Finite Group Databasebeta

← #29683751

Subgroups — #29683751

Order 15616.

1570 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459148C211yesmaximal, characteristic
2#21454382C211yesmaximal, characteristic, complemented
3C2×#16839780C211yesmaximal, characteristic, complemented
4C22×#14519704C211yesmaximal, characteristic, complemented
5C22×#14519703C211yesmaximal, characteristic, complemented
6#117×Dic61C211yesmaximal, characteristic, complemented
7#21446649C214yesmaximal, complemented
8#21446645C214yesmaximal, complemented
9C2×C61×#421C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12#16840340—22no—
13#16840336C412yescomplemented
14#16840333C412yescomplemented
15#16840329C412yescomplemented
16#16840326C412yescomplemented
17#16840321—22no—
18#16840308C2214yes—
19#16840273—22no—
20#16840269—22no—
21C2×#14519736C2211yescharacteristic
22C22×C4×Dic61C2211yescharacteristic
23#16839790—22no—
24#16839788—22no—
25#16839785C2214yescomplemented
26#16839780—24no—
27C2×#14519705C2214yescomplemented
28C2×#14519704—24no—
29C2×#14519704C2212yescomplemented
30C2×#14519704C2212yescomplemented
31C2×#14519703—24no—
32C2×#14519703C2212yescomplemented
33C2×#14519703C2212yescomplemented
34C4⋊C4×Dic61—22no—
35C22⋊C4×Dic61—22no—
36#16839608C2212yescomplemented
37C42×Dic61C2212yescomplemented
38C22×#41×C61C2211yescharacteristic
39C22×M16×C61C2211yescharacteristic
40C2×#117×C61C2211yescharacteristic
41C61×#421—24no—
42C2×#104×C61C2214yescomplemented
43C2×#11758222—21no—
44C2×#11758222—21no—
45C2×#11758222—22no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222C2×C412yes—
50C2×#11758222C2312yes—
51C22×Dic122C2×C411yescharacteristic
52C22×D244C2×C411yescharacteristic
53C22×C22⋊D61—21no—
54C23×Dic61—21no—
55C22×C4×D61—21no—
56C2×#11758194—21no—
57C2×#11758194—22no—
58C2×#11758194—22no—
59C2×#11758194C2312yes—
60C22×#92418—21no—
61#14519775C2×C414yescomplemented
62#14519773C2×C414yescomplemented
63#14519766C2×C414yescomplemented
64#14519764C2×C414yescomplemented
65#14519760—24no—
66#14519758—24no—
67#14519756—24no—
68#14519753—24no—
69#14519751—24no—
70#14519749—28no—
71#14519748—24no—
72#14519746—24no—
73#14519745—24no—
74C4×C22⋊D61—28no—
75#14519736C2×C412yes—
76#14519736C2×C412yes—
77#14519736D412yes—
78#14519736D412yes—
79C2×C4⋊Dic61D412yes—
80C2×C22⋊Dic61D412yes—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—22no—
83C2×C4×Dic61—24no—
84C2×C4×Dic61D414yescomplemented
85C2×C4×Dic61C2311yescharacteristic
86C2×C4×Dic61C2311yescharacteristic
87C2×C4×Dic61C2312yes—
88#14519706—28no—
89#14519705—28no—
90#14519704—24no—
91#14519704—24no—
92#14519703—24no—
93#14519703—24no—
94#14519684—24no—
95C2×#41×C61—21no—
96C2×#41×C61—21no—
97C2×#41×C61—22no—
98C2×#41×C61—22no—
99C2×#41×C61—22no—
100C2×#41×C61—22no—
101C2×#41×C61C2×C412yes—
102C2×#41×C61C2312yes—
103C22×Q8×C61C2×C411yescharacteristic
104C22×D4×C61—21no—
105C22×D4×C61C2×C411yescharacteristic
106C23×C4×C61—21no—
107C23×C4×C61C2311yescharacteristic
