Finite Group Databasebeta

← #29606927

Subgroups — #29606927

Order 15616.

1614 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839687C211yesmaximal, characteristic
2C2×#16839658C211yesmaximal, characteristic, complemented
3C2×#16839634C212yesmaximal
4C2×C22⋊C4×Dic61C211yesmaximal, characteristic
5#21444088C211yesmaximal, characteristic, complemented
6#21444082C211yesmaximal, characteristic, complemented
7#21444077C212yesmaximal, complemented
8#21444066C212yesmaximal
9#21444050C212yesmaximal, complemented
10#21444044C211yesmaximal, characteristic, complemented
11C61×#1439C211yesmaximal, characteristic
12C24×Dic61C2211yescharacteristic
13C22×C4⋊Dic61C2211yescharacteristic
14C22×#11758196C2212yes—
15C22×#11758196C2212yes—
16C22×C22⋊Dic61C2211yescharacteristic
17C22×C22⋊Dic61C2212yes—
18C22×C4×Dic61C2211yescharacteristic
19C22×C4×Dic61C2211yescharacteristic
20#16839687C418yescomplemented
21#16839658—24no—
22#16839634—28no—
23C22⋊C4×Dic61—24no—
24C2×#14519684C2212yes—
25C2×#14519684C2212yes—
26C2×#14519684C2212yes—
27C2×#14519684C2212yes—
28C2×#14519684C2211yescharacteristic
29C2×#14519684C2211yescharacteristic
30C2×#14519684C2212yes—
31C2×#14519684C2212yes—
32C2×#14519684C2211yescharacteristic
33C2×#14519684C2212yes—
34C22×C4⋊C4×C61C2211yescharacteristic
35C22×C22⋊C4×C61C2211yescharacteristic
36C22×C22⋊C4×C61C2212yes—
37C2×#95×C61C2211yescharacteristic
38C2×#95×C61C2212yes—
39C23×Dic61—22no—
40C23×Dic61—22no—
41C23×Dic61—42no—
42C23×Dic61—42no—
43C23×Dic61D414yes—
44C23×Dic61C2312yes—
45C23×Dic61C2312yes—
46C23×Dic61C2311yescharacteristic
47C23×Dic61C2311yescharacteristic
48C23×Dic61C2311yescharacteristic
49C23×Dic61C2311yescharacteristic
50C2×C4⋊Dic61—22no—
51C2×C4⋊Dic61—22no—
52C2×C4⋊Dic61C2×C414yescomplemented
53C2×#11758196—24no—
54C2×#11758196—24no—
55C2×#11758196—24no—
56C2×#11758196—24no—
57C2×#11758196—42no—
58C2×#11758196—42no—
59C2×C22⋊Dic61—24no—
60C2×C22⋊Dic61—22no—
61C2×C22⋊Dic61—24no—
62C2×C22⋊Dic61—22no—
63C2×C22⋊Dic61—22no—
64C2×C22⋊Dic61C2×C418yes—
65C2×C4×Dic61—22no—
66C2×C4×Dic61—42no—
67C2×C4×Dic61—42no—
68C2×C4×Dic61D414yes—
69#14519684—22no—
70#14519684—22no—
71#14519684—24no—
72#14519684—24no—
73#14519684—24no—
74#14519684—24no—
75#14519684—42no—
76#14519684—42no—
77#14519684C2×C414yescomplemented
78#14519684C2×C418yescomplemented
79C25×C61C2311yescharacteristic
80C23×C4×C61C2312yes—
81C23×C4×C61C2312yes—
82C23×C4×C61C2311yescharacteristic
83C23×C4×C61C2311yescharacteristic
84C2×C4⋊C4×C61—22no—
85C2×C4⋊C4×C61—42no—
86C2×C22⋊C4×C61—22no—
87C2×C22⋊C4×C61—22no—
88C2×C22⋊C4×C61—24no—
89C2×C22⋊C4×C61—42no—
90C2×C22⋊C4×C61—42no—
91C2×C22⋊C4×C61C2×C414yes—
92#95×C61—22no—
93#95×C61—42no—
94C24×C61—21no—
95C24×C61—21no—
96C24×C61—21no—
97C24×C61—21no—
98C24×C61—22no—
99C24×C61—22no—
100C24×C61—22no—
101C24×C61—22no—
102C24×C61C22×C412yes—
103C24×C61C2×D412yes—
104C24×C61#4112yes—
105C24×C61C2411yescharacteristic
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—21no—
109C22×C4×C61—21no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61C22×C412yes—
127C22×C4×C61C22×C412yes—
128C22×C4×C61C2×D412yes—
129C22×C4×C61#4112yes—
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61—21no—
