Finite Group Databasebeta

← #29721374

Subgroups — #29721374

Order 15616.

746 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840058C211yesmaximal, characteristic, complemented
2#21451640C211yesmaximal, characteristic
3#21448501C211yesmaximal, characteristic
4#21447247C211yesmaximal, characteristic
5#21447242C211yesmaximal, characteristic
6#21447023C211yesmaximal, characteristic
7#21447016C211yesmaximal, characteristic
8#21446770C211yesmaximal, characteristic, complemented
9#21446589C211yesmaximal, characteristic, complemented
10#21446562C211yesmaximal, characteristic, complemented
11#21446552C211yesmaximal, characteristic, complemented
12#21446509C211yesmaximal, characteristic, complemented
13#21446495C211yesmaximal, characteristic, complemented
14#21445835C211yesmaximal, characteristic, complemented
15C2×C61×#433C211yesmaximal, characteristic
16C2×#14519777C2211yescharacteristic
17C2×#14519756C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19#16840058—24no—
20C2×#14519711C2211yescharacteristic
21C2×#14519711C2211yescharacteristic
22C2×#14519711C2211yescharacteristic
23C2×#14519709C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28#16839701—21no—
29#16839698—21no—
30#16839682—21no—
31#16839674—21no—
32#16839668C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839660C2211yescharacteristic
35#16839660C2211yescharacteristic
36#16839657—21no—
37#16839656—21no—
38#16839650—21no—
39#16839643—21no—
40#16839628C2211yescharacteristic
41#16839622C2211yescharacteristic
42#16839622C2211yescharacteristic
43#16839622C2211yescharacteristic
44C4×C4⋊Dic61C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53C2×#117×C61C2211yescharacteristic
54C61×#433—24no—
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×C4⋊C8×C61C2211yescharacteristic
57C2×C22⋊C8×C61C2211yescharacteristic
58C2×C22⋊C8×C61C2211yescharacteristic
59C2×#97×C61C2211yescharacteristic
60C2×C4×C8×C61C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122C2311yescharacteristic
63#14519777—22no—
64#14519777—22no—
65#14519756—22no—
66#14519756—22no—
67C4×Dic122—22no—
68C4×Dic122—22no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88#14519711—22no—
89#14519711—22no—
90#14519711—22no—
91#14519709—22no—
92#14519702—22no—
93#14519702—22no—
94#14519702—22no—
95#14519702—22no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107C23×C4×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110C22×C8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112#117×C61—22no—
113#117×C61—22no—
114C2×C4⋊C4×C61D411yescharacteristic
115C2×C4⋊C4×C61D411yescharacteristic
116C2×C22⋊C4×C61D411yescharacteristic
117C2×C22⋊C4×C61D411yescharacteristic
118C2×C42×C61C2311yescharacteristic
119C2×C42×C61C2311yescharacteristic
120C4⋊C8×C61—22no—
121C4⋊C8×C61D414yes—
122C22⋊C8×C61—22no—
123C22⋊C8×C61—22no—
124#97×C61—22no—
125C4×C8×C61—22no—
126C24×C61C2×D411yescharacteristic
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2411yescharacteristic
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C2×Dic122—21no—
148C2×Dic122—21no—
149C2×Dic122—21no—
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122—42no—
153C2×Dic122—81no—
154C2×Dic122—81no—
155C2×Dic122#4112yes—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61C2×D412yes—
167C2×C8×C61C2×D412yes—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—22no—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—22no—
172C42×C61—21no—
173C42×C61—21no—
174C42×C61C2×D412yes—
175C42×C61#4112yes—
176C22⋊Dic61—24no—
177C22⋊Dic61—81no—
178C22⋊Dic61—81no—
