Finite Group Databasebeta

← #29601545

Subgroups — #29601545

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×#16839654C212yesmaximal
2C2×C4×#11758198C211yesmaximal, characteristic
3C2×#16839609C211yesmaximal, characteristic
4C2×#16839607C211yesmaximal, characteristic, complemented
5#21444057C212yesmaximal
6#21444053C212yesmaximal, complemented
7#21444049C211yesmaximal, characteristic, complemented
8#21444043C211yesmaximal, characteristic, complemented
9#21444040C211yesmaximal, characteristic
10C4×#14519684C212yesmaximal, complemented
11C4×#95×C61C211yesmaximal, characteristic, complemented
12C23×C4×D61C2211yescharacteristic
13C22×C4⋊Dic61C2212yes—
14C22×#11758198C2212yes—
15C22×#11758198C2211yescharacteristic
16C22×#11758198C2212yes—
17C22×#11758198C2211yescharacteristic
18C22×C4×Dic61C2212yes—
19C22×C4×Dic61C2211yescharacteristic
20C22×C4×Dic61C2212yes—
21#16839654C4116yescomplemented
22C4×#11758198C418yescomplemented
23#16839609—24no—
24#16839607—24no—
25C2×#14519684C2212yes—
26C2×#14519684C2212yes—
27C2×#14519684C2211yescharacteristic
28C2×#14519684C2211yescharacteristic
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2211yescharacteristic
32C2×#14519684C2211yescharacteristic
33C2×#14519684C2212yes—
34C22×C42×C61C2212yes—
35C22×C42×C61C2211yescharacteristic
36C2×#95×C61C2211yescharacteristic
37C2×#95×C61C2212yes—
38C2×#95×C61C2211yescharacteristic
39C24×D61C2311yescharacteristic
40C23×Dic61C2312yes—
41C23×Dic61C2312yes—
42C23×Dic61C2311yescharacteristic
43C23×Dic61C2311yescharacteristic
44C23×Dic61C2311yescharacteristic
45C22×C4×D61—22no—
46C22×C4×D61—22no—
47C22×C4×D61—24no—
48C22×C4×D61—42no—
49C22×C4×D61C2×C414yes—
50C2×C4⋊Dic61—42no—
51C2×C4⋊Dic61—42no—
52C2×C4⋊Dic61C2×C418yescomplemented
53C2×#11758198—24no—
54C2×#11758198—24no—
55C2×#11758198—22no—
56C2×#11758198—22no—
57C2×#11758198—42no—
58C2×#11758198—42no—
59C2×#11758198C2×C414yes—
60C2×#11758198C2×C418yes—
61C2×#11758198C2×C418yes—
62C2×#11758198C2×C414yes—
63C2×#11758198C2×C418yes—
64C2×C4×Dic61—22no—
65C2×C4×Dic61—24no—
66C2×C4×Dic61—22no—
67C2×C4×Dic61—24no—
68C2×C4×Dic61—24no—
69C2×C4×Dic61—24no—
70C2×C4×Dic61—42no—
71C2×C4×Dic61C2×C414yes—
72C2×C4×Dic61C2×C418yescomplemented
73#14519684—22no—
74#14519684—24no—
75#14519684—22no—
76#14519684—24no—
77#14519684—24no—
78#14519684—42no—
79#14519684C2×C414yes—
80#14519684C2×C414yes—
81#14519684C2×C418yescomplemented
82C23×C4×C61C2312yes—
83C23×C4×C61C2311yescharacteristic
84C23×C4×C61C2312yes—
85C23×C4×C61C2311yescharacteristic
86C23×C4×C61C2311yescharacteristic
87C2×C42×C61—24no—
88C2×C42×C61—24no—
89C2×C42×C61—42no—
90C2×C42×C61—42no—
91C2×C42×C61C2×C414yes—
92#95×C61—22no—
93#95×C61C2×C418yescomplemented
94#95×C61D414yes—
95C24×C61C2411yescharacteristic
96C23×D61—21no—
97C23×D61—22no—
98C23×D61—21no—
99C23×D61—21no—
100C23×D61—21no—
101C23×D61—22no—
102C23×D61—22no—
103C23×D61—21no—
104C23×D61—21no—
105C23×D61C22×C412yes—
