Finite Group Databasebeta

← #29647307

Subgroups — #29647307

Order 15616.

992 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840314C211yesmaximal, characteristic
2#21454448C211yesmaximal, characteristic
3#21448210C211yesmaximal, characteristic
4#21444368C214yesmaximal, complemented
5#21444353C211yesmaximal, characteristic
6#21444352C211yesmaximal, characteristic
7#21444351C211yesmaximal, characteristic
8#21444316C214yesmaximal, complemented
9C61×#1511C211yesmaximal, characteristic
10#16840314—21no—
11#16840314—21no—
12#16840314—21no—
13#16840314—21no—
14#16840314C412yescomplemented
15#16840314C412yescomplemented
16#16840314C2212yes—
17#16840314C2212yes—
18C2×#14519776C411yescharacteristic, complemented
19C2×#14519775C411yescharacteristic, complemented
20C2×#14519767—21no—
21C2×#14519756C2211yescharacteristic
22C2×#14519754—21no—
23C2×Q8×Dic61C411yescharacteristic, complemented
24C2×#14519742—21no—
25C2×#14519742C411yescharacteristic, complemented
26C2×#14519733—21no—
27C22×C4×Dic61C2211yescharacteristic
28#16839816—21no—
29#16839816C2212yes—
30#16839812—21no—
31#16839812C2212yes—
32#16839785—21no—
33#16839726C2211yescharacteristic
34M16×Dic61—21no—
35#16839629C2212yes—
36#16839626—21no—
37#16839625C2211yescharacteristic
38#16839620—21no—
39#16839613C2212yes—
40C22⋊C4×Dic61—21no—
41#16839607—21no—
42#16839500C2214yes—
43#16839499C2214yes—
44#16839498C2214yes—
45C2×C22⋊Q8×C61C2211yescharacteristic
46C61×#407C2211yescharacteristic
47C61×#381C2211yescharacteristic
48C61×#328C2214yes—
49C22×Dic122—21no—
50C22×Dic122C2×C411yescharacteristic
51C23×Dic61D411yescharacteristic
52C23×Dic61D411yescharacteristic
53C2×#11758194—21no—
54#14519776—21no—
55#14519776—21no—
56#14519776C2×C412yes—
57#14519776C2×C412yes—
58#14519775—21no—
59#14519775—21no—
60#14519775C2×C412yes—
61#14519775C2×C412yes—
62#14519767—22no—
63#14519767—22no—
64#14519767—22no—
65#14519767—22no—
66#14519756—21no—
67#14519756—21no—
68#14519756C2×C412yes—
69#14519756C2312yes—
70#14519754—22no—
71#14519754—22no—
72#14519754—22no—
73#14519754—22no—
74Q8×Dic61—21no—
75Q8×Dic61—21no—
76Q8×Dic61C2×C412yes—
77Q8×Dic61C2×C412yes—
78#14519742—22no—
79#14519742—22no—
80#14519742—21no—
81#14519742—21no—
82#14519742—22no—
83#14519742—22no—
84#14519742C2×C412yes—
85#14519742C2×C412yes—
86#14519733—22no—
87#14519733—22no—
88#14519733—22no—
89#14519733—22no—
90C2×C4⋊Dic61—21no—
91C2×C4⋊Dic61C2×C411yescharacteristic
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—21no—
96C2×#11758196—21no—
97C2×#11758196—22no—
98C2×C22⋊Dic61—21no—
99C2×C22⋊Dic61—21no—
100C2×C4×Dic61—21no—
101C2×C4×Dic61—21no—
102C2×C4×Dic61—22no—
103C2×C4×Dic61C2×C411yescharacteristic
104C2×C4×Dic61C2×C411yescharacteristic
105C2×C4×Dic61C2×C411yescharacteristic
106C2×C4×Dic61C2×C411yescharacteristic
107C2×C4×Dic61D411yescharacteristic
108C2×C4×Dic61D411yescharacteristic
109C2×C4×Dic61D411yescharacteristic
110C2×C4×Dic61D411yescharacteristic
111C2×C4×Dic61D411yescharacteristic
112C2×C4×Dic61D411yescharacteristic
113C2×C4×Dic61C2311yescharacteristic
114C2×C4×Dic61C2311yescharacteristic
115#14519706—22no—
116#14519706—22no—
117C22⋊#92418—21no—
118C22⋊#92418C2312yes—
119#14519687—22no—
120C8×Dic61—22no—
121#14519684—21no—
122#14519684—22no—
123#14519684—22no—
124#14519684C2312yes—
125C22×Q8×C61C2×C411yescharacteristic
126C23×C4×C61C2311yescharacteristic