179C22⋊Dic61—81no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—42no—
194C4⋊Dic61#4112yes—
195#11758196—24no—
196#11758196—42no—
197#11758196—81no—
198#11758196—81no—
199#11758196—81no—
200C4×Dic61—42no—
201C4×Dic61—81no—
202C23×C61—21no—
203C23×C61—21no—
204C23×C61—21no—
205C23×C61—21no—
206C23×C61—21no—
207C23×C61—21no—
208C23×C61—21no—
209C23×C61C22×D411yescharacteristic
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61#12112yes—
228C2×C4×C61#12112yes—
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C612−1+411yescharacteristic
234C2×C4×C612−1+411yescharacteristic
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259Dic122—22no—
260Dic122—22no—
261Dic122—41no—
262Dic122—41no—
263Dic122—84no—
264C8×C61—22no—
265C8×C61—22no—
266C8×C61—41no—
267C8×C61—41no—
268#24714—611nomaximal, Sylow
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61C2×#12111yescharacteristic
277C22×C61#52511yescharacteristic
278C22×C61#53611yescharacteristic
279C22×C61#57611yescharacteristic
280C22×C61#57711yescharacteristic
281C22×C61#57811yescharacteristic
282C22×C61#57811yescharacteristic
283C4×C61—22no—
284C4×C61—22no—
285C4×C61—41no—
286C4×C61—41no—
287C4×C61—41no—
288C4×C61#54711yescharacteristic
289C4×C61#54811yescharacteristic
290C4×C61#55211yescharacteristic
291C4×C61#55611yescharacteristic
292Dic61—42no—
293Dic61—42no—
294Dic61—81no—
295Dic61—81no—
296Dic61—81no—
297Dic61—81no—
298Dic61—81no—
299Dic61—81no—
300C2×#477—611no—
301C2×#433D6111yescharacteristic
302#2628—611no—
303#2323—611no—
304#2076—611no—
305#2075—611no—
306#2036—611no—
307#2035—611no—
308#1988—611no—
309#1949—611no—
310#1944—611no—
311#1942—611no—
312#1935—611no—
313#1933—611no—
314#1805—611no—
315C2×C61—81no—
316C2×C61#241511yescharacteristic
317C2×C61#305911yescharacteristic
318C2×C61#306211yescharacteristic
319C2×C61#312111yescharacteristic
320C2×C61#312411yescharacteristic
321C2×C61#324211yescharacteristic
322C2×C61#324311yescharacteristic
323C2×C4⋊Q8—611no—
324C2×C22⋊Q8—611no—
325C2×C4×Q8—611no—
326C2×#117C2×D6111yescharacteristic
327#477—1224no—
328#433—24no—
329C2×#107—611no—
330C2×#107—611no—
331C2×#107—611no—
332C2×#106—611no—
333C2×C4⋊C8C2×D6111yescharacteristic
334C2×C4⋊C8C2×D6111yescharacteristic
335C2×#103—611no—
336C2×#103—611no—
337C2×#103—611no—
338C2×#103—611no—
339C2×C22⋊C8C2×D6111yescharacteristic
340C2×C22⋊C8C2×D6111yescharacteristic
341C2×#97C2×D6111yescharacteristic
342C2×C4×C8C2×D6111yescharacteristic
343#400—1221no—
344#399—1221no—
345#395—1221no—
346#393—1221no—
347#391—611no—
348#389—611no—
349#389—611no—
350#389—611no—
351#389—1221no—
352#388—1221no—
353#388—1221no—
354#387—1221no—
355#382—611no—
356#380—611no—
357#380—611no—
358#380—611no—
359C4×C4⋊C4—611no—
360#340—611no—
361#340—611no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367#340—611no—
368C61#2471411yescharacteristic, complemented, Sylow
369C22×Q8—611no—
370C22×Q8—1221no—
371C23×C4C22×D6111yescharacteristic
372C22×C8C22×D6111yescharacteristic
373C22×C8C22×D6111yescharacteristic
374C22×C8C22×D6111yescharacteristic
375C22×C8C22×D6111yescharacteristic
376C4⋊Q8—1222no—
377C4⋊Q8—1222no—
378C22⋊Q8—1222no—
379C22⋊Q8—1222no—
380C4×Q8—1222no—
381C4×Q8—1222no—
382#117—22no—
383#117—22no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4D24411yescharacteristic
398C2×C4⋊C4D24411yescharacteristic
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C22⋊C4—1221no—
402C2×C22⋊C4—1221no—
403C2×C22⋊C4D24411yescharacteristic
404C2×C22⋊C4D24411yescharacteristic
405C2×C42—1221no—
406C2×C42—1221no—
