Finite Group Databasebeta

← #29688128

Subgroups — #29688128

Order 15616.

826 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840122C211yesmaximal, characteristic
2#21448567C211yesmaximal, characteristic, complemented
3#21448530C211yesmaximal, characteristic
4#21446780C211yesmaximal, characteristic, complemented
5#21446630C211yesmaximal, characteristic
6#21446516C211yesmaximal, characteristic
7#21446509C211yesmaximal, characteristic, complemented
8#21446502C211yesmaximal, characteristic
9#21446495C211yesmaximal, characteristic, complemented
10#21445953C211yesmaximal, characteristic, complemented
11#21445650C211yesmaximal, characteristic, complemented
12#21445645C211yesmaximal, characteristic
13C4×#14519702C211yesmaximal, characteristic, complemented
14C4×#14519699C211yesmaximal, characteristic
15C61×#1726C211yesmaximal, characteristic, complemented
16C2×#14519777C2211yescharacteristic
17C2×#14519765C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19C2×C4×D244C2211yescharacteristic
20#16840122C418yescomplemented
21C2×#14519702C2211yescharacteristic
22C2×#14519702C2211yescharacteristic
23C2×#14519702C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28C2×#14519699C2211yescharacteristic
29#16839668C2211yescharacteristic
30#16839666—21no—
31#16839666—21no—
32#16839664C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839660C2211yescharacteristic
35#16839654—21no—
36#16839654—21no—
37#16839645—21no—
38#16839645—21no—
39#16839636—21no—
40#16839636—21no—
41#16839632C2211yescharacteristic
42#16839628C2211yescharacteristic
43C4×C4⋊Dic61C2211yescharacteristic
44C4×C4⋊Dic61C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53C2×#97×C61C2211yescharacteristic
54C2×#97×C61C2211yescharacteristic
55C2×#97×C61C2211yescharacteristic
56C2×C4×C8×C61C2211yescharacteristic
57C2×C4×C8×C61C2211yescharacteristic
58C2×C4×C8×C61C2211yescharacteristic
59C61×#376C2211yescharacteristic
60C22×Dic122—21no—
61C22×Dic122C2311yescharacteristic
62C22×D244C2311yescharacteristic
63C22×C4×D61—21no—
64#14519777—22no—
65#14519777C2×C414yescomplemented
66#14519765—22no—
67#14519765C2×C414yes—
68C4×Dic122—22no—
69C4×Dic122—22no—
70C4×D244—22no—
71C4×D244—22no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92#14519702—22no—
93#14519702—22no—
94#14519702C2×C414yescomplemented
95#14519702C2×C414yescomplemented
96#14519699—22no—
97#14519699—22no—
98#14519699C2×C414yes—
99#14519699C2×C414yes—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108C22×C8×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110C22×C8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C2×C42×C61C2311yescharacteristic
113C2×C42×C61C2311yescharacteristic
114C2×C42×C61C2311yescharacteristic
115#97×C61—22no—
116#97×C61—22no—
117#97×C61C2×C414yescomplemented
118C4×C8×C61—22no—
119C4×C8×C61—22no—
120C4×C8×C61—22no—
121#95×C61D411yescharacteristic
122#95×C61D411yescharacteristic
123#95×C61D411yescharacteristic
124#95×C61D411yescharacteristic
125C23×D61—21no—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2411yescharacteristic
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244C22×C412yes—
146C2×C4×D61—42no—
147C2×C4×D61—81no—
148C2×C4×D61—81no—
149C2×Dic122—21no—
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122—24no—
155C2×Dic122—81no—
156C2×Dic122—81no—
157C2×Dic122C22×C412yes—
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×C61C22×C412yes—
167C2×C8×C61C22×C412yes—
168C2×C8×C61#4112yes—
169C2×C8×C61#4112yes—
170C42×C61—21no—
171C42×C61—21no—
172C42×C61C22×C412yes—
173C42×C61#4112yes—
174C42×C61#4112yes—
175C42×C61#4112yes—
176#11758198—24no—
177#11758198—24no—
178#11758198—42no—
179#11758198—81no—
180#11758198—81no—
181C4⋊Dic61—24no—
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—81no—
193C4⋊Dic61—81no—
194C4⋊Dic61C22×C412yes—
195C4⋊Dic61C22×C412yes—
196#11758196—42no—
197#11758196—81no—
198#11758196—81no—
199C4×Dic61—42no—
