Finite Group Databasebeta

← #29684097

Subgroups — #29684097

Order 15616.

1694 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#124×D61C211yesmaximal, characteristic
2C2×#16839784C211yesmaximal, characteristic, complemented
3C2×#16839769C212yesmaximal, complemented
4C2×#16839765C212yesmaximal
5C4×#14519735C211yesmaximal, characteristic, complemented
6#21446752C211yesmaximal, characteristic, complemented
7#21446751C211yesmaximal, characteristic, complemented
8#21446750C212yesmaximal, complemented
9#21446749C211yesmaximal, characteristic, complemented
10#21446748C212yesmaximal, complemented
11C61×#1983C211yesmaximal, characteristic, complemented
12C22×Q8×D61C2211yescharacteristic
13C22×D4×D61C2211yescharacteristic
14C22×#11758189C2212yes—
15C2×#14519766C2211yescharacteristic
16C2×#14519765C2211yescharacteristic
17C2×#14519764C2211yescharacteristic
18C2×#14519763—21no—
19C2×#14519763—21no—
20#124×D61—24no—
21#124×D61C418yescomplemented
22C2×#14519735C2212yes—
23C2×C22⋊C4×D61—21no—
24C2×C22⋊C4×D61—21no—
25C2×C42×D61C2211yescharacteristic
26#16839784—24no—
27C2×C4⋊#92418C2211yescharacteristic, complemented
28C2×#14519706C2212yescomplemented
29#16839769—28no—
30#16839765—28no—
31C2×#14519702C2212yescomplemented
32C2×#14519701C2212yescomplemented
33C2×#14519700C2212yescomplemented
34C2×#14519699C2212yes—
35C2×#14519698C2212yes—
36C2×#14519697C2212yes—
37C4⋊C4×Dic61C2212yescomplemented
38C4×C4⋊Dic61C2211yescharacteristic, complemented
39C4×#11758198—21no—
40#16839609—21no—
41#16839607—21no—
42C42×Dic61—21no—
43C2×#124×C61C2211yescharacteristic
44C2×C4⋊C8×C61C2211yescharacteristic, complemented
45C2×#103×C61C2212yescomplemented
46C2×#102×C61C2212yes—
47C4×C4⋊C4×C61C2211yescharacteristic, complemented
48C24×D61—21no—
49C22×Dic122—21no—
50C22×Dic122C2311yescharacteristic
51C2×Q8×D61—22no—
52C2×Q8×D61—24no—
53C2×Q8×D61—24no—
54C2×Q8×D61C2×C414yes—
55C22×D244C2311yescharacteristic
56C22×C22⋊D61—21no—
57C2×D4×D61—24no—
58C2×D4×D61—22no—
59C2×D4×D61—24no—
60C2×D4×D61C2×C414yes—
61C22×C4×D61—21no—
62C22×C4×D61D411yescharacteristic
63C22×C4×D61D411yescharacteristic
64C22×C4×D61C2311yescharacteristic
65C2×#11758189—24no—
66C2×#11758189—24no—
67C2×#11758189—24no—
68C22×#92418C2312yes—
69#14519766—22no—
70#14519766C2×C414yes—
71#14519765—22no—
72#14519765C2×C414yes—
73#14519764—22no—
74#14519764C2×C414yes—
75#14519763—28no—
76#14519763—28no—
77#14519735—24no—
78#14519735D418yes—
79C2×C4⋊Dic61C2312yes—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×#11758198—22no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C22⋊C4×D61—28no—
88C22⋊C4×D61—28no—
89C2×C4×Dic61—22no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—22no—
92C2×C4×Dic61D411yescharacteristic
93C2×C4×Dic61D411yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95C42×D61C2×C414yes—
96C42×D61D414yes—
97C4⋊#92418—22no—
98#14519706—24no—
99#14519702—24no—
100#14519701—24no—
101#14519700—24no—
102#14519699—24no—
103#14519698—24no—
104#14519697—24no—
105#14519684—21no—
106#14519684—22no—
107C22×Q8×C61C2311yescharacteristic
108C22×D4×C61C2311yescharacteristic
109C22×C8×C61C2312yes—
110#124×C61—22no—
111#124×C61C2×C414yes—
112C2×C4⋊C4×C61C2312yes—
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61—21no—
115C2×C42×C61—21no—
116C2×C42×C61C2311yescharacteristic
