Finite Group Databasebeta

← #29712167

Subgroups — #29712167

Order 15616.

1690 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840416C211yesmaximal, characteristic, complemented
2C2×#16839896C211yesmaximal, characteristic, complemented
3C22×#14519698C211yesmaximal, characteristic, complemented
4#21446955C212yesmaximal, complemented
5#21446815C212yesmaximal, complemented
6#21446245C211yesmaximal, characteristic, complemented
7#21446170C212yesmaximal, complemented
8#21446137C212yesmaximal, complemented
9#21446112C211yesmaximal, characteristic, complemented
10#21445903C211yesmaximal, characteristic, complemented
11C61×#2408C211yesmaximal, characteristic, complemented
12C22×#11758224—21no—
13C22×#11758199C2212yes—
14C23×#92418C2211yescharacteristic
15#16840416—24no—
16#16840416—24no—
17C2×#14519777C2211yescharacteristic, complemented
18C2×#14519761—21no—
19C2×#14519761—21no—
20C2×#14519757—21no—
21C2×#14519751C2211yescharacteristic
22C2×#14519751C2211yescharacteristic, complemented
23C2×C4×D244C2211yescharacteristic, complemented
24C22×C4⋊Dic61C2211yescharacteristic
25C22×C4⋊Dic61C2211yescharacteristic, complemented
26#16839896—24no—
27C2×#14519698—24no—
28C2×#14519698C2212yes—
29C2×#14519698C2212yes—
30C2×#14519698C2212yes—
31C2×#14519697C2212yescomplemented
32C2×C22⋊#92418C2211yescharacteristic
33C2×C22⋊#92418C2212yescomplemented
34#16839524C2212yescomplemented
35#16839524C2212yescomplemented
36#16839524C2212yescomplemented
37#16839517C2212yescomplemented
38C23×D4×C61C2211yescharacteristic
39C2×#121×C61C2212yes—
40C2×#120×C61—21no—
41C2×C4×D4×C61C2211yescharacteristic
42C61×#394—21no—
43C61×#392—21no—
44C61×#390C2212yes—
45C61×#380C2211yescharacteristic
46C61×#379—21no—
47C2×#11758224—28no—
48C2×#11758224—44no—
49C2×#11758224—44no—
50C22×Dic122—21no—
51C22×D244C2311yescharacteristic
52C22×C22⋊D61—21no—
53C22×C22⋊D61—21no—
54C23×Dic61—21no—
55C23×Dic61—21no—
56C22×C4×D61—21no—
57C2×#11758199—24no—
58C2×#11758199—24no—
59C2×#11758199—42no—
60C22×#92418—22no—
61C22×#92418—42no—
62C22×#92418C2312yes—
63C22×#92418C2312yes—
64#14519777—22no—
65#14519777—22no—
66#14519761—28no—
67#14519761—28no—
68#14519757—28no—
69#14519751—22no—
70#14519751—22no—
71#14519751—22no—
72#14519751D414yescomplemented
73C4×D244—22no—
74C4×D244—22no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—22no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—22no—
80C2×C4⋊Dic61—22no—
81C2×C4⋊Dic61D414yescomplemented
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88#14519698—24no—
89#14519698—24no—
90#14519698—24no—
91#14519697—42no—
92C22⋊#92418—22no—
93C22⋊#92418—42no—
94C25×C61—21no—
95C22×D4×C61—24no—
96C22×D4×C61—22no—
97C22×D4×C61—22no—
98C22×D4×C61—22no—
99C22×D4×C61—21no—
100C22×D4×C61—42no—
101C22×D4×C61D411yescharacteristic
102C22×D4×C61C2312yes—
103C23×C4×C61C2311yescharacteristic
104C23×C4×C61C2311yescharacteristic
105#121×C61—24no—
106#121×C61—42no—
107#120×C61—44no—
108C4×D4×C61—22no—
109C4×D4×C61—22no—
110C2×C4⋊C4×C61D411yescharacteristic
111C2×C4⋊C4×C61C2312yes—
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61—22no—
114C2×C22⋊C4×C61—22no—
115C2×C22⋊C4×C61—22no—
116C2×C22⋊C4×C61D411yescharacteristic
117C2×C22⋊C4×C61D411yescharacteristic
118C2×C42×C61C2311yescharacteristic
119#95×C61—21no—
120C24×C61—22no—
121C24×C61—44no—
122C24×C61—42no—
123C24×C61—42no—
124C24×C61—42no—
125C24×C61—82no—
126C24×C61C2×D411yescharacteristic
127C24×C61C2×D411yescharacteristic