200C4×Dic61—81no—
201C4×Dic61—81no—
202C23×C61C22×D411yescharacteristic
203C22×D61—22no—
204C22×D61—41no—
205C22×D61—41no—
206C22×D61—41no—
207C22×D61—41no—
208C22×D61—41no—
209C22×D61—41no—
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—21no—
224C2×C4×C61C4×D412yes—
225C2×C4×C61C4×D412yes—
226C2×C4×C61#12412yes—
227C2×C4×C61#12412yes—
228C2×C4×C61C23×C411yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×Dic61—22no—
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—
255D244—22no—
256D244—22no—
257D244—41no—
258D244—41no—
259C4×D61—84no—
260Dic122—22no—
261Dic122—22no—
262Dic122—41no—
263Dic122—41no—
264Dic122—84no—
265C8×C61—22no—
266C8×C61—22no—
267C8×C61—22no—
268C8×C61—22no—
269#21654—611nomaximal, Sylow
270C22×C61C2×C4×D411yescharacteristic
271C22×C61C2×#12411yescharacteristic
272C22×C61#53211yescharacteristic
273C22×C61#57611yescharacteristic
274C22×C61#57711yescharacteristic
275C22×C61#57711yescharacteristic
276C22×C61#57811yescharacteristic
277C2×D61—42no—
278C2×D61—42no—
279C2×D61—42no—
280C2×D61—81no—
281C2×D61—81no—
282C2×D61—81no—
283C2×D61—81no—
284C4×C61—22no—
285C4×C61—22no—
286C4×C61—22no—
287C4×C61—22no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61C4×#4111yescharacteristic
291C4×C61C4×#4111yescharacteristic
292C4×C61#52911yescharacteristic
293C4×C61#52911yescharacteristic
294Dic61—42no—
295Dic61—42no—
296Dic61—42no—
297Dic61—81no—
298Dic61—81no—
299Dic61—81no—
300Dic61—81no—
301C2×#489—611no—
302#2331—611no—
303#2326—611no—
304#1990—611no—
305#1958—611no—
306#1936—611no—
307#1935—611no—
308#1934—611no—
309#1933—611no—
310#1836—611no—
311#1763—611no—
312#1762—611no—
313C4×#103—611no—
314C4×#102—611no—
315#1726D6111yescharacteristic, complemented
316C2×C61C4×#12411yescharacteristic
317C2×C61#294611yescharacteristic
318C2×C61#294711yescharacteristic
319C2×C61#313611yescharacteristic
320C2×C61#313711yescharacteristic
321C2×C61#317311yescharacteristic
322C2×C61#317511yescharacteristic
323D61—82no—
324C2×C4⋊Q8—611no—
325C2×#124—611no—
326C2×C4×Q8—611no—
327C2×C4×D4—611no—
328#489—618no—
329C2×#103—611no—
330C2×#103—611no—
331C2×#103—611no—
332C2×#103—611no—
333C2×#102—611no—
334C2×#102—611no—
335C2×#102—611no—
336C2×#102—611no—
337C2×#97C2×D6111yescharacteristic
338C2×#97C2×D6111yescharacteristic
339C2×#97C2×D6111yescharacteristic
340C2×C4×C8C2×D6111yescharacteristic
341C2×C4×C8C2×D6111yescharacteristic
342C2×C4×C8C2×D6111yescharacteristic
343#391—611no—
344#391—1221no—
345#391—1221no—
346#390—611no—
347#389—611no—
348#389—611no—
349#388—1221no—
350#388—1221no—
351#387—1221no—
352#387—1221no—
353#385—1221no—
354#385—1221no—
355#384—611no—
356#382—611no—
357C4×C4⋊C4—611no—
358C4×C4⋊C4—611no—
359#376C2×D6111yescharacteristic
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#2165411yescharacteristic, complemented, Sylow
369C22×Q8—611no—
370C22×Q8—1221no—
371C22×D4—611no—
372C23×C4—1221no—
373C22×C8C22×D6111yescharacteristic
374C22×C8C22×D6111yescharacteristic
375C22×C8C22×D6111yescharacteristic
376C22×C8C22×D6111yescharacteristic
377C4⋊Q8—614no—
378C4⋊Q8—1222no—
379#124—614no—
380#124—1222no—
381C4×Q8—1222no—
382C4×Q8—1222no—
383C4×D4—1222no—
384C4×D4—1222no—
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×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C22⋊C4—1221no—
402C2×C42—1221no—
403C2×C42—1221no—
404C2×C42—1221no—
405C2×C42C22×D6111yescharacteristic
406C2×C42C22×D6111yescharacteristic
407C2×C42C22×D6111yescharacteristic
408#103—614no—
409#103—614no—
410#103—1222no—
411#103—1222no—
412#102—614no—
413#102—614no—
414#102—1222no—
415#102—1222no—
416#97—22no—
417#97—22no—
418#97C4×D6114yescomplemented
419C4×C8—22no—
420C4×C8—22no—
421C4×C8—22no—
422#95—1221no—
423#95—1221no—
424#95—1221no—
425#95—1221no—
426#95—1221no—
427#95—1221no—
428#95—1221no—
429#95—1221no—
430#95D24411yescharacteristic
431#95D24411yescharacteristic
432#95D24411yescharacteristic
