Finite Group Databasebeta

← #29602636

Subgroups — #29602636

Order 15616.

1630 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839655C212yesmaximal
2#21444076C212yesmaximal, complemented
3#21444064C212yesmaximal
4#21444053C212yesmaximal, complemented
5#21444049C211yesmaximal, characteristic, complemented
6#21444049C211yesmaximal, characteristic, complemented
7#21444046C211yesmaximal, characteristic, complemented
8#21444042C211yesmaximal, characteristic
9#21444042C211yesmaximal, characteristic
10C4×#14519684C211yesmaximal, characteristic, complemented
11C2×C4×C22⋊C4×C61C211yesmaximal, characteristic
12C22×C4⋊Dic61C2212yes—
13C22×C4⋊Dic61C2212yes—
14C22×C22⋊Dic61C2212yes—
15C22×C22⋊Dic61C2212yes—
16C22×C4×Dic61C2212yes—
17#16839655—28no—
18C2×#14519684C2212yes—
19C2×#14519684C2212yes—
20C2×#14519684C2211yescharacteristic
21C2×#14519684C2211yescharacteristic
22C2×#14519684C2212yes—
23C2×#14519684C2211yescharacteristic
24C2×#14519684C2211yescharacteristic
25C2×#14519684C2212yes—
26C2×#14519684C2212yes—
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C24×C4×C61C2211yescharacteristic
30C22×C22⋊C4×C61C2211yescharacteristic
31C22×C22⋊C4×C61C2211yescharacteristic
32C22×C42×C61C2211yescharacteristic
33C22×C42×C61C2211yescharacteristic
34C4×C22⋊C4×C61C418yescomplemented
35C2×#95×C61C2211yescharacteristic
36C2×#95×C61C2211yescharacteristic
37C23×Dic61C2312yes—
38C23×Dic61C2312yes—
39C23×Dic61C2312yes—
40C23×Dic61C2312yes—
41C2×C4⋊Dic61—24no—
42C2×C4⋊Dic61—24no—
43C2×C4⋊Dic61—24no—
44C2×C4⋊Dic61—24no—
45C2×C4⋊Dic61—42no—
46C2×C4⋊Dic61—42no—
47C2×C22⋊Dic61—24no—
48C2×C22⋊Dic61—24no—
49C2×C22⋊Dic61—24no—
50C2×C22⋊Dic61—24no—
51C2×C22⋊Dic61—42no—
52C2×C22⋊Dic61—42no—
53C2×C4×Dic61—24no—
54C2×C4×Dic61—24no—
55C2×C4×Dic61—42no—
56#14519684—24no—
57#14519684—24no—
58#14519684—22no—
59#14519684—22no—
60#14519684—24no—
61#14519684—22no—
62#14519684—22no—
63#14519684—24no—
64#14519684—42no—
65#14519684—42no—
66#14519684—42no—
67C25×C61C2311yescharacteristic
68C23×C4×C61—22no—
69C23×C4×C61—22no—
70C23×C4×C61—42no—
71C23×C4×C61—42no—
72C23×C4×C61C2×C414yes—
73C23×C4×C61C2311yescharacteristic
74C23×C4×C61C2311yescharacteristic
75C23×C4×C61C2311yescharacteristic
76C23×C4×C61C2311yescharacteristic
77C23×C4×C61C2311yescharacteristic
78C23×C4×C61C2311yescharacteristic
79C2×C22⋊C4×C61—22no—
80C2×C22⋊C4×C61—22no—
81C2×C22⋊C4×C61C2×C414yes—
82C2×C22⋊C4×C61C2×C414yes—
83C2×C22⋊C4×C61D414yes—
84C2×C22⋊C4×C61Q814yes—
85C2×C42×C61—42no—
86C2×C42×C61—42no—
87C2×C42×C61C2×C414yes—
88C2×C42×C61C2×C414yes—
89#95×C61C2×C414yes—
90#95×C61C2×C414yes—
91C24×C61—21no—
92C24×C61—21no—
93C24×C61—21no—
94C24×C61—21no—
95C24×C61—22no—
96C24×C61—22no—
97C24×C61—22no—
98C24×C61—22no—
99C24×C61C22×C412yes—
100C24×C61C2×D412yes—
101C24×C61C2×Q812yes—
102C24×C61C2411yescharacteristic
103C22×C4×C61—22no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—28no—
120C22×C4×C61—44no—
121C22×C4×C61—44no—
122C22×C4×C61—48no—
123C22×C4×C61—48no—
124C22×C4×C61C22×C412yes—
125C22×C4×C61C22×C412yes—
126C22×C4×C61C22×C412yes—
127C22×C4×C61C22×C412yes—
