Finite Group Databasebeta

← #30614653

Subgroups — #30614653

Order 15616.

886 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840786C211yesmaximal, characteristic
2C2×#16840786C211yesmaximal, characteristic
3#21461619C211yesmaximal, characteristic, complemented
4#21461619C211yesmaximal, characteristic, complemented
5#21461618C211yesmaximal, characteristic, complemented
6#21461618C211yesmaximal, characteristic, complemented
7#21461615C211yesmaximal, characteristic, complemented
8#21461614C211yesmaximal, characteristic, complemented
9#21461611C211yesmaximal, characteristic, complemented
10C4⋊C4×#92420C211yesmaximal, characteristic, complemented
11#21461549C211yesmaximal, characteristic
12#21461549C211yesmaximal, characteristic
13#21461545C211yesmaximal, characteristic
14C4×C22⋊#92420C211yesmaximal, characteristic
15#21448298C211yesmaximal, characteristic
16#16840786C418yescomplemented
17#16840786C418yescomplemented
18C2×#14519844C2211yescharacteristic
19C2×#14519844C2211yescharacteristic
20C2×C4⋊#92420C2211yescharacteristic
21C2×C4⋊#92420C2211yescharacteristic
22C2×C22⋊#92420C2211yescharacteristic
23C2×C22⋊#92420C2211yescharacteristic
24C2×C22⋊#92420C2211yescharacteristic
25C2×C22⋊#92420C2211yescharacteristic
26C2×#14519832C2211yescharacteristic
27C2×#14519832C2211yescharacteristic
28C2×#14519832C2211yescharacteristic
29C2×#14519832C2211yescharacteristic
30C2×#14519832C2211yescharacteristic
31C2×#14519832C2211yescharacteristic
32C2×C4×#92420C2211yescharacteristic
33C2×C4×#92420C2211yescharacteristic
34C2×C4×#92420C2211yescharacteristic
35C2×C4×#92420C2211yescharacteristic
36C2×C4×#92420C2211yescharacteristic
37C2×C4×#92420C2211yescharacteristic
38#16840628C2211yescharacteristic
39#16840628C2211yescharacteristic
40#16840628C2211yescharacteristic
41#16840628C2211yescharacteristic
42#16840628C2211yescharacteristic
43#16840628C2211yescharacteristic
44#16840628C2211yescharacteristic
45#16840628C2211yescharacteristic
46C2×#14519733C2211yescharacteristic
47C2×#14519733C2211yescharacteristic
48C22×C4×Dic61C2211yescharacteristic
49#16839659C411yescharacteristic
50#16839659C411yescharacteristic
51#16839657C411yescharacteristic
52#16839657C411yescharacteristic
53#16839637C411yescharacteristic
54#16839635C411yescharacteristic
55#16839634C411yescharacteristic
56C4⋊C4×Dic61C2211yescharacteristic
57C4⋊C4×Dic61C2211yescharacteristic
58#16839613C2211yescharacteristic
59#16839613C2211yescharacteristic
60C61×#385C411yescharacteristic
61C22×#92420C2311yescharacteristic
62C22×#92420C2311yescharacteristic
63C22×#92420C2311yescharacteristic
64C22×#92420C2311yescharacteristic
65C22×#92420C2311yescharacteristic
66C22×#92420C2311yescharacteristic
67C22×#92420C2311yescharacteristic
68C22×#92420C2311yescharacteristic
69#14519844C2×C414yescomplemented
70#14519844C2×C414yescomplemented
71C4⋊#92420C2×C414yescomplemented
72C4⋊#92420C2×C414yescomplemented
73C22⋊#92420C2×C414yes—
