Finite Group Databasebeta

← #29699888

Subgroups — #29699888

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×#16839864C211yesmaximal, characteristic
2#21448782C211yesmaximal, characteristic, complemented
3#21448766C211yesmaximal, characteristic
4#21446700C211yesmaximal, characteristic
5#21446673C211yesmaximal, characteristic
6#21446461C211yesmaximal, characteristic, complemented
7#21446454C211yesmaximal, characteristic, complemented
8#21446447C211yesmaximal, characteristic
9#21446440C211yesmaximal, characteristic
10#21445987C211yesmaximal, characteristic, complemented
11#21445830C211yesmaximal, characteristic
12#21445825C211yesmaximal, characteristic, complemented
13#21445716C211yesmaximal, characteristic, complemented
14C4×#14519711C211yesmaximal, characteristic, complemented
15C61×#1771C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19C2×#14519735—21no—
20#16839864C418yescomplemented
21C2×#14519711C2211yescharacteristic
22C2×#14519711C2211yescharacteristic
23C2×#14519711C2211yescharacteristic
24C2×#14519709C2211yescharacteristic
25C2×#14519709C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519699C2211yescharacteristic
30C2×#14519699C2211yescharacteristic
31C2×#14519699C2211yescharacteristic
32#16839668C2211yescharacteristic
33#16839668C2211yescharacteristic
34#16839664C2211yescharacteristic
35#16839664C2211yescharacteristic
36#16839654—21no—
37#16839654—21no—
38#16839643—21no—
39#16839643—21no—
40#16839632C2211yescharacteristic
41#16839629—21no—
42#16839628C2211yescharacteristic
43#16839619—21no—
44C4×C4⋊Dic61C2211yescharacteristic
45C4×C4⋊Dic61C2211yescharacteristic
46C4×C4⋊Dic61C2211yescharacteristic
47C4×C4⋊Dic61C2211yescharacteristic
48#16839607—21no—
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53C2×C4⋊C8×C61C2211yescharacteristic
54C2×#97×C61C2211yescharacteristic
55C2×#97×C61C2211yescharacteristic
56C2×C4×C8×C61C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C22×Dic122C2311yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C4×D61—21no—
63C2×#11758190—22no—
64C2×#11758190—22no—
65C2×#11758190C2×C414yes—
66C4×Dic122—22no—
67C4×Dic122C2×C414yes—
68C4×D244—22no—
69C4×D244C2×C414yes—
70#14519735—44no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90#14519711—22no—
91#14519711—41no—
92#14519711C2×C414yes—
93#14519709—22no—
94#14519709—41no—
95#14519702—41no—
96#14519702—41no—
97#14519702C2×C414yes—
98#14519699—41no—
99#14519699—41no—
100#14519699C2×C414yes—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111C22×C8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C2×C4⋊C4×C61D411yescharacteristic
116C2×C4⋊C4×C61D411yescharacteristic
117C2×C42×C61C2311yescharacteristic
118C2×C42×C61C2311yescharacteristic
119C2×C42×C61C2311yescharacteristic
120C4⋊C8×C61—22no—
121#97×C61—22no—
122#97×C61C2×C414yes—
123C4×C8×C61—22no—
124#95×C61D411yescharacteristic
125#95×C61D411yescharacteristic
126C23×D61—21no—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2411yescharacteristic
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244C22×C412yes—
147C2×C4×D61—24no—
148C2×C4×D61—42no—
149C2×C4×D61—42no—
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122C22×C412yes—
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×C61C22×C412yes—
165C2×C8×C61C22×C412yes—
166C2×C8×C61#4112yes—
167C2×C8×C61#4112yes—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—41no—
170C42×C61—21no—
171C42×C61—21no—
172C42×C61—21no—
173C42×C61—21no—
174C42×C61C22×C412yes—
175C42×C61#4112yes—
176#11758198—24no—
177#11758198—42no—
178#11758198—42no—
179#11758198—81no—
180#11758198—81no—
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—42no—
