Finite Group Databasebeta

← #29647388

Subgroups — #29647388

Order 15616.

1058 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#533×D61C211yesmaximal, characteristic
2#431×D61C211yesmaximal, characteristic
3#21448225C211yesmaximal, characteristic
4#21444389C214yesmaximal
5#21444384C211yesmaximal, characteristic, complemented
6#21444383C211yesmaximal, characteristic, complemented
7#21444382C211yesmaximal, characteristic, complemented
8#21444316C214yesmaximal, complemented
9C61×#1518C211yesmaximal, characteristic, complemented
10#16840319C2211yescharacteristic
11#16840318—21no—
12#16840317C411yescharacteristic, complemented
13#16840317C411yescharacteristic, complemented
14#16840316—21no—
15#16840316C412yescomplemented
16#16840315—21no—
17#16840314—21no—
18#16840314C2212yes—
19C4⋊Q8×D61C411yescharacteristic, complemented
20C4⋊Q8×D61C411yescharacteristic, complemented
21#125×D61—21no—
22C22⋊Q8×D61—21no—
23C22⋊Q8×D61C2212yes—
24C4×Q8×D61—21no—
25C4×Q8×D61C412yescomplemented
26C2×#14519735—21no—
27#117×D61—21no—
28C2×C42×D61C2211yescharacteristic
29#16839815C2211yescharacteristic
30#16839814—21no—
31#16839813—21no—
32#16839812C2212yes—
33C4⋊C8×D61—21no—
34C22⋊C8×D61C2212yes—
35C4×C8×D61—21no—
36#16839631—21no—
37#16839628C2211yescharacteristic
38#16839621—21no—
39#16839621C2212yes—
40#16839613—21no—
41#16839613C2212yes—
42#16839500C2214yes—
43#16839499C2214yes—
44#16839498C2214yes—
45C61×#533C2211yescharacteristic
46C61×#431C2211yescharacteristic
47C61×#382C2211yescharacteristic
48C61×#328C2214yes—
49C2×Q8×D61—21no—
50C2×Q8×D61C2×C412yes—
51C22×C4×D61D411yescharacteristic
52C22×C4×D61D411yescharacteristic
53C22×C4×D61C2311yescharacteristic
54C2×C8×D61—22no—
55#14519777C2×C411yescharacteristic
56#14519777C2×C411yescharacteristic
57#14519776—21no—
58#14519776—21no—
59#14519776—21no—
60#14519776C2×C412yes—
61#14519775—21no—
62#14519775C2×C412yes—
63#14519769—21no—
64#14519768—21no—
65#14519768—21no—
66#14519767—22no—
67#14519767—22no—
68#14519757—21no—
69#14519757C2×C412yes—
70#14519756—21no—
71#14519756C2312yes—
72#14519755—22no—
73#14519755—22no—
74#14519754—22no—
75#14519754—22no—
76#14519753—21no—
77#14519753C2×C412yes—
78Q8×Dic61—21no—
79Q8×Dic61C2×C412yes—
80C4×Dic122—21no—
81C4×Dic122—22no—
82C4×Dic122—22no—
83C4×Dic122C2×C412yes—
84#14519742—21no—
85#14519742—22no—
86#14519742—22no—
87#14519742C2×C412yes—
88#14519736—21no—
89#14519735—21no—
90#14519735—21no—
91#14519735—44no—
92#14519734—22no—
93#14519733—22no—
94C2×C4⋊Dic61—21no—
95C2×#11758196—22no—
96C2×#11758196—22no—
97C4⋊C4×D61—21no—
98C4⋊C4×D61—21no—
99C4⋊C4×D61—21no—
100C4⋊C4×D61—22no—
101C4⋊C4×D61—22no—
102C4⋊C4×D61C2×C412yes—
103C2×#11758198—22no—
104C22⋊C4×D61—22no—
105C2×C4×Dic61—22no—
106C2×C4×Dic61D411yescharacteristic
107C2×C4×Dic61D411yescharacteristic
108C2×C4×Dic61C2311yescharacteristic
