Finite Group Databasebeta

← #29681761

Subgroups — #29681761

Order 15616.

1818 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
1C22×#14519785C211yesmaximal, characteristic, complemented
2C2×#16840340C211yesmaximal, characteristic
3C2×#16839758C211yesmaximal, characteristic, complemented
4#21448412C211yesmaximal, characteristic, complemented
5#21446254C212yesmaximal, complemented
6#21446249C212yesmaximal, complemented
7#21446244C212yesmaximal, complemented
8#21446239C212yesmaximal, complemented
9#21445539C211yesmaximal, characteristic, complemented
10C2×#16839524C211yesmaximal, characteristic, complemented
11C2×C61×#417C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C2×#14519785—24no—
14C2×#14519785—24no—
15C2×#14519785—24no—
16C22×#11758189C2212yes—
17C23×#92418C2211yescharacteristic
18C22×C8×D61C2212yes—
19#16840340—24no—
20#16840340C418yescomplemented
21C2×#14519756—21no—
22C2×#14519753C2212yes—
23C2×#14519751—21no—
24C2×#14519749—22no—
25C2×#14519748C2212yes—
26C2×#14519736C2211yescharacteristic
27C22×C4⋊Dic61C2211yescharacteristic
28#16839758—24no—
29C2×#14519701C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519698C2211yescharacteristic
33#16839667—21no—
34#16839663—21no—
35#16839641—22no—
36#16839619C2212yes—
37#16839619C2212yes—
38#16839525C2212yes—
39#16839525C2212yes—
40#16839524C2212yes—
41#16839524C2212yes—
42C22×#41×C61C2211yescharacteristic
43C22×M16×C61C2211yescharacteristic
44C22×C4⋊C4×C61C2211yescharacteristic
45C61×#417—24no—
46C2×#103×C61C2212yes—
47C2×#102×C61C2212yes—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222—22no—
51C2×#11758222—22no—
52C2×#11758222C2×C414yes—
53C2×#11758222D414yes—
54C22×Dic122D411yescharacteristic
55C22×D244D411yescharacteristic
56C22×C22⋊D61—22no—
57C22×C22⋊D61D412yes—
58C23×Dic61—21no—
59C22×C4×D61C2312yes—
60#14519785—232no—
61C2×#11758189—24no—
62C2×#11758189—24no—
63C2×#11758189—24no—
64C22×#92418—22no—
65C22×#92418—22no—
66C22×#92418D414yescomplemented
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69C2×C8×D61—24no—
70C2×C8×D61—24no—
71C2×C8×D61—24no—
72#14519756—28no—
73#14519753—24no—
74#14519753C2×C418yescomplemented
75#14519751—28no—
76#14519749—216no—
77#14519748—24no—
78#14519748C2×C418yescomplemented
79#14519736—22no—
80#14519736—22no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61—22no—
83C2×C4⋊Dic61C2×C414yes—
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2312yes—
87C2×#11758196—22no—
88C2×C22⋊Dic61—22no—
89C2×C22⋊Dic61—21no—
90C2×#11758198—22no—
91C2×#11758198—22no—
92C2×#11758198—22no—
93C2×C4×Dic61—21no—
94#14519701—22no—
95#14519701—22no—
96#14519698—22no—
97#14519698—22no—
98#14519684—22no—
99#14519684—22no—
100C2×#41×C61—22no—
101C2×#41×C61—22no—
102C2×#41×C61—22no—
103C2×#41×C61—22no—
104C2×#41×C61—22no—
105C2×#41×C61C2×C414yes—
106C22×Q8×C61C2311yescharacteristic
107C22×D4×C61—21no—
108C22×D4×C61C2311yescharacteristic
109C23×C4×C61—21no—
110C23×C4×C61—21no—
