Finite Group Databasebeta

← #29743114

Subgroups — #29743114

Order 15616.

848 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840062C211yesmaximal, characteristic
2C2×C4×#11758200C211yesmaximal, characteristic
3#21449680C211yesmaximal, characteristic, complemented
4#21447370C211yesmaximal, characteristic, complemented
5#21447301C211yesmaximal, characteristic, complemented
6#21447197C211yesmaximal, characteristic, complemented
7#21447157C211yesmaximal, characteristic, complemented
8#21446544C211yesmaximal, characteristic, complemented
9#21446378C211yesmaximal, characteristic, complemented
10#21446153C211yesmaximal, characteristic
11#21446153C211yesmaximal, characteristic
12#21446080C211yesmaximal, characteristic
13#21446072C211yesmaximal, characteristic, complemented
14#21446066C211yesmaximal, characteristic
15C61×#2349C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×#14519769C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19#16840062—24no—
20C4×#11758200—24no—
21C2×#14519710C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519700C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519698C2211yescharacteristic
33C2×C4×#92418C2211yescharacteristic
34#16839703—21no—
35#16839693—21no—
36#16839668C2211yescharacteristic
37#16839668C2211yescharacteristic
38#16839660C2211yescharacteristic
39#16839660C2211yescharacteristic
40#16839657—21no—
41#16839644—21no—
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50C2×C4×D4×C61C2211yescharacteristic
51C2×C4×D4×C61C2211yescharacteristic
52C2×C4×D4×C61C2211yescharacteristic
53C61×#389C2211yescharacteristic
54C61×#389C2211yescharacteristic
55C61×#385—21no—
56C61×#385—21no—
57C61×#382C2211yescharacteristic
58C61×#380—21no—
59C4×C4⋊C4×C61C2211yescharacteristic
60C2×#95×C61—21no—
61C22×Dic122C2311yescharacteristic
62C2×#11758200—22no—
63C2×#11758200—22no—
64C2×#11758200Q814yes—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519769—22no—
70#14519769—41no—
71C4×Dic122—22no—
72C4×Dic122—41no—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×C4×Dic61—21no—
82#14519710—22no—
83#14519710—41no—
84#14519708—22no—
85#14519708—22no—
86#14519708—22no—
87C4⋊#92418—22no—
88#14519700—22no—
89#14519700—22no—
90#14519700—41no—
91#14519698—22no—
92#14519698—22no—
93#14519698D414yes—
94C4×#92418—22no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101C22×D4×C61C2311yescharacteristic
102C23×C4×C61—21no—
103C23×C4×C61—21no—
104C23×C4×C61—21no—
105C4×D4×C61—22no—
106C4×D4×C61—22no—
107C4×D4×C61—22no—
108C4×D4×C61—22no—
109C4×D4×C61—41no—
110C4×D4×C61—41no—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61D411yescharacteristic
114C2×C4⋊C4×C61D411yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C22⋊C4×C61—21no—
119C2×C22⋊C4×C61—21no—
120C2×C22⋊C4×C61—21no—
121C2×C42×C61C2311yescharacteristic
122C2×C42×C61C2311yescharacteristic
123C2×C42×C61C2311yescharacteristic
124#95×C61—21no—
125#95×C61—21no—
126#95×C61—21no—
127#95×C61—21no—
128#95×C61D411yescharacteristic
129#95×C61D411yescharacteristic
130C24×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61C2×D412yes—
135C2×D4×C61C2×Q812yes—
136C2×D4×C61#4112yes—
137C22×C4×C61—21no—
138C22×C4×C61—24no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—42no—
142C22×C4×C61—42no—
143C22×C4×C61—42no—
144C22×C4×C61—42no—
145C22×C4×C61—81no—
146C22×C4×C61—81no—
147C22×C4×C61—81no—
148C22×C4×C61—81no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2411yescharacteristic
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122C2×Q812yes—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—41no—
173C4⋊C4×C61—81no—
174C4⋊C4×C61—81no—
175C4⋊C4×C61#4112yes—
176C22⋊C4×C61—24no—
177C22⋊C4×C61—42no—
178C22⋊C4×C61—42no—
179C42×C61—21no—
180C42×C61—21no—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61—21no—
185#11758200—28no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61C2×D412yes—
