Finite Group Databasebeta

← #29707990

Subgroups — #29707990

Order 15616.

1458 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×#14519797C211yesmaximal, characteristic
2C2×#16839789C211yesmaximal, characteristic, complemented
3C2×#16839763C211yesmaximal, characteristic, complemented
4C2×#16839761C211yesmaximal, characteristic, complemented
5#21448407C211yesmaximal, characteristic, complemented
6#21447141C212yesmaximal, complemented
7#21447136C212yesmaximal, complemented
8#21447134C212yesmaximal, complemented
9#21447129C212yesmaximal, complemented
10C2×#16839524C211yesmaximal, characteristic, complemented
11C61×#2311C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C2×#14519797—24no—
14C2×#14519797—24no—
15C2×#14519797—24no—
16C22×#11758202C2211yescharacteristic
17C22×#11758200C2211yescharacteristic
18C22×#11758201C2211yescharacteristic
19C22×#11758199C2211yescharacteristic
20C22×#11758194C2211yescharacteristic
21C23×#92418C2211yescharacteristic
22C22×C4⋊Dic61C2211yescharacteristic
23#16839789—24no—
24C2×C4⋊#92418C2212yes—
25C2×C4⋊#92418C2212yes—
26#16839763—24no—
27#16839761—24no—
28C2×#14519701C2212yes—
29C2×#14519700C2212yes—
30C2×#14519698C2212yes—
31C2×#14519697C2212yes—
32#16839668C2212yes—
33#16839664C2212yes—
34#16839639—22no—
35#16839619—21no—
36#16839619—21no—
37#16839524C2212yes—
38#16839524C2212yes—
39C22×#41×C61C2211yescharacteristic
40C2×#117×C61C2211yescharacteristic
41C22×C4⋊C4×C61C2211yescharacteristic
42C61×#391C2212yes—
43C61×#390C2212yes—
44C61×#386—22no—
45C61×#380—21no—
46C61×#380—21no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222—22no—
51C2×#11758222—22no—
52C2×#11758222D414yes—
53C22×Dic122C2311yescharacteristic
54C22×D244C2311yescharacteristic
55C22×C22⋊D61—21no—
56C23×Dic61—21no—
57C22×C4×D61—21no—
58#14519797—232no—
59C2×#11758202—22no—
60C2×#11758202—22no—
61C2×#11758202—22no—
62C2×#11758200—22no—
63C2×#11758200—22no—
64C2×#11758200—22no—
65C2×#11758201—22no—
66C2×#11758201—22no—
67C2×#11758201—22no—
68C2×#11758199—22no—
69C2×#11758199—22no—
70C2×#11758199—22no—
71C2×#11758194—22no—
72C2×#11758194—22no—
73C2×#11758194—22no—
74C22×#92418—22no—
75C22×#92418—22no—
76C22×#92418D414yes—
77C22×#92418C2312yes—
78C22×#92418C2311yescharacteristic
79C22×#92418C2311yescharacteristic
80C2×C4⋊Dic61—22no—
81C2×C4⋊Dic61—22no—
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61C2312yes—
84C2×#11758198—22no—
85C2×#11758198—22no—
86C4⋊#92418—24no—
87C4⋊#92418—24no—
88#14519701—24no—
89#14519700—24no—
90#14519698—24no—
91#14519697—24no—
92#14519684—22no—
93#14519684—22no—
94C2×#41×C61—22no—
95C2×#41×C61—22no—
96C2×#41×C61—22no—
97C2×#41×C61—22no—
98C2×#41×C61—22no—
99C2×#41×C61D414yes—
100C22×Q8×C61C2311yescharacteristic
101C22×D4×C61—21no—
102C22×D4×C61C2311yescharacteristic
103C23×C4×C61—21no—
104C23×C4×C61—21no—
105C23×C4×C61C2311yescharacteristic
106#117×C61—22no—
