Finite Group Databasebeta

← #29699870

Subgroups — #29699870

Order 15616.

810 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839862C211yesmaximal, characteristic
2#21448792C211yesmaximal, characteristic
3#21448778C211yesmaximal, characteristic, complemented
4#21446701C211yesmaximal, characteristic
5#21446674C211yesmaximal, characteristic
6#21446458C211yesmaximal, characteristic, complemented
7#21446451C211yesmaximal, characteristic, complemented
8#21446449C211yesmaximal, characteristic
9#21446442C211yesmaximal, characteristic
10#21445985C211yesmaximal, characteristic, complemented
11#21445835C211yesmaximal, characteristic, complemented
12#21445715C211yesmaximal, characteristic, complemented
13#21445711C211yesmaximal, characteristic, complemented
14#21445696C211yesmaximal, characteristic, complemented
15C61×#1802C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18#16839862C418yescomplemented
19C2×#14519710C2211yescharacteristic
20C2×#14519710C2211yescharacteristic
21C2×#14519710C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519688C2211yescharacteristic
29C2×#14519687C2211yescharacteristic
30C2×C8×Dic61C2211yescharacteristic
31#16839667C2211yescharacteristic
32#16839667C2211yescharacteristic
33#16839665C2211yescharacteristic
34#16839665C2211yescharacteristic
35#16839655—21no—
36#16839655—21no—
37#16839643—21no—
38#16839643—21no—
39#16839628C2211yescharacteristic
40#16839627—21no—
41#16839625—21no—
42C4⋊C4×Dic61—21no—
43C4⋊C4×Dic61C2211yescharacteristic
44C4⋊C4×Dic61C2211yescharacteristic
45C4⋊C4×Dic61C2211yescharacteristic
46C4⋊C4×Dic61C2211yescharacteristic
47C4×C4⋊Dic61C2211yescharacteristic
48C4×C22⋊Dic61—21no—
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×C4×D4×C61C2211yescharacteristic
54C2×C4⋊C8×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C61×#384C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C22×Dic122C2311yescharacteristic
61C2×#11758200—22no—
62C2×#11758200—22no—
63C2×#11758200C2×C414yes—
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66C4×Dic122—22no—
67C4×Dic122C2×C414yes—
68C2×C4⋊Dic61D411yescharacteristic
69C2×C4⋊Dic61D411yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×C22⋊Dic61—21no—
75C2×C22⋊Dic61—21no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61C2311yescharacteristic
84C2×C4×Dic61C2311yescharacteristic
85#14519710—22no—
86#14519710—41no—
87#14519710C2×C414yes—
88#14519708—22no—
89#14519708—41no—
90#14519702—41no—
91#14519702—41no—
92#14519700C2×C414yes—
93#14519698C2×C414yes—
94#14519688C2×C414yes—
95#14519687—22no—
96C8×Dic61—22no—
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—
107#14519684D411yescharacteristic
108#14519684D411yescharacteristic
109C22×D4×C61C2311yescharacteristic
110C23×C4×C61—21no—
111C22×C8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C4×D4×C61—22no—
114C4×D4×C61C2×C414yes—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C22⋊C4×C61—21no—
121C2×C42×C61C2311yescharacteristic
122C4⋊C8×C61—22no—
123#102×C61—41no—
124#102×C61—41no—
125C24×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61C22×C412yes—
132C22×C4×C61—21no—
133C22×C4×C61—24no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—81no—
137C22×C4×C61—81no—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2411yescharacteristic
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C2×Dic122—21no—
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122C22×C412yes—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—42no—
167C4⋊C4×C61—81no—
168C4⋊C4×C61C22×C412yes—
169C22⋊C4×C61—24no—
170C42×C61C22×C412yes—
171C42×C61#4112yes—
172C22⋊Dic61—42no—
173C22⋊Dic61—42no—
174C22⋊Dic61—81no—
175C22⋊Dic61—81no—
176#11758200—28no—
177C4⋊Dic61—21no—
178C4⋊Dic61—22no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—41no—
184C4⋊Dic61C22×C412yes—
