Finite Group Databasebeta

← #29707499

Subgroups — #29707499

Order 15616.

1610 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840274C211yesmaximal, characteristic
2C2×#16839798C211yesmaximal, characteristic, complemented
3C2×#16839759C211yesmaximal, characteristic, complemented
4C2×#16839756C211yesmaximal, characteristic, complemented
5C22×#14519689C211yesmaximal, characteristic, complemented
6#21448198C211yesmaximal, characteristic, complemented
7#21447025C212yesmaximal, complemented
8#21447018C212yesmaximal, complemented
9#21447011C212yesmaximal, complemented
10#21447004C212yesmaximal, complemented
11C61×#1735C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C23×C4×D61C2211yescharacteristic
14C23×#92418C2211yescharacteristic
15#16840274—24no—
16#16840274—24no—
17C2×#14519761—21no—
18C2×#14519757C2212yes—
19C2×#14519752C2212yes—
20C2×#14519747—21no—
21C2×C4×D244—22no—
22C2×#14519736C2211yescharacteristic
23#16839798—24no—
24C2×#14519710C2212yescomplemented
25C2×#14519708C2212yescomplemented
26#16839759—24no—
27#16839756—24no—
28C2×#14519702C2212yescomplemented
29C2×#14519700C2211yescharacteristic, complemented
30C2×#14519700C2211yescharacteristic, complemented
31C2×#14519699C2212yescomplemented
32C2×#14519697C2211yescharacteristic, complemented
33C2×#14519697C2211yescharacteristic, complemented
34C2×#14519689—24no—
35C2×#14519689C2212yes—
36C2×#14519689C2212yes—
37#16839623—21no—
38#16839619C2212yes—
39#16839619C2212yes—
40#16839617—21no—
41C4×#11758198—22no—
42C23×C8×C61C2211yescharacteristic
43C2×#117×C61C2211yescharacteristic
44C2×#117×C61C2211yescharacteristic
45C61×#340C2212yescomplemented
46C61×#340C2212yescomplemented
47C24×D61D411yescharacteristic
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222—22no—
51C2×#11758222—22no—
52C2×#11758222—22no—
53C2×#11758222D414yes—
54C22×Dic122C2311yescharacteristic
55C22×D244—22no—
56C22×D244C2311yescharacteristic
57C22×C22⋊D61—21no—
58C23×Dic61D411yescharacteristic
59C22×C4×D61—21no—
60C22×C4×D61—22no—
61C22×C4×D61—22no—
62C22×C4×D61—22no—
63C22×C4×D61—22no—
64C22×C4×D61—22no—
65C22×C4×D61—22no—
66C22×C4×D61D412yes—
67C22×C4×D61C2312yes—
68C22×#92418—22no—
69C22×#92418—22no—
70C22×#92418D414yescomplemented
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73#14519761—28no—
74#14519757—24no—
75#14519757—24no—
76#14519752—24no—
77#14519752—24no—
78#14519747—28no—
79C4×D244—216no—
80#14519736—22no—
81#14519736—22no—
82C2×C4⋊Dic61—22no—
83C2×C4⋊Dic61C2312yes—
84C2×C22⋊Dic61—21no—
85C2×#11758198—22no—
86C2×#11758198—22no—
87C2×#11758198—22no—
88C2×#11758198—22no—
89C2×C4×Dic61—21no—
90#14519710—24no—
91#14519708—24no—
92#14519702—24no—
93#14519700—22no—
94#14519700—22no—
95#14519699—24no—
96#14519697—22no—
97#14519697—22no—
98#14519689—24no—
99#14519689—24no—
100#14519684—22no—
101#14519684—22no—
102C23×C4×C61C2311yescharacteristic
103C22×C8×C61—22no—
104C22×C8×C61—22no—
105C22×C8×C61—22no—
106C22×C8×C61C2312yes—
107#117×C61—22no—
108#117×C61—22no—
109#117×C61—22no—
110#117×C61D414yes—
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2312yes—
114C2×C22⋊C4×C61—21no—
115C2×C22⋊C4×C61—21no—
116C2×C42×C61—21no—
117C2×C42×C61—21no—
118C24×C61C2×D411yescharacteristic
119C23×D61—22no—
120C23×D61—24no—
121C23×D61—22no—
122C23×D61—22no—
123C23×D61—21no—
124C23×D61—42no—
125C23×D61C2×D412yes—
126C22×C4×C61—22no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2×D412yes—
132C22×C4×C61C2×D412yes—
133C22×C4×C61#4112yes—
134C22×C4×C61#4112yes—
135C22×C4×C61#4112yes—
