Finite Group Databasebeta

← #29709849

Subgroups — #29709849

Order 15616.

1210 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839801C211yesmaximal, characteristic, complemented
2C2×#16839799C211yesmaximal, characteristic, complemented
3C2×#16839765C211yesmaximal, characteristic
4C2×#102×D61C211yesmaximal, characteristic
5#21448444C211yesmaximal, characteristic, complemented
6#21448398C211yesmaximal, characteristic
7#21447352C211yesmaximal, characteristic, complemented
8#21447349C211yesmaximal, characteristic, complemented
9#21447346C211yesmaximal, characteristic, complemented
10#21447345C211yesmaximal, characteristic, complemented
11#21447343C211yesmaximal, characteristic, complemented
12#21447338C211yesmaximal, characteristic, complemented
13#21447337C211yesmaximal, characteristic, complemented
14#21445514C211yesmaximal, characteristic, complemented
15C61×#2092C211yesmaximal, characteristic, complemented
16C22×D4×D61C2211yescharacteristic
17C22×#11758189C2211yescharacteristic
18C22×C8×D61C2211yescharacteristic
19C2×#14519735C2211yescharacteristic
20C2×#14519735C2211yescharacteristic
21C2×#14519734C2211yescharacteristic
22C2×C4⋊C4×D61C2211yescharacteristic
23#16839801—24no—
24#16839799—24no—
25C2×#14519711C2211yescharacteristic, complemented
26C2×#14519711C2211yescharacteristic, complemented
27C2×#14519710C2211yescharacteristic, complemented
28C2×#14519708C2211yescharacteristic, complemented
29C2×#14519708C2211yescharacteristic, complemented
30C2×#14519708C2211yescharacteristic, complemented
31#16839765—24no—
32C2×#14519699C2211yescharacteristic
33C2×#14519699C2211yescharacteristic
34C2×#14519698C2211yescharacteristic
35C2×#14519698C2211yescharacteristic
36C2×#14519697C2211yescharacteristic
37C2×#14519697C2211yescharacteristic
38#102×D61—24no—
39#16839665C2211yescharacteristic
40#16839664C2211yescharacteristic
41#16839663C2211yescharacteristic
42#16839662—22no—
43#16839659—22no—
44#16839636—22no—
45#16839632C2211yescharacteristic, complemented
46#16839631C2211yescharacteristic, complemented
47#16839627C2211yescharacteristic, complemented
48#16839617—22no—
49#16839517C2211yescharacteristic, complemented
50#16839516C2211yescharacteristic, complemented
51#16839516C2211yescharacteristic, complemented
52C2×#107×C61C2211yescharacteristic, complemented
53C2×#106×C61C2211yescharacteristic, complemented
54C2×#102×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C61×#390C2211yescharacteristic
57C61×#382C2211yescharacteristic, complemented
58C61×#336C2211yescharacteristic, complemented
59C24×D61—21no—
60C22×D244C2311yescharacteristic
61C22×C22⋊D61—21no—
62C2×D4×D61—22no—
63C2×D4×D61—22no—
64C2×D4×D61—22no—
65C2×D4×D61—22no—
66C2×D4×D61—22no—
67C2×D4×D61Q814yes—
68C22×C4×D61—21no—
69C22×C4×D61C2311yescharacteristic
70C2×#11758189—22no—
71C2×#11758189—22no—
72C2×#11758189D414yes—
73C22×#92418C2311yescharacteristic
74C22×#92418C2311yescharacteristic
75C2×C8×D61—22no—
76C2×C8×D61—22no—
77C2×C8×D61D414yes—
78#14519735—22no—
79#14519735—22no—
80#14519735—22no—
81#14519735—22no—
82#14519734—22no—
83#14519734—22no—
84C2×C4⋊Dic61—22no—
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×#11758196—21no—
89C2×#11758196—22no—
90C2×#11758196D412yes—
91C4⋊C4×D61—22no—
92C4⋊C4×D61—22no—
93C2×C22⋊Dic61—22no—
94C2×#11758198—21no—
95C2×#11758198—21no—
96C2×#11758198—22no—
97C2×#11758198—22no—
98C2×#11758198D412yes—
99C2×C4×Dic61—21no—
100C2×C4×Dic61—21no—
101C2×C4×Dic61C2311yescharacteristic
102#14519711—22no—
103#14519711—22no—
104#14519710—22no—
105#14519708—22no—
106#14519708—22no—
107#14519708—22no—
108#14519699—22no—
109#14519699—22no—
110#14519698—22no—
111#14519698—22no—
112#14519697—22no—
113#14519697—22no—
114#14519684—22no—
115C22×D4×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C22⋊C4×C61—22no—
122C2×C42×C61C2311yescharacteristic
123#107×C61—22no—
124#106×C61—22no—
125#102×C61—22no—
126#102×C61—22no—
127#95×C61—22no—
128C24×C61—21no—
129C23×D61—21no—
130C23×D61—24no—
131C23×D61—42no—
132C23×D61—42no—
133C23×D61—82no—
134C23×D61—82no—
135C23×D61C2×D411yescharacteristic
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61C2×Q812yes—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2411yescharacteristic
147C2×C22⋊D61—24no—
148C2×C22⋊D61—42no—
149C2×C22⋊D61—42no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61C2×D412yes—
154C22×Dic61C2×D411yescharacteristic
155D4×D61—28no—
156D4×D61—28no—
