Finite Group Databasebeta

← #29647308

Subgroups — #29647308

Order 15616.

992 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#533×D61C211yesmaximal, characteristic
2#21454447C211yesmaximal, characteristic
3#21448215C211yesmaximal, characteristic
4#21444368C214yesmaximal, complemented
5#21444350C214yesmaximal
6#21444319C211yesmaximal, characteristic, complemented
7#21444318C211yesmaximal, characteristic, complemented
8#21444317C211yesmaximal, characteristic, complemented
9C61×#1503C211yesmaximal, 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
26#117×D61—21no—
27C2×C42×D61C2211yescharacteristic
28#16839818—21no—
29#16839816C2212yes—
30#16839815C2211yescharacteristic
31#16839813—21no—
32#16839784—21no—
33#16839724C2212yes—
34C4×#11758189—21no—
35#16839629C2212yes—
36#16839627—21no—
37#16839624C2212yes—
38#16839619—21no—
39C4×C4⋊Dic61C2211yescharacteristic
40C4×#11758198—21no—
41#16839608—21no—
42#16839500C2214yes—
43#16839499C2214yes—
44#16839498C2214yes—
45C61×#533C2211yescharacteristic
46C61×#431C2211yescharacteristic
47C4×C4⋊C4×C61C2211yescharacteristic
48C61×#328C2214yes—
49C2×Q8×D61—21no—
50C2×Q8×D61C2×C412yes—
51C22×C4×D61D411yescharacteristic
52C22×C4×D61D411yescharacteristic
53C22×C4×D61C2311yescharacteristic
54C2×#11758189—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#14519734—22no—
92#14519733—22no—
93C2×C4⋊Dic61—21no—
94C4⋊C4×D61—21no—
95C4⋊C4×D61—21no—
96C4⋊C4×D61—21no—
97C4⋊C4×D61—22no—
98C4⋊C4×D61—22no—
99C4⋊C4×D61C2×C412yes—
100C2×#11758198—22no—
101C22⋊C4×D61—22no—
102C2×C4×Dic61—21no—
103C2×C4×Dic61D411yescharacteristic
104C2×C4×Dic61D411yescharacteristic
105C2×C4×Dic61C2311yescharacteristic
106C42×D61—22no—
107C42×D61C2×C411yescharacteristic
108C42×D61C2×C411yescharacteristic
109C42×D61C2×C411yescharacteristic
110C42×D61C2×C411yescharacteristic
111C42×D61D411yescharacteristic
112C42×D61D411yescharacteristic
113C42×D61D411yescharacteristic
114C42×D61D411yescharacteristic
115C4⋊#92418—21no—
116#14519706—22no—
117C22⋊#92418C2312yes—
118#14519689—22no—
119#14519687—22no—
120C4×#92418—21no—
121#14519684—22no—
122#14519684—22no—
123#14519684—22no—
124#14519684C2312yes—
125C4⋊Q8×C61C2×C411yescharacteristic
126C4⋊Q8×C61C2×C411yescharacteristic
127#125×C61—21no—
128C22⋊Q8×C61—21no—
129C22⋊Q8×C61C2312yes—
130C4×Q8×C61—21no—
131C4×Q8×C61C2×C412yes—
132#117×C61—21no—
133C2×C4⋊C4×C61—21no—
134C2×C42×C61—21no—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61—21no—
137C22⋊C8×C61C2312yes—
138C4×C8×C61—21no—
139#95×C61C2312yes—
140C23×D61C2×D411yescharacteristic
141C2×Q8×C61—21no—
142C2×Q8×C61C22×C412yes—
143C22×C4×C61—22no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2411yescharacteristic
147C22×Dic61—22no—
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151Q8×D61—44no—
152Q8×D61—44no—
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—42no—
158C2×C4×D61—42no—