111C23×C4×C61C2311yescharacteristic
112C2×M16×C61—22no—
113C2×M16×C61—22no—
114C2×M16×C61D414yescomplemented
115C22×C8×C61C2312yes—
116C2×C4⋊C4×C61—22no—
117C2×C4⋊C4×C61—22no—
118C2×C4⋊C4×C61—21no—
119C2×C4⋊C4×C61C2312yes—
120#103×C61—24no—
121#102×C61—24no—
122C24×C61—21no—
123C24×C61C2×D411yescharacteristic
124C23×D61C2×D412yes—
125#41×C61—28no—
126#41×C61—28no—
127#41×C61—44no—
128#41×C61—44no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61C22×C412yes—
135C2×D4×C61—24no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—42no—
142C2×D4×C61—42no—
143C2×D4×C61C22×C412yes—
144C22×C4×C61—21no—
145C22×C4×C61—24no—
146C22×C4×C61—22no—
147C22×C4×C61—42no—
148C22×C4×C61—42no—
149C22×C4×C61—42no—
150C22×C4×C61—81no—
151C22×C4×C61—81no—
152C22×C4×C61C22×C412yes—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D412yes—
155C22×C4×C61C2×D412yes—
156C22×C4×C61C2×D412yes—
157C22×C4×C61#4112yes—
158C22×C4×C61#4112yes—
159C22×C4×C61C2411yescharacteristic
160C2×C22⋊D61—24no—
161C2×C22⋊D61—24no—
162C2×C22⋊D61—28no—
163C2×C22⋊D61—82no—
164C2×C22⋊D61—82no—
165C2×C22⋊D61C22⋊C418yes—
166C22×Dic61—22no—
167C22×Dic61—24no—
168C22×Dic61—21no—
169C22×Dic61—21no—
170C22×Dic61—81no—
171C22×Dic61—81no—
172C22×Dic61C2×D412yes—
173#11758222—28no—
174#11758222—28no—
175#11758222—28no—
176#11758222—28no—
177C2×D244—22no—
178C2×D244—22no—
179C2×D244C22⋊C414yes—
180C2×C4×D61—22no—
181C2×C4×D61—22no—
182C2×C4×D61—22no—
183C2×C4×D61—22no—
184C2×C4×D61C22×C414yes—
185C2×C4×D61C2×D414yes—
186C2×Dic122—22no—
187C2×Dic122—22no—
188C2×Dic122C22⋊C414yes—
189M16×C61—28no—
190C2×C8×C61—22no—
191C2×C8×C61—22no—
192C2×C8×C61C2×D414yes—
193C4⋊C4×C61—22no—
194C4⋊C4×C61—22no—
195C4⋊C4×C61—42no—
196C22⋊Dic61—28no—
197C22⋊Dic61—42no—
198#11758198—28no—
199#11758198—82no—
200#11758198—82no—
201C4⋊Dic61—24no—
202C4⋊Dic61—22no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61C22×C414yes—
208#11758196—28no—
209C4×Dic61—42no—
210C2×#92418—28no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418C2×D412yes—
216C2×#92418C2×D412yes—
217#11758189—216no—
218C8×D61—216no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—21no—
222C23×C61—21no—
223C23×C61—22no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61—41no—
230C23×C61C2×C22⋊C412yes—
231C23×C61#12112yes—
232C23×C61#12112yes—
233C23×C61C22×D411yescharacteristic
234C22×D61—22no—
235C22×D61—22no—
236C22×D61—22no—
237C22×D61—22no—
238C22×D61—22no—
239C22×D61—22no—
240C22×D61C2×C22⋊C414yes—
241Q8×C61—22no—
242Q8×C61—22no—
243Q8×C61—41no—
244Q8×C61—41no—
245D4×C61—22no—
246D4×C61—22no—
247D4×C61—48no—
248D4×C61—41no—
249D4×C61—41no—
250C2×C4×C61—22no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—48no—
254C2×C4×C61—42no—
255C2×C4×C61—42no—
256C2×C4×C61—42no—
257C2×C4×C61—84no—
258C2×C4×C61C2×C22⋊C412yes—
259C2×C4×C61C4×D414yes—
260C2×C4×C61C4×D414yes—
261C2×C4×C61#12014yes—
262C2×C4×C61#12112yes—
263C2×C4×C61#12112yes—
264C2×C4×C61#12314yes—
265C2×C4×C61C23×C411yescharacteristic
