Finite Group Databasebeta

← #29687249

Subgroups — #29687249

Order 15616.

876 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840100C211yesmaximal, characteristic
2C4×#14519736C211yesmaximal, characteristic
3#21446457C211yesmaximal, characteristic, complemented
4#21446450C211yesmaximal, characteristic, complemented
5#21446408C211yesmaximal, characteristic, complemented
6#21446401C211yesmaximal, characteristic, complemented
7#21446132C211yesmaximal, characteristic
8#21446016C211yesmaximal, characteristic
9#21445911C211yesmaximal, characteristic
10#21445717C211yesmaximal, characteristic, complemented
11#21445717C211yesmaximal, characteristic, complemented
12#21445649C211yesmaximal, characteristic, complemented
13#21445649C211yesmaximal, characteristic, complemented
14#21445581C211yesmaximal, characteristic
15C4×C22⋊Q8×C61C211yesmaximal, characteristic
16C2×#14519736C2211yescharacteristic
17C2×#14519736C2211yescharacteristic
18#16840100C418yescomplemented
19C2×#14519710C2211yescharacteristic
20C2×#14519710C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×C22⋊#92418C2211yescharacteristic
28C2×C22⋊#92418C2211yescharacteristic
29C2×#14519688C2211yescharacteristic
30C2×#14519688C2211yescharacteristic
31C4×C4⋊Dic61C2211yescharacteristic
32C4×C4⋊Dic61C2211yescharacteristic
33C4×C4⋊Dic61C2211yescharacteristic
34C4×C4⋊Dic61C2211yescharacteristic
35C4×C22⋊Dic61—21no—
36C4×C22⋊Dic61—21no—
37#16839608—21no—
38C42×Dic61—21no—
39#16839524C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839517C2211yescharacteristic
48#16839517C2211yescharacteristic
49C2×C22⋊Q8×C61C2211yescharacteristic
50C2×C4×Q8×C61C2211yescharacteristic
51C22×C42×C61C2211yescharacteristic
52C61×#391C2211yescharacteristic
53C61×#389C2211yescharacteristic
54C61×#387—21no—
55C61×#385—21no—
56C61×#380C2211yescharacteristic
57C4×C4⋊C4×C61—21no—
58C4×C4⋊C4×C61C2211yescharacteristic
59C4×C22⋊C4×C61—21no—
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64#14519736—22no—
65#14519736—22no—
66#14519736—22no—
67#14519736C2×C414yes—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C22⋊Dic61—21no—
73C2×C22⋊Dic61—21no—
74C2×C4×Dic61—21no—
75C2×C4×Dic61—21no—
76C2×C4×Dic61—21no—
77C2×C4×Dic61—21no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61—21no—
80#14519710—22no—
81#14519710C2×C414yescomplemented
82#14519708—22no—
83#14519708C2×C414yescomplemented
84#14519701—22no—
85#14519701—22no—
86#14519701C2×C414yescomplemented
87#14519701C2×C414yescomplemented
88C22⋊#92418—22no—
89C22⋊#92418C2×C414yes—
90#14519688—22no—
91#14519688—22no—
92#14519684—21no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96C22×Q8×C61C2311yescharacteristic
97C23×C4×C61D411yescharacteristic
98C23×C4×C61D411yescharacteristic
99C23×C4×C61C2311yescharacteristic
100C22⋊Q8×C61—22no—
101C22⋊Q8×C61C2×C414yes—
102C4×Q8×C61—22no—
103C4×Q8×C61—22no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61—21no—
112C2×C42×C61—22no—
113C2×C42×C61—22no—
114C2×C42×C61—21no—
115C2×C42×C61—21no—
116C2×C42×C61D411yescharacteristic
117C2×C42×C61D411yescharacteristic
118C2×C42×C61C2311yescharacteristic
119C2×C42×C61C2311yescharacteristic
120#95×C61—21no—
121#95×C61—21no—
122#95×C61—21no—
123#95×C61—21no—
124#95×C61—21no—
125C24×C61C2×D411yescharacteristic
126C2×Q8×C61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61C22×C412yes—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—41no—
144C22×C4×C61—41no—
145C22×C4×C61—41no—
146C22×C4×C61C22×C412yes—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61#4112yes—
153C22×C4×C61C2411yescharacteristic
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C4⋊C4×C61—24no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—42no—
165C4⋊C4×C61—42no—
166C4⋊C4×C61C22×C412yes—
167C22⋊C4×C61—24no—
168C22⋊C4×C61—42no—
169C42×C61—22no—
170C42×C61—22no—
171C42×C61—21no—
172C42×C61—21no—
173C42×C61—42no—