107#117×C61—22no—
108C2×C4⋊C4×C61—22no—
109C2×C4⋊C4×C61—22no—
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61D411yescharacteristic
112C2×C4⋊C4×C61D411yescharacteristic
113C2×C4⋊C4×C61C2312yes—
114C2×C22⋊C4×C61—22no—
115C2×C22⋊C4×C61—22no—
116C2×C22⋊C4×C61D412yes—
117C2×C42×C61C2312yes—
118#95×C61—22no—
119#95×C61—22no—
120C24×C61—21no—
121C24×C61C2×D411yescharacteristic
122C23×D61—21no—
123#41×C61—28no—
124#41×C61—28no—
125#41×C61—44no—
126#41×C61—44no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61C2×D412yes—
133C2×D4×C61—24no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—42no—
140C2×D4×C61—42no—
141C2×D4×C61C2×D412yes—
142C22×C4×C61—21no—
143C22×C4×C61—24no—
144C22×C4×C61—22no—
145C22×C4×C61—42no—
146C22×C4×C61—42no—
147C22×C4×C61—42no—
148C22×C4×C61—81no—
149C22×C4×C61—81no—
150C22×C4×C61C2×D412yes—
151C22×C4×C61C2×D412yes—
152C22×C4×C61C2×D412yes—
153C22×C4×C61C2×D412yes—
154C22×C4×C61C2×D412yes—
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61#4112yes—
157C22×C4×C61#4112yes—
158C22×C4×C61#4112yes—
159C22×C4×C61C2411yescharacteristic
160C2×C22⋊D61—24no—
161C2×C22⋊D61—42no—
162C2×C22⋊D61—42no—
163C22×Dic61—21no—
164C22×Dic61—22no—
165C22×Dic61—42no—
166C22×Dic61—81no—
167C22×Dic61—81no—
168#11758222—28no—
169#11758222—28no—
170#11758222—44no—
171#11758222—44no—
172C2×D244—21no—
173C2×D244—21no—
174C2×D244—21no—
175C2×D244—21no—
176C2×D244—21no—
177C2×D244C2×D412yes—
178C2×C4×D61—24no—
179C2×C4×D61—42no—
180C2×C4×D61—42no—
181C2×Dic122—21no—
182C2×Dic122—21no—
183C2×Dic122—21no—
184C2×Dic122—21no—
185C2×Dic122—21no—
186C2×Dic122C2×D412yes—
187C4⋊C4×C61—22no—
188C4⋊C4×C61—22no—
189C4⋊C4×C61—22no—
190C4⋊C4×C61—22no—
191C4⋊C4×C61—42no—
192C22⋊C4×C61—24no—
193C22⋊C4×C61—82no—
194C22⋊C4×C61—82no—
195C42×C61—22no—
196C42×C61—22no—
197#11758202—28no—
198#11758201—28no—
199#11758200—28no—
200#11758199—28no—
201#11758198—82no—
202#11758198—82no—
203C4⋊Dic61—22no—
204C4⋊Dic61—22no—
205C4⋊Dic61—42no—
206#11758194—44no—
207C2×#92418—28no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—22no—
213C2×#92418—22no—
214C2×#92418—22no—
215C2×#92418C2×D412yes—
216C2×#92418C2×D412yes—
217C23×C61—21no—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—22no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61#12112yes—
229C23×C61#12112yes—
230C23×C61#12112yes—
231C23×C61C22×D411yescharacteristic
232C22×D61—22no—
233C22×D61—41no—
234C22×D61—41no—
235C22×D61—41no—
236C22×D61—41no—
237C22×D61—41no—
238C22×D61—41no—
239Q8×C61—22no—
240Q8×C61—22no—
241Q8×C61—41no—
242Q8×C61—41no—
243D4×C61—22no—
244D4×C61—22no—
245D4×C61—48no—
246D4×C61—41no—
247D4×C61—41no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—22no—
254C2×C4×C61—22no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—48no—
258C2×C4×C61—42no—
259C2×C4×C61—42no—
260C2×C4×C61—42no—
261C2×C4×C61—84no—
262C2×C4×C61#12014yes—
