Finite Group Databasebeta

← #29680901

Subgroups — #29680901

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×#16839802C211yesmaximal, characteristic, complemented
3C2×#16839780C211yesmaximal, characteristic
4C2×#16839761C211yesmaximal, characteristic
5#21446051C214yesmaximal, complemented
6#21446044C214yesmaximal, complemented
7#21445558C211yesmaximal, characteristic, complemented
8#21445529C211yesmaximal, characteristic, complemented
9C61×#2317C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12C22×#11758194C2211yescharacteristic
13#16840243C2212yes—
14#16840243C2212yes—
15#16840242—21no—
16#16840242—21no—
17#16840239—21no—
18#16840236—21no—
19#16840235—22no—
20C4×#11758222—22no—
21#16840230—22no—
22C2×#14519736C2211yescharacteristic
23#117×D61—22no—
24#16839802—22no—
25#16839802C2214yescomplemented
26C2×#14519710—21no—
27C2×#14519708—21no—
28#16839780C418yescomplemented
29C2×#14519705C2214yes—
30#16839761—22no—
31#16839761C2214yes—
32C2×#14519700—21no—
33C2×#14519697—21no—
34#16839531C2212yescomplemented
35#16839531C2212yescomplemented
36#16839522C2214yescomplemented
37C2×#117×C61C2211yescharacteristic
38C2×#117×C61C2211yescharacteristic
39C22×C4⋊C4×C61C2211yescharacteristic
40C61×#389—22no—
41C61×#385—22no—
42C61×#384C2212yescomplemented
43C61×#382C2212yescomplemented
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—21no—
59C2×#11758194—22no—
60C2×#11758194—22no—
61C2×#11758194C2×C414yescomplemented
62C2×#11758194C2312yes—
63C2×#11758194C2312yes—
64C22×#92418—21no—
65C22×#92418—21no—
66C4×Dic122—22no—
67C4×Dic122—22no—
68#14519742—22no—
69#14519742—22no—
70C4×D244—22no—
71C4×D244—22no—
72#14519739—22no—
73#14519739—22no—
74C4×C22⋊D61—24no—
75C4×C22⋊D61—24no—
76#14519737—24no—
77#14519737—24no—
78#14519736—21no—
79#14519736—21no—
80#14519736D412yes—
81#14519736C2312yes—
82#14519735—22no—
83#14519735—22no—
84#14519734—24no—
85#14519733—24no—
86C2×C4⋊Dic61D411yescharacteristic
87C2×C4⋊Dic61D411yescharacteristic
88C4⋊C4×D61—22no—
89C4⋊C4×D61—22no—
90C2×C22⋊Dic61—21no—
91C22⋊C4×D61—24no—
92C2×C4×Dic61—21no—
93C42×D61—24no—
94#14519710—24no—
95#14519708—24no—
96#14519705C2×C4116yescomplemented
97#14519700—24no—
98#14519697—24no—
99C23×C4×C61—21no—
100C23×C4×C61C2311yescharacteristic
101#117×C61—21no—
102#117×C61—21no—
103#117×C61—21no—
104#117×C61—21no—
105#117×C61—22no—
106#117×C61C2×C414yescomplemented
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61—24no—
114C2×C4⋊C4×C61C2312yes—
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C22⋊C4×C61—21no—
118C2×C22⋊C4×C61D412yes—
119C2×C42×C61D412yes—
120C2×C42×C61C2312yes—
121#95×C61—24no—
122C24×C61C2×D411yescharacteristic
123C23×D61—21no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—41no—
127C22×C4×C61—41no—
128C22×C4×C61—82no—
129C22×C4×C61C22×C412yes—
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2×D412yes—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×Q812yes—
138C22×C4×C61#4112yes—
139C22×C4×C61#4112yes—
140C22×C4×C61C2411yescharacteristic
141C2×C22⋊D61—24no—
142C2×C22⋊D61—22no—
143C2×C22⋊D61—24no—
144C2×C22⋊D61—22no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
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—
172C4⋊C4×C61—24no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—41no—
180C4⋊C4×C61—41no—
181C4⋊C4×C61—84no—
182C22⋊C4×C61—24no—
183C22⋊C4×C61—22no—
184C22⋊C4×C61—22no—
185C42×C61—22no—
186C42×C61—22no—
187C42×C61—22no—
188C42×C61C22×C414yes—
189C22⋊Dic61—22no—
190C22⋊Dic61—22no—
191#11758198—24no—
192#11758198—24no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61C2×D412yes—