117C4⋊C8×C61—22no—
118#103×C61—24no—
119#102×C61—24no—
120#95×C61—21no—
121C24×C61—21no—
122C23×D61—21no—
123C23×D61—24no—
124C23×D61—42no—
125C23×D61—44no—
126C23×D61—42no—
127C23×D61—44no—
128C23×D61C2×D411yescharacteristic
129C2×Q8×C61—21no—
130C2×Q8×C61—22no—
131C2×Q8×C61—22no—
132C2×Q8×C61C22×C412yes—
133C2×D4×C61—22no—
134C2×D4×C61—21no—
135C2×D4×C61—22no—
136C2×D4×C61C22×C412yes—
137C22×C4×C61—21no—
138C22×C4×C61—22no—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2411yescharacteristic
142C2×C22⋊D61—24no—
143C2×C22⋊D61—44no—
144C22×Dic61—21no—
145C22×Dic61—22no—
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149Q8×D61—216no—
150Q8×D61—44no—
151Q8×D61—44no—
152D4×D61—216no—
153D4×D61—44no—
154D4×D61—44no—
155C2×D244—22no—
156C2×D244—22no—
157C2×D244—21no—
158C2×D244C22×C412yes—
159C2×C4×D61—24no—
160C2×C4×D61—42no—
161C2×C4×D61—41no—
162C2×C4×D61—41no—
163C2×C4×D61—44no—
164C2×C4×D61C22⋊C414yes—
165C2×C4×D61C22⋊C414yes—
166C2×C4×D61C22×C412yes—
167C2×C4×D61C2×D414yes—
168C2×C4×D61C2×D412yes—
169C2×C4×D61#4114yes—
170C2×Dic122—22no—
171C2×Dic122—22no—
172C2×Dic122—21no—
173C2×Dic122—24no—
174C2×Dic122—44no—
175C2×Dic122C22×C412yes—
176C2×C8×C61—22no—
177C2×C8×C61—22no—
178C2×C8×C61—22no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61C2×D414yes—
181C22⋊C4×C61—24no—
182C22⋊C4×C61—24no—
183C42×C61—42no—
184C42×C61C22×C412yes—
185C42×C61C2×D412yes—
186C22⋊Dic61—24no—
187C22⋊Dic61—24no—
188#11758198—28no—
189#11758198—24no—
190#11758198—24no—
191#11758198—24no—
192#11758198—24no—
193C4⋊Dic61—22no—
194C4⋊Dic61C2×D414yes—
195C4×Dic61—28no—
196C4×Dic61—44no—
197C4×Dic61—42no—
198C4×Dic61C22⋊C414yes—
199C4×Dic61C22⋊C414yes—
200C4×Dic61C22×C412yes—
201C4×Dic61C2×D412yes—
202C2×#92418—22no—
203C2×#92418—22no—
204C2×#92418—22no—
205#11758189—48no—
206C23×C61—22no—
207C23×C61—42no—
208C23×C61—41no—
209C23×C61—42no—
210C23×C61—41no—
211C23×C61C22×D411yescharacteristic
212C22×D61—21no—
213C22×D61—22no—
214C22×D61—21no—
215C22×D61—22no—
216C22×D61—48no—
217C22×D61—48no—
218C22×D61—42no—
219C22×D61—41no—
220C22×D61—42no—
221C22×D61—41no—
222C22×D61—48no—
223C22×D61—88no—
224C22×D61—88no—
225C22×D61C2×C22⋊C412yes—
226C22×D61#12114yes—
227Q8×C61—24no—
228Q8×C61—41no—
229Q8×C61—41no—
230D4×C61—24no—
231D4×C61—41no—
232D4×C61—41no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—24no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61C2×C22⋊C412yes—
242C2×C4×C61C2×C22⋊C412yes—
243C2×C4×C61C23×C411yescharacteristic
244C2×C4×C61C22×D412yes—
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C2×#4112yes—
247C22⋊D61—48no—
248C2×Dic61—22no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—22no—
252C2×Dic61—24no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61C2×C22⋊C412yes—
257C2×Dic61C2×C22⋊C412yes—
258C2×Dic61C2×C22⋊C412yes—
259C2×Dic61#12114yes—
260D244—24no—
261D244—41no—
262D244—41no—
263C4×D61—44no—
264C4×D61—44no—
265C4×D61—48no—
266C4×D61C4×D418yes—
267C4×D61#12414yes—
268C4×D61#12714yes—
269Dic122—24no—
270Dic122—41no—
271Dic122—41no—
272Dic122—48no—
273C8×C61—42no—
274#92418—42no—
275#21261—611nomaximal, Sylow
276C22×C61—42no—
277C22×C61—42no—
278C22×C61—42no—
279C22×C61—82no—
280C22×C61—82no—
281C22×C61C22×C22⋊C411yescharacteristic
282C22×C61C2×#12112yes—
283C22×C61#57511yescharacteristic
284C22×C61#57512yes—
