Finite Group Databasebeta

← #29606921

Subgroups — #29606921

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×#16839660C211yesmaximal, characteristic, complemented
3C2×#16839635C212yesmaximal
4#21444086C211yesmaximal, characteristic, complemented
5#21444083C211yesmaximal, characteristic, complemented
6#21444074C212yesmaximal, complemented
7#21444066C212yesmaximal
8#21444064C211yesmaximal, characteristic
9#21444052C212yesmaximal, complemented
10#21444046C211yesmaximal, characteristic, complemented
11C2×C4×C22⋊C4×C61C211yesmaximal, characteristic
12C22×C4⋊Dic61C2211yescharacteristic
13C22×C4⋊Dic61C2211yescharacteristic
14C22×#11758196C2212yes—
15C22×#11758196C2212yes—
16C22×C22⋊Dic61C2212yes—
17C22×C22⋊Dic61C2212yes—
18#16839687C418yescomplemented
19#16839660—24no—
20#16839635—28no—
21C2×#14519684C2212yes—
22C2×#14519684C2212yes—
23C2×#14519684C2212yes—
24C2×#14519684C2211yescharacteristic
25C2×#14519684C2212yes—
26C2×#14519684C2212yes—
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2211yescharacteristic
30C2×#14519684C2212yes—
31C24×C4×C61C2211yescharacteristic
32C22×C22⋊C4×C61C2211yescharacteristic
33C22×C22⋊C4×C61C2211yescharacteristic
34C22×C42×C61C2211yescharacteristic
35C22×C42×C61C2211yescharacteristic
36C4×C22⋊C4×C61—24no—
37C2×#95×C61C2211yescharacteristic
38C2×#95×C61C2211yescharacteristic
39C23×Dic61C2312yes—
40C23×Dic61C2312yes—
41C23×Dic61C2312yes—
42C23×Dic61C2312yes—
43C2×C4⋊Dic61—22no—
44C2×C4⋊Dic61—22no—
45C2×C4⋊Dic61—22no—
46C2×C4⋊Dic61—42no—
47C2×C4⋊Dic61C2×C414yescomplemented
48C2×#11758196—24no—
49C2×#11758196—24no—
50C2×#11758196—24no—
51C2×#11758196—24no—
52C2×#11758196—42no—
53C2×#11758196—42no—
54C2×C22⋊Dic61—24no—
55C2×C22⋊Dic61—24no—
56C2×C22⋊Dic61—24no—
57C2×C22⋊Dic61—42no—
58C2×C22⋊Dic61—42no—
59C2×C22⋊Dic61C2×C418yes—
60#14519684—24no—
61#14519684—24no—
62#14519684—24no—
63#14519684—24no—
64#14519684—22no—
65#14519684—42no—
66#14519684—42no—
67#14519684—42no—
68#14519684C2×C414yescomplemented
69#14519684C2×C418yescomplemented
70C25×C61C2311yescharacteristic
71C23×C4×C61—22no—
72C23×C4×C61—22no—
73C23×C4×C61—42no—
74C23×C4×C61—42no—
75C23×C4×C61D414yes—
76C23×C4×C61C2311yescharacteristic
77C23×C4×C61C2311yescharacteristic
78C23×C4×C61C2311yescharacteristic
79C23×C4×C61C2311yescharacteristic
80C23×C4×C61C2311yescharacteristic
81C23×C4×C61C2311yescharacteristic
82C2×C22⋊C4×C61—22no—
83C2×C22⋊C4×C61—22no—
84C2×C22⋊C4×C61—22no—
85C2×C22⋊C4×C61—22no—
86C2×C22⋊C4×C61—22no—
87C2×C22⋊C4×C61C2×C414yes—
88C2×C42×C61—22no—
89C2×C42×C61—42no—
90C2×C42×C61—42no—
91C2×C42×C61D414yes—
92#95×C61—22no—
93#95×C61—22no—
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—22no—
109C22×C4×C61—22no—
110C22×C4×C61—21no—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
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×C61—28no—
127C22×C4×C61—48no—
128C22×C4×C61—44no—
129C22×C4×C61—44no—
130C22×C4×C61—48no—
131C22×C4×C61C22×C412yes—
132C22×C4×C61C22×C412yes—
133C22×C4×C61C2×D412yes—
134C22×C4×C61C2×D412yes—
135C22×C4×C61C2×D412yes—
136C22×C4×C61C2×D412yes—
137C22×C4×C61C2×D412yes—
138C22×C4×C61#4112yes—
139C22×C4×C61#4112yes—
140C22×C4×C61#4112yes—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—22no—
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—22no—
157C22×Dic61—22no—
158C22×Dic61—22no—
159C22×Dic61—22no—
160C22×Dic61—22no—
161C22×Dic61—22no—
162C22×Dic61—22no—
163C22×Dic61—22no—
164C22×Dic61—22no—
165C22×Dic61—22no—
166C22×Dic61—22no—
167C22×Dic61C22×C414yes—
168C22×Dic61C22×C414yes—
169C22⋊C4×C61—28no—