108C2×M16×C61—21no—
109C2×M16×C61—22no—
110C2×M16×C61—22no—
111C2×M16×C61C2312yes—
112C22×C8×C61—21no—
113#117×C61—22no—
114#117×C61D414yescomplemented
115C2×C4⋊C4×C61—21no—
116C2×C22⋊C4×C61—21no—
117C2×C42×C61C2312yes—
118#104×C61—28no—
119C24×C61—21no—
120C24×C61C2×D411yescharacteristic
121C23×D61—21no—
122#41×C61—24no—
123#41×C61—24no—
124#41×C61—22no—
125#41×C61—22no—
126#41×C61—22no—
127#41×C61—22no—
128#41×C61—42no—
129#41×C61—42no—
130#41×C61—42no—
131#41×C61—42no—
132C2×Q8×C61—21no—
133C2×Q8×C61—22no—
134C2×Q8×C61—22no—
135C2×Q8×C61C22×C412yes—
136C2×D4×C61—22no—
137C2×D4×C61—22no—
138C2×D4×C61—21no—
139C2×D4×C61—22no—
140C2×D4×C61—22no—
141C2×D4×C61—42no—
142C2×D4×C61—42no—
143C2×D4×C61C22×C412yes—
144C22×C4×C61—21no—
145C22×C4×C61—22no—
146C22×C4×C61—22no—
147C22×C4×C61—22no—
148C22×C4×C61—42no—
149C22×C4×C61—42no—
150C22×C4×C61C22×C411yescharacteristic
151C22×C4×C61C22×C411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D412yes—
156C22×C4×C61C2×D412yes—
157C22×C4×C61#4112yes—
158C22×C4×C61#4112yes—
159C22×C4×C61C2411yescharacteristic
160C2×C22⋊D61—22no—
161C2×C22⋊D61—22no—
162C2×C22⋊D61—28no—
163C2×C22⋊D61—42no—
164C2×C22⋊D61—42no—
165C22×Dic61—21no—
166C22×Dic61—21no—
167C22×Dic61—24no—
168C22×Dic61—21no—
169C22×Dic61—44no—
170C22×Dic61C2×D414yes—
171#11758222—24no—
172#11758222—24no—
173#11758222—22no—
174#11758222—22no—
175#11758222—22no—
176#11758222—22no—
177#11758222—42no—
178#11758222—42no—
179#11758222—42no—
180#11758222—42no—
181C2×D244—22no—
182C2×D244—22no—
183C2×D244—21no—
184C2×D244C22×C412yes—
185C2×C4×D61—22no—
186C2×C4×D61—22no—
187C2×C4×D61—42no—
188C2×C4×D61—42no—
189C2×Dic122—22no—
190C2×Dic122—22no—
191C2×Dic122—21no—
192C2×Dic122C22×C412yes—
193M16×C61—22no—
194M16×C61—22no—
195M16×C61—42no—
196C2×C8×C61—22no—
197C2×C8×C61—42no—
198C4⋊C4×C61—24no—
199C22⋊C4×C61—24no—
200C42×C61—22no—
201C42×C61C2×D414yes—
202C22⋊Dic61—28no—
203C22⋊Dic61C22⋊C414yes—
204C22⋊Dic61C22⋊C414yes—
205#11758198—28no—
206C4⋊Dic61C22⋊C414yes—
207C4⋊Dic61C22⋊C414yes—
208#11758196—28no—
209C4×Dic61—24no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C4×Dic61—48no—
213C4×Dic61C22×C414yes—
214C4×Dic61C2×D412yes—
215C4×Dic61C2×D412yes—
216C4×Dic61C2×D414yes—
217#11758194—22no—
218#11758194—22no—
219#11758194—42no—
220C2×#92418—22no—
221C2×#92418—42no—
222C23×C61—22no—
223C23×C61—42no—
224C23×C61—42no—
225C23×C61—82no—
226C23×C61C2×C22⋊C411yescharacteristic
227C23×C61C2×C22⋊C411yescharacteristic
228C23×C61#12112yes—
229C23×C61#12112yes—
230C23×C61C22×D411yescharacteristic
231C22×D61—22no—
232C22×D61—42no—
233C22×D61—82no—
234Q8×C61—22no—
235Q8×C61—22no—
236Q8×C61—42no—
237D4×C61—22no—
238D4×C61—22no—
239D4×C61—42no—
240D4×C61—42no—
241D4×C61—44no—
242D4×C61—42no—
243C2×C4×C61—22no—
244C2×C4×C61—24no—
245C2×C4×C61—44no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61—42no—
249C2×C4×C61—44no—
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61C4×D412yes—
253C2×C4×C61C4×D412yes—
254C2×C4×C61C4×D412yes—
