Finite Group Databasebeta

← #29601543

Subgroups — #29601543

Order 15616.

1906 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839650C212yesmaximal
2C2×C4×C22⋊Dic61C211yesmaximal, characteristic
3C2×#16839607C211yesmaximal, characteristic, complemented
4#21444056C212yesmaximal
5#21444052C212yesmaximal, complemented
6#21444047C211yesmaximal, characteristic, complemented
7#21444043C211yesmaximal, characteristic, complemented
8#21444042C211yesmaximal, characteristic
9C4×#14519684C212yesmaximal, complemented
10#95×Dic61C211yesmaximal, characteristic, complemented
11C2×C4×C22⋊C4×C61C211yesmaximal, characteristic
12C22×#11758196C2212yes—
13C22×C22⋊Dic61C2211yescharacteristic
14C22×C22⋊Dic61C2212yes—
15C22×C22⋊Dic61C2211yescharacteristic
16C22×C4×Dic61C2212yes—
17C22×C4×Dic61C2211yescharacteristic
18C22×C4×Dic61C2212yes—
19C22×C4×Dic61C2211yescharacteristic
20#16839650C4116yescomplemented
21C4×C22⋊Dic61C418yescomplemented
22#16839607—24no—
23C2×#14519684C2211yescharacteristic
24C2×#14519684C2212yes—
25C2×#14519684C2212yes—
26C2×#14519684C2211yescharacteristic
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2211yescharacteristic
30C2×#14519684C2211yescharacteristic
31C2×#14519684C2212yes—
32C2×#14519684C2212yes—
33C24×C4×C61C2211yescharacteristic
34C22×C22⋊C4×C61C2212yes—
35C22×C42×C61C2212yes—
36C4×C22⋊C4×C61—24no—
37C2×#95×C61C2211yescharacteristic
38C2×#95×C61C2211yescharacteristic
39C23×Dic61C2312yes—
40C23×Dic61C2311yescharacteristic
41C23×Dic61C2312yes—
42C23×Dic61C2311yescharacteristic
43C23×Dic61C2311yescharacteristic
44C23×Dic61C2311yescharacteristic
45C2×#11758196—42no—
46C2×#11758196—42no—
47C2×#11758196C2×C418yescomplemented
48C2×C22⋊Dic61—22no—
49C2×C22⋊Dic61—22no—
50C2×C22⋊Dic61—42no—
51C2×C22⋊Dic61—42no—
52C2×C22⋊Dic61C2×C414yes—
53C2×C22⋊Dic61C2×C418yes—
54C2×C22⋊Dic61C2×C414yes—
55C2×C22⋊Dic61C2×C418yes—
56C2×C4×Dic61—22no—
57C2×C4×Dic61—24no—
58C2×C4×Dic61—22no—
59C2×C4×Dic61—24no—
60C2×C4×Dic61—24no—
61C2×C4×Dic61—24no—
62C2×C4×Dic61—42no—
63C2×C4×Dic61—42no—
64C2×C4×Dic61C2×C414yes—
65C2×C4×Dic61C2×C414yes—
66C2×C4×Dic61C2×C418yescomplemented
67#14519684—22no—
68#14519684—24no—
69#14519684—22no—
70#14519684—24no—
71#14519684—24no—
72#14519684—42no—
73#14519684C2×C414yes—
74#14519684C2×C414yes—
75#14519684C2×C418yescomplemented
76#14519684C2×C418yescomplemented
77C25×C61C2311yescharacteristic
78C23×C4×C61—22no—
79C23×C4×C61—22no—
80C23×C4×C61—24no—
81C23×C4×C61—42no—
82C23×C4×C61C2×C414yes—
83C23×C4×C61C2312yes—
84C23×C4×C61C2311yescharacteristic
85C23×C4×C61C2312yes—
86C23×C4×C61C2311yescharacteristic
87C2×C22⋊C4×C61—24no—
88C2×C22⋊C4×C61—24no—
89C2×C22⋊C4×C61C2×C418yes—
90C2×C42×C61—24no—
91C2×C42×C61—24no—
92C2×C42×C61—42no—
93#95×C61—22no—
94#95×C61D414yes—
95C24×C61—21no—
96C24×C61—22no—
97C24×C61—22no—
98C24×C61—21no—
99C24×C61—21no—
100C24×C61—21no—
101C24×C61—22no—