74C22⋊#92420C2×C414yes—
75C22⋊#92420C2×C414yes—
76C22⋊#92420C2×C414yes—
77#14519832—41no—
78#14519832—41no—
79#14519832—41no—
80#14519832—41no—
81#14519832C2×C414yescomplemented
82#14519832C2×C414yescomplemented
83C4×#92420—41no—
84C4×#92420—41no—
85C4×#92420—41no—
86C4×#92420—41no—
87C4×#92420C2×C414yescomplemented
88C4×#92420C2×C414yescomplemented
89C23×Dic61C2×C411yescharacteristic
90C23×Dic61C2311yescharacteristic
91#14519733—41no—
92#14519733—41no—
93#14519733C2×C414yes—
94#14519733C2×C414yes—
95C2×C4⋊Dic61C2×C411yescharacteristic
96C2×C4⋊Dic61C2×C411yescharacteristic
97C2×#11758196C2×C411yescharacteristic
98C2×#11758196C2×C411yescharacteristic
99C2×#11758196C2×C411yescharacteristic
100C2×#11758196C2×C411yescharacteristic
101C2×#11758196C2×C411yescharacteristic
102C2×#11758196C2×C411yescharacteristic
103C2×#11758196C2×C411yescharacteristic
104C2×#11758196C2×C411yescharacteristic
105C2×C22⋊Dic61C2×C411yescharacteristic
106C2×C22⋊Dic61C2×C411yescharacteristic
107C2×C4×Dic61—41no—
108C2×C4×Dic61—41no—
109C2×C4×Dic61C2×C411yescharacteristic
110C2×C4×Dic61C2×C411yescharacteristic
111C2×C4×Dic61C2311yescharacteristic
112C2×C4×Dic61C2311yescharacteristic
113C2×C4×Dic61C2311yescharacteristic
114C2×C4×Dic61C2311yescharacteristic
115C2×C4×Dic61C2311yescharacteristic
116C2×C4×Dic61C2311yescharacteristic
117#14519684C2×C411yescharacteristic
118#14519684C2×C411yescharacteristic
119#14519684C2×C411yescharacteristic
120#14519684C2×C411yescharacteristic
121#14519684C2×C411yescharacteristic
122#14519684C2×C411yescharacteristic
123C23×C4×C61C2×C411yescharacteristic
124C2×C4⋊C4×C61C2×C411yescharacteristic
125C2×C4⋊C4×C61C2×C411yescharacteristic
126C2×C22⋊C4×C61C2×C411yescharacteristic
127C2×C22⋊C4×C61C2×C411yescharacteristic
128#95×C61C2×C411yescharacteristic
129#95×C61C2×C411yescharacteristic
130C24×C61C22×C411yescharacteristic
131C22×C4×C61—22no—
132C22×C4×C61—41no—
133C22×C4×C61—41no—
134C22×C4×C61C22×C411yescharacteristic
135C22×C4×C61C22×C411yescharacteristic
136C22×C4×C61C22×C411yescharacteristic
137C22×C4×C61C22×C411yescharacteristic
138C22×C4×C61C22×C411yescharacteristic
139C22×C4×C61C22×C411yescharacteristic
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—22no—
145C22×Dic61—41no—
146C22×Dic61—41no—
147C22×Dic61C22×C411yescharacteristic
148C22×Dic61C22×C411yescharacteristic
149C22×Dic61C22×C411yescharacteristic
150C22×Dic61C22×C411yescharacteristic
151C22×Dic61C22×C411yescharacteristic
152C22×Dic61C22×C411yescharacteristic
153C22×Dic61C22×C412yes—
154C22×Dic61C22×C412yes—
155C22×Dic61C22×C411yescharacteristic
156C22×Dic61C2411yescharacteristic
157C2×#92420—21no—
158C2×#92420—21no—
159C2×#92420—21no—
160C2×#92420—21no—
161C2×#92420—21no—
162C2×#92420—21no—
163C2×#92420—21no—
164C2×#92420—21no—
165C2×#92420—21no—
166C2×#92420—21no—
167C2×#92420—21no—
168C2×#92420—21no—
169C2×#92420—21no—
170C2×#92420—21no—
171C2×#92420—21no—
172C2×#92420—21no—
173C2×#92420C22×C412yes—
