Finite Group Databasebeta

← #29828846

Subgroups — #29828846

Order 15616.

1468 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11758222C211yesmaximal, characteristic
2C2×#16839969C211yesmaximal, characteristic, complemented
3C2×#16839964C211yesmaximal, characteristic, complemented
4C2×#16839922C212yesmaximal
5#21446163C212yesmaximal, complemented
6#21446156C212yesmaximal, complemented
7#21446132C212yesmaximal, complemented
8#21445821C211yesmaximal, characteristic, complemented
9#21445816C211yesmaximal, characteristic, complemented
10#21445799C211yesmaximal, characteristic, complemented
11C61×#2857C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C22×#11758202C2212yes—
14C22×#11758201C2212yes—
15C4×#11758222—24no—
16C4×#11758222—24no—
17C2×C4×Dic122C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19C2×C4×D244C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21C2×C4×C22⋊D61—21no—
22C2×C4×C22⋊D61—21no—
23C2×#14519736C2211yescharacteristic
24C2×#14519734—21no—
25C2×C42×D61—21no—
26#16839969—24no—
27#16839964—24no—
28#16839922—28no—
29C2×C4⋊#92418C2211yescharacteristic
30C2×C4⋊#92418C2211yescharacteristic
31C2×#14519701C2212yes—
32C2×#14519701C2212yes—
33C2×#14519700C2212yes—
34C2×#14519697C2212yes—
35C2×C22⋊#92418C2212yes—
36#16839524C2212yes—
37#16839524C2212yes—
38#16839517C2212yes—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×C22⋊Q8×C61C2212yes—
41C22×C42×C61C2211yescharacteristic
42C61×#391C2212yes—
43C61×#389—21no—
44C61×#389—21no—
45C61×#384C2211yescharacteristic
46C61×#380—21no—
47C61×#380—21no—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222—22no—
51C2×#11758222—22no—
52C2×#11758222—22no—
53C2×#11758222D414yes—
54C22×Dic122C2311yescharacteristic
55C22×D244C2311yescharacteristic
56C22×C22⋊D61—21no—
57C22×C4×D61—21no—
58C22×C4×D61—21no—
59C22×C4×D61—21no—
60C2×#11758202—24no—
61C2×#11758202—24no—
62C2×#11758202—24no—
63C2×#11758201—24no—
64C2×#11758201—24no—
65C2×#11758201—24no—
66C22×#92418C2312yes—
67C22×#92418C2312yes—
68C4×Dic122—22no—
69C4×Dic122—22no—
70C4×Dic122—22no—
71C4×Dic122D414yes—
72C4×D244—22no—
73C4×D244—22no—
74C4×D244—22no—
75C4×D244D414yes—
76C4×C22⋊D61—28no—
77C4×C22⋊D61—28no—
78#14519736—22no—
79#14519736—22no—
80#14519734—28no—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×C22⋊Dic61—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91C42×D61—28no—
92C4⋊#92418—22no—
93C4⋊#92418—22no—
94#14519701—24no—
95#14519701—24no—
96#14519700—24no—
97#14519697—24no—
98C22⋊#92418—24no—
99C22×Q8×C61C2312yes—
100C23×C4×C61D411yescharacteristic
101C23×C4×C61D411yescharacteristic
102C23×C4×C61C2311yescharacteristic
103C4⋊Q8×C61—22no—
104C4⋊Q8×C61—22no—
105C22⋊Q8×C61—24no—
106C22⋊Q8×C61—24no—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61—22no—
112C2×C4⋊C4×C61C2312yes—
113C2×C22⋊C4×C61—22no—
114C2×C42×C61—22no—
115C2×C42×C61—22no—
116C2×C42×C61D411yescharacteristic
117C2×C42×C61D411yescharacteristic
118C2×C42×C61C2311yescharacteristic
119C2×C42×C61C2311yescharacteristic
120#95×C61—22no—
121#95×C61—22no—
122C24×C61C2×D411yescharacteristic
123C23×D61—21no—
124C2×Q8×C61—22no—
125C2×Q8×C61—22no—
126C2×Q8×C61—22no—
127C2×Q8×C61—22no—
128C2×Q8×C61—22no—
129C2×Q8×C61—22no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—41no—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61#4112yes—
149C22×C4×C61C2411yescharacteristic
