Finite Group Databasebeta

← #29713324

Subgroups — #29713324

Order 15616.

1072 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840316C211yesmaximal, characteristic
2C2×#16840106C211yesmaximal, characteristic
3C2×#16840102C211yesmaximal, characteristic
4#21448300C211yesmaximal, characteristic
5#21447150C211yesmaximal, characteristic, complemented
6#21447146C211yesmaximal, characteristic, complemented
7#21447064C211yesmaximal, characteristic, complemented
8#21447063C211yesmaximal, characteristic, complemented
9#21446130C211yesmaximal, characteristic
10#21446128C211yesmaximal, characteristic
11#21446086C211yesmaximal, characteristic, complemented
12#21446076C211yesmaximal, characteristic, complemented
13#21446073C211yesmaximal, characteristic, complemented
14#21445886C211yesmaximal, characteristic
15C61×#1865C211yesmaximal, characteristic, complemented
16#16840316—24no—
17#16840316—24no—
18C2×#14519776C2211yescharacteristic
19C2×#14519775C2211yescharacteristic
20C2×#14519767—21no—
21C2×#14519757C2211yescharacteristic
22C2×#14519755—21no—
23C2×#14519753C2211yescharacteristic
24C2×C4×Dic122—21no—
25C2×C4×Dic122C2211yescharacteristic
26C2×#14519735C2211yescharacteristic
27C2×#14519735C2211yescharacteristic
28C2×#14519734—21no—
29C2×C42×D61C2211yescharacteristic
30#16840106—24no—
31#16840102—24no—
32C2×#14519711C2211yescharacteristic
33C2×#14519710C2211yescharacteristic
34C2×#14519708C2211yescharacteristic
35C2×#14519702C2211yescharacteristic
36C2×#14519702C2211yescharacteristic
37C2×#14519701C2211yescharacteristic
38C2×#14519701C2211yescharacteristic
39C2×#14519700C2211yescharacteristic
40C2×#14519700C2211yescharacteristic
41C2×#14519689C2211yescharacteristic
42C2×#14519689C2211yescharacteristic
43#16839661C2211yescharacteristic
44#16839660C2211yescharacteristic
45#16839658C2211yescharacteristic
46#16839636—21no—
47#16839633—21no—
48C4⋊C4×Dic61—21no—
49#16839613—21no—
50#16839525C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839517C2211yescharacteristic
54#16839516C2211yescharacteristic
55#16839516C2211yescharacteristic
56C2×C4×Q8×C61C2211yescharacteristic
57C2×#106×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C2×#103×C61C2211yescharacteristic
60C61×#389C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C61×#336C2211yescharacteristic
63C22×Dic122—21no—
64C22×Dic122C2311yescharacteristic
65C22×C4×D61D411yescharacteristic
66C22×C4×D61D411yescharacteristic
67C22×C4×D61C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70#14519776—22no—
71#14519776—22no—
72#14519775—22no—
73#14519775—22no—
74#14519767—28no—
75#14519757—22no—
76#14519757D414yes—
77#14519755—28no—
78#14519753—22no—
79#14519753D414yes—
80C4×Dic122—28no—
81C4×Dic122—22no—
82C4×Dic122—22no—
83#14519735—22no—
84#14519735—22no—
85#14519735—22no—
86#14519735—22no—
87#14519734—28no—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61—21no—
90C2×C4⋊Dic61C2311yescharacteristic
91C2×C4⋊Dic61C2311yescharacteristic
92C2×C4⋊Dic61C2311yescharacteristic
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—21no—
96C2×#11758196—21no—
97C2×#11758196—21no—
98C2×#11758196—21no—
99C2×#11758196—21no—
100C2×#11758198—21no—
101C2×#11758198—21no—
102C2×#11758198—21no—
103C2×#11758198—21no—
104C2×C4×Dic61—21no—
105C2×C4×Dic61—21no—
106C2×C4×Dic61—21no—
107C2×C4×Dic61D411yescharacteristic
108C2×C4×Dic61D411yescharacteristic
109C2×C4×Dic61C2311yescharacteristic
110C42×D61—22no—
111C42×D61—22no—
112#14519711—22no—
113#14519710—22no—
114#14519708—22no—
115#14519702—22no—
116#14519702—22no—
117#14519701—22no—
118#14519701—22no—
119#14519700—22no—
