C2×Dic61—82no—
261C2×Dic61—82no—
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—
274C8×C61—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#38512yes—
296C4×C61#38512yes—
297C4×C61C22×C4⋊C411yescharacteristic
298C4×C61C2×C4×D411yescharacteristic
299C4×C61C2×C22⋊Q811yescharacteristic
300C4×C61C2×#12311yescharacteristic
301Dic61—24no—
302Dic61—44no—
303Dic61—42no—
304Dic61—42no—
305Dic61—82no—
306#3428—611no—
307C2×#428—611no—
308C2×#421—611no—
309C2×#417—611no—
310#2317—611no—
311#1860—614no—
312#1859—614no—
313#1742D6111yescharacteristic, complemented
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×M16C2×D6111yescharacteristic
325C22×M16C2×D6111yescharacteristic
326#520—612no—
327#520—612no—
328#520—1221no—
329#520—1221no—
330#519—1221no—
331#518—1221no—
332#518—1222no—
333C4×#41—1222no—
334C4×#41—1222no—
335C2×#117—611no—
336C2×#117—611no—
337C2×#117—1222no—
338C2×#117C2×D6111yescharacteristic
339C22×C4⋊C4—611no—
340#428—614no—
341#428—1222no—
342C2×#107—1221no—
343C2×#106—1221no—
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#343C2×D6112yescomplemented
355#343C2×D6112yescomplemented
356#339—614no—
357C61#2086511yescharacteristic, complemented, Sylow
358C2×#41—614no—
359C2×#41—612no—
360C2×#41—614no—
361C2×#41—612no—
362C2×#41—1221no—
363C2×#41—1222no—
364C2×#41—1221no—
365C2×#41—1222no—
366C22×Q8—611no—
367C22×D4—611no—
368C22×D4—1221no—
369C23×C4—1221no—
370C23×C4—1221no—
371C23×C4C22×D6111yescharacteristic
372C2×M16—21no—
373C2×M16—22no—
374C2×M16—21no—
375C2×M16—22no—
376C2×M16—22no—
377C2×M16C4×D6114yescomplemented
378C2×M16C22×D6112yes—
379C2×M16C22×D6112yes—
380C22×C8—21no—
381C22×C8—21no—
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—21no—
397#117—21no—
398#117—614no—
399#117—612no—
400#117—612no—
401#117—1224no—
402#117—1224no—
403#117—1222no—
404#117—1222no—
405#117—1221no—
406#117—1222no—
407#117—1221no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—612no—
411C2×C4⋊C4—1222no—
412C2×C4⋊C4—1222no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1224no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4C22×D6111yescharacteristic
421C2×C4⋊C4C22×D6111yescharacteristic
422C2×C22⋊C4—1224no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4D24412yes—
426C2×C42—612no—
427C2×C42—1224no—
428C2×C42—1221no—
429C2×C42D24412yes—
430#107—1224no—
431#106—1224no—
432#104—6116no—
433#103—1224no—
434#102—1224no—
435#95—1224no—
436C24—1221no—
437C24C2×D24411yescharacteristic
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—1224no—
461C22×C4—1224no—
462C22×C4—1221no—
463C22×C4—1222no—
464C22×C4—1224no—
465C22×C4—1222no—
466C22×C4—1224no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—4882no—
472C22×C4—2441no—
473C22×C4—2441no—
474C22×C4C2×C4×D6112yes—
475C22×C4C2×D24412yes—
476C22×C4C2×D24412yes—
477C22×C4C2×D24411yescharacteristic
478C22×C4#1175822212yes—
479C22×C4D4×D6111yescharacteristic
480C22×C4D4×D6111yescharacteristic
481C22×C4D4×D6111yescharacteristic
482C22×C4D4×D6111yescharacteristic
483C22×C4#1175822412yes—
484C22×C4Q8×D6112yes—
485C22×C4C23×D6111yescharacteristic
486M16—22no—
487M16—24no—
488M16—22no—
489M16—22no—
490M16—42no—
491M16C2×C4×D6114yes—
492C2×C8—22no—
493C2×C8—24no—
494C2×C8—22no—
495C2×C8—42no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—612no—