150C2×C22⋊D61—24no—
151C2×C22⋊D61—24no—
152C2×C22⋊D61—24no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156#11758222—28no—
157#11758222—28no—
158#11758222—28no—
159#11758222—28no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244C2×D412yes—
164C2×D244C2×D412yes—
165C2×D244#4112yes—
166C2×C4×D61—24no—
167C2×C4×D61—24no—
168C2×C4×D61—24no—
169C2×C4×D61—24no—
170C2×C4×D61—24no—
171C2×C4×D61—42no—
172C2×C4×D61—42no—
173C2×C4×D61—42no—
174C2×C4×D61—42no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122C2×D412yes—
179C2×Dic122C2×D412yes—
180C2×Dic122#4112yes—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—42no—
184C4⋊C4×C61—42no—
185C4⋊C4×C61—42no—
186C4⋊C4×C61—44no—
187C4⋊C4×C61—42no—
188C22⋊C4×C61—44no—
189C42×C61—22no—
190C42×C61—22no—
191C42×C61—21no—
192C42×C61—21no—
193C42×C61C2×D412yes—
194C42×C61C2×D412yes—
195C22⋊Dic61—24no—
196#11758202—48no—
197#11758201—48no—
198#11758198—24no—
199#11758198—24no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61C2×D412yes—
203C4⋊Dic61C2×D412yes—
204#11758196—24no—
205#11758196—24no—
206C4×Dic61—24no—
207C4×Dic61—24no—
208C4×Dic61—24no—
209C2×#92418—22no—
210C2×#92418—22no—
211C2×#92418—22no—
212C2×#92418—22no—
213C2×#92418—22no—
214C2×#92418—22no—
215C23×C61—21no—
216C23×C61—21no—
217C23×C61—21no—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61#12112yes—
222C23×C61C22×D411yescharacteristic
223C22×D61—22no—
224C22×D61—22no—
225C22×D61—22no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—41no—
229C22×D61—41no—
230Q8×C61—42no—
231Q8×C61—42no—
232Q8×C61—42no—
233Q8×C61—42no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—44no—
248C2×C4×C61—44no—
249C2×C4×C61—42no—
250C2×C4×C61—42no—
251C2×C4×C61—42no—
252C2×C4×C61—42no—
253C2×C4×C61—42no—
254C2×C4×C61—42no—
255C2×C4×C61#12012yes—
256C2×C4×C61#12012yes—
257C2×C4×C61#12112yes—
258C2×C4×C61#13512yes—
259C2×C4×C61#13712yes—
260C2×C4×C61#13712yes—
261C2×C4×C61C22×D411yescharacteristic
262C2×C4×C61C22×D411yescharacteristic
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C22⋊D61—216no—
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277D244—22no—
278D244—22no—
279D244—22no—
280D244#12114yes—
281C4×D61—216no—
282C4×D61—48no—
283C4×D61—48no—
284Dic122—22no—
285Dic122—22no—
286Dic122—22no—
287Dic122#12114yes—
288#92418—42no—
289#92418—42no—
290#32907—611nomaximal, Sylow
291C22×C61—22no—
292C22×C61—22no—
293C22×C61—22no—
294C22×C61—41no—
295C22×C61—41no—
296C22×C61—41no—
297C22×C61—41no—
298C22×C61C2×#12011yescharacteristic
299C22×C61C2×#12111yescharacteristic
300C22×C61#52511yescharacteristic
301C22×C61C2×#13511yescharacteristic
302C22×C61C2×#13711yescharacteristic
303C22×C61C2×#13711yescharacteristic
304C22×C61#57511yescharacteristic
305C2×D61—24no—
306C2×D61—42no—
307C2×D61—42no—
308C2×D61—42no—
309C2×D61—42no—
310C2×D61—42no—
311C2×D61—42no—
312C4×C61—22no—
313C4×C61—22no—
314C4×C61—22no—
315C4×C61—41no—
316C4×C61—82no—
317C4×C61—82no—
318C4×C61#45214yescomplemented
319C4×C61C2×#12111yescharacteristic
320C4×C61C2×#12111yescharacteristic
321C4×C61#52511yescharacteristic
322C4×C61#52511yescharacteristic
323Dic61—24no—
324Dic61—42no—
325Dic61—42no—
326C2×C4×#41—611no—
327C2×#462—611no—
328C2×#461—611no—
329C2×#452—612no—
330#2857D6111yescharacteristic, complemented
331#1878—612no—
332#1877—612no—
333#1873—612no—
334#1803—611no—
335#1802—611no—
336#1797—611no—