285C22×C61#57511yescharacteristic
286C2×D61—22no—
287C2×D61—22no—
288C2×D61—24no—
289C2×D61—24no—
290C2×D61—42no—
291C2×D61—42no—
292C2×D61—42no—
293C2×D61—816no—
294C2×D61—82no—
295C2×D61—82no—
296C2×D61#39014yes—
297C4×C61—22no—
298C4×C61—22no—
299C4×C61—44no—
300C4×C61—42no—
301C4×C61C2×C4×D412yes—
302C4×C61C2×#12411yescharacteristic
303C4×C61C2×#12711yescharacteristic
304Dic61—22no—
305Dic61—22no—
306Dic61—44no—
307Dic61—42no—
308Dic61#39014yes—
309C22×#124—611no—
310C2×#423—611no—
311C2×#419—612no—
312C2×#418—612no—
313C4×#117—611no—
314#1983—611no—
315#1983—612no—
316#1983—612no—
317#1983—611no—
318#1983—611no—
319#1983D6111yescharacteristic, complemented
320C2×C61—82no—
321C2×C61C2×#39011yescharacteristic
322C2×C61#289611yescharacteristic
323C2×C61#289611yescharacteristic
324C2×C61#304012yes—
325C2×C61#304012yes—
326D61—24no—
327D61—82no—
328C23×Q8—611no—
329C23×D4—611no—
330C22×M16—612no—
331C2×#124—611no—
332C2×#124—611no—
333C2×#124—618no—
334C2×#124—611no—
335C2×#124—1221no—
336C2×#124—1221no—
337C2×#124—1224no—
338C2×#124C2×D6111yescharacteristic
339C2×#117—612no—
340C22×C22⋊C4—1221no—
341C22×C22⋊C4—1221no—
342C22×C42—611no—
343#423—1224no—
344C2×C4⋊C8—612no—
345C2×C4⋊C8—611no—
346C2×C4⋊C8C2×D6111yescharacteristic, complemented
347#419—1228no—
348#418—1228no—
349C2×#103—612no—
350C2×#103—612no—
351C2×#103—612no—
352C2×#103C2×D6112yescomplemented
353C2×#102—612no—
354C2×#102—612no—
355C2×#102—612no—
356C2×#102C2×D6112yes—
357C4×C4⋊C4—612no—
358C4×C4⋊C4—611no—
359C4×C4⋊C4C2×D6111yescharacteristic, complemented
360C4×C22⋊C4—1221no—
361C4×C22⋊C4—1221no—
362#376—1221no—
363C43—1221no—
364C61#2126111yescharacteristic, complemented, Sylow
365C25—1221no—
366C22×Q8—614no—
367C22×Q8—611no—
368C22×Q8—1221no—
369C22×Q8—1222no—
370C22×Q8—1224no—
371C22×Q8—1224no—
372C22×Q8C22×D6111yescharacteristic
373C22×D4—614no—
374C22×D4—611no—
375C22×D4—1224no—
376C22×D4—1224no—
377C22×D4—1222no—
378C22×D4—1221no—
379C22×D4C22×D6111yescharacteristic
380C23×C4—611no—
381C23×C4—611no—
382C23×C4—611no—
383C23×C4—1221no—
384C2×M16—1224no—
385C2×M16—1224no—
386C2×M16—1224no—
387C22×C8—612no—
388C22×C8C22×D6112yes—
389#124—22no—
390#124—614no—
391#124—614no—
392#124—614no—
393#124—1228no—
394#124—1228no—
395#124—1222no—
396#124—1222no—
397#124—1222no—
398#124C4×D6114yes—
399#117—618no—
400#117—1224no—
401C2×C4⋊C4—612no—
402C2×C4⋊C4C22×D6112yes—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—1228no—
406C2×C22⋊C4—1228no—
407C2×C22⋊C4—1221no—
408C2×C22⋊C4—1221no—
409C2×C22⋊C4—1222no—
410C2×C22⋊C4—1221no—
411C2×C22⋊C4—1221no—
412C2×C22⋊C4—1221no—
413C2×C22⋊C4—1221no—
414C2×C42—21no—
415C2×C42—611no—
416C2×C42—611no—
417C2×C42—611no—
418C2×C42—614no—
419C2×C42—614no—
420C2×C42—1222no—
421C2×C42—1222no—
422C2×C42—1221no—
423C2×C42C22×D6111yescharacteristic
424C4⋊C8—22no—
425C4⋊C8—1224no—
426C4⋊C8—1222no—
427#103—24no—
428#103—1224no—
429#103—1224no—
430#103—1224no—
431#102—24no—
432#102—1224no—
433#102—1224no—
434#102—1224no—
435#95—21no—
436#95—1222no—
437#95—1221no—
438C24—21no—
439C24—611no—
440C24—1224no—
441C24—1221no—
442C24—2444no—
443C24—2442no—
444C24—2444no—
445C24—2442no—
446C2×Q8—21no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—612no—
450C2×Q8—1224no—
451C2×Q8—12216no—
452C2×Q8—1221no—
453C2×Q8—1222no—
454C2×Q8—1222no—
