Finite Group Databasebeta

← #30602433

Subgroups — #30602433

Order 15616.

1406 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#407C211yesmaximal, characteristic
2C2×#16840704C211yesmaximal, characteristic
3C2×#16840641C214yesmaximal, complemented
4C22×M16×D61C211yesmaximal, characteristic
5C2×D61⋊M16C2212yes—
6C22×C4⋊#6923C2212yes—
7C22×C22⋊#6923C412yes—
8#16840734C412yescomplemented
9#16840733—21no—
10#16840733—21no—
11#16840732C412yescomplemented
12D61⋊(C22⋊C8)—21no—
13D61⋊(C22⋊C8)—21no—
14#16840704—21no—
15#16840704—21no—
16#16840704—21no—
17#16840704—21no—
18C2×(C2×#6923)⋊C4C412yes—
19#16840641—28no—
20C23×C4×D61C2211yescharacteristic
21C22×#11758194C411yescharacteristic, complemented
22C22×#11758189—21no—
23C2×M16×D61—28no—
24C2×M16×D61—21no—
25C2×M16×D61—21no—
26C2×M16×D61C412yescomplemented
27C2×M16×D61C2212yes—
28C22×C8×D61—21no—
29C22×M16×C61C411yescharacteristic, complemented
30C23×#6923C2×C414yes—
31C2×#11758213—22no—
32D61⋊M16—24no—
33D61⋊M16—24no—
34C2×#11758217C2×C414yes—
35C2×D61⋊C8—22no—
36C2×C4⋊#6923—22no—
37C2×C4⋊#6923—22no—
38C2×C4⋊#6923—24no—
39C2×C4⋊#6923—24no—
40C2×C22⋊#6923—22no—
41C2×C22⋊#6923—22no—
42C2×C22⋊#6923—44no—
43#14519834—22no—
44#14519834—22no—
45C22⋊#92420—22no—
46C22⋊#92420—22no—
47(C2×#6923)⋊C4—22no—
48(C2×#6923)⋊C4—22no—
49C24×D61C2×C411yescharacteristic
50C23×Dic61C2×C411yescharacteristic
51C22×C4×D61—24no—
52C22×C4×D61—24no—
53C22×C4×D61C2×C411yescharacteristic
54C22×C4×D61C2×C411yescharacteristic
55C22×C4×D61C2×C411yescharacteristic
56C22×C4×D61D411yescharacteristic
57C22×C4×D61D411yescharacteristic
58C22×C4×D61D411yescharacteristic
59C22×C4×D61D411yescharacteristic
60C22×C4×D61D411yescharacteristic
61C22×C4×D61D411yescharacteristic
62C22×C4×D61Q811yescharacteristic
63C22×C4×D61Q811yescharacteristic
64C22×C4×D61C2311yescharacteristic
65C2×#11758194—21no—
66C2×#11758194—24no—
67C2×#11758194C2×C412yes—
68C2×#11758189—28no—
69C2×#11758189—22no—
70C2×#11758189—22no—
71M16×D61—28no—
72M16×D61—28no—
73M16×D61—24no—
74M16×D61—24no—
75M16×D61—24no—
76M16×D61D418yes—
77C22×#92418—21no—
78C2×C8×D61—28no—
79C2×C8×D61—22no—
80C2×C8×D61—22no—
81C23×C4×C61C2×C411yescharacteristic
82C2×M16×C61—21no—
83C2×M16×C61—24no—
84C2×M16×C61C2×C412yes—
85C22×C8×C61—21no—
86C24×C61C4211yescharacteristic
87C23×D61—22no—
88C23×D61—22no—
89C23×D61—24no—
90C23×D61—42no—
91C23×D61C4211yescharacteristic
92C23×D61C22⋊C411yescharacteristic
93C23×D61C22⋊C411yescharacteristic
94C23×D61C4⋊C411yescharacteristic
95C23×D61C4⋊C411yescharacteristic
96C23×D61C22×C411yescharacteristic
97C22×#6923—24no—
98C22×#6923—28no—
99C22×#6923—44no—
100C22×#6923—44no—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61C4211yescharacteristic