185#11758196—24no—
186C4×Dic61—24no—
187C4×Dic61—21no—
188C4×Dic61—21no—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61—81no—
192C4×Dic61—81no—
193C4×Dic61—81no—
194C4×Dic61—81no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418C22×C412yes—
198C2×#92418C22×C412yes—
199C2×#92418#4112yes—
200C2×#92418#4112yes—
201C23×C61—22no—
202C23×C61—41no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61C22×D411yescharacteristic
209D4×C61—22no—
210D4×C61—22no—
211D4×C61—41no—
212D4×C61—41no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—22no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—84no—
227C2×C4×C61C4×D412yes—
228C2×C4×C61C4×D412yes—
229C2×C4×C61C23×C411yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61H(5,𝔽2)11yescharacteristic
234C2×C4×C612−1+411yescharacteristic
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
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#13712yes—
259Dic122—22no—
260Dic122—22no—
261Dic122—41no—
262Dic122—41no—
263C8×C61—41no—
264C8×C61—41no—
265#92418—22no—
266#92418#11714yes—
267#22819—611nomaximal, Sylow
268C22×C61—42no—
269C22×C61—42no—
270C22×C61—42no—
271C22×C61—81no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61—81no—
275C22×C61C2×C4×D411yescharacteristic
276C22×C61#54711yescharacteristic
277C22×C61#55011yescharacteristic
278C22×C61C2×#13711yescharacteristic
279C22×C61#57511yescharacteristic
280C22×C61#57611yescharacteristic
281C22×C61#57711yescharacteristic
282C4×C61—22no—
283C4×C61—22no—
284C4×C61—42no—
285C4×C61—81no—
286C4×C61—81no—
287C4×C61C2×#11711yescharacteristic
288C4×C61#52011yescharacteristic
289C4×C61#55511yescharacteristic
290C4×C61#55611yescharacteristic
291Dic61—22no—
292Dic61—41no—
293Dic61—41no—
294Dic61—41no—
295Dic61—42no—
296Dic61—42no—
297Dic61—81no—
298Dic61—81no—
299C2×#440—611no—
300#2357—611no—
301#2355—611no—
302#1973—611no—
303#1968—611no—
304#1928—611no—
305#1927—611no—
306#1926—611no—
307#1925—611no—
308#1847—611no—
309#1805—611no—
310#1802D6111yescharacteristic
311#1776—611no—
312C4×#107—611no—
313#1771—611no—
314C2×C61—82no—
315C2×C61#235411yescharacteristic
316C2×C61C4×#13711yescharacteristic
317C2×C61#294611yescharacteristic
318C2×C61#334611yescharacteristic
319C2×C61#335011yescharacteristic
320C2×C61#340011yescharacteristic
321C2×C61#340311yescharacteristic
322C22×SD16—611no—
323C2×C4×Q8—611no—
324C2×C4×D4C2×D6111yescharacteristic
325#440—618no—
326C2×#107—611no—
327C2×#107—611no—
328C2×#107—611no—
329C2×#106—611no—
330C2×#106—611no—
331C2×C4⋊C8C2×D6111yescharacteristic
332C2×#103—611no—
333C2×#103—611no—
334C2×#103—611no—
335C2×#102—611no—
336C2×#102C2×D6111yescharacteristic
337C2×#102C2×D6111yescharacteristic
338C2×#97—611no—
339C2×#97—611no—
340C2×C4×C8—611no—
341#391—611no—
342#391—611no—
343#390—611no—
344#390—611no—
345#388—1221no—
346#388—1221no—
347#387—1221no—
348#387—1221no—
349#384C2×D6111yescharacteristic
350#382—611no—
351#382—1221no—
352#381—1221no—
353C4×C4⋊C4—611no—
354C4×C4⋊C4—611no—
355C4×C4⋊C4—611no—
356C4×C4⋊C4—611no—
357C4×C4⋊C4—611no—
358C4×C4⋊C4—1221no—
359C4×C22⋊C4—1221no—
360#340—611no—
361#340—611no—
362#340C2×D6111yescharacteristic
363#340C2×D6111yescharacteristic
364#336—611no—
365#336—611no—
366C61#2281911yescharacteristic, complemented, Sylow
367C22×Q8—611no—
368C22×D4C22×D6111yescharacteristic
369C23×C4—21no—
370C2×SD16—614no—
371C2×SD16—1222no—
372C2×SD16—1222no—
373C22×C8—611no—
374C22×C8—611no—
375C22×C8C22×D6111yescharacteristic
376C22×C8C22×D6111yescharacteristic
377C4×Q8—614no—
378C4×Q8—1222no—
379C4×D4—22no—
380C4×D4C4×D6114yes—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4C22×D6111yescharacteristic
390C2×C4⋊C4C22×D6111yescharacteristic
391C2×C4⋊C4C22×D6111yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—1221no—
394C2×C22⋊C4—1221no—
395C2×C22⋊C4—1221no—
