Finite Group Databasebeta

← #29707988

Subgroups — #29707988

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×#16839790C211yesmaximal, characteristic, complemented
3C2×#16839764C211yesmaximal, characteristic, complemented
4#21448412C211yesmaximal, characteristic, complemented
5#21448411C211yesmaximal, characteristic, complemented
6#21447140C212yesmaximal, complemented
7#21447135C212yesmaximal, complemented
8#21447133C212yesmaximal, complemented
9#21447128C212yesmaximal, complemented
10C2×#16839524C211yesmaximal, characteristic, complemented
11C2×C61×#417C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C2×#14519797—24no—
14C2×#14519797—24no—
15C2×#14519797—24no—
16C22×#11758202C2211yescharacteristic
17C22×#11758200C2211yescharacteristic
18C22×#11758201C2211yescharacteristic
19C22×#11758199C2211yescharacteristic
20C23×#92418C2211yescharacteristic
21C2×#14519736C2211yescharacteristic
22C22×C4⋊Dic61C2211yescharacteristic
23#16839790—24no—
24C2×#14519706C2212yes—
25C2×#14519706C2212yes—
26#16839764—24no—
27C2×#14519702C2212yes—
28C2×#14519699C2212yes—
29#16839669C2212yes—
30#16839667C2212yes—
31#16839665C2212yes—
32#16839663C2212yes—
33#16839642—22no—
34#16839641—22no—
35#16839622—21no—
36#16839622—21no—
37#16839619—21no—
38#16839619—21no—
39#16839524C2212yes—
40#16839524C2212yes—
41C22×#41×C61C2211yescharacteristic
42C22×M16×C61C2211yescharacteristic
43C22×C4⋊C4×C61C2211yescharacteristic
44C61×#417—24no—
45C2×#103×C61C2212yes—
46C2×#102×C61C2212yes—
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—
71C22×#92418—22no—
72C22×#92418—22no—
73C22×#92418D414yes—
74C22×#92418C2311yescharacteristic
75C22×#92418C2311yescharacteristic
76#14519736—22no—
77#14519736—22no—
78C2×C4⋊Dic61—22no—
79C2×C4⋊Dic61—22no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61D411yescharacteristic
82C2×C4⋊Dic61D411yescharacteristic
83C2×C4⋊Dic61C2312yes—
84C2×C22⋊Dic61—22no—
85C2×C22⋊Dic61—22no—
86C2×C22⋊Dic61D412yes—
87C2×#11758198—22no—
88C2×#11758198—22no—
89C2×C4×Dic61C2312yes—
90#14519706—24no—
91#14519706—24no—
92#14519702—24no—
93#14519699—24no—
94#14519684—22no—
95#14519684—22no—
96#14519684—22no—
97#14519684—22no—
98C2×#41×C61—22no—
99C2×#41×C61—22no—
100C2×#41×C61—22no—
101C2×#41×C61—22no—
102C2×#41×C61—22no—
103C2×#41×C61D414yes—
104C22×Q8×C61C2311yescharacteristic
105C22×D4×C61—21no—
106C22×D4×C61C2311yescharacteristic
107C23×C4×C61—21no—
108C23×C4×C61—21no—
109C23×C4×C61C2311yescharacteristic
110C2×M16×C61—22no—
111C2×M16×C61—22no—
112C2×M16×C61—22no—
113C22×C8×C61C2312yes—
114C2×C4⋊C4×C61—22no—
115C2×C4⋊C4×C61—22no—
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61C2312yes—
118#103×C61—24no—
119#102×C61—24no—
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×D411yescharacteristic
154C22×C4×C61#4112yes—
155C22×C4×C61#4112yes—
156C22×C4×C61#4112yes—
157C22×C4×C61C2411yescharacteristic
158C2×C22⋊D61—24no—
159C2×C22⋊D61—42no—
160C2×C22⋊D61—42no—
161C22×Dic61—21no—
162C22×Dic61—22no—
163C22×Dic61—42no—
164C22×Dic61—81no—
165C22×Dic61—81no—
166C22×Dic61C2×D412yes—
167C22×Dic61C2×D412yes—
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—
187M16×C61—44no—
