Finite Group Databasebeta

← #29602717

Subgroups — #29602717

Order 15616.

2078 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519733C211yesmaximal, characteristic
2C2×#16839623C211yesmaximal, characteristic
3C22×#14519684C211yesmaximal, characteristic
4#21444085C212yesmaximal, complemented
5#21444085C212yesmaximal, complemented
6#21444045C212yesmaximal, complemented
7#21444044C212yesmaximal, complemented
8#21444042C211yesmaximal, characteristic
9#21444041C211yesmaximal, characteristic
10#21444041C211yesmaximal, characteristic
11C61×C22⋊#95C211yesmaximal, characteristic
12C24×Dic61C2211yescharacteristic
13C24×Dic61C2211yescharacteristic
14C2×#14519733—24no—
15C2×#14519733—24no—
16C2×#14519733C418yescomplemented
17C22×#11758196C2211yescharacteristic
18C22×#11758196C2211yescharacteristic
19C22×C22⋊Dic61C2211yescharacteristic
20C22×C22⋊Dic61C2211yescharacteristic
21C22×C4×Dic61C2211yescharacteristic
22C22×C4×Dic61C2211yescharacteristic
23#16839623—24no—
24C2×#14519684—24no—
25C2×#14519684C2212yes—
26C2×#14519684C2212yes—
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2212yes—
32C2×#14519684C2212yes—
33C2×#14519684C2212yes—
34C2×#14519684C2212yes—
35C24×C4×C61C2211yescharacteristic
36C22×C22⋊C4×C61C2211yescharacteristic
37C22×C22⋊C4×C61C2211yescharacteristic
38C2×#95×C61C2212yes—
39C2×#95×C61C2212yes—
40C23×Dic61—22no—
41C23×Dic61—22no—
42C23×Dic61—22no—
43C23×Dic61—22no—
44C23×Dic61—22no—
45C23×Dic61—24no—
46C23×Dic61—22no—
47C23×Dic61—22no—
48C23×Dic61—42no—
49C23×Dic61C2×C414yes—
50C23×Dic61D414yes—
51C23×Dic61D414yes—
52C23×Dic61C2312yes—
53C23×Dic61C2311yescharacteristic
54C23×Dic61C2311yescharacteristic
55C23×Dic61C2312yes—
56C23×Dic61C2311yescharacteristic
57C23×Dic61C2311yescharacteristic
58#14519733—232no—
59C2×#11758196—22no—
60C2×#11758196—22no—
61C2×#11758196—22no—
62C2×#11758196—22no—
63C2×#11758196C2×C414yescomplemented
64C2×#11758196C2×C414yescomplemented
65C2×C22⋊Dic61—22no—
66C2×C22⋊Dic61—22no—
67C2×C22⋊Dic61—22no—
68C2×C22⋊Dic61—22no—
69C2×C22⋊Dic61—22no—
70C2×C22⋊Dic61C2×C414yes—
71C2×C4×Dic61—22no—
72C2×C4×Dic61—22no—
73C2×C4×Dic61—22no—
74C2×C4×Dic61—22no—
75C2×C4×Dic61C2×C414yescomplemented
76C2×C4×Dic61C2×C414yescomplemented
77#14519684—24no—
78#14519684—24no—
79#14519684—24no—
80#14519684—24no—
81#14519684—24no—
82#14519684—24no—
83#14519684—24no—
84#14519684—42no—
85#14519684—42no—
86#14519684—42no—
87C25×C61C2311yescharacteristic
88C23×C4×C61—22no—
89C23×C4×C61—24no—
90C23×C4×C61—22no—
91C23×C4×C61—22no—
92C23×C4×C61—42no—
93C23×C4×C61C2312yes—
94C23×C4×C61C2311yescharacteristic
95C23×C4×C61C2311yescharacteristic
96C23×C4×C61C2312yes—
97C2×C22⋊C4×C61—22no—
98C2×C22⋊C4×C61—22no—
99C2×C22⋊C4×C61—22no—
100C2×C22⋊C4×C61—22no—
101C2×C22⋊C4×C61—22no—
102C2×C22⋊C4×C61C2×C414yes—
103#95×C61—24no—
104#95×C61—42no—
105C24×C61—22no—
106C24×C61—22no—
107C24×C61—22no—
108C24×C61—22no—
109C24×C61C22×C412yes—
110C24×C61C2×D412yes—