174C42×C61—42no—
175C42×C61#4112yes—
176C42×C61#4112yes—
177C22⋊Dic61—24no—
178C22⋊Dic61—24no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61C22×C412yes—
184C4⋊Dic61C22×C412yes—
185C4⋊Dic61#4112yes—
186C4⋊Dic61#4112yes—
187C4×Dic61—24no—
188C4×Dic61—24no—
189C4×Dic61—42no—
190C4×Dic61—42no—
191C4×Dic61—42no—
192C4×Dic61—42no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418C22×C412yes—
204C2×#92418C22×C412yes—
205C23×C61—21no—
206C23×C61—21no—
207C23×C61—21no—
208C23×C61—21no—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61C4×D412yes—
212C23×C61C22×D411yescharacteristic
213Q8×C61—22no—
214Q8×C61—22no—
215Q8×C61—41no—
216Q8×C61—41no—
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×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—44no—
235C2×C4×C61—44no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61C4×D412yes—
247C2×C4×C61#12312yes—
248C2×C4×C61#12312yes—
249C2×C4×C61#13712yes—
250C2×C4×C61C23×C411yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268#92418—22no—
269#92418—22no—
270#92418—41no—
271#92418—41no—
272#21583—611nomaximal, Sylow
273C22×C61—22no—
274C22×C61—22no—
275C22×C61—22no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61C2×C4×D411yescharacteristic
281C22×C61C2×#12311yescharacteristic
282C22×C61#52511yescharacteristic
283C22×C61C2×#13711yescharacteristic
284C22×C61#57511yescharacteristic
285C22×C61#57511yescharacteristic
286C22×C61#57511yescharacteristic
287C4×C61—22no—
288C4×C61—22no—
289C4×C61—22no—
290C4×C61—22no—
291C4×C61—41no—
292C4×C61—42no—
293C4×C61—42no—
294C4×C61—81no—
295C4×C61—81no—
296C4×C61C4×#4111yescharacteristic
297C4×C61C4×#4111yescharacteristic
298C4×C61#52911yescharacteristic
299C4×C61#52911yescharacteristic
300Dic61—42no—
301Dic61—42no—
302Dic61—42no—
303Dic61—42no—
304C2×#485—611no—
305C4×C22⋊Q8D6111yescharacteristic
306C4×#117—611no—
307#1928—611no—
308#1927—611no—
309#1921—611no—
310#1920—611no—
311#1873—611no—
312#1854—611no—
313#1825—611no—
314#1776—611no—
315#1776—611no—
316#1763—611no—
317#1763—611no—
318#1749—611no—
319C2×C61—42no—
320C2×C61C4×#12311yescharacteristic
321C2×C61#294311yescharacteristic
322C2×C61C4×#13711yescharacteristic
323C2×C61#309311yescharacteristic
324C2×C61#309511yescharacteristic
325C2×C61#311111yescharacteristic
326C2×C61#311611yescharacteristic
327C2×C22⋊Q8C2×D6111yescharacteristic
328C2×C4×Q8C2×D6111yescharacteristic
329C2×#117—611no—
330C2×#117—611no—
331C22×C42C2×D6111yescharacteristic
332#485—618no—
333C2×#107—611no—
334C2×#107—611no—
335C2×#106—611no—
336C2×#106—611no—
337C2×#103—611no—
338C2×#103—611no—
339C2×#103—611no—
340C2×#103—611no—
341C2×C22⋊C8—611no—
342C2×C22⋊C8—611no—
343C2×#97—611no—
344C2×#97—611no—
345#391C2×D6111yescharacteristic
346#389C2×D6111yescharacteristic
347#387—21no—
348#385—21no—
349#380C2×D6111yescharacteristic
350C4×C4⋊C4—21no—
351C4×C4⋊C4—611no—
352C4×C4⋊C4—611no—
353C4×C4⋊C4—611no—
354C4×C4⋊C4—611no—
355C4×C4⋊C4C2×D6111yescharacteristic
356C4×C22⋊C4—21no—
357C4×C22⋊C4—1221no—
358C4×C22⋊C4—1221no—
359#376—1221no—
360C43—1221no—
361#340—611no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367#340—611no—
368#340—611no—
369#336—611no—
370#336—611no—
371C61#2158311yescharacteristic, complemented, Sylow
372C22×Q8C22×D6111yescharacteristic
373C23×C4C22⋊D6111yescharacteristic
374C23×C4C22⋊D6111yescharacteristic
375C23×C4C22×D6111yescharacteristic
376C22×C8—611no—
377C22×C8—611no—
378C22×C8—611no—
379C22×C8—611no—
380C22⋊Q8—22no—
381C22⋊Q8C4×D6114yes—
382C4×Q8—22no—
383C4×Q8—22no—
384#117—614no—
385#117—1222no—
386#117—1222no—
387#117—1222no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—611no—
392C2×C4⋊C4—611no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4C22×D6111yescharacteristic
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×C42—22no—
403C2×C42—22no—
404C2×C42—21no—
405C2×C42—21no—
406C2×C42—1221no—
