Finite Group Databasebeta

← #30620530

Subgroups — #30620530

Order 15616.

1212 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840781C211yesmaximal, characteristic, complemented
2C2×#16840733C211yesmaximal, characteristic
3#21461893C212yesmaximal, complemented
4#21461888C212yesmaximal, complemented
5#21461765C212yesmaximal, complemented
6#21461711C211yesmaximal, characteristic, complemented
7#21461674C211yesmaximal, characteristic, complemented
8#21461642C211yesmaximal, characteristic
9#21461560C212yesmaximal, complemented
10#21461546C211yesmaximal, characteristic, complemented
11C4×C22⋊C4×D61C211yesmaximal, characteristic
12C22×#11758213C2212yes—
13#16840781—24no—
14C2×C4⋊#92420C2212yescomplemented
15C2×C4⋊#92420C2212yescomplemented
16#16840733—24no—
17C2×#14519834C2212yes—
18C2×#14519834C2211yescharacteristic
19C2×#14519834C2211yescharacteristic
20C2×#14519834C2211yescharacteristic
21C2×#14519834C2211yescharacteristic
22C2×C22⋊#92420C2212yes—
23C2×C22⋊#92420C2212yes—
24C2×#14519832C2212yescomplemented
25C2×C4×#92420C2212yescomplemented
26#16840628C2212yescomplemented
27#16840628C2212yescomplemented
28#16840628C2212yescomplemented
29#16840628C2212yescomplemented
30C23×C4×D61C2211yescharacteristic
31C2×C22⋊C4×D61C411yescharacteristic
32C2×C22⋊C4×D61C411yescharacteristic
33C2×C42×D61C2211yescharacteristic
34C2×C42×D61C2211yescharacteristic
35C22⋊C4×Dic61C2211yescharacteristic
36C22⋊C4×Dic61C2211yescharacteristic
37C4×#11758198C411yescharacteristic
38C4×#11758198C411yescharacteristic
39C4×C22⋊Dic61C411yescharacteristic
40#16839609C2211yescharacteristic
41#16839609C2211yescharacteristic
42#95×D61C411yescharacteristic
43#95×D61C411yescharacteristic
44C4×C22⋊C4×C61C411yescharacteristic
45C2×#11758213—24no—
46C2×#11758213—24no—
47C2×#11758213—42no—
48C22×#92420C2312yes—
49C22×#92420C2312yes—
50C22×#92420C2312yes—
51C22×#92420C2312yes—
52C4⋊#92420—24no—
53C4⋊#92420—42no—
54#14519834—22no—
55#14519834—22no—
56#14519834—24no—
57#14519834—22no—
58#14519834D414yes—
59C22⋊#92420—24no—
60C22⋊#92420—42no—
61#14519832—42no—
62C4×#92420—24no—
63C24×D61C2×C411yescharacteristic
64C23×Dic61C2311yescharacteristic
65C22×C4×D61—22no—
66C22×C4×D61—24no—
67C22×C4×D61—22no—
68C22×C4×D61—42no—
69C22×C4×D61C2×C411yescharacteristic
70C22×C4×D61C2×C411yescharacteristic
71C22×C4×D61C2×C411yescharacteristic
72C22×C4×D61C2×C411yescharacteristic
73C22×C4×D61C2×C411yescharacteristic
74C22×C4×D61C2×C411yescharacteristic
75C22×C4×D61C2311yescharacteristic
76C22×C4×D61C2311yescharacteristic
77C2×C22⋊Dic61C2×C411yescharacteristic
78C2×C22⋊Dic61C2×C411yescharacteristic
79C2×#11758198C2×C411yescharacteristic
80C2×#11758198C2×C411yescharacteristic
81C2×#11758198C2×C411yescharacteristic
82C2×#11758198C2×C411yescharacteristic
83C22⋊C4×D61—24no—
84C22⋊C4×D61—24no—
85C2×C4×Dic61C2×C411yescharacteristic