111C24×C61C2×D412yes—
112C24×C61#4112yes—
113C24×C61#4112yes—
114C24×C61#4112yes—
115C24×C61#4112yes—
116C24×C61C2411yescharacteristic
117C22×C4×C61—28no—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61—48no—
140C22×C4×C61—48no—
141C22×C4×C61—44no—
142C22×C4×C61—44no—
143C22×C4×C61C22×C412yes—
144C22×C4×C61C22×C412yes—
145C22×C4×C61#4114yes—
146C22×Dic61—28no—
147C22×Dic61—28no—
148C22×Dic61—28no—
149C22×Dic61—28no—
150C22×Dic61—28no—
151C22×Dic61—28no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—28no—
161C22×Dic61—22no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61—21no—
165C22×Dic61—21no—
166C22×Dic61—22no—
167C22×Dic61—22no—
168C22×Dic61—22no—
169C22×Dic61—22no—
170C22×Dic61—22no—
171C22×Dic61—22no—
172C22×Dic61—22no—
173C22×Dic61—22no—
174C22×Dic61—22no—
175C22×Dic61—22no—
176C22×Dic61—22no—
177C22×Dic61—22no—
178C22×Dic61—22no—
179C22×Dic61—22no—
180C22×Dic61—22no—
181C22×Dic61—22no—
182C22×Dic61—48no—
183C22×Dic61—48no—
184C22×Dic61—44no—
185C22×Dic61—44no—
186C22×Dic61C22⋊C4116yes—
187C22×Dic61C22×C412yes—
188C22×Dic61C22×C412yes—
189C22×Dic61C22×C412yes—
190C22×Dic61C22×C412yes—
191C22×Dic61C2×D412yes—
192C22×Dic61C2×D412yes—
193C22×Dic61C2×D412yes—
194C22×Dic61C2×D412yes—
195C22×Dic61#4114yes—
196C22⋊C4×C61—28no—
197C22⋊C4×C61—44no—
198C22⋊Dic61—28no—
199C22⋊Dic61—44no—
200#11758196—28no—
201#11758196—28no—
202C4×Dic61—28no—
203C4×Dic61—28no—
204C23×C61—24no—
205C23×C61—24no—
206C23×C61—22no—
207C23×C61—22no—
208C23×C61—24no—
209C23×C61—22no—
210C23×C61—22no—
211C23×C61—24no—
212C23×C61—24no—
213C23×C61—24no—
214C23×C61—24no—
215C23×C61—24no—
216C23×C61—24no—
217C23×C61—24no—
218C23×C61—24no—
219C23×C61—24no—
220C23×C61C2×C22⋊C414yes—
221C23×C61#11714yes—
222C23×C61#11714yes—
223C23×C61#12314yes—
224C23×C61#12314yes—
225C23×C61#12314yes—
226C23×C61#12314yes—
227C23×C61C23×C411yescharacteristic
228C23×C61C22×D411yescharacteristic
229C23×C61C22×D411yescharacteristic
230C23×C61C2×#4111yescharacteristic
231C23×C61C2×#4111yescharacteristic
232C23×C61C2×#4111yescharacteristic
233C23×C61C2×#4111yescharacteristic
234C23×C61C2×#4112yes—
235C23×C61C2×#4112yes—
236C23×C61H(5,𝔽2)12yes—
237C23×C612−1+412yes—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—24no—
241C2×C4×C61—24no—
242C2×C4×C61—24no—
243C2×C4×C61—24no—
244C2×C4×C61—24no—
245C2×C4×C61—416no—
246C2×C4×C61—42no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—42no—
250C2×C4×C61—41no—
251C2×C4×C61—42no—
252C2×C4×C61—41no—
253C2×C4×C61—42no—
254C2×C4×C61—42no—
255C2×C4×C61—42no—
256C2×C4×C61—42no—
257C2×C4×C61—42no—
258C2×C4×C61—42no—
259C2×C4×C61—42no—
260C2×C4×C61—42no—
261C2×C4×C61—42no—
262C2×C4×C61—42no—
263C2×Dic61—22no—
264C2×Dic61—22no—
265C2×Dic61—22no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—22no—
274C2×Dic61—22no—
275C2×Dic61—232no—
276C2×Dic61—22no—
277C2×Dic61—22no—
278C2×Dic61—24no—
279C2×Dic61—24no—
280C2×Dic61—24no—
281C2×Dic61—24no—