188C2×C8×C61—22no—
189C2×C8×C61—22no—
190C2×C8×C61—22no—
191C4⋊C4×C61—22no—
192C4⋊C4×C61—22no—
193C4⋊C4×C61—42no—
194C22⋊Dic61—24no—
195C22⋊Dic61—82no—
196C22⋊Dic61—82no—
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—22no—
206C4⋊Dic61—22no—
207C4⋊Dic61—42no—
208C4×Dic61—22no—
209C4×Dic61—22no—
210C2×#92418—28no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
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—41no—
250C2×C4×C61—41no—
251C2×C4×C61—48no—
252C2×C4×C61—42no—
253C2×C4×C61—42no—
254C2×C4×C61—42no—
255C2×C4×C61—84no—
256C2×C4×C61#12014yes—
257C2×C4×C61#12112yes—
258C2×C4×C61#12112yes—
259C2×C4×C61#12112yes—
260C2×C4×C61#12314yes—
261C2×C4×C61#12314yes—
262C2×C4×C61#12414yes—
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C2×C4×C61C2×#4111yescharacteristic
269C22⋊D61—48no—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—22no—
274C2×Dic61—22no—
275C2×Dic61—22no—
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—
292C8×C61—42no—
293#92418—22no—
294#92418—22no—
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—82no—
326C4×C61#39414yes—
327C4×C61C2×#12011yescharacteristic
328C4×C61C2×#12311yescharacteristic
329C4×C61C2×#12311yescharacteristic
330C4×C61C2×#12411yescharacteristic
331Dic61—42no—
332Dic61—42no—
333Dic61—42no—
334Dic61—82no—
335C22×#135—611no—
336C2×#424—611no—
337C2×#417—611no—
338C2×#417D6111yescharacteristic, complemented
339#2311—611no—
340#2311—611no—
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×M16C2×D6111yescharacteristic
366C23×C8—611no—
367C2×#117—611no—
368C22×C4⋊C4—611no—
369C22×C4⋊C4C2×D6111yescharacteristic
370#424—1224no—
371C2×C4⋊C8—612no—
372C2×C4⋊C8—612no—
373#417—24no—
374#417—1224no—
375C2×#103—612no—
376C2×#103C2×D6112yes—
377C2×#102—612no—
378C2×#102C2×D6112yes—
379#391—612no—
380#391—612no—
381#390—612no—
382#390—612no—
383#386—1222no—
384#386—1222no—
385#380—1221no—
386#380—1221no—
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×#41C22⋊D6114yes—
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—22no—
429C2×M16—22no—
430C2×M16—22no—
431C22×C8—614no—
432C22×C8—611no—
433C22×C8—611no—
434C22×C8—1222no—
435C22×C8—1222no—
436C22×C8C22×D6112yes—
437#117—1222no—
438#117—1222no—
439C2×C4⋊C4—22no—
440C2×C4⋊C4—22no—
441C2×C4⋊C4—21no—
442C2×C4⋊C4—611no—
443C2×C4⋊C4—611no—
444C2×C4⋊C4—612no—
445C2×C4⋊C4—1222no—
446C2×C4⋊C4—1222no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4C22×D6112yes—
449C2×C22⋊C4—612no—
450C2×C22⋊C4—1222no—
451C2×C22⋊C4—1222no—
452C2×C22⋊C4—1222no—
453C2×C22⋊C4—1222no—
454C2×C42—612no—
455C4⋊C8—1224no—
456C4⋊C8—1224no—
457#103—24no—
458#103—1224no—
459#102—24no—
460#102—1224no—
461#95—1222no—
462#95—1222no—
463#95—1222no—
464#95—1222no—
465C24—21no—
466C24—1221no—
467C24D4×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×C22⋊D6112yes—
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×C22⋊D6112yes—
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—612no—
515C22×C4—612no—
516C22×C4—1221no—
517C22×C4—1224no—
518C22×C4—1222no—
519C22×C4—4881no—
520C22×C4—4881no—
521C22×C4—2442no—
522C22×C4—2442no—
523C22×C4—2442no—
524C22×C4#1175822212yes—
525C22×C4#1175822212yes—