133C22×Dic61—21no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—21no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61—22no—
156C22×Dic61—28no—
157C22×Dic61—48no—
158C22×Dic61—44no—
159C22×Dic61—44no—
160C22×Dic61—48no—
161C22×Dic61C22×C414yes—
162C22×Dic61C22×C414yes—
163C22×Dic61C2×D412yes—
164C22×Dic61C2×D412yes—
165C22×Dic61C2×D412yes—
166C22×Dic61C2×D412yes—
167C22×Dic61#4112yes—
168C22×Dic61#4112yes—
169C4⋊C4×C61—44no—
170C22⋊C4×C61—28no—
171C22⋊C4×C61—48no—
172C22⋊Dic61—28no—
173C22⋊Dic61—216no—
174C4⋊Dic61—28no—
175#11758196—48no—
176#11758196—48no—
177C4×Dic61—44no—
178C4×Dic61—44no—
179C23×C61—22no—
180C23×C61—22no—
181C23×C61—24no—
182C23×C61—24no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—22no—
187C23×C61—24no—
188C23×C61—24no—
189C23×C61—24no—
190C23×C61—24no—
191C23×C61—41no—
192C23×C61—41no—
193C23×C61—41no—
194C23×C61—41no—
195C23×C61—42no—
196C23×C61—42no—
197C23×C61—42no—
198C23×C61—42no—
199C23×C61—42no—
200C23×C61—42no—
201C23×C61C4×D414yes—
202C23×C61C23×C411yescharacteristic
203C23×C61C22×D411yescharacteristic
204C23×C61C22×D411yescharacteristic
205C23×C61C2×#4111yescharacteristic
206C23×C61C2×#4111yescharacteristic
207C23×C61C2×#4111yescharacteristic
208C23×C61C2×#4111yescharacteristic
209C23×C61H(5,𝔽2)12yes—
210C23×C61H(5,𝔽2)12yes—
211C23×C612−1+412yes—
212C23×C612−1+412yes—
213C2×C4×C61—24no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—24no—
217C2×C4×C61—22no—
218C2×C4×C61—24no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61C4×D414yes—
237C2×Dic61—22no—
238C2×Dic61—24no—
239C2×Dic61—22no—
240C2×Dic61—24no—
241C2×Dic61—24no—
242C2×Dic61—24no—
243C2×Dic61—24no—
244C2×Dic61—24no—
245C2×Dic61—24no—
246C2×Dic61—24no—
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—24no—
252C2×Dic61—22no—
253C2×Dic61—24no—
254C2×Dic61—416no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61—42no—
258C2×Dic61—42no—
259C2×Dic61—42no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—42no—
264C2×Dic61—41no—
265C2×Dic61—42no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61—42no—
270C2×Dic61—42no—
271C2×Dic61#12114yes—
272#14313—611nomaximal, Sylow
273C22×C61—28no—
274C22×C61—24no—
275C22×C61—42no—
276C22×C61—42no—
277C22×C61—44no—
278C22×C61—42no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61—42no—
282C22×C61—44no—
283C22×C61—44no—
284C22×C61C2×C4×D411yescharacteristic
285C22×C61C2×C4×D411yescharacteristic
286C22×C61C4×#4111yescharacteristic
287C22×C61#52012yes—
288C22×C61#52012yes—
289C22×C61C2×#12111yescharacteristic
290C22×C61#52511yescharacteristic
291C22×C61#52911yescharacteristic
292C22×C61#53211yescharacteristic
293C22×C61#53411yescharacteristic
294C22×C61#53412yes—
295C22×C61#53511yescharacteristic
296C22×C61#53611yescharacteristic
297C22×C61#53611yescharacteristic
298C22×C61#53612yes—
299C22×C61#53812yes—
300C22×C61#53912yes—
301C22×C61#53912yes—
302C22×C61#53912yes—
303C22×C61#53912yes—
304C22×C61#54112yes—
305C22×C61#54212yes—
306C22×C61#54212yes—
307C4×C61—24no—
308C4×C61—42no—
309C4×C61—82no—
310C4×C61—82no—
311Dic61—24no—
312Dic61—44no—
313Dic61—42no—
314Dic61—42no—
315Dic61—42no—
316Dic61—44no—