106C23×D61C22×C414yes—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
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×C61C22×C414yes—
125C22×C4×C61C22×C414yes—
126C22×C4×C61C22×C412yes—
127C22×C4×C61C22×C412yes—
128C22×C4×C61C22×C412yes—
129C22×C4×C61C22×C414yes—
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2×D414yes—
132C22×C4×C61#4112yes—
133C22×C4×C61#4112yes—
134C22×C4×C61#4112yes—
135C22×C4×C61#4114yes—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—21no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—21no—
150C22×Dic61—22no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—22no—
154C22×Dic61—21no—
155C22×Dic61—22no—
156C22×Dic61—22no—
157C22×Dic61—22no—
158C22×Dic61—22no—
159C22×Dic61—21no—
160C22×Dic61C22×C412yes—
161C22×Dic61C22×C414yes—
162C22×Dic61C22×C414yes—
163C22×Dic61C22×C412yes—
164C22×Dic61C22×C414yes—
165C22×Dic61C22×C412yes—
166C22×Dic61#4114yes—
167C2×C4×D61—28no—
168C2×C4×D61—28no—
169C2×C4×D61—44no—
170C2×C4×D61—48no—
171C2×C4×D61—48no—
172C42×C61—44no—
173C42×C61—48no—
174#11758198—28no—
175#11758198—216no—
176#11758198—48no—
177#11758198C42116yescomplemented
178C4⋊Dic61—48no—
179C4×Dic61—28no—
180C4×Dic61—216no—
181C4×Dic61—48no—
182C23×C61C23×C411yescharacteristic
183C23×C61C23×C412yes—
184C23×C61C22×D411yescharacteristic
185C23×C61C2×#4112yes—
186C23×C61C2×#4111yescharacteristic
187C23×C61C2×#4111yescharacteristic
188C23×C61C2×#4111yescharacteristic
189C23×C61C2×#4112yes—
190C23×C61H(5,𝔽2)12yes—
191C23×C61H(5,𝔽2)11yescharacteristic
192C23×C612−1+411yescharacteristic
193C22×D61—22no—
194C22×D61—22no—
195C22×D61—24no—
196C22×D61—22no—
197C22×D61—24no—
198C22×D61—24no—
199C22×D61—22no—
200C22×D61—24no—
201C22×D61—24no—
202C22×D61—22no—
203C22×D61—22no—
204C22×D61—24no—
205C22×D61—42no—
206C22×D61—41no—
207C22×D61—41no—
208C22×D61—42no—
209C22×D61—42no—
210C22×D61—42no—
211C22×D61—42no—
212C22×D61—42no—
213C22×D61—41no—
214C22×D61—41no—
215C22×D61C2×C4214yes—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—24no—
219C2×C4×C61—24no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—24no—
224C2×C4×C61—24no—
225C2×C4×C61—22no—
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—24no—
234C2×C4×C61—24no—
235C2×C4×C61—24no—
236C2×C4×C61—24no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61C2×C4214yes—
242C2×C4×C61#11714yes—
243C2×C4×C61C4×D418yes—
244C2×C4×C61C4×D414yes—
245C2×Dic61—22no—
246C2×Dic61—24no—
247C2×Dic61—22no—
248C2×Dic61—22no—
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—42no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61—42no—
270C2×Dic61—41no—
271C2×Dic61—42no—
272C2×Dic61—42no—
273C2×Dic61—41no—
274C2×Dic61C2×C4214yes—
275C2×Dic61#11714yes—
276C4×D61—416no—
277#13816—611nomaximal, Sylow
278C22×C61C22×C4211yescharacteristic
279C22×C61C2×#11711yescharacteristic
280C22×C61C2×#11711yescharacteristic
281C22×C61C2×C4×D411yescharacteristic
282C22×C61C2×C4×D412yes—
283C22×C61C4×#4112yes—
284C22×C61C4×#4111yescharacteristic
