Finite Group Databasebeta

← #29707494

Subgroups — #29707494

Order 15616.

1610 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840273C211yesmaximal, characteristic
2C2×#16839798C211yesmaximal, characteristic, complemented
3C2×#16839756C211yesmaximal, characteristic, complemented
4C2×#16839756C211yesmaximal, characteristic, complemented
5C22×C22⋊#92418C211yesmaximal, characteristic, complemented
6#21447022C212yesmaximal, complemented
7#21447015C212yesmaximal, complemented
8#21447008C212yesmaximal, complemented
9#21447001C212yesmaximal, complemented
10#21445535C211yesmaximal, characteristic, complemented
11C61×#2280C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C23×#92418C2211yescharacteristic
14C23×#92418C2211yescharacteristic
15#16840273—24no—
16#16840273—24no—
17C2×#14519760—21no—
18C2×#14519756C2212yes—
19C2×#14519751C2212yes—
20C2×#14519746—21no—
21C2×C4×C22⋊D61—22no—
22C2×#14519736C2211yescharacteristic
23#16839798—24no—
24C2×#14519710C2212yescomplemented
25C2×#14519708C2212yescomplemented
26#16839756—24no—
27#16839756—24no—
28C2×#14519700C2211yescharacteristic, complemented
29C2×#14519700C2212yescomplemented
30C2×#14519700C2211yescharacteristic, complemented
31C2×#14519697C2211yescharacteristic, complemented
32C2×#14519697C2212yescomplemented
33C2×#14519697C2211yescharacteristic, complemented
34C2×C22⋊#92418—24no—
35C2×C22⋊#92418C2212yes—
36C2×C22⋊#92418C2212yes—
37#16839524C2212yescomplemented
38#16839524C2212yescomplemented
39C24×C4×C61C2211yescharacteristic
40C2×#117×C61C2211yescharacteristic
41C2×#117×C61C2211yescharacteristic
42C61×#381—21no—
43C61×#380C2212yes—
44C61×#380C2212yes—
45C61×#379—21no—
46C4×C22⋊C4×C61—22no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222—22no—
51C2×#11758222—22no—
52C2×#11758222D414yes—
53C22×Dic122C2311yescharacteristic
54C22×D244C2311yescharacteristic
55C22×C22⋊D61—22no—
56C22×C22⋊D61—21no—
57C22×C4×D61—21no—
58C22×#92418—22no—
59C22×#92418—22no—
60C22×#92418—22no—
61C22×#92418—22no—
62C22×#92418—22no—
63C22×#92418D414yescomplemented
64C22×#92418C2312yes—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67#14519760—28no—
68#14519756—24no—
69#14519756—24no—
70#14519751—24no—
71#14519751—24no—
72#14519746—28no—
73C4×C22⋊D61—216no—
74#14519736—22no—
75#14519736D414yes—
76C2×C4⋊Dic61C2312yes—
77C2×#11758196—22no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—22no—
80C2×#11758198—22no—
81C2×C4×Dic61—21no—
82#14519710—24no—
83#14519708—24no—
84#14519700—24no—
85#14519700—22no—
86#14519700—22no—
87#14519697—24no—
88#14519697—22no—
89#14519697—22no—
90C22⋊#92418—24no—
91C22⋊#92418—24no—
92C25×C61D411yescharacteristic
93C23×C4×C61—22no—
94C23×C4×C61—22no—
95C23×C4×C61—22no—
96C23×C4×C61—22no—
97C23×C4×C61—22no—
98C23×C4×C61—22no—
99C23×C4×C61D412yes—
100C23×C4×C61D411yescharacteristic
101C23×C4×C61C2312yes—
102C23×C4×C61C2311yescharacteristic
103#117×C61—22no—
104#117×C61—22no—
105#117×C61—22no—
106#117×C61—22no—
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C4⋊C4×C61C2312yes—
110C2×C22⋊C4×C61—22no—
111C2×C22⋊C4×C61—22no—
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61—21no—
114C2×C42×C61—21no—
115C2×C42×C61—21no—
116#95×C61—22no—
117#95×C61—22no—
118C24×C61—22no—
119C24×C61—24no—
120C24×C61—22no—
121C24×C61—22no—
122C24×C61—42no—
123C24×C61C2×D412yes—
124C24×C61C2×D411yescharacteristic
125C23×D61—21no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—24no—