255C2×C4×C61C4×D412yes—
256C2×C4×C61#12112yes—
257C2×C4×C61#12112yes—
258C2×C4×C61#12412yes—
259C2×C4×C61#12412yes—
260C2×C4×C61#12712yes—
261C2×C4×C61#12712yes—
262C2×C4×C61C23×C411yescharacteristic
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C22⋊D61—42no—
269C22⋊D61—42no—
270C22⋊D61—44no—
271C22⋊D61—88no—
272C2×Dic61—24no—
273C2×Dic61—22no—
274C2×Dic61—22no—
275C2×Dic61—22no—
276C2×Dic61—44no—
277C2×Dic61—48no—
278C2×Dic61—42no—
279C2×Dic61—42no—
280C2×Dic61C2×C22⋊C418yes—
281D244—22no—
282D244—22no—
283D244—42no—
284C4×D61—42no—
285C4×D61—42no—
286C4×D61—44no—
287Dic122—22no—
288Dic122—22no—
289Dic122—42no—
290C8×C61—42no—
291#92418—42no—
292#21210—611nomaximal, Sylow
293C22×C61—22no—
294C22×C61—22no—
295C22×C61—22no—
296C22×C61—42no—
297C22×C61—42no—
298C22×C61—44no—
299C22×C61—42no—
300C22×C61—84no—
301C22×C61#39012yes—
302C22×C61#39012yes—
303C22×C61C22×C22⋊C411yescharacteristic
304C22×C61C2×#12111yescharacteristic
305C22×C61C2×#12111yescharacteristic
306C2×D61—44no—
307C2×D61—42no—
308C2×D61—84no—
309C4×C61—44no—
310C4×C61—42no—
311C4×C61#39012yes—
312C4×C61#39012yes—
313C4×C61C2×C4×D411yescharacteristic
314C4×C61C2×C4×D411yescharacteristic
315C4×C61C2×#12411yescharacteristic
316C4×C61C2×#12711yescharacteristic
317Dic61—24no—
318Dic61—28no—
319Dic61—42no—
320Dic61—42no—
321#3449—611no—
322#2905—611no—
323C2×#421—611no—
324C2×#421D6111yescharacteristic, complemented
325C22×#104—611no—
326C22×#104—611no—
327C4×#117—611no—
328#1963—614no—
329#1962—614no—
330C2×C61—21no—
331C2×C61—21no—
332C2×C61—22no—
333C2×C61—22no—
334C2×C61—82no—
335C2×C61C2×#39011yescharacteristic
336C2×C61#290112yes—
337D61—82no—
338C22×#41—611no—
339C22×#41C2×D6111yescharacteristic
340C22×M16—611no—
341C22×M16C2×D6111yescharacteristic
342#537—1222no—
343#536—612no—
344#536—612no—
345#535—612no—
346#535—612no—
347#534—1222no—
348#532—614no—
349#525—1222no—
350#525—1222no—
351C2×#117—611no—
352C2×#117C2×D6111yescharacteristic
353C22×C42—611no—
354#424—1222no—
355#424—1222no—
356#423—614no—
357#421—24no—
358#421—1224no—
359C2×#104—612no—
360C2×#104—612no—
361C2×#104—612no—
362C2×#104—612no—
363C2×#104—614no—
364C2×#104—1224no—
365C2×#104—1224no—
366C2×#104C2×D6114yescomplemented
367C4×C4⋊C4—1222no—
368C4×C22⋊C4—1222no—
369#376—612no—
370C43—612no—
371C61#2121011yescharacteristic, complemented, Sylow
372C2×#41—21no—
373C2×#41—21no—
374C2×#41—22no—
375C2×#41—22no—
376C2×#41—22no—
377C2×#41—22no—
378C2×#41—612no—
379C2×#41—612no—
380C2×#41—1221no—
381C2×#41—1221no—
382C2×#41—1222no—
383C2×#41—1222no—
384C2×#41—1222no—
385C2×#41—1222no—
386C2×#41C4×D6112yes—
387C2×#41C22×D6112yes—
388C22×Q8—611no—
389C22×Q8C4×D6111yescharacteristic
390C22×D4—21no—
391C22×D4—611no—
392C22×D4—1221no—
393C22×D4C4×D6111yescharacteristic
394C23×C4—21no—
395C23×C4—1221no—
396C23×C4—1221no—
397C23×C4C22×D6111yescharacteristic
398C2×M16—21no—
399C2×M16—22no—
400C2×M16—22no—
401C2×M16—612no—
402C2×M16—1221no—
403C2×M16—1222no—
404C2×M16—1222no—