120#14519700—22no—
121#14519689—22no—
122#14519689—22no—
123#14519684—21no—
124#14519684—21no—
125#14519684—21no—
126C22×Q8×C61C2311yescharacteristic
127C22×C8×C61C2311yescharacteristic
128C22×C8×C61C2311yescharacteristic
129C4×Q8×C61—22no—
130C4×Q8×C61—22no—
131C2×C4⋊C4×C61—21no—
132C2×C4⋊C4×C61—21no—
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C4⋊C4×C61C2311yescharacteristic
135C2×C4⋊C4×C61C2311yescharacteristic
136C2×C42×C61—21no—
137C2×C42×C61C2311yescharacteristic
138#106×C61—22no—
139#103×C61—22no—
140#103×C61—22no—
141#95×C61—21no—
142C23×D61C2×D411yescharacteristic
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61C2×D412yes—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2411yescharacteristic
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×C4×D61—21no—
166C2×C4×D61—21no—
167C2×C4×D61—21no—
168C2×C4×D61—21no—
169C2×C4×D61—22no—
170C2×C4×D61—22no—
171C2×C4×D61—22no—
172C2×C4×D61—41no—
173C2×C4×D61C2×D412yes—
174C2×C4×D61C2×D412yes—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122—24no—
180C2×Dic122—21no—
181C2×Dic122—42no—
182C2×Dic122—42no—
183C2×Dic122C2×D412yes—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C4⋊C4×C61—24no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—42no—
196C4⋊C4×C61C2×D412yes—
197C4⋊C4×C61#4112yes—
198C42×C61—24no—
199C42×C61—21no—
200C42×C61—21no—
201#11758198—24no—
202#11758198—24no—
203#11758198—24no—
204#11758198—42no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—24no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—42no—
211C4⋊Dic61C2×D412yes—
212C4⋊Dic61#4112yes—
213#11758196—24no—
214#11758196—24no—
215#11758196—24no—
216#11758196—24no—
217#11758196—24no—
218#11758196—42no—
219#11758196—42no—
220C4×Dic61—24no—
221C4×Dic61—24no—
222C4×Dic61—21no—
223C4×Dic61—21no—
224C4×Dic61—22no—
225C4×Dic61—22no—
226C4×Dic61—42no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418—21no—
231C2×#92418—21no—
232C2×#92418—21no—
233C23×C61C22×D411yescharacteristic
234C22×D61—21no—
235C22×D61—21no—
236C22×D61—21no—
237C22×D61—21no—
238C22×D61—21no—
239C22×D61—21no—
240C22×D61#12012yes—
241Q8×C61—22no—
242Q8×C61—22no—
243Q8×C61—41no—
244Q8×C61—41no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61#13712yes—
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C612−1+411yescharacteristic
268C2×C4×C612−1+411yescharacteristic
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—22no—
276C2×Dic61—22no—
277C2×Dic61—22no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61#12012yes—
288C2×Dic61#13512yes—
289C2×Dic61#13712yes—
290C4×D61—28no—
291C4×D61—22no—
292C4×D61—22no—
293C4×D61—22no—
294C4×D61—22no—
295C4×D61—44no—
296Dic122—22no—
297Dic122—22no—
298Dic122—48no—
299Dic122—41no—
300Dic122—41no—
301C8×C61—41no—
302C8×C61—41no—
303#92418—41no—
304#92418—41no—
305#24065—611nomaximal, Sylow
306C22×C61C2×#12011yescharacteristic
307C22×C61#52511yescharacteristic
308C22×C61#53611yescharacteristic
309C22×C61C2×#13511yescharacteristic
310C22×C61C2×#13711yescharacteristic
311C22×C61C2×#13711yescharacteristic
312C22×C61#57511yescharacteristic
313C2×D61—22no—
314C2×D61—22no—
315C2×D61—22no—
316C2×D61—41no—
317C2×D61—41no—
318C2×D61—41no—
319C2×D61—41no—
320C4×C61—22no—
321C4×C61—42no—
322C4×C61—42no—
323C4×C61—42no—
324C4×C61—41no—
325C4×C61—81no—
326C4×C61#54711yescharacteristic
327C4×C61#54711yescharacteristic
328C4×C61#54711yescharacteristic
329C4×C61#54711yescharacteristic
330Dic61—22no—
331Dic61—22no—
332Dic61—22no—
333Dic61—42no—
334Dic61—42no—
335Dic61—42no—
336Dic61—81no—
337C2×#533—611no—
338C2×#485—611no—
339C2×#485—611no—
