Finite Group Databasebeta

← #29647563

Subgroups — #29647563

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#21454854C211yesmaximal, characteristic, complemented
3#21444386C212yesmaximal, complemented
4#21444373C212yesmaximal, complemented
5#21444360C212yesmaximal, complemented
6#21444347C212yesmaximal, complemented
7#21444342C211yesmaximal, characteristic, complemented
8#21444326C211yesmaximal, characteristic, complemented
9#21444313C211yesmaximal, characteristic, complemented
10#21444303C211yesmaximal, characteristic, complemented
11C61×#2319C211yesmaximal, 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#16839840—21no—
29#16839834—22no—
30D4×#92418—21no—
31#16839819C2211yescharacteristic, complemented
32#16839815C2211yescharacteristic, complemented
33#16839726C2212yescomplemented
34C2×C22⋊#92418C2212yescomplemented
35#16839498C2211yescharacteristic, complemented
36#16839498C2212yescomplemented
37#16839498C2211yescharacteristic, complemented
38#16839495C2211yescharacteristic
39#16839495C2212yes—
40#16839495C2211yescharacteristic
41#16839495C2212yescomplemented
42#16839495C2212yescomplemented
43#16839495C2212yescomplemented
44#16839495C2212yescomplemented
45C2×C4×D4×C61C2211yescharacteristic
46C61×#390—21no—
47C61×#389—21no—
48C61×#388—22no—
49C61×#381C2212yescomplemented
50C61×#380C2212yescomplemented
51C4×C4⋊C4×C61C2211yescharacteristic, complemented
52C61×#376C2211yescharacteristic, complemented
53C2×#11758222—21no—
54C2×#11758222C2×C412yes—
55C22×D244C2×C412yes—
56C22×C22⋊D61—22no—
57C2×D4×D61—21no—
58C2×#11758194—22no—
59C22×#92418—22no—
60#14519777C2×C411yescharacteristic
61#14519774C2×C411yescharacteristic
62#14519773—21no—
63#14519772—22no—
64#14519771—21no—
65#14519769C2×C411yescharacteristic
66#14519768—21no—
67#14519765—21no—
68#14519765C2×C411yescharacteristic
69#14519763—22no—
70#14519762—22no—
71#14519760—22no—
72#14519760—21no—
73#14519759—22no—
74#14519756C2×C412yes—
75#14519756C2311yescharacteristic
76#14519754—22no—
77#14519752—21no—
78#14519751—22no—
79#14519751—21no—
80#14519751—22no—
81#14519751C2×C412yes—
82#14519751C2×C412yes—
83#14519751C2×C412yes—
84#14519751C2311yescharacteristic
85#14519751C2312yes—
86#14519750—22no—
87#14519750—22no—
88#14519749—22no—
89#14519748—21no—
90#14519747—22no—
91#14519746—22no—
92#14519745—22no—
93C4×Dic122C2×C411yescharacteristic
94C4×D244—21no—
95C4×D244C2×C412yes—
96C4×D244C2×C411yescharacteristic
97D4×Dic61—21no—
98#14519739—21no—
99C4×C22⋊D61—22no—
100C4×C22⋊D61—21no—
101C4×C22⋊D61—22no—
102#14519737—22no—
103#14519736C2×C412yes—
104#14519735—21no—
105#14519734—21no—
106C2×C4⋊Dic61C2×C412yes—
107C2×#11758198—22no—
108C22⋊C4×D61—22no—
109C4⋊#92418—21no—
110C4⋊#92418—21no—
111C22⋊#92418—22no—
112C22⋊#92418—22no—
113C22⋊#92418C2311yescharacteristic
114C22⋊#92418C2312yes—
115C22⋊#92418C2311yescharacteristic
116#14519688—21no—
117C4×#92418—21no—
118C22×D4×C61D411yescharacteristic
119C23×C4×C61C2312yes—
120C4×D4×C61C2×C412yes—
121C4×D4×C61C2×C411yescharacteristic
122C4×D4×C61C2×C411yescharacteristic
123C4×D4×C61D411yescharacteristic
124C4×D4×C61D411yescharacteristic
125C4×D4×C61D412yes—
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61—22no—
128C2×C4⋊C4×C61D411yescharacteristic
129C2×C22⋊C4×C61—22no—
130C2×C22⋊C4×C61—22no—
131C2×C22⋊C4×C61D412yes—
132C2×C42×C61—21no—
133C2×C42×C61—21no—
134C2×C42×C61C2311yescharacteristic
135#95×C61—21no—
136#95×C61—22no—
137#95×C61C2311yescharacteristic
138#95×C61C2311yescharacteristic
139#95×C61C2312yes—
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—21no—
147C22×C4×C61—21no—
148C22×C4×C61—22no—
149C22×C4×C61—82no—
150C22×C4×C61C22×C412yes—
151C22×C4×C61C22×C412yes—
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D412yes—
155C22×C4×C61C2×D412yes—
156C22×C4×C61C2×D412yes—
157C22×C4×C61C2411yescharacteristic
158C2×C22⋊D61—22no—
159C2×C22⋊D61—21no—
160C2×C22⋊D61—22no—
161C2×C22⋊D61—22no—
162C2×C22⋊D61—22no—
163C2×C22⋊D61—22no—
164C2×C22⋊D61—82no—
165C2×C22⋊D61—82no—
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
182C4⋊C4×C61—42no—
183C4⋊C4×C61—82no—
184C4⋊C4×C61C22⋊C411yescharacteristic
185C4⋊C4×C61C22⋊C411yescharacteristic
186C4⋊C4×C61C22⋊C412yes—
187C22⋊C4×C61—82no—
188C22⋊C4×C61—82no—
189C22⋊C4×C61C22⋊C412yes—
190C22⋊C4×C61C22⋊C412yes—
191C22⋊C4×C61C22⋊C412yes—
192C22⋊C4×C61C22⋊C412yes—
193C42×C61—42no—
194C42×C61—81no—
195C42×C61C22×C411yescharacteristic
196C42×C61C22×C411yescharacteristic
197C42×C61C2×D411yescharacteristic
198C42×C61C2×D411yescharacteristic
199C22⋊Dic61—22no—
200C22⋊Dic61—22no—
201C22⋊Dic61—21no—
202#11758198—21no—
203#11758198—22no—
204#11758198—22no—
205#11758198—21no—
206#11758198—22no—
207#11758198—22no—
208#11758198—21no—
209#11758198—21no—
210#11758198—22no—
211C4⋊Dic61—22no—
212C4⋊Dic61—21no—
213C4⋊Dic61C22×C411yescharacteristic
214C4⋊Dic61C22×C412yes—
215C4⋊Dic61C22×C411yescharacteristic
216#11758196—21no—
217#11758196—21no—
218#11758196—22no—
219#11758196—21no—
220#11758196—21no—
221C4×Dic61—21no—
222C4×Dic61—22no—
223#11758194—82no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—22no—
227C2×#92418—82no—
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×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—42no—