102C24×C61—21no—
103C24×C61—21no—
104C24×C61C22×C414yes—
105C24×C61C22×C412yes—
106C24×C61C2411yescharacteristic
107C22×C4×C61—21no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—21no—
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—28no—
123C22×C4×C61—28no—
124C22×C4×C61—44no—
125C22×C4×C61—48no—
126C22×C4×C61—48no—
127C22×C4×C61C22×C414yes—
128C22×C4×C61C22×C412yes—
129C22×C4×C61C22×C414yes—
130C22×C4×C61C22×C412yes—
131C22×C4×C61C2×D412yes—
132C22×C4×C61C2×D414yes—
133C22×C4×C61#4114yes—
134C22×C4×C61#4114yes—
135C22×C4×C61#4112yes—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—22no—
145C22×Dic61—21no—
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61—21no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—22no—
158C22×Dic61—22no—
159C22×Dic61—22no—
160C22×Dic61—22no—
161C22×Dic61—21no—
162C22×Dic61C22×C412yes—
163C22×Dic61C22×C414yes—
164C22×Dic61C22×C414yes—
165C22×Dic61C22×C412yes—
166C22×Dic61C22×C412yes—
167C22×Dic61C22×C414yes—
168C22×Dic61C22×C414yes—
169C22×Dic61C22×C412yes—
170C22×Dic61#4112yes—
171C22×Dic61#4112yes—
172C22⋊C4×C61—216no—
173C42×C61—48no—
174C22⋊Dic61—28no—
175C22⋊Dic61—48no—
176C22⋊Dic61C42116yescomplemented
177#11758196—48no—
178C4×Dic61—28no—
179C4×Dic61—216no—
180C4×Dic61—44no—
181C4×Dic61—48no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61—24no—
185C23×C61—22no—
186C23×C61—24no—
187C23×C61—24no—
188C23×C61—22no—
189C23×C61—24no—
190C23×C61—24no—
191C23×C61—22no—
192C23×C61—22no—
193C23×C61—24no—
194C23×C61—42no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61—42no—
198C23×C61—42no—
199C23×C61—42no—
200C23×C61—42no—
201C23×C61—42no—
202C23×C61—41no—
203C23×C61—41no—
204C23×C61C2×C4214yes—
205C23×C61C23×C411yescharacteristic
206C23×C61C23×C412yes—
207C23×C61C22×D411yescharacteristic
208C23×C61C2×#4112yes—
209C23×C61C2×#4111yescharacteristic
210C23×C61C2×#4111yescharacteristic
211C23×C61C2×#4111yescharacteristic
212C23×C61C2×#4112yes—
213C23×C61H(5,𝔽2)12yes—
214C23×C61H(5,𝔽2)11yescharacteristic
215C23×C612−1+411yescharacteristic
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—24no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—24no—
222C2×C4×C61—24no—
223C2×C4×C61—24no—
224C2×C4×C61—24no—
225C2×C4×C61—24no—
226C2×C4×C61—24no—
227C2×C4×C61—24no—
228C2×C4×C61—24no—
229C2×C4×C61—24no—
230C2×C4×C61—24no—
231C2×C4×C61—24no—
232C2×C4×C61—24no—
233C2×C4×C61—416no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61#11714yes—
239C2×C4×C61C4×D418yes—
240C2×C4×C61C4×D414yes—
241C2×Dic61—22no—
242C2×Dic61—24no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—24no—
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—24no—
249C2×Dic61—24no—
250C2×Dic61—22no—
251C2×Dic61—24no—
252C2×Dic61—24no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—24no—
256C2×Dic61—24no—
257C2×Dic61—24no—
258C2×Dic61—24no—