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C22×D411yescharacteristic
268C2×C4×C61C22×D411yescharacteristic
269C2×C4×C61C2×#4111yescharacteristic
270C2×C4×C61C2×#4111yescharacteristic
271C22⋊D61—216no—
272C22⋊D61—88no—
273C2×Dic61—22no—
274C2×Dic61—22no—
275C2×Dic61—24no—
276C2×Dic61—42no—
277C2×Dic61—42no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—84no—
285C2×Dic61C2×C22⋊C414yes—
286D244—28no—
287C4×D61—24no—
288C4×D61—24no—
289C4×D61—24no—
290C4×D61—24no—
291Dic122—28no—
292C8×C61—24no—
293#92418—22no—
294#92418—22no—
295#20976—611nomaximal, Sylow
296C22×C61—22no—
297C22×C61—22no—
298C22×C61—22no—
299C22×C61—22no—
300C22×C61—22no—
301C22×C61—22no—
302C22×C61—42no—
303C22×C61—42no—
304C22×C61—42no—
305C22×C61—81no—
306C22×C61—81no—
307C22×C61—81no—
308C22×C61—81no—
309C22×C61#38614yes—
310C22×C61C22×C22⋊C411yescharacteristic
311C22×C61C2×#12111yescharacteristic
312C22×C61C2×#12111yescharacteristic
313C22×C61#57611yescharacteristic
314C22×C61#57711yescharacteristic
315C22×C61#57711yescharacteristic
316C22×C61#57811yescharacteristic
317C2×D61—24no—
318C2×D61—24no—
319C2×D61—24no—
320C2×D61—42no—
321C2×D61—42no—
322C2×D61—42no—
323C2×D61—42no—
324C4×C61—42no—
325C4×C61—82no—
326C4×C61#38614yes—
327C4×C61C2×C4×D411yescharacteristic
328C4×C61C2×C4×D411yescharacteristic
329C4×C61C2×#12011yescharacteristic
330C4×C61C2×#12311yescharacteristic
331Dic61—24no—
332Dic61—44no—
333Dic61—81no—
334Dic61—81no—
335C22×#131—611no—
336C2×#537—611no—
337C2×#417D6111yescharacteristic, complemented
338C2×#416—611no—
339#2311—611no—
340#1895—612no—
341#1894—612no—
342#1893—612no—
343#1892—612no—
344#1736—611no—
345C2×#340—611no—
346C2×C61—24no—
347C2×C61—82no—
348C2×C61C2×#38611yescharacteristic
349C2×C61#289711yescharacteristic
350C2×C61#289811yescharacteristic
351C2×C61#306011yescharacteristic
352C2×C61#306111yescharacteristic
353C2×C61#306211yescharacteristic
354C2×C61#306311yescharacteristic
355D61—44no—
356C22×#41—611no—
357C22×#41C2×D6111yescharacteristic
358C2×#131—1224no—
359C2×#131—1224no—
360C2×#131—1224no—
361C22×M16—612no—
362C22×M16C2×D6111yescharacteristic
363C23×C8—611no—
364C23×C8—612no—
365#537—618no—
366#537—1224no—
367C2×C22⋊Q8—612no—
368C2×C22⋊Q8—1221no—
369C2×#121—612no—
370C2×#121—1221no—
371C2×#121—1222no—
372C2×#117—611no—
373C22×C4⋊C4—611no—
374C22×C4⋊C4C2×D6111yescharacteristic
375#417—24no—
376#416—1224no—
377C2×#103—611no—
378C2×#103—611no—
379C2×#103C2×D6112yes—
380C2×#102—611no—
381C2×#102—611no—
382C2×#102C2×D6112yes—
383#391—1221no—
384#390—1221no—
385#386—1222no—
386#380—612no—
387#380—612no—
388#340—612no—
389#340—612no—
390#340—612no—
391#340—612no—
392C61#2097611yescharacteristic, complemented, Sylow
393C2×#41—22no—
394C2×#41—22no—
395C2×#41—22no—
396C2×#41—22no—
397C2×#41—22no—
398C2×#41—614no—
399C2×#41—614no—
400C2×#41—1222no—
401C2×#41—1222no—
402C2×#41—1222no—
403C2×#41—1222no—
404C2×#41C4×D6114yes—
405C22×Q8—611no—
406C22×Q8C22×D6111yescharacteristic