407C2×C42C22×D6111yescharacteristic
408C2×C42C22×D6111yescharacteristic
409#107—1222no—
410#107—1222no—
411#107—1222no—
412#106—1222no—
413C4⋊C8—22no—
414C4⋊C8C22⋊D6114yes—
415#103—1222no—
416#103—1222no—
417#103—1222no—
418#103—1222no—
419C22⋊C8—22no—
420C22⋊C8—22no—
421#97—22no—
422C4×C8—22no—
423#95—1221no—
424#95—1221no—
425#95—1221no—
426#95—1221no—
427#95—1221no—
428#95—1221no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434C24C2×D24411yescharacteristic
435C2×Q8—612no—
436C2×Q8—1221no—
437C2×Q8—1221no—
438C2×Q8—1221no—
439C2×Q8—1221no—
440C2×Q8—1221no—
441C2×Q8—4881no—
442C2×Q8—4881no—
443C2×Q8—2442no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4C2×D24411yescharacteristic
459C22×C4C2×D24411yescharacteristic
460C22×C4C2×D24411yescharacteristic
461C22×C4C2×D24411yescharacteristic
462C22×C4C2×D24411yescharacteristic
463C22×C4C23×D6111yescharacteristic
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8C2×C22⋊D6112yes—
475C2×C8C2×C22⋊D6112yes—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—612no—
479C4⋊C4—1224no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—4881no—
494C4⋊C4—4881no—
495C4⋊C4—4881no—
496C4⋊C4—2442no—
497C4⋊C4—2442no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—1224no—
501C22⋊C4—4881no—
502C22⋊C4—4881no—
503C22⋊C4—4881no—
504C42—21no—
505C42—21no—
506C42—4881no—
507C42—2442no—
508C42#1175822212yes—
509C42C2×C22⋊D6112yes—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23C22×D24411yescharacteristic
518Q8—1222no—
519Q8—1222no—
520Q8—4884no—
521Q8—2441no—
522Q8—2441no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4#1451975112yes—
565C2×C4#1451975112yes—
566C2×C4C22×C22⋊D6111yescharacteristic
567C2×C4#41×D6111yescharacteristic
568C2×C4#41×D6111yescharacteristic
569C2×C4C2×#1175822211yescharacteristic
570C2×C4#1451982411yescharacteristic
571C2×C4#1451982411yescharacteristic
572C8—22no—
573C8—22no—
574C8—41no—
575C8—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22C2×#1451975111yescharacteristic
584C22#1684027411yescharacteristic
585C22#1684033811yescharacteristic
586C22#1684056711yescharacteristic
587C22#1684057411yescharacteristic
588C22#1684057911yescharacteristic
589C22#1684057911yescharacteristic
590C4—22no—
591C4—22no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—4881no—
596C4—4881no—
597C4—4881no—
598C4—4881no—
599C4—4881no—
600C4—4881no—
601C4—2442no—
602C4—2442no—
603C4#1684041111yescharacteristic
604C4#1684041911yescharacteristic
605C4#1684045111yescharacteristic
606C4#1684048011yescharacteristic
607C2—81no—
608C2#2144924211yescharacteristic
609C2#2145547411yescharacteristic
610C2#2145549911yescharacteristic
611C2#2145605011yescharacteristic
612C2#2145608311yescharacteristic
613C2#2145708911yescharacteristic
614C2#2145710411yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#433C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenterC2×#433D61
hypocenterC61#24714
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#433

Complements

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

NH ≅ G/Ntype
1. C2×#16840058C2semidirect
8. #21446770C2semidirect
9. #21446589C2semidirect
10. #21446562C2semidirect
11. #21446552C2semidirect
12. #21446509C2semidirect
13. #21446495C2semidirect
14. #21445835C2semidirect
368. C61#24714semidirect

Maximal subgroups

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

subgroupindex
C2×#168400582
#214516402
#214485012
#214472472
#214472422
#214470232
#214470162
#214467702
#214465892
#214465622
#214465522
#214465092
#214464952
#214458352
C2×C61×#4332
#2471461