104C22×C4×C61C22⋊C411yescharacteristic
105C22×C4×C61C22⋊C411yescharacteristic
106C22×C4×C61C4⋊C411yescharacteristic
107C22×C4×C61C4⋊C411yescharacteristic
108C22×C4×C61C22×C411yescharacteristic
109C22×Dic61—24no—
110C22×Dic61C4211yescharacteristic
111C22×Dic61C22⋊C411yescharacteristic
112C22×Dic61C22⋊C411yescharacteristic
113C22×Dic61C4⋊C411yescharacteristic
114C22×Dic61C4⋊C411yescharacteristic
115C22×Dic61C22×C411yescharacteristic
116C2×C4×D61—24no—
117C2×C4×D61—22no—
118C2×C4×D61—22no—
119C2×C4×D61—24no—
120C2×C4×D61—22no—
121C2×C4×D61—22no—
122C2×C4×D61—22no—
123C2×C4×D61—22no—
124C2×C4×D61—24no—
125C2×C4×D61—24no—
126C2×C4×D61—24no—
127C2×C4×D61—24no—
128C2×C4×D61—22no—
129C2×C4×D61—22no—
130C2×C4×D61—22no—
131C2×C4×D61—22no—
132C2×C4×D61C22⋊C411yescharacteristic
133C2×C4×D61C22⋊C411yescharacteristic
134C2×C4×D61C22⋊C411yescharacteristic
135C2×C4×D61C22⋊C411yescharacteristic
136C2×C4×D61C22⋊C411yescharacteristic
137C2×C4×D61C22⋊C411yescharacteristic
138C2×C4×D61C4⋊C411yescharacteristic
139C2×C4×D61C4⋊C411yescharacteristic
140C2×C4×D61C4⋊C411yescharacteristic
141C2×C4×D61C4⋊C411yescharacteristic
142C2×C4×D61C4⋊C411yescharacteristic
143C2×C4×D61C4⋊C411yescharacteristic
144C2×C4×D61C2×D414yes—
145C2×C4×D61C2×D411yescharacteristic
146C2×C4×D61C2×D411yescharacteristic
147C2×C4×D61C2×D411yescharacteristic
148C2×C4×D61C2×Q811yescharacteristic
149C2×C4×D61#4114yes—
150C22⋊#6923—44no—
151C22⋊#6923—44no—
152#11758217—28no—
153C2×#92420—24no—
154C4⋊#6923—42no—
155C4⋊#6923—44no—
156C4⋊#6923—42no—
157#11758213—42no—
158#11758213—42no—
159D61⋊C8—44no—
160M16×C61—24no—
161M16×C61—22no—
162M16×C61C22⋊C414yescomplemented
163C2×C8×C61—24no—
164C2×C8×C61—22no—
165#11758194—24no—
166#11758194—22no—
167#11758194C22⋊C414yescomplemented
168C2×#92418—24no—
169C2×#92418—22no—
170#11758189—28no—
171#11758189—28no—
172C8×D61—28no—
173C8×D61—28no—
174C23×C61—22no—
175C23×C61—22no—
176C23×C61—41no—
177C23×C61#9511yescharacteristic
178C23×C61#9511yescharacteristic
179C23×C61C2×C4211yescharacteristic
180C22×D61—22no—
181C22×D61—21no—
182C22×D61—21no—
183C22×D61—22no—
184C22×D61—22no—
185C22×D61—22no—
186C22×D61—22no—
187C22×D61—22no—
188C22×D61—22no—
189C22×D61—22no—
190C22×D61—22no—
191C22×D61—22no—
192C22×D61—21no—
193C22×D61—21no—
194C22×D61—22no—
195C22×D61—41no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61—48no—
199C22×D61—48no—
200C22×D61#9511yescharacteristic
201C22×D61#9511yescharacteristic
202C22×D61#10012yes—
203C22×D61#10112yes—
204C22×D61C2×C22⋊C411yescharacteristic
205C22×D61C2×C4⋊C411yescharacteristic
206C22×D61C4×D414yes—
207C2×#6923—44no—
208C2×#6923—44no—
209C2×#6923—44no—
210C2×#6923—88no—