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×D61C22⋊C411yescharacteristic
167C2×C4×D61C22×C411yescharacteristic
168C2×C4×D61C22×C411yescharacteristic
169C2×C4×D61C2×D411yescharacteristic
170C2×C4×D61C2×D411yescharacteristic
171C2×Dic122—21no—
172C2×Dic122—22no—
173C2×Dic122—22no—
174C2×Dic122C22×C412yes—
175C2×C8×C61—22no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—42no—
182C4⋊C4×C61C22×C412yes—
183C22⋊C4×C61—22no—
184C42×C61—22no—
185C42×C61—42no—
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—
209C4×Dic61—22no—
210C4×Dic61—22no—
211C4×Dic61—22no—
212C4×Dic61—42no—
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
225C2×#92418—22no—
226#11758189—44no—
227C23×C61C22×D411yescharacteristic
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—22no—
231C22×D61—41no—
232C22×D61C2×C22⋊C411yescharacteristic
233C22×D61C2×C22⋊C411yescharacteristic
234Q8×C61—42no—
235Q8×C61—42no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—41no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61C2×C22⋊C411yescharacteristic
244C2×C4×C61C2×C22⋊C411yescharacteristic
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C2×C4×C61#13712yes—
248C2×C4×C61#13712yes—
249C2×C4×C61C23×C411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—41no—
256C2×Dic61—41no—
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
266C4×D61—22no—
267C4×D61—22no—
268C4×D61—22no—
269C4×D61—22no—
270C4×D61—22no—
271C4×D61—22no—
272C4×D61—22no—
273C4×D61—22no—
274C4×D61—41no—
275C4×D61—41no—
276C4×D61—41no—
277C4×D61—41no—
278C4×D61—44no—
279C4×D61—44no—
280Dic122—44no—
281Dic122—44no—
282Dic122—42no—
283Dic122—42no—
284C8×C61—42no—
285#92418—42no—
286#17562—611nomaximal, Sylow
287C22×C61—21no—
288C22×C61—21no—
289C22×C61C22×C22⋊C411yescharacteristic
290C22×C61C2×#13711yescharacteristic
291C22×C61C2×#13711yescharacteristic
292C2×D61—21no—
293C2×D61—21no—
294C2×D61—21no—
295C2×D61—21no—
296C2×D61—21no—
297C2×D61—42no—
298C2×D61—42no—
299C2×D61#41012yes—
300C4×C61—21no—
301C4×C61—21no—
302C4×C61—21no—
303C4×C61—42no—
304C4×C61—42no—
305C4×C61—82no—
306C4×C61#41912yes—
307C4×C61#41912yes—
308C4×C61#42112yes—
309Dic61—21no—
310Dic61—21no—
311Dic61—21no—
312Dic61—21no—
313Dic61—21no—
314Dic61—21no—
315Dic61—42no—
316Dic61—42no—
317Dic61—82no—
318C2×#533—611no—
319#2920—611no—
320#2283—611no—
321#1515—614no—
322#1511—614no—
323#1503—611no—
324#1503—611no—
325#1503—611no—
326#1503D6111yescharacteristic, complemented
327C2×C61—21no—
328C2×C61—21no—
329C2×C61C2×#41011yescharacteristic
330C2×C61C2×#41911yescharacteristic
331C2×C61C2×#42111yescharacteristic
332D61—22no—
333D61—41no—
334#533—611no—
335#533—611no—
336#533—612no—
337#533—612no—
338#533—611no—
339#533—1221no—
340#533—1221no—
341#533—1221no—
342#533—1221no—
343#533C2×D6111yescharacteristic
344C2×C4⋊Q8—611no—
345C2×C4⋊Q8—611no—
346C2×#125—1221no—
347C2×C22⋊Q8—612no—
348C2×C22⋊Q8—1221no—
349C2×C4×Q8—612no—
350C2×C4×Q8—1221no—
