Finite Group Databasebeta

← #29699890

Subgroups — #29699890

Order 15616.

838 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×#16839863C211yesmaximal, characteristic
2#21448783C211yesmaximal, characteristic, complemented
3#21448767C211yesmaximal, characteristic
4#21446702C211yesmaximal, characteristic
5#21446463C211yesmaximal, characteristic, complemented
6#21446450C211yesmaximal, characteristic, complemented
7#21446449C211yesmaximal, characteristic
8#21446436C211yesmaximal, characteristic
9#21445984C211yesmaximal, characteristic, complemented
10#21445827C211yesmaximal, characteristic
11#21445822C211yesmaximal, characteristic, complemented
12#21445719C211yesmaximal, characteristic, complemented
13C4×#14519711C211yesmaximal, characteristic, complemented
14#21445693C211yesmaximal, characteristic, complemented
15C61×#1968C211yesmaximal, characteristic
16C2×Q8×Dic61C2211yescharacteristic
17C2×D4×Dic61C2211yescharacteristic
18C2×#14519733—21no—
19#16839863C418yescomplemented
20C2×#14519711C2211yescharacteristic
21C2×#14519711C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519687C2211yescharacteristic
30C2×#14519687C2211yescharacteristic
31#16839669C2211yescharacteristic
32#16839665C2211yescharacteristic
33#16839656—21no—
34#16839650—21no—
35#16839647—21no—
36#16839645—21no—
37#16839631C2211yescharacteristic
38#16839629—21no—
39#16839627C2211yescharacteristic
40#16839620—21no—
41C4⋊C4×Dic61C2211yescharacteristic
42C4⋊C4×Dic61C2211yescharacteristic
43C4×#11758196C2211yescharacteristic
44C4×C4⋊Dic61C2211yescharacteristic
45C4×C4⋊Dic61C2211yescharacteristic
46#16839607—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×#11758196D411yescharacteristic
76C2×#11758196D411yescharacteristic
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61C2311yescharacteristic
86C2×C4×Dic61C2311yescharacteristic
87#14519711—22no—
88#14519711C2×C414yes—
89#14519710—41no—
90#14519708—41no—
91#14519706—22no—
92#14519701—41no—
93#14519701C2×C414yes—
94#14519698—41no—
95#14519698C2×C414yes—
96#14519687—22no—
97#14519687C2×C414yes—
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#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—41no—
185#11758196—42no—
186#11758196—81no—
187C4×Dic61—21no—
188C4×Dic61—21no—
189C4×Dic61—24no—
190C4×Dic61—42no—
191C4×Dic61—42no—
192C4×Dic61—42no—
193C4×Dic61—42no—
194C4×Dic61C22×C412yes—
195C4×Dic61#4112yes—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418C22×C412yes—
202C23×C61—22no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61C22×D411yescharacteristic
210Q8×C61—22no—
211Q8×C61—22no—
212Q8×C61—41no—
213Q8×C61—41no—
214D4×C61—22no—
215D4×C61—22no—
216D4×C61—41no—
217D4×C61—41no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—22no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
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×C61AGL(1,ℤ/8)12yes—
233C2×C4×C61C23×C411yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61H(5,𝔽2)11yescharacteristic
238C2×C4×C612−1+411yescharacteristic
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—48no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61C4×D412yes—
263C2×Dic61C4×D412yes—
264C8×C61—41no—
265C8×C61#11714yescomplemented
266#92418—22no—
267#92418—41no—
268#22820—611nomaximal, Sylow
269C22×C61—42no—
270C22×C61—42no—
271C22×C61—42no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61—81no—
275C22×C61—81no—
276C22×C61C2×C4×D411yescharacteristic
277C22×C61#54711yescharacteristic
278C22×C61#55011yescharacteristic
279C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C61#57511yescharacteristic
281C22×C61#57711yescharacteristic
282C22×C61#57811yescharacteristic
283C4×C61—22no—
284C4×C61—41no—
285C4×C61—42no—
286C4×C61—81no—
287C4×C61—81no—
288C4×C61C2×#11711yescharacteristic
289C4×C61#52011yescharacteristic
290C4×C61#55511yescharacteristic
291C4×C61#55611yescharacteristic
292Dic61—22no—
293Dic61—22no—
294Dic61—41no—
295Dic61—41no—
296Dic61—42no—
297Dic61—42no—
298Dic61—81no—
299Dic61—81no—
300C2×#440—611no—
301#2356—611no—
302#2354—611no—
303#1973—611no—
304#1968D6111yescharacteristic
305#1928—611no—
306#1927—611no—
307#1926—611no—
308#1925—611no—
309#1847—611no—
310#1804—611no—
311#1803—611no—
312#1776—611no—
313C4×#107—611no—
314#1771—611no—
315C2×C61—82no—
316C2×C61#235411yescharacteristic
317C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
318C2×C61#294711yescharacteristic
319C2×C61#334311yescharacteristic
320C2×C61#334811yescharacteristic
321C2×C61#339711yescharacteristic
322C2×C61#340211yescharacteristic
323C22×SD16C2×D6111yescharacteristic
324C2×C4×Q8—611no—
325C2×C4×D4—611no—
326C2×#117—1221no—
327#440—618no—
328C2×#107—611no—
329C2×#107—611no—
330C2×#107—611no—
331C2×#106—611no—
332C2×#106C2×D6111yescharacteristic
333C2×C4⋊C8—611no—
334C2×#103—611no—
335C2×#103—611no—
336C2×#103C2×D6111yescharacteristic
337C2×#102—611no—
338C2×#102—611no—
339C2×#102C2×D6111yescharacteristic
340C2×#97—611no—
341C2×#97—611no—
342C2×C4×C8C2×D6111yescharacteristic
