Finite Group Databasebeta

← #30614577

Subgroups — #30614577

Order 15616.

1314 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#407C211yesmaximal, characteristic
2C2×C4×C22⋊#6923C211yesmaximal, characteristic
3#21461562C214yesmaximal, complemented
4#21461559C214yesmaximal, complemented
5#21461547C212yesmaximal, complemented
6C22⋊C8×#6923C212yesmaximal, complemented
7#407×D61C211yesmaximal, characteristic
8C2×D61⋊M16C2212yes—
9C22×C22⋊#6923C412yes—
10C22×C4×#6923C2211yescharacteristic
11C22×C4×#6923C2211yescharacteristic
12#16840734C412yescomplemented
13#16840733C412yescomplemented
14#16840733C412yescomplemented
15#16840732C412yescomplemented
16D61⋊(C22⋊C8)C2212yes—
17D61⋊(C22⋊C8)C2212yes—
18C2×#14519831C2214yescomplemented
19C2×C8×#6923C2214yescomplemented
20C4×C22⋊#6923C412yes—
21C4×C22⋊#6923C412yes—
22C4×C22⋊#6923C2212yes—
23C4×C22⋊#6923C2212yes—
24C2×(C2×#6923)⋊C4C412yes—
25(C2×#6923)⋊C8C2214yescomplemented
26(C2×#6923)⋊C8C2214yescomplemented
27C23×C4×D61C2211yescharacteristic
28C2×M16×D61C2212yes—
29#16839726C411yescharacteristic, complemented
30#16839725C412yescomplemented
31#16839724C412yescomplemented
32#16839724C412yescomplemented
33C22⋊C8×D61C2212yes—
34C22⋊C8×D61C2212yes—
35C61×#407C411yescharacteristic, complemented
36C23×#6923C2×C412yes—
37C23×#6923C2×C412yes—
38C2×#11758213C2×C414yescomplemented
39D61⋊M16—42no—
40D61⋊M16—42no—
41C2×#11758217C2×C414yescomplemented
42C2×D61⋊C8C2314yes—
43C2×C22⋊#6923—28no—
44C2×C22⋊#6923C2×C414yes—
45C2×C22⋊#6923C2×C414yes—
46C2×C4×#6923—42no—
47C2×C4×#6923—42no—
48C2×C4×#6923C2×C412yes—
49C2×C4×#6923C2×C412yes—
50C2×C4×#6923C2312yes—
51C2×C4×#6923C2312yes—
52#14519834C2×C414yes—
53#14519834C2×C414yescomplemented
54C22⋊#92420C2×C414yes—
55C22⋊#92420C2×C414yescomplemented
56#14519831—44no—
57C8×#6923—44no—
58(C2×#6923)⋊C4C2×C414yes—
59(C2×#6923)⋊C4C2×C414yes—
60C24×D61C2×C411yescharacteristic
61C23×Dic61C2×C411yescharacteristic
62C22×C4×D61—42no—
63C22×C4×D61—42no—
64C22×C4×D61C2×C411yescharacteristic
65C22×C4×D61C2×C411yescharacteristic
66C22×C4×D61C2×C411yescharacteristic
67C22×C4×D61C2×C411yescharacteristic
68C22×C4×D61C2×C411yescharacteristic
69C22×C4×D61C2×C411yescharacteristic
70C22×C4×D61C2×C411yescharacteristic
71C22×C4×D61C2×C411yescharacteristic
72C22×C4×D61C2×C411yescharacteristic
73C22×C4×D61C2311yescharacteristic
74C22×C4×D61C2311yescharacteristic
75C22×C4×D61C2311yescharacteristic
76C2×#11758194C2×C412yescomplemented
77C2×#11758189C2×C414yescomplemented
78M16×D61—42no—
79M16×D61—42no—
80C2×C8×D61C2314yes—
81C22⋊#92418C2×C412yes—
82C22⋊#92418C2×C412yescomplemented
83#14519689C2×C414yes—
84#14519689C2×C414yescomplemented
85C23×C4×C61C2×C411yescharacteristic
86C2×M16×C61C2×C412yescomplemented
87C22⋊C8×C61C2×C412yes—
88C22⋊C8×C61C2×C412yescomplemented
