Finite Group Databasebeta

← #29699410

Subgroups — #29699410

Order 15616.

894 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839863C211yesmaximal, characteristic
2#21448783C211yesmaximal, characteristic, complemented
3D4×#11758196C211yesmaximal, characteristic
4#21446506C211yesmaximal, characteristic, complemented
5#21446498C211yesmaximal, characteristic, complemented
6#21446410C211yesmaximal, characteristic, complemented
7#21446391C211yesmaximal, characteristic, complemented
8#21446155C211yesmaximal, characteristic
9#21445951C211yesmaximal, characteristic, complemented
10#21445827C211yesmaximal, characteristic
11#21445822C211yesmaximal, characteristic, complemented
12#21445690C211yesmaximal, characteristic, complemented
13#21445646C211yesmaximal, characteristic
14C4×#14519698C211yesmaximal, characteristic
15C61×#1877C211yesmaximal, characteristic
16C2×Q8×Dic61C2211yescharacteristic
17C2×D4×Dic61C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19C22×#11758196—21no—
20#16839863C418yescomplemented
21C2×#14519711C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519687C2211yescharacteristic
28C2×#14519687C2211yescharacteristic
29C2×C4×#92418C2211yescharacteristic
30#16839669C2211yescharacteristic
31#16839660C2211yescharacteristic
32#16839658C2211yescharacteristic
33#16839647—21no—
34#16839645—21no—
35#16839635—21no—
36#16839634—21no—
37#16839631C2211yescharacteristic
38#16839629—21no—
39#16839627C2211yescharacteristic
40#16839620—21no—
41C4⋊C4×Dic61C2211yescharacteristic
42C4×#11758196C2211yescharacteristic
43C4×C4⋊Dic61C2211yescharacteristic
44#16839607—21no—
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53C22×SD16×C61C2211yescharacteristic
54C2×C4×D4×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C61×#391C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C23×Dic61—21no—
61C23×Dic61—21no—
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64Q8×Dic61—22no—
65Q8×Dic61C2×C414yescomplemented
66D4×Dic61—22no—
67D4×Dic61—22no—
68D4×Dic61—41no—
69D4×Dic61C2×C414yes—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—44no—
80C2×#11758196D411yescharacteristic
81C2×#11758196D411yescharacteristic
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89#14519711C2×C414yescomplemented
90#14519706—22no—
91#14519701—41no—
92#14519701C2×C414yescomplemented
93#14519698—22no—
94#14519698C2×C414yes—
95#14519687—41no—
96#14519687C2×C414yescomplemented
97C4×#92418—41no—
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
110C23×C4×C61—21no—
111C2×SD16×C61—22no—
112C2×SD16×C61—22no—
113C2×SD16×C61C2×C414yes—
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116C4×D4×C61—22no—
117C4×D4×C61—41no—
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C22⋊C4×C61—21no—
121C2×C42×C61C2311yescharacteristic
122#103×C61—41no—
123#102×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×C61C22×C412yes—
137C2×D4×C61#4112yes—
138C2×D4×C61#4112yes—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—42no—
143C22×C4×C61—42no—
144C22×C4×C61—81no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2411yescharacteristic
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—24no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—42no—
154C22×Dic61—42no—
155C22×Dic61—42no—
156C22×Dic61—42no—
157C22×Dic61—81no—
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162SD16×C61—28no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61C22×C412yes—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C22⋊C4×C61—42no—
174C42×C61—21no—
175C42×C61—21no—
176C22⋊Dic61—24no—
177C22⋊Dic61—42no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—24no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61C22×C412yes—
186C4⋊Dic61C22×C412yes—
187#11758196—22no—
188#11758196—41no—
189#11758196—42no—
190#11758196—42no—
191#11758196—42no—
192#11758196—81no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C4×Dic61—24no—
196C4×Dic61—42no—
197C4×Dic61—42no—
198C4×Dic61C22×C412yes—
199C4×Dic61#4112yes—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418C22×C412yes—
