Finite Group Databasebeta

← #29699737

Subgroups — #29699737

Order 15616.

858 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×Dic61C211yesmaximal, characteristic
2#21448775C211yesmaximal, characteristic, complemented
3#21448767C211yesmaximal, characteristic
4#21446675C211yesmaximal, characteristic
5#21446456C211yesmaximal, characteristic, complemented
6#21446450C211yesmaximal, characteristic, complemented
7#21446449C211yesmaximal, characteristic
8#21446443C211yesmaximal, characteristic
9#21445974C211yesmaximal, characteristic, complemented
10#21445822C211yesmaximal, characteristic, complemented
11#21445817C211yesmaximal, characteristic
12#106×Dic61C211yesmaximal, characteristic, complemented
13C4×#14519709C211yesmaximal, characteristic, complemented
14C8×#11758196C211yesmaximal, characteristic, complemented
15C61×#1968C211yesmaximal, characteristic
16C2×Q8×Dic61C2211yescharacteristic
17C2×D4×Dic61C2211yescharacteristic
18C2×#14519733—21no—
19SD16×Dic61C418yescomplemented
20C2×#14519709C2211yescharacteristic
21C2×#14519709C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×C8×Dic61C2211yescharacteristic
30C2×C8×Dic61C2211yescharacteristic
31#16839669C2211yescharacteristic
32#16839665C2211yescharacteristic
33#16839656—21no—
34#16839650—21no—
35#16839647—21no—
36#16839645—21no—
37#16839631C2211yescharacteristic
38#16839631C2211yescharacteristic
39#16839626—21no—
40#16839620—21no—
41C4⋊C4×Dic61C2211yescharacteristic
42C4⋊C4×Dic61C2211yescharacteristic
43C4×#11758196C2211yescharacteristic
44C4×C4⋊Dic61C2211yescharacteristic
45C4×C4⋊Dic61C2211yescharacteristic
46#16839613—21no—
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53C22×SD16×C61C2211yescharacteristic
54C2×#106×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×C4×C8×C61C2211yescharacteristic
58C61×#391C2211yescharacteristic
59C61×#390C2211yescharacteristic
60C23×Dic61—21no—
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63Q8×Dic61—22no—
64Q8×Dic61C2×C414yes—
65D4×Dic61—22no—
66D4×Dic61C2×C414yes—
67#14519733—44no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196D411yescharacteristic
78C2×#11758196D411yescharacteristic
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89#14519709—22no—
90#14519709C2×C414yes—
91#14519708—41no—
92#14519708—41no—
93#14519706—22no—
94#14519701—41no—
95#14519701C2×C414yes—
96#14519698—41no—
97#14519698C2×C414yes—
98C8×Dic61—22no—
99C8×Dic61C2×C414yes—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684D411yescharacteristic
107#14519684D411yescharacteristic
108C22×Q8×C61C2311yescharacteristic
109C22×D4×C61C2311yescharacteristic
110C2×SD16×C61—22no—
111C2×SD16×C61—22no—
112C2×SD16×C61C2×C414yes—
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C22⋊C4×C61—21no—
118C2×C22⋊C4×C61—21no—
119C2×C42×C61C2311yescharacteristic
120#106×C61—22no—
121#103×C61—41no—
122#102×C61—41no—
123C4×C8×C61—22no—
124#95×C61—21no—
125#95×C61—21no—
126C24×C61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61C22×C412yes—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61C22×C412yes—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2411yescharacteristic
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—24no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—42no—
151C22×Dic61—42no—
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156SD16×C61—28no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61C22×C412yes—
161C2×C8×C61#4112yes—
162C2×C8×C61#4112yes—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C22⋊C4×C61—42no—
168C22⋊C4×C61—81no—
169C42×C61—21no—
170C42×C61—21no—
171C22⋊Dic61—24no—
172C22⋊Dic61—42no—
173C22⋊Dic61—81no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—24no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61C22×C412yes—
182C4⋊Dic61C22×C412yes—
183#11758196—22no—
184#11758196—42no—
185#11758196—42no—
186#11758196—81no—
187#11758196—81no—
188#11758196SD1614yescomplemented
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61—24no—