526C22×C4D4×D6112yes—
527C22×C4D4×D6112yes—
528C22×C4D4×D6111yescharacteristic
529C22×C4#1175822412yes—
530C22×C4C2×C22⋊D6112yes—
531C22×C4C23×D6111yescharacteristic
532Q16—1228no—
533SD16—1228no—
534SD16—1228no—
535D8—1228no—
536M16—44no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—612no—
541C2×C8—612no—
542C2×C8—1221no—
543C2×C8—1221no—
544C2×C8—1221no—
545C2×C8—1221no—
546C2×C8—1228no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—42no—
550C4⋊C4—1222no—
551C4⋊C4—1222no—
552C4⋊C4—1222no—
553C4⋊C4—1222no—
554C4⋊C4—2442no—
555C22⋊C4—1224no—
556C22⋊C4—4882no—
557C22⋊C4—4882no—
558C22⋊C4—4882no—
559C22⋊C4—4882no—
560C42—1222no—
561C42—1222no—
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#1451974912yes—
581C23#1451975012yes—
582C23#1451975012yes—
583C23C2×D4×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—41no—
604C2×C4—41no—
605C2×C4—48no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—84no—
610C2×C4—1222no—
611C2×C4—1222no—
612C2×C4—1222no—
613C2×C4—1222no—
614C2×C4—1222no—
615C2×C4—1222no—
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#1451974514yes—
625C2×C4#1451974912yes—
626C2×C4#1451975012yes—
627C2×C4#1451975012yes—
628C2×C4#1451975914yes—
629C2×C4#1451976014yes—
630C2×C4#1451976314yes—
631C2×C4C2×D4×D6111yescharacteristic
632C2×C4C2×D4×D6111yescharacteristic
633C2×C4C22×C22⋊D6111yescharacteristic
634C2×C4C2×#1175822411yescharacteristic
635C2×C4C2×#1175822211yescharacteristic
636C2×C4C2×#1175822211yescharacteristic
637C8—42no—
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—4881no—
654C22—4881no—
655C22—4881no—
656C22—4881no—
657C22—2442no—
658C22—2442no—
659C22—2442no—
660C22#1683967714yes—
661C22C2×#1451974911yescharacteristic
662C22C2×#1451975011yescharacteristic
663C22C2×#1451975011yescharacteristic
664C22#1684056611yescharacteristic
665C22#1684057111yescharacteristic
666C22#1684057211yescharacteristic
667C22#1684057811yescharacteristic
668C4—42no—
669C4—82no—
670C4—4882no—
671C4—2442no—
672C4—2442no—
673C4—2442no—
674C4#1683967714yes—
675C4C2×#1451974511yescharacteristic
676C4C2×#1451975911yescharacteristic
677C4C2×#1451976011yescharacteristic
678C4C2×#1451976311yescharacteristic
679C2—24no—
680C2—82no—
681C2—4882no—
682C2C2×#1683967711yescharacteristic
683C2#2145535711yescharacteristic
684C2#2145536211yescharacteristic
685C2#2145536811yescharacteristic
686C2#2145537311yescharacteristic
687C2#2145548011yescharacteristic
688C2#2145548811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#417C2
socleC23×C61C22×D4
prime radicalC22×#14519797C2
hypercenterC2×#417D61
hypocenterC61#23543
LCSG ▷ C22×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
2. C2×#16839790C2semidirect
3. C2×#16839764C2semidirect
4. #21448412C2semidirect
5. #21448411C2semidirect
6. #21447140C2semidirect
7. #21447135C2semidirect
8. #21447133C2semidirect
9. #21447128C2semidirect
10. C2×#16839524C2semidirect
11. C2×C61×#417C2semidirect
338. C2×#417D61semidirect
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×#168397902
C2×#168397642
#214484122
#214484112
#214471402
#214471352
#214471332
#214471282
C2×#168395242
C2×C61×#4172
#2354361