130C22×C4×C61—24no—
131C22×C4×C61—24no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—44no—
142C22×C4×C61—44no—
143C22×C4×C61—44no—
144C22×C4×C61—44no—
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D412yes—
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2×D412yes—
150C22×C4×C61#4112yes—
151C22×C4×C61#4112yes—
152C22×C4×C61#4112yes—
153C22×C4×C61C2411yescharacteristic
154C2×C22⋊D61—28no—
155C2×C22⋊D61—24no—
156C2×C22⋊D61—44no—
157C2×C22⋊D61—42no—
158C2×C22⋊D61—42no—
159C2×C22⋊D61—44no—
160C22×Dic61—21no—
161C22×Dic61—22no—
162#11758222—28no—
163#11758222—28no—
164#11758222—44no—
165#11758222—44no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×D244—21no—
170C2×D244—21no—
171C2×D244C2×D412yes—
172C2×C4×D61—24no—
173C2×C4×D61—42no—
174C2×C4×D61—42no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122C2×D412yes—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C22⋊C4×C61—44no—
188C22⋊C4×C61—42no—
189C22⋊C4×C61—42no—
190C22⋊C4×C61—44no—
191C42×C61—42no—
192C42×C61—42no—
193C22⋊Dic61—28no—
194C22⋊Dic61—24no—
195#11758198—28no—
196C4⋊Dic61—22no—
197C4⋊Dic61C2×D414yes—
198#11758196—28no—
199C4×Dic61—24no—
200C2×#92418—28no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—22no—
206C2×#92418—22no—
207C2×#92418—22no—
208C2×#92418—44no—
209C2×#92418C2×D412yes—
210C2×#92418C2×D412yes—
211C23×C61—22no—
212C23×C61—22no—
213C23×C61—28no—
214C23×C61—22no—
215C23×C61—22no—
216C23×C61—22no—
217C23×C61—22no—
218C23×C61—22no—
219C23×C61—22no—
220C23×C61—48no—
221C23×C61—48no—
222C23×C61—44no—
223C23×C61—44no—
224C23×C61#12012yes—
225C23×C61#12112yes—
226C23×C61#12312yes—
227C23×C61#13514yes—
228C23×C61C22×D411yescharacteristic
229C22×D61—22no—
230C22×D61—41no—
231C22×D61—41no—
232C22×D61—42no—
233C22×D61—42no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—44no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61—48no—
243C2×C4×C61—42no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61#12012yes—
251C2×C4×C61#12114yes—
252C2×C4×C61#12114yes—
253C2×C4×C61#12112yes—
254C2×C4×C61#12314yes—
255C2×C4×C61#12312yes—
256C2×C4×C61#12414yes—
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C22⋊D61—416no—
264C22⋊D61—48no—
265C2×Dic61—24no—
266C2×Dic61—22no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—42no—
270C2×Dic61—42no—
271D244—22no—
272D244—22no—
273D244—41no—
274D244—41no—
275C4×D61—48no—
276Dic122—22no—
277Dic122—22no—
278Dic122—41no—
279Dic122—41no—
280#92418—22no—
281#92418—22no—
282#92418—42no—
283#23479—611nomaximal, Sylow
284C22×C61—24no—
285C22×C61—24no—
286C22×C61—22no—
287C22×C61—22no—
288C22×C61—24no—
289C22×C61—24no—
290C22×C61—24no—
291C22×C61—416no—
292C22×C61—42no—
293C22×C61—42no—
294C22×C61—42no—
295C22×C61—42no—
296C22×C61#39414yes—
297C22×C61C2×#12011yescharacteristic
298C22×C61C2×#12111yescharacteristic
299C22×C61C2×#12311yescharacteristic
300C22×C61C2×#13512yes—
301C22×C61#57512yes—
302C2×D61—42no—
303C2×D61—44no—
304C2×D61—82no—
305C2×D61—81no—
306C2×D61—81no—
307C4×C61—42no—
308C4×C61—82no—
309C4×C61—81no—
310C4×C61—81no—
311C4×C61#39414yes—
312C4×C61C2×#12111yescharacteristic
313C4×C61C2×#12111yescharacteristic
314C4×C61C2×#12311yescharacteristic
315C4×C61C2×#12411yescharacteristic
316Dic61—42no—
317Dic61—44no—
318C2×#525—611no—
319C2×#427—611no—
320C2×#416—611no—
321C2×#416—611no—
322C22×C22⋊C8—611no—
323#2280D6111yescharacteristic, complemented
324#2036—612no—
325#2035—612no—
326#2034—612no—
327#2033—612no—