211C2×C4×C61—22no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61#9511yescharacteristic
219C2×C4×C61#9511yescharacteristic
220C2×C4×C61C2×C22⋊C412yes—
221C2×C4×C61C2×C22⋊C411yescharacteristic
222C2×C4×C61C2×C4⋊C411yescharacteristic
223C2×C4×C61#11712yes—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61#9511yescharacteristic
230C2×Dic61#9511yescharacteristic
231C2×Dic61C2×C22⋊C411yescharacteristic
232C2×Dic61C2×C4⋊C411yescharacteristic
233C2×Dic61C4×D414yes—
234C4×D61—22no—
235C4×D61—22no—
236C4×D61—22no—
237C4×D61—22no—
238C4×D61—21no—
239C4×D61—21no—
240C4×D61—22no—
241C4×D61—22no—
242C4×D61—21no—
243C4×D61—21no—
244C4×D61—22no—
245C4×D61—22no—
246C4×D61—22no—
247C4×D61—22no—
248C4×D61#12012yes—
249C4×D61#12012yes—
250C4×D61#12114yes—
251C4×D61C22⋊Q814yes—
252C4×D61#12312yes—
253C4×D61#12312yes—
254#92420—44no—
255C8×C61—24no—
256#92418—24no—
257#10739—611nomaximal, Sylow
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—21no—
263C22×C61—21no—
264C22×C61—21no—
265C22×C61—44no—
266C22×C61#34312yescomplemented
267C22×C61C2×#9511yescharacteristic
268C22×C61C4×C22⋊C412yes—
269C2×D61—21no—
270C2×D61—21no—
271C2×D61—21no—
272C2×D61—21no—
273C2×D61—24no—
274C2×D61—24no—
275C2×D61—24no—
276C2×D61—24no—
277C2×D61—21no—
278C2×D61—21no—
279C2×D61—21no—
280C2×D61—44no—
281C2×D61—44no—
282C2×D61—44no—
283C2×D61—42no—
284C2×D61—42no—
285C2×D61#34312yescomplemented
286C2×D61#38512yes—
287C2×D61#38612yes—
288C2×D61C2×#10011yescharacteristic
289C2×D61C2×#10111yescharacteristic
290C2×D61#41312yes—
291#6923—84no—
292C4×C61—21no—
293C4×C61—21no—
294C4×C61#37911yescharacteristic
295C4×C61#38012yes—
296C4×C61#38111yescharacteristic
297Dic61—21no—
298Dic61—21no—
299Dic61#38512yes—
300Dic61#38612yes—
301C23×M16—611no—
302C2×#407—611no—
303C2×#380—611no—
304C2×#343—614no—
305C2×C61—22no—
306C2×C61—22no—
307C2×C61—41no—
308C2×C61C22⋊#9511yescharacteristic
309C2×C61C2×#34311yescharacteristic
310C2×C61#174211yescharacteristic
311D61—22no—
312D61—22no—
313D61—41no—
314D61#190012yes—
315D61#190112yes—
316C24×C4—611no—
317C22×M16—612no—
318C22×M16—612no—
319C22×M16—611no—
320C22×M16—612no—
321C22×M16—1221no—
322C22×M16—1228no—
323C22×M16—1221no—
324C22×M16—1221no—
325C22×M16#692311yescharacteristic, complemented
326C23×C8—1221no—
327C22×C4⋊C4—612no—
328C22×C22⋊C4—612no—
329#407—612no—
330#407—612no—
331#407—1221no—
332#407—1221no—
333C2×C22⋊C8—1221no—
334C2×C22⋊C8—1221no—
335#380—1221no—
336#380—1221no—
337#380—1221no—
338#380—1221no—
339C2×#95—612no—
340#343—1228no—
341C61#1073911yescharacteristic, complemented, Sylow
342C25—611no—
343C23×C4—611no—
344C23×C4—611no—
345C23×C4—611no—
346C23×C4—611no—
347C23×C4—611no—
348C23×C4—611no—
349C23×C4—611no—