405C2×M16C22×D6112yes—
406C22×C8—21no—
407C22×C8—1221no—
408C4⋊Q8—614no—
409#127—614no—
410#124—614no—
411#124—614no—
412#123—1224no—
413#123—1224no—
414C22⋊Q8—1224no—
415C22⋊Q8—1224no—
416#121—1224no—
417#121—1224no—
418#121—1228no—
419#120—1224no—
420#120—1224no—
421C4×D4—1228no—
422#117—22no—
423#117—612no—
424#117—612no—
425#117—612no—
426#117—612no—
427#117C22⋊D6114yescomplemented
428C2×C4⋊C4—21no—
429C2×C4⋊C4—612no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—612no—
432C2×C42—612no—
433C2×C42—614no—
434C2×C42—611no—
435C2×C42—611no—
436C2×C42—1222no—
437C2×C42—1221no—
438C2×C42—1224no—
439C2×C42C22×D6112yes—
440C4⋊C8—1228no—
441#104—28no—
442#104—1224no—
443#104—1224no—
444#104—1224no—
445#104—1224no—
446#104—1228no—
447#95—1224no—
448C24—21no—
449C24—1221no—
450C24D4×D6111yescharacteristic
451#41—22no—
452#41—22no—
453#41—22no—
454#41—22no—
455#41—24no—
456#41—24no—
457#41—42no—
458#41—42no—
459#41—42no—
460#41—42no—
461#41—1222no—
462#41—1222no—
463#41—1222no—
464#41—1222no—
465#41—1224no—
466#41—1224no—
467#41—2442no—
468#41—2442no—
469#41—2442no—
470#41—2442no—
471C2×Q8—21no—
472C2×Q8—22no—
473C2×Q8—22no—
474C2×Q8—612no—
475C2×Q8—1221no—
476C2×Q8—1222no—
477C2×Q8—1222no—
478C2×Q8C2×C4×D6112yes—
479C2×D4—22no—
480C2×D4—22no—
481C2×D4—21no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—42no—
485C2×D4—42no—
486C2×D4—612no—
487C2×D4—1228no—
488C2×D4—1222no—
489C2×D4—1222no—
490C2×D4—1221no—
491C2×D4—1222no—
492C2×D4—1222no—
493C2×D4—2442no—
494C2×D4—2442no—
495C2×D4C2×C4×D6112yes—
496C22×C4—22no—
497C22×C4—21no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—614no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4—1224no—
507C22×C4—1222no—
508C22×C4—1222no—
509C22×C4—2444no—
510C22×C4—2442no—
511C22×C4—2442no—
512C22×C4C2×C4×D6111yescharacteristic
513C22×C4C2×C4×D6111yescharacteristic
514C22×C4#1175822212yes—
515C22×C4D4×D6112yes—
516C22×C4D4×D6111yescharacteristic
517C22×C4D4×D6111yescharacteristic
518C22×C4D4×D6111yescharacteristic
519C22×C4#1175822412yes—
520C22×C4C2×C22⋊D6112yes—
521C22×C4C23×D6111yescharacteristic
522M16—22no—
523M16—22no—
524M16—42no—
525M16—1222no—
526M16—1222no—
527M16—2442no—
528C2×C8—22no—
529C2×C8—42no—
530C2×C8—1222no—
531C2×C8—2442no—
532C4⋊C4—24no—
533C4⋊C4—614no—
534C4⋊C4—614no—
535C4⋊C4—1228no—
536C22⋊C4—24no—
537C22⋊C4—614no—
538C22⋊C4—614no—
539C22⋊C4—1228no—
540C22⋊C4—1228no—
541C42—22no—
542C42—614no—
543C42—614no—
544C42—612no—
545C42—612no—
546C42—1221no—
547C42—1221no—
548C42—1224no—
549C42—2448no—
550C42C2×C22⋊D6114yes—
551C23—22no—
552C23—42no—
553C23—42no—
554C23—82no—
555C23—1222no—
556C23—4882no—
557C23—2442no—
558C23C22⋊C4×D6111yescharacteristic
559C23C22⋊C4×D6111yescharacteristic
560C23#1451974912yes—
561C23#1451975012yes—
562C23C2×D4×D6111yescharacteristic
563Q8—22no—
564Q8—22no—
565Q8—42no—
566Q8—1222no—
567Q8—1222no—
568Q8—2442no—
569D4—22no—
570D4—22no—
571D4—42no—
572D4—42no—
573D4—44no—
574D4—42no—
575D4—1222no—