157D4×D61—44no—
158D4×D61—44no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244C2×Q812yes—
165C2×C4×D61—42no—
166C2×C4×D61—82no—
167C2×C4×D61C2×D412yes—
168C2×C4×D61C2×D412yes—
169C2×C4×D61C2×Q812yes—
170C2×C4×D61#4112yes—
171C2×C4×D61#4112yes—
172C2×C4×D61#4112yes—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61C2×D412yes—
178C2×C8×C61C2×D412yes—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C22⋊C4×C61—82no—
186C42×C61—21no—
187C42×C61—21no—
188C22⋊Dic61—82no—
189#11758198—24no—
190#11758198—42no—
191#11758198—42no—
192#11758198—82no—
193#11758198—82no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—82no—
201#11758196—24no—
202#11758196—42no—
203#11758196—82no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—42no—
207C4×Dic61—42no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418C2×D412yes—
213C2×#92418C2×D412yes—
214#11758189—28no—
215C8×D61—28no—
216C23×C61—22no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—42no—
220C23×C61—42no—
221C23×C61C22×D411yescharacteristic
222C22×D61—22no—
223C22×D61—22no—
224C22×D61—22no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—42no—
228C22×D61—42no—
229C22×D61—48no—
230C22×D61—88no—
231C22×D61—88no—
232C22×D61—84no—
233C22×D61—84no—
234C22×D61C22⋊Q812yes—
235C22×D61#12312yes—
236C22×D61#12712yes—
237D4×C61—22no—
238D4×C61—22no—
239D4×C61—41no—
240D4×C61—41no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—42no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×Q811yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C22⋊D61—48no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—42no—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61C22⋊Q812yes—
267C2×Dic61#12312yes—
268C2×Dic61#12712yes—
269D244—22no—
270D244—22no—
271D244—41no—
272D244—41no—
273C4×D61—84no—
274C4×D61C22⋊Q814yes—
275C4×D61C22⋊Q814yes—
276C4×D61#12314yes—
277C4×D61#12314yes—
278C8×C61—22no—
279C8×C61—22no—
280#92418—22no—
281#92418—22no—
282#23735—611nomaximal, Sylow
283C22×C61—44no—
284C22×C61—42no—
285C22×C61—82no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61C2×C22⋊Q811yescharacteristic
289C22×C61C2×#12311yescharacteristic
290C22×C61C2×#12711yescharacteristic
291C22×C61#57612yes—
292C22×C61#57712yes—
293C2×D61—24no—
294C2×D61—24no—
295C2×D61—22no—
296C2×D61—22no—
297C2×D61—44no—
298C2×D61—42no—
299C2×D61—816no—
300C2×D61—82no—
301C2×D61—81no—
302C2×D61—81no—
303C2×D61#39914yes—
304C4×C61—42no—
305C4×C61—81no—
306C4×C61—81no—
307C4×C61—81no—
308C4×C61C2×C22⋊Q811yescharacteristic
309C4×C61C2×C22⋊Q811yescharacteristic
310C4×C61C2×#12311yescharacteristic
311C4×C61C2×#12311yescharacteristic
312Dic61—42no—
313Dic61—81no—
314Dic61—81no—
315Dic61—81no—
316Dic61#39914yes—
317C2×#428—611no—
318C2×#427—611no—
319C2×#418—611no—
320C22×#102—611no—
321#2317—611no—
322#2309—611no—
323#2094—611no—
324#2094—611no—
325#2093—611no—
326#2093—611no—
327#2092—611no—
328#2092—611no—
329#2092—611no—
330#2092D6111yescharacteristic, complemented
331#1729—611no—
332C2×C61—82no—
333C2×C61C2×#39911yescharacteristic
334C2×C61#308112yes—
335C2×C61#309612yes—
336C2×C61#315411yescharacteristic
337C2×C61#315511yescharacteristic
338D61—24no—
339D61—82no—
340C23×D4—611no—
341C22×M16—611no—
342C23×C8—611no—
343C2×#117—611no—
344C2×#117—611no—
345C2×#117—611no—
346C22×C4⋊C4—611no—
347#428—1224no—
348#427—1224no—
349C2×#107—611no—
350C2×#107—611no—
351C2×#107—611no—
352C2×#107C2×D6111yescharacteristic, complemented
353C2×#106—611no—
354C2×#106—611no—
355C2×#106—611no—
356C2×#106C2×D6111yescharacteristic, complemented
357#418—1224no—
358C2×#102—611no—
359C2×#102—611no—
360C2×#102—611no—
361C2×#102—611no—
362C2×#102—611no—
363C2×#102—611no—
364C2×#102—1224no—
365C2×#102C2×D6111yescharacteristic
366C2×#102C2×D6111yescharacteristic
367#390—611no—
368#390—611no—
369#390—611no—
370#390—1222no—
371#390C2×D6111yescharacteristic
372#389—1222no—
373#385—1222no—
374#384—611no—
375#384—611no—
376#382—611no—
377#382C2×D6111yescharacteristic, complemented
378#379—1222no—
379#336—611no—
380#336—611no—
381#336—611no—
382#336C2×D6111yescharacteristic, complemented
383C61#2373511yescharacteristic, complemented, Sylow
384C25—1221no—
385C22×D4—614no—
386C22×D4—611no—
387C22×D4—1222no—
388C22×D4—1222no—
389C22×D4—1222no—
390C22×D4—1222no—
391C22×D4—1222no—
392C22×D4—1221no—
393C22×D4C22×D6111yescharacteristic
394C23×C4—611no—
395C23×C4—1221no—
396C2×M16—614no—
397C2×M16—1222no—
398C2×M16—1222no—
