Finite Group Databasebeta

← #29647569

Subgroups — #29647569

Order 15616.

1046 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459851C211yesmaximal, characteristic
2#21448462C211yesmaximal, characteristic, complemented
3#21444388C212yesmaximal, complemented
4#21444376C212yesmaximal, complemented
5#21444363C212yesmaximal, complemented
6#21444349C212yesmaximal, complemented
7#21444345C211yesmaximal, characteristic, complemented
8#21444328C211yesmaximal, characteristic, complemented
9#21444315C211yesmaximal, characteristic, complemented
10#21444306C211yesmaximal, characteristic, complemented
11C61×#2974C211yesmaximal, characteristic, complemented
12#16840477C411yescharacteristic, complemented
13#16840469C411yescharacteristic, complemented
14#16840460—21no—
15#16840458C412yescomplemented
16#16840433C411yescharacteristic, complemented
17#16840408C412yescomplemented
18#16840402—22no—
19D4×D244C411yescharacteristic, complemented
20D4×C22⋊D61—21no—
21#16840363—21no—
22#16840345—22no—
23#16840312C2211yescharacteristic
24#16840296C2211yescharacteristic
25#16840273C2212yes—
26C2×#14519751C2212yes—
27#16840234—21no—
28#16839665—21no—
29#16839661—21no—
30#16839655—22no—
31#16839625C2212yescomplemented
32#16839622C2212yescomplemented
33C4×C4⋊Dic61C2211yescharacteristic, complemented
34#16839608C2211yescharacteristic, complemented
35#16839500C2211yescharacteristic, complemented
36#16839500C2212yescomplemented
37#16839500C2211yescharacteristic, complemented
38#16839497C2211yescharacteristic
39#16839497C2212yes—
40#16839497C2211yescharacteristic
41#16839497C2212yescomplemented
42#16839497C2212yescomplemented
43#16839497C2212yescomplemented
44#16839497C2212yescomplemented
45C2×C4×D4×C61C2211yescharacteristic
46C61×#436—21no—
47C61×#435—22no—
48C8×D4×C61—21no—
49C61×#432C2211yescharacteristic, complemented
50C61×#431C2211yescharacteristic, complemented
51C61×#407C2212yescomplemented
52C2×C22⋊C8×C61C2212yescomplemented
53C2×#11758222—21no—
54C2×#11758222C2×C412yes—
55C22×D244C2×C412yes—
56C22×C22⋊D61—22no—
57C2×D4×D61—21no—
58#14519777C2×C411yescharacteristic
59#14519774C2×C411yescharacteristic
60#14519773—21no—
61#14519772—22no—
62#14519771—21no—
63#14519769C2×C411yescharacteristic
64#14519768—21no—
65#14519765—21no—
66#14519765C2×C411yescharacteristic
67#14519763—22no—
68#14519762—22no—
69#14519760—22no—
70#14519760—21no—
71#14519759—22no—
72#14519756C2×C412yes—
73#14519756C2311yescharacteristic
74#14519754—22no—
75#14519752—21no—
76#14519751—22no—
77#14519751—21no—
78#14519751—22no—
79#14519751C2×C412yes—
80#14519751C2×C412yes—
81#14519751C2×C412yes—
82#14519751C2311yescharacteristic
83#14519751C2312yes—
84#14519750—22no—
85#14519750—22no—
86#14519749—22no—
87#14519748—21no—
88#14519747—22no—
89#14519746—22no—
90#14519745—22no—
91C4×Dic122C2×C411yescharacteristic
92C4×D244—21no—
93C4×D244C2×C412yes—
94C4×D244C2×C411yescharacteristic
95D4×Dic61—21no—
96#14519739—21no—
97C4×C22⋊D61—22no—
98C4×C22⋊D61—21no—
99C4×C22⋊D61—22no—
100#14519737—22no—
101#14519736C2×C412yes—
102#14519735—21no—
103#14519734—21no—
104C2×C4⋊Dic61—21no—
105C2×C4⋊Dic61—22no—
106C2×C4⋊Dic61C2×C412yes—
107C2×C22⋊Dic61—22no—
108C2×C22⋊Dic61—22no—
109C2×#11758198—22no—
110C22⋊C4×D61—22no—
111C2×C4×Dic61—21no—
112C2×C4×Dic61—21no—
113#14519684—21no—
114#14519684—22no—
115#14519684C2311yescharacteristic
116#14519684C2311yescharacteristic
117#14519684C2312yes—
118C22×D4×C61D411yescharacteristic
119C23×C4×C61C2312yes—
120C2×M16×C61—22no—
121C22×C8×C61—22no—
122C4×D4×C61C2×C412yes—
123C4×D4×C61C2×C411yescharacteristic
124C4×D4×C61C2×C411yescharacteristic
125C4×D4×C61D411yescharacteristic
126C4×D4×C61D412yes—