196C4⋊Dic61C2×D412yes—
197#11758196—24no—
198#11758196—24no—
199C4×Dic61—24no—
200C4×Dic61—24no—
201C4×Dic61—22no—
202C4×Dic61—22no—
203#11758194—24no—
204#11758194—22no—
205#11758194—22no—
206#11758194—22no—
207#11758194—42no—
208#11758194C22×C414yes—
209C2×#92418—24no—
210C2×#92418—22no—
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—24no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—44no—
232C2×C4×C61—42no—
233C2×C4×C61—82no—
234C2×C4×C61—82no—
235C2×C4×C61C2×C4⋊C412yes—
236C2×C4×C61C2×C4⋊C412yes—
237C2×C4×C61C4×D412yes—
238C2×C4×C61C4×D412yes—
239C2×C4×C61C4×D412yes—
240C2×C4×C61#12011yescharacteristic
241C2×C4×C61#12011yescharacteristic
242C2×C4×C61C22⋊Q812yes—
243C2×C4×C61C22⋊Q812yes—
244C2×C4×C61C22⋊Q812yes—
245C2×C4×C61#12312yes—
246C2×C4×C61#12312yes—
247C2×C4×C61C23×C411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×Q811yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C22⋊D61—24no—
254C22⋊D61—24no—
255C22⋊D61—28no—
256C2×Dic61—22no—
257C2×Dic61—24no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—81no—
261C2×Dic61—81no—
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—
273#92418—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—44no—
294C4×C61—42no—
295C4×C61—42no—
296C4×C61—82no—
297C4×C61#38512yes—
298C4×C61#38512yes—
299C4×C61C22×C4⋊C411yescharacteristic
300C4×C61C2×C4×D411yescharacteristic
301C4×C61C2×C22⋊Q811yescharacteristic
302C4×C61C2×#12311yescharacteristic
303Dic61—24no—
304Dic61—42no—
305Dic61—42no—
306#3428—611no—
307C2×#428—611no—
308C2×#421—611no—
309C2×#417—611no—
310#2317D6111yescharacteristic, complemented
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×M16—611no—
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—1222no—
337C2×#117C2×D6111yescharacteristic
338C2×#117C2×D6111yescharacteristic
339C22×C4⋊C4C2×D6111yescharacteristic
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—22no—
351#385—22no—
352#384C2×D6112yescomplemented
353#382C2×D6112yescomplemented
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—614no—
373C2×M16—612no—
374C2×M16—612no—
375C2×M16—1221no—
376C2×M16—1222no—
377C2×M16—1221no—
378C2×M16—1222no—
379C2×M16—1222no—
380C22×C8—1221no—
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—22no—
395#117—21no—
396#117—21no—
397#117—21no—
398#117—21no—
399#117—612no—
400#117—612no—
401#117—1224no—
402#117—1224no—
403#117—1222no—
404#117—1222no—
405#117—1221no—
406#117—1221no—
407#117C4×D6114yescomplemented
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—24no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—1222no—
418C2×C4⋊C4—1222no—
419C2×C4⋊C4C22×D6112yes—
420C2×C4⋊C4C22×D6111yescharacteristic
421C2×C4⋊C4C22×D6111yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1224no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4C22⋊D6112yes—
426C2×C42—1224no—
427C2×C42—1221no—
428C2×C42C22⋊D6112yes—
429C2×C42C22×D6112yes—
430#107—1224no—
431#106—1224no—
432#104—6116no—
433#103—1224no—
434#102—1224no—
435#95—24no—
436C24—1221no—
437C24C2×C22⋊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—22no—
461C22×C4—22no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—82no—
465C22×C4—1224no—
466C22×C4—1224no—
467C22×C4—1221no—
468C22×C4—1222no—
469C22×C4—1224no—
470C22×C4—1222no—
471C22×C4—1224no—
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4C2×C4×D6112yes—
475C22×C4#1175822212yes—
476C22×C4D4×D6111yescharacteristic
477C22×C4D4×D6111yescharacteristic
478C22×C4D4×D6111yescharacteristic
479C22×C4D4×D6111yescharacteristic
480C22×C4Q8×D6112yes—
481C22×C4#1175822612yes—