89C24×C61C4211yescharacteristic
90C23×D61—24no—
91C23×D61—41no—
92C23×D61—41no—
93C23×D61—42no—
94C23×D61C4211yescharacteristic
95C23×D61C4211yescharacteristic
96C23×D61C4211yescharacteristic
97C23×D61C22×C411yescharacteristic
98C23×D61C22×C411yescharacteristic
99C23×D61C22×C411yescharacteristic
100C22×#6923—24no—
101C22×#6923—42no—
102C22×#6923—42no—
103C22×#6923—42no—
104C22×#6923—42no—
105C22×#6923M1618yescomplemented
106C22×#6923C22×C414yes—
107C22×#6923C22×C414yes—
108C22×C4×C61—42no—
109C22×C4×C61C4211yescharacteristic
110C22×C4×C61C4211yescharacteristic
111C22×C4×C61C4211yescharacteristic
112C22×C4×C61C22×C411yescharacteristic
113C22×C4×C61C22×C411yescharacteristic
114C22×C4×C61C22×C411yescharacteristic
115C22×Dic61—42no—
116C22×Dic61C4211yescharacteristic
117C22×Dic61C4211yescharacteristic
118C22×Dic61C4211yescharacteristic
119C22×Dic61C22×C411yescharacteristic
120C22×Dic61C22×C411yescharacteristic
121C22×Dic61C22×C411yescharacteristic
122C2×C4×D61—22no—
123C2×C4×D61—22no—
124C2×C4×D61—22no—
125C2×C4×D61—22no—
126C2×C4×D61—22no—
127C2×C4×D61—22no—
128C2×C4×D61—42no—
129C2×C4×D61—42no—
130C2×C4×D61—42no—
131C2×C4×D61—42no—
132C2×C4×D61—42no—
133C2×C4×D61—42no—
134C2×C4×D61—41no—
135C2×C4×D61—41no—
136C2×C4×D61—41no—
137C2×C4×D61—41no—
138C2×C4×D61—41no—
139C2×C4×D61—41no—
140C2×C4×D61C4211yescharacteristic
141C2×C4×D61C4211yescharacteristic
142C2×C4×D61C4211yescharacteristic
143C2×C4×D61C4211yescharacteristic
144C2×C4×D61C4211yescharacteristic
145C2×C4×D61C4211yescharacteristic
146C2×C4×D61C22×C411yescharacteristic
147C2×C4×D61C22×C411yescharacteristic
148C2×C4×D61C22×C411yescharacteristic
149C2×C4×D61C22×C411yescharacteristic
150C2×C4×D61C22×C411yescharacteristic
151C2×C4×D61C22×C411yescharacteristic
152C2×C4×D61C22×C411yescharacteristic
153C2×C4×D61C22×C411yescharacteristic
154C2×C4×D61C22×C411yescharacteristic
155C2×C4×D61C2411yescharacteristic
156C22⋊#6923—28no—
157C22⋊#6923—28no—
158#11758217—44no—
159C2×#92420C22×C418yes—
160C4×#6923—24no—
161C4×#6923—24no—
162C4×#6923—42no—
163C4×#6923—42no—
164#11758213—44no—
165D61⋊C8—24no—
166D61⋊C8—24no—
167M16×C61—42no—
168C2×C8×C61C22×C414yes—
169#11758194—42no—
170C2×#92418C22×C414yes—
171#11758189—44no—
172C8×D61—24no—
173C8×D61—24no—
174C23×C61—22no—
175C23×C61—41no—
176C23×C61—41no—
177C23×C61C2×C4211yescharacteristic
178C23×C61C2×C4211yescharacteristic
179C23×C61C2×C4211yescharacteristic
180C22×D61—21no—
181C22×D61—21no—
182C22×D61—22no—
183C22×D61—22no—
184C22×D61—21no—
185C22×D61—21no—
186C22×D61—21no—
187C22×D61—21no—
188C22×D61—22no—
189C22×D61—22no—
190C22×D61—22no—
191C22×D61—22no—
192C22×D61—22no—
193C22×D61—22no—
194C22×D61—41no—
195C22×D61—41no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61—41no—
199C22×D61—41no—
200C22×D61—48no—