328#1735—611no—
329C2×C61—24no—
330C2×C61—44no—
331C2×C61C2×#39411yescharacteristic
332C2×C61#300212yes—
333C2×C61#305011yescharacteristic
334C2×C61#305411yescharacteristic
335C2×C61#309112yes—
336D61—82no—
337C22×#41—611no—
338C24×C4C2×D6111yescharacteristic
339C23×C8—611no—
340C23×C8—611no—
341#525—1224no—
342#525—1224no—
343C2×#123—1221no—
344C2×C22⋊Q8—612no—
345C2×#121—612no—
346C2×#120—1221no—
347C2×C4×D4—1222no—
348C2×#117—611no—
349C2×#117C2×D6111yescharacteristic
350C2×#117C2×D6111yescharacteristic
351#427—1224no—
352C2×#107—612no—
353C2×#106—612no—
354#416—1224no—
355#416—1224no—
356C2×#103—611no—
357C2×#103—612no—
358C2×#103—611no—
359C2×#102—611no—
360C2×#102—612no—
361C2×#102—611no—
362C2×C22⋊C8—612no—
363C2×C22⋊C8—612no—
364C2×C22⋊C8—1224no—
365#381—21no—
366#380C2×D6112yes—
367#380C2×D6112yes—
368#379—21no—
369C4×C22⋊C4—22no—
370#340—612no—
371#340—612no—
372C61#2347911yescharacteristic, complemented, Sylow
373C25C22⋊D6111yescharacteristic
374C2×#41—614no—
375C2×#41—1222no—
376C2×#41—1222no—
377C2×#41—1222no—
378C2×#41—1222no—
379C2×#41—1222no—
380C22×Q8—611no—
381C22×D4—611no—
382C22×D4—1222no—
383C22×D4—1221no—
384C23×C4—22no—
385C23×C4—22no—
386C23×C4—22no—
387C23×C4—22no—
388C23×C4—22no—
389C23×C4—22no—
390C23×C4—1221no—
391C23×C4C22⋊D6111yescharacteristic
392C23×C4C22⋊D6112yes—
393C23×C4C22×D6112yes—
394C23×C4C22×D6111yescharacteristic
395C22×C8—614no—
396C22×C8—611no—
397C22×C8—611no—
398C22×C8—612no—
399C22×C8—1222no—
400C22×C8—1222no—
401C22×C8—1222no—
402C22×C8—1222no—
403C22×C8—1222no—
404#123—1228no—
405C22⋊Q8—1224no—
406C22⋊Q8—1224no—
407#121—1224no—
408#121—1224no—
409#120—1228no—
410C4×D4—12216no—
411#117—22no—
412#117—22no—
413#117—22no—
414#117—22no—
415#117—614no—
416#117—1222no—
417C2×C4⋊C4—612no—
418C2×C4⋊C4—1222no—
419C2×C4⋊C4C22×D6111yescharacteristic
420C2×C4⋊C4C22×D6111yescharacteristic
421C2×C4⋊C4C22×D6112yes—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—22no—
425C2×C22⋊C4—22no—
426C2×C22⋊C4—1221no—
427C2×C22⋊C4—1222no—
428C2×C22⋊C4—1222no—
429C2×C42—21no—
430C2×C42—21no—
431C2×C42—1221no—
432#107—1224no—
433#106—1224no—
434#103—1222no—
435#103—1222no—
436#103—1224no—
437#102—1222no—
438#102—1222no—
439#102—1224no—
440C22⋊C8—1224no—
441C22⋊C8—1224no—
442#95—22no—
443#95—22no—
444C24—22no—
445C24—24no—
446C24—22no—
447C24—22no—
448C24—42no—
449C24—1221no—
450C24C2×C22⋊D6112yes—
451C24C2×C22⋊D6111yescharacteristic
452#41—1228no—
453#41—1228no—
454#41—2444no—
455#41—2444no—
456C2×Q8—612no—
457C2×Q8—1221no—
458C2×Q8—1221no—
459C2×Q8—1221no—
460C2×Q8—1221no—
461C2×Q8—1221no—
462C2×D4—612no—
463C2×D4—1228no—
464C2×D4—1224no—
465C2×D4—1221no—
466C2×D4—1221no—
467C2×D4—1221no—
468C2×D4—1221no—
469C2×D4—1221no—
470C2×D4—2444no—
471C2×D4—2442no—
472C2×D4—2442no—
473C2×D4—2444no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—22no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—1222no—
494C22×C4—1221no—
495C22×C4—1224no—
496C22×C4—2442no—
497C22×C4—2442no—
498C22×C4#1175822212yes—
499C22×C4#1175822212yes—
500C22×C4D4×D6112yes—
501C22×C4D4×D6112yes—
502C22×C4D4×D6112yes—
503C22×C4#1175822412yes—
504C22×C4C2×C22⋊D6112yes—
505C22×C4C2×C22⋊D6111yescharacteristic
506C22×C4C23×D6111yescharacteristic
507C2×C8—612no—
508C2×C8—612no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1228no—
514C2×C8—1222no—
515C2×C8—1222no—
516C2×C8—1222no—
