Finite Group Databasebeta

← #29647527

Subgroups — #29647527

Order 15616.

994 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840314C211yesmaximal, characteristic
2C2×#16839816C211yesmaximal, characteristic
3#21448214C211yesmaximal, characteristic
4#21444392C211yesmaximal, characteristic
5#21444391C211yesmaximal, characteristic
6#21444390C211yesmaximal, characteristic
7#21444368C214yesmaximal, complemented
8#21444329C214yesmaximal, complemented
9C61×#1520C211yesmaximal, 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#16839816—21no—
30#16839816C2212yes—
31#16839816C2212yes—
32C2×#14519706—21no—
33C2×C22⋊#92418C2211yescharacteristic
34C2×#14519687—21no—
35#16839629C2212yes—
36#16839627—21no—
37#16839623—21no—
38#16839622C2211yescharacteristic
39C4⋊C4×Dic61—21no—
40C22⋊C4×Dic61—21no—
41#16839607C2212yes—
42#16839500C2214yes—
43#16839499C2214yes—
44#16839498C2214yes—
45C2×C22⋊Q8×C61C2211yescharacteristic
46C2×C22⋊C8×C61C2211yescharacteristic
47C61×#380C2211yescharacteristic
48C61×#328C2214yes—
49C22×Dic122—21no—
50C22×Dic122C2×C411yescharacteristic
51C23×Dic61D411yescharacteristic
52C23×Dic61D411yescharacteristic
53C22×#92418—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⋊Dic61—21no—
92C2×C4⋊Dic61C2×C411yescharacteristic
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—21no—
96C2×#11758196—21no—
97C2×#11758196—21no—
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—
120#14519687—22no—
121#14519684—22no—
122#14519684—22no—
123#14519684—22no—
124#14519684C2312yes—
125C22×Q8×C61C2×C411yescharacteristic
126C23×C4×C61C2311yescharacteristic
127C22×C8×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×C61—21no—
134C2×C4⋊C4×C61C2×C411yescharacteristic
135C2×C22⋊C4×C61—21no—
136C2×C22⋊C4×C61—21no—
137C22⋊C8×C61—21no—
138C22⋊C8×C61C2312yes—
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—22no—
155C22×Dic61—22no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—22no—
159C22×Dic61—42no—
160C22×Dic61C22⋊C411yescharacteristic
161C22×Dic61C22⋊C411yescharacteristic
162C22×Dic61C22⋊C411yescharacteristic
163C22×Dic61C22⋊C411yescharacteristic
164C22×Dic61C22⋊C411yescharacteristic
165C22×Dic61C22⋊C411yescharacteristic
166C22×Dic61C22⋊C411yescharacteristic
167C22×Dic61C22⋊C411yescharacteristic
168C22×Dic61C2×D411yescharacteristic
169C22×Dic61C2×D411yescharacteristic
170C22×Dic61C2×D411yescharacteristic
171C22×Dic61C2×D411yescharacteristic
172C2×Dic122—21no—
173C2×Dic122—22no—
174C2×Dic122—22no—
175C2×Dic122—44no—
176C2×Dic122—42no—
177C2×Dic122C22×C412yes—
178C2×C8×C61—22no—
179C2×C8×C61—42no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—42no—
184C4⋊C4×C61C22×C412yes—
185C22⋊C4×C61—22no—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—42no—
188C22⋊Dic61—22no—
189C22⋊Dic61—22no—
190C22⋊Dic61—42no—
191C4⋊Dic61—21no—
192C4⋊Dic61—22no—
193C4⋊Dic61—22no—
194C4⋊Dic61—42no—
195C4⋊Dic61C22×C412yes—
196#11758196—22no—
197#11758196—22no—
198#11758196—22no—
199#11758196—22no—
200#11758196—22no—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206C4×Dic61—22no—
207C4×Dic61—22no—
208C4×Dic61—22no—
209C4×Dic61—22no—
210C4×Dic61—44no—
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×Dic61C22×C411yescharacteristic
222C4×Dic61C22×C411yescharacteristic
223C4×Dic61C2×D411yescharacteristic
224C4×Dic61C2×D411yescharacteristic
225C4×Dic61C2×D411yescharacteristic
226C4×Dic61C2×D411yescharacteristic
227C2×#92418—22no—
228C2×#92418—42no—
229C23×C61—41no—
230C23×C61—41no—
231C23×C61C2×C22⋊C411yescharacteristic
232C23×C61C2×C22⋊C411yescharacteristic
233C23×C61#13712yes—
234C23×C61#13712yes—
235C23×C61C22×D411yescharacteristic
236Q8×C61—42no—
237Q8×C61—42no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61—42no—