192C4⋊Dic61—81no—
193C4⋊Dic61C22×C412yes—
194C4⋊Dic61C22×C412yes—
195#11758196—24no—
196C4×Dic61—24no—
197C4×Dic61—42no—
198C4×Dic61—42no—
199C4×Dic61—42no—
200C4×Dic61—42no—
201#11758190—28no—
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×C61—21no—
225C2×C4×C61C4×D412yes—
226C2×C4×C61C4×D412yes—
227C2×C4×C61AGL(1,ℤ/8)12yes—
228C2×C4×C61C23×C411yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61H(5,𝔽2)11yescharacteristic
233C2×C4×C612−1+411yescharacteristic
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—48no—
260Dic122—22no—
261Dic122—22no—
262Dic122—41no—
263Dic122—41no—
264C8×C61—22no—
265C8×C61—41no—
266C8×C61—41no—
267C8×C61#11714yescomplemented
268#22820—611nomaximal, Sylow
269C22×C61C2×C4×D411yescharacteristic
270C22×C61#54711yescharacteristic
271C22×C61#55011yescharacteristic
272C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
273C22×C61#57511yescharacteristic
274C22×C61#57711yescharacteristic
275C22×C61#57811yescharacteristic
276C2×D61—42no—
277C2×D61—42no—
278C2×D61—42no—
279C2×D61—81no—
280C2×D61—81no—
281C2×D61—81no—
282C2×D61—81no—
283C4×C61—22no—
284C4×C61—22no—
285C4×C61—22no—
286C4×C61—41no—
287C4×C61—41no—
288C4×C61—41no—
289C4×C61C2×#11711yescharacteristic
290C4×C61#52011yescharacteristic
291C4×C61#55511yescharacteristic
292C4×C61#55611yescharacteristic
293Dic61—42no—
294Dic61—42no—
295Dic61—42no—
296Dic61—81no—
297Dic61—81no—
298Dic61—81no—
299Dic61—81no—
300C2×#440—611no—
301#2356—611no—
302#2354—611no—
303#1973—611no—
304#1968—611no—
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#1771D6111yescharacteristic, complemented
315C2×C61#235411yescharacteristic
316C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
317C2×C61#294711yescharacteristic
318C2×C61#334311yescharacteristic
319C2×C61#334811yescharacteristic
320C2×C61#339711yescharacteristic
321C2×C61#340211yescharacteristic
322D61—82no—
323C22×SD16—611no—
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×#106—611no—
333C2×C4⋊C8C2×D6111yescharacteristic
334C2×#103—611no—
335C2×#103—611no—
336C2×#103—611no—
337C2×#102—611no—
338C2×#102—611no—
339C2×#102—611no—
340C2×#97C2×D6111yescharacteristic
341C2×#97C2×D6111yescharacteristic
342C2×C4×C8C2×D6111yescharacteristic
343#391—611no—
344#391—611no—
345#390—611no—
346#390—611no—
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⋊C4C2×D6111yescharacteristic
360#376—1221no—
361#340—611no—
362#340—611no—
363#340—611no—
364#340—611no—
365#336C2×D6111yescharacteristic
366#336C2×D6111yescharacteristic
367C61#2282011yescharacteristic, complemented, Sylow
368C22×Q8—611no—
369C22×D4—611no—
370C23×C4—1221no—
371C2×SD16—614no—
372C2×SD16—1222no—
373C2×SD16—1222no—
374C22×C8C22×D6111yescharacteristic
375C22×C8C22×D6111yescharacteristic
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⋊C4D24411yescharacteristic
393C2×C4⋊C4D24411yescharacteristic
394C2×C22⋊C4—1221no—
395C2×C22⋊C4—1221no—
396C2×C22⋊C4—1221no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C42—1221no—
400C2×C42—1221no—
401C2×C42—1221no—
402C2×C42—1221no—
403C2×C42—1221no—
404C2×C42C22×D6111yescharacteristic
405C2×C42C22×D6111yescharacteristic
406C2×C42C22×D6111yescharacteristic
407#107—614no—
408#107—1222no—
409#107—2441no—
410#106—1222no—
411#106—2441no—
412C4⋊C8—22no—
413#103—614no—
414#103—2441no—
415#103—2441no—
416#102—614no—
417#102—2441no—
418#102—2441no—
419#97—22no—
420#97C4×D6114yes—
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#95—1221no—
431#95—1221no—
432#95D24411yescharacteristic
433#95D24411yescharacteristic
434C24—1221no—
435C2×Q8—612no—
436C2×Q8—1221no—
437C2×Q8—1221no—
438C2×Q8—1221no—