174C2×#92420C22×C412yes—
175C2×#92420C22×C412yes—
176C2×#92420C22×C412yes—
177C2×#92420C22×C412yes—
178C2×#92420C22×C412yes—
179C2×#92420C22×C412yes—
180C2×#92420C22×C412yes—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—41no—
183C22⋊C4×C61—41no—
184C22⋊C4×C61C4214yes—
185C22⋊Dic61—41no—
186C22⋊Dic61C4214yes—
187C4⋊Dic61—22no—
188C4⋊Dic61—41no—
189#11758196—22no—
190#11758196—22no—
191#11758196—41no—
192#11758196—41no—
193#11758196—41no—
194#11758196—41no—
195#11758196C4214yes—
196#11758196C4214yes—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C4×Dic61—21no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—41no—
206C4×Dic61—41no—
207C4×Dic61C22×C412yes—
208C4×Dic61C22×C412yes—
209C4×Dic61C22×C412yes—
210C4×Dic61C22×C412yes—
211C23×C61—21no—
212C23×C61—21no—
213C23×C61—21no—
214C23×C61—21no—
215C23×C61—21no—
216C23×C61—21no—
217C23×C61C2×C4212yes—
218C23×C61C23×C411yescharacteristic
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—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—44no—
235C2×C4×C61C2×C4212yes—
236C2×C4×C61C2×C4212yes—
237C2×C4×C61#13112yes—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—22no—
247C2×Dic61—22no—
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—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—44no—
260C2×Dic61C2×C4212yes—
261C2×Dic61C2×C4212yes—
262C2×Dic61C2×C4212yes—
263C2×Dic61#13112yes—
264C2×Dic61C23×C411yescharacteristic
265C2×Dic61C23×C411yescharacteristic
266C2×Dic61H(5,𝔽2)11yescharacteristic
267C2×Dic61H(5,𝔽2)11yescharacteristic
268C2×Dic612−1+411yescharacteristic
269C2×Dic612−1+411yescharacteristic
270#92420—22no—
271#92420—22no—
272#92420—22no—
273#92420—22no—
274#92420—22no—
275#92420—22no—
276#92420—22no—
277#92420—22no—
278#20541—611nomaximal, Sylow
279C22×C61—22no—
280C22×C61—22no—
281C22×C61—22no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61C22×C4211yescharacteristic
287C22×C61#51811yescharacteristic
288C22×C61#51911yescharacteristic
289C22×C61C2×#13111yescharacteristic
290C22×C61C2×#13111yescharacteristic
291C22×C61#56811yescharacteristic
292C22×C61#56811yescharacteristic
293C4×C61—22no—
294C4×C61—22no—
295C4×C61—22no—
296C4×C61—41no—
297C4×C61—41no—
298C4×C61—41no—
299Dic61—22no—
300Dic61—22no—
301Dic61—22no—
302Dic61—22no—
303Dic61—41no—
304Dic61—41no—
305Dic61—41no—
306Dic61#52011yescharacteristic
307Dic61#52011yescharacteristic
308Dic61#52011yescharacteristic
309Dic61#52011yescharacteristic
310C2×#433—611no—
311C2×#433—611no—
312#2296—611no—
313#1771—611no—
314#1771—611no—
315#1771—611no—
316#1771—611no—
317#1770—611no—
318#1770—611no—
319C8×C4⋊C4—611no—
320C8×C4⋊C4—611no—
321#1750—611no—
322#1750—611no—
323#1749—611no—
324C4×C22⋊C8—611no—
325C2×C61—42no—
326C2×C61#227411yescharacteristic
327C2×C61C4×#13111yescharacteristic
328C2×C61#287511yescharacteristic
329C2×C61#297611yescharacteristic
330C2×C61#297611yescharacteristic
331C2×C61#298511yescharacteristic