109C42×D61—22no—
110C42×D61C2×C411yescharacteristic
111C42×D61C2×C411yescharacteristic
112C42×D61C2×C411yescharacteristic
113C42×D61C2×C411yescharacteristic
114C42×D61D411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61D411yescharacteristic
118C4⋊#92418—21no—
119#14519706—22no—
120C22⋊#92418C2312yes—
121#14519689—22no—
122C4×#92418—21no—
123C8×Dic61—22no—
124#14519684—21no—
125#14519684C2312yes—
126C4⋊Q8×C61C2×C411yescharacteristic
127C4⋊Q8×C61C2×C411yescharacteristic
128#125×C61—21no—
129C22⋊Q8×C61—21no—
130C22⋊Q8×C61C2312yes—
131C4×Q8×C61—21no—
132C4×Q8×C61C2×C412yes—
133#117×C61—21no—
134C2×C4⋊C4×C61—21no—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61—21no—
137C22⋊C8×C61C2312yes—
138C4×C8×C61—21no—
139#95×C61—21no—
140#95×C61C2312yes—
141C23×D61C2×D411yescharacteristic
142C2×Q8×C61—21no—
143C2×Q8×C61C22×C412yes—
144C22×C4×C61—22no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2411yescharacteristic
148C22×Dic61—22no—
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152Q8×D61—44no—
153Q8×D61—44no—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61—42no—
157C2×C4×D61—42no—
158C2×C4×D61C22⋊C411yescharacteristic
159C2×C4×D61C22⋊C411yescharacteristic
160C2×C4×D61C22⋊C411yescharacteristic
161C2×C4×D61C22⋊C411yescharacteristic
162C2×C4×D61C22⋊C411yescharacteristic
163C2×C4×D61C22⋊C411yescharacteristic
164C2×C4×D61C22⋊C411yescharacteristic
165C2×C4×D61C22⋊C411yescharacteristic
166C2×C4×D61SD1614yes—
167C2×C4×D61Q1614yes—
168C2×C4×D61C22×C411yescharacteristic
169C2×C4×D61C22×C411yescharacteristic
170C2×C4×D61C2×D411yescharacteristic
171C2×C4×D61C2×D411yescharacteristic
172C2×Dic122—21no—
173C2×Dic122—22no—
174C2×Dic122—22no—
175C2×Dic122C22×C412yes—
176C2×C8×C61—22no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—42no—
183C4⋊C4×C61C22×C412yes—
184C22⋊C4×C61—22no—
185C42×C61—22no—
186C42×C61C22×C411yescharacteristic
187C42×C61C22×C411yescharacteristic
188C42×C61C2×D411yescharacteristic
189C42×C61C2×D411yescharacteristic
190C22⋊Dic61—22no—
191#11758198—22no—
192#11758198—22no—
193#11758198—44no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—22no—
198C4⋊Dic61—22no—
199C4⋊Dic61—42no—
200C4⋊Dic61C22×C412yes—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207#11758196—22no—
208#11758196—22no—
209#11758196—44no—
210#11758196—44no—
211C4×Dic61—22no—
212C4×Dic61—22no—
213C4×Dic61—22no—
214C4×Dic61—44no—
215C4×Dic61C22⋊C411yescharacteristic
216C4×Dic61C22⋊C411yescharacteristic
217C4×Dic61C22⋊C411yescharacteristic
218C4×Dic61C22⋊C411yescharacteristic
219C4×Dic61C22⋊C411yescharacteristic
220C4×Dic61C22⋊C411yescharacteristic
221C4×Dic61C22⋊C411yescharacteristic
222C4×Dic61C22⋊C411yescharacteristic
223C4×Dic61C22×C411yescharacteristic
224C4×Dic61C22×C411yescharacteristic
225C4×Dic61C2×D411yescharacteristic