201C22×D61—48no—
202C22×D61C2×C4211yescharacteristic
203C22×D61C2×C4211yescharacteristic
204C22×D61C2×C4211yescharacteristic
205C22×D61C2×M1614yes—
206C22×D61C23×C411yescharacteristic
207C2×#6923—24no—
208C2×#6923—24no—
209C2×#6923—24no—
210C2×#6923—24no—
211C2×#6923—24no—
212C2×#6923—48no—
213C2×#6923—48no—
214C2×#6923C2×M1618yes—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61C2×C4211yescharacteristic
222C2×C4×C61C2×C4211yescharacteristic
223C2×C4×C61C2×C4211yescharacteristic
224C2×C4×C61C23×C411yescharacteristic
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61C2×C4211yescharacteristic
232C2×Dic61C2×C4211yescharacteristic
233C2×Dic61C2×C4211yescharacteristic
234C2×Dic61C23×C411yescharacteristic
235C4×D61—21no—
236C4×D61—21no—
237C4×D61—21no—
238C4×D61—21no—
239C4×D61—21no—
240C4×D61—21no—
241C4×D61—22no—
242C4×D61—22no—
243C4×D61—22no—
244C4×D61—22no—
245C4×D61—22no—
246C4×D61—22no—
247C4×D61—41no—
248C4×D61—41no—
249C4×D61—41no—
250C4×D61—41no—
251C4×D61—41no—
252C4×D61—41no—
253C4×D61H(5,𝔽2)12yes—
254C4×D61H(5,𝔽2)12yes—
255#92420—28no—
256C8×C61—24no—
257#92418—24no—
258#20524—611nomaximal, Sylow
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—21no—
263C22×C61—21no—
264C22×C61—21no—
265C22×C61—21no—
266C22×C61—21no—
267C22×C61—44no—
268C22×C61C4×M1612yes—
269C22×C61C22×C4211yescharacteristic
270C2×D61—21no—
271C2×D61—21no—
272C2×D61—24no—
273C2×D61—24no—
274C2×D61—21no—
275C2×D61—21no—
276C2×D61—21no—
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—42no—
286C2×D61—42no—
287C2×D61C4×M1612yes—
288C2×D61#51811yescharacteristic
289C2×D61#51811yescharacteristic
290C2×D61C22×M1612yes—
291C2×D61#56812yes—
292#6923—28no—
293#6923—44no—
294C4×C61—21no—
295C4×C61—21no—
296C4×C61—21no—
297C4×C61#51812yes—
298Dic61—21no—
299Dic61—21no—
300Dic61—21no—
301Dic61#51812yes—
302C2×#407—611no—
303C2×#407—611no—
304C2×C4×C22⋊C4—611no—
305#1753—614no—
306#1753—614no—
307#1749—612no—
308C4×C22⋊C8—612no—
309C2×C61—22no—
310C2×C61—41no—
311C2×C61—41no—
312C2×C61#227311yescharacteristic
313C2×C61C2×C4×M1611yescharacteristic
314C2×C61#287311yescharacteristic
315D61—22no—
316D61—41no—
317D61—41no—
318D61#297312yes—
319D61#297512yes—
320C24×C4—611no—
321C22×M16—612no—
322C22×M16—612no—
323C22×C22⋊C4—612no—
324C22×C42—611no—
325C22×C42—611no—
326#407—612no—
327#407—612no—
328#407—612no—
329#407—612no—
330#407—611no—
331#407—612no—
332#407—612no—
333#407—612no—
334#407#692311yescharacteristic, complemented
335C2×C22⋊C8—612no—
336C2×C22⋊C8—612no—
337C2×C22⋊C8—612no—
338C2×C22⋊C8—612no—
339C2×#97—614no—
340C2×C4×C8—614no—
341C4×C22⋊C4—612no—
342C4×C22⋊C4—612no—
343C4×C22⋊C4—612no—
344C4×C22⋊C4—612no—
345C2×#95—612no—
346#336—614no—
347#336—614no—
348C61#2052411yescharacteristic, complemented, Sylow
349C25—611no—