433#95D24411yescharacteristic
434C24—1221no—
435C2×Q8—612no—
436C2×Q8—1221no—
437C2×Q8—1224no—
438C2×Q8—1221no—
439C2×Q8—1221no—
440C2×Q8—1221no—
441C2×Q8—1221no—
442C2×Q8—4881no—
443C2×Q8—4881no—
444C2×D4—612no—
445C2×D4—1221no—
446C2×D4—1221no—
447C2×D4—1221no—
448C2×D4—1221no—
449C2×D4—1221no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—4881no—
458C22×C4—4881no—
459C22×C4—2442no—
460C22×C4C2×D24411yescharacteristic
461C22×C4C2×D24411yescharacteristic
462C22×C4C2×D24411yescharacteristic
463C22×C4C2×D24411yescharacteristic
464C22×C4C2×D24411yescharacteristic
465C22×C4C2×D24411yescharacteristic
466C22×C4C23×D6111yescharacteristic
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8C2×C4×D6112yes—
476C2×C8C2×C4×D6112yes—
477C2×C8#1175822212yes—
478C2×C8#1175822212yes—
479C4⋊C4—612no—
480C4⋊C4—612no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1224no—
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—4881no—
493C4⋊C4—4881no—
494C4⋊C4—4881no—
495C4⋊C4—4881no—
496C4⋊C4—2442no—
497C22⋊C4—1224no—
498C22⋊C4—1224no—
499C22⋊C4—4881no—
500C22⋊C4—4881no—
501C22⋊C4—2442no—
502C42—21no—
503C42—21no—
504C42—4881no—
505C42—4881no—
506C42—2442no—
507C42C2×C4×D6112yes—
508C42#1175822212yes—
509C42#1175822212yes—
510C42#1175822212yes—
511C23—1222no—
512C23—2441no—
513C23—2441no—
514C23—2441no—
515C23—2441no—
516C23—2441no—
517C23—2441no—
518C23C22×D24411yescharacteristic
519Q8—1222no—
520Q8—1222no—
521Q8—4884no—
522Q8—2441no—
523Q8—2441no—
524D4—1222no—
525D4—1222no—
526D4—2441no—
527D4—2441no—
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—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—4884no—
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×C4C4×D24412yes—
565C2×C4C4×D24412yes—
566C2×C4#1451976512yes—
567C2×C4#1451976512yes—
568C2×C4C22×C4×D6111yescharacteristic
569C2×C4C2×#1175822211yescharacteristic
570C2×C4C2×#1175822211yescharacteristic
571C2×C4C2×#1175822211yescharacteristic
572C2×C4C2×#1175822211yescharacteristic
573C2×C4C2×#1175822211yescharacteristic
574C8—22no—
575C8—22no—
576C8—22no—
577C8—22no—
578C22—4881no—
579C22—4881no—
580C22—4881no—
581C22—4881no—
582C22—2442no—
583C22—2442no—
584C22—2442no—
585C22C2×C4×D24411yescharacteristic
586C22C2×#1451976511yescharacteristic
587C22#1684031211yescharacteristic
588C22#1684056711yescharacteristic
589C22#1684057411yescharacteristic
590C22#1684057411yescharacteristic
591C22#1684057911yescharacteristic
592C4—22no—
593C4—22no—
594C4—22no—
595C4—22no—
596C4—41no—
597C4—41no—
598C4—4881no—
599C4—4881no—
600C4—4881no—
601C4—4881no—
602C4—2442no—
603C4—2442no—
604C4—2442no—
605C4C4×#1175822211yescharacteristic
606C4C4×#1175822211yescharacteristic
607C4#1684029611yescharacteristic
608C4#1684029611yescharacteristic
609C2—4882no—
610C2C4×#1451976511yescharacteristic
611C2#2145465311yescharacteristic
612C2#2145466411yescharacteristic
613C2#2145617511yescharacteristic
614C2#2145618611yescharacteristic
615C2#2145643611yescharacteristic
616C2#2145645811yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1726C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1726D61
hypocenterC61#21654
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ #376 ◁ #1726

Complements

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

NH ≅ G/Ntype
2. #21448567C2semidirect
4. #21446780C2semidirect
7. #21446509C2semidirect
9. #21446495C2semidirect
10. #21445953C2semidirect
11. #21445650C2semidirect
13. C4×#14519702C2semidirect
15. C61×#1726C2semidirect
20. #16840122C4semidirect
65. #14519777C2×C4semidirect
94. #14519702C2×C4semidirect
95. #14519702C2×C4semidirect
117. #97×C61C2×C4semidirect
315. #1726D61semidirect
368. C61#21654semidirect
418. #97C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401222
#214485672
#214485302
#214467802
#214466302
#214465162
#214465092
#214465022
#214464952
#214459532
#214456502
#214456452
C4×#145197022
C4×#145196992
C61×#17262
#2165461