226C4×Dic61C2×D411yescharacteristic
227C2×#92418—22no—
228C8×D61—44no—
229C23×C61C22×D411yescharacteristic
230C22×D61—41no—
231C22×D61—41no—
232C22×D61C2×C22⋊C411yescharacteristic
233C22×D61C2×C22⋊C411yescharacteristic
234C22×D61C2×SD1612yes—
235C22×D61C2×Q1612yes—
236Q8×C61—42no—
237Q8×C61—42no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×SD1612yes—
250C2×C4×C61C2×Q1612yes—
251C2×C4×C61C23×C411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—42no—
258C2×Dic61—42no—
259C2×Dic61—42no—
260C2×Dic61C2×C22⋊C411yescharacteristic
261C2×Dic61C2×C22⋊C411yescharacteristic
262C2×Dic61C2×C22⋊C411yescharacteristic
263C2×Dic61C2×C22⋊C411yescharacteristic
264C2×Dic61C2×C22⋊C411yescharacteristic
265C2×Dic61C2×C22⋊C411yescharacteristic
266C2×Dic61C2×SD1612yes—
267C2×Dic61C2×Q1612yes—
268C4×D61—22no—
269C4×D61—22no—
270C4×D61—22no—
271C4×D61—22no—
272C4×D61—41no—
273C4×D61—41no—
274C4×D61—41no—
275C4×D61—41no—
276C4×D61—44no—
277C4×D61—44no—
278C4×D61#10314yes—
279C4×D61#10314yes—
280C4×D61#10314yes—
281C4×D61#10314yes—
282Dic122—42no—
283Dic122—44no—
284Dic122—44no—
285Dic122—42no—
286C8×C61—42no—
287#92418—42no—
288#17575—611nomaximal, Sylow
289C22×C61—21no—
290C22×C61—21no—
291C22×C61C22×C22⋊C411yescharacteristic
292C22×C61C22×SD1611yescharacteristic
293C22×C61C22×Q1611yescharacteristic
294C2×D61—21no—
295C2×D61—21no—
296C2×D61—21no—
297C2×D61—21no—
298C2×D61—42no—
299C2×D61—42no—
300C2×D61C2×#10312yes—
301C2×D61C2×#10312yes—
302C4×C61—21no—
303C4×C61—21no—
304C4×C61—21no—
305C4×C61—21no—
306C4×C61—42no—
307C4×C61—42no—
308C4×C61—82no—
309C4×C61C2×#10312yes—
310C4×C61C2×#10312yes—
311Dic61—21no—
312Dic61—21no—
313Dic61—42no—
314Dic61—42no—
315Dic61—82no—
316Dic61#41012yes—
317Dic61C2×#10312yes—
318Dic61C2×#10312yes—
319Dic61#42112yes—
320C2×#533—611no—
321C2×#431—611no—
322#2285—611no—
323#1520—614no—
324#1518—611no—
325#1518—611no—
326#1518—611no—
327#1518D6111yescharacteristic, complemented
328#1503—614no—
329C2×C61—21no—
330C2×C61—21no—
331C2×C61C2×#41011yescharacteristic
332C2×C61C22×#10311yescharacteristic
333C2×C61C2×#42111yescharacteristic
334D61—22no—
335D61—22no—
336#533—611no—
337#533—611no—
338#533—612no—
339#533—612no—
340#533—611no—
341#533—1221no—
342#533—1221no—
343#533—1221no—
344#533—1221no—
345#533C2×D6111yescharacteristic
346C2×C4⋊Q8—611no—
347C2×C4⋊Q8—611no—
348C2×#125—1221no—
349C2×C22⋊Q8—612no—
350C2×C22⋊Q8—1221no—
351C2×C4×Q8—612no—
352C2×C4×Q8—1221no—
353C2×#117—1221no—
354C2×#117—1221no—
355C22×C42—611no—
356#431—611no—
357#431—612no—
358#431—1221no—
359#431—1221no—
360#431C2×D6111yescharacteristic
361C2×C4⋊C8—1221no—
362C2×C22⋊C8—612no—
363C2×C4×C8—1221no—
364#384—1221no—
365#382—611no—
366#382C2×D6111yescharacteristic
367#380—612no—
368#380—1221no—
369C4×C4⋊C4—612no—
370C4×C4⋊C4—1221no—
371#328—614no—
372#328—614no—