127C2×M16×C61—21no—
128C22⋊Q8×C61—21no—
129C22⋊Q8×C61—21no—
130C22⋊Q8×C61C2×C412yes—
131C22⋊Q8×C61C2312yes—
132C2×C4⋊C4×C61—21no—
133C2×C4⋊C4×C61C2×C411yescharacteristic
134C2×C22⋊C4×C61—21no—
135C2×C22⋊C4×C61—21no—
136C22⋊C8×C61—21no—
137C22⋊C8×C61C2312yes—
138#95×C61—21no—
139#95×C61C2312yes—
140C24×C61C2×D411yescharacteristic
141C2×Q8×C61—21no—
142C2×Q8×C61—42no—
143C2×Q8×C61C22×C412yes—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—22no—
147C22×C4×C61—42no—
148C22×C4×C61C22×C411yescharacteristic
149C22×C4×C61C22×C411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2411yescharacteristic
154C22×Dic61—24no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—22no—
158C22×Dic61—42no—
159C22×Dic61C22⋊C411yescharacteristic
160C22×Dic61C22⋊C411yescharacteristic
161C22×Dic61C22⋊C411yescharacteristic
162C22×Dic61C22⋊C411yescharacteristic
163C22×Dic61C22⋊C411yescharacteristic
164C22×Dic61C22⋊C411yescharacteristic
165C22×Dic61C22⋊C411yescharacteristic
166C22×Dic61C22⋊C411yescharacteristic
167C22×Dic61C2×D411yescharacteristic
168C22×Dic61C2×D411yescharacteristic
169C22×Dic61C2×D411yescharacteristic
170C22×Dic61C2×D411yescharacteristic
171C2×Dic122—21no—
172C2×Dic122—22no—
173C2×Dic122—22no—
174C2×Dic122—44no—
175C2×Dic122—42no—
176C2×Dic122C22×C412yes—
177M16×C61—42no—
178C2×C8×C61—22no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61C22×C412yes—
183C22⋊C4×C61—22no—
184C22⋊C4×C61—22no—
185C22⋊C4×C61—42no—
186C22⋊Dic61—22no—
187C22⋊Dic61—22no—
188C22⋊Dic61—42no—
189C4⋊Dic61—21no—
190C4⋊Dic61—22no—
191C4⋊Dic61—22no—
192C4⋊Dic61C22×C412yes—
193#11758196—22no—
194#11758196—22no—
195#11758196—22no—
196#11758196—22no—
197#11758196—22no—
198#11758196—22no—
199#11758196—22no—
200#11758196—22no—
201#11758196—22no—
202#11758196—22no—
203#11758196—44no—
204C4×Dic61—22no—
205C4×Dic61—22no—
206C4×Dic61—22no—
207C4×Dic61—22no—
208C4×Dic61—44no—
209C4×Dic61C22⋊C411yescharacteristic
210C4×Dic61C22⋊C411yescharacteristic
211C4×Dic61C22⋊C411yescharacteristic
212C4×Dic61C22⋊C411yescharacteristic
213C4×Dic61C22⋊C411yescharacteristic
214C4×Dic61C22⋊C411yescharacteristic
215C4×Dic61C22⋊C411yescharacteristic
216C4×Dic61C22⋊C411yescharacteristic
217C4×Dic61C22×C411yescharacteristic
218C4×Dic61C22×C411yescharacteristic
219C4×Dic61C22×C411yescharacteristic
220C4×Dic61C22×C411yescharacteristic
221C4×Dic61C2×D411yescharacteristic
222C4×Dic61C2×D411yescharacteristic
223C4×Dic61C2×D411yescharacteristic
224C4×Dic61C2×D411yescharacteristic
225#11758194—42no—
226C2×#92418—22no—
227C23×C61—22no—
228C23×C61—22no—
229C23×C61—41no—
230C23×C61C2×C22⋊C411yescharacteristic
231C23×C61C2×C22⋊C411yescharacteristic
232C23×C61C22×D411yescharacteristic
233Q8×C61—42no—
234Q8×C61—42no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C23×C411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—42no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61—42no—
270C2×Dic61C2×C22⋊C411yescharacteristic
271C2×Dic61C2×C22⋊C411yescharacteristic
272C2×Dic61C2×C22⋊C411yescharacteristic
273C2×Dic61C2×C22⋊C411yescharacteristic
274C2×Dic61C2×C22⋊C411yescharacteristic
275C2×Dic61C2×C22⋊C411yescharacteristic
276C2×Dic61C2×C22⋊C411yescharacteristic
277C2×Dic61C2×C22⋊C411yescharacteristic
278C2×Dic61#13714yes—
279Dic122—44no—
280Dic122—42no—
281Dic122—44no—
282Dic122—42no—
283C8×C61—42no—