256C2×C4×C61—42no—
257C2×C4×C61—41no—
258C2×C4×C61—42no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61C2×C22⋊C412yes—
262C2×C4×C61C2×C22⋊C412yes—
263C2×C4×C61C2×C22⋊C411yescharacteristic
264C2×C4×C61C2×C22⋊C411yescharacteristic
265C2×C4×C61C2×C22⋊C411yescharacteristic
266C2×C4×C61C2×C22⋊C411yescharacteristic
267C2×C4×C61C23×C411yescharacteristic
268C2×C4×C61C22×D411yescharacteristic
269C22⋊D61—44no—
270C22⋊D61—42no—
271C22⋊D61—41no—
272C22⋊D61—42no—
273C22⋊D61—41no—
274C22⋊D61—82no—
275C22⋊D61—82no—
276C22⋊D61—82no—
277C2×Dic61—21no—
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—22no—
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—
292#92418—41no—
293#92418—42no—
294#92418—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—81no—
324C4×C61—82no—
325C4×C61—81no—
326C4×C61#42112yes—
327Dic61—41no—
328Dic61—42no—
329Dic61—41no—
330Dic61—41no—
331#3519—611no—
332#2974—611no—
333#2319D6111yescharacteristic, complemented
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—1221no—
369#435—1222no—
370C8×D4—1221no—
371#432—611no—
372#431—611no—
373#407—612no—
374C2×C22⋊C8—612no—
375#390—21no—
376#389—21no—
377#388—22no—
378#381C2×D6112yescomplemented
379#380C2×D6112yescomplemented
380C4×C4⋊C4C2×D6111yescharacteristic, complemented
381#376C2×D6111yescharacteristic, complemented
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×D4C22⋊D6111yescharacteristic
399C23×C4C22×D6112yes—
400C2×M16—1222no—
401C22×C8—1222no—
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—1222no—
413#123—1221no—
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—1222no—
428#121—1221no—
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—1222no—
441C4×D4—1221no—
442C4×D4—1221no—
443C4×D4—1222no—
444C4×D4—1221no—
445C4×D4C4×D6111yescharacteristic
446C4×D4C4×D6112yes—
447C4×D4C4×D6111yescharacteristic
448C4×D4D24411yescharacteristic
449C4×D4D24412yes—
450C4×D4D24411yescharacteristic
451#117—612no—
452#117—1221no—
453#117—1221no—
454C2×C4⋊C4—21no—
455C2×C4⋊C4—22no—
456C2×C4⋊C4—612no—
457C2×C4⋊C4C22⋊D6111yescharacteristic
458C2×C22⋊C4—22no—
459C2×C22⋊C4—22no—
460C2×C22⋊C4—1222no—
461C2×C22⋊C4—1222no—
462C2×C22⋊C4C22⋊D6112yes—
463C2×C42—21no—
464C2×C42—21no—
465C2×C42C22×D6111yescharacteristic
466C4⋊C8—1221no—
467C4⋊C8—1221no—
468C22⋊C8—611no—
469C22⋊C8—612no—
470C22⋊C8—611no—
471C22⋊C8—1222no—
472C22⋊C8—1222no—
473#97—1221no—
474C4×C8—1221no—
475#95—21no—
476#95—22no—
477#95C22×D6111yescharacteristic
478#95C22×D6111yescharacteristic
479#95C22×D6112yes—
480C24—1222no—
481C24C2×C22⋊D6112yes—
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—21no—
507C22×C4—21no—
508C22×C4—22no—
509C22×C4—82no—
510C22×C4—1222no—
511C22×C4—1222no—
512C22×C4—1222no—
513C22×C4—1221no—
514C22×C4C2×C4×D6112yes—
515C22×C4C2×C4×D6112yes—
516C22×C4C2×D24412yes—
517C22×C4C2×D24412yes—
518C22×C4C2×C22⋊D6111yescharacteristic
519C22×C4C2×C22⋊D6111yescharacteristic
520C22×C4C2×C22⋊D6112yes—
521C22×C4C23×D6111yescharacteristic
522M16—4882no—
523C2×C8—1221no—
524C2×C8—1221no—
525C2×C8—1222no—
526C2×C8—4882no—
527C4⋊C4—42no—
528C4⋊C4—82no—
529C4⋊C4—611no—
530C4⋊C4—612no—
531C4⋊C4—611no—
532C4⋊C4—1221no—
533C4⋊C4—1222no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1222no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4#1175819811yescharacteristic
540C4⋊C4#1175819811yescharacteristic
541C4⋊C4#1175819812yes—
542C22⋊C4—82no—
543C22⋊C4—82no—
544C22⋊C4—1221no—
545C22⋊C4—1222no—
546C22⋊C4—1222no—
547C22⋊C4—1221no—
548C22⋊C4—1222no—
549C22⋊C4—1222no—
550C22⋊C4—1221no—
551C22⋊C4—1222no—
552C22⋊C4—1221no—
553C22⋊C4—1222no—
554C22⋊C4—1221no—
555C22⋊C4—1222no—
556C22⋊C4#1175819812yes—
557C22⋊C4#1175819812yes—
558C22⋊C4#1175819812yes—
559C22⋊C4#1175819812yes—
560C42—42no—
561C42—81no—
562C42—1222no—
563C42—1221no—
564C42C2×C4×D6111yescharacteristic
565C42C2×C4×D6111yescharacteristic
566C42C2×D24411yescharacteristic
567C42C2×D24411yescharacteristic
568C23—22no—
569C23—22no—
570C23—42no—
571C23—42no—
572C23—1221no—
573C23—1222no—
574C23—4882no—
575C23—4882no—
576C23—2444no—
577C23C2×#1175819812yes—
578C23C2×#1175819812yes—
579C23C22×C22⋊D6111yescharacteristic
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—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—1221no—
617C2×C4—1222no—
618C2×C4—1221no—
619C2×C4—1221no—
620C2×C4—4882no—
621C2×C4—4882no—
622C2×C4—4882no—
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×D24411yescharacteristic
632C8—2441no—
633C8—2442no—
634C8—2441no—
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#1684056511yescharacteristic
654C22#1684057011yescharacteristic
655C4—21no—
656C4—21no—
657C4—21no—
658C4—21no—
659C4—21no—
660C4—22no—
661C4—22no—
662C4—81no—
663C4—82no—
664C4—81no—
665C4—2441no—
666C4—2442no—
667C4—2441no—
668C4—2441no—
669C4#1683978012yes—
670C2—21no—
671C2—21no—
672C2—42no—
673C2—42no—
674C2—4881no—
675C2—4882no—
676C2#2145427911yescharacteristic
677C2#2145434211yescharacteristic
678C2C2×#1683978011yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2319C2
socleC22×C61C22×C22⋊C4
prime radical#21459851C2
cosocleC22×C4×C61C24
hypercenter#2319D61
hypocenterC61#17599
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2319