373#328—614no—
374#328C2×D6114yes—
375C61#1757511yescharacteristic, complemented, Sylow
376C22×Q8—612no—
377C22×Q8—1221no—
378C23×C4—611no—
379C23×C4—611no—
380C23×C4—611no—
381C22×C8—1222no—
382C4⋊Q8—612no—
383C4⋊Q8—612no—
384C4⋊Q8—611no—
385C4⋊Q8—611no—
386C4⋊Q8—1221no—
387C4⋊Q8—1221no—
388C4⋊Q8—1221no—
389C4⋊Q8—1221no—
390C4⋊Q8C4×D6111yescharacteristic
391C4⋊Q8C4×D6111yescharacteristic
392#125—21no—
393#125—1222no—
394#125—1222no—
395#125—1221no—
396#125—1221no—
397#125—1221no—
398C22⋊Q8—21no—
399C22⋊Q8—612no—
400C22⋊Q8—612no—
401C22⋊Q8—612no—
402C22⋊Q8—1222no—
403C22⋊Q8—1222no—
404C22⋊Q8—1222no—
405C22⋊Q8—1222no—
406C22⋊Q8—1221no—
407C22⋊Q8—1221no—
408C22⋊Q8—1221no—
409C22⋊Q8C22×D6112yes—
410C4×Q8—21no—
411C4×Q8—612no—
412C4×Q8—612no—
413C4×Q8—612no—
414C4×Q8—1222no—
415C4×Q8—1222no—
416C4×Q8—1222no—
417C4×Q8—1222no—
418C4×Q8—1221no—
419C4×Q8—1221no—
420C4×Q8—1221no—
421C4×Q8C4×D6112yes—
422#117—21no—
423#117—1222no—
424#117—1222no—
425#117—1221no—
426#117—1221no—
427#117—1221no—
428#117—2444no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—612no—
431C2×C4⋊C4—1221no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1222no—
434C2×C4⋊C4—1222no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1222no—
437C2×C4⋊C4—1222no—
438C2×C4⋊C4—1221no—
439C2×C22⋊C4—1222no—
440C2×C22⋊C4—1222no—
441C2×C42—611no—
442C2×C42—611no—
443C2×C42—611no—
444C2×C42—611no—
445C2×C42—611no—
446C2×C42—611no—
447C2×C42—611no—
448C2×C42—611no—
449C2×C42—611no—
450C2×C42—611no—
451C2×C42—611no—
452C2×C42—1222no—
453C2×C42—1222no—
454C2×C42C22×D6111yescharacteristic
455C4⋊C8—21no—
456C4⋊C8—1222no—
457C4⋊C8—1221no—
458C22⋊C8—612no—
459C22⋊C8—1222no—
460C22⋊C8C22×D6112yes—
461C4×C8—21no—
462C4×C8—1222no—
463C4×C8—1221no—
464#95—21no—
465#95—612no—
466#95—1221no—
467#95C22×D6112yes—
468C24—611no—
469C2×Q8—21no—
470C2×Q8—612no—
471C2×Q8—1222no—
472C2×Q8—1222no—
473C2×Q8—1221no—
474C2×Q8—2444no—
475C2×Q8—2444no—
476C2×Q8C2×C4×D6112yes—
477C22×C4—22no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—614no—
484C22×C4—614no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—611no—
490C22×C4—611no—
491C22×C4—611no—
492C22×C4—611no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—1222no—
498C22×C4—2442no—
499C22×C4—2442no—
500C22×C4D4×D6111yescharacteristic
501C22×C4D4×D6111yescharacteristic
502C22×C4C23×D6111yescharacteristic
503C2×C8—22no—
504C2×C8—1222no—
505C2×C8—2444no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—42no—
512C4⋊C4—612no—
513C4⋊C4—1222no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—1222no—
518C4⋊C4—1222no—
519C4⋊C4—1222no—
520C4⋊C4—1222no—
521C4⋊C4—1222no—
522C4⋊C4—1222no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—2444no—
527C4⋊C4—2444no—
528C4⋊C4—2442no—
529C4⋊C4C2×C4×D6112yes—
530C22⋊C4—22no—
531C22⋊C4—1222no—
532C22⋊C4—1222no—