396C2×C22⋊C4—1221no—
397C2×C42—611no—
398C2×C42—611no—
399C2×C42—1221no—
400C2×C42—1221no—
401C2×C42—1221no—
402C2×C42—1221no—
403C2×C42—1221no—
404C2×C42C22×D6111yescharacteristic
405#107—614no—
406#107—1222no—
407#107—2441no—
408#106—1222no—
409#106—2441no—
410C4⋊C8—22no—
411#103—614no—
412#103—2441no—
413#103—2441no—
414#102—41no—
415#102—41no—
416#102—614no—
417#97—614no—
418#97—1222no—
419C4×C8—1222no—
420#95—611no—
421#95—611no—
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—
432C24—21no—
433C2×Q8—612no—
434C2×Q8—1221no—
435C2×Q8—1221no—
436C2×Q8—1221no—
437C2×Q8—1221no—
438C2×Q8—1221no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4C2×C4×D6112yes—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—81no—
450C22×C4—81no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4D4×D6111yescharacteristic
460C22×C4D4×D6111yescharacteristic
461C22×C4C23×D6111yescharacteristic
462SD16—1228no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—612no—
470C2×C8—612no—
471C2×C8—612no—
472C2×C8—612no—
473C2×C8—1221no—
474C2×C8—1221no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—42no—
481C4⋊C4—81no—
482C4⋊C4—612no—
483C4⋊C4—1222no—
484C4⋊C4—1221no—
485C4⋊C4—1224no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—2441no—
491C4⋊C4C2×C4×D6112yes—
492C22⋊C4—24no—
493C22⋊C4—4881no—
494C22⋊C4—4881no—
495C22⋊C4—2442no—
496C22⋊C4—2442no—
497C42—1221no—
498C42—1221no—
499C42—1221no—
500C42—1221no—
501C42—1224no—
502C42—4881no—
503C42—4881no—
504C42—4881no—
505C42—4881no—
506C42C2×C4×D6112yes—
507C42#1175822412yes—
508C23—22no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23C2×D4×D6111yescharacteristic
516Q8—1222no—
517Q8—1222no—
518Q8—2441no—
519Q8—2441no—
520D4—22no—
521D4—22no—
522D4—41no—
523D4—41no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—84no—
538C2×C4—612no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1221no—
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#1451973712yes—
563C2×C4#1451973712yes—
564C2×C4C22×C4×D6111yescharacteristic
565C2×C4#41×D6111yescharacteristic
566C2×C4#41×D6111yescharacteristic
567C2×C4C2×#1175822411yescharacteristic
568C2×C4#1451982211yescharacteristic
569C2×C4#1451982511yescharacteristic
570C8—41no—
571C8—41no—
572C8—614no—
573C8—1222no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22C2×#1451973711yescharacteristic
582C22#1684041411yescharacteristic
583C22#1684043011yescharacteristic
584C22#137×D6111yescharacteristic
585C22#1684055811yescharacteristic
586C22#1684056311yescharacteristic
587C22#1684056811yescharacteristic
588C4—22no—
589C4—22no—
590C4—42no—
591C4—81no—
592C4—81no—
593C4—1222no—
594C4—4881no—
595C4—4881no—
596C4—2442no—
597C4—2442no—
598C4—2441no—
599C4—2441no—
600C4—2441no—
601C4#117×D6111yescharacteristic
602C4#1684024311yescharacteristic
603C4#1684047611yescharacteristic
604C4#1684048311yescharacteristic
605C2—82no—
606C2#2144876311yescharacteristic
607C2#2145464311yescharacteristic
608C2#2145464511yescharacteristic
609C2#2145830111yescharacteristic
610C2#2145837111yescharacteristic
611C2#2145886711yescharacteristic
612C2#2145890211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1802C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenter#1802D61
hypocenterC61#22819
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
3. #21448778C2semidirect
6. #21446458C2semidirect
7. #21446451C2semidirect
10. #21445985C2semidirect
11. #21445835C2semidirect
12. #21445715C2semidirect
13. #21445711C2semidirect
14. #21445696C2semidirect
18. #16839862C4semidirect
366. C61#22819semidirect

Maximal subgroups

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

subgroupindex
C2×#168398622
#214487922
#214487782
#214467012
#214466742
#214464582
#214464512
#214464492
#214464422
#214459852
#214458352
#214457152
#214457112
#214456962
C61×#18022
#2281961