206C23×C61—22no—
207C23×C61—22no—
208C23×C61—22no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—41no—
212C23×C61—41no—
213C23×C61C22×D411yescharacteristic
214Q8×C61—22no—
215Q8×C61—22no—
216Q8×C61—41no—
217Q8×C61—41no—
218D4×C61—22no—
219D4×C61—22no—
220D4×C61—22no—
221D4×C61#11714yes—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—22no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
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×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—84no—
237C2×C4×C61AGL(1,ℤ/8)12yes—
238C2×C4×C61C23×C411yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61H(5,𝔽2)11yescharacteristic
243C2×C4×C612−1+411yescharacteristic
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—22no—
249C2×Dic61—22no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—48no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—84no—
268C2×Dic61C4×D412yes—
269C2×Dic61C4×D412yes—
270C8×C61—22no—
271C8×C61—41no—
272#92418—22no—
273#92418—41no—
274#22778—611nomaximal, Sylow
275C22×C61—24no—
276C22×C61—42no—
277C22×C61—42no—
278C22×C61—42no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61—42no—
282C22×C61C2×C4×D411yescharacteristic
283C22×C61#54611yescharacteristic
284C22×C61#54811yescharacteristic
285C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C61#57511yescharacteristic
287C22×C61#57711yescharacteristic
288C22×C61#57811yescharacteristic
289C4×C61—22no—
290C4×C61—42no—
291C4×C61—41no—
292C4×C61—81no—
293C4×C61—81no—
294C4×C61C2×#11711yescharacteristic
295C4×C61#52011yescharacteristic
296C4×C61#55211yescharacteristic
297C4×C61#55311yescharacteristic
298Dic61—22no—
299Dic61—22no—
300Dic61—42no—
301Dic61—42no—
302Dic61—41no—
303Dic61—41no—
304Dic61—81no—
305Dic61—81no—
306C2×#440—611no—
307#2356—611no—
308D4×C4⋊C4—611no—
309#1935—611no—
310#1933—611no—
311#1921—611no—
312#1918—611no—
313#1877—611no—
314#1877D6111yescharacteristic
315#1836—611no—
316#1804—611no—
317#1803—611no—
318#1771—611no—
319#1762—611no—
320C4×#102—611no—
321C2×C61—44no—
322C2×C61#235211yescharacteristic
323C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
324C2×C61#294711yescharacteristic
325C2×C61#331011yescharacteristic
326C2×C61#331511yescharacteristic
327C2×C61#336711yescharacteristic
328C2×C61#337611yescharacteristic
329C22×SD16C2×D6111yescharacteristic
330C2×C4×Q8—611no—
331C2×C4×D4—611no—
332C2×C4×D4—611no—
333C2×C4×D4C2×D6111yescharacteristic
334C22×C4⋊C4—1221no—
335#440—618no—
336C2×#107—611no—
337C2×C4⋊C8—611no—
338C2×#103—611no—
339C2×#103—611no—
340C2×#103C2×D6111yescharacteristic
341C2×#102—611no—
342C2×#102—611no—
343C2×#102C2×D6111yescharacteristic
344C2×#97—611no—
345C2×#97—611no—
346C2×C4×C8—611no—
347#391—611no—
348#391C2×D6111yescharacteristic
349#389—611no—
350#389—611no—
351#387—1221no—
352#387—1221no—
353#385—1221no—
354#385—1221no—
355#384—611no—
356#383—1221no—
357#382—611no—
358#380—1221no—
359C4×C4⋊C4—611no—
360C4×C4⋊C4—611no—
361C4×C4⋊C4—611no—
362#376—1221no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367#340—611no—
368#340—611no—
369#340—611no—
370#340C2×D6111yescharacteristic
371#336—611no—
372#336C2×D6111yescharacteristic
373C61#2277811yescharacteristic, complemented, Sylow
374C22×Q8C22×D6111yescharacteristic
375C22×D4C22×D6111yescharacteristic
376C23×C4—21no—
377C23×C4—1221no—
378C23×C4—1221no—
379C2×SD16—22no—
380C2×SD16—22no—
381C2×SD16C4×D6114yes—
382C22×C8—611no—
383C22×C8—611no—
384C22×C8C22×D6111yescharacteristic
385C22×C8C22×D6111yescharacteristic
386C4×Q8—614no—
387C4×Q8—1222no—
388C4×D4—22no—
389C4×D4—41no—
390C4×D4—614no—
391C4×D4—1222no—
392C4×D4—1222no—
393C4×D4—2441no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—2444no—
406C2×C4⋊C4C22×D6111yescharacteristic
407C2×C4⋊C4C22×D6111yescharacteristic
408C2×C22⋊C4—21no—
409C2×C22⋊C4—1221no—
410C2×C22⋊C4—1221no—
411C2×C42—611no—
412C2×C42—611no—
413C2×C42—1221no—
414C2×C42—1221no—
415C2×C42—1221no—
416C2×C42C22×D6111yescharacteristic
417#107—614no—
418C4⋊C8—1222no—
419#103—41no—