337C2×C61—42no—
338C2×C61#284711yescharacteristic
339C2×C61C2×#45211yescharacteristic
340C2×C61#300711yescharacteristic
341C2×C61#303511yescharacteristic
342C2×C61#303911yescharacteristic
343C2×C61#310711yescharacteristic
344C2×C61#311611yescharacteristic
345D61—44no—
346C22×#41—611no—
347C22×Q16—612no—
348C22×SD16—612no—
349C2×C4⋊Q8C2×D6111yescharacteristic
350C2×C22⋊Q8C2×D6112yes—
351C4×#41—1224no—
352C4×#41—1224no—
353C2×C4×Q8—611no—
354C2×C4×Q8—611no—
355C2×C4×D4—611no—
356C2×C4×D4—611no—
357C2×C4×D4—1221no—
358C2×C4×D4—1221no—
359C2×#117—611no—
360C2×#117—1221no—
361C22×C42—1221no—
362C22×C42C2×D6111yescharacteristic
363#462—1224no—
364#461—1224no—
365#452—1228no—
366C2×C4⋊C8—611no—
367C2×C4⋊C8—611no—
368C2×#103—612no—
369C2×#103—612no—
370C2×#103—612no—
371C2×#102—612no—
372C2×C22⋊C8—612no—
373#391C2×D6112yes—
374#389—21no—
375#389—21no—
376#384C2×D6111yescharacteristic
377#380—21no—
378#380—21no—
379#340—612no—
380#340—612no—
381#336—612no—
382C61#3290711yescharacteristic, complemented, Sylow
383C2×#41—614no—
384C2×#41—1222no—
385C2×#41—1222no—
386C2×#41—1222no—
387C2×#41—1222no—
388C2×#41—1222no—
389C22×Q8—611no—
390C22×Q8C22×D6112yes—
391C22×D4—611no—
392C22×D4—1221no—
393C23×C4—1221no—
394C23×C4—1221no—
395C23×C4—1221no—
396C23×C4C22⋊D6111yescharacteristic
397C23×C4C22⋊D6111yescharacteristic
398C23×C4C22×D6111yescharacteristic
399C2×Q16—1224no—
400C2×Q16—1224no—
401C2×Q16—1224no—
402C2×SD16—1224no—
403C2×SD16—1224no—
404C2×SD16—1224no—
405C22×C8—612no—
406C22×C8—612no—
407C4⋊Q8—22no—
408C4⋊Q8—22no—
409C22⋊Q8—24no—
410C22⋊Q8—24no—
411C4×Q8—614no—
412C4×Q8—1222no—
413C4×Q8—1222no—
414C4×Q8—1222no—
415C4×D4—614no—
416C4×D4—1222no—
417C4×D4—1222no—
418C4×D4—1228no—
419C4×D4—1228no—
420C4×D4—1222no—
421#117—1228no—
422#117—1222no—
423#117—1222no—
424C2×C4⋊C4—22no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—1221no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4C22×D6112yes—
434C2×C22⋊C4—22no—
435C2×C22⋊C4—1221no—
436C2×C22⋊C4—1221no—
437C2×C22⋊C4—1221no—
438C2×C42—22no—
439C2×C42—22no—
440C2×C42—1228no—
441C2×C42—1221no—
442C2×C42—1221no—
443C2×C42—1221no—
444C2×C42C22⋊D6111yescharacteristic
445C2×C42C22⋊D6111yescharacteristic
446C2×C42C22×D6111yescharacteristic
447C2×C42C22×D6111yescharacteristic
448C4⋊C8—1222no—
449C4⋊C8—1222no—
450#103—1224no—
451#103—1224no—
452#103—1224no—
453#102—1224no—
454C22⋊C8—1224no—
455#95—22no—
456#95—22no—
457C24—1221no—
458C24C2×C22⋊D6111yescharacteristic
459#41—1228no—
460#41—1228no—
461#41—1228no—
462#41—1228no—
463C2×Q8—22no—
464C2×Q8—22no—
465C2×Q8—22no—
466C2×Q8—22no—
467C2×Q8—22no—
468C2×Q8—22no—
469C2×Q8—612no—
470C2×Q8—612no—
471C2×Q8—612no—
472C2×Q8—1221no—
473C2×Q8—1221no—
474C2×Q8—1221no—
475C2×D4—612no—
476C2×D4—612no—
477C2×D4—612no—
478C2×D4—1224no—
479C2×D4—1224no—
480C2×D4—1224no—
481C2×D4—1221no—
482C2×D4—1221no—
483C2×D4—1221no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—41no—
496C22×C4—1224no—
497C22×C4—1224no—
498C22×C4—1224no—
499C22×C4—1221no—
500C22×C4—1224no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1224no—
504C22×C4—2442no—
505C22×C4—2442no—
506C22×C4—2442no—
507C22×C4—2442no—
508C22×C4D4×D6112yes—
509C22×C4#1175822412yes—
510C22×C4C2×C22⋊D6111yescharacteristic
511C22×C4C2×C22⋊D6111yescharacteristic
512C22×C4C2×C22⋊D6111yescharacteristic
513C22×C4C2×C22⋊D6111yescharacteristic