263C2×C4×C61#12112yes—
264C2×C4×C61#12112yes—
265C2×C4×C61#12112yes—
266C2×C4×C61#12314yes—
267C2×C4×C61#12314yes—
268C2×C4×C61#12414yes—
269C2×C4×C61C22×D411yescharacteristic
270C2×C4×C61C22×D411yescharacteristic
271C2×C4×C61C22×D411yescharacteristic
272C2×C4×C61C2×#4111yescharacteristic
273C2×C4×C61C2×#4111yescharacteristic
274C2×C4×C61C2×#4111yescharacteristic
275C22⋊D61—48no—
276C2×Dic61—22no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—42no—
280C2×Dic61—42no—
281C2×Dic61—42no—
282C2×Dic61—84no—
283D244—22no—
284D244—22no—
285D244—41no—
286D244—41no—
287C4×D61—48no—
288Dic122—22no—
289Dic122—22no—
290Dic122—41no—
291Dic122—41no—
292#92418—22no—
293#92418—22no—
294#92418—42no—
295#23543—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#39414yes—
310C22×C61C2×#12111yescharacteristic
311C22×C61C2×#12111yescharacteristic
312C22×C61C2×#12111yescharacteristic
313C22×C61#57611yescharacteristic
314C22×C61#57711yescharacteristic
315C22×C61#57711yescharacteristic
316C22×C61#57811yescharacteristic
317C2×D61—42no—
318C2×D61—42no—
319C2×D61—42no—
320C2×D61—81no—
321C2×D61—81no—
322C2×D61—81no—
323C2×D61—81no—
324C4×C61—42no—
325C4×C61—42no—
326C4×C61—42no—
327C4×C61—82no—
328C4×C61#39414yes—
329C4×C61C2×#12011yescharacteristic
330C4×C61C2×#12311yescharacteristic
331C4×C61C2×#12311yescharacteristic
332C4×C61C2×#12411yescharacteristic
333Dic61—42no—
334Dic61—82no—
335C22×#135—611no—
336C2×#424—611no—
337C2×#417—611no—
338C2×#417—611no—
339#2311—611no—
340#2311D6111yescharacteristic, complemented
341#2056—612no—
342#2055—612no—
343#2055—612no—
344#2054—612no—
345C2×#340—611no—
346C2×C61—24no—
347C2×C61—82no—
348C2×C61C2×#39411yescharacteristic
349C2×C61#304511yescharacteristic
350C2×C61#304511yescharacteristic
351C2×C61#304611yescharacteristic
352C2×C61#304611yescharacteristic
353C2×C61#306011yescharacteristic
354C2×C61#306111yescharacteristic
355D61—82no—
356C22×#41—611no—
357C22×#41C2×D6111yescharacteristic
358C2×#135—1224no—
359C2×#135—1224no—
360C2×#135—1224no—
361C22×Q16—611no—
362C22×SD16—611no—
363C22×SD16—611no—
364C22×D8—611no—
365C22×M16—611no—
366C23×C8—611no—
367C2×#117C2×D6111yescharacteristic
368C22×C4⋊C4—611no—
369C22×C4⋊C4C2×D6111yescharacteristic
370#424—1224no—
371C2×C4⋊C8—612no—
372C2×C4⋊C8—612no—
373#417—1224no—
374#417—1224no—
375C2×#103—612no—
376C2×#103—612no—
377C2×#102—612no—
378C2×#102—612no—
379#391—612no—
380#391C2×D6112yes—
381#390—612no—
382#390C2×D6112yes—
383#386—22no—
384#386—1222no—
385#380—21no—
386#380—21no—
387#380—1221no—
388#380—1221no—
389#340—612no—
390#340—612no—
391C61#2354311yescharacteristic, complemented, Sylow
392C2×#41—22no—
393C2×#41—22no—
394C2×#41—22no—
395C2×#41—22no—
396C2×#41—22no—
397C2×#41—614no—
398C2×#41—1222no—
399C2×#41—1222no—
400C2×#41—1222no—
401C2×#41—1222no—
402C2×#41—1222no—
403C2×#41D24414yes—
404C22×Q8—611no—
405C22×Q8C22×D6111yescharacteristic