455C2×Q8—2444no—
456C2×Q8—2444no—
457C2×Q8—2444no—
458C2×Q8C2×C4×D6112yes—
459C2×D4—22no—
460C2×D4—21no—
461C2×D4—22no—
462C2×D4—612no—
463C2×D4—12216no—
464C2×D4—1224no—
465C2×D4—1222no—
466C2×D4—1221no—
467C2×D4—1222no—
468C2×D4—2444no—
469C2×D4—2444no—
470C2×D4—2444no—
471C2×D4C2×C4×D6112yes—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—611no—
475C22×C4—612no—
476C22×C4—612no—
477C22×C4—614no—
478C22×C4—614no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—614no—
482C22×C4—614no—
483C22×C4—1224no—
484C22×C4—1222no—
485C22×C4—1221no—
486C22×C4—2441no—
487C22×C4—2442no—
488C22×C4—2441no—
489C22×C4—2444no—
490C22×C4D4×D6111yescharacteristic
491C22×C4D4×D6111yescharacteristic
492C22×C4C23×D6111yescharacteristic
493M16—2448no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—1222no—
498C2×C8—1222no—
499C2×C8—1222no—
500C4⋊C4—22no—
501C4⋊C4—614no—
502C4⋊C4—1222no—
503C4⋊C4D4×D6114yes—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—1224no—
507C22⋊C4—1224no—
508C22⋊C4—1228no—
509C22⋊C4—1224no—
510C22⋊C4—1224no—
511C22⋊C4—1224no—
512C22⋊C4—1224no—
513C42—42no—
514C42—612no—
515C42—612no—
516C42—614no—
517C42—614no—
518C42—1228no—
519C42—2442no—
520C42—2444no—
521C42C2×C4×D6112yes—
522C42D4×D6112yes—
523C23—22no—
524C23—42no—
525C23—41no—
526C23—42no—
527C23—41no—
528C23—612no—
529C23—614no—
530C23—1221no—
531C23—1222no—
532C23—1221no—
533C23—1222no—
534C23—4888no—
535C23—4888no—
536C23—2448no—
537C23—2448no—
538C23—2448no—
539C23—2442no—
540C23—2441no—
541C23—2442no—
542C23—2441no—
543C23C2×D4×D6111yescharacteristic
544Q8—24no—
545Q8—41no—
546Q8—41no—
547Q8—1224no—
548Q8—2448no—
549Q8—2441no—
550Q8—2441no—
551D4—24no—
552D4—41no—
553D4—41no—
554D4—1224no—
555D4—2448no—
556D4—2441no—
557D4—2441no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—612no—
567C2×C4—614no—
568C2×C4—618no—
569C2×C4—614no—
570C2×C4—614no—
571C2×C4—612no—
572C2×C4—612no—
573C2×C4—1221no—
574C2×C4—1222no—
575C2×C4—1221no—
576C2×C4—1222no—
577C2×C4—1224no—
578C2×C4—2444no—
579C2×C4—2444no—
580C2×C4—2448no—
581C2×C4—2442no—
582C2×C4—2442no—
583C2×C4—2442no—
584C2×C4C22⋊C4×D6112yes—
585C2×C4C22⋊C4×D6112yes—
586C2×C4C22×C4×D6111yescharacteristic
587C2×C4C2×D4×D6112yes—
588C2×C4C2×D4×D6111yescharacteristic
589C2×C4#41×D6112yes—
590C8—42no—
591C8—2442no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—82no—
596C22—82no—
597C22—614no—
598C22—1224no—
599C22—1224no—
600C22—1222no—
601C22—1222no—
602C22—48816no—
603C22—4882no—
604C22—4882no—
605C22—2442no—
606C22—2442no—
607C22—2442no—
608C22C2×C22⋊C4×D6111yescharacteristic
609C22#121×D6112yes—
610C22#1684055811yescharacteristic
611C22#1684055912yes—
612C22#1684056111yescharacteristic
613C4—22no—
614C4—22no—
615C4—42no—
616C4—44no—
617C4—614no—
618C4—1222no—
619C4—1222no—
620C4—2442no—
621C4—2444no—
622C4C4×D4×D6112yes—
623C4#124×D6111yescharacteristic
624C4#127×D6111yescharacteristic
625C2—82no—
626C2—1224no—
627C2—4882no—
628C2#390×D6111yescharacteristic
629C2#2145433411yescharacteristic
630C2#2145433511yescharacteristic
631C2#2145530812yes—
632C2#2145531012yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1983C2
socleC23×C61C22×D4
prime radicalC22×D4×D61C22
cosocleC22×C4×C61C24
hypercenter#1983D61
hypocenterC61#21261
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1983