407C22×D4—21no—
408C22×D4—612no—
409C22×D4—611no—
410C22×D4—1222no—
411C22×D4C22×D6111yescharacteristic
412C23×C4—21no—
413C23×C4—21no—
414C23×C4—612no—
415C23×C4—1221no—
416C23×C4C22×D6111yescharacteristic
417#131—12232no—
418C2×M16—22no—
419C2×M16—22no—
420C2×M16—1224no—
421C2×M16—1224no—
422C2×M16—1224no—
423C2×M16C22⋊D6114yescomplemented
424C22×C8—614no—
425C22×C8—611no—
426C22×C8—611no—
427C22×C8—1224no—
428C22×C8—1224no—
429C22×C8—1222no—
430C22×C8—1222no—
431C22×C8—1224no—
432C22×C8C22×D6112yes—
433C22⋊Q8—618no—
434C22⋊Q8—1228no—
435C22⋊Q8—1224no—
436#121—618no—
437#121—1228no—
438#121—1224no—
439#121—12216no—
440#117—1222no—
441#117—1222no—
442C2×C4⋊C4—22no—
443C2×C4⋊C4—22no—
444C2×C4⋊C4—21no—
445C2×C4⋊C4—614no—
446C2×C4⋊C4—612no—
447C2×C4⋊C4—611no—
448C2×C4⋊C4—611no—
449C2×C4⋊C4—1222no—
450C2×C4⋊C4—1221no—
451C2×C4⋊C4—1222no—
452C2×C4⋊C4C22×D6112yes—
453C2×C22⋊C4—1222no—
454C2×C22⋊C4—1222no—
455C2×C22⋊C4—1222no—
456C2×C22⋊C4—1222no—
457C2×C22⋊C4—1221no—
458C2×C42—1221no—
459#103—24no—
460#103—1222no—
461#103—1222no—
462#102—24no—
463#102—1222no—
464#102—1222no—
465#95—1222no—
466#95—1222no—
467C24—21no—
468C24—612no—
469C24D4×D6111yescharacteristic
470#41—28no—
471#41—28no—
472#41—44no—
473#41—44no—
474#41—1228no—
475#41—1228no—
476#41—1228no—
477#41—1228no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—614no—
484C2×Q8—1222no—
485C2×Q8—1222no—
486C2×Q8C2×C4×D6112yes—
487C2×D4—24no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—42no—
494C2×D4—42no—
495C2×D4—618no—
496C2×D4—614no—
497C2×D4—1224no—
498C2×D4—1224no—
499C2×D4—1228no—
500C2×D4—1222no—
501C2×D4—1222no—
502C2×D4—4882no—
503C2×D4—4882no—
504C2×D4C2×C4×D6112yes—
505C22×C4—21no—
506C22×C4—24no—
507C22×C4—22no—
508C22×C4—42no—
509C22×C4—42no—
510C22×C4—42no—
511C22×C4—81no—
512C22×C4—81no—
513C22×C4—612no—
514C22×C4—614no—
515C22×C4—614no—
516C22×C4—1222no—
517C22×C4—1222no—
518C22×C4—1222no—
519C22×C4—1222no—
520C22×C4—1224no—
521C22×C4—1222no—
522C22×C4—1221no—
523C22×C4—1221no—
524C22×C4—4881no—
525C22×C4—4881no—
526C22×C4C2×C4×D6112yes—
527C22×C4#1175822212yes—
528C22×C4D4×D6112yes—
529C22×C4D4×D6112yes—
530C22×C4D4×D6111yescharacteristic
531C22×C4#1175822412yes—
532C22×C4C2×C22⋊D6112yes—
533C22×C4C23×D6111yescharacteristic
534M16—28no—
535M16—12216no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—612no—
539C2×C8—612no—
540C2×C8—12216no—
541C2×C8—1221no—
542C2×C8—1221no—
543C2×C8—1221no—
544C2×C8—1221no—
545C2×C8—1228no—
546C2×C8C2×C22⋊D6114yes—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—42no—
550C4⋊C4—614no—
551C4⋊C4—1228no—
552C4⋊C4—1222no—
553C4⋊C4—1224no—
554C4⋊C4—1221no—
555C4⋊C4—1221no—
556C4⋊C4—1221no—
557C4⋊C4—1221no—
558C22⋊C4—1228no—
559C22⋊C4—1228no—
560C22⋊C4—4882no—
561C22⋊C4—4882no—
562C22⋊C4—2442no—
563C42—2442no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—22no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—614no—
576C23—1222no—
577C23—1222no—
578C23—1222no—