127C4×D4×C61D411yescharacteristic
128C2×C4⋊C4×C61D411yescharacteristic
129C2×C22⋊C4×C61D412yes—
130C2×C42×C61C2311yescharacteristic
131C4⋊C8×C61—21no—
132C4⋊C8×C61—21no—
133C22⋊C8×C61—22no—
134C22⋊C8×C61—22no—
135C22⋊C8×C61C2311yescharacteristic
136C22⋊C8×C61C2312yes—
137C22⋊C8×C61C2311yescharacteristic
138#97×C61—21no—
139C4×C8×C61—21no—
140C24×C61C2×D412yes—
141C23×D61—22no—
142C2×D4×C61—42no—
143C2×D4×C61C22⋊C411yescharacteristic
144C2×D4×C61C22⋊C411yescharacteristic
145C2×D4×C61C22⋊C412yes—
146C22×C4×C61—82no—
147C22×C4×C61C22×C412yes—
148C22×C4×C61C22×C412yes—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D412yes—
152C22×C4×C61C2×D412yes—
153C22×C4×C61C2×D412yes—
154C22×C4×C61C2411yescharacteristic
155C2×C22⋊D61—22no—
156C2×C22⋊D61—21no—
157C2×C22⋊D61—22no—
158C2×C22⋊D61—22no—
159C2×C22⋊D61—22no—
160C2×C22⋊D61—22no—
161C2×C22⋊D61—82no—
162C2×C22⋊D61—82no—
163C22×Dic61—22no—
164C22×Dic61—21no—
165C22×Dic61—21no—
166C22×Dic61—22no—
167D4×D61—44no—
168#11758222—82no—
169#11758222—81no—
170#11758222—81no—
171C2×D244—21no—
172C2×D244—22no—
173C2×D244—21no—
174C2×D244—82no—
175C2×D244C22×C411yescharacteristic
176C2×D244C22×C412yes—
177C2×C4×D61—21no—
178C2×C4×D61—22no—
179C2×C4×D61—22no—
180C2×Dic122—21no—
181C2×Dic122C22×C411yescharacteristic
182M16×C61—82no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61—22no—
186C2×C8×C61—82no—
187C4⋊C4×C61C22⋊C411yescharacteristic
188C4⋊C4×C61C22⋊C411yescharacteristic
189C4⋊C4×C61C22⋊C412yes—
190C22⋊C4×C61C22⋊C412yes—
191C22⋊C4×C61C22⋊C412yes—
192C22⋊C4×C61C22⋊C412yes—
193C22⋊C4×C61C22⋊C412yes—
194C42×C61C22×C411yescharacteristic
195C42×C61C22×C411yescharacteristic
196C42×C61C2×D411yescharacteristic
197C42×C61C2×D411yescharacteristic
198C22⋊Dic61—22no—
199C22⋊Dic61—22no—
200C22⋊Dic61—21no—
201C22⋊Dic61—82no—
202C22⋊Dic61—82no—
203#11758198—21no—
204#11758198—22no—
205#11758198—22no—
206#11758198—21no—
207#11758198—22no—
208#11758198—22no—
209#11758198—21no—
210#11758198—21no—
211#11758198—22no—
212C4⋊Dic61—22no—
213C4⋊Dic61—21no—
214C4⋊Dic61—42no—
215C4⋊Dic61—82no—
216C4⋊Dic61C22×C411yescharacteristic
217C4⋊Dic61C22×C412yes—
218C4⋊Dic61C22×C411yescharacteristic
219#11758196—21no—
220#11758196—21no—
221#11758196—22no—
222#11758196—21no—
223#11758196—21no—
224C4×Dic61—21no—
225C4×Dic61—22no—
226C4×Dic61—42no—
227C4×Dic61—81no—
228C23×C61—22no—
229C23×C61—22no—
230C23×C61—42no—
231C23×C61—42no—
232C23×C61C2×C22⋊C412yes—
233C23×C61C2×C22⋊C412yes—
234C23×C61C22×D411yescharacteristic
235C22×D61—21no—
236C22×D61—22no—
237C22×D61—44no—
238C22×D61—82no—
239C22×D61—82no—
240D4×C61—22no—
241D4×C61—22no—
242D4×C61—42no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—22no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61—42no—
251C2×C4×C61—41no—
252C2×C4×C61C2×C22⋊C412yes—
253C2×C4×C61C2×C22⋊C412yes—
254C2×C4×C61C2×C22⋊C411yescharacteristic
255C2×C4×C61C2×C22⋊C411yescharacteristic
256C2×C4×C61C2×C22⋊C411yescharacteristic
257C2×C4×C61C2×C22⋊C411yescharacteristic
258C2×C4×C61C23×C411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C22⋊D61—44no—
261C22⋊D61—42no—
262C22⋊D61—41no—
263C22⋊D61—42no—
264C22⋊D61—41no—
265C22⋊D61—82no—
266C22⋊D61—82no—
267C22⋊D61—82no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—22no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—42no—
275C2×Dic61—41no—
276C2×Dic61—42no—
277C2×Dic61—41no—
278C2×Dic61—42no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—82no—
282D244—41no—
283D244—42no—
284D244—42no—
285D244—82no—
286D244—81no—
287C4×D61—42no—
288C4×D61—82no—
289C4×D61—82no—
290Dic122—41no—
291Dic122—81no—
292C8×C61—41no—
293C8×C61—42no—
294C8×C61—41no—
295#17599—611nomaximal, Sylow
296C22×C61—21no—