86C2×C4×Dic61C2×C411yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89C2×C4×Dic61C2311yescharacteristic
90C2×C4×Dic61C2311yescharacteristic
91C42×D61—22no—
92C42×D61—22no—
93C42×D61—42no—
94#14519684C2×C411yescharacteristic
95#14519684C2×C411yescharacteristic
96#14519684C2×C411yescharacteristic
97#14519684C2×C411yescharacteristic
98#14519684C2×C411yescharacteristic
99#14519684C2×C411yescharacteristic
100C23×C4×C61C2×C411yescharacteristic
101C2×C22⋊C4×C61C2×C411yescharacteristic
102C2×C22⋊C4×C61C2×C411yescharacteristic
103C2×C42×C61C2×C411yescharacteristic
104C2×C42×C61C2×C411yescharacteristic
105#95×C61C2×C411yescharacteristic
106#95×C61C2×C411yescharacteristic
107C24×C61C22×C411yescharacteristic
108C23×D61—22no—
109C23×D61—22no—
110C23×D61—22no—
111C23×D61—22no—
112C23×D61—22no—
113C23×D61—41no—
114C23×D61—41no—
115C23×D61C22×C411yescharacteristic
116C23×D61C22×C411yescharacteristic
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—41no—
120C22×C4×C61C22×C411yescharacteristic
121C22×C4×C61C22×C411yescharacteristic
122C22×C4×C61C22×C411yescharacteristic
123C22×C4×C61C22×C411yescharacteristic
124C22×C4×C61C22×C411yescharacteristic
125C22×C4×C61C22×C411yescharacteristic
126C22×Dic61—21no—
127C22×Dic61—22no—
128C22×Dic61—21no—
129C22×Dic61—22no—
130C22×Dic61C22×C411yescharacteristic
131C22×Dic61C22×C411yescharacteristic
132C22×Dic61C22×C411yescharacteristic
133C22×Dic61C22×C411yescharacteristic
134C22×Dic61C22×C411yescharacteristic
135C22×Dic61C2411yescharacteristic
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—21no—
139C2×C4×D61—22no—
140C2×C4×D61—21no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61—22no—
151C2×C4×D61—22no—
152C2×C4×D61—48no—
153C2×C4×D61—48no—
154C2×C4×D61—41no—
155C2×C4×D61—41no—
156C2×C4×D61—41no—
157C2×C4×D61—41no—
158C2×C4×D61—41no—
159C2×C4×D61—41no—
160C2×C4×D61M1614yescomplemented
161C2×C4×D61C2×D412yes—
162C2×C4×D61#4114yes—
163C2×C4×D61#4112yes—
164C2×#92420—22no—
165C2×#92420—22no—
166C2×#92420—22no—
167C2×#92420—22no—
168C2×#92420—22no—
169C2×#92420—22no—
170C2×#92420—22no—
171C2×#92420—22no—
172C2×#92420—22no—
173C2×#92420—22no—
174C2×#92420—22no—
175C2×#92420C2×D414yes—
176#11758213—48no—
177C22⋊C4×C61—22no—
178C22⋊C4×C61M1614yescomplemented
179C42×C61—22no—
180C42×C61—41no—
181C22⋊Dic61—22no—
182C22⋊Dic61—22no—
183#11758198—22no—
184#11758198—22no—
185#11758198—22no—
186#11758198—22no—
187C4×Dic61—22no—
188C4×Dic61—21no—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61—22no—
193C4×Dic61—22no—
194C4×Dic61—41no—
195C23×C61—21no—
196C23×C61—21no—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61—21no—
200C23×C61—21no—
201C23×C61C2×M1612yes—
202C23×C61C23×C411yescharacteristic
203C22×D61—21no—
204C22×D61—21no—
205C22×D61—21no—
206C22×D61—21no—
207C22×D61—28no—
208C22×D61—21no—
209C22×D61—21no—
210C22×D61—21no—
211C22×D61—21no—
212C22×D61—21no—