259C2×Dic61—24no—
260C2×Dic61—24no—
261C2×Dic61—22no—
262C2×Dic61—42no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—42no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61—41no—
270C2×Dic61—42no—
271C2×Dic61—42no—
272C2×Dic61—42no—
273C2×Dic61—41no—
274C2×Dic61C2×C4214yes—
275C2×Dic61C2×C4214yes—
276C2×Dic61#11714yes—
277#13816—611nomaximal, Sylow
278C22×C61—24no—
279C22×C61—24no—
280C22×C61—24no—
281C22×C61—42no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61—42no—
285C22×C61—44no—
286C22×C61—44no—
287C22×C61—44no—
288C22×C61—44no—
289C22×C61C22×C4211yescharacteristic
290C22×C61C2×#11711yescharacteristic
291C22×C61C2×#11711yescharacteristic
292C22×C61C2×C4×D411yescharacteristic
293C22×C61C2×C4×D412yes—
294C22×C61C4×#4112yes—
295C22×C61C4×#4112yes—
296C22×C61C4×#4111yescharacteristic
297C22×C61C4×#4112yes—
298C22×C61#51812yes—
299C22×C61#51811yescharacteristic
300C22×C61#52012yes—
301C22×C61#52011yescharacteristic
302C22×C61#54612yes—
303C22×C61#54711yescharacteristic
304C22×C61#55011yescharacteristic
305C22×C61#55112yes—
306C22×C61#55112yes—
307C22×C61#55212yes—
308C22×C61#55312yes—
309C22×C61#55511yescharacteristic
310C22×C61#55512yes—
311C22×C61#55611yescharacteristic
312C4×C61—24no—
313C4×C61—24no—
314C4×C61—28no—
315C4×C61—44no—
316Dic61—24no—
317Dic61—24no—
318Dic61—24no—
319Dic61—42no—
320Dic61—44no—
321Dic61—44no—
322C2×#388—612no—
323C2×C4×C22⋊C4—611no—
324C2×C4×C22⋊C4D6111yescharacteristic
325C2×#376—611no—
326#1437—612no—
327#1436—612no—
328C4⋊#95—611no—
329#1434—611no—
330C22⋊#95—611no—
331C4×#95—612no—
332C4×#95—611no—
333C2×C61—44no—
334C2×C61C4×#11711yescharacteristic
335C2×C61C42×D411yescharacteristic
336C2×C61#227511yescharacteristic
337C2×C61#234211yescharacteristic
338C2×C61#234611yescharacteristic
339C2×C61#235311yescharacteristic
340C2×C61#235411yescharacteristic
341C2×C61#235912yes—
342C2×C61#236312yes—
343C2×C61#237212yes—
344C2×C61#237312yes—
345C24×C4C2×D6111yescharacteristic
346C22×C4⋊C4—612no—
347C22×C22⋊C4—611no—
348C22×C22⋊C4—611no—
349C22×C22⋊C4—612no—
350C22×C22⋊C4C2×D6112yes—
351C22×C42—612no—
352C22×C42—611no—
353C22×C42—611no—
354C22×C42—612no—
355C22×C42C2×D6112yes—
356#388—6116no—
357C4×C22⋊C4—24no—
358C4×C22⋊C4—618no—
359#376—1224no—
360C2×#95—611no—
361C2×#95—612no—
362C2×#95—612no—
363C2×#95—611no—
364C2×#95—611no—
365C2×#95—612no—
366C2×#95—612no—
367C2×#95—611no—
368C2×#95—612no—
369C2×#95—612no—
370C2×#95C2×D6111yescharacteristic
371C2×#95C2×D6111yescharacteristic
372C61#1381611yescharacteristic, complemented, Sylow
373C25C22×D6111yescharacteristic
374C23×C4—22no—
375C23×C4—24no—
376C23×C4—22no—
377C23×C4—42no—
378C23×C4—611no—
379C23×C4—611no—
380C23×C4—611no—
381C23×C4—612no—
382C23×C4—612no—
383C23×C4—611no—
384C23×C4C4×D6114yes—
385C23×C4C22×D6112yes—
386C23×C4C22×D6112yes—
387C23×C4C22×D6111yescharacteristic
388C23×C4C22×D6111yescharacteristic