399C22×C8—611no—
400C22×C8—611no—
401C22×C8—614no—
402C22×C8—1222no—
403C22×C8—1222no—
404C22×C8C22×D6111yescharacteristic
405C22×C8C22×D6111yescharacteristic
406#117—1222no—
407#117—1222no—
408#117—1222no—
409#117—1222no—
410#117—1222no—
411#117—1222no—
412C2×C4⋊C4—612no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—1222no—
417C2×C4⋊C4—1222no—
418C2×C4⋊C4—1222no—
419C2×C4⋊C4—1222no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4C22×D6111yescharacteristic
422C2×C4⋊C4C22×D6111yescharacteristic
423C2×C4⋊C4C22×D6111yescharacteristic
424C2×C22⋊C4—22no—
425C2×C22⋊C4—612no—
426C2×C22⋊C4—1222no—
427C2×C22⋊C4—1222no—
428C2×C22⋊C4—1222no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1221no—
431C2×C42—611no—
432C2×C42—1221no—
433C2×C42—1221no—
434C2×C42C22×D6111yescharacteristic
435#107—22no—
436#107—1222no—
437#107—1222no—
438#107—1222no—
439#106—22no—
440#106—1222no—
441#106—1222no—
442#106—1222no—
443#102—22no—
444#102—22no—
445#102—1222no—
446#102—1222no—
447#102—1222no—
448#102—1222no—
449#102—1222no—
450#102—1222no—
451#95—22no—
452#95—1222no—
453C24—21no—
454C24—611no—
455C24—1224no—
456C24—1221no—
457C24—4882no—
458C24—4882no—
459C24—2442no—
460C24—2442no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—612no—
467C2×D4—1228no—
468C2×D4—1228no—
469C2×D4—1224no—
470C2×D4—1221no—
471C2×D4—1221no—
472C2×D4—1221no—
473C2×D4—1221no—
474C2×D4—1221no—
475C2×D4—2444no—
476C2×D4—2444no—
477C2×D4—2442no—
478C2×D4—2442no—
479C2×D4Q8×D6112yes—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—611no—
484C22×C4—612no—
485C22×C4—612no—
486C22×C4—612no—
487C22×C4—612no—
488C22×C4—612no—
489C22×C4—612no—
490C22×C4—612no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—4882no—
495C22×C4—2442no—
496C22×C4D4×D6112yes—
497C22×C4C23×D6111yescharacteristic
498M16—1228no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—612no—
504C2×C8—612no—
505C2×C8—1228no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8D4×D6112yes—
511C2×C8D4×D6112yes—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—1224no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—4882no—
526C4⋊C4—4882no—
527C4⋊C4—2442no—
528C22⋊C4—82no—
529C22⋊C4—1224no—
530C22⋊C4—4882no—
531C22⋊C4—4882no—
532C22⋊C4—4882no—
533C22⋊C4—2442no—
534C22⋊C4—2442no—
535C42—21no—
536C42—21no—
537C42—1221no—
538C42—1221no—
539C42—2442no—
540C42—2442no—
541C23—22no—
542C23—41no—
543C23—41no—
544C23—42no—
545C23—42no—
546C23—612no—
547C23—612no—
548C23—612no—
549C23—1222no—
550C23—1222no—
551C23—1222no—
552C23—4888no—
553C23—4888no—
554C23—4884no—
555C23—4884no—
556C23—2448no—
557C23—2441no—
558C23—2441no—
559C23—2442no—
560C23—2442no—
561C23C2×D4×D6111yescharacteristic
562D4—22no—
563D4—22no—
564D4—41no—
565D4—41no—
566D4—1222no—
567D4—1222no—
568D4—2448no—
569D4—2441no—
570D4—2441no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—612no—
581C2×C4—612no—
582C2×C4—612no—
583C2×C4—614no—
584C2×C4—614no—
585C2×C4—614no—
586C2×C4—614no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—4884no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2442no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4C2×D4×D6111yescharacteristic
598C2×C4C2×D4×D6111yescharacteristic
599C2×C4C2×Q8×D6111yescharacteristic
600C2×C4#41×D6111yescharacteristic
601C2×C4#41×D6111yescharacteristic
602C2×C4#41×D6111yescharacteristic
603C8—22no—
604C8—22no—
605C8—1222no—
606C8—1222no—
607C22—44no—
608C22—42no—
609C22—82no—
610C22—81no—
611C22—81no—
612C22—614no—
613C22—1224no—
614C22—1224no—
615C22—1222no—
616C22—1222no—
617C22—48816no—
618C22—4882no—
619C22—4881no—
620C22—4881no—
621C22—2444no—
622C22—2442no—
623C22C22⋊Q8×D6111yescharacteristic
624C22#123×D6111yescharacteristic
625C22#127×D6111yescharacteristic
626C22#1684056412yes—
627C22#1684057212yes—
628C4—42no—
629C4—81no—
630C4—81no—
631C4—81no—
632C4—614no—
633C4—4881no—
634C4—4881no—
635C4—4881no—
636C4—2442no—
637C4C22⋊Q8×D6111yescharacteristic
638C4C22⋊Q8×D6111yescharacteristic
639C4#123×D6111yescharacteristic
640C4#123×D6111yescharacteristic
641C2—82no—
642C2—1224no—
643C2—4882no—
644C2#399×D6111yescharacteristic
645C2#2145564712yes—
646C2#2145577412yes—
647C2#2145629211yescharacteristic
648C2#2145630111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2092C2
socleC23×C61C22×D4
prime radicalC22×D4×D61C22
hypercenter#2092D61
hypocenterC61#23735
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2092

