Finite Group Databasebeta

← #29681819

Subgroups — #29681819

Order 15616.

2030 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840331C211yesmaximal, characteristic
2C2×#16840236C211yesmaximal, characteristic, complemented
3C2×#16839735C211yesmaximal, characteristic, complemented
4C2×#16839735C211yesmaximal, characteristic, complemented
5#21448399C211yesmaximal, characteristic, complemented
6#21446271C218yesmaximal, complemented
7#21445552C211yesmaximal, characteristic, complemented
8C61×#1740C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10C23×D244C2211yescharacteristic
11#16840331—22no—
12#16840331—22no—
13#16840331—22no—
14#16840331—22no—
15C2×#14519774C412yescomplemented
16C2×#14519765C412yescomplemented
17C2×#14519752C414yescomplemented
18C2×#14519747C2214yes—
19#16840236—24no—
20#16840236—24no—
21C2×C4×D244—22no—
22C2×#14519739—22no—
23C2×#14519736C2211yescharacteristic
24C2×#14519736C2211yescharacteristic
25C2×C22⋊C4×D61C2214yes—
26#16839735—24no—
27#16839735—24no—
28C2×#14519691C2214yes—
29C2×#14519691C2214yes—
30#16839664—22no—
31#16839663—22no—
32#16839617C2214yes—
33#16839529C2214yes—
34C2×#117×C61C2211yescharacteristic
35C2×#117×C61C2211yescharacteristic
36C2×#117×C61C2211yescharacteristic
37C61×#342C2214yes—
38C24×D61C2312yes—
39C2×#11758222—22no—
40C2×#11758222—22no—
41C2×#11758222—21no—
42C2×#11758222—21no—
43C2×#11758222—21no—
44C2×#11758222—21no—
45C2×#11758222—22no—
46C2×#11758222—22no—
47C22×Dic122C2×C411yescharacteristic, complemented
48C22×D244—22no—
49C22×D244—22no—
50C22×D244—22no—
51C22×D244—22no—
52C22×D244C2×C412yes—
53C22×D244C2×C412yes—
54C22×D244C2×C411yescharacteristic, complemented
55C22×D244C2×C418yescomplemented
56C22×D244D412yes—
57C22×D244D412yes—
58C22×D244D412yes—
59C22×D244D412yes—
60C22×C22⋊D61C2312yes—
61C22×C4×D61—24no—
62C22×C4×D61C2×C412yescomplemented
63#14519774—24no—
64#14519774—24no—
65#14519765—24no—
66#14519765—24no—
67#14519752—28no—
68#14519752—28no—
69#14519747—28no—
70#14519747—28no—
71C4×D244—216no—
72#14519739—216no—
73#14519736—22no—
74#14519736—22no—
75#14519736—22no—
76#14519736D414yescomplemented
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—21no—
79C2×C22⋊Dic61C2312yes—
80C2×C22⋊Dic61C2312yes—
81C2×#11758198—24no—
82C2×#11758198—28no—
83C2×#11758198C2×C418yescomplemented
84C22⋊C4×D61—28no—
85C22⋊C4×D61—28no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88#14519691—28no—
89#14519691—28no—
90C23×C4×C61C2311yescharacteristic
91#117×C61—22no—
92#117×C61—21no—
93#117×C61—21no—
94#117×C61—21no—
95#117×C61—21no—
96#117×C61—22no—
97#117×C61—22no—
98#117×C61D414yescomplemented
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61C2×C412yescomplemented
102C2×C22⋊C4×C61C2312yes—
103C2×C22⋊C4×C61C2312yes—
104C2×C22⋊C4×C61C2312yes—
105C2×C42×C61—21no—
106C2×C42×C61—21no—
107C2×C42×C61C2×C412yescomplemented