297C22×C61—21no—
298C22×C61—22no—
299C22×C61—22no—
300C22×C61—22no—
301C22×C61—22no—
302C22×C61—22no—
303C22×C61—22no—
304C22×C61—82no—
305C22×C61—82no—
306C22×C61C22×C22⋊C411yescharacteristic
307C22×C61#57611yescharacteristic
308C22×C61#57711yescharacteristic
309C2×D61—42no—
310C2×D61—41no—
311C2×D61—42no—
312C2×D61—41no—
313C2×D61—41no—
314C2×D61—42no—
315C2×D61—164no—
316C4×C61—21no—
317C4×C61—21no—
318C4×C61—22no—
319C4×C61—22no—
320C4×C61—21no—
321C4×C61—21no—
322C4×C61—21no—
323C4×C61#42112yes—
324Dic61—41no—
325Dic61—42no—
326Dic61—41no—
327Dic61—41no—
328Dic61—81no—
329Dic61—82no—
330Dic61—81no—
331#3519—611no—
332#2974D6111yescharacteristic, complemented
333#2319—611no—
334#1519—612no—
335#1516—612no—
336#1513—612no—
337#1510—612no—
338#1509—611no—
339#1505—611no—
340#1502—611no—
341#1500—611no—
342C2×C61—21no—
343C2×C61—21no—
344C2×C61—42no—
345C2×C61—42no—
346C2×C61#288311yescharacteristic
347C2×C61#289711yescharacteristic
348C2×C61C2×#42111yescharacteristic
349D61—82no—
350D61—81no—
351#555—611no—
352#554—611no—
353#553—612no—
354#553—1221no—
355#550—611no—
356#546—612no—
357#546—1222no—
358D42—611no—
359D42—1221no—
360#540—1221no—
361#538—1222no—
362#532—611no—
363#529—611no—
364#525—612no—
365C2×#121—612no—
366#518—1221no—
367C2×C4×D4C2×D6111yescharacteristic
368#436—21no—
369#435—22no—
370C8×D4—21no—
371#432C2×D6111yescharacteristic, complemented
372#431C2×D6111yescharacteristic, complemented
373#407C2×D6112yescomplemented
374C2×C22⋊C8C2×D6112yescomplemented
375#390—1221no—
376#389—1221no—
377#388—1222no—
378#381—612no—
379#380—612no—
380C4×C4⋊C4—611no—
381#376—611no—
382#328—611no—
383#328—611no—
384#328—612no—
385#327—611no—
386#327—611no—
387#327—612no—
388#327—612no—
389#327—612no—
390#327—612no—
391#327—612no—
392C61#1759911yescharacteristic, complemented, Sylow
393C2×#41—612no—
394C2×#41—1221no—
395C22×D4—612no—
396C22×D4—1222no—
397C22×D4—1221no—
398C22×D4D24411yescharacteristic
399C23×C4C22×D6112yes—
400C2×M16—22no—
401C22×C8—22no—
402C4⋊Q8—611no—
403#127—611no—
404#127—1221no—
405#126—1222no—
406#126—1221no—
407#125—611no—
408#125—1221no—
409#124—611no—
410#124—1222no—
411#124—1221no—
412#123—1221no—
413#123—1222no—
414#123—1222no—
415#123—1222no—
416C22⋊Q8—612no—
417C22⋊Q8—611no—
418C22⋊Q8—1222no—
419#121—612no—
420#121—612no—
421#121—612no—
422#121—611no—
423#121—612no—
424#121—1222no—
425#121—1221no—
426#121—1222no—
427#121—1221no—
428#121—1222no—
429#121—1221no—
430#121—1222no—
431#121—1222no—
432#120—1222no—
433#120—1222no—
434#120—1222no—
435C4×Q8—611no—
436C4×D4—611no—
437C4×D4—612no—
438C4×D4—1221no—
439C4×D4—1222no—
440C4×D4—1221no—
441C4×D4—1222no—
442C4×D4—1221no—
443C4×D4—1222no—
444C4×D4—1221no—
445C4×D4C4×D6111yescharacteristic
446C4×D4C4×D6112yes—
447C4×D4C4×D6111yescharacteristic
448C4×D4C22⋊D6111yescharacteristic
449C4×D4C22⋊D6112yes—
450C4×D4C22⋊D6111yescharacteristic
451#117—612no—
452#117—1221no—
453#117—1221no—
454C2×C4⋊C4—612no—
455C2×C4⋊C4—1222no—
456C2×C4⋊C4—1221no—
457C2×C4⋊C4D24411yescharacteristic
458C2×C22⋊C4—1222no—
459C2×C22⋊C4—1222no—
460C2×C22⋊C4—1222no—
461C2×C22⋊C4—1222no—
462C2×C22⋊C4D24412yes—
463C2×C42—1221no—
464C2×C42—1221no—
465C2×C42C22×D6111yescharacteristic
466C4⋊C8—21no—
467C4⋊C8—21no—
468C22⋊C8—22no—
469C22⋊C8—22no—
470C22⋊C8C22×D6111yescharacteristic
471C22⋊C8C22×D6112yes—
472C22⋊C8C22×D6111yescharacteristic
473#97—21no—
474C4×C8—21no—
475#95—611no—
476#95—611no—
477#95—612no—
478#95—1221no—
479#95—1222no—
480C24—1222no—
481C24C2×D24412yes—
482#41—4881no—
483#41—4881no—
484#41—4882no—
485C2×Q8—611no—
486C2×Q8—1221no—
487C2×D4—42no—
488C2×D4—611no—
489C2×D4—612no—
490C2×D4—1222no—
491C2×D4—1222no—
492C2×D4—1222no—