350C23×C4—612no—
351C23×C4—611no—
352C23×C4—611no—
353C23×C4—611no—
354C23×C4—611no—
355C23×C4—611no—
356C23×C4—611no—
357C23×C4—611no—
358C23×C4—611no—
359C23×C4—611no—
360C23×C4—611no—
361C23×C4—611no—
362C23×C4—611no—
363C23×C4—611no—
364C23×C4—612no—
365C23×C4—2442no—
366C23×C4—2442no—
367C23×C4C2×#692311yescharacteristic
368C2×M16—614no—
369C2×M16—614no—
370C2×M16—612no—
371C2×M16—614no—
372C2×M16—2442no—
373C2×M16—2442no—
374C2×M16—2442no—
375C2×M16—2442no—
376C2×M16C2×#692312yescomplemented
377C22×C8—614no—
378C22×C8—614no—
379C2×C22⋊C4—614no—
380C2×C22⋊C4—614no—
381C2×C22⋊C4—1228no—
382C2×C42—612no—
383C2×C42—612no—
384C2×C42—612no—
385C2×C42—612no—
386C2×C42—2442no—
387C2×C42—2442no—
388C22⋊C8—614no—
389C22⋊C8—614no—
390C22⋊C8—614no—
391C22⋊C8—614no—
392C22⋊C8—612no—
393C22⋊C8—612no—
394C22⋊C8—614no—
395C22⋊C8—614no—
396C22⋊C8C2×#692312yes—
397C22⋊C8C2×#692312yescomplemented
398#97—2444no—
399C4×C8—2444no—
400#95—614no—
401#95—614no—
402C24—611no—
403C24—611no—
404C24—611no—
405C24—611no—
406C24—611no—
407C24—611no—
408C24—1224no—
409C24—2441no—
410C24—2441no—
411C24—2442no—
412C24C4×#692311yescharacteristic
413C22×C4—42no—
414C22×C4—611no—
415C22×C4—614no—
416C22×C4—611no—
417C22×C4—611no—
418C22×C4—611no—
419C22×C4—618no—
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—611no—
436C22×C4—611no—
437C22×C4—611no—
438C22×C4—614no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4—1224no—
446C22×C4—2442no—
447C22×C4—2442no—
448C22×C4—2442no—
449C22×C4—2442no—
450C22×C4—2442no—
451C22×C4—2442no—
452C22×C4—2441no—
453C22×C4—2441no—
454C22×C4—2441no—
455C22×C4—2441no—
456C22×C4—2441no—
457C22×C4—2441no—
458C22×C4—2442no—
459C22×C4—2442no—
460C22×C4—2442no—
461C22×C4—2442no—
462C22×C4—2442no—
463C22×C4C4×#692311yescharacteristic
464C22×C4C4×#692311yescharacteristic
465C22×C4C4×#692311yescharacteristic
466C22×C4C22×#692311yescharacteristic
467C22×C4C22×#692311yescharacteristic
468C22×C4C22×#692311yescharacteristic
469M16—42no—
470M16—2444no—
471M16—2442no—
472M16—2444no—
473M16—2444no—
474C2×C8—618no—
475C2×C8—614no—
476C2×C8—1224no—
477C2×C8—1224no—
478C2×C8—1224no—
479C2×C8—1224no—
480C2×C8C22×#692314yes—
481C22⋊C4—1228no—
482C22⋊C4—1228no—
483C42—1224no—
484C42—1224no—
485C42—2442no—
486C42—2442no—
487C23—22no—
488C23—41no—
489C23—41no—
490C23—611no—
491C23—614no—
492C23—611no—
493C23—611no—
494C23—611no—
495C23—1221no—
496C23—1221no—
497C23—1222no—
498C23—1222no—
499C23—1222no—
500C23—1221no—
501C23—1221no—
502C23—1221no—
503C23—1221no—
504C23—1222no—
505C23—1222no—
506C23—1222no—
507C23—1222no—
508C23—1222no—
509C23—2441no—
510C23—2441no—
511C23—2441no—
512C23—2441no—
513C23—2441no—
514C23—2441no—
515C23—2448no—
516C23—2448no—
517C23C2×C4×#692311yescharacteristic
518C23C2×C4×#692311yescharacteristic