579C23—1222no—
580C23—1222no—
581C23—1222no—
582C23C22⋊C4×D6112yes—
583C23#1451974812yes—
584C23#1451975012yes—
585C23C2×D4×D6111yescharacteristic
586Q8—22no—
587Q8—22no—
588Q8—41no—
589Q8—41no—
590Q8—1228no—
591D4—22no—
592D4—22no—
593D4—48no—
594D4—41no—
595D4—41no—
596D4—12216no—
597D4—1228no—
598D4—4888no—
599C2×C4—22no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—48no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—84no—
607C2×C4—614no—
608C2×C4—1222no—
609C2×C4—1222no—
610C2×C4—1224no—
611C2×C4—1224no—
612C2×C4—1224no—
613C2×C4—1224no—
614C2×C4—1224no—
615C2×C4—4884no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2442no—
623C2×C4—2442no—
624C2×C4C22⋊C4×D6112yes—
625C2×C4#1451973714yes—
626C2×C4C4×C22⋊D6114yes—
627C2×C4#1451974514yes—
628C2×C4#1451974812yes—
629C2×C4#1451975012yes—
630C2×C4#1451975914yes—
631C2×C4C22×C4×D6111yescharacteristic
632C2×C4C2×D4×D6111yescharacteristic
633C2×C4C2×D4×D6111yescharacteristic
634C2×C4C22×C22⋊D6111yescharacteristic
635C2×C4C2×#1175822411yescharacteristic
636C2×C4C2×#1175822211yescharacteristic
637C8—24no—
638C8—1222no—
639C8—1222no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—1224no—
654C22—1224no—
655C22—1224no—
656C22—2442no—
657C22—2442no—
658C22—2442no—
659C22—2442no—
660C22#1683963814yes—
661C22C2×C22⋊C4×D6111yescharacteristic
662C22C2×#1451974811yescharacteristic
663C22C2×#1451975011yescharacteristic
664C22#1684056611yescharacteristic
665C22#1684057111yescharacteristic
666C22#1684057211yescharacteristic
667C22#1684057811yescharacteristic
668C4—42no—
669C4—82no—
670C4—1224no—
671C4—4881no—
672C4—4881no—
673C4—2444no—
674C4#1683963814yes—
675C4C2×#1451973711yescharacteristic
676C4C2×C4×C22⋊D6111yescharacteristic
677C4C2×#1451974511yescharacteristic
678C4C2×#1451975911yescharacteristic
679C2—24no—
680C2—82no—
681C2—2444no—
682C2C2×#1683963811yescharacteristic
683C2#2145434311yescharacteristic
684C2#2145435011yescharacteristic
685C2#2145548011yescharacteristic
686C2#2145548811yescharacteristic
687C2#2145549311yescharacteristic
688C2#2145550411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#417C2
socleC23×C61C22×D4
prime radicalC2×#16840340C2
cosocleC22×C4×C61C24
hypercenterC2×#417D61
hypocenterC61#20976
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
1. C22×#14519785C2semidirect
3. C2×#16839758C2semidirect
4. #21448412C2semidirect
5. #21446254C2semidirect
6. #21446249C2semidirect
7. #21446244C2semidirect
8. #21446239C2semidirect
9. #21445539C2semidirect
10. C2×#16839524C2semidirect
11. C2×C61×#417C2semidirect
20. #16840340C4semidirect
66. C22×#92418D4semidirect
74. #14519753C2×C4semidirect
78. #14519748C2×C4semidirect
114. C2×M16×C61D4semidirect
337. C2×#417D61semidirect
392. C61#20976semidirect
423. C2×M16C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145197852
C2×#168403402
C2×#168397582
#214484122
#214462542
#214462492
#214462442
#214462392
#214455392
C2×#168395242
C2×C61×#4172
#2097661