Complements

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

NH ≅ G/Ntype
2. #21454854C2semidirect
3. #21444386C2semidirect
4. #21444373C2semidirect
5. #21444360C2semidirect
6. #21444347C2semidirect
7. #21444342C2semidirect
8. #21444326C2semidirect
9. #21444313C2semidirect
10. #21444303C2semidirect
11. C61×#2319C2semidirect
12. #16840477C4semidirect
13. #16840469C4semidirect
15. #16840458C4semidirect
16. #16840433C4semidirect
17. #16840408C4semidirect
19. D4×D244C4semidirect
31. #16839819C22semidirect
32. #16839815C22semidirect
33. #16839726C22semidirect
34. C2×C22⋊#92418C22semidirect
35. #16839498C22semidirect
36. #16839498C22semidirect
37. #16839498C22semidirect
41. #16839495C22semidirect
42. #16839495C22semidirect
43. #16839495C22semidirect
44. #16839495C22semidirect
49. C61×#381C22semidirect
50. C61×#380C22semidirect
51. C4×C4⋊C4×C61C22semidirect
52. C61×#376C22semidirect
333. #2319D61semidirect
378. #381C2×D61semidirect
379. #380C2×D61semidirect
380. C4×C4⋊C4C2×D61semidirect
381. #376C2×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
#214548542
#214443862
#214443732
#214443602
#214443472
#214443422
#214443262
#214443132
#214443032
C61×#23192
#1759961