284#92418—42no—
285#17562—611nomaximal, Sylow
286C22×C61—21no—
287C22×C61—21no—
288C22×C61—21no—
289C22×C61—21no—
290C22×C61—21no—
291C22×C61—21no—
292C22×C61—21no—
293C22×C61—44no—
294C22×C61#41012yes—
295C22×C61C22×C22⋊C411yescharacteristic
296C22×C61C2×#13712yes—
297C4×C61—21no—
298C4×C61—21no—
299C4×C61—21no—
300C4×C61—21no—
301C4×C61—42no—
302C4×C61—42no—
303C4×C61—82no—
304Dic61—21no—
305Dic61—21no—
306Dic61—21no—
307Dic61—21no—
308Dic61—21no—
309Dic61—42no—
310Dic61—42no—
311Dic61—82no—
312Dic61#41912yes—
313Dic61#41912yes—
314Dic61#42112yes—
315C2×#533—611no—
316#2920—611no—
317#2283—611no—
318#1515—614no—
319#1511—611no—
320#1511—611no—
321#1511—611no—
322#1511D6111yescharacteristic
323#1503—614no—
324C2×C61—22no—
325C2×C61—22no—
326C2×C61—41no—
327C2×C61C2×#41011yescharacteristic
328C2×C61C2×#41911yescharacteristic
329C2×C61C2×#42111yescharacteristic
330#533—612no—
331#533—612no—
332#533—612no—
333#533—612no—
334#533—1221no—
335#533—1221no—
336#533—1221no—
337#533—1221no—
338C2×C4⋊Q8—611no—
339C2×C4⋊Q8—611no—
340C2×#125—1221no—
341C2×C22⋊Q8—611no—
342C2×C22⋊Q8—1221no—
343C2×C22⋊Q8C2×D6111yescharacteristic
344C2×C4×Q8—611no—
345C2×C4×Q8—611no—
346C2×C4×Q8—1221no—
347C2×#117—1221no—
348C22×C42—611no—
349#432—612no—
350#432—1221no—
351#431—612no—
352#431—1221no—
353#423—1221no—
354#407—611no—
355#407C2×D6111yescharacteristic
356C4×M16—1221no—
357#383—612no—
358#382—1221no—
359#381—611no—
360#381C2×D6111yescharacteristic
361#380—1221no—
362C4×C4⋊C4—612no—
363C4×C22⋊C4—1221no—
364#376—1221no—
365#328—614no—
366#328—614no—
367#328—614no—
368#328C2×D6114yes—
369C61#1756211yescharacteristic, complemented, Sylow
370C22×Q8—611no—
371C22×Q8—1221no—
372C22×Q8C4×D6111yescharacteristic
373C23×C4—611no—
374C23×C4—611no—
375C23×C4C22×D6111yescharacteristic
376C2×M16—21no—
377C2×M16—1221no—
378C4⋊Q8—612no—
379C4⋊Q8—612no—
380C4⋊Q8—612no—
381C4⋊Q8—612no—
382C4⋊Q8—1221no—
383C4⋊Q8—1221no—
384C4⋊Q8—1221no—
385C4⋊Q8—1221no—
386#125—1222no—
387#125—1222no—
388#125—1222no—
389#125—1222no—
390C22⋊Q8—21no—
391C22⋊Q8—21no—
392C22⋊Q8—612no—
393C22⋊Q8—612no—
394C22⋊Q8—1222no—
395C22⋊Q8—1222no—
396C22⋊Q8—1222no—
397C22⋊Q8—1222no—
398C22⋊Q8—1221no—
399C22⋊Q8—1221no—
400C22⋊Q8C4×D6112yes—
401C22⋊Q8C22×D6112yes—
402C4×Q8—612no—
403C4×Q8—612no—
404C4×Q8—612no—
405C4×Q8—612no—
406C4×Q8—1221no—
407C4×Q8—1221no—
408C4×Q8—1222no—
409C4×Q8—1222no—
410C4×Q8—1222no—
411C4×Q8—1222no—
412C4×Q8—1221no—
413C4×Q8—1221no—
414#117—1222no—
415#117—1222no—
416#117—1222no—
417#117—1222no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1222no—
427C2×C4⋊C4C4×D6111yescharacteristic
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—1221no—
431C2×C22⋊C4—1221no—
432C2×C42—611no—
433C2×C42—611no—
434C2×C42—611no—
435C2×C42—611no—
436C2×C42—611no—
437C2×C42—611no—
438C2×C42—611no—
439C2×C42—611no—
440C2×C42—611no—
441C2×C42—611no—
442C2×C42—611no—
443C2×C42—611no—
444C2×C42—1221no—
445C2×C42—1221no—
446C2×C42—1222no—
447C4⋊C8—1222no—
448C4⋊C8—1222no—
449C22⋊C8—21no—
450C22⋊C8—612no—
451C22⋊C8—1221no—
452C22⋊C8C22×D6112yes—
453#97—1222no—
454C4×C8—1222no—
455#95—21no—
456#95—612no—
457#95—1222no—
458#95—1222no—
459#95—1221no—
460#95C22×D6112yes—
461C24D4×D6111yescharacteristic
462C2×Q8—21no—
463C2×Q8—42no—
464C2×Q8—612no—
465C2×Q8—1222no—
466C2×Q8—1222no—
467C2×Q8—1221no—
468C2×Q8—2442no—
469C2×Q8—2444no—