128C23×D61—21no—
129C2×D4×C61—22no—
130C2×D4×C61—22no—
131C2×D4×C61—28no—
132C2×D4×C61—22no—
133C2×D4×C61—22no—
134C2×D4×C61—22no—
135C2×D4×C61—22no—
136C2×D4×C61—22no—
137C2×D4×C61—48no—
138C2×D4×C61—44no—
139C2×D4×C61—44no—
140C2×D4×C61—44no—
141C2×D4×C61—44no—
142C2×D4×C61—42no—
143C2×D4×C61—82no—
144C2×D4×C61D814yescomplemented
145C2×D4×C61#4114yescomplemented
146C22×C4×C61—22no—
147C22×C4×C61—22no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—22no—
151C22×C4×C61—22no—
152C22×C4×C61—22no—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D412yes—
157C22×C4×C61C2×D412yes—
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2411yescharacteristic
160C2×C22⋊D61—24no—
161C2×C22⋊D61—24no—
162C2×C22⋊D61—42no—
163C2×C22⋊D61—42no—
164C2×C22⋊D61—42no—
165C2×C22⋊D61—42no—
166C22×Dic61—24no—
167C22×Dic61—24no—
168C22×Dic61—21no—
169C22×Dic61—21no—
170C22×Dic61—21no—
171C22×Dic61—42no—
172C22×Dic61—42no—
173C22×Dic61—42no—
174C22×Dic61—42no—
175#11758224—432no—
176C2×D244—22no—
177C2×D244—22no—
178C2×D244—21no—
179C2×D244C2×D412yes—
180C2×C4×D61—24no—
181C2×C4×D61—42no—
182C2×C4×D61—42no—
183C2×Dic122—24no—
184C2×Dic122—42no—
185C2×Dic122—42no—
186C4⋊C4×C61—22no—
187C4⋊C4×C61—22no—
188C4⋊C4×C61—22no—
189C22⋊C4×C61—22no—
190C22⋊C4×C61—22no—
191C22⋊C4×C61—44no—
192C22⋊C4×C61—42no—
193C22⋊C4×C61—44no—
194C22⋊C4×C61—82no—
195C42×C61—21no—
196C42×C61—21no—
197#11758199—48no—
198#11758198—24no—
199#11758198—24no—
200#11758198—24no—
201C4⋊Dic61—24no—
202C4⋊Dic61—24no—
203C4⋊Dic61—24no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61C2×D412yes—
208C4⋊Dic61C2×D412yes—
209C4⋊Dic61C2×D412yes—
210C2×#92418—22no—
211C2×#92418—22no—
212C2×#92418—22no—
213C2×#92418—22no—
214C2×#92418—22no—
215C2×#92418—22no—
216C2×#92418—44no—
217C23×C61—22no—
218C23×C61—22no—
219C23×C61—21no—
220C23×C61—22no—
221C23×C61—21no—
222C23×C61—22no—
223C23×C61—24no—
224C23×C61—48no—
225C23×C61—42no—
226C23×C61—42no—
227C23×C61—42no—
228C23×C61—42no—
229C23×C61—42no—
230C23×C61—42no—
231C23×C61—88no—
232C23×C61—88no—
233C23×C61—84no—
234C23×C61—84no—
235C23×C61#12012yes—
236C23×C61C2×D812yes—
237C23×C61C2×D812yes—
238C23×C61AGL(1,ℤ/8)12yes—
239C23×C61C22×D411yescharacteristic
240C22×D61—22no—
241C22×D61—42no—
242C22×D61—41no—
243C22×D61—41no—
244C22×D61—42no—
245D4×C61—28no—
246D4×C61—24no—
247D4×C61—24no—
248D4×C61—48no—
249D4×C61—42no—
250D4×C61—42no—
251D4×C61—88no—
252C2×C4×C61—21no—
253C2×C4×C61—22no—
254C2×C4×C61—21no—
255C2×C4×C61—22no—
256C2×C4×C61—22no—
257C2×C4×C61—24no—
258C2×C4×C61—21no—
259C2×C4×C61—21no—
260C2×C4×C61—22no—
261C2×C4×C61—22no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—42no—
265C2×C4×C61—42no—
266C2×C4×C61—42no—
267C2×C4×C61—42no—
268C2×C4×C61#12012yes—
269C2×C4×C61C2×D812yes—
270C2×C4×C61C22×D411yescharacteristic
271C2×C4×C61C22×D411yescharacteristic
272C2×C4×C61C2×#4112yes—
273C2×C4×C61H(5,𝔽2)12yes—
274C22⋊D61—48no—
275C22⋊D61—48no—
276C2×Dic61—22no—
277C2×Dic61—22no—
278C2×Dic61—22no—
279C2×Dic61—48no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—48no—
287D244—24no—
288D244—41no—
289D244—41no—
290C4×D61—48no—
291Dic122—48no—
292#92418—42no—
293#92418—42no—
294#23960—611nomaximal, Sylow
295C22×C61—22no—
296C22×C61—22no—
297C22×C61—24no—
298C22×C61—24no—
299C22×C61—22no—
300C22×C61—24no—
301C22×C61—41no—
302C22×C61—41no—
303C22×C61—42no—
304C22×C61—44no—
305C22×C61—44no—
306C22×C61—44no—
307C22×C61—816no—
308C22×C61—82no—