192C4×Dic61—42no—
193C4×Dic61—42no—
194C4×Dic61—81no—
195C4×Dic61—81no—
196C4×Dic61C22×C412yes—
197C4×Dic61#4112yes—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418C22×C412yes—
204C23×C61—22no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61C22×D411yescharacteristic
212Q8×C61—22no—
213Q8×C61—22no—
214Q8×C61—41no—
215Q8×C61—41no—
216D4×C61—22no—
217D4×C61—22no—
218D4×C61—41no—
219D4×C61—41no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—22no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61C2×SD1612yes—
235C2×C4×C61C23×C411yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61H(5,𝔽2)11yescharacteristic
240C2×C4×C612−1+411yescharacteristic
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—22no—
246C2×Dic61—22no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—48no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61C4×D412yes—
262C2×Dic61C4×D412yes—
263C2×Dic61C2×SD1612yes—
264C2×Dic61C2×SD1612yes—
265C2×Dic61#13512yes—
266C8×C61—41no—
267C8×C61#11714yescomplemented
268#92418—22no—
269#92418—41no—
270#22805—611nomaximal, Sylow
271C22×C61—42no—
272C22×C61—42no—
273C22×C61—42no—
274C22×C61—81no—
275C22×C61—81no—
276C22×C61—81no—
277C22×C61—81no—
278C22×C61C2×C4×D411yescharacteristic
279C22×C61#54711yescharacteristic
280C22×C61#55011yescharacteristic
281C22×C61C22×SD1611yescharacteristic
282C22×C61C2×#13511yescharacteristic
283C22×C61#57711yescharacteristic
284C22×C61#57711yescharacteristic
285C4×C61—22no—
286C4×C61—41no—
287C4×C61—42no—
288C4×C61—81no—
289C4×C61—81no—
290C4×C61C2×#11711yescharacteristic
291C4×C61#52011yescharacteristic
292C4×C61#55511yescharacteristic
293C4×C61#55611yescharacteristic
294Dic61—22no—
295Dic61—22no—
296Dic61—41no—
297Dic61—42no—
298Dic61—42no—
299Dic61—81no—
300Dic61—81no—
301Dic61C4×SD1614yes—
302C2×C4×SD16—611no—
303#2355—611no—
304#2354—611no—
305#1968—611no—
306#1968D6111yescharacteristic
307#1927—611no—
308#1927—611no—
309#1926—611no—
310#1926—611no—
311#1844—611no—
312#1803—611no—
313#1802—611no—
314C4×#106—611no—
315C4×#106—611no—
316C8×C4⋊C4—611no—
317C2×C61—82no—
318C2×C61#235411yescharacteristic
319C2×C61C2×C4×SD1611yescharacteristic
320C2×C61#295211yescharacteristic
321C2×C61#333511yescharacteristic
322C2×C61#334011yescharacteristic
323C2×C61#339211yescharacteristic
324C2×C61#339411yescharacteristic
325C22×SD16C2×D6111yescharacteristic
326C2×C4×Q8—611no—
327C2×C4×D4—611no—
328C2×#117—1221no—
329C4×SD16—618no—
330C2×#106—611no—
331C2×#106—611no—
332C2×#106—611no—
333C2×#106—611no—
334C2×#106C2×D6111yescharacteristic
335C2×C4⋊C8—611no—
336C2×#103—611no—
337C2×#103—611no—
338C2×#103C2×D6111yescharacteristic
339C2×#102—611no—
340C2×#102—611no—
341C2×#102C2×D6111yescharacteristic
342C2×C4×C8—611no—
343C2×C4×C8—611no—
344C2×C4×C8C2×D6111yescharacteristic
345#391—611no—
346#391C2×D6111yescharacteristic
347#390—611no—
348#390C2×D6111yescharacteristic
349#388—1221no—
350#388—1221no—
351#387—1221no—
352#387—1221no—
353#384—611no—
354#384—611no—
355#382—1221no—
356#380—1221no—
357C4×C4⋊C4—611no—
358C4×C4⋊C4—611no—
359C4×C4⋊C4—611no—
360C4×C4⋊C4—611no—
361C4×C4⋊C4—611no—
362C4×C4⋊C4—1221no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367#336—611no—
368#336—611no—
369C61#2280511yescharacteristic, complemented, Sylow
370C22×Q8C22×D6111yescharacteristic
371C22×D4C22×D6111yescharacteristic
372C23×C4—1221no—
373C2×SD16—22no—
374C2×SD16—22no—
375C2×SD16C4×D6114yes—
376C22×C8—611no—
377C22×C8—611no—
378C22×C8C22×D6111yescharacteristic
379C22×C8C22×D6111yescharacteristic
380C4×Q8—614no—
381C4×Q8—1222no—
382C4×D4—614no—
383C4×D4—1222no—
384#117—2444no—
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⋊C4C22×D6111yescharacteristic
397C2×C4⋊C4C22×D6111yescharacteristic
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—1221no—
401C2×C22⋊C4—1221no—
402C2×C22⋊C4—1221no—
403C2×C42—611no—
404C2×C42—611no—
405C2×C42—1221no—
406C2×C42—1221no—
407C2×C42—1221no—
408C2×C42—1221no—
409C2×C42—1221no—