317C2×#397—611no—
318C2×#389—611no—
319C2×#385—612no—
320C2×C4×C22⋊C4—611no—
321#1445—611no—
322#1444—611no—
323#1442—612no—
324#1440—612no—
325#1439D6111yescharacteristic
326#1436—612no—
327#1434—611no—
328C2×C61—44no—
329C2×C61C4×#12111yescharacteristic
330C2×C61#232212yes—
331C2×C61#232211yescharacteristic
332C2×C61#232611yescharacteristic
333C2×C61#233412yes—
334C2×C61#261711yescharacteristic
335C2×C61#262111yescharacteristic
336C2×C61#263311yescharacteristic
337C2×C61#263811yescharacteristic
338C2×C61#265212yes—
339C2×C61#266312yes—
340C24×C4—611no—
341C22×C4⋊C4—611no—
342C22×C4⋊C4—612no—
343C22×C4⋊C4—612no—
344C22×C4⋊C4C2×D6111yescharacteristic
345C22×C22⋊C4—612no—
346C22×C22⋊C4—611no—
347C22×C22⋊C4C2×D6111yescharacteristic
348C22×C22⋊C4C2×D6112yes—
349C22×C42—611no—
350C22×C42—611no—
351#397—618no—
352#389—1224no—
353#385—1228no—
354C4×C22⋊C4—1224no—
355C2×#95—611no—
356C2×#95—612no—
357C2×#95—612no—
358C2×#95—612no—
359C2×#95—612no—
360C2×#95—611no—
361C2×#95—612no—
362C2×#95—611no—
363C2×#95—612no—
364C2×#95—612no—
365C2×#95C2×D6111yescharacteristic
366C2×#95C2×D6112yes—
367C61#1431311yescharacteristic, complemented, Sylow
368C25C22×D6111yescharacteristic
369C23×C4—611no—
370C23×C4—612no—
371C23×C4—612no—
372C23×C4—611no—
373C23×C4—611no—
374C23×C4—611no—
375C23×C4—614no—
376C23×C4—1222no—
377C23×C4—1222no—
378C23×C4—2442no—
379C23×C4—2442no—
380C23×C4C22×D6112yes—
381C23×C4C22×D6112yes—
382C23×C4C22×D6111yescharacteristic
383C23×C4C22×D6111yescharacteristic
384C2×C4⋊C4—22no—
385C2×C4⋊C4—42no—
386C2×C4⋊C4—614no—
387C2×C4⋊C4—1222no—
388C2×C4⋊C4—1222no—
389C2×C4⋊C4—1224no—
390C2×C4⋊C4—1224no—
391C2×C4⋊C4—1224no—
392C2×C4⋊C4—1224no—
393C2×C4⋊C4—2442no—
394C2×C4⋊C4—2442no—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—24no—
398C2×C22⋊C4—42no—
399C2×C22⋊C4—42no—
400C2×C22⋊C4—618no—
401C2×C22⋊C4—1222no—
402C2×C22⋊C4—1222no—
403C2×C22⋊C4—1224no—
404C2×C22⋊C4—1222no—
405C2×C22⋊C4—1224no—
406C2×C22⋊C4C4×D6114yes—
407C2×C42—614no—
408C2×C42—1222no—
409C2×C42—2442no—
410C2×C42—2442no—
411#95—22no—
412#95—42no—
413#95—614no—
414#95—618no—
415#95—1222no—
416#95—1224no—
417#95—1224no—
418#95—1224no—
419#95—1224no—
420#95—1222no—
421#95—2442no—
422#95—2442no—
423C24—21no—
424C24—21no—
425C24—21no—
426C24—21no—
427C24—22no—
428C24—22no—
429C24—22no—
430C24—22no—
431C24C2×C4×D6112yes—
432C24D4×D6112yes—
433C24#1175822612yes—
434C24C23×D6111yescharacteristic
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—612no—
456C22×C4—612no—
457C22×C4—612no—
458C22×C4—614no—
459C22×C4—614no—
460C22×C4—612no—
461C22×C4—612no—
462C22×C4—612no—
463C22×C4—1228no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1221no—
473C22×C4—1222no—
474C22×C4—1221no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1221no—
479C22×C4—1221no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—1222no—
488C22×C4—1222no—
489C22×C4—1222no—
490C22×C4—2448no—
491C22×C4—2444no—
492C22×C4—2444no—
493C22×C4—2448no—
494C22×C4C2×C4×D6112yes—
495C22×C4C2×C4×D6112yes—
496C22×C4D4×D6112yes—
497C22×C4#1175822412yes—
498C4⋊C4—44no—
499C4⋊C4—1228no—
500C4⋊C4—2448no—
501C4⋊C4—2448no—
502C22⋊C4—28no—
503C22⋊C4—48no—
504C22⋊C4—1228no—