533C22⋊C4—1222no—
534C22⋊C4—2444no—
535C42—22no—
536C42—611no—
537C42—611no—
538C42—611no—
539C42—611no—
540C42—611no—
541C42—611no—
542C42—611no—
543C42—611no—
544C42—611no—
545C42—611no—
546C42—611no—
547C42—611no—
548C42—1222no—
549C42—1222no—
550C42—1222no—
551C42—2444no—
552C42C2×C4×D6111yescharacteristic
553C42C2×C4×D6111yescharacteristic
554C42D4×D6111yescharacteristic
555C42D4×D6111yescharacteristic
556C23—611no—
557C23—611no—
558C23—612no—
559C23—612no—
560C23—2441no—
561C23—2441no—
562C23C2×D4×D6111yescharacteristic
563Q8—42no—
564Q8—42no—
565Q8—2442no—
566Q8—2442no—
567Q8—2444no—
568Q8—2444no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—611no—
577C2×C4—611no—
578C2×C4—611no—
579C2×C4—611no—
580C2×C4—612no—
581C2×C4—612no—
582C2×C4—614no—
583C2×C4—614no—
584C2×C4—614no—
585C2×C4—614no—
586C2×C4—611no—
587C2×C4—611no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—1222no—
591C2×C4—1222no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2444no—
601C2×C4—2444no—
602C2×C4—2442no—
603C2×C4—2442no—
604C2×C4—2442no—
605C2×C4C22⋊C4×D6111yescharacteristic
606C2×C4C22⋊C4×D6111yescharacteristic
607C2×C4C22⋊C4×D6111yescharacteristic
608C2×C4C22⋊C4×D6111yescharacteristic
609C2×C4SD16×D6112yes—
610C2×C4Q16×D6112yes—
611C2×C4C22×C4×D6111yescharacteristic
612C2×C4C2×D4×D6111yescharacteristic
613C8—42no—
614C8—2442no—
615C22—21no—
616C22—21no—
617C22—612no—
618C22—612no—
619C22—1221no—
620C22—1221no—
621C22—1221no—
622C22—1221no—
623C22—2442no—
624C22—2442no—
625C22C2×C22⋊C4×D6111yescharacteristic
626C22C2×SD16×D6111yescharacteristic
627C22C2×Q16×D6111yescharacteristic
628C4—21no—
629C4—21no—
630C4—21no—
631C4—21no—
632C4—42no—
633C4—42no—
634C4—82no—
635C4—612no—
636C4—612no—
637C4—612no—
638C4—612no—
639C4—1221no—
640C4—1221no—
641C4—4882no—
642C4—2442no—
643C4—2442no—
644C4#103×D6112yes—
645C4#103×D6112yes—
646C2—21no—
647C2—21no—
648C2—1222no—
649C2—1222no—
650C2#410×D6111yescharacteristic
651C2C2×#103×D6111yescharacteristic
652C2#421×D6111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1518C2
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1518D61
hypocenterC61#17575
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1518

Complements

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

NH ≅ G/Ntype
5. #21444384C2semidirect
6. #21444383C2semidirect
7. #21444382C2semidirect
8. #21444316C2semidirect
9. C61×#1518C2semidirect
12. #16840317C4semidirect
13. #16840317C4semidirect
15. #16840316C4semidirect
19. C4⋊Q8×D61C4semidirect
20. C4⋊Q8×D61C4semidirect
25. C4×Q8×D61C4semidirect
327. #1518D61semidirect
375. C61#17575semidirect

Maximal subgroups

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

subgroupindex
#533×D612
#431×D612
#214482252
#214443892
#214443842
#214443832
#214443822
#214443162
C61×#15182
#1757561