389C2×C4⋊C4—618no—
390C2×C4⋊C4—2442no—
391C2×C4⋊C4—2442no—
392C2×C22⋊C4—24no—
393C2×C22⋊C4—24no—
394C2×C22⋊C4—614no—
395C2×C22⋊C4—618no—
396C2×C22⋊C4—614no—
397C2×C22⋊C4—618no—
398C2×C22⋊C4—1222no—
399C2×C22⋊C4—1222no—
400C2×C22⋊C4—2442no—
401C2×C22⋊C4—2442no—
402C2×C22⋊C4C4×D6118yes—
403C2×C42—24no—
404C2×C42—24no—
405C2×C42—42no—
406C2×C42—614no—
407C2×C42—618no—
408C2×C42—614no—
409C2×C42—1222no—
410C2×C42—1224no—
411C2×C42—1222no—
412C2×C42—1224no—
413C2×C42—1224no—
414C2×C42—1224no—
415C2×C42—2442no—
416C2×C42—2442no—
417#95—22no—
418#95—614no—
419#95—614no—
420#95—618no—
421#95—618no—
422#95—1224no—
423#95—1222no—
424#95—1224no—
425#95—1222no—
426#95—1224no—
427#95—2442no—
428#95C22⋊D6114yes—
429C24—21no—
430C24—21no—
431C24—21no—
432C24—22no—
433C24—22no—
434C24—21no—
435C24—22no—
436C24—21no—
437C24—21no—
438C24C2×C4×D6114yes—
439C24C2×C4×D6112yes—
440C24C23×D6111yescharacteristic
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
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—22no—
456C22×C4—28no—
457C22×C4—28no—
458C22×C4—44no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—612no—
462C22×C4—614no—
463C22×C4—612no—
464C22×C4—612no—
465C22×C4—614no—
466C22×C4—612no—
467C22×C4—612no—
468C22×C4—614no—
469C22×C4—612no—
470C22×C4—614no—
471C22×C4—1221no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1221no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1221no—
478C22×C4—1221no—
479C22×C4—1222no—
480C22×C4—1221no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—1222no—
488C22×C4—1222no—
489C22×C4—1221no—
490C22×C4—1222no—
491C22×C4—1222no—
492C22×C4—1222no—
493C22×C4—1221no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1221no—
497C22×C4C2×C4×D6114yes—
498C22×C4C2×C4×D6112yes—
499C22×C4C2×C4×D6114yes—
500C22×C4C2×C4×D6112yes—
501C22×C4#1175822212yes—
502C22×C4#1175822214yes—
503C22×C4#1175822414yes—
504C22×C4C2×C22⋊D6112yes—
505C22×C4C2×C22⋊D6114yes—
506C4⋊C4—2448no—
507C22⋊C4—216no—
508C22⋊C4—6116no—
509C22⋊C4—1228no—
510C22⋊C4—2448no—
511C42—48no—
512C42—12216no—
513C42—1228no—
514C42—2444no—
515C42—2448no—
516C23—22no—
517C23—22no—
518C23—24no—
519C23—22no—
520C23—24no—
521C23—24no—
522C23—22no—
523C23—24no—
524C23—24no—
525C23—22no—
526C23—24no—
527C23—22no—
528C23—42no—
529C23—42no—
530C23—41no—
531C23—42no—
532C23—41no—
533C23—42no—
534C23—41no—
535C23—41no—
536C23—42no—
537C23—42no—
538C23C42×D6114yes—
539C23C22×C4×D6111yescharacteristic
540C23C22×C4×D6112yes—
541C23C22×C22⋊D6111yescharacteristic
542C23#41×D6111yescharacteristic
543C23#41×D6111yescharacteristic
544C23C2×#1175822412yes—
545C23C2×#1175822212yes—
546C23C2×#1175822211yescharacteristic
547C23#1451982212yes—
548C23#1451982311yescharacteristic
549C23#1451982411yescharacteristic
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—416no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—614no—