136C22×C4×C61C2411yescharacteristic
137C2×C22⋊D61—24no—
138C2×C22⋊D61—42no—
139C2×C22⋊D61—42no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—21no—
144C22×Dic61—22no—
145C22×Dic61C2×D412yes—
146#11758222—28no—
147#11758222—28no—
148#11758222—44no—
149#11758222—44no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—28no—
155C2×D244—21no—
156C2×D244—44no—
157C2×D244—44no—
158C2×D244C2×D412yes—
159C2×C4×D61—24no—
160C2×C4×D61—24no—
161C2×C4×D61—24no—
162C2×C4×D61—24no—
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×C4×D61—22no—
166C2×C4×D61—22no—
167C2×C4×D61—22no—
168C2×C4×D61—22no—
169C2×C4×D61—44no—
170C2×C4×D61—44no—
171C2×C4×D61—42no—
172C2×C4×D61—44no—
173C2×C4×D61—42no—
174C2×C4×D61—44no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122C2×D412yes—
181C2×C8×C61—22no—
182C2×C8×C61—22no—
183C2×C8×C61—22no—
184C2×C8×C61—44no—
185C4⋊C4×C61—22no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61C2×D414yes—
191C22⋊C4×C61—24no—
192C22⋊C4×C61—42no—
193C42×C61—24no—
194C42×C61—42no—
195C22⋊Dic61—42no—
196#11758198—28no—
197#11758198—28no—
198#11758198—44no—
199#11758198—44no—
200C4⋊Dic61—28no—
201C4⋊Dic61—22no—
202C4⋊Dic61—22no—
203C4×Dic61—42no—
204C2×#92418—28no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418C2×D412yes—
210C2×#92418C2×D412yes—
211C23×C61—22no—
212C23×C61—22no—
213C23×C61#12012yes—
214C23×C61#12112yes—
215C23×C61#12312yes—
216C23×C61C22×D411yescharacteristic
217C22×D61—28no—
218C22×D61—22no—
219C22×D61—22no—
220C22×D61—22no—
221C22×D61—22no—
222C22×D61—22no—
223C22×D61—22no—
224C22×D61—22no—
225C22×D61—48no—
226C22×D61—48no—
227C22×D61—44no—
228C22×D61—44no—
229C22×D61—41no—
230C22×D61—41no—
231C22×D61—42no—
232C22×D61—42no—
233C22×D61#13514yes—
234C2×C4×C61—24no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61#12012yes—
242C2×C4×C61#12114yes—
243C2×C4×C61#12114yes—
244C2×C4×C61#12112yes—
245C2×C4×C61#12314yes—
246C2×C4×C61#12312yes—
247C2×C4×C61#12414yes—
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C22⋊D61—48no—
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—44no—
262C2×Dic61—42no—
263C2×Dic61—42no—
264C2×Dic61—42no—
265D244—22no—
266D244—22no—
267D244—416no—
268D244—41no—
269D244—41no—
270C4×D61—42no—
271C4×D61—42no—
272C4×D61—42no—
273C4×D61—42no—
274C4×D61—48no—
275C4×D61—48no—
276Dic122—22no—
277Dic122—22no—
278Dic122—41no—
279Dic122—41no—
280C8×C61—42no—
281#92418—22no—
282#92418—22no—
283#23479—611nomaximal, Sylow
284C22×C61—24no—
285C22×C61—24no—
286C22×C61—22no—
287C22×C61—22no—
288C22×C61#39414yes—
289C22×C61C2×#12011yescharacteristic
290C22×C61C2×#12111yescharacteristic
291C22×C61C2×#12311yescharacteristic
292C22×C61C2×#13512yes—
293C22×C61#57512yes—
294C2×D61—24no—
295C2×D61—24no—
296C2×D61—24no—
297C2×D61—416no—
298C2×D61—42no—
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—42no—
303C2×D61—44no—
304C2×D61—82no—
305C2×D61—81no—
306C2×D61—81no—
307C4×C61—44no—
308C4×C61—81no—
309C4×C61—81no—
310C4×C61#39414yes—
311C4×C61C2×#12111yescharacteristic
312C4×C61C2×#12111yescharacteristic
313C4×C61C2×#12311yescharacteristic
314C4×C61C2×#12411yescharacteristic
315Dic61—42no—
316Dic61—42no—
317Dic61—82no—
318C2×#525—611no—
319C2×#427—611no—
320C2×#416—611no—
321C2×#416—611no—
322C22×C22⋊C8—611no—
323#2280—611no—
324#2036—612no—
325#2035—612no—
326#2034—612no—
327#2033—612no—
328#1735D6111yescharacteristic, complemented
329C2×C61—24no—
330C2×C61C2×#39411yescharacteristic
331C2×C61#300212yes—
332C2×C61#305011yescharacteristic
333C2×C61#305411yescharacteristic