493C2×D4—1222no—
494C2×D4—1221no—
495C2×D4—1222no—
496C2×D4—1221no—
497C2×D4—1222no—
498C2×D4—1221no—
499C2×D4—4882no—
500C2×D4—4882no—
501C2×D4—4882no—
502C2×D4—2444no—
503C2×D4#1175819811yescharacteristic
504C2×D4#1175819811yescharacteristic
505C2×D4#1175819812yes—
506C22×C4—82no—
507C22×C4—1222no—
508C22×C4—1222no—
509C22×C4—1222no—
510C22×C4—1221no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—1222no—
514C22×C4C2×C4×D6112yes—
515C22×C4C2×C4×D6112yes—
516C22×C4C2×D24411yescharacteristic
517C22×C4C2×D24411yescharacteristic
518C22×C4C2×D24412yes—
519C22×C4C2×C22⋊D6112yes—
520C22×C4C2×C22⋊D6112yes—
521C22×C4C23×D6111yescharacteristic
522M16—82no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—22no—
526C2×C8—82no—
527C4⋊C4—611no—
528C4⋊C4—612no—
529C4⋊C4—611no—
530C4⋊C4—1221no—
531C4⋊C4—1222no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—1222no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—4882no—
538C4⋊C4—2442no—
539C4⋊C4#1175819811yescharacteristic
540C4⋊C4#1175819811yescharacteristic
541C4⋊C4#1175819812yes—
542C22⋊C4—1221no—
543C22⋊C4—1222no—
544C22⋊C4—1222no—
545C22⋊C4—1221no—
546C22⋊C4—1222no—
547C22⋊C4—1222no—
548C22⋊C4—1221no—
549C22⋊C4—1222no—
550C22⋊C4—1221no—
551C22⋊C4—1222no—
552C22⋊C4—1221no—
553C22⋊C4—1222no—
554C22⋊C4—4882no—
555C22⋊C4—4882no—
556C22⋊C4#1175819812yes—
557C22⋊C4#1175819812yes—
558C22⋊C4#1175819812yes—
559C22⋊C4#1175819812yes—
560C42—1222no—
561C42—1221no—
562C42—4881no—
563C42—2442no—
564C42C2×C4×D6111yescharacteristic
565C42C2×C4×D6111yescharacteristic
566C42C2×C22⋊D6111yescharacteristic
567C42C2×C22⋊D6111yescharacteristic
568C23—22no—
569C23—22no—
570C23—42no—
571C23—42no—
572C23—1221no—
573C23—1222no—
574C23—4882no—
575C23—4882no—
576C23—2444no—
577C23C2×#1175819812yes—
578C23C2×#1175819812yes—
579C23C22×D24411yescharacteristic
580Q8—4881no—
581Q8—2441no—
582D4—22no—
583D4—22no—
584D4—42no—
585D4—4882no—
586D4—4881no—
587D4—4882no—
588D4—4882no—
589D4—4882no—
590D4—2441no—
591D4—2442no—
592D4—2444no—
593D4—2441no—
594D4—2442no—
595D4—2441no—
596D4—2442no—
597D4—2442no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—1221no—
608C2×C4—1222no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—4882no—
612C2×C4—4882no—
613C2×C4—4882no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2442no—
618C2×C4—2441no—
619C2×C4—2442no—
620C2×C4—2442no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2442no—
624C2×C4C2×#1175819812yes—
625C2×C4C2×#1175819811yescharacteristic
626C2×C4C2×#1175819811yescharacteristic
627C2×C4C2×#1175819811yescharacteristic
628C2×C4C2×#1175819812yes—
629C2×C4C2×#1175819811yescharacteristic
630C2×C4C22×C4×D6111yescharacteristic
631C2×C4C22×C22⋊D6111yescharacteristic
632C8—41no—
633C8—42no—
634C8—41no—
635C22—21no—
636C22—21no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—82no—
644C22—82no—
645C22—9764no—
646C22—2442no—
647C22—2441no—
648C22—2441no—
649C22—2442no—
650C22—2442no—
651C22—2441no—
652C22C22×#1175819811yescharacteristic
653C22#1684056711yescharacteristic
654C22#1684057411yescharacteristic
655C4—21no—
656C4—21no—
657C4—21no—
658C4—21no—
659C4—21no—
660C4—22no—
661C4—22no—
662C4—4881no—
663C4—4882no—
664C4—4881no—
665C4—2441no—
666C4—2442no—
667C4—2441no—
668C4—2441no—
669C4#1683977812yes—
670C2—21no—
671C2—21no—
672C2—42no—
673C2—42no—
674C2—4881no—
675C2—4882no—
676C2#2145427911yescharacteristic
677C2#2145434611yescharacteristic
678C2C2×#1683977811yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2974C2
socleC22×C61C22×C22⋊C4
prime radical#21459851C2
cosocleC22×C4×C61C24
hypercenter#2974D61
hypocenterC61#17599
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2974