482C22×C4C2×C22⋊D6112yes—
483C22×C4C2×C22⋊D6112yes—
484C22×C4C2×C22⋊D6111yescharacteristic
485C22×C4C23×D6111yescharacteristic
486M16—614no—
487M16—1222no—
488M16—1224no—
489M16—1222no—
490M16—1222no—
491M16—2442no—
492C2×C8—1222no—
493C2×C8—1224no—
494C2×C8—1222no—
495C2×C8—2442no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—84no—
506C4⋊C4—612no—
507C4⋊C4—612no—
508C4⋊C4—1224no—
509C4⋊C4—1224no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C22⋊C4—24no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—1224no—
516C22⋊C4—1224no—
517C22⋊C4—1222no—
518C22⋊C4—1222no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—1224no—
523C42—1224no—
524C42—1222no—
525C42—1222no—
526C42C2×C4×D6114yes—
527C23—41no—
528C23—41no—
529C23—1224no—
530C23—1222no—
531C23—4881no—
532C23—4881no—
533C23C4×C22⋊D6112yes—
534C23#1451974511yescharacteristic
535C23#1451974511yescharacteristic
536C23#1451975312yes—
537C23C22×C22⋊D6111yescharacteristic
538Q8—614no—
539Q8—1222no—
540Q8—1224no—
541D4—614no—
542D4—1224no—
543D4—1224no—
544D4—1228no—
545D4—1222no—
546D4—1224no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—1222no—
559C2×C4—1224no—
560C2×C4—1224no—
561C2×C4—1224no—
562C2×C4—1228no—
563C2×C4—1222no—
564C2×C4—1222no—
565C2×C4—4884no—
566C2×C4—4884no—
567C2×C4—4881no—
568C2×C4—4881no—
569C2×C4C4⋊C4×D6112yes—
570C2×C4C4⋊C4×D6112yes—
571C2×C4C4×C22⋊D6112yes—
572C2×C4#1451973912yes—
573C2×C4#1451973912yes—
574C2×C4#1451974511yescharacteristic
575C2×C4#1451974511yescharacteristic
576C2×C4#1451975312yes—
577C2×C4#1451975512yes—
578C2×C4#1451975512yes—
579C2×C4#1451976212yes—
580C2×C4#1451976212yes—
581C2×C4C22×C4×D6111yescharacteristic
582C2×C4C2×D4×D6111yescharacteristic
583C2×C4C2×D4×D6111yescharacteristic
584C2×C4C2×Q8×D6111yescharacteristic
585C2×C4C2×#1175822211yescharacteristic
586C2×C4C2×#1175822611yescharacteristic
587C8—1224no—
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×C4×C22⋊D6111yescharacteristic
604C22C2×#1451974511yescharacteristic
605C22C2×#1451975311yescharacteristic
606C4—44no—
607C4—42no—
608C4—42no—
609C4—82no—
610C4—1224no—
611C4—2442no—
612C4—2442no—
613C4#1683963312yes—
614C4#1683963312yes—
615C4C2×C4⋊C4×D6111yescharacteristic
616C4C2×#1451973911yescharacteristic
617C4C2×#1451975511yescharacteristic
618C4C2×#1451976211yescharacteristic
619C2—21no—
620C2—22no—
621C2—21no—
622C2—22no—
623C2—2444no—
624C2C2×#1683963311yescharacteristic
625C2#2145519011yescharacteristic
626C2#2145519711yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2317C2
socleC22×C61C2×#120
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenter#2317D61
hypocenterC61#20865
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2317

Complements

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

NH ≅ G/Ntype
2. C2×#16839802C2semidirect
5. #21446051C2semidirect
6. #21446044C2semidirect
7. #21445558C2semidirect
8. #21445529C2semidirect
9. C61×#2317C2semidirect
25. #16839802C22semidirect
28. #16839780C4semidirect
34. #16839531C22semidirect
35. #16839531C22semidirect
36. #16839522C22semidirect
42. C61×#384C22semidirect
43. C61×#382C22semidirect
61. C2×#11758194C2×C4semidirect
96. #14519705C2×C4semidirect
106. #117×C61C2×C4semidirect
155. #11758222C4⋊C4semidirect
160. C2×D244C4⋊C4semidirect
170. C2×Dic122C4⋊C4semidirect
310. #2317D61semidirect
352. #384C2×D61semidirect
353. #382C2×D61semidirect
357. C61#20865semidirect
407. #117C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214590202
C2×#168398022
C2×#168397802
C2×#168397612
#214460512
#214460442
#214455582
#214455292
C61×#23172
#2086561