128C22×C4×C61C22×C412yes—
129C22×C4×C61C22×C412yes—
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2×D412yes—
132C22×C4×C61C2×Q812yes—
133C22×C4×C61C2×Q812yes—
134C22×C4×C61#4112yes—
135C22×C4×C61#4112yes—
136C22×C4×C61#4112yes—
137C22×C4×C61#4112yes—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
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#4114yes—
165C22×Dic61#4114yes—
166C22⋊C4×C61—28no—
167C22⋊C4×C61C4⋊C4116yescomplemented
168C42×C61—44no—
169C42×C61—44no—
170C22⋊Dic61—48no—
171C22⋊Dic61—48no—
172C4⋊Dic61—48no—
173C4⋊Dic61—48no—
174C4×Dic61—48no—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61—22no—
178C23×C61—22no—
179C23×C61—22no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—24no—
186C23×C61—24no—
187C23×C61—24no—
188C23×C61—24no—
189C23×C61—42no—
190C23×C61—42no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61—42no—
195C23×C61—42no—
196C23×C61—42no—
197C23×C61C2×C4⋊C414yes—
198C23×C61C23×C411yescharacteristic
199C23×C61C22×D411yescharacteristic
200C23×C61C22×Q811yescharacteristic
201C23×C61C2×#4111yescharacteristic
202C23×C61C2×#4111yescharacteristic
203C23×C61C2×#4111yescharacteristic
204C23×C61C2×#4111yescharacteristic
205C23×C61C2×#4112yes—
206C23×C61C2×#4112yes—
207C23×C61H(5,𝔽2)12yes—
208C23×C61H(5,𝔽2)12yes—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—24no—
218C2×C4×C61—24no—
219C2×C4×C61—24no—
220C2×C4×C61—24no—
221C2×C4×C61—24no—
222C2×C4×C61—24no—
223C2×C4×C61—24no—
224C2×C4×C61—24no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—416no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61C2×C4⋊C414yes—
237C2×C4×C61C2×C4⋊C414yes—
238C2×C4×C61#11714yes—
239C2×C4×C61#11714yes—
240C2×Dic61—24no—
241C2×Dic61—24no—
242C2×Dic61—24no—
243C2×Dic61—24no—
244C2×Dic61—24no—
245C2×Dic61—24no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61—42no—
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—
268#13948—611nomaximal, Sylow
269C22×C61—24no—
270C22×C61—24no—
271C22×C61—24no—
272C22×C61—42no—
273C22×C61—44no—
274C22×C61—44no—
275C22×C61—42no—
276C22×C61—42no—
277C22×C61—42no—
278C22×C61—44no—
279C22×C61—44no—
280C22×C61C22×C4⋊C411yescharacteristic
281C22×C61C2×#11711yescharacteristic
282C22×C61C2×#11711yescharacteristic
283C22×C61C4×#4111yescharacteristic
284C22×C61C4×#4111yescharacteristic
285C22×C61#51811yescharacteristic
286C22×C61#51811yescharacteristic
287C22×C61#52912yes—
288C22×C61#52912yes—
289C22×C61#53211yescharacteristic
290C22×C61#53211yescharacteristic
291C22×C61#53311yescharacteristic
292C22×C61#53311yescharacteristic
293C22×C61#53812yes—
294C22×C61#53812yes—
295C22×C61#54612yes—
296C22×C61#54612yes—
297C22×C61#55112yes—
298C22×C61#55312yes—
299C22×C61#55312yes—
300C22×C61#55412yes—
301C22×C61#55412yes—
302C22×C61#55912yes—
303C4×C61—24no—
304C4×C61—24no—
305C4×C61—24no—
306C4×C61—24no—
307C4×C61—42no—
308C4×C61—42no—
309Dic61—44no—
310Dic61—44no—
311Dic61—82no—
312Dic61—82no—
313C2×#388—612no—
314C2×C4×C22⋊C4D6111yescharacteristic
315#1442—612no—
316#1439—612no—
317#1436—612no—
318C4⋊#95—611no—
319C4⋊#95—611no—
320#1434—611no—