340#2296—611no—
341#2058—611no—
342#2057—611no—
343#2043—611no—
344#2043—611no—
345#1873—611no—
346#1873—611no—
347#1865—611no—
348#1865D6111yescharacteristic, complemented
349#1863—611no—
350#1863—611no—
351#1817—611no—
352C2×C61#240611yescharacteristic
353C2×C61#300211yescharacteristic
354C2×C61#300411yescharacteristic
355C2×C61#310711yescharacteristic
356C2×C61#311611yescharacteristic
357C2×C61#323111yescharacteristic
358C2×C61#323711yescharacteristic
359D61—42no—
360#533—1224no—
361#533—1224no—
362C2×C4⋊Q8—611no—
363C2×C4⋊Q8—611no—
364C2×#125—1221no—
365C2×C22⋊Q8—611no—
366C2×C22⋊Q8—611no—
367C2×C22⋊Q8—1221no—
368C2×C4×Q8—611no—
369C2×C4×Q8—1221no—
370C2×C4×Q8C2×D6111yescharacteristic
371C2×#117—611no—
372C2×#117—611no—
373C2×#117—1221no—
374C22×C42—611no—
375#485—1224no—
376#485—1224no—
377C2×#107—611no—
378C2×#107—611no—
379C2×#106—611no—
380C2×#106C2×D6111yescharacteristic
381C2×#103—611no—
382C2×#103—611no—
383C2×#103—611no—
384C2×#103—611no—
385C2×#103—611no—
386C2×#103—611no—
387C2×#103C2×D6111yescharacteristic
388C2×#103C2×D6111yescharacteristic
389C2×C22⋊C8—611no—
390C2×C22⋊C8—611no—
391#389—611no—
392#389—611no—
393#389—611no—
394#389C2×D6111yescharacteristic
395#385—1221no—
396#385—1221no—
397C4×C4⋊C4—1221no—
398C4×C4⋊C4—1221no—
399#340—611no—
400#340—611no—
401#340—611no—
402#340C2×D6111yescharacteristic
403#336—611no—
404#336—611no—
405#336—611no—
406#336C2×D6111yescharacteristic
407C61#2406511yescharacteristic, complemented, Sylow
408C22×Q8—611no—
409C22×Q8—1221no—
410C22×Q8C22×D6111yescharacteristic
411C23×C4—611no—
412C23×C4—611no—
413C23×C4—611no—
414C22×C8—611no—
415C22×C8—611no—
416C22×C8C22×D6111yescharacteristic
417C22×C8C22×D6111yescharacteristic
418C4⋊Q8—1222no—
419C4⋊Q8—1222no—
420C4⋊Q8—1222no—
421C4⋊Q8—1222no—
422#125—1228no—
423C22⋊Q8—614no—
424C22⋊Q8—614no—
425C22⋊Q8—1222no—
426C22⋊Q8—1228no—
427C22⋊Q8—1222no—
428C4×Q8—22no—
429C4×Q8—22no—
430C4×Q8—1222no—
431C4×Q8—1222no—
432C4×Q8—1228no—
433#117—1228no—
434#117—1222no—
435#117—1222no—
436#117—1222no—
437#117—1222no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—21no—
440C2×C4⋊C4—611no—
441C2×C4⋊C4—611no—
442C2×C4⋊C4—611no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4—1221no—
449C2×C4⋊C4—1221no—
450C2×C4⋊C4—1221no—
451C2×C4⋊C4—1221no—
452C2×C4⋊C4C22×D6111yescharacteristic
453C2×C4⋊C4C22×D6111yescharacteristic
454C2×C4⋊C4C22×D6111yescharacteristic
455C2×C22⋊C4—1221no—
456C2×C22⋊C4—1221no—
457C2×C22⋊C4—1221no—
458C2×C22⋊C4—1221no—
459C2×C42—21no—
460C2×C42—611no—
461C2×C42—611no—
462C2×C42—611no—
463C2×C42—1222no—
464C2×C42—1222no—
465C2×C42—1221no—
466C2×C42—1221no—
467C2×C42—1221no—
468C2×C42C22×D6111yescharacteristic
469#107—1222no—
470#107—1222no—
471#106—22no—
472#106—1222no—
473#103—22no—
474#103—22no—
475#103—1222no—
476#103—1222no—
477#103—1222no—
478#103—1222no—
479#103—1222no—
480#103—1222no—
481C22⋊C8—1222no—
482C22⋊C8—1222no—
483#95—21no—
484#95—1221no—
485#95—1221no—
486#95—1221no—
487C24—611no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—21no—
492C2×Q8—21no—
493C2×Q8—612no—
494C2×Q8—1224no—
495C2×Q8—1221no—
496C2×Q8—1221no—
497C2×Q8—1221no—
498C2×Q8—1221no—
499C2×Q8—1221no—
500C2×Q8—2442no—
501C2×Q8—2442no—
502C2×Q8D4×D6112yes—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—611no—
508C22×C4—611no—
509C22×C4—611no—
510C22×C4—612no—
511C22×C4—612no—
512C22×C4—1222no—
513C22×C4—1222no—
514C22×C4—1222no—
515C22×C4—1222no—
516C22×C4—1221no—
517C22×C4—1221no—
518C22×C4—1221no—
519C22×C4—1221no—
520C22×C4—1222no—
521C22×C4—1221no—
522C22×C4—1221no—
523C22×C4—1221no—
524C22×C4—1221no—