343#391—611no—
344#391C2×D6111yescharacteristic
345#390—611no—
346#390C2×D6111yescharacteristic
347#388—1221no—
348#388—1221no—
349#387—1221no—
350#387—1221no—
351#384—611no—
352#383—1221no—
353#382—611no—
354#380—1221no—
355C4×C4⋊C4—611no—
356C4×C4⋊C4—611no—
357C4×C4⋊C4—611no—
358C4×C4⋊C4—611no—
359C4×C4⋊C4—611no—
360#376—1221no—
361#340—611no—
362#340—611no—
363#340—611no—
364#340—611no—
365#336—611no—
366#336—611no—
367C61#2282011yescharacteristic, complemented, Sylow
368C22×Q8C22×D6111yescharacteristic
369C22×D4C22×D6111yescharacteristic
370C23×C4—1221no—
371C2×SD16—22no—
372C2×SD16—22no—
373C2×SD16C4×D6114yes—
374C22×C8—611no—
375C22×C8—611no—
376C22×C8C22×D6111yescharacteristic
377C22×C8C22×D6111yescharacteristic
378C4×Q8—614no—
379C4×Q8—1222no—
380C4×D4—614no—
381C4×D4—1222no—
382#117—2444no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4C22×D6111yescharacteristic
393C2×C4⋊C4C22×D6111yescharacteristic
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—1221no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C42—611no—
400C2×C42—611no—
401C2×C42—1221no—
402C2×C42—1221no—
403C2×C42—1221no—
404C2×C42—1221no—
405C2×C42—1221no—
406C2×C42C22×D6111yescharacteristic
407#107—614no—
408#107—1222no—
409#107—2441no—
410#106—22no—
411#106—2441no—
412C4⋊C8—1222no—
413#103—41no—
414#103—614no—
415#103—2441no—
416#102—41no—
417#102—614no—
418#102—2441no—
419#97—614no—
420#97—1222no—
421C4×C8—22no—
422#95—21no—
423#95—21no—
424#95—611no—
425#95—611no—
426#95—1221no—
427#95—1221no—
428#95—1221no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434C24—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8C2×C4×D6112yes—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4C2×C4×D6112yes—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—1224no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—2442no—
460C22×C4—2442no—
461C22×C4D4×D6111yescharacteristic
462C22×C4D4×D6111yescharacteristic
463C22×C4C23×D6111yescharacteristic
464SD16—28no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—612no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8C2×C4×D6112yes—
475C2×C8#1175822412yes—
476C2×C8#1175822412yes—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—612no—
482C4⋊C4—612no—
483C4⋊C4—1224no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1222no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—4881no—
492C4⋊C4—2441no—
493C4⋊C4—2442no—
494C22⋊C4—42no—
495C22⋊C4—81no—
496C22⋊C4—1224no—
497C22⋊C4—4881no—
498C22⋊C4—2442no—
499C42—21no—
500C42—21no—
501C42—612no—
502C42—612no—
503C42—1224no—
504C42—1221no—
505C42—1221no—
506C42—2442no—
507C42—2442no—
508C42—2442no—
509C42—2442no—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23C2×D4×D6111yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—41no—
521Q8—41no—
522D4—22no—
523D4—22no—
524D4—41no—
525D4—41no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—612no—
541C2×C4—612no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—2448no—
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—2441no—
565C2×C4#1451980412yes—
566C2×C4C22×C4×D6111yescharacteristic
567C2×C4#41×D6111yescharacteristic
568C2×C4C2×#1175822411yescharacteristic
569C2×C4C2×#1175822411yescharacteristic
570C2×C4#1451982211yescharacteristic
571C2×C4#1451982411yescharacteristic
572C8—41no—
573C8—1222no—
574C8—2441no—
575C8#1451973314yescomplemented
576C22—42no—
577C22—42no—
578C22—42no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22C4×D4×D6111yescharacteristic
584C22#1684041211yescharacteristic
585C22#1684043211yescharacteristic
586C22C2×#1451980411yescharacteristic
587C22#1684055911yescharacteristic
588C22#1684056811yescharacteristic
589C22#1684057611yescharacteristic
590C4—22no—
591C4—42no—
592C4—41no—
593C4—81no—
594C4—81no—
595C4—1222no—
596C4—1222no—
597C4—4881no—
598C4—4881no—
599C4—2442no—
600C4—2442no—
601C4—2441no—
602C4—2441no—
603C4C2×#1451973311yescharacteristic
604C4#1684024111yescharacteristic
605C4#1684047611yescharacteristic
606C4#1684048111yescharacteristic
607C2—82no—
608C2#2144876711yescharacteristic
609C2C4×#1451980411yescharacteristic
610C2#2145465511yescharacteristic
611C2#2145826511yescharacteristic
612C2#2145834011yescharacteristic
613C2#2145884411yescharacteristic
614C2#2145890111yescharacteristic
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#22820
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. #21448783C2semidirect
5. #21446463C2semidirect
6. #21446450C2semidirect
9. #21445984C2semidirect
11. #21445822C2semidirect
12. #21445719C2semidirect
13. C4×#14519711C2semidirect
14. #21445693C2semidirect
19. #16839863C4semidirect
265. C8×C61#117semidirect
367. C61#22820semidirect
575. C8#14519733semidirect

Maximal subgroups

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

subgroupindex
C2×#168398632
#214487832
#214487672
#214467022
#214464632
#214464502
#214464492
#214464362
#214459842
#214458272
#214458222
#214457192
C4×#145197112
#214456932
C61×#19682
#2282061