Complements

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

NH ≅ G/Ntype
2. #21448462C2semidirect
3. #21444388C2semidirect
4. #21444376C2semidirect
5. #21444363C2semidirect
6. #21444349C2semidirect
7. #21444345C2semidirect
8. #21444328C2semidirect
9. #21444315C2semidirect
10. #21444306C2semidirect
11. C61×#2974C2semidirect
12. #16840477C4semidirect
13. #16840469C4semidirect
15. #16840458C4semidirect
16. #16840433C4semidirect
17. #16840408C4semidirect
19. D4×D244C4semidirect
31. #16839625C22semidirect
32. #16839622C22semidirect
33. C4×C4⋊Dic61C22semidirect
34. #16839608C22semidirect
35. #16839500C22semidirect
36. #16839500C22semidirect
37. #16839500C22semidirect
41. #16839497C22semidirect
42. #16839497C22semidirect
43. #16839497C22semidirect
44. #16839497C22semidirect
49. C61×#432C22semidirect
50. C61×#431C22semidirect
51. C61×#407C22semidirect
52. C2×C22⋊C8×C61C22semidirect
332. #2974D61semidirect
371. #432C2×D61semidirect
372. #431C2×D61semidirect
373. #407C2×D61semidirect
374. C2×C22⋊C8C2×D61semidirect
392. C61#17599semidirect

Maximal subgroups

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

subgroupindex
#214598512
#214484622
#214443882
#214443762
#214443632
#214443492
#214443452
#214443282
#214443152
#214443062
C61×#29742
#1759961