108C24×C61C2411yescharacteristic
109C23×D61—22no—
110C23×D61—22no—
111C23×D61—22no—
112C23×D61—22no—
113C23×D61—84no—
114C23×D61C22×C412yes—
115C23×D61C22×C414yes—
116C23×D61C2×D414yes—
117C23×D61C2×D414yes—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61C22×C414yes—
121C22×C4×C61C22×C411yescharacteristic
122C22×C4×C61C22×C411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2×D412yes—
129C22×C4×C61#4112yes—
130C22×C4×C61#4112yes—
131C2×C22⋊D61—22no—
132C2×C22⋊D61—22no—
133C2×C22⋊D61—22no—
134C2×C22⋊D61—22no—
135C2×C22⋊D61—22no—
136C2×C22⋊D61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61C22×C412yes—
140#11758222—42no—
141#11758222—42no—
142#11758222—42no—
143#11758222—42no—
144#11758222—42no—
145#11758222—42no—
146#11758222—42no—
147#11758222—42no—
148C2×D244—24no—
149C2×D244—24no—
150C2×D244—24no—
151C2×D244—24no—
152C2×D244—24no—
153C2×D244—24no—
154C2×D244—24no—
155C2×D244—24no—
156C2×D244—24no—
157C2×D244—24no—
158C2×D244—24no—
159C2×D244—24no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—22no—
163C2×D244—22no—
164C2×D244—216no—
165C2×D244—88no—
166C2×D244C22⋊C414yes—
167C2×D244C22⋊C414yes—
168C2×D244C22⋊C414yes—
169C2×D244C22⋊C414yes—
170C2×C4×D61—22no—
171C2×C4×D61—22no—
172C2×C4×D61—24no—
173C2×C4×D61—24no—
174C2×C4×D61—216no—
175C2×C4×D61—88no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—22no—
179C2×Dic122—22no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—24no—
183C4⋊C4×C61—42no—
184C22⋊C4×C61—22no—
185C22⋊C4×C61—22no—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—22no—
188C22⋊C4×C61—22no—
189C22⋊C4×C61C2×D414yes—
190C42×C61—22no—
191C42×C61—22no—
192C42×C61—24no—
193C42×C61—42no—
194C22⋊Dic61—22no—
195C22⋊Dic61—22no—
196C22⋊Dic61—22no—
197C22⋊Dic61C2×D414yes—
198#11758198—216no—
199#11758198—216no—
200#11758198—88no—
201C4⋊Dic61—24no—
202C4⋊Dic61—42no—
203C4×Dic61—24no—
204C4×Dic61—42no—
205C23×C61—42no—
206C23×C61C23×C411yescharacteristic
207C23×C61C22×D411yescharacteristic
208C23×C61C22×D411yescharacteristic
209C23×C61C22×D411yescharacteristic
210C23×C61C22×D411yescharacteristic
211C23×C61C2×#4111yescharacteristic
212C23×C61C2×#4111yescharacteristic
213C22×D61—24no—
214C22×D61—24no—
215C22×D61—24no—
216C22×D61—24no—
217C22×D61—24no—
218C22×D61—24no—
219C22×D61—22no—
220C22×D61—22no—
221C22×D61—22no—
222C22×D61—48no—
223C22×D61—48no—
224C22×D61—48no—
225C22×D61—44no—
226C22×D61—84no—
227C22×D61—84no—
228C22×D61—84no—
229C22×D61—84no—
230C22×D61C2×C22⋊C418yes—
231C2×C4×C61—24no—
232C2×C4×C61—24no—
233C2×C4×C61—44no—
234C2×C4×C61—44no—
235C2×C4×C61—44no—
236C2×C4×C61—42no—
237C2×C4×C61C2×C22⋊C411yescharacteristic
238C2×C4×C61C2×C22⋊C411yescharacteristic
239C2×C4×C61C2×C22⋊C411yescharacteristic
240C2×C4×C61C2×C22⋊C411yescharacteristic
241C2×C4×C61C4×D412yes—
242C2×C4×C61C4×D412yes—
243C2×C4×C61C4×D412yes—
244C2×C4×C61C4×D412yes—