282C2×Dic61—24no—
283C2×Dic61—416no—
284C2×Dic61—42no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—42no—
288C2×Dic61—41no—
289C2×Dic61—42no—
290C2×Dic61—41no—
291C2×Dic61—42no—
292C2×Dic61—42no—
293C2×Dic61—42no—
294C2×Dic61—42no—
295C2×Dic61—42no—
296C2×Dic61—42no—
297C2×Dic61—42no—
298C2×Dic61—42no—
299C2×Dic61—42no—
300C2×Dic61—42no—
301C2×Dic61C2×C22⋊C414yes—
302C2×Dic61C2×C22⋊C414yes—
303#13958—611nomaximal, Sylow
304C22×C61—24no—
305C22×C61—28no—
306C22×C61—28no—
307C22×C61—24no—
308C22×C61—24no—
309C22×C61—24no—
310C22×C61—24no—
311C22×C61—28no—
312C22×C61—28no—
313C22×C61—24no—
314C22×C61#38118yes—
315C22×C61C22×C22⋊C411yescharacteristic
316C22×C61C2×#11711yescharacteristic
317C22×C61C2×#11711yescharacteristic
318C22×C61C4×#4112yes—
319C22×C61#52012yes—
320C22×C61C2×#12311yescharacteristic
321C22×C61C2×#12311yescharacteristic
322C22×C61C2×#12311yescharacteristic
323C22×C61C2×#12311yescharacteristic
324C22×C61#52511yescharacteristic
325C22×C61#52511yescharacteristic
326C22×C61#53212yes—
327C22×C61#53511yescharacteristic
328C22×C61#53611yescharacteristic
329C22×C61#53712yes—
330C22×C61#55112yes—
331C22×C61#55112yes—
332C22×C61#55112yes—
333C22×C61#55112yes—
334C22×C61#55212yes—
335C22×C61#55212yes—
336C22×C61#55212yes—
337C22×C61#55212yes—
338C4×C61—42no—
339C4×C61—44no—
340C4×C61—42no—
341C4×C61—82no—
342Dic61—24no—
343Dic61—24no—
344Dic61—42no—
345Dic61—44no—
346Dic61—42no—
347Dic61—82no—
348C22×#117—611no—
349C2×#381—611no—
350C22×#95—611no—
351#1445—612no—
352#1445—612no—
353#1434—612no—
354#1434—612no—
355C22⋊#95—611no—
356C22⋊#95—611no—
357C22⋊#95—611no—
358C22⋊#95D6111yescharacteristic
359C2×C61—28no—
360C2×C61C2×#38111yescharacteristic
361C2×C61#229711yescharacteristic
362C2×C61#231111yescharacteristic
363C2×C61#234312yes—
364C2×C61#234312yes—
365C2×C61#248211yescharacteristic
366C2×C61#248211yescharacteristic
367C2×C61#248411yescharacteristic
368C2×C61#248411yescharacteristic
369C2×C61#249812yes—
370C2×C61#249812yes—
371C24×C4—611no—
372C24×C4—611no—
373C24×C4C2×D6111yescharacteristic
374C2×#117—618no—
375C2×#117—1224no—
376C2×#117—1224no—
377C22×C4⋊C4—611no—
378C22×C4⋊C4—611no—
379C22×C22⋊C4—611no—
380C22×C22⋊C4—611no—
381C22×C22⋊C4C2×D6111yescharacteristic
382C22×C22⋊C4C2×D6111yescharacteristic
383C22×C42—611no—
384C22×C42—611no—
385#381—1224no—
386C2×#95—612no—
387C2×#95—612no—
388C2×#95—612no—
389C2×#95—612no—
390C2×#95—612no—
391C2×#95—612no—
392C2×#95—612no—
393C2×#95—612no—
394C2×#95—612no—
395C2×#95—612no—
396C2×#95—1224no—
397C2×#95C2×D6112yes—
398C2×#95C2×D6112yes—
399C61#1395811yescharacteristic, complemented, Sylow
400C25C22×D6111yescharacteristic
401C23×C4—22no—
402C23×C4—24no—
403C23×C4—22no—
404C23×C4—22no—
405C23×C4—42no—
406C23×C4—614no—
407C23×C4—614no—
408C23×C4—614no—
409C23×C4—611no—
410C23×C4—611no—
411C23×C4—611no—
412C23×C4—611no—
413C23×C4—612no—
414C23×C4—612no—
415C23×C4—1224no—
416C23×C4—1222no—