Complements

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

NH ≅ G/Ntype
1. C2×#16839801C2semidirect
2. C2×#16839799C2semidirect
5. #21448444C2semidirect
7. #21447352C2semidirect
8. #21447349C2semidirect
9. #21447346C2semidirect
10. #21447345C2semidirect
11. #21447343C2semidirect
12. #21447338C2semidirect
13. #21447337C2semidirect
14. #21445514C2semidirect
15. C61×#2092C2semidirect
25. C2×#14519711C22semidirect
26. C2×#14519711C22semidirect
27. C2×#14519710C22semidirect
28. C2×#14519708C22semidirect
29. C2×#14519708C22semidirect
30. C2×#14519708C22semidirect
45. #16839632C22semidirect
46. #16839631C22semidirect
47. #16839627C22semidirect
49. #16839517C22semidirect
50. #16839516C22semidirect
51. #16839516C22semidirect
52. C2×#107×C61C22semidirect
53. C2×#106×C61C22semidirect
57. C61×#382C22semidirect
58. C61×#336C22semidirect
330. #2092D61semidirect
352. C2×#107C2×D61semidirect
356. C2×#106C2×D61semidirect
377. #382C2×D61semidirect
382. #336C2×D61semidirect
383. C61#23735semidirect

Maximal subgroups

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

subgroupindex
C2×#168398012
C2×#168397992
C2×#168397652
C2×#102×D612
#214484442
#214483982
#214473522
#214473492
#214473462
#214473452
#214473432
#214473382
#214473372
#214455142
C61×#20922
#2373561