245C2×C4×C61#12012yes—
246C2×C4×C61#12012yes—
247C2×C4×C61#12112yes—
248C2×C4×C61#12112yes—
249C2×C4×C61#12112yes—
250C2×C4×C61#12112yes—
251C2×C4×C61#12312yes—
252C2×C4×C61#12312yes—
253C22⋊D61—42no—
254C22⋊D61—42no—
255C22⋊D61—42no—
256C22⋊D61—42no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—24no—
261C2×Dic61—44no—
262C2×Dic61—44no—
263C2×Dic61—42no—
264D244—28no—
265D244—28no—
266D244—28no—
267D244—28no—
268D244—42no—
269D244—42no—
270D244—816no—
271C4×D61—44no—
272C4×D61—44no—
273C4×D61—816no—
274Dic122—42no—
275Dic122—42no—
276#20985—611nomaximal, Sylow
277C22×C61—22no—
278C22×C61—22no—
279C22×C61—22no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61C22×C22⋊C411yescharacteristic
285C22×C61C2×C4×D411yescharacteristic
286C22×C61C2×C4×D411yescharacteristic
287C22×C61C2×#12011yescharacteristic
288C22×C61C2×#12111yescharacteristic
289C22×C61C2×#12111yescharacteristic
290C22×C61C2×#12311yescharacteristic
291C2×D61—216no—
292C2×D61—28no—
293C2×D61—48no—
294C2×D61—48no—
295C2×D61—44no—
296C2×D61—42no—
297C2×D61—84no—
298C2×D61—84no—
299C2×D61—84no—
300C2×D61—84no—
301C2×D61—82no—
302C2×D61—82no—
303C4×C61—44no—
304C4×C61—44no—
305C4×C61—82no—
306C4×C61#38612yes—
307C4×C61#38612yes—
308C4×C61#38612yes—
309C4×C61#38612yes—
310Dic61—44no—
311Dic61—42no—
312Dic61—82no—
313C2×#535—611no—
314C2×#518—611no—
315C2×#410—611no—
316C2×#410—611no—
317#2309—611no—
318#1898—618no—
319#1740—611no—
320#1740D6111yescharacteristic, complemented
321C2×C61—21no—
322C2×C61—21no—
323C2×C61—21no—
324C2×C61—21no—
325C2×C61—21no—
326C2×C61—21no—
327C2×C61C2×#38611yescharacteristic
328C2×C61#288312yes—
329D61—48no—
330D61—82no—
331C22×#41—611no—
332C23×D4—611no—
333#535—1222no—
334#535—1222no—
335#535—1222no—
336#535—1222no—
337C2×#127—612no—
338C2×#124—612no—
339C2×#121—614no—
340C2×#120—614no—
341#518—1224no—
342#518—1224no—
343C2×C4×D4—1222no—
344C2×C4×D4—1222no—
345C2×#117—611no—
346C2×#117—611no—
347C2×#117C2×D6111yescharacteristic
348C2×#117C2×D6111yescharacteristic
349C2×#117C2×D6111yescharacteristic
350C22×C22⋊C4—614no—
351#410—1224no—
352#410—1224no—
353C2×#99—614no—
354C2×#99—614no—
355#390—1222no—
356#390—1222no—
357#379—614no—
358#342—614no—
359#342C2×D6114yes—
360C61#2098511yescharacteristic, complemented, Sylow
361C25—612no—
362C2×#41—1222no—
363C2×#41—1222no—
364C2×#41—1221no—
365C2×#41—1221no—
366C2×#41—1221no—
367C2×#41—1221no—
368C2×#41—1222no—
369C2×#41—1222no—
370C22×Q8—611no—
371C22×D4—612no—
372C22×D4—612no—
373C22×D4—612no—
374C22×D4—611no—
375C22×D4—618no—
376C22×D4—612no—
377C22×D4—612no—
378C22×D4—612no—
379C22×D4—612no—
380C22×D4—1222no—
381C22×D4—1222no—
382C22×D4—1222no—
383C22×D4—1222no—
384C23×C4—612no—
385C23×C4—1224no—
386C23×C4C22×D6111yescharacteristic
387#127—1224no—
388#127—1224no—
389#124—1224no—
390#124—1224no—
391#121—1228no—
392#121—1228no—
393#120—1228no—
394#120—1228no—