417C23×C4—1222no—
418C23×C4—1222no—
419C23×C4—1222no—
420C23×C4—1222no—
421C23×C4—1222no—
422C23×C4—1222no—
423C23×C4—2442no—
424C23×C4C22×D6112yes—
425C23×C4C22×D6111yescharacteristic
426C23×C4C22×D6111yescharacteristic
427C23×C4C22×D6112yes—
428#117—12232no—
429C2×C4⋊C4—614no—
430C2×C4⋊C4—614no—
431C2×C4⋊C4—1222no—
432C2×C4⋊C4—1222no—
433C2×C4⋊C4—1222no—
434C2×C4⋊C4—1222no—
435C2×C22⋊C4—22no—
436C2×C22⋊C4—22no—
437C2×C22⋊C4—22no—
438C2×C22⋊C4—22no—
439C2×C22⋊C4—22no—
440C2×C22⋊C4—614no—
441C2×C22⋊C4—1222no—
442C2×C22⋊C4—1222no—
443C2×C22⋊C4—1222no—
444C2×C22⋊C4—1222no—
445C2×C22⋊C4—1222no—
446C2×C22⋊C4C4×D6114yes—
447C2×C42—614no—
448C2×C42—614no—
449C2×C42—1222no—
450C2×C42—1222no—
451C2×C42—1222no—
452C2×C42—1222no—
453#95—24no—
454#95—42no—
455#95—1224no—
456#95—1224no—
457#95—1224no—
458#95—1224no—
459#95—1224no—
460#95—1224no—
461#95—1224no—
462#95—2442no—
463#95—2442no—
464#95—2442no—
465C24—22no—
466C24—22no—
467C24—22no—
468C24—22no—
469C24C2×C4×D6112yes—
470C24#1175822212yes—
471C24#1175822212yes—
472C24D4×D6112yes—
473C24D4×D6112yes—
474C24#1175822412yes—
475C24#1175822612yes—
476C24C23×D6111yescharacteristic
477C22×C4—28no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—48no—
500C22×C4—48no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—612no—
504C22×C4—612no—
505C22×C4—612no—
506C22×C4—612no—
507C22×C4—612no—
508C22×C4—612no—
509C22×C4—6116no—
510C22×C4—612no—
511C22×C4—612no—
512C22×C4—614no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—1221no—
518C22×C4—1221no—
519C22×C4—1221no—
520C22×C4—1221no—
521C22×C4—1228no—
522C22×C4—1228no—
523C22×C4—1228no—
524C22×C4—1228no—
525C22×C4—1228no—
526C22×C4—1228no—
527C22×C4—1228no—
528C22×C4—1222no—
529C22×C4—1221no—
530C22×C4—1221no—
531C22×C4—1221no—
532C22×C4—1221no—
533C22×C4—1222no—
534C22×C4—1222no—
535C22×C4—1222no—
536C22×C4—1222no—
537C22×C4—1222no—
538C22×C4—1222no—
539C22×C4—1222no—
540C22×C4—1222no—
541C22×C4—1222no—
542C22×C4—1222no—
543C22×C4—1222no—
544C22×C4—1222no—
545C22×C4—1222no—
546C22×C4—1222no—
547C22×C4—1222no—
548C22×C4—1222no—
549C22×C4—2448no—
550C22×C4—2448no—
551C22×C4—2444no—
552C22×C4—2444no—
553C22×C4C2×C4×D6112yes—
554C22×C4C2×C4×D6112yes—
555C22×C4#1175822414yes—
556C4⋊C4—1228no—
557C4⋊C4—1228no—
558C22⋊C4—28no—
559C22⋊C4—44no—
560C22⋊C4—1228no—
561C22⋊C4—2444no—
562C42—1228no—
563C42—1228no—
564C23—24no—
565C23—24no—
566C23—24no—
567C23—24no—
568C23—24no—
569C23—22no—
570C23—22no—
571C23—24no—
572C23—22no—
573C23—22no—
574C23—24no—
575C23—24no—
576C23—24no—
577C23—24no—
578C23—24no—
579C23—24no—
580C23C22⋊C4×D6114yes—
581C23#1451973414yes—
582C23#1451973514yes—
583C23#1451975914yes—
584C23#1451975914yes—
585C23#1451976214yes—
586C23#1451976214yes—
587C23C22×C4×D6111yescharacteristic
588C23C2×D4×D6111yescharacteristic
589C23C2×D4×D6111yescharacteristic
590C23#41×D6112yes—
591C23C2×#1175822411yescharacteristic
592C23C2×#1175822412yes—
593C23C2×#1175822211yescharacteristic