501C4⋊C4—612no—
502C4⋊C4—1224no—
503C4⋊C4—1224no—
504C4⋊C4—1224no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1222no—
508C4⋊C4—1222no—
509C4⋊C4—4884no—
510C4⋊C4—2441no—
511C4⋊C4—2441no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—1224no—
515C22⋊C4—1224no—
516C22⋊C4—1222no—
517C22⋊C4—1224no—
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—
533C23C4×D24412yes—
534C23#1451974711yescharacteristic
535C23#1451974711yescharacteristic
536C23#1451975712yes—
537C23C22×D24411yescharacteristic
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—42no—
550C2×C4—42no—
551C2×C4—1222no—
552C2×C4—1224no—
553C2×C4—1224no—
554C2×C4—1224no—
555C2×C4—1228no—
556C2×C4—1224no—
557C2×C4—1222no—
558C2×C4—1222no—
559C2×C4—4884no—
560C2×C4—4884no—
561C2×C4—4882no—
562C2×C4—4882no—
563C2×C4—4881no—
564C2×C4—4881no—
565C2×C4—2444no—
566C2×C4—2442no—
567C2×C4—2442no—
568C2×C4—2442no—
569C2×C4C4⋊C4×D6112yes—
570C2×C4C4⋊C4×D6112yes—
571C2×C4#1451973712yes—
572C2×C4#1451973712yes—
573C2×C4C4×D24412yes—
574C2×C4#1451974711yescharacteristic
575C2×C4#1451974711yescharacteristic
576C2×C4#1451975512yes—
577C2×C4#1451975512yes—
578C2×C4#1451975712yes—
579C2×C4#1451975912yes—
580C2×C4#1451975912yes—
581C2×C4C22×C4×D6111yescharacteristic
582C2×C4C2×D4×D6111yescharacteristic
583C2×C4C2×D4×D6111yescharacteristic
584C2×C4C2×Q8×D6111yescharacteristic
585C2×C4C2×#1175822411yescharacteristic
586C2×C4C2×#1175822211yescharacteristic
587C8—24no—
588C8—42no—
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#1683963612yes—
602C22#1683963612yes—
603C22C2×C4×D24411yescharacteristic
604C22C2×#1451974711yescharacteristic
605C22C2×#1451975711yescharacteristic
606C4—42no—
607C4—42no—
608C4—1224no—
609C4—4882no—
610C4—2444no—
611C4—2442no—
612C4—2442no—
613C4#1683963612yes—
614C4#1683963612yes—
615C4C2×C4⋊C4×D6111yescharacteristic
616C4C2×#1451973711yescharacteristic
617C4C2×#1451975511yescharacteristic
618C4C2×#1451975911yescharacteristic
619C2—21no—
620C2—22no—
621C2—21no—
622C2—22no—
623C2—2444no—
624C2C2×#1683963611yescharacteristic
625C2#2145519311yescharacteristic
626C2#2145520011yescharacteristic
subgroupG/H
centerC22C2×#14519747
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1742C2
socleC22×C61C2×#120
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenter#1742D61
hypocenterC61#20865
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1742

Complements

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

NH ≅ G/Ntype
2. C2×#16839804C2semidirect
5. #21448439C2semidirect
6. #21446055C2semidirect
7. #21446048C2semidirect
8. #21445530C2semidirect
9. C61×#1742C2semidirect
25. #16839804C22semidirect
28. #16839778C4semidirect
36. #16839631C22semidirect
37. #16839627C22semidirect
38. #16839523C22semidirect
42. C61×#343C22semidirect
43. C61×#343C22semidirect
72. #14519736C2×C4semidirect
99. #14519703C2×C4semidirect
109. C2×M16×C61C2×C4semidirect
155. #11758222C4⋊C4semidirect
160. C2×D244C4⋊C4semidirect
170. C2×Dic122C4⋊C4semidirect
313. #1742D61semidirect
354. #343C2×D61semidirect
355. #343C2×D61semidirect
357. C61#20865semidirect
377. 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×#168398042
C2×#168397782
C2×#168397642
#214484392
#214460552
#214460482
#214455302
C61×#17422
#2086561