406C22×D4—21no—
407C22×D4—611no—
408C22×D4—1221no—
409C22×D4C22×D6111yescharacteristic
410C23×C4—21no—
411C23×C4—21no—
412C23×C4—1221no—
413C23×C4—1221no—
414C23×C4C22×D6111yescharacteristic
415#135—12232no—
416C2×Q16—1222no—
417C2×Q16—1222no—
418C2×Q16—1222no—
419C2×SD16—1222no—
420C2×SD16—1222no—
421C2×SD16—1222no—
422C2×SD16—1222no—
423C2×SD16—1222no—
424C2×SD16—1222no—
425C2×D8—1222no—
426C2×D8—1222no—
427C2×D8—1222no—
428C2×M16—1222no—
429C2×M16—1222no—
430C2×M16—1222no—
431C22×C8—614no—
432C22×C8—611no—
433C22×C8—611no—
434C22×C8—612no—
435C22×C8—1222no—
436C22×C8—1222no—
437#117—22no—
438#117—22no—
439C2×C4⋊C4—22no—
440C2×C4⋊C4—22no—
441C2×C4⋊C4—21no—
442C2×C4⋊C4—612no—
443C2×C4⋊C4—1222no—
444C2×C4⋊C4—1222no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4C22⋊D6111yescharacteristic
447C2×C4⋊C4C22⋊D6111yescharacteristic
448C2×C4⋊C4C22×D6112yes—
449C2×C22⋊C4—22no—
450C2×C22⋊C4—22no—
451C2×C22⋊C4—1222no—
452C2×C22⋊C4—1222no—
453C2×C22⋊C4C22⋊D6112yes—
454C2×C42C22×D6112yes—
455C4⋊C8—1224no—
456C4⋊C8—1224no—
457#103—1224no—
458#103—1224no—
459#102—1224no—
460#102—1224no—
461#95—22no—
462#95—22no—
463#95—1222no—
464#95—1222no—
465C24—21no—
466C24—1221no—
467C24C2×C22⋊D6111yescharacteristic
468#41—28no—
469#41—28no—
470#41—44no—
471#41—44no—
472#41—1228no—
473#41—1228no—
474#41—2444no—
475#41—2444no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—612no—
482C2×Q8—1221no—
483C2×Q8—1221no—
484C2×Q8—1221no—
485C2×Q8—1221no—
486C2×Q8—1221no—
487C2×Q8C2×D24412yes—
488C2×D4—24no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—42no—
495C2×D4—42no—
496C2×D4—612no—
497C2×D4—1224no—
498C2×D4—1221no—
499C2×D4—1221no—
500C2×D4—1221no—
501C2×D4—1221no—
502C2×D4—1221no—
503C2×D4—2442no—
504C2×D4—2442no—
505C2×D4C2×D24412yes—
506C22×C4—21no—
507C22×C4—24no—
508C22×C4—22no—
509C22×C4—42no—
510C22×C4—42no—
511C22×C4—42no—
512C22×C4—81no—
513C22×C4—81no—
514C22×C4—1221no—
515C22×C4—1224no—
516C22×C4—1222no—
517C22×C4—4881no—
518C22×C4—4881no—
519C22×C4—2442no—
520C22×C4—2442no—
521C22×C4—2442no—
522C22×C4C2×D24412yes—
523C22×C4#1175822212yes—
524C22×C4D4×D6112yes—
525C22×C4D4×D6112yes—
526C22×C4#1175822412yes—
527C22×C4#1175822412yes—
528C22×C4C2×C22⋊D6112yes—
529C22×C4C2×C22⋊D6112yes—
530C22×C4C2×C22⋊D6111yescharacteristic
531C22×C4C23×D6111yescharacteristic
532Q16—1228no—
533SD16—1228no—
534SD16—1228no—
535D8—1228no—
536M16—2444no—
537C2×C8—612no—
538C2×C8—612no—
539C2×C8—1221no—
540C2×C8—1221no—
541C2×C8—1221no—
542C2×C8—1221no—
543C2×C8—1228no—
544C2×C8—1222no—
545C2×C8—1222no—
546C2×C8—1222no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—42no—
552C4⋊C4—1222no—
553C4⋊C4—1222no—
554C4⋊C4—2442no—
555C22⋊C4—24no—
556C22⋊C4—82no—
557C22⋊C4—82no—
558C22⋊C4—4882no—
559C22⋊C4—4882no—
560C42—22no—
561C42—22no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—22no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—1222no—