594C23C2×#1175822211yescharacteristic
595C23C2×#1175822611yescharacteristic
596C23#1451982212yes—
597C23#1451982412yes—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—416no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—614no—
624C2×C4—614no—
625C2×C4—1222no—
626C2×C4—1222no—
627C2×C4—1222no—
628C2×C4—1222no—
629C2×C4—1222no—
630C2×C4—1222no—
631C2×C4—1222no—
632C2×C4—1222no—
633C2×C4—1222no—
634C2×C4—1222no—
635C2×C4—1222no—
636C2×C4—1222no—
637C2×C4—12232no—
638C2×C4—1222no—
639C2×C4—1222no—
640C2×C4—1224no—
641C2×C4—1224no—
642C2×C4—1224no—
643C2×C4—1224no—
644C2×C4—1224no—
645C2×C4—2442no—
646C2×C4—2442no—
647C2×C4—2442no—
648C2×C4—2442no—
649C2×C4—2442no—
650C2×C4—2442no—
651C2×C4—2442no—
652C2×C4—24416no—
653C2×C4—2442no—
654C2×C4—2441no—
655C2×C4—2441no—
656C2×C4—2442no—
657C2×C4—2441no—
658C2×C4—2442no—
659C2×C4—2441no—
660C2×C4—2442no—
661C2×C4—2442no—
662C2×C4—2442no—
663C22—28no—
664C22—28no—
665C22—24no—
666C22—24no—
667C22—24no—
668C22—24no—
669C22—24no—
670C22—24no—
671C22—28no—
672C22—28no—
673C22#1683962418yes—
674C22C2×C22⋊C4×D6111yescharacteristic
675C22C2×#1451973411yescharacteristic
676C22C2×#1451973511yescharacteristic
677C22C4×#1175822412yes—
678C22#1684024112yes—
679C22C2×#1451975911yescharacteristic
680C22C2×#1451975911yescharacteristic
681C22C2×#1451976211yescharacteristic
682C22C2×#1451976211yescharacteristic
683C22#1684026811yescharacteristic
684C22#1684027011yescharacteristic
685C22#1684030712yes—
686C22#1684032711yescharacteristic
687C22#1684033411yescharacteristic
688C22#1684033912yes—
689C22#1684043812yes—
690C22#1684043912yes—
691C22#1684044112yes—
692C22#1684044312yes—
693C22#1684044512yes—
694C22#1684044912yes—
695C22#1684045212yes—
696C22#1684045412yes—
697C4—44no—
698C4—42no—
699C4—42no—
700C4—82no—
701C4—1224no—
702C4—1224no—
703C4—4882no—
704C4—2444no—
705C4—2442no—
706C4—2442no—
707C2—28no—
708C2C2×#1683962411yescharacteristic
709C2#2144830411yescharacteristic
710C2#2144840611yescharacteristic
711C2#2144867312yes—
712C2#2144867712yes—
713C2#2145009011yescharacteristic
714C2#2145009211yescharacteristic
715C2#2145010311yescharacteristic
716C2#2145010511yescharacteristic
717C2#2145025912yes—
718C2#2145026112yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×C22⋊#95C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenterC22⋊#95D61
hypocenterC61#13958
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
4. #21444085C2semidirect
5. #21444085C2semidirect
6. #21444045C2semidirect
7. #21444044C2semidirect
16. C2×#14519733C4semidirect
63. C2×#11758196C2×C4semidirect
64. C2×#11758196C2×C4semidirect
75. C2×C4×Dic61C2×C4semidirect
76. C2×C4×Dic61C2×C4semidirect
399. C61#13958semidirect

Maximal subgroups

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

subgroupindex
C22×#145197332
C2×#168396232
C22×#145196842
#214440852
#214440452
#214440442
#214440422
#214440412
C61×C22⋊#952
#1395861