332C2×C61#298511yescharacteristic
333C2×#117—611no—
334C2×#117—611no—
335C22×C42—611no—
336#433—618no—
337#433—618no—
338C2×C4⋊C8—611no—
339C2×C4⋊C8—611no—
340C2×C4⋊C8—611no—
341C2×C4⋊C8—611no—
342C2×C22⋊C8—611no—
343C2×C22⋊C8—611no—
344C2×C22⋊C8—611no—
345C2×C22⋊C8—611no—
346C2×#97—611no—
347C2×#97—611no—
348C2×#97—611no—
349C2×#97—611no—
350C2×#97—611no—
351C2×#97—611no—
352C2×C4×C8—611no—
353C2×C4×C8—611no—
354C2×C4×C8—611no—
355C2×C4×C8—611no—
356C2×C4×C8—611no—
357C2×C4×C8—611no—
358#389—611no—
359#389—611no—
360#389—611no—
361#389—611no—
362#385—611no—
363#385—611no—
364#385—611no—
365#385#692311yescharacteristic
366C4×C4⋊C4—611no—
367C4×C4⋊C4—611no—
368C4×C4⋊C4—611no—
369C4×C4⋊C4—611no—
370#336—611no—
371#336—611no—
372#336—611no—
373#336—611no—
374#336—611no—
375#336—611no—
376#336—611no—
377#336—611no—
378C61#2054111yescharacteristic, complemented, Sylow
379C23×C4—611no—
380C23×C4—611no—
381C23×C4C2×#692311yescharacteristic
382C22×C8—611no—
383C22×C8—611no—
384C22×C8—611no—
385C22×C8—611no—
386C22×C8—611no—
387C22×C8—611no—
388C22×C8—611no—
389C22×C8—611no—
390#117—614no—
391#117—614no—
392#117—2441no—
393#117—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—611no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4C2×#692311yescharacteristic
405C2×C4⋊C4C2×#692311yescharacteristic
406C2×C22⋊C4—611no—
407C2×C22⋊C4—611no—
408C2×C22⋊C4C2×#692311yescharacteristic
409C2×C22⋊C4C2×#692311yescharacteristic
410C2×C42—611no—
411C2×C42—611no—
412C2×C42—611no—
413C2×C42—611no—
414C2×C42—611no—
415C2×C42—611no—
416C2×C42—611no—
417C2×C42—611no—
418C2×C42—2441no—
419C2×C42—2441no—
420C4⋊C8—614no—
421C4⋊C8—614no—
422C4⋊C8—614no—
423C4⋊C8—614no—
424C22⋊C8—614no—
425C22⋊C8—614no—
426C22⋊C8—614no—
427C22⋊C8—614no—
428#97—614no—
429#97—614no—
430#97—2441no—
431#97—2441no—
432#97—2441no—
433#97—2441no—
434C4×C8—614no—
435C4×C8—614no—
436C4×C8—2441no—
437C4×C8—2441no—
438C4×C8—2441no—
439C4×C8—2441no—
440#95—611no—
441#95—611no—
442#95—611no—
443#95—611no—
444#95—611no—
445#95—611no—
446#95C2×#692311yescharacteristic
447#95C2×#692311yescharacteristic
448C24C22×#692311yescharacteristic
449C22×C4—22no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—611no—
453C22×C4—612no—
454C22×C4—612no—
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—1222no—
467C22×C4—2441no—
468C22×C4—2441no—
469C22×C4C22×#692311yescharacteristic
470C22×C4C22×#692311yescharacteristic
471C22×C4C22×#692311yescharacteristic
472C22×C4C22×#692311yescharacteristic
473C22×C4C22×#692311yescharacteristic
474C22×C4C22×#692311yescharacteristic
475C2×C8—612no—
476C2×C8—612no—
477C2×C8—612no—
478C2×C8—612no—
479C2×C8—612no—
480C2×C8—612no—
481C2×C8—612no—
482C2×C8—612no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8—1221no—
496C2×C8—1221no—
497C2×C8—1221no—
498C2×C8—1221no—
499C4⋊C4—22no—
500C4⋊C4—41no—
501C4⋊C4—614no—
502C4⋊C4—614no—
503C4⋊C4—1222no—
504C4⋊C4—1222no—
505C4⋊C4—1222no—