309C22×C61—82no—
310C22×C61—82no—
311C22×C61—82no—
312C22×C61#44714yes—
313C22×C61C2×#12011yescharacteristic
314C22×C61#54612yes—
315C22×C61C22×D811yescharacteristic
316C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
317C22×C61#57712yes—
318C2×D61—42no—
319C2×D61—42no—
320C2×D61—42no—
321C2×D61—82no—
322C2×D61—82no—
323C4×C61—22no—
324C4×C61—22no—
325C4×C61—41no—
326C4×C61—41no—
327C4×C61—82no—
328C4×C61#52511yescharacteristic
329C4×C61#53511yescharacteristic
330C4×C61#54612yes—
331Dic61—42no—
332Dic61—42no—
333Dic61—42no—
334C2×#547—611no—
335C2×#447—611no—
336C22×#102—611no—
337#2408D6111yescharacteristic, complemented
338#2023—612no—
339#1999—612no—
340#1893—611no—
341#1879—612no—
342#1874—612no—
343#1869—611no—
344#1823—611no—
345C2×C61—24no—
346C2×C61—42no—
347C2×C61—84no—
348C2×C61#240411yescharacteristic
349C2×C61C2×#44711yescharacteristic
350C2×C61#301011yescharacteristic
351C2×C61#329412yes—
352C2×C61#330612yes—
353D61—82no—
354C22×#41—1221no—
355C23×D4C2×D6111yescharacteristic
356C22×D8—612no—
357C23×C8—611no—
358#547—1224no—
359#547—1224no—
360C2×C4⋊Q8—611no—
361C2×#123—1221no—
362C2×#123—1221no—
363C2×C22⋊Q8—1221no—
364C2×#121—611no—
365C2×#121—611no—
366C2×#121C2×D6112yes—
367C2×#120—21no—
368C2×C4×D4—611no—
369C2×C4×D4C2×D6111yescharacteristic
370C22×C4⋊C4—611no—
371C22×C4⋊C4—611no—
372#447—1224no—
373C2×#102—612no—
374C2×#102—612no—
375C2×#102—612no—
376C2×#102—612no—
377C2×#102—1224no—
378C2×C22⋊C8—611no—
379C2×C22⋊C8—612no—
380#394—21no—
381#392—21no—
382#390C2×D6112yes—
383#380C2×D6111yescharacteristic
384#379—21no—
385#340—612no—
386#340—612no—
387#340—612no—
388#336—612no—
389C61#2396011yescharacteristic, complemented, Sylow
390C25—21no—
391C2×#41—1228no—
392C2×#41—2444no—
393C2×#41—2444no—
394C22×Q8—1221no—
395C22×D4—24no—
396C22×D4—22no—
397C22×D4—22no—
398C22×D4—22no—
399C22×D4—21no—
400C22×D4—42no—
401C22×D4—611no—
402C22×D4—1221no—
403C22×D4—1221no—
404C22×D4C22⋊D6111yescharacteristic
405C22×D4C22×D6112yes—
406C23×C4—1221no—
407C23×C4—1221no—
408C23×C4—1221no—
409C23×C4C22×D6111yescharacteristic
410C23×C4C22×D6111yescharacteristic
411C2×D8—1224no—
412C2×D8—1224no—
413C2×D8—2442no—
414C22×C8—612no—
415C22×C8—612no—
416C22×C8—1222no—
417C22×C8—2442no—
418C4⋊Q8—1222no—
419C4⋊Q8—1222no—
420#123—1228no—
421#123—1228no—
422C22⋊Q8—1228no—
423#121—24no—
424#121—42no—
425#121—614no—
426#121—1222no—
427#121—1222no—
428#121—1222no—
429#120—44no—
430C4×D4—22no—
431C4×D4—22no—
432C4×D4—1222no—
433C4×D4—1222no—
434C2×C4⋊C4—614no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—611no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—1222no—
439C2×C4⋊C4—1222no—
440C2×C4⋊C4—1222no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4C22⋊D6111yescharacteristic
445C2×C4⋊C4C22×D6112yes—
446C2×C22⋊C4—21no—
447C2×C22⋊C4—22no—
448C2×C22⋊C4—22no—
449C2×C22⋊C4—22no—
450C2×C22⋊C4—1221no—
451C2×C22⋊C4—1221no—
452C2×C22⋊C4—1221no—
453C2×C22⋊C4C22⋊D6111yescharacteristic
454C2×C22⋊C4C22⋊D6111yescharacteristic
455C2×C42C22×D6111yescharacteristic
456#102—1224no—
457#102—1224no—
458#102—1224no—
459#102—2442no—
460C22⋊C8—1222no—
461C22⋊C8—2442no—
462#95—21no—
463C24—22no—
464C24—44no—
465C24—42no—
466C24—42no—
467C24—42no—
468C24—82no—
469C24—1221no—
470C24C2×C22⋊D6111yescharacteristic
471C24C2×C22⋊D6111yescharacteristic
472#41—24432no—
473C2×Q8—1224no—
474C2×Q8—2442no—
475C2×Q8—2442no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—28no—