213C22×D61—21no—
214C22×D61—44no—
215C22×D61—44no—
216C22×D61—44no—
217C22×D61—44no—
218C22×D61—44no—
219C22×D61—44no—
220C22×D61#11712yes—
221C22×D61C4×D412yes—
222C22×D61C2×M1612yes—
223C22×D61#13112yes—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—44no—
238C2×C4×C61#11712yes—
239C2×C4×C61C4×D412yes—
240C2×C4×C61C2×M1612yes—
241C2×C4×C61C2×M1612yes—
242C2×C4×C61C2×M1612yes—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—42no—
258C2×Dic61—42no—
259C2×Dic61C2×M1612yes—
260C2×Dic61C22×D411yescharacteristic
261C2×Dic61C2×#4112yes—
262C2×Dic61C2×#4111yescharacteristic
263C2×Dic61H(5,𝔽2)12yes—
264C4×D61—28no—
265C4×D61—24no—
266C4×D61—24no—
267C4×D61—24no—
268C4×D61—24no—
269C4×D61—44no—
270C4×D61—44no—
271C4×D61—44no—
272C4×D61—44no—
273C4×D61—44no—
274#92420—24no—
275#92420—42no—
276#92420—42no—
277#92420—42no—
278#22553—611nomaximal, Sylow
279C22×C61—22no—
280C22×C61—22no—
281C22×C61—22no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61C2×#11711yescharacteristic
287C22×C61C2×C4×D411yescharacteristic
288C22×C61#51811yescharacteristic
289C22×C61C22×M1611yescharacteristic
290C22×C61C22×M1611yescharacteristic
291C22×C61C2×#13111yescharacteristic
292C22×C61#56811yescharacteristic
293C2×D61—22no—
294C2×D61—22no—
295C2×D61—22no—
296C2×D61—22no—
297C2×D61—22no—
298C2×D61—22no—
299C2×D61—22no—
300C2×D61—22no—
301C2×D61—22no—
302C2×D61—416no—
303C2×D61—41no—
304C2×D61—41no—
305C2×D61—41no—
306C2×D61—41no—
307C2×D61#43514yes—
308C4×C61—22no—
309C4×C61—22no—
310C4×C61—22no—
311C4×C61—22no—
312C4×C61—22no—
313C4×C61—41no—
314Dic61—22no—
315Dic61—41no—
316Dic61—41no—
317Dic61—41no—
318Dic61—41no—
319Dic61#54612yes—
320Dic61#55112yes—
321C2×#431—611no—
322C2×#407—611no—
323C2×C4×C22⋊C4—611no—
324#1924—612no—
325#1923—612no—
326#1851—612no—
327#1817—611no—
328#1797—611no—
329#1779—611no—
330#1753—612no—
331#1749—611no—
332C2×C61—42no—
333C2×C61#234211yescharacteristic
334C2×C61#292011yescharacteristic
335C2×C61#292411yescharacteristic
336C2×C61C2×#43511yescharacteristic
337C2×C61D4×M1611yescharacteristic
338C2×C61#297411yescharacteristic
339C2×C61#297511yescharacteristic
340D61—24no—
341D61—42no—
342C24×C4—611no—
343C22×M16—612no—
344C22×C22⋊C4—611no—
345C22×C22⋊C4—611no—
346C22×C42—611no—
347C22×C42—611no—
348#431—1224no—
349C2×C4⋊C8—612no—
350C2×C4⋊C8—612no—
351#407—1224no—
352C2×C22⋊C8—611no—
353C2×C22⋊C8—611no—
354C2×C22⋊C8—611no—
355C2×C22⋊C8—611no—
356C2×C22⋊C8—612no—
357C2×C22⋊C8—612no—
358C2×C22⋊C8—612no—
359C2×#97—612no—
360C2×C4×C8—612no—
361C4×C22⋊C4—611no—
362C4×C22⋊C4—611no—
363C4×C22⋊C4—611no—
364C4×C22⋊C4—611no—
365C4×C22⋊C4—611no—
366C4×C22⋊C4—611no—
367C4×C22⋊C4—611no—
368C4×C22⋊C4#692311yescharacteristic
369C2×#95—611no—
370C2×#95—611no—
371#336—612no—
372#336—612no—
373#336—612no—
374#336—612no—
375C61#2255311yescharacteristic, complemented, Sylow
376C25—611no—