420#103—614no—
421#103—2441no—
422#102—22no—
423#102—614no—
424#102—1222no—
425#97—614no—
426#97—2441no—
427C4×C8—2441no—
428#95—21no—
429#95—21no—
430#95—611no—
431#95—611no—
432#95—1221no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437#95—1221no—
438#95—1221no—
439#95—1221no—
440C24—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8C2×C4×D6112yes—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4C2×C4×D6112yes—
451C2×D4#1175822412yes—
452C2×D4#1175822412yes—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—81no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—1224no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—4881no—
469C22×C4—2442no—
470C22×C4—2442no—
471C22×C4—2442no—
472C22×C4—2442no—
473C22×C4D4×D6111yescharacteristic
474C22×C4D4×D6111yescharacteristic
475C22×C4C23×D6111yescharacteristic
476SD16—28no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—612no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8C2×C4×D6112yes—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—612no—
494C4⋊C4—612no—
495C4⋊C4—1224no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1222no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—4881no—
504C4⋊C4—2441no—
505C4⋊C4—2442no—
506C4⋊C4—2442no—
507C4⋊C4—2442no—
508C22⋊C4—42no—
509C22⋊C4—1224no—
510C22⋊C4—2442no—
511C42—21no—
512C42—21no—
513C42—612no—
514C42—612no—
515C42—1224no—
516C42—1221no—
517C42—1221no—
518C42—2442no—
519C42—2442no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23C2×D4×D6111yescharacteristic
528Q8—22no—
529Q8—22no—
530Q8—41no—
531Q8—41no—
532D4—22no—
533D4—22no—
534D4—22no—
535D4#1451973314yes—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—84no—
551C2×C4—612no—
552C2×C4—612no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1222no—
558C2×C4—1222no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—4884no—
566C2×C4—2448no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4#1451980412yes—
578C2×C4C22×C4×D6111yescharacteristic
579C2×C4#41×D6111yescharacteristic
580C2×C4C2×#1175822411yescharacteristic
581C2×C4C2×#1175822411yescharacteristic
582C2×C4#1451982211yescharacteristic
583C2×C4#1451982411yescharacteristic
584C8—22no—
585C8—41no—
586C8—1222no—
587C8—2441no—
588C22—24no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22C4×D4×D6111yescharacteristic
596C22#1684039911yescharacteristic
597C22#1684041811yescharacteristic
598C22C2×#1451980411yescharacteristic
599C22#1684055911yescharacteristic
600C22#1684057211yescharacteristic
601C22#1684057811yescharacteristic
602C4—22no—
603C4—42no—
604C4—41no—
605C4—81no—
606C4—81no—
607C4—1222no—
608C4—1222no—
609C4—4881no—
610C4—4881no—
611C4—2442no—
612C4—2442no—
613C4—2441no—
614C4—2441no—
615C4C2×#1451973311yescharacteristic
616C4#1684024111yescharacteristic
617C4#1684044911yescharacteristic
618C4#1684045511yescharacteristic
619C2—44no—
620C2#2144874611yescharacteristic
621C2C4×#1451980411yescharacteristic
622C2#2145465911yescharacteristic
623C2#2145784411yescharacteristic
624C2#2145791911yescharacteristic
625C2#2145854111yescharacteristic
626C2#2145865011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1877C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenter#1877D61
hypocenterC61#22778
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1877

Complements

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

NH ≅ G/Ntype
2. #21448783C2semidirect
4. #21446506C2semidirect
5. #21446498C2semidirect
6. #21446410C2semidirect
7. #21446391C2semidirect
9. #21445951C2semidirect
11. #21445822C2semidirect
12. #21445690C2semidirect
20. #16839863C4semidirect
65. Q8×Dic61C2×C4semidirect
89. #14519711C2×C4semidirect
92. #14519701C2×C4semidirect
96. #14519687C2×C4semidirect
373. C61#22778semidirect

Maximal subgroups

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

subgroupindex
C2×#168398632
#214487832
D4×#117581962
#214465062
#214464982
#214464102
#214463912
#214461552
#214459512
#214458272
#214458222
#214456902
#214456462
C4×#145196982
C61×#18772
#2277861