351C2×#117—1221no—
352C22×C42—611no—
353#432—612no—
354#432—1221no—
355#431—611no—
356#431—1221no—
357#431C2×D6111yescharacteristic
358#423—1221no—
359#407—612no—
360C4×M16—1221no—
361#383—612no—
362#382—1221no—
363#381—612no—
364#380—1221no—
365C4×C4⋊C4—611no—
366C4×C4⋊C4C2×D6111yescharacteristic
367C4×C22⋊C4—1221no—
368#376—1221no—
369#328—614no—
370#328—614no—
371#328—614no—
372#328C2×D6114yes—
373C61#1756211yescharacteristic, complemented, Sylow
374C22×Q8—612no—
375C22×Q8—1221no—
376C23×C4—611no—
377C23×C4—611no—
378C23×C4—611no—
379C2×M16—1222no—
380C4⋊Q8—612no—
381C4⋊Q8—612no—
382C4⋊Q8—611no—
383C4⋊Q8—611no—
384C4⋊Q8—1221no—
385C4⋊Q8—1221no—
386C4⋊Q8—1221no—
387C4⋊Q8—1221no—
388C4⋊Q8C4×D6111yescharacteristic
389C4⋊Q8C4×D6111yescharacteristic
390#125—21no—
391#125—1222no—
392#125—1222no—
393#125—1221no—
394#125—1221no—
395#125—1221no—
396C22⋊Q8—21no—
397C22⋊Q8—612no—
398C22⋊Q8—612no—
399C22⋊Q8—612no—
400C22⋊Q8—1222no—
401C22⋊Q8—1222no—
402C22⋊Q8—1222no—
403C22⋊Q8—1222no—
404C22⋊Q8—1221no—
405C22⋊Q8—1221no—
406C22⋊Q8—1221no—
407C22⋊Q8C22×D6112yes—
408C4×Q8—21no—
409C4×Q8—612no—
410C4×Q8—612no—
411C4×Q8—612no—
412C4×Q8—1222no—
413C4×Q8—1222no—
414C4×Q8—1222no—
415C4×Q8—1222no—
416C4×Q8—1221no—
417C4×Q8—1221no—
418C4×Q8—1221no—
419C4×Q8C4×D6112yes—
420#117—21no—
421#117—1222no—
422#117—1222no—
423#117—1221no—
424#117—1221no—
425#117—1221no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—612no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4—1222no—
431C2×C4⋊C4—1222no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C22⋊C4—1222no—
435C2×C22⋊C4—1222no—
436C2×C42—21no—
437C2×C42—611no—
438C2×C42—611no—
439C2×C42—611no—
440C2×C42—611no—
441C2×C42—611no—
442C2×C42—611no—
443C2×C42—611no—
444C2×C42—611no—
445C2×C42—611no—
446C2×C42—611no—
447C2×C42—611no—
448C2×C42—1222no—
449C2×C42—1221no—
450C2×C42C22×D6111yescharacteristic
451C4⋊C8—21no—
452C4⋊C8—1222no—
453C4⋊C8—1221no—
454C22⋊C8—612no—
455C22⋊C8—1222no—
456C22⋊C8C22×D6112yes—
457#97—1222no—
458C4×C8—21no—
459C4×C8—1221no—
460#95—612no—
461#95—1222no—
462#95—1222no—
463#95—1222no—
464#95C22×D6112yes—
465C24—611no—
466C2×Q8—21no—
467C2×Q8—612no—
468C2×Q8—1222no—
469C2×Q8—1222no—
470C2×Q8—1221no—
471C2×Q8—2444no—
472C2×Q8—2444no—
473C2×Q8C2×C4×D6112yes—
474C22×C4—22no—
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—611no—
488C22×C4—611no—
489C22×C4—611no—
490C22×C4—1222no—
491C22×C4—1222no—
492C22×C4—1222no—
493C22×C4—1222no—
494C22×C4—1222no—
495C22×C4—2442no—
496C22×C4—2442no—
497C22×C4D4×D6111yescharacteristic
498C22×C4D4×D6111yescharacteristic
499C22×C4C23×D6111yescharacteristic
500M16—2444no—
501C2×C8—22no—
502C2×C8—1222no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—42no—
509C4⋊C4—612no—
510C4⋊C4—1222no—
511C4⋊C4—1222no—
512C4⋊C4—1222no—
513C4⋊C4—1222no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—1222no—
518C4⋊C4—1222no—