505C22⋊C4—12216no—
506C42—2444no—
507C42—2444no—
508C23—22no—
509C23—22no—
510C23—24no—
511C23—24no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23#1451973914yes—
531C23C22×C4×D6111yescharacteristic
532C23C2×D4×D6111yescharacteristic
533C23C2×D4×D6111yescharacteristic
534C23#41×D6111yescharacteristic
535C23#41×D6111yescharacteristic
536C23C2×#1175822411yescharacteristic
537C23C2×#1175822611yescharacteristic
538C23#1451982212yes—
539C23#1451982212yes—
540C23#1451982412yes—
541C23#1451982412yes—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—614no—
566C2×C4—1222no—
567C2×C4—1224no—
568C2×C4—1222no—
569C2×C4—1224no—
570C2×C4—1224no—
571C2×C4—1224no—
572C2×C4—1224no—
573C2×C4—1224no—
574C2×C4—1224no—
575C2×C4—1224no—
576C2×C4—1222no—
577C2×C4—1222no—
578C2×C4—1222no—
579C2×C4—1222no—
580C2×C4—1224no—
581C2×C4—1222no—
582C2×C4—1224no—
583C2×C4—24416no—
584C2×C4—2442no—
585C2×C4—2442no—
586C2×C4—2442no—
587C2×C4—2442no—
588C2×C4—2442no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2442no—
593C2×C4—2441no—
594C2×C4—2442no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4—2442no—
598C2×C4—2442no—
599C2×C4—2442no—
600C2×C4#1451973714yes—
601C22—28no—
602C22—24no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—44no—
608C22—44no—
609C22—42no—
610C22—44no—
611C22—42no—
612C22C2×#1451973711yescharacteristic
613C22C2×#1451973911yescharacteristic
614C22#1684023011yescharacteristic
615C22#1684024112yes—
616C22#1684024112yes—
617C22#121×D6111yescharacteristic
618C22#1684027511yescharacteristic
619C22#1684029811yescharacteristic
620C22#1684030711yescharacteristic
621C22#1684032212yes—
622C22#1684032211yescharacteristic
623C22#1684032511yescharacteristic
624C22#1684033211yescharacteristic
625C22#1684033412yes—
626C22#1684033411yescharacteristic
627C22#1684034412yes—
628C22#1684035512yes—
629C22#1684035712yes—
630C22#1684036112yes—
631C22#1684036112yes—
632C22#1684036912yes—
633C22#1684037612yes—
634C22#1684037912yes—
635C4—24no—
636C4—42no—
637C4—82no—
638C4—82no—
639C4—1224no—
640C4—2442no—
641C4—2444no—
642C4—2444no—
643C4—2442no—
644C4—2442no—
645C2—44no—
646C2#2144832511yescharacteristic
647C2#2144848712yes—
648C2#2144849311yescharacteristic
649C2#2144852911yescharacteristic
650C2#2144859212yes—
651C2#2145155211yescharacteristic
652C2#2145158411yescharacteristic
653C2#2145169311yescharacteristic
654C2#2145174211yescharacteristic
655C2#2145187012yes—
656C2#2145196612yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1439C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenter#1439D61
hypocenterC61#14313
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
2. C2×#16839658C2semidirect
5. #21444088C2semidirect
6. #21444082C2semidirect
7. #21444077C2semidirect
9. #21444050C2semidirect
10. #21444044C2semidirect
20. #16839687C4semidirect
52. C2×C4⋊Dic61C2×C4semidirect
77. #14519684C2×C4semidirect
78. #14519684C2×C4semidirect
367. C61#14313semidirect

Maximal subgroups

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

subgroupindex
C2×#168396872
C2×#168396582
C2×#168396342
C2×C22⋊C4×Dic612
#214440882
#214440822
#214440772
#214440662
#214440502
#214440442
C61×#14392
#1431361