350C23×C4—611no—
351C23×C4—611no—
352C23×C4—611no—
353C23×C4—611no—
354C23×C4—611no—
355C23×C4—611no—
356C23×C4—614no—
357C23×C4—1224no—
358C23×C4—1224no—
359C23×C4C2×#692311yescharacteristic
360C2×M16—21no—
361C2×M16—24no—
362C2×M16—612no—
363C2×M16—618no—
364C2×M16—614no—
365C2×M16—1222no—
366C2×M16—1222no—
367C2×M16—1228no—
368C2×M16—1224no—
369C2×M16—1224no—
370C2×M16—1228no—
371C2×M16—1228no—
372C2×M16—1224no—
373C2×M16—1221no—
374C2×M16—1224no—
375C2×M16—1222no—
376C2×M16—1224no—
377C2×M16—1224no—
378C2×M16C2×#692312yes—
379C22×C8—21no—
380C22×C8—1222no—
381C22×C8—1222no—
382C22×C8—1228no—
383C22×C8—1221no—
384C22×C8—1222no—
385C2×C4⋊C4—1224no—
386C2×C4⋊C4—1224no—
387C2×C4⋊C4—1222no—
388C2×C4⋊C4—1222no—
389C2×C22⋊C4—1222no—
390C2×C22⋊C4—1222no—
391C2×C22⋊C4—2444no—
392C22⋊C8—1222no—
393C22⋊C8—1222no—
394C22⋊C8—1222no—
395C22⋊C8—1222no—
396#95—1222no—
397#95—1222no—
398C24—611no—
399C24—611no—
400C24—611no—
401C24—611no—
402C24—611no—
403C24—611no—
404C24—1222no—
405C24—1224no—
406C24—1222no—
407C24—2442no—
408C24C4×#692311yescharacteristic
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—611no—
412C22×C4—611no—
413C22×C4—614no—
414C22×C4—611no—
415C22×C4—611no—
416C22×C4—611no—
417C22×C4—614no—
418C22×C4—611no—
419C22×C4—611no—
420C22×C4—611no—
421C22×C4—611no—
422C22×C4—611no—
423C22×C4—611no—
424C22×C4—611no—
425C22×C4—611no—
426C22×C4—611no—
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—611no—
430C22×C4—611no—
431C22×C4—611no—
432C22×C4—611no—
433C22×C4—611no—
434C22×C4—611no—
435C22×C4—1222no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1224no—
440C22×C4—1224no—
441C22×C4—1224no—
442C22×C4—1224no—
443C22×C4—1224no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1224no—
449C22×C4—1224no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1228no—
453C22×C4—1224no—
454C22×C4—2444no—
455C22×C4—2444no—
456C22×C4C4×#692311yescharacteristic
457C22×C4C4⋊#692311yescharacteristic
458C22×C4C4⋊#692311yescharacteristic
459C22×C4C22⋊#692311yescharacteristic
460C22×C4C22⋊#692311yescharacteristic
461C22×C4C22×#692311yescharacteristic
462M16—24no—
463M16—22no—
464M16—614no—
465M16—1228no—
466M16—1228no—
467M16—1224no—
468M16—1222no—
469M16—1228no—
470M16—2442no—
471M16—2442no—
472M16C22⋊#692314yescomplemented
473C2×C8—24no—
474C2×C8—22no—
475C2×C8—1228no—
476C2×C8—1228no—
477C2×C8—1224no—
478C2×C8—1222no—
479C2×C8—1224no—
480C2×C8—2444no—
481C4⋊C4—2444no—
482C4⋊C4—2442no—
483C4⋊C4—2442no—
484C22⋊C4—2444no—
485C22⋊C4—2444no—
486C23—22no—
487C23—22no—
488C23—41no—
489C23—611no—
490C23—612no—
491C23—612no—
492C23—611no—
493C23—611no—
494C23—614no—
495C23—611no—
496C23—1222no—
497C23—1221no—
498C23—1222no—