519C23C2×C4×#692311yescharacteristic
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—611no—
527C2×C4—612no—
528C2×C4—612no—
529C2×C4—618no—
530C2×C4—611no—
531C2×C4—611no—
532C2×C4—611no—
533C2×C4—1222no—
534C2×C4—1222no—
535C2×C4—1222no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—1224no—
546C2×C4—1224no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—1222no—
550C2×C4—1224no—
551C2×C4—1224no—
552C2×C4—1224no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4—2448no—
563C2×C4—2448no—
564C2×C4C2×C4×#692311yescharacteristic
565C2×C4C2×C4×#692311yescharacteristic
566C2×C4C2×C4×#692311yescharacteristic
567C2×C4C23×#692311yescharacteristic
568C8—24no—
569C8—1228no—
570C8—1224no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—44no—
580C22—612no—
581C22—612no—
582C22—611no—
583C22—611no—
584C22—612no—
585C22—1224no—
586C22—1224no—
587C22—1221no—
588C22—1221no—
589C22—1221no—
590C22—1221no—
591C22—1221no—
592C22—1221no—
593C22—1221no—
594C22—1221no—
595C22—2444no—
596C22—2442no—
597C22—2442no—
598C22—2442no—
599C22—2442no—
600C22—2444no—
601C22—2444no—
602C22M16×#692312yes—
603C22C22×C4×#692311yescharacteristic
604C4—21no—
605C4—21no—
606C4—21no—
607C4—612no—
608C4—1228no—
609C4—1221no—
610C4—1221no—
611C4—1221no—
612C4—2444no—
613C4#1684082912yes—
614C2—22no—
615C2—41no—
616C2—41no—
617C2—612no—
618C2—612no—
619C2—1222no—
620C2—2441no—
621C2—2441no—
622C2#2146225511yescharacteristic
623C2C2×M16×#692311yescharacteristic
624C2#2146238211yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×#6923
commutator, socleC22×C61C22×C42
FittingC61×#407C4
prime radicalC24×D61C2×C4
cosocleC2×C4×D61C24
hypercenter#407#6923
hypocenterC61#20524
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
3. #21461562C2semidirect
4. #21461559C2semidirect
5. #21461547C2semidirect
6. C22⋊C8×#6923C2semidirect
12. #16840734C4semidirect
13. #16840733C4semidirect
14. #16840733C4semidirect
15. #16840732C4semidirect
18. C2×#14519831C22semidirect
19. C2×C8×#6923C22semidirect
25. (C2×#6923)⋊C8C22semidirect
26. (C2×#6923)⋊C8C22semidirect
29. #16839726C4semidirect
30. #16839725C4semidirect
31. #16839724C4semidirect
32. #16839724C4semidirect
35. C61×#407C4semidirect
38. C2×#11758213C2×C4semidirect
41. C2×#11758217C2×C4semidirect
53. #14519834C2×C4semidirect
55. C22⋊#92420C2×C4semidirect
76. C2×#11758194C2×C4semidirect
77. C2×#11758189C2×C4semidirect
82. C22⋊#92418C2×C4semidirect
84. #14519689C2×C4semidirect
86. C2×M16×C61C2×C4semidirect
88. C22⋊C8×C61C2×C4semidirect
105. C22×#6923M16semidirect
334. #407#6923semidirect
348. C61#20524semidirect
376. C2×M16C2×#6923semidirect
397. C22⋊C8C2×#6923semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4072
C2×C4×C22⋊#69232
#214615622
#214615592
#214615472
C22⋊C8×#69232
#407×D612
#2052461