410C2×C42C22×D6111yescharacteristic
411#106—22no—
412#106—614no—
413#106—1222no—
414#106—2441no—
415#106—2441no—
416C4⋊C8—1222no—
417#103—41no—
418#103—614no—
419#103—2441no—
420#102—41no—
421#102—614no—
422#102—2441no—
423C4×C8—22no—
424C4×C8—614no—
425C4×C8—1222no—
426#95—21no—
427#95—21no—
428#95—611no—
429#95—611no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436C24—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C2×C4×D6112yes—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4C2×C4×D6112yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—1224no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—2442no—
462C22×C4—2442no—
463C22×C4D4×D6111yescharacteristic
464C22×C4D4×D6111yescharacteristic
465C22×C4C23×D6111yescharacteristic
466SD16—28no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—612no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8C2×C4×D6112yes—
477C2×C8#1175822412yes—
478C2×C8#1175822412yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—614no—
484C4⋊C4—612no—
485C4⋊C4—612no—
486C4⋊C4—1222no—
487C4⋊C4—1224no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—4881no—
495C4⋊C4—4881no—
496C4⋊C4—2442no—
497C4⋊C4—2442no—
498C22⋊C4—42no—
499C22⋊C4—81no—
500C22⋊C4—1224no—
501C22⋊C4—4881no—
502C22⋊C4—2442no—
503C42—21no—
504C42—21no—
505C42—612no—
506C42—612no—
507C42—1224no—
508C42—1221no—
509C42—1221no—
510C42—4881no—
511C42—4881no—
512C42—2442no—
513C42—2442no—
514C23—22no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C2×D4×D6111yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—41no—
525Q8—41no—
526D4—22no—
527D4—22no—
528D4—41no—
529D4—41no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—612no—
545C2×C4—612no—
546C2×C4—612no—
547C2×C4—612no—
548C2×C4—612no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1222no—
554C2×C4—1222no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—2448no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4SD16×D6112yes—
570C2×C4C22×C4×D6111yescharacteristic
571C2×C4#41×D6111yescharacteristic
572C2×C4C2×#1175822411yescharacteristic
573C2×C4C2×#1175822411yescharacteristic
574C2×C4#1451982211yescharacteristic
575C2×C4#1451982411yescharacteristic
576C8—41no—
577C8—1222no—
578C8—2441no—
579C8#1451973314yescomplemented
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22C4×D4×D6111yescharacteristic
588C22#1684041211yescharacteristic
589C22#1684043211yescharacteristic
590C22C2×SD16×D6111yescharacteristic
591C22#135×D6111yescharacteristic
592C22#1684056811yescharacteristic
593C22#1684056911yescharacteristic
594C4—22no—
595C4—41no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—614no—
600C4—1222no—
601C4—1222no—
602C4—4881no—
603C4—4881no—
604C4—2441no—
605C4—2442no—
606C4—2442no—
607C4C2×#1451973311yescharacteristic
608C4#1684024111yescharacteristic
609C4#1684047611yescharacteristic
610C4#1684048111yescharacteristic
611C2—82no—
612C2#2144876711yescharacteristic
613C2C4×SD16×D6111yescharacteristic
614C2#2145469711yescharacteristic
615C2#2145817111yescharacteristic
616C2#2145822711yescharacteristic
617C2#2145879711yescharacteristic
618C2#2145881211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1968C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenter#1968D61
hypocenterC61#22805
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1968

Complements

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

NH ≅ G/Ntype
2. #21448775C2semidirect
5. #21446456C2semidirect
6. #21446450C2semidirect
9. #21445974C2semidirect
10. #21445822C2semidirect
12. #106×Dic61C2semidirect
13. C4×#14519709C2semidirect
14. C8×#11758196C2semidirect
19. SD16×Dic61C4semidirect
188. #11758196SD16semidirect
267. C8×C61#117semidirect
369. C61#22805semidirect
579. C8#14519733semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×Dic612
#214487752
#214487672
#214466752
#214464562
#214464502
#214464492
#214464432
#214459742
#214458222
#214458172
#106×Dic612
C4×#145197092
C8×#117581962
C61×#19682
#2280561