395C4×D4—12216no—
396C4×D4—12216no—
397#117—22no—
398#117—21no—
399#117—21no—
400#117—21no—
401#117—21no—
402#117—22no—
403#117—22no—
404#117—614no—
405#117—1222no—
406#117—1222no—
407#117—1222no—
408#117D24414yescomplemented
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4C4×D6112yescomplemented
414C2×C22⋊C4—612no—
415C2×C22⋊C4—618no—
416C2×C22⋊C4—612no—
417C2×C22⋊C4—1228no—
418C2×C22⋊C4—1224no—
419C2×C22⋊C4—1228no—
420C2×C22⋊C4—1228no—
421C2×C22⋊C4C22×D6112yes—
422C2×C22⋊C4C22×D6112yes—
423C2×C22⋊C4C22×D6112yes—
424C2×C42—21no—
425C2×C42—21no—
426C2×C42—1221no—
427C2×C42—1221no—
428C2×C42C4×D6112yescomplemented
429#99—1228no—
430#99—1228no—
431C24—614no—
432C24—614no—
433C24—614no—
434C24—612no—
435C24—1222no—
436C24—1222no—
437C24—1222no—
438C24—1222no—
439C24—4884no—
440C24C23×D6111yescharacteristic
441#41—2442no—
442#41—2442no—
443#41—2442no—
444#41—2442no—
445#41—2442no—
446#41—2442no—
447#41—2442no—
448#41—2442no—
449C2×Q8—1221no—
450C2×Q8—1221no—
451C2×Q8—1222no—
452C2×Q8—1222no—
453C2×D4—614no—
454C2×D4—614no—
455C2×D4—614no—
456C2×D4—614no—
457C2×D4—12216no—
458C2×D4—1222no—
459C2×D4—1222no—
460C2×D4—1222no—
461C2×D4—1222no—
462C2×D4—1222no—
463C2×D4—1222no—
464C2×D4—1221no—
465C2×D4—1221no—
466C2×D4—1224no—
467C2×D4—1224no—
468C2×D4—1222no—
469C2×D4—1224no—
470C2×D4—1224no—
471C2×D4—1222no—
472C2×D4—1224no—
473C2×D4—1224no—
474C2×D4—1224no—
475C2×D4—1224no—
476C2×D4—1224no—
477C2×D4—1224no—
478C2×D4—1224no—
479C2×D4—1224no—
480C2×D4—4888no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—612no—
484C22×C4—12216no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—1222no—
488C22×C4—1224no—
489C22×C4—1224no—
490C22×C4—1222no—
491C22×C4—4888no—
492C22×C4C2×C4×D6114yes—
493C22×C4C2×C4×D6111yescharacteristic
494C22×C4C2×C4×D6111yescharacteristic
495C22×C4C2×D24412yes—
496C22×C4#1175822212yes—
497C22×C4D4×D6111yescharacteristic
498C22×C4D4×D6111yescharacteristic
499C22×C4D4×D6111yescharacteristic
500C22×C4D4×D6111yescharacteristic
501C22×C4D4×D6112yes—
502C22×C4#1175822612yes—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—24no—
506C4⋊C4—42no—
507C4⋊C4—1224no—
508C4⋊C4—2442no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—614no—
515C22⋊C4—12216no—
516C22⋊C4—12216no—
517C22⋊C4—1222no—
518C22⋊C4—1222no—
519C22⋊C4—1222no—
520C22⋊C4—4888no—
521C22⋊C4C2×D24414yes—
522C42—22no—
523C42—22no—
524C42—24no—
525C42—42no—
526C42—1224no—
527C42—2442no—
528C23—42no—
529C23—618no—
530C23—1224no—
531C23—1224no—
532C23—1224no—
533C23—1224no—
534C23—1224no—
535C23—1224no—
536C23—1222no—
537C23—1222no—
538C23—1222no—
539C23—4884no—
540C23—4884no—
541C23—4884no—
542C23—4884no—
543C23—2448no—
544C23—2448no—
545C23—2448no—
546C23—2444no—
547C23C22×C4×D6111yescharacteristic
548C23C2×D4×D6111yescharacteristic
549C23C2×D4×D6111yescharacteristic
550C23C2×D4×D6111yescharacteristic