574C23—2441no—
575C23—2441no—
576C23—2441no—
577C23—2441no—
578C23—2441no—
579C23—2441no—
580C23#1451974812yes—
581C23#1451974912yes—
582C23#1451975112yes—
583C23C22×C22⋊D6111yescharacteristic
584Q8—22no—
585Q8—22no—
586Q8—41no—
587Q8—41no—
588Q8—1222no—
589Q8—1222no—
590Q8—2441no—
591Q8—2441no—
592D4—22no—
593D4—22no—
594D4—48no—
595D4—41no—
596D4—41no—
597D4—1222no—
598D4—1222no—
599D4—2448no—
600D4—2441no—
601D4—2441no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—48no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—84no—
616C2×C4—1222no—
617C2×C4—4884no—
618C2×C4—2442no—
619C2×C4—2442no—
620C2×C4—2442no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2448no—
624C2×C4#1451974714yes—
625C2×C4#1451974812yes—
626C2×C4#1451974912yes—
627C2×C4#1451975112yes—
628C2×C4#1451975914yes—
629C2×C4#1451976114yes—
630C2×C4#1451976314yes—
631C2×C4C2×D4×D6111yescharacteristic
632C2×C4C2×D4×D6111yescharacteristic
633C2×C4C22×D24411yescharacteristic
634C2×C4C2×#1175822411yescharacteristic
635C2×C4C2×#1175822411yescharacteristic
636C2×C4C2×#1175822211yescharacteristic
637C8—1222no—
638C8—1222no—
639C8—2442no—
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—4881no—
654C22—4881no—
655C22—4881no—
656C22—4881no—
657C22—2442no—
658C22—2442no—
659C22—2442no—
660C22#1683967814yes—
661C22C2×#1451974811yescharacteristic
662C22C2×#1451974911yescharacteristic
663C22C2×#1451975111yescharacteristic
664C22#1684056511yescharacteristic
665C22#1684057011yescharacteristic
666C22#1684057311yescharacteristic
667C22#1684057711yescharacteristic
668C4—42no—
669C4—42no—
670C4—42no—
671C4—82no—
672C4—4882no—
673C4—2442no—
674C4#1683967814yes—
675C4C2×#1451974711yescharacteristic
676C4C2×#1451975911yescharacteristic
677C4C2×#1451976111yescharacteristic
678C4C2×#1451976311yescharacteristic
679C2—24no—
680C2—82no—
681C2—4882no—
682C2C2×#1683967811yescharacteristic
683C2#2145535411yescharacteristic
684C2#2145535811yescharacteristic
685C2#2145536511yescharacteristic
686C2#2145536911yescharacteristic
687C2#2145547811yescharacteristic
688C2#2145548611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2311C2
socleC23×C61C22×D4
prime radicalC22×#14519797C2
hypercenter#2311D61
hypocenterC61#23543
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. C2×#16839789C2semidirect
3. C2×#16839763C2semidirect
4. C2×#16839761C2semidirect
5. #21448407C2semidirect
6. #21447141C2semidirect
7. #21447136C2semidirect
8. #21447134C2semidirect
9. #21447129C2semidirect
10. C2×#16839524C2semidirect
11. C61×#2311C2semidirect
340. #2311D61semidirect
391. C61#23543semidirect

Maximal subgroups

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

subgroupindex
C22×#145197972
C2×#168397892
C2×#168397632
C2×#168397612
#214484072
#214471412
#214471362
#214471342
#214471292
C2×#168395242
C61×#23112
#2354361