170C22⋊C4×C61—28no—
171C42×C61—44no—
172C42×C61—44no—
173C22⋊Dic61—216no—
174C22⋊Dic61—48no—
175C4⋊Dic61—28no—
176C4⋊Dic61—44no—
177#11758196—48no—
178#11758196—48no—
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—22no—
219C2×C4×C61—24no—
220C2×C4×C61—24no—
221C2×C4×C61—22no—
222C2×C4×C61—24no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—24no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—24no—
230C2×C4×C61—416no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61C4×D414yes—
243C2×C4×C61#12114yes—
244C2×Dic61—24no—
245C2×Dic61—24no—
246C2×Dic61—24no—
247C2×Dic61—24no—
248C2×Dic61—24no—
249C2×Dic61—24no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61—42no—
258C2×Dic61—42no—
259C2×Dic61—42no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263C2×Dic61—42no—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61—42no—
270C2×Dic61—42no—
271C2×Dic61—42no—
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—24no—
309C4×C61—42no—
310C4×C61—42no—
311C4×C61—42no—
312C4×C61—42no—
313Dic61—44no—
314Dic61—44no—
315Dic61—82no—
316Dic61—82no—
317C2×#397—611no—
318C2×#389—611no—
319C2×#385—612no—
320C2×C4×C22⋊C4D6111yescharacteristic
321#1445—611no—
322#1444—611no—
323#1442—612no—
324#1440—612no—
325#1439—611no—
326#1436—612no—
327#1434—611no—
328C2×C61—44no—
329C2×C61C4×#12111yescharacteristic
330C2×C61#232211yescharacteristic
331C2×C61#232212yes—
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×C4C2×D6111yescharacteristic
341C22×C4⋊C4—611no—
342C22×C4⋊C4—612no—
343C22×C4⋊C4—612no—
344C22×C4⋊C4—611no—
345C22×C22⋊C4—612no—
346C22×C22⋊C4—612no—
347C22×C22⋊C4C2×D6111yescharacteristic
348C22×C22⋊C4C2×D6111yescharacteristic
349C22×C42C2×D6111yescharacteristic
350C22×C42C2×D6111yescharacteristic
351#397—618no—
352#389—1224no—
353#385—1228no—
354C4×C22⋊C4—24no—
355C2×#95—611no—
356C2×#95—612no—
357C2×#95—612no—
358C2×#95—611no—
359C2×#95—612no—
360C2×#95—612no—
361C2×#95—612no—
362C2×#95—612no—
363C2×#95—612no—
364C2×#95—612no—
365C2×#95C2×D6111yescharacteristic
366C2×#95C2×D6111yescharacteristic
367C61#1431311yescharacteristic, complemented, Sylow
368C25C22×D6111yescharacteristic
369C23×C4—22no—
370C23×C4—22no—
371C23×C4—42no—
372C23×C4—42no—
373C23×C4—612no—
374C23×C4—612no—
375C23×C4—612no—
376C23×C4—612no—
377C23×C4D24414yes—
378C23×C4C22×D6111yescharacteristic
379C23×C4C22×D6111yescharacteristic
380C23×C4C22×D6111yescharacteristic
381C23×C4C22×D6111yescharacteristic
382C23×C4C22×D6111yescharacteristic
383C23×C4C22×D6111yescharacteristic
384C2×C4⋊C4—614no—
385C2×C4⋊C4—1222no—
386C2×C4⋊C4—1222no—
387C2×C4⋊C4—1224no—
388C2×C4⋊C4—1224no—
389C2×C4⋊C4—1224no—
390C2×C4⋊C4—1224no—
391C2×C4⋊C4—1222no—
392C2×C4⋊C4—2442no—
393C2×C4⋊C4—2442no—
394C2×C4⋊C4—2442no—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—22no—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—618no—
401C2×C22⋊C4—1224no—
402C2×C22⋊C4—1224no—
403C2×C22⋊C4—1224no—
404C2×C22⋊C4—2442no—
405C2×C22⋊C4—2442no—
406C2×C22⋊C4C4×D6114yes—
407C2×C42—22no—
408C2×C42—42no—
409C2×C42—42no—
410C2×C42C22⋊D6114yes—
411#95—22no—
412#95—22no—
413#95—614no—
414#95—618no—
415#95—1224no—
416#95—1224no—
417#95—1224no—
418#95—1222no—
419#95—1224no—
420#95—2442no—
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—
432C24C2×D24412yes—
433C24#1175822212yes—
434C24C23×D6111yescharacteristic
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
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—28no—
456C22×C4—48no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—48no—