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61C2×C22⋊C411yescharacteristic
253C2×C4×C61C23×C411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—22no—
265C2×Dic61—22no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—42no—
272C2×Dic61—42no—
273C2×Dic61—42no—
274C2×Dic61—42no—
275C2×Dic61—42no—
276C2×Dic61C2×C22⋊C411yescharacteristic
277C2×Dic61C2×C22⋊C411yescharacteristic
278C2×Dic61C2×C22⋊C411yescharacteristic
279C2×Dic61C2×C22⋊C411yescharacteristic
280C2×Dic61C2×C22⋊C411yescharacteristic
281C2×Dic61C2×C22⋊C411yescharacteristic
282C2×Dic61C2×C22⋊C411yescharacteristic
283C2×Dic61C2×C22⋊C411yescharacteristic
284Dic122—44no—
285Dic122—42no—
286Dic122—44no—
287Dic122—42no—
288C8×C61—42no—
289#92418—42no—
290#17596—611nomaximal, Sylow
291C22×C61—21no—
292C22×C61—21no—
293C22×C61—21no—
294C22×C61—21no—
295C22×C61—21no—
296C22×C61—21no—
297C22×C61—42no—
298C22×C61—42no—
299C22×C61#41912yes—
300C22×C61#41912yes—
301C22×C61C22×C22⋊C411yescharacteristic
302C22×C61C2×#13711yescharacteristic
303C22×C61C2×#13711yescharacteristic
304C4×C61—21no—
305C4×C61—21no—
306C4×C61—21no—
307C4×C61—42no—
308C4×C61—42no—
309C4×C61—82no—
310C4×C61#41012yes—
311Dic61—21no—
312Dic61—21no—
313Dic61—21no—
314Dic61—21no—
315Dic61—21no—
316Dic61—21no—
317Dic61—21no—
318Dic61—42no—
319Dic61—42no—
320Dic61—82no—
321Dic61#42112yes—
322C2×#533—611no—
323C2×#432—611no—
324#2283—611no—
325#1520—611no—
326#1520—611no—
327#1520—611no—
328#1520D6111yescharacteristic
329#1515—614no—
330#1506—614no—
331C2×C61—21no—
332C2×C61—21no—
333C2×C61—22no—
334C2×C61—22no—
335C2×C61C2×#41011yescharacteristic
336C2×C61C2×#41911yescharacteristic
337C2×C61C2×#42111yescharacteristic
338#533—612no—
339#533—612no—
340#533—612no—
341#533—612no—
342#533—1221no—
343#533—1221no—
344#533—1221no—
345#533—1221no—
346C2×C4⋊Q8—611no—
347C2×C4⋊Q8—611no—
348C2×#125—1221no—
349C2×C22⋊Q8—611no—
350C2×C22⋊Q8—1221no—
351C2×C22⋊Q8C2×D6111yescharacteristic
352C2×C4×Q8—611no—
353C2×C4×Q8—611no—
354C2×C4×Q8—1221no—
355C2×#117—1221no—
356C22×C42—611no—
357#432—612no—
358#432—612no—
359#432—1221no—
360#432—1221no—
361C2×C4⋊C8—1221no—
362C2×C22⋊C8—611no—
363C2×C22⋊C8C2×D6111yescharacteristic
364C2×#97—1221no—
365#383—612no—
366#382—1221no—
367#381—1221no—
368#380—611no—
369#380C2×D6111yescharacteristic
370C4×C4⋊C4—1221no—
371C4×C22⋊C4—1221no—
372#376—612no—
373#328—614no—
374#328—614no—
375#328—614no—
376#328C2×D6114yes—
377C61#1759611yescharacteristic, complemented, Sylow
378C22×Q8—611no—
379C22×Q8—1221no—
380C22×Q8C4×D6111yescharacteristic
381C23×C4—611no—
382C23×C4—611no—
383C23×C4C22×D6111yescharacteristic
384C22×C8—21no—
385C22×C8—1221no—
386C4⋊Q8—612no—
387C4⋊Q8—612no—
388C4⋊Q8—612no—
389C4⋊Q8—612no—
390C4⋊Q8—1221no—
391C4⋊Q8—1221no—
392C4⋊Q8—1221no—
393C4⋊Q8—1221no—
394#125—1222no—
395#125—1222no—
396#125—1222no—
397#125—1222no—
398C22⋊Q8—21no—
399C22⋊Q8—21no—
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⋊Q8C4×D6112yes—
409C22⋊Q8C22×D6112yes—
410C4×Q8—612no—
411C4×Q8—612no—
412C4×Q8—612no—
413C4×Q8—612no—
414C4×Q8—1221no—
415C4×Q8—1221no—
416C4×Q8—1222no—
417C4×Q8—1222no—
418C4×Q8—1222no—
419C4×Q8—1222no—
420C4×Q8—1221no—
421C4×Q8—1221no—
422#117—1222no—
423#117—1222no—
424#117—1222no—
425#117—1222no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4—1221no—
431C2×C4⋊C4—1221no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4C4×D6111yescharacteristic
437C2×C22⋊C4—21no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
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—611no—
453C2×C42—1221no—
454C2×C42—1221no—
455C2×C42—1222no—
456C4⋊C8—1222no—
457C4⋊C8—1222no—
458C22⋊C8—21no—
459C22⋊C8—612no—
460C22⋊C8—1221no—
461C22⋊C8C22×D6112yes—
462#97—1222no—