479C2×D4—22no—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—48no—
485C2×D4—44no—
486C2×D4—44no—
487C2×D4—44no—
488C2×D4—44no—
489C2×D4—42no—
490C2×D4—82no—
491C2×D4—612no—
492C2×D4—1224no—
493C2×D4—1224no—
494C2×D4—1221no—
495C2×D4—1222no—
496C2×D4—1222no—
497C2×D4—2442no—
498C2×D4—2442no—
499C2×D4—2442no—
500C2×D4—2442no—
501C2×D4#1175819914yescomplemented
502C2×D4#1175822414yescomplemented
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—1224no—
511C22×C4—1224no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—1224no—
516C22×C4—2442no—
517C22×C4—2442no—
518C22×C4—2442no—
519C22×C4—2442no—
520C22×C4—2442no—
521C22×C4—2442no—
522C22×C4D4×D6112yes—
523C22×C4D4×D6112yes—
524C22×C4C2×C22⋊D6111yescharacteristic
525C22×C4C2×C22⋊D6111yescharacteristic
526C22×C4C2×C22⋊D6111yescharacteristic
527C22×C4C2×C22⋊D6111yescharacteristic
528C22×C4C23×D6111yescharacteristic
529D8—2448no—
530C2×C8—1222no—
531C2×C8—1222no—
532C2×C8—1222no—
533C2×C8—1222no—
534C2×C8—1222no—
535C2×C8—1222no—
536C2×C8—2444no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—612no—
541C4⋊C4—612no—
542C4⋊C4—612no—
543C4⋊C4—1224no—
544C4⋊C4—1224no—
545C4⋊C4—1221no—
546C4⋊C4—1221no—
547C4⋊C4—1221no—
548C4⋊C4—1224no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—44no—
552C22⋊C4—42no—
553C22⋊C4—44no—
554C22⋊C4—82no—
555C22⋊C4—1224no—
556C22⋊C4—1224no—
557C22⋊C4—1224no—
558C42—21no—
559C42—21no—
560C23—22no—
561C23—22no—
562C23—21no—
563C23—21no—
564C23—22no—
565C23—22no—
566C23—24no—
567C23—48no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—88no—
575C23—88no—
576C23—84no—
577C23—84no—
578C23—1222no—
579C23—2441no—
580C23—2441no—
581C23—2442no—
582C23—2442no—
583C23#1451974512yes—
584C23C2×#1175819912yes—
585C23C2×#1175819912yes—
586C23#1451980212yes—
587C23C22×C22⋊D6111yescharacteristic
588Q8—2448no—
589D4—28no—
590D4—24no—
591D4—24no—
592D4—48no—
593D4—42no—
594D4—42no—
595D4—88no—
596D4—1224no—
597D4—2448no—
598D4—2448no—
599D4—2441no—
600D4—2441no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—1222no—
618C2×C4—1222no—
619C2×C4—1222no—
620C2×C4—2448no—
621C2×C4—2448no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2448no—
629C2×C4#1451974512yes—
630C2×C4C2×#1175819912yes—
631C2×C4C2×D4×D6111yescharacteristic
632C2×C4C2×D4×D6111yescharacteristic
633C2×C4C2×#1175822412yes—
634C2×C4#1451982212yes—
635C8—2442no—
636C8—2442no—
637C22—24no—
638C22—22no—
639C22—24no—
640C22—22no—
641C22—22no—
642C22—24no—
643C22—42no—
644C22—41no—
645C22—41no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—816no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—4882no—
655C22—4882no—
656C22—2442no—
657C22—2442no—
658C22—2442no—
659C22#1683989514yes—
660C22C2×#1451974511yescharacteristic
661C22#1684040612yes—
662C22C22×#1175819911yescharacteristic
663C22C2×#1451980211yescharacteristic
664C22#1684057312yes—
665C4—22no—
666C4—22no—
667C4—41no—
668C4—41no—
669C4—82no—
670C4—2442no—
671C4—2442no—
672C4—2442no—
673C4#1684026811yescharacteristic
674C4#1684032711yescharacteristic
675C4#1684039912yes—
676C2—24no—
677C2—42no—
678C2—84no—
679C2—4882no—
680C2#2144913511yescharacteristic
681C2C2×#1683989511yescharacteristic
682C2#2145510311yescharacteristic
683C2#2145763212yes—
684C2#2145778012yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2408C2
socleC23×C61C22×D4
hypercenter#2408D61
hypocenterC61#23960
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2408