460C22×C4—614no—
461C22×C4—614no—
462C22×C4—1222no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1222no—
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×C4C2×C4×D6112yes—
489C22×C4C2×C4×D6112yes—
490C22×C4C2×D24412yes—
491C22×C4C2×D24412yes—
492C22×C4#1175822212yes—
493C22×C4#1175822212yes—
494C22×C4#1175822212yes—
495C22×C4C2×C22⋊D6112yes—
496C22×C4C2×C22⋊D6112yes—
497C22×C4C2×C22⋊D6112yes—
498C4⋊C4—1228no—
499C4⋊C4—2448no—
500C4⋊C4—2444no—
501C4⋊C4—2448no—
502C22⋊C4—28no—
503C22⋊C4—28no—
504C22⋊C4—12216no—
505C22⋊C4—2448no—
506C42—44no—
507C42—44no—
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—
530C23C4×D24414yes—
531C23C22×C4×D6111yescharacteristic
532C23C22×C22⋊D6111yescharacteristic
533C23C22×D24411yescharacteristic
534C23C2×#1175822211yescharacteristic
535C23C2×#1175822211yescharacteristic
536C23C2×#1175822211yescharacteristic
537C23C2×#1175822211yescharacteristic
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—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—416no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—1224no—
572C2×C4—1224no—
573C2×C4—1224no—
574C2×C4—1224no—
575C2×C4—1224no—
576C2×C4—1224no—
577C2×C4—2442no—
578C2×C4—2442no—
579C2×C4—2442no—
580C2×C4—2442no—
581C2×C4—2442no—
582C2×C4—2442no—
583C2×C4—2442no—
584C2×C4—2442no—
585C2×C4—2442no—
586C2×C4—2442no—
587C2×C4—2442no—
588C2×C4—2442no—
589C2×C4—2442no—
590C2×C4—2442no—
591C2×C4—2442no—
592C2×C4—2442no—
593C2×C4—2442no—
594C2×C4—2442no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4—2442no—
598C2×C4—2442no—
599C2×C4C4×C22⋊D6114yes—
600C2×C4#1451975114yes—
601C22—28no—
602C22—24no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—44no—
608C22—44no—
609C22—42no—
610C22—44no—
611C22—42no—
612C22C2×C4×C22⋊D6111yescharacteristic
613C22C2×C4×D24411yescharacteristic
614C22C4×#1175822211yescharacteristic
615C22#1684024112yes—
616C22#1684024112yes—
617C22C2×#1451975111yescharacteristic
618C22#1684027311yescharacteristic
619C22#1684029611yescharacteristic
620C22#1684031211yescharacteristic
621C22#1684032111yescharacteristic
622C22#1684032412yes—
623C22#1684032911yescharacteristic
624C22#1684033311yescharacteristic
625C22#1684033311yescharacteristic
626C22#1684033812yes—
627C22#1684035012yes—
628C22#1684035412yes—
629C22#1684035412yes—
630C22#1684035412yes—
631C22#1684035412yes—
632C22#1684037412yes—
633C22#1684037712yes—
634C22#1684037712yes—
635C4—24no—
636C4—24no—
637C4—42no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—4882no—
642C4—4882no—
643C4—2444no—
644C4—2444no—
645C2—44no—
646C2C4×#1451975111yescharacteristic
647C2#2144848411yescharacteristic
648C2#2144849012yes—
649C2#2144853111yescharacteristic
650C2#2144859612yes—
651C2#2145155311yescharacteristic
652C2#2145158511yescharacteristic
653C2#2145168711yescharacteristic
654C2#2145173611yescharacteristic
655C2#2145186112yes—
656C2#2145196712yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C4×C22⋊C4×C61C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenterC2×C4×C22⋊C4D61
hypocenterC61#14313
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#16839660C2semidirect
4. #21444086C2semidirect
5. #21444083C2semidirect
6. #21444074C2semidirect
9. #21444052C2semidirect
10. #21444046C2semidirect
18. #16839687C4semidirect
47. C2×C4⋊Dic61C2×C4semidirect
68. #14519684C2×C4semidirect
69. #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×#168396602
C2×#168396352
#214440862
#214440832
#214440742
#214440662
#214440642
#214440522
#214440462
C2×C4×C22⋊C4×C612
#1431361