321C22⋊#95—611no—
322C22⋊#95—611no—
323C4×#95—611no—
324C2×C61—44no—
325C2×C61#228511yescharacteristic
326C2×C61#229811yescharacteristic
327C2×C61#231211yescharacteristic
328C2×C61#234911yescharacteristic
329C2×C61#234911yescharacteristic
330C2×C61#235311yescharacteristic
331C2×C61#235711yescharacteristic
332C2×C61#255712yes—
333C2×C61#255912yes—
334C2×C61#258712yes—
335C2×C61#258912yes—
336C24×C4C2×D6111yescharacteristic
337C22×C4⋊C4—612no—
338C22×C4⋊C4—612no—
339C22×C22⋊C4—612no—
340C22×C22⋊C4—612no—
341C22×C22⋊C4C2×D6111yescharacteristic
342C22×C22⋊C4C2×D6111yescharacteristic
343C22×C42—612no—
344C22×C42C2×D6111yescharacteristic
345C22×C42C2×D6111yescharacteristic
346#388—1228no—
347C4×C22⋊C4Dic6118yescomplemented
348C2×#95—611no—
349C2×#95—611no—
350C2×#95—612no—
351C2×#95—611no—
352C2×#95—611no—
353C2×#95—612no—
354C2×#95—612no—
355C2×#95—612no—
356C2×#95—612no—
357C2×#95—612no—
358C2×#95—612no—
359C2×#95C2×D6111yescharacteristic
360C2×#95C2×D6111yescharacteristic
361C61#1394811yescharacteristic, complemented, Sylow
362C25C22×D6111yescharacteristic
363C23×C4—22no—
364C23×C4—22no—
365C23×C4—42no—
366C23×C4—42no—
367C23×C4—612no—
368C23×C4—612no—
369C23×C4—612no—
370C23×C4—612no—
371C23×C4C2×Dic6114yes—
372C23×C4C22×D6111yescharacteristic
373C23×C4C22×D6111yescharacteristic
374C23×C4C22×D6111yescharacteristic
375C23×C4C22×D6111yescharacteristic
376C23×C4C22×D6111yescharacteristic
377C23×C4C22×D6111yescharacteristic
378C2×C4⋊C4—1224no—
379C2×C4⋊C4—1224no—
380C2×C4⋊C4—1224no—
381C2×C4⋊C4—1224no—
382C2×C4⋊C4—2442no—
383C2×C4⋊C4—2442no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—1224no—
387C2×C22⋊C4—1224no—
388C2×C22⋊C4—1224no—
389C2×C22⋊C4—1224no—
390C2×C22⋊C4—2442no—
391C2×C22⋊C4—2442no—
392C2×C22⋊C4Dic12214yes—
393C2×C22⋊C4D24414yes—
394C2×C22⋊C4C2×Dic6114yes—
395C2×C22⋊C4C2×Dic6114yes—
396C2×C42—42no—
397C2×C42—42no—
398C2×C42—1224no—
399C2×C42—1224no—
400C2×C42—2442no—
401C2×C42C2×Dic6114yes—
402C2×C42C2×Dic6114yes—
403#95—1224no—
404#95—1224no—
405#95—1224no—
406#95—1222no—
407#95—1222no—
408#95—1224no—
409#95—1222no—
410#95—1222no—
411#95—2442no—
412#95—2442no—
413#95—2442no—
414#95C2×Dic6114yes—
415#95C2×Dic6114yes—
416C24—22no—
417C24—22no—
418C24—21no—
419C24—21no—
420C24—21no—
421C24—21no—
422C24—22no—
423C24—22no—
424C24C2×Dic12212yes—
425C24C2×D24412yes—
426C24C22×Dic6112yes—
427C24C23×D6111yescharacteristic
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—28no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—614no—
450C22×C4—614no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—1222no—
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×C4C2×Dic12212yes—
478C22×C4C2×Dic12212yes—
479C22×C4C2×D24412yes—
480C22×C4C2×D24412yes—
481C22×C4#1175822212yes—
482C22×C4#1175822212yes—
483C22×C4#1175822212yes—
484C22×C4#1175822212yes—
485C22×C4C22×Dic6112yes—
486C22×C4C22×Dic6112yes—
487C22×C4C22×Dic6112yes—
488C22×C4C22×Dic6112yes—
489C22×C4C22×Dic6112yes—
490C22×C4C22×Dic6112yes—
491C4⋊C4—2448no—
492C4⋊C4—2448no—
493C22⋊C4—28no—
494C22⋊C4—2448no—
495C22⋊C4—2448no—