192#11758196—42no—
193#11758196—42no—
194#11758196—42no—
195#11758196—42no—
196#11758196—42no—
197C4×Dic61—42no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418C2×D412yes—
209C2×#92418C2×Q812yes—
210C23×C61—22no—
211C23×C61—22no—
212C23×C61—22no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61C22×D411yescharacteristic
218D4×C61—22no—
219D4×C61—22no—
220D4×C61—22no—
221D4×C61C22⋊Q814yes—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—22no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—84no—
249C2×C4×C61—84no—
250C2×C4×C61—84no—
251C2×C4×C61#13512yes—
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×Q811yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61H(5,𝔽2)11yescharacteristic
257C2×C4×C612−1+411yescharacteristic
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270Dic122—22no—
271Dic122—22no—
272Dic122—41no—
273Dic122—41no—
274#92418—22no—
275#92418—22no—
276#92418—41no—
277#92418—41no—
278#26378—611nomaximal, Sylow
279C22×C61—24no—
280C22×C61—42no—
281C22×C61—42no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61—42no—
285C22×C61—42no—
286C22×C61#52511yescharacteristic
287C22×C61D4211yescharacteristic
288C22×C61D4×Q811yescharacteristic
289C22×C61C2×#13511yescharacteristic
290C22×C61#57511yescharacteristic
291C22×C61#57711yescharacteristic
292C22×C61#57711yescharacteristic
293C4×C61—22no—
294C4×C61—41no—
295C4×C61—41no—
296C4×C61—42no—
297C4×C61—41no—
298C4×C61—41no—
299C4×C61—41no—
300C4×C61—81no—
301C4×C61—81no—
302C4×C61C2×C22⋊Q811yescharacteristic
303C4×C61#53911yescharacteristic
304C4×C61#54711yescharacteristic
305C4×C61#55411yescharacteristic
306Dic61—42no—
307Dic61—42no—
308Dic61—81no—
309Dic61—81no—
310C2×#478—611no—
311C2×C4×SD16—611no—
312#2453—611no—
313#2349D6111yescharacteristic
314#2097—611no—
315#2085—611no—
316#2066—611no—
317#2059—611no—
318#1941—611no—
319#1916—611no—
320#1877—611no—
321#1877—611no—
322#1864—611no—
323#1863—611no—
324#1862—611no—
325C2×C61—44no—
326C2×C61#244911yescharacteristic
327C2×C61#310211yescharacteristic
328C2×C61#312311yescharacteristic
329C2×C61#326711yescharacteristic
330C2×C61#328211yescharacteristic
331C2×C61#338111yescharacteristic
332C2×C61#338811yescharacteristic
333C22×SD16—611no—
334C2×#125—611no—
335C2×C4×Q8—611no—
336C2×C4×D4C2×D6111yescharacteristic
337C2×C4×D4C2×D6111yescharacteristic
338C2×C4×D4C2×D6111yescharacteristic
339#478—1224no—
340C4×SD16—1224no—
341C2×#107—611no—
342C2×#107—611no—
343C2×#106—611no—
344C2×#106—611no—
345C2×#106—611no—
346C2×C4⋊C8—611no—
347C2×#103—611no—
348C2×#103—611no—
349C2×#103—611no—
350C2×#102—611no—
351C2×#102—611no—
352C2×#102—611no—
353C2×C4×C8—611no—
354#400—1221no—
355#398—1221no—
356#391—611no—
357#391—611no—
358#389—611no—
359#389—611no—
360#389—1221no—
361#389C2×D6111yescharacteristic
362#389C2×D6111yescharacteristic
363#387—1221no—
364#385—21no—
365#385—21no—
366#382C2×D6111yescharacteristic
367#380—21no—
368C4×C4⋊C4C2×D6111yescharacteristic
369C2×#95—21no—
370#340—611no—
371#340—611no—
372#340—611no—
373#340—611no—
374#336—611no—
375#336—611no—
376#336—611no—
377#336—611no—
378C61#2637811yescharacteristic, complemented, Sylow
379C22×Q8—611no—
380C22×D4C22×D6111yescharacteristic
381C23×C4—21no—
382C23×C4—21no—
383C23×C4—21no—
384C2×SD16—614no—
385C2×SD16—1222no—
386C2×SD16—1222no—
387C22×C8—611no—
388C22×C8—611no—
389C22×C8—611no—
390C22×C8—611no—
391#125—1222no—
392#125—2441no—
393C4×Q8—1222no—
394C4×Q8—2441no—
395C4×D4—22no—
396C4×D4—22no—
397C4×D4—22no—
398C4×D4—22no—
399C4×D4—41no—
400C4×D4—41no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4C22⋊D6111yescharacteristic
412C2×C4⋊C4C22⋊D6111yescharacteristic
413C2×C4⋊C4C22×D6111yescharacteristic
414C2×C4⋊C4C22×D6111yescharacteristic
415C2×C4⋊C4C22×D6111yescharacteristic
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C42—1221no—