573C2×C4—614no—
574C2×C4—614no—
575C2×C4—1224no—
576C2×C4—1222no—
577C2×C4—1224no—
578C2×C4—1222no—
579C2×C4—1222no—
580C2×C4—1224no—
581C2×C4—1224no—
582C2×C4—1222no—
583C2×C4—1222no—
584C2×C4—1224no—
585C2×C4—1222no—
586C2×C4—1224no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1224no—
590C2×C4—1224no—
591C2×C4—1224no—
592C2×C4—1224no—
593C2×C4—1224no—
594C2×C4—1224no—
595C2×C4—1222no—
596C2×C4—2442no—
597C2×C4—2441no—
598C2×C4—2442no—
599C2×C4—2442no—
600C2×C4—2442no—
601C2×C4—2441no—
602C2×C4—2442no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2442no—
606C2×C4—2442no—
607C2×C4—2442no—
608C2×C4#1451973314yes—
609C2×C4C4×C22⋊D6118yes—
610C2×C4C4×C22⋊D6114yes—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—42no—
615C22—42no—
616C22—44no—
617C22—44no—
618C22—42no—
619C22—44no—
620C22—44no—
621C22—42no—
622C22C2×C42×D6111yescharacteristic
623C22#117×D6111yescharacteristic
624C22C2×#1451973311yescharacteristic
625C22C2×C4×C22⋊D6111yescharacteristic
626C22C2×C4×C22⋊D6112yes—
627C22C4×#1175822412yes—
628C22C4×#1175822412yes—
629C22C4×#1175822211yescharacteristic
630C22C4×#1175822212yes—
631C22#1684023412yes—
632C22#1684023511yescharacteristic
633C22#1684024112yes—
634C22#1684024211yescharacteristic
635C22#1684040612yes—
636C22#1684041111yescharacteristic
637C22#1684043111yescharacteristic
638C22#1684043812yes—
639C22#1684044112yes—
640C22#1684045212yes—
641C22#1684046212yes—
642C22#1684047511yescharacteristic
643C22#1684047712yes—
644C22#1684048011yescharacteristic
645C4—28no—
646C4—24no—
647C4—24no—
648C4—44no—
649C4—1224no—
650C4—1224no—
651C4—1224no—
652C4—2442no—
653C4—2444no—
654C4—2444no—
655C2—44no—
656C2C4×#1451973311yescharacteristic
657C2C42×C22⋊D6111yescharacteristic
658C2#2144817811yescharacteristic
659C2#2144866411yescharacteristic
660C2#2144870211yescharacteristic
661C2#2144876211yescharacteristic
662C2#2144876911yescharacteristic
663C2#2144880512yes—
664C2#2144885312yes—
665C2#2144892712yes—
666C2#2144894112yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC22×C61C22×C42
FittingC2×C4×C22⋊C4×C61C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenterC2×C4×C22⋊C4D61
hypocenterC61#13816
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#16839607C2semidirect
5. #21444052C2semidirect
6. #21444047C2semidirect
7. #21444043C2semidirect
9. C4×#14519684C2semidirect
10. #95×Dic61C2semidirect
20. #16839650C4semidirect
21. C4×C22⋊Dic61C4semidirect
47. C2×#11758196C2×C4semidirect
66. C2×C4×Dic61C2×C4semidirect
75. #14519684C2×C4semidirect
76. #14519684C2×C4semidirect
176. C22⋊Dic61C42semidirect
372. C61#13816semidirect

Maximal subgroups

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

subgroupindex
C2×#168396502
C2×C4×C22⋊Dic612
C2×#168396072
#214440562
#214440522
#214440472
#214440432
#214440422
C4×#145196842
#95×Dic612
C2×C4×C22⋊C4×C612
#1381661