377C23×C4—611no—
378C23×C4—611no—
379C23×C4—611no—
380C23×C4—611no—
381C23×C4—611no—
382C23×C4—611no—
383C23×C4—611no—
384C23×C4—611no—
385C23×C4—611no—
386C23×C4—1224no—
387C23×C4—1222no—
388C23×C4—1222no—
389C23×C4—2442no—
390C23×C4C2×#692311yescharacteristic
391C2×M16—1224no—
392C2×M16—1224no—
393C2×M16—2442no—
394C22×C8—612no—
395C22×C8—612no—
396C22×C8—612no—
397C22×C8—612no—
398C2×C22⋊C4—611no—
399C2×C22⋊C4—611no—
400C2×C22⋊C4—611no—
401C2×C22⋊C4—611no—
402C2×C22⋊C4—611no—
403C2×C22⋊C4—611no—
404C2×C22⋊C4—1224no—
405C2×C22⋊C4—1224no—
406C2×C22⋊C4C2×#692311yescharacteristic
407C2×C22⋊C4C2×#692311yescharacteristic
408C2×C42—611no—
409C2×C42—611no—
410C2×C42—611no—
411C2×C42—611no—
412C2×C42—611no—
413C2×C42—611no—
414C2×C42—1222no—
415C2×C42—1222no—
416C2×C42—2442no—
417C2×C42C2×#692311yescharacteristic
418C2×C42C2×#692311yescharacteristic
419C4⋊C8—1224no—
420C4⋊C8—2442no—
421C22⋊C8—614no—
422C22⋊C8—1224no—
423C22⋊C8—1224no—
424C22⋊C8—1222no—
425C22⋊C8—1222no—
426C22⋊C8—1222no—
427C22⋊C8—2442no—
428#97—2442no—
429C4×C8—1224no—
430#95—611no—
431#95—611no—
432#95—611no—
433#95—611no—
434#95—611no—
435#95—611no—
436#95C2×#692311yescharacteristic
437#95C2×#692311yescharacteristic
438C24—611no—
439C24—611no—
440C24—1222no—
441C24—1222no—
442C24—1222no—
443C24—1222no—
444C24—1222no—
445C24—2441no—
446C24—2441no—
447C24C22×#692311yescharacteristic
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—41no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—614no—
454C22×C4—612no—
455C22×C4—612no—
456C22×C4—614no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—1222no—
462C22×C4—1221no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—1221no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4—1221no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4—1221no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—2441no—
482C22×C4—2441no—
483C22×C4—2441no—
484C22×C4—2441no—
485C22×C4—2441no—
486C22×C4—2441no—
487C22×C4—2448no—
488C22×C4—2448no—
489C22×C4C22×#692311yescharacteristic
490C22×C4C22×#692311yescharacteristic
491C22×C4C22×#692311yescharacteristic
492C22×C4C22×#692311yescharacteristic
493C22×C4C22×#692311yescharacteristic
494C22×C4C22×#692311yescharacteristic
495M16—2448no—
496C2×C8—614no—
497C2×C8—1222no—
498C2×C8—1222no—
499C2×C8—1222no—
500C2×C8—1222no—
501C2×C8—1222no—
502C2×C8—1222no—
503C2×C8—1222no—
504C2×C8—1222no—
505C2×C8—1222no—
506C2×C8—1222no—
507C2×C8—1222no—
508C22⋊C4—22no—
509C22⋊C4—1222no—
510C22⋊C4—1222no—
511C22⋊C4—1222no—
512C22⋊C4—1222no—
513C22⋊C4—1222no—
514C22⋊C4—1222no—
515C22⋊C4#1175821314yescomplemented
516C42—22no—
517C42—41no—
518C42—1221no—
519C42—1222no—
520C42—1221no—
521C42—1222no—
522C42—1221no—
523C42—1221no—
524C42—1222no—
525C42—2441no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—612no—
533C23—612no—
534C23—612no—