514C22×C4C2×C22⋊D6111yescharacteristic
515C22×C4C23×D6111yescharacteristic
516Q16—2448no—
517SD16—2448no—
518C2×C8—1222no—
519C2×C8—1222no—
520C2×C8—1222no—
521C2×C8—1222no—
522C2×C8—1222no—
523C2×C8—1222no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—44no—
530C4⋊C4—42no—
531C4⋊C4—612no—
532C4⋊C4—612no—
533C4⋊C4—1224no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1224no—
537C22⋊C4—44no—
538C22⋊C4—1224no—
539C22⋊C4—1224no—
540C22⋊C4—1224no—
541C42—21no—
542C42—21no—
543C42—22no—
544C42—22no—
545C42—1224no—
546C42—1224no—
547C42—1224no—
548C42D4×D6112yes—
549C42D4×D6112yes—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—1222no—
557C23—1222no—
558C23—1222no—
559C23—2441no—
560C23—2441no—
561C23—2441no—
562C23—2441no—
563C23#1451974812yes—
564C23C22×C22⋊D6111yescharacteristic
565Q8—42no—
566Q8—42no—
567Q8—42no—
568Q8—42no—
569Q8—614no—
570Q8—1222no—
571Q8—1222no—
572Q8—1222no—
573D4—614no—
574D4—12216no—
575D4—1222no—
576D4—1222no—
577D4—1222no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—12216no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—1222no—
603C2×C4—1222no—
604C2×C4—1222no—
605C2×C4—2448no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2448no—
611C2×C4#1451974512yes—
612C2×C4#1451974512yes—
613C2×C4#1451974812yes—
614C2×C4#1451979712yes—
615C2×C4#1451980812yes—
616C2×C4#1451980812yes—
617C2×C4C2×D4×D6111yescharacteristic
618C2×C4C2×D4×D6111yescharacteristic
619C2×C4C2×D4×D6111yescharacteristic
620C2×C4#41×D6111yescharacteristic
621C2×C4#41×D6111yescharacteristic
622C2×C4C2×#1175822411yescharacteristic
623C8—2442no—
624C8—2442no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—1224no—
633C22—2442no—
634C22—2442no—
635C22—2442no—
636C22—2442no—
637C22—2442no—
638C22—2442no—
639C22C2×#1451974511yescharacteristic
640C22C2×#1451974811yescharacteristic
641C22#1684026911yescharacteristic
642C22C2×#1451979711yescharacteristic
643C22C2×#1451980811yescharacteristic
644C22C2×#1451980811yescharacteristic
645C22#1684056011yescharacteristic
646C4—22no—
647C4—22no—
648C4—22no—
649C4—41no—
650C4—82no—
651C4—82no—
652C4—1224no—
653C4—2442no—
654C4—2442no—
655C4#1683992214yescomplemented
656C4#121×D6111yescharacteristic
657C4#121×D6111yescharacteristic
658C4#1684027211yescharacteristic
659C4#1684027211yescharacteristic
660C2—42no—
661C2—2444no—
662C2#2145409811yescharacteristic
663C2C2×#1683992211yescharacteristic
664C2#2145507711yescharacteristic
665C2#2145526811yescharacteristic
666C2#2145530211yescharacteristic
667C2#2145588411yescharacteristic
668C2#2145597511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2857C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
hypercenter#2857D61
hypocenterC61#32907
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2857

Complements

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

NH ≅ G/Ntype
2. C2×#16839969C2semidirect
3. C2×#16839964C2semidirect
5. #21446163C2semidirect
6. #21446156C2semidirect
7. #21446132C2semidirect
8. #21445821C2semidirect
9. #21445816C2semidirect
10. #21445799C2semidirect
11. C61×#2857C2semidirect
318. C4×C61#452semidirect
330. #2857D61semidirect
382. C61#32907semidirect
655. C4#16839922semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117582222
C2×#168399692
C2×#168399642
C2×#168399222
#214461632
#214461562
#214461322
#214458212
#214458162
#214457992
C61×#28572
#3290761