463#97—1222no—
464#95—612no—
465#95—1222no—
466#95—1222no—
467#95—1222no—
468#95C22×D6112yes—
469C24D4×D6111yescharacteristic
470C2×Q8—21no—
471C2×Q8—42no—
472C2×Q8—612no—
473C2×Q8—1222no—
474C2×Q8—1222no—
475C2×Q8—1221no—
476C2×Q8—2442no—
477C2×Q8—2444no—
478C2×Q8C2×C4×D6112yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—42no—
483C22×C4—611no—
484C22×C4—611no—
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—1221no—
498C22×C4—1221no—
499C22×C4—1222no—
500C22×C4—2442no—
501C22×C4C2×C4×D6111yescharacteristic
502C22×C4C2×C4×D6111yescharacteristic
503C22×C4D4×D6111yescharacteristic
504C22×C4D4×D6111yescharacteristic
505C22×C4D4×D6111yescharacteristic
506C22×C4C23×D6111yescharacteristic
507C2×C8—22no—
508C2×C8—42no—
509C2×C8—1222no—
510C2×C8—2442no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—42no—
515C4⋊C4—612no—
516C4⋊C4—1222no—
517C4⋊C4—1222no—
518C4⋊C4—1222no—
519C4⋊C4—1222no—
520C4⋊C4—1222no—
521C4⋊C4—1222no—
522C4⋊C4—1222no—
523C4⋊C4—1222no—
524C4⋊C4—1222no—
525C4⋊C4—1222no—
526C4⋊C4—1222no—
527C4⋊C4—1221no—
528C4⋊C4—1222no—
529C4⋊C4—2442no—
530C4⋊C4C2×C4×D6112yes—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—42no—
534C22⋊C4—1222no—
535C22⋊C4—1222no—
536C22⋊C4—2442no—
537C42—611no—
538C42—611no—
539C42—611no—
540C42—611no—
541C42—611no—
542C42—611no—
543C42—611no—
544C42—611no—
545C42—611no—
546C42—611no—
547C42—611no—
548C42—611no—
549C42—611no—
550C42—611no—
551C42—611no—
552C42—611no—
553C42—1222no—
554C42—1222no—
555C42—1222no—
556C42—1222no—
557C42—2444no—
558C23—41no—
559C23—41no—
560C23C22⋊C4×D6111yescharacteristic
561C23C22⋊C4×D6111yescharacteristic
562C23#1451980612yes—
563C23#1451980912yes—
564C23C2×D4×D6111yescharacteristic
565Q8—42no—
566Q8—42no—
567Q8—2442no—
568Q8—2444no—
569Q8—2444no—
570Q8—2442no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—611no—
585C2×C4—611no—
586C2×C4—611no—
587C2×C4—611no—
588C2×C4—611no—
589C2×C4—611no—
590C2×C4—611no—
591C2×C4—611no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1222no—
604C2×C4—1222no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2442no—
609C2×C4—2442no—
610C2×C4—2442no—
611C2×C4—2442no—
612C2×C4—2442no—
613C2×C4C22⋊C4×D6111yescharacteristic
614C2×C4C22⋊C4×D6111yescharacteristic
615C2×C4C22×C4×D6111yescharacteristic
616C2×C4C2×D4×D6111yescharacteristic
617C8—42no—
618C8—2442no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—42no—
626C22—42no—
627C22#1683976912yes—
628C22#1683976912yes—
629C22C2×C22⋊C4×D6111yescharacteristic
630C22C2×#1451980611yescharacteristic
631C22C2×#1451980911yescharacteristic
632C4—21no—
633C4—21no—
634C4—21no—
635C4—42no—
636C4—42no—
637C4—82no—
638C4—612no—
639C4—1221no—
640C4—1221no—
641C4—1221no—
642C4—1221no—
643C4—1221no—
644C4—1221no—
645C4—1221no—
646C4—4882no—
647C4—2442no—
648C4—2442no—
649C4#1683973412yes—
650C2—21no—
651C2—21no—
652C2—22no—
653C2—22no—
654C2C2×#1683973411yescharacteristic
655C2C2×#1683976911yescharacteristic
656C2#421×D6111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1520C2
socleC22×C61C22×C22⋊C4
prime radicalC24×C61C2×D4
cosocleC22×C4×C61C24
hypercenter#1520D61
hypocenterC61#17596
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1520

Complements

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

NH ≅ G/Ntype
7. #21444368C2semidirect
8. #21444329C2semidirect
14. #16840314C4semidirect
15. #16840314C4semidirect
18. C2×#14519776C4semidirect
19. C2×#14519775C4semidirect
23. C2×Q8×Dic61C4semidirect
25. C2×#14519742C4semidirect
377. C61#17596semidirect

Maximal subgroups

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

subgroupindex
C2×#168403142
C2×#168398162
#214482142
#214443922
#214443912
#214443902
#214443682
#214443292
C61×#15202
#1759661