506C4⋊C4—2441no—
507C4⋊C4—2441no—
508C4⋊C4—2441no—
509C4⋊C4—2441no—
510C4⋊C4—2441no—
511C22⋊C4—41no—
512C22⋊C4—614no—
513C22⋊C4—2441no—
514C22⋊C4C4×#692314yes—
515C42—612no—
516C42—612no—
517C42—612no—
518C42—612no—
519C42—1221no—
520C42—1221no—
521C42—1221no—
522C42—1221no—
523C42—1221no—
524C42—1221no—
525C42—1221no—
526C42—1221no—
527C42—2441no—
528C42—2441no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23C2×C4×#692312yes—
536C23C23×#692311yescharacteristic
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—44no—
553C2×C4—611no—
554C2×C4—611no—
555C2×C4—611no—
556C2×C4—611no—
557C2×C4—611no—
558C2×C4—611no—
559C2×C4—612no—
560C2×C4—612no—
561C2×C4—612no—
562C2×C4—612no—
563C2×C4—1222no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—1222no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1221no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—1221no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—2444no—
585C2×C4C2×C4×#692312yes—
586C2×C4C2×C4×#692312yes—
587C2×C4#1451986412yes—
588C8—1222no—
589C8—1222no—
590C8—1222no—
591C8—1222no—
592C8—1222no—
593C8—1222no—
594C8—1222no—
595C8—1222no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22C22×C4×#692311yescharacteristic
604C22#1684082911yescharacteristic
605C22#1684083011yescharacteristic
606C22C2×#1451986411yescharacteristic
607C22D61⋊#13111yescharacteristic
608C22#1684084011yescharacteristic
609C22#1684084111yescharacteristic
610C4—22no—
611C4—22no—
612C4—22no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—611no—
617C4—611no—
618C4—611no—
619C4—611no—
620C4—1222no—
621C4—1222no—
622C4—1222no—
623C4—1222no—
624C4—2441no—
625C4—2441no—
626C4—2441no—
627C2—42no—
628C2#2146225711yescharacteristic
629C2C4×#1451986411yescharacteristic
630C2#2146239011yescharacteristic
631C2#2146268511yescharacteristic
632C2#2146268711yescharacteristic
633C2#2146272311yescharacteristic
634C2#2146272511yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6923
commutatorC22×C61C22×C42
FittingC61×#385C4
socleC23×C61C23×C4
prime radicalC24×C61C22×C4
cosocleC22×Dic61C24
hypercenter#385#6923
hypocenterC61#20541
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
3. #21461619C2semidirect
4. #21461619C2semidirect
5. #21461618C2semidirect
6. #21461618C2semidirect
7. #21461615C2semidirect
8. #21461614C2semidirect
9. #21461611C2semidirect
10. C4⋊C4×#92420C2semidirect
16. #16840786C4semidirect
17. #16840786C4semidirect
69. #14519844C2×C4semidirect
70. #14519844C2×C4semidirect
71. C4⋊#92420C2×C4semidirect
72. C4⋊#92420C2×C4semidirect
81. #14519832C2×C4semidirect
82. #14519832C2×C4semidirect
87. C4×#92420C2×C4semidirect
88. C4×#92420C2×C4semidirect
378. C61#20541semidirect

Maximal subgroups

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

subgroupindex
C2×#168407862
#214616192
#214616182
#214616152
#214616142
#214616112
C4⋊C4×#924202
#214615492
#214615452
C4×C22⋊#924202
#214482982
#2054161