407C2×C42—1221no—
408C2×C42—1221no—
409C2×C42—1221no—
410C2×C42—1221no—
411C2×C42—1221no—
412C2×C42C22⋊D6111yescharacteristic
413C2×C42C22⋊D6111yescharacteristic
414C2×C42C22×D6111yescharacteristic
415C2×C42C22×D6111yescharacteristic
416#107—614no—
417#107—1222no—
418#106—614no—
419#106—1222no—
420#103—614no—
421#103—614no—
422#103—1222no—
423#103—1222no—
424C22⋊C8—614no—
425C22⋊C8—1222no—
426#97—1222no—
427#97—1222no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437C24C2×C22⋊D6111yescharacteristic
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8C2×C4×D6112yes—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4C2×C4×D6112yes—
463C22×C4#1175822212yes—
464C22×C4C2×C22⋊D6111yescharacteristic
465C22×C4C2×C22⋊D6111yescharacteristic
466C22×C4C2×C22⋊D6111yescharacteristic
467C22×C4C2×C22⋊D6111yescharacteristic
468C22×C4C2×C22⋊D6111yescharacteristic
469C22×C4C23×D6111yescharacteristic
470C2×C8—612no—
471C2×C8—612no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1221no—
482C4⋊C4—21no—
483C4⋊C4—24no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—612no—
491C4⋊C4—612no—
492C4⋊C4—612no—
493C4⋊C4—612no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4C2×C4×D6112yes—
499C22⋊C4—24no—
500C22⋊C4—42no—
501C22⋊C4—1224no—
502C22⋊C4—1224no—
503C42—21no—
504C42—21no—
505C42—22no—
506C42—22no—
507C42—42no—
508C42—42no—
509C42—1224no—
510C42—1224no—
511C42—2442no—
512C42—2442no—
513C42—2442no—
514C42—2442no—
515C42#1175822412yes—
516C42#1175822412yes—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23C4×C22⋊D6112yes—
524C23C22×C22⋊D6111yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—41no—
528Q8—41no—
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—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4C4×C22⋊D6112yes—
571C2×C4#1451975812yes—
572C2×C4#1451975812yes—
573C2×C4#1451980812yes—
574C2×C4C22×C4×D6111yescharacteristic
575C2×C4#41×D6111yescharacteristic
576C2×C4#41×D6111yescharacteristic
577C2×C4C2×#1175822411yescharacteristic
578C2×C4C2×#1175822411yescharacteristic
579C2×C4C2×#1175822211yescharacteristic
580C8—1222no—
581C8—1222no—
582C8—2441no—
583C8—2441no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22C2×C4×C22⋊D6111yescharacteristic
592C22C2×#1451975811yescharacteristic
593C22#1684026911yescharacteristic
594C22C2×#1451980811yescharacteristic
595C22#1684056011yescharacteristic
596C22#1684056011yescharacteristic
597C22#1684056011yescharacteristic
598C4—22no—
599C4—22no—
600C4—22no—
601C4—22no—
602C4—42no—
603C4—41no—
604C4—42no—
605C4—81no—
606C4—81no—
607C4—2442no—
608C4—2442no—
609C4—2442no—
610C4—2442no—
611C4C4×#1175822411yescharacteristic
612C4C4×#1175822411yescharacteristic
613C4#1684029211yescharacteristic
614C4#1684029211yescharacteristic
615C2—42no—
616C2C4×#1451975811yescharacteristic
617C2#2145461711yescharacteristic
618C2C4×#1451980811yescharacteristic
619C2#2145574811yescharacteristic
620C2#2145576311yescharacteristic
621C2#2145592211yescharacteristic
622C2#2145597511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×C22⋊Q8×C61C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
cosocleC22×C4×C61C24
hypercenterC4×C22⋊Q8D61
hypocenterC61#21583
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #21446457C2semidirect
4. #21446450C2semidirect
5. #21446408C2semidirect
6. #21446401C2semidirect
10. #21445717C2semidirect
11. #21445717C2semidirect
12. #21445649C2semidirect
13. #21445649C2semidirect
18. #16840100C4semidirect
81. #14519710C2×C4semidirect
83. #14519708C2×C4semidirect
86. #14519701C2×C4semidirect
87. #14519701C2×C4semidirect
371. C61#21583semidirect

Maximal subgroups

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

subgroupindex
C2×#168401002
C4×#145197362
#214464572
#214464502
#214464082
#214464012
#214461322
#214460162
#214459112
#214457172
#214456492
#214455812
C4×C22⋊Q8×C612
#2158361