334C2×C61#309112yes—
335D61—44no—
336D61—82no—
337C22×#41—611no—
338C24×C4—611no—
339C23×C8—611no—
340C23×C8C2×D6111yescharacteristic
341#525—1224no—
342#525—1224no—
343C2×#123—1221no—
344C2×C22⋊Q8—612no—
345C2×#121—612no—
346C2×#120—1221no—
347C2×C4×D4—1222no—
348C2×#117—611no—
349C2×#117C2×D6111yescharacteristic
350C2×#117C2×D6111yescharacteristic
351#427—1224no—
352C2×#107—612no—
353C2×#106—612no—
354#416—1224no—
355#416—1224no—
356C2×#103—612no—
357C2×#103—611no—
358C2×#103—611no—
359C2×#102—612no—
360C2×#102—611no—
361C2×#102—611no—
362C2×C22⋊C8—612no—
363C2×C22⋊C8—612no—
364C2×C22⋊C8—1224no—
365#381—1221no—
366#380—612no—
367#380—612no—
368#379—1221no—
369C4×C22⋊C4—1222no—
370#340C2×D6112yescomplemented
371#340C2×D6112yescomplemented
372C61#2347911yescharacteristic, complemented, Sylow
373C25—611no—
374C2×#41—614no—
375C2×#41—1222no—
376C2×#41—1222no—
377C2×#41—1222no—
378C2×#41—1222no—
379C2×#41—1222no—
380C22×Q8—611no—
381C22×D4—611no—
382C22×D4—1222no—
383C22×D4—1221no—
384C23×C4—611no—
385C23×C4—612no—
386C23×C4—612no—
387C23×C4—1222no—
388C23×C4—1222no—
389C23×C4—1222no—
390C23×C4—1222no—
391C23×C4—1222no—
392C23×C4—1221no—
393C23×C4—1222no—
394C23×C4C22×D6111yescharacteristic
395C22×C8—22no—
396C22×C8—22no—
397C22×C8—22no—
398C22×C8—614no—
399C22×C8—611no—
400C22×C8—611no—
401C22×C8—1222no—
402C22×C8—1222no—
403C22×C8C22×D6112yes—
404#123—1228no—
405C22⋊Q8—1224no—
406C22⋊Q8—1224no—
407#121—1224no—
408#121—1224no—
409#120—1228no—
410C4×D4—12216no—
411#117—22no—
412#117—22no—
413#117—22no—
414#117—1222no—
415#117—1222no—
416#117D24414yes—
417C2×C4⋊C4—612no—
418C2×C4⋊C4—1222no—
419C2×C4⋊C4C22×D6111yescharacteristic
420C2×C4⋊C4C22×D6111yescharacteristic
421C2×C4⋊C4C22×D6112yes—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—1222no—
425C2×C22⋊C4—1222no—
426C2×C22⋊C4—1222no—
427C2×C22⋊C4—1222no—
428C2×C22⋊C4—1221no—
429C2×C42—21no—
430C2×C42—21no—
431C2×C42—1221no—
432#107—1224no—
433#106—1224no—
434#103—1222no—
435#103—1222no—
436#103—1224no—
437#102—1222no—
438#102—1222no—
439#102—1224no—
440C22⋊C8—1224no—
441C22⋊C8—1224no—
442#95—1222no—
443#95—1222no—
444C24—612no—
445C24—1224no—
446C24—1222no—
447C24—1221no—
448C24—1222no—
449C24—1222no—
450C24—2442no—
451C24D4×D6111yescharacteristic
452#41—1228no—
453#41—1228no—
454#41—2444no—
455#41—2444no—
456C2×Q8—612no—
457C2×Q8—1221no—
458C2×Q8—1221no—
459C2×Q8—1221no—
460C2×Q8—1221no—
461C2×Q8—1221no—
462C2×D4—612no—
463C2×D4—1228no—
464C2×D4—1224no—
465C2×D4—1221no—
466C2×D4—1221no—
467C2×D4—1221no—
468C2×D4—1221no—
469C2×D4—1221no—
470C2×D4—2444no—
471C2×D4—2444no—
472C2×D4—2442no—
473C2×D4—2442no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—612no—
478C22×C4—1221no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—1222no—
488C22×C4—1224no—
489C22×C4—1224no—
490C22×C4—1224no—
491C22×C4—1224no—
492C22×C4—1222no—
493C22×C4—2444no—
494C22×C4—2442no—
495C22×C4—2444no—
496C22×C4—2444no—
497C22×C4—2442no—
498C22×C4—2444no—
499C22×C4C2×D24412yes—
500C22×C4#1175822212yes—
501C22×C4#1175822212yes—
502C22×C4D4×D6112yes—
503C22×C4D4×D6112yes—
504C22×C4D4×D6111yescharacteristic
505C22×C4#1175822612yes—
506C22×C4C23×D6111yescharacteristic
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—44no—
511C2×C8—612no—
512C2×C8—612no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1228no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—1228no—
524C4⋊C4—1222no—
525C4⋊C4—1222no—
526C4⋊C4C2×D24414yes—
527C22⋊C4—24no—
528C22⋊C4—42no—
529C22⋊C4—1228no—
530C22⋊C4—1228no—