439C2×Q8—1221no—
440C2×Q8—1221no—
441C2×D4—612no—
442C2×D4—1221no—
443C2×D4—1221no—
444C2×D4—1221no—
445C2×D4—1221no—
446C2×D4—1221no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1224no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—2442no—
456C22×C4—2442no—
457C22×C4C2×D24411yescharacteristic
458C22×C4C2×D24411yescharacteristic
459C22×C4C2×D24411yescharacteristic
460C22×C4C2×D24411yescharacteristic
461C22×C4C2×D24411yescharacteristic
462C22×C4C2×D24411yescharacteristic
463C22×C4C23×D6111yescharacteristic
464SD16—1228no—
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×C8C2×C4×D6112yes—
474C2×C8C2×C4×D6112yes—
475C2×C8#1175822412yes—
476C2×C8#1175822612yes—
477C4⋊C4—22no—
478C4⋊C4—41no—
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—2442no—
494C22⋊C4—1224no—
495C22⋊C4—4881no—
496C22⋊C4—4881no—
497C22⋊C4—2442no—
498C22⋊C4—2442no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—1224no—
504C42—2442no—
505C42—2442no—
506C42—2442no—
507C42—2442no—
508C42C2×C4×D6112yes—
509C42#1175822212yes—
510C23—1222no—
511C23—2441no—
512C23—2441no—
513C23—2441no—
514C23—2441no—
515C23—2441no—
516C23—2441no—
517C23C22×D24411yescharacteristic
518Q8—1222no—
519Q8—1222no—
520Q8—2441no—
521Q8—2441no—
522D4—1222no—
523D4—1222no—
524D4—2441no—
525D4—2441no—
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—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—2448no—
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×C4C4×D24412yes—
564C2×C4C4×D24412yes—
565C2×C4#1451980512yes—
566C2×C4C22×C4×D6111yescharacteristic
567C2×C4C2×#1175822411yescharacteristic
568C2×C4C2×#1175822211yescharacteristic
569C2×C4C2×#1175822611yescharacteristic
570C2×C4#1451982311yescharacteristic
571C2×C4#1451982411yescharacteristic
572C8—22no—
573C8—41no—
574C8—41no—
575C8#1451973514yescomplemented
576C22—4881no—
577C22—4881no—
578C22—4881no—
579C22—4881no—
580C22—2442no—
581C22—2442no—
582C22—2442no—
583C22C2×C4×D24411yescharacteristic
584C22#1684041611yescharacteristic
585C22#1684043411yescharacteristic
586C22C2×#1451980511yescharacteristic
587C22#1684056211yescharacteristic
588C22#1684057411yescharacteristic
589C22#1684057911yescharacteristic
590C4—22no—
591C4—22no—
592C4—22no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—4881no—
597C4—4881no—
598C4—4881no—
599C4—4881no—
600C4—2442no—
601C4—2442no—
602C4—2442no—
603C4C2×#1451973511yescharacteristic
604C4#1684024211yescharacteristic
605C4#1684047711yescharacteristic
606C4#1684048511yescharacteristic
607C2—4882no—
608C2#2144876611yescharacteristic
609C2C4×#1451980511yescharacteristic
610C2#2145466411yescharacteristic
611C2#2145826111yescharacteristic
612C2#2145833611yescharacteristic
613C2#2145884111yescharacteristic
614C2#2145890011yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1771C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1771D61
hypocenterC61#22820
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1771

Complements

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

NH ≅ G/Ntype
2. #21448782C2semidirect
6. #21446461C2semidirect
7. #21446454C2semidirect
10. #21445987C2semidirect
12. #21445825C2semidirect
13. #21445716C2semidirect
14. C4×#14519711C2semidirect
15. C61×#1771C2semidirect
20. #16839864C4semidirect
267. C8×C61#117semidirect
314. #1771D61semidirect
367. C61#22820semidirect
575. C8#14519735semidirect

Maximal subgroups

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

subgroupindex
C2×#168398642
#214487822
#214487662
#214467002
#214466732
#214464612
#214464542
#214464472
#214464402
#214459872
#214458302
#214458252
#214457162
C4×#145197112
C61×#17712
#2282061