525C22×C4—2441no—
526C22×C4D4×D6111yescharacteristic
527C22×C4D4×D6111yescharacteristic
528C22×C4C23×D6111yescharacteristic
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—1221no—
536C2×C8—1221no—
537C2×C8—1221no—
538C2×C8—1221no—
539C2×C8—1221no—
540C2×C8—1221no—
541C4⋊C4—24no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—42no—
547C4⋊C4—612no—
548C4⋊C4—612no—
549C4⋊C4—1224no—
550C4⋊C4—1224no—
551C4⋊C4—1224no—
552C4⋊C4—1224no—
553C4⋊C4—1224no—
554C4⋊C4—1224no—
555C4⋊C4—1221no—
556C4⋊C4—1221no—
557C4⋊C4—1221no—
558C4⋊C4—1221no—
559C4⋊C4—2442no—
560C4⋊C4—2442no—
561C4⋊C4—2442no—
562C4⋊C4#1175822212yes—
563C4⋊C4D4×D6112yes—
564C22⋊C4—1224no—
565C22⋊C4—1224no—
566C22⋊C4—1224no—
567C22⋊C4—2442no—
568C42—24no—
569C42—21no—
570C42—21no—
571C42—1222no—
572C42—1222no—
573C42—1224no—
574C42—1224no—
575C42—1221no—
576C42—1221no—
577C42—2442no—
578C23—612no—
579C23—1221no—
580C23—1221no—
581C23—1221no—
582C23—1221no—
583C23—1221no—
584C23—1221no—
585C23C2×D4×D6111yescharacteristic
586Q8—22no—
587Q8—22no—
588Q8—41no—
589Q8—41no—
590Q8—1222no—
591Q8—1222no—
592Q8—2448no—
593Q8—2441no—
594Q8—2441no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—612no—
613C2×C4—612no—
614C2×C4—612no—
615C2×C4—1228no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—1221no—
620C2×C4—1221no—
621C2×C4—1221no—
622C2×C4—1222no—
623C2×C4—1222no—
624C2×C4—1222no—
625C2×C4—1222no—
626C2×C4—1222no—
627C2×C4—1222no—
628C2×C4—1222no—
629C2×C4—2444no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4—2441no—
635C2×C4—2441no—
636C2×C4—2441no—
637C2×C4—2441no—
638C2×C4—2441no—
639C2×C4#1451980612yes—
640C2×C4C2×D4×D6111yescharacteristic
641C2×C4C2×D4×D6111yescharacteristic
642C2×C4#41×D6111yescharacteristic
643C2×C4C2×#1175822211yescharacteristic
644C2×C4#1451982411yescharacteristic
645C2×C4#1451982511yescharacteristic
646C8—41no—
647C8—41no—
648C8—2441no—
649C8—2441no—
650C22—1222no—
651C22—1222no—
652C22—1222no—
653C22—2441no—
654C22—2441no—
655C22—2441no—
656C22—2441no—
657C22#120×D6111yescharacteristic
658C22#1684027111yescharacteristic
659C22#1684033511yescharacteristic
660C22#135×D6111yescharacteristic
661C22#137×D6111yescharacteristic
662C22C2×#1451980611yescharacteristic
663C22#1684056111yescharacteristic
664C4—22no—
665C4—42no—
666C4—42no—
667C4—41no—
668C4—42no—
669C4—81no—
670C4—1222no—
671C4—1222no—
672C4—1222no—
673C4—4881no—
674C4—2442no—
675C4—2442no—
676C4—2442no—
677C4#1684041011yescharacteristic
678C4#1684041311yescharacteristic
679C4#1684041411yescharacteristic
680C4#1684041711yescharacteristic
681C2—2442no—
682C2#2144915211yescharacteristic
683C2#2145504011yescharacteristic
684C2#2145505011yescharacteristic
685C2#2145588611yescharacteristic
686C2#2145597911yescharacteristic
687C2#2145697711yescharacteristic
688C2#2145703211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1865C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#1865D61
hypocenterC61#24065
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1865

Complements

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

NH ≅ G/Ntype
5. #21447150C2semidirect
6. #21447146C2semidirect
7. #21447064C2semidirect
8. #21447063C2semidirect
11. #21446086C2semidirect
12. #21446076C2semidirect
13. #21446073C2semidirect
15. C61×#1865C2semidirect
348. #1865D61semidirect
407. C61#24065semidirect

Maximal subgroups

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

subgroupindex
C2×#168403162
C2×#168401062
C2×#168401022
#214483002
#214471502
#214471462
#214470642
#214470632
#214461302
#214461282
#214460862
#214460762
#214460732
#214458862
C61×#18652
#2406561