531C22⋊C4—2444no—
532C22⋊C4—2444no—
533C22⋊C4—2442no—
534C42—24no—
535C42—42no—
536C42—2442no—
537C23—22no—
538C23—22no—
539C23—614no—
540C23—1228no—
541C23—1222no—
542C23—1222no—
543C23—1222no—
544C23—1222no—
545C23—1222no—
546C23—1222no—
547C23—1222no—
548C23—2448no—
549C23—2448no—
550C23—2444no—
551C23—2441no—
552C23—2441no—
553C23—2442no—
554C23—2442no—
555C23—2444no—
556C23#1451974712yes—
557C23#1451975012yes—
558C23#1451976212yes—
559C23C2×D4×D6111yescharacteristic
560Q8—1222no—
561Q8—1222no—
562Q8—2441no—
563Q8—2441no—
564D4—1222no—
565D4—1222no—
566D4—24416no—
567D4—2448no—
568D4—2441no—
569D4—2441no—
570C2×C4—24no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—1222no—
578C2×C4—1222no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—2442no—
582C2×C4—2442no—
583C2×C4—2442no—
584C2×C4—2444no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2448no—
588C2×C4—2448no—
589C2×C4—2442no—
590C2×C4—2442no—
591C2×C4—2442no—
592C2×C4—2442no—
593C2×C4#1451974712yes—
594C2×C4#1451975012yes—
595C2×C4#1451975014yes—
596C2×C4#1451975214yes—
597C2×C4#1451976212yes—
598C2×C4#1451976214yes—
599C2×C4#1451976514yes—
600C2×C4C2×D4×D6111yescharacteristic
601C2×C4C2×D4×D6111yescharacteristic
602C2×C4C22×D24411yescharacteristic
603C2×C4C2×#1175822211yescharacteristic
604C2×C4C2×#1175822211yescharacteristic
605C2×C4C2×#1175822611yescharacteristic
606C8—42no—
607C8—1222no—
608C8—1222no—
609C22—22no—
610C22—22no—
611C22—24no—
612C22—24no—
613C22—1224no—
614C22—1224no—
615C22—1224no—
616C22—4881no—
617C22—4881no—
618C22—4882no—
619C22—24416no—
620C22—2442no—
621C22—2444no—
622C22—2442no—
623C22—2442no—
624C22—2442no—
625C22—2442no—
626C22#1683968014yes—
627C22C2×#1451974711yescharacteristic
628C22C2×#1451975011yescharacteristic
629C22C2×#1451976211yescharacteristic
630C22#135×D6112yes—
631C22#1684055912yes—
632C4—44no—
633C4—81no—
634C4—81no—
635C4—4882no—
636C4—2442no—
637C4—2442no—
638C4#1683968014yes—
639C4C2×#1451975011yescharacteristic
640C4C2×#1451975211yescharacteristic
641C4C2×#1451976211yescharacteristic
642C4C2×#1451976511yescharacteristic
643C2—24no—
644C2—4882no—
645C2—2444no—
646C2C2×#1683968011yescharacteristic
647C2#2145503912yes—
648C2#2145539911yescharacteristic
649C2#2145542711yescharacteristic
650C2#2145573612yes—
subgroupG/H
centerC22×C4D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1735C2
socleC23×C61C22×D4
prime radicalC2×#16840274C2
hypercenter#1735D61
hypocenterC61#23479
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ #1735

Complements

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

NH ≅ G/Ntype
2. C2×#16839798C2semidirect
3. C2×#16839759C2semidirect
4. C2×#16839756C2semidirect
5. C22×#14519689C2semidirect
6. #21448198C2semidirect
7. #21447025C2semidirect
8. #21447018C2semidirect
9. #21447011C2semidirect
10. #21447004C2semidirect
11. C61×#1735C2semidirect
24. C2×#14519710C22semidirect
25. C2×#14519708C22semidirect
28. C2×#14519702C22semidirect
29. C2×#14519700C22semidirect
30. C2×#14519700C22semidirect
31. C2×#14519699C22semidirect
32. C2×#14519697C22semidirect
33. C2×#14519697C22semidirect
45. C61×#340C22semidirect
46. C61×#340C22semidirect
70. C22×#92418D4semidirect
328. #1735D61semidirect
370. #340C2×D61semidirect
371. #340C2×D61semidirect
372. C61#23479semidirect

Maximal subgroups

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

subgroupindex
C2×#168402742
C2×#168397982
C2×#168397592
C2×#168397562
C22×#145196892
#214481982
#214470252
#214470182
#214470112
#214470042
C61×#17352
#2347961