Complements

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

NH ≅ G/Ntype
2. C2×#16839784C2semidirect
3. C2×#16839769C2semidirect
5. C4×#14519735C2semidirect
6. #21446752C2semidirect
7. #21446751C2semidirect
8. #21446750C2semidirect
9. #21446749C2semidirect
10. #21446748C2semidirect
11. C61×#1983C2semidirect
21. #124×D61C4semidirect
27. C2×C4⋊#92418C22semidirect
28. C2×#14519706C22semidirect
31. C2×#14519702C22semidirect
32. C2×#14519701C22semidirect
33. C2×#14519700C22semidirect
37. C4⋊C4×Dic61C22semidirect
38. C4×C4⋊Dic61C22semidirect
44. C2×C4⋊C8×C61C22semidirect
45. C2×#103×C61C22semidirect
47. C4×C4⋊C4×C61C22semidirect
319. #1983D61semidirect
346. C2×C4⋊C8C2×D61semidirect
352. C2×#103C2×D61semidirect
359. C4×C4⋊C4C2×D61semidirect
364. C61#21261semidirect

Maximal subgroups

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

subgroupindex
C2×#124×D612
C2×#168397842
C2×#168397692
C2×#168397652
C4×#145197352
#214467522
#214467512
#214467502
#214467492
#214467482
C61×#19832
#2126161