470C2×Q8C2×C4×D6112yes—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—42no—
475C22×C4—611no—
476C22×C4—611no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—1224no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1222no—
491C22×C4—2442no—
492C22×C4C2×C4×D6111yescharacteristic
493C22×C4C2×C4×D6111yescharacteristic
494C22×C4D4×D6111yescharacteristic
495C22×C4D4×D6111yescharacteristic
496C22×C4D4×D6111yescharacteristic
497C22×C4C23×D6111yescharacteristic
498M16—42no—
499M16—2442no—
500C2×C8—22no—
501C2×C8—1222no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—612no—
506C4⋊C4—1222no—
507C4⋊C4—1222no—
508C4⋊C4—1222no—
509C4⋊C4—1222no—
510C4⋊C4—1222no—
511C4⋊C4—1222no—
512C4⋊C4—1222no—
513C4⋊C4—1222no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—1221no—
518C4⋊C4—1222no—
519C4⋊C4—2444no—
520C4⋊C4C2×C4×D6112yes—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—42no—
524C22⋊C4—1222no—
525C22⋊C4—1222no—
526C22⋊C4—2442no—
527C42—611no—
528C42—611no—
529C42—611no—
530C42—611no—
531C42—611no—
532C42—611no—
533C42—611no—
534C42—611no—
535C42—611no—
536C42—611no—
537C42—611no—
538C42—611no—
539C42—611no—
540C42—611no—
541C42—611no—
542C42—611no—
543C42—1222no—
544C42—1222no—
545C42—1222no—
546C42—1222no—
547C42—2444no—
548C23—22no—
549C23—22no—
550C23—41no—
551C23C22⋊C4×D6111yescharacteristic
552C23C22⋊C4×D6111yescharacteristic
553C23C2×D4×D6111yescharacteristic
554Q8—42no—
555Q8—42no—
556Q8—2442no—
557Q8—2442no—
558Q8—2444no—
559Q8—2444no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—611no—
574C2×C4—611no—
575C2×C4—611no—
576C2×C4—614no—
577C2×C4—611no—
578C2×C4—611no—
579C2×C4—611no—
580C2×C4—611no—
581C2×C4—611no—
582C2×C4—1222no—
583C2×C4—1222no—
584C2×C4—1222no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—1222no—
591C2×C4—1222no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4—2442no—
598C2×C4—2442no—
599C2×C4—2442no—
600C2×C4C22⋊C4×D6111yescharacteristic
601C2×C4C22⋊C4×D6111yescharacteristic
602C2×C4C22×C4×D6111yescharacteristic
603C2×C4C2×D4×D6111yescharacteristic
604C8—42no—
605C8—2442no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—44no—
614C22#1683973412yes—
615C22C2×C22⋊C4×D6111yescharacteristic
616C22#137×D6112yes—
617C4—21no—
618C4—21no—
619C4—21no—
620C4—21no—
621C4—42no—
622C4—42no—
623C4—82no—
624C4—612no—
625C4—612no—
626C4—612no—
627C4—1221no—
628C4—1221no—
629C4—1221no—
630C4—1221no—
631C4—1221no—
632C4—4882no—
633C4—2442no—
634C4—2442no—
635C2—22no—
636C2—22no—
637C2—41no—
638C2C2×#1683973411yescharacteristic
639C2#419×D6111yescharacteristic
640C2#421×D6111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1511C2
socleC22×C61C22×C22⋊C4
prime radicalC24×C61C2×D4
cosocleC22×C4×C61C24
hypercenter#1511D61
hypocenterC61#17562
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1511

Complements

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

NH ≅ G/Ntype
4. #21444368C2semidirect
8. #21444316C2semidirect
14. #16840314C4semidirect
15. #16840314C4semidirect
18. C2×#14519776C4semidirect
19. C2×#14519775C4semidirect
23. C2×Q8×Dic61C4semidirect
25. C2×#14519742C4semidirect
369. C61#17562semidirect

Maximal subgroups

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

subgroupindex
C2×#168403142
#214544482
#214482102
#214443682
#214443532
#214443522
#214443512
#214443162
C61×#15112
#1756261