576D4—1222no—
577D4—4888no—
578D4—2442no—
579D4—2442no—
580D4—2444no—
581D4—2442no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—618no—
590C2×C4—1224no—
591C2×C4—1222no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—2444no—
595C2×C4—2448no—
596C2×C4—2442no—
597C2×C4—2442no—
598C2×C4—2442no—
599C2×C4—2442no—
600C2×C4—2444no—
601C2×C4C22⋊C4×D6111yescharacteristic
602C2×C4C22⋊C4×D6111yescharacteristic
603C2×C4#1451973712yes—
604C2×C4#1451973712yes—
605C2×C4C4×C22⋊D6112yes—
606C2×C4C4×C22⋊D6112yes—
607C2×C4#1451974912yes—
608C2×C4#1451975012yes—
609C2×C4#1451976312yes—
610C2×C4#1451976312yes—
611C2×C4#1451977312yes—
612C2×C4#1451977312yes—
613C2×C4C22×C4×D6111yescharacteristic
614C2×C4C2×D4×D6111yescharacteristic
615C2×C4C2×D4×D6111yescharacteristic
616C2×C4C22×C22⋊D6111yescharacteristic
617C2×C4C2×#1175822411yescharacteristic
618C2×C4C2×#1175822211yescharacteristic
619C8—42no—
620C8—2442no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—42no—
625C22—42no—
626C22—44no—
627C22—42no—
628C22—84no—
629C22—4884no—
630C22—2444no—
631C22—2442no—
632C22#1683966212yes—
633C22#1683966212yes—
634C22C2×C22⋊C4×D6111yescharacteristic
635C22C2×#1451974911yescharacteristic
636C22C2×#1451975011yescharacteristic
637C4—44no—
638C4—42no—
639C4—1224no—
640C4—1228no—
641C4—2442no—
642C4—2442no—
643C4#1683966212yes—
644C4#1683966212yes—
645C4C2×#1451973711yescharacteristic
646C4C2×C4×C22⋊D6111yescharacteristic
647C4C2×#1451976311yescharacteristic
648C4C2×#1451977311yescharacteristic
649C2—21no—
650C2—21no—
651C2—22no—
652C2—22no—
653C2—82no—
654C2—4882no—
655C2C2×#1683966211yescharacteristic
656C2#2145436512yes—
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#421C2
socleC22×C61C22×C22⋊C4
prime radical#21459148C2
cosocleC22×C4×C61C24
hypercenterC2×#421D61
hypocenterC61#21210
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#421

Complements

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

NH ≅ G/Ntype
2. #21454382C2semidirect
3. C2×#16839780C2semidirect
4. C22×#14519704C2semidirect
5. C22×#14519703C2semidirect
6. #117×Dic61C2semidirect
7. #21446649C2semidirect
8. #21446645C2semidirect
9. C2×C61×#421C2semidirect
13. #16840336C4semidirect
14. #16840333C4semidirect
15. #16840329C4semidirect
16. #16840326C4semidirect
25. #16839785C22semidirect
27. C2×#14519705C22semidirect
29. C2×#14519704C22semidirect
30. C2×#14519704C22semidirect
32. C2×#14519703C22semidirect
33. C2×#14519703C22semidirect
36. #16839608C22semidirect
37. C42×Dic61C22semidirect
42. C2×#104×C61C22semidirect
61. #14519775C2×C4semidirect
62. #14519773C2×C4semidirect
63. #14519766C2×C4semidirect
64. #14519764C2×C4semidirect
84. C2×C4×Dic61D4semidirect
114. #117×C61D4semidirect
324. C2×#421D61semidirect
366. C2×#104C2×D61semidirect
371. C61#21210semidirect
427. #117C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214591482
#214543822
C2×#168397802
C22×#145197042
C22×#145197032
#117×Dic612
#214466492
#214466452
C2×C61×#4212
#2121061