496C22⋊C4C4⋊Dic61116yescomplemented
497C42—44no—
498C42—44no—
499C42—2448no—
500C23—24no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—24no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23C2×C4⋊Dic6114yes—
523C23C23×Dic6111yescharacteristic
524C23C22×D24411yescharacteristic
525C23C22×Dic12211yescharacteristic
526C23#41×D6112yes—
527C23#41×D6112yes—
528C23C2×#1175822211yescharacteristic
529C23C2×#1175822211yescharacteristic
530C23C2×#1175822211yescharacteristic
531C23C2×#1175822211yescharacteristic
532C23#1451982212yes—
533C23#1451982212yes—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—416no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—1224no—
562C2×C4—1224no—
563C2×C4—1224no—
564C2×C4—1224no—
565C2×C4—1224no—
566C2×C4—1224no—
567C2×C4—2442no—
568C2×C4—2442no—
569C2×C4—2442no—
570C2×C4—2442no—
571C2×C4—2442no—
572C2×C4—2442no—
573C2×C4—2442no—
574C2×C4—2442no—
575C2×C4—2442no—
576C2×C4—2442no—
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×C4C2×C4⋊Dic6114yes—
590C2×C4C2×C4⋊Dic6114yes—
591C2×C4#1451973614yes—
592C2×C4#1451973614yes—
593C22—24no—
594C22—24no—
595C22—24no—
596C22—44no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—44no—
601C22—44no—
602C22—42no—
603C22—44no—
604C22C22×C4⋊Dic6111yescharacteristic
605C22C2×#1451973611yescharacteristic
606C22C2×#1451973611yescharacteristic
607C22#41×Dic6111yescharacteristic
608C22#41×Dic6111yescharacteristic
609C22#1684023711yescharacteristic
610C22#1684023711yescharacteristic
611C22#1684029012yes—
612C22#1684029012yes—
613C22#1684031211yescharacteristic
614C22#1684031211yescharacteristic
615C22#1684031911yescharacteristic
616C22#1684031911yescharacteristic
617C22#1684035012yes—
618C22#1684035012yes—
619C22#1684040712yes—
620C22#1684040712yes—
621C22#1684044112yes—
622C22#1684046112yes—
623C22#1684046212yes—
624C22#1684047112yes—
625C22#1684047112yes—
626C22#1684049512yes—
627C4—24no—
628C4—24no—
629C4—24no—
630C4—24no—
631C4—42no—
632C4—42no—
633C4—4882no—
634C4—4882no—
635C4—2444no—
636C4—2444no—
637C2—44no—
638C2#2144822611yescharacteristic
639C2#2144831711yescharacteristic
640C2#2144841711yescharacteristic
641C2#2144872911yescharacteristic
642C2#2144872911yescharacteristic
643C2#2144875911yescharacteristic
644C2#2144879211yescharacteristic
645C2#2145091212yes—
646C2#2145094212yes—
647C2#2145123412yes—
648C2#2145126412yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C4×C22⋊C4×C61C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenterC2×C4×C22⋊C4D61
hypocenterC61#13948
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. #21444076C2semidirect
4. #21444053C2semidirect
5. #21444049C2semidirect
6. #21444049C2semidirect
7. #21444046C2semidirect
10. C4×#14519684C2semidirect
34. C4×C22⋊C4×C61C4semidirect
167. C22⋊C4×C61C4⋊C4semidirect
347. C4×C22⋊C4Dic61semidirect
361. C61#13948semidirect
496. C22⋊C4C4⋊Dic61semidirect

Maximal subgroups

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

subgroupindex
C2×#168396552
#214440762
#214440642
#214440532
#214440492
#214440462
#214440422
C4×#145196842
C2×C4×C22⋊C4×C612
#1394861