517C2×C8—2444no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—22no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—614no—
525C4⋊C4—1228no—
526C4⋊C4—1222no—
527C22⋊C4—44no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—44no—
531C22⋊C4—1228no—
532C22⋊C4—1228no—
533C22⋊C4—1224no—
534C42—42no—
535C42—42no—
536C42—1224no—
537C23—22no—
538C23—22no—
539C23—28no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—48no—
547C23—48no—
548C23—44no—
549C23—44no—
550C23—1222no—
551C23—2441no—
552C23—2441no—
553C23—2442no—
554C23—2442no—
555C23#1451974512yes—
556C23#1451974912yes—
557C23#1451976012yes—
558C23#1451979714yes—
559C23C22×C22⋊D6111yescharacteristic
560Q8—1222no—
561Q8—1222no—
562Q8—2441no—
563Q8—2441no—
564D4—1222no—
565D4—1222no—
566D4—24416no—
567D4—2448no—
568D4—2441no—
569D4—2441no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—48no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—1224no—
587C2×C4—1222no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2442no—
591C2×C4—2442no—
592C2×C4—2448no—
593C2×C4#1451974512yes—
594C2×C4#1451974912yes—
595C2×C4#1451975014yes—
596C2×C4#1451975014yes—
597C2×C4#1451975914yes—
598C2×C4#1451976012yes—
599C2×C4#1451976314yes—
600C2×C4C2×D4×D6111yescharacteristic
601C2×C4C2×D4×D6111yescharacteristic
602C2×C4C2×D4×D6111yescharacteristic
603C2×C4C2×#1175822411yescharacteristic
604C2×C4C2×#1175822211yescharacteristic
605C2×C4C2×#1175822211yescharacteristic
606C8—1222no—
607C8—1222no—
608C8—2442no—
609C22—22no—
610C22—22no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—416no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—4881no—
622C22—4881no—
623C22—4882no—
624C22—2442no—
625C22—2444no—
626C22#1683967714yes—
627C22C2×#1451974511yescharacteristic
628C22C2×#1451974911yescharacteristic
629C22C2×#1451976011yescharacteristic
630C22C2×#1451979712yes—
631C22#1684056012yes—
632C4—42no—
633C4—81no—
634C4—81no—
635C4—82no—
636C4—2442no—
637C4—2444no—
638C4#1683967714yes—
639C4C2×#1451975011yescharacteristic
640C4C2×#1451975011yescharacteristic
641C4C2×#1451975911yescharacteristic
642C4C2×#1451976311yescharacteristic
643C2—24no—
644C2—44no—
645C2—4882no—
646C2C2×#1683967711yescharacteristic
647C2#2145503512yes—
648C2#2145539711yescharacteristic
649C2#2145542511yescharacteristic
650C2#2145573212yes—
subgroupG/H
centerC22×C4C2×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2280C2
socleC23×C61C22×D4
prime radicalC2×#16840273C2
hypercenter#2280D61
hypocenterC61#23479
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ #2280

Complements

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

NH ≅ G/Ntype
2. C2×#16839798C2semidirect
3. C2×#16839756C2semidirect
4. C2×#16839756C2semidirect
5. C22×C22⋊#92418C2semidirect
6. #21447022C2semidirect
7. #21447015C2semidirect
8. #21447008C2semidirect
9. #21447001C2semidirect
10. #21445535C2semidirect
11. C61×#2280C2semidirect
24. C2×#14519710C22semidirect
25. C2×#14519708C22semidirect
28. C2×#14519700C22semidirect
29. C2×#14519700C22semidirect
30. C2×#14519700C22semidirect
31. C2×#14519697C22semidirect
32. C2×#14519697C22semidirect
33. C2×#14519697C22semidirect
37. #16839524C22semidirect
38. #16839524C22semidirect
63. C22×#92418D4semidirect
323. #2280D61semidirect
372. C61#23479semidirect

Maximal subgroups

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

subgroupindex
C2×#168402732
C2×#168397982
C2×#168397562
C22×C22⋊#924182
#214470222
#214470152
#214470082
#214470012
#214455352
C61×#22802
#2347961