499C23—1221no—
500C23—1222no—
501C23—1221no—
502C23—1222no—
503C23—1222no—
504C23—1222no—
505C23—1222no—
506C23—1221no—
507C23—1222no—
508C23—1222no—
509C23—1222no—
510C23—1222no—
511C23—2441no—
512C23—2441no—
513C23—2441no—
514C23—2448no—
515C23—2448no—
516C23(C2×#6923)⋊C411yescharacteristic
517C23(C2×#6923)⋊C411yescharacteristic
518C23C2×C4×#692311yescharacteristic
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—611no—
527C2×C4—611no—
528C2×C4—614no—
529C2×C4—614no—
530C2×C4—614no—
531C2×C4—612no—
532C2×C4—612no—
533C2×C4—612no—
534C2×C4—612no—
535C2×C4—611no—
536C2×C4—611no—
537C2×C4—1222no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—1222no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—4888no—
557C2×C4—2444no—
558C2×C4—2444no—
559C2×C4—2444no—
560C2×C4(C2×#6923)⋊C411yescharacteristic
561C2×C4(C2×#6923)⋊C411yescharacteristic
562C2×C4C2×C22⋊#692312yes—
563C2×C4C2×C22⋊#692311yescharacteristic
564C2×C4C2×C4⋊#692311yescharacteristic
565C2×C4#1451985612yes—
566C8—24no—
567C8—1224no—
568C8—2444no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—44no—
577C22—612no—
578C22—611no—
579C22—611no—
580C22—612no—
581C22—612no—
582C22—612no—
583C22—1224no—
584C22—1224no—
585C22—1224no—
586C22—1224no—
587C22—1221no—
588C22—1221no—
589C22—1221no—
590C22—1221no—
591C22—1221no—
592C22—1221no—
593C22—1221no—
594C22—2444no—
595C22—2444no—
596C22—2442no—
597C22—2442no—
598C22—2444no—
599C22#1684064112yescomplemented
600C22C2×(C2×#6923)⋊C411yescharacteristic
601C22C4×C22⋊#692312yes—
602C4—21no—
603C4—21no—
604C4—612no—
605C4—612no—
606C4—1221no—
607C4—1221no—
608C4—4884no—
609C4#1684070311yescharacteristic
610C4#1684070412yes—
611C4#1684070511yescharacteristic
612C2—22no—
613C2—22no—
614C2—41no—
615C2—612no—
616C2—612no—
617C2—1222no—
618C2—1222no—
619C2—2441no—
620C2#2146099311yescharacteristic
621C2C2×#1684064111yescharacteristic
622C2#2146152311yescharacteristic
subgroupG/H
centerC22C2×(C2×#6923)⋊C4
commutatorC23×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC22×M16×C61C4
socleC22×C61C2×#95
prime radicalC24×D61C2×C4
cosocleC22×C4×D61C23
hypercenterC22×M16#6923
hypocenterC61#10739
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
3. C2×#16840641C2semidirect
8. #16840734C4semidirect
11. #16840732C4semidirect
21. C22×#11758194C4semidirect
26. C2×M16×D61C4semidirect
29. C22×M16×C61C4semidirect
162. M16×C61C22⋊C4semidirect
167. #11758194C22⋊C4semidirect
266. C22×C61#343semidirect
285. C2×D61#343semidirect
325. C22×M16#6923semidirect
341. C61#10739semidirect
472. M16C22⋊#6923semidirect
599. C22#16840641semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4072
C2×#168407042
C2×#168406412
C22×M16×D612
#1073961