535C23—612no—
536C23—1228no—
537C23—1221no—
538C23—1221no—
539C23—1221no—
540C23—1221no—
541C23—1221no—
542C23—1221no—
543C23—1221no—
544C23—1221no—
545C23—1221no—
546C23—1221no—
547C23—2444no—
548C23—2444no—
549C23—2444no—
550C23—2444no—
551C23—2444no—
552C23—2444no—
553C23C2×#1175821312yes—
554C23C23×#692311yescharacteristic
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—44no—
569C2×C4—612no—
570C2×C4—611no—
571C2×C4—611no—
572C2×C4—612no—
573C2×C4—612no—
574C2×C4—1224no—
575C2×C4—1224no—
576C2×C4—1224no—
577C2×C4—1224no—
578C2×C4—1228no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—2444no—
594C2×C4—2444no—
595C2×C4—2444no—
596C2×C4—2444no—
597C2×C4—2444no—
598C2×C4—2442no—
599C2×C4—2442no—
600C2×C4#1451985612yes—
601C2×C4D4×#692312yes—
602C2×C4D61⋊M1612yes—
603C2×C4C2×#1175821312yes—
604C2×C4C2×#1175821312yes—
605C8—1224no—
606C8—2442no—
607C8—2442no—
608C8—2442no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—614no—
617C22—1222no—
618C22—1222no—
619C22—1222no—
620C22—1222no—
621C22—1222no—
622C22—1222no—
623C22—1222no—
624C22—1222no—
625C22—1222no—
626C22—24416no—
627C22—2441no—
628C22—2441no—
629C22—2441no—
630C22—2441no—
631C22C2×#1451985611yescharacteristic
632C22C2×D4×#692311yescharacteristic
633C22#1684082911yescharacteristic
634C22C2×D61⋊M1611yescharacteristic
635C22C22×#1175821311yescharacteristic
636C22D61⋊#13111yescharacteristic
637C22#1684084111yescharacteristic
638C4—22no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—22no—
643C4—41no—
644C4—612no—
645C4—612no—
646C4—1222no—
647C4—2441no—
648C4—2441no—
649C4—2441no—
650C4—2441no—
651C2—42no—
652C2—1224no—
653C2—2442no—
654C2#2146232511yescharacteristic
655C2#2146249611yescharacteristic
656C2#2146251311yescharacteristic
657C2D61⋊#43511yescharacteristic
658C2D4×#1175821311yescharacteristic
659C2#2146267911yescharacteristic
660C2#2146268211yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC4×C22⋊C4×C61C4
prime radicalC24×D61C2×C4
cosocleC22×Dic61C24
hypercenterC4×C22⋊C4#6923
hypocenterC61#22553
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#16840781C2semidirect
3. #21461893C2semidirect
4. #21461888C2semidirect
5. #21461765C2semidirect
6. #21461711C2semidirect
7. #21461674C2semidirect
9. #21461560C2semidirect
10. #21461546C2semidirect
14. C2×C4⋊#92420C22semidirect
15. C2×C4⋊#92420C22semidirect
24. C2×#14519832C22semidirect
25. C2×C4×#92420C22semidirect
26. #16840628C22semidirect
27. #16840628C22semidirect
28. #16840628C22semidirect
29. #16840628C22semidirect
160. C2×C4×D61M16semidirect
178. C22⋊C4×C61M16semidirect
375. C61#22553semidirect
515. C22⋊C4#11758213semidirect

Maximal subgroups

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

subgroupindex
C2×#168407812
C2×#168407332
#214618932
#214618882
#214617652
#214617112
#214616742
#214616422
#214615602
#214615462
C4×C22⋊C4×D612
#2255361