551C23C22×D24411yescharacteristic
552C23C2×#1175822211yescharacteristic
553C23C2×#1175822611yescharacteristic
554Q8—2442no—
555Q8—2442no—
556D4—1228no—
557D4—1228no—
558D4—1228no—
559D4—1228no—
560D4—48816no—
561D4—2442no—
562D4—2442no—
563D4—2442no—
564D4—2442no—
565D4—2442no—
566D4—2442no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—1222no—
576C2×C4—1224no—
577C2×C4—48816no—
578C2×C4—2444no—
579C2×C4—2442no—
580C2×C4—2444no—
581C2×C4—2444no—
582C2×C4—2444no—
583C2×C4C22⋊C4×D6111yescharacteristic
584C2×C4C22⋊C4×D6111yescharacteristic
585C2×C4C22⋊C4×D6111yescharacteristic
586C2×C4C22⋊C4×D6111yescharacteristic
587C2×C4#1451973912yes—
588C2×C4#1451973912yes—
589C2×C4C4×D24412yes—
590C2×C4C4×D24412yes—
591C2×C4#1451974712yes—
592C2×C4#1451974712yes—
593C2×C4#1451975012yes—
594C2×C4#1451975012yes—
595C2×C4#1451975212yes—
596C2×C4#1451975212yes—
597C2×C4#1451976212yes—
598C2×C4#1451976212yes—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—1228no—
607C22—12216no—
608C22—4882no—
609C22—4882no—
610C22—4884no—
611C22—4884no—
612C22—4884no—
613C22—4884no—
614C22—2448no—
615C22—2448no—
616C22—2442no—
617C22—2444no—
618C22C2×C22⋊C4×D6111yescharacteristic
619C22C2×#1451973911yescharacteristic
620C22C2×C4×D24411yescharacteristic
621C22C2×#1451974711yescharacteristic
622C22C2×#1451975011yescharacteristic
623C22C2×#1451975211yescharacteristic
624C22C2×#1451976211yescharacteristic
625C4—44no—
626C4—44no—
627C4—82no—
628C4—4882no—
629C4—2444no—
630C4—2442no—
631C4#1683964012yes—
632C4#1683964012yes—
633C4#1683964012yes—
634C4#1683964012yes—
635C2—21no—
636C2—21no—
637C2—21no—
638C2—21no—
639C2—21no—
640C2—21no—
641C2—4882no—
642C2—2448no—
643C2C2×#1683964011yescharacteristic
644C2#2145427812yes—
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC61×#1740C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840331C2
cosocleC24×C61C24
hypercenter#1740D61
hypocenterC61#20985
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1740

Complements

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

NH ≅ G/Ntype
2. C2×#16840236C2semidirect
3. C2×#16839735C2semidirect
4. C2×#16839735C2semidirect
5. #21448399C2semidirect
6. #21446271C2semidirect
7. #21445552C2semidirect
8. C61×#1740C2semidirect
15. C2×#14519774C4semidirect
16. C2×#14519765C4semidirect
17. C2×#14519752C4semidirect
47. C22×Dic122C2×C4semidirect
54. C22×D244C2×C4semidirect
55. C22×D244C2×C4semidirect
62. C22×C4×D61C2×C4semidirect
76. #14519736D4semidirect
83. C2×#11758198C2×C4semidirect
98. #117×C61D4semidirect
101. C2×C4⋊C4×C61C2×C4semidirect
107. C2×C42×C61C2×C4semidirect
320. #1740D61semidirect
360. C61#20985semidirect
408. #117D244semidirect
413. C2×C4⋊C4C4×D61semidirect
428. C2×C42C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403312
C2×#168402362
C2×#168397352
#214483992
#214462712
#214455522
C61×#17402
#2098561