Complements

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

NH ≅ G/Ntype
1. C2×#16840416C2semidirect
2. C2×#16839896C2semidirect
3. C22×#14519698C2semidirect
4. #21446955C2semidirect
5. #21446815C2semidirect
6. #21446245C2semidirect
7. #21446170C2semidirect
8. #21446137C2semidirect
9. #21446112C2semidirect
10. #21445903C2semidirect
11. C61×#2408C2semidirect
17. C2×#14519777C22semidirect
22. C2×#14519751C22semidirect
23. C2×C4×D244C22semidirect
25. C22×C4⋊Dic61C22semidirect
31. C2×#14519697C22semidirect
33. C2×C22⋊#92418C22semidirect
34. #16839524C22semidirect
35. #16839524C22semidirect
36. #16839524C22semidirect
37. #16839517C22semidirect
72. #14519751D4semidirect
81. C2×C4⋊Dic61D4semidirect
144. C2×D4×C61D8semidirect
145. C2×D4×C61#41semidirect
337. #2408D61semidirect
389. C61#23960semidirect
501. C2×D4#11758199semidirect
502. C2×D4#11758224semidirect

Maximal subgroups

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

subgroupindex
C2×#168404162
C2×#168398962
C22×#145196982
#214469552
#214468152
#214462452
#214461702
#214461372
#214461122
#214459032
C61×#24082
#2396061