420C2×C42C22×D6111yescharacteristic
421C2×C42C22×D6111yescharacteristic
422C2×C42C22×D6111yescharacteristic
423#107—1222no—
424#107—2441no—
425#106—1222no—
426#106—1222no—
427#106—1222no—
428C4⋊C8—1222no—
429#103—1222no—
430#103—1222no—
431#103—2441no—
432#102—614no—
433#102—1222no—
434#102—1222no—
435C4×C8—1222no—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—21no—
440#95—1221no—
441#95—1221no—
442#95—1221no—
443#95—1221no—
444#95—1221no—
445#95—1221no—
446#95C22⋊D6111yescharacteristic
447#95C22⋊D6111yescharacteristic
448C24—21no—
449C2×Q8—612no—
450C2×Q8—1221no—
451C2×Q8—1221no—
452C2×Q8—1221no—
453C2×Q8—1221no—
454C2×Q8—1221no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4#1175822212yes—
459C2×D4D4×D6112yes—
460C2×D4Q8×D6112yes—
461C22×C4—21no—
462C22×C4—24no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—81no—
470C22×C4—81no—
471C22×C4—81no—
472C22×C4—81no—
473C22×C4—1221no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4C2×C22⋊D6111yescharacteristic
478C22×C4C2×C22⋊D6111yescharacteristic
479C22×C4C2×C22⋊D6111yescharacteristic
480C22×C4C2×C22⋊D6111yescharacteristic
481C22×C4C2×C22⋊D6111yescharacteristic
482C22×C4C2×C22⋊D6111yescharacteristic
483C22×C4C23×D6111yescharacteristic
484SD16—1228no—
485C2×C8—612no—
486C2×C8—612no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8—1221no—
496C2×C8—1221no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—41no—
504C4⋊C4—81no—
505C4⋊C4—81no—
506C4⋊C4—612no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—2442no—
513C4⋊C4—2442no—
514C4⋊C4—2442no—
515C4⋊C4—2442no—
516C4⋊C4—2442no—
517C4⋊C4#1175822212yes—
518C22⋊C4—24no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—2442no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23C22×C22⋊D6111yescharacteristic
536Q8—1222no—
537Q8—1222no—
538Q8—2441no—
539Q8—2441no—
540D4—22no—
541D4—22no—
542D4—22no—
543D4#1451975514yes—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—84no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4#1451979712yes—
586C2×C4C2×D4×D6111yescharacteristic
587C2×C4C2×Q8×D6111yescharacteristic
588C2×C4C2×#1175822211yescharacteristic
589C2×C4C2×#1175822211yescharacteristic
590C2×C4#1451982211yescharacteristic
591C2×C4#1451982511yescharacteristic
592C8—1222no—
593C8—1222no—
594C8—2441no—
595C8—2441no—
596C22—24no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22#1684027311yescharacteristic
604C22D4×C22⋊D6111yescharacteristic
605C22Q8×C22⋊D6111yescharacteristic
606C22C2×#1451979711yescharacteristic
607C22#1684056011yescharacteristic
608C22#1684057311yescharacteristic
609C22#1684057311yescharacteristic
610C4—22no—
611C4—41no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4—42no—
616C4—41no—
617C4—81no—
618C4—81no—
619C4—4881no—
620C4—4881no—
621C4—2442no—
622C4—2442no—
623C4C2×#1451975511yescharacteristic
624C4#1684035911yescharacteristic
625C4#1684041311yescharacteristic
626C4#1684046711yescharacteristic
627C2—44no—
628C2#2144962811yescharacteristic
629C2#2145583711yescharacteristic
630C2#2145607111yescharacteristic
631C2#2145734011yescharacteristic
632C2#2145752211yescharacteristic
633C2#2145869411yescharacteristic
634C2#2145876411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2349C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2349D61
hypocenterC61#26378
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
3. #21449680C2semidirect
4. #21447370C2semidirect
5. #21447301C2semidirect
6. #21447197C2semidirect
7. #21447157C2semidirect
8. #21446544C2semidirect
9. #21446378C2semidirect
13. #21446072C2semidirect
378. C61#26378semidirect

Maximal subgroups

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

subgroupindex
C2×#168400622
C2×C4×#117582002
#214496802
#214473702
#214473012
#214471972
#214471572
#214465442
#214463782
#214461532
#214460802
#214460722
#214460662
C61×#23492
#2637861