519C4⋊C4—1222no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—2442no—
524C4⋊C4C2×C4×D6112yes—
525C22⋊C4—22no—
526C22⋊C4—1222no—
527C22⋊C4—1222no—
528C22⋊C4—1222no—
529C22⋊C4—2444no—
530C42—22no—
531C42—42no—
532C42—611no—
533C42—611no—
534C42—611no—
535C42—611no—
536C42—611no—
537C42—611no—
538C42—611no—
539C42—611no—
540C42—611no—
541C42—611no—
542C42—611no—
543C42—611no—
544C42—1222no—
545C42—1222no—
546C42—1222no—
547C42—2442no—
548C42C2×C4×D6111yescharacteristic
549C42C2×C4×D6111yescharacteristic
550C42D4×D6111yescharacteristic
551C42D4×D6111yescharacteristic
552C23—611no—
553C23—611no—
554C23—1222no—
555C23—1221no—
556C23—1221no—
557C23—2441no—
558C23C2×D4×D6111yescharacteristic
559Q8—42no—
560Q8—42no—
561Q8—2442no—
562Q8—2444no—
563Q8—2444no—
564Q8—2442no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—611no—
573C2×C4—611no—
574C2×C4—611no—
575C2×C4—611no—
576C2×C4—611no—
577C2×C4—611no—
578C2×C4—1222no—
579C2×C4—1222no—
580C2×C4—1221no—
581C2×C4—1221no—
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—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2444no—
595C2×C4—2444no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2442no—
599C2×C4—2442no—
600C2×C4—2442no—
601C2×C4C22⋊C4×D6111yescharacteristic
602C2×C4C22⋊C4×D6111yescharacteristic
603C2×C4C22⋊C4×D6111yescharacteristic
604C2×C4C22⋊C4×D6111yescharacteristic
605C2×C4#1451980612yes—
606C2×C4#1451980912yes—
607C2×C4C22×C4×D6111yescharacteristic
608C2×C4C2×D4×D6111yescharacteristic
609C8—42no—
610C8—2442no—
611C22—21no—
612C22—21no—
613C22—612no—
614C22—1221no—
615C22—1221no—
616C22—1221no—
617C22—1221no—
618C22—1221no—
619C22—2442no—
620C22—2442no—
621C22C2×C22⋊C4×D6111yescharacteristic
622C22C2×#1451980611yescharacteristic
623C22C2×#1451980911yescharacteristic
624C4—21no—
625C4—21no—
626C4—21no—
627C4—42no—
628C4—42no—
629C4—82no—
630C4—1221no—
631C4—1221no—
632C4—1221no—
633C4—1221no—
634C4—1221no—
635C4—1221no—
636C4—4882no—
637C4—2442no—
638C4—2442no—
639C4#1683976912yes—
640C4#1683976912yes—
641C4#1683977712yes—
642C2—21no—
643C2—21no—
644C2—1222no—
645C2—2441no—
646C2#410×D6111yescharacteristic
647C2C2×#1683976911yescharacteristic
648C2C2×#1683977711yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1503C2
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1503D61
hypocenterC61#17562
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
4. #21444368C2semidirect
6. #21444319C2semidirect
7. #21444318C2semidirect
8. #21444317C2semidirect
9. C61×#1503C2semidirect
12. #16840317C4semidirect
13. #16840317C4semidirect
15. #16840316C4semidirect
19. C4⋊Q8×D61C4semidirect
20. C4⋊Q8×D61C4semidirect
25. C4×Q8×D61C4semidirect
326. #1503D61semidirect
373. C61#17562semidirect

Maximal subgroups

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

subgroupindex
#533×D612
#214544472
#214482152
#214443682
#214443502
#214443192
#214443182
#214443172
C61×#15032
#1756261