285C22×C61C4×#4112yes—
286C22×C61C4×#4112yes—
287C22×C61#51812yes—
288C22×C61#51811yescharacteristic
289C22×C61#52012yes—
290C22×C61#52011yescharacteristic
291C22×C61#54612yes—
292C22×C61#54711yescharacteristic
293C22×C61#55011yescharacteristic
294C22×C61#55112yes—
295C22×C61#55112yes—
296C22×C61#55212yes—
297C22×C61#55312yes—
298C22×C61#55512yes—
299C22×C61#55511yescharacteristic
300C22×C61#55611yescharacteristic
301C2×D61—24no—
302C2×D61—24no—
303C2×D61—24no—
304C2×D61—42no—
305C2×D61—42no—
306C2×D61—42no—
307C2×D61—42no—
308C2×D61—44no—
309C2×D61—44no—
310C2×D61—44no—
311C2×D61—44no—
312C4×C61—24no—
313C4×C61—24no—
314C4×C61—28no—
315C4×C61—24no—
316C4×C61—44no—
317Dic61—24no—
318Dic61—24no—
319Dic61—42no—
320Dic61—44no—
321Dic61—44no—
322C2×#388—612no—
323C2×C4×C22⋊C4—611no—
324C2×C4×C22⋊C4—611no—
325C2×#376—611no—
326#1437—612no—
327#1436—612no—
328C4⋊#95—611no—
329#1434—611no—
330C22⋊#95—611no—
331C4×#95—612no—
332C4×#95D6111yescharacteristic, complemented
333C2×C61C4×#11711yescharacteristic
334C2×C61C42×D411yescharacteristic
335C2×C61#227511yescharacteristic
336C2×C61#234211yescharacteristic
337C2×C61#234611yescharacteristic
338C2×C61#235311yescharacteristic
339C2×C61#235411yescharacteristic
340C2×C61#235912yes—
341C2×C61#236312yes—
342C2×C61#237212yes—
343C2×C61#237312yes—
344D61—44no—
345C24×C4—611no—
346C22×C4⋊C4—612no—
347C22×C22⋊C4—611no—
348C22×C22⋊C4—611no—
349C22×C22⋊C4—612no—
350C22×C22⋊C4—612no—
351C22×C42—612no—
352C22×C42—611no—
353C22×C42—612no—
354C22×C42C2×D6112yes—
355C22×C42C2×D6111yescharacteristic
356#388—6116no—
357C4×C22⋊C4—618no—
358C4×C22⋊C4—1224no—
359#376—1224no—
360C2×#95—611no—
361C2×#95—612no—
362C2×#95—611no—
363C2×#95—612no—
364C2×#95—612no—
365C2×#95—611no—
366C2×#95—611no—
367C2×#95—612no—
368C2×#95—612no—
369C2×#95C2×D6111yescharacteristic
370C2×#95C2×D6112yes—
371C2×#95C2×D6111yescharacteristic
372C61#1381611yescharacteristic, complemented, Sylow
373C25—611no—
374C23×C4—611no—
375C23×C4—611no—
376C23×C4—612no—
377C23×C4—612no—
378C23×C4—611no—
379C23×C4—614no—
380C23×C4—1224no—
381C23×C4—1222no—
382C23×C4—1222no—
383C23×C4—2442no—
384C23×C4C22×D6112yes—
385C23×C4C22×D6111yescharacteristic
386C23×C4C22×D6112yes—
387C23×C4C22×D6111yescharacteristic
388C23×C4C22×D6111yescharacteristic
389C2×C4⋊C4—618no—
390C2×C4⋊C4—2442no—
391C2×C4⋊C4—2442no—
392C2×C22⋊C4—614no—
393C2×C22⋊C4—618no—
394C2×C22⋊C4—614no—
395C2×C22⋊C4—618no—
396C2×C22⋊C4—618no—
397C2×C22⋊C4—1224no—
398C2×C22⋊C4—1224no—
399C2×C22⋊C4—1222no—
400C2×C22⋊C4—1222no—
401C2×C22⋊C4—2442no—
402C2×C22⋊C4—2442no—
403C2×C42—24no—
404C2×C42—24no—
405C2×C42—42no—
406C2×C42—42no—
407C2×C42—614no—
408C2×C42—618no—
409C2×C42—1222no—
410C2×C42—1224no—
411C2×C42—1222no—
412C2×C42—1224no—
413C2×C42—1224no—
414C2×C42—1224no—
415C2×C42—2442no—
416C2×C42C4×D6114yes—
417#95—22no—
418#95—614no—
419#95—614no—
420#95—618no—
421#95—1224no—
422#95—1222no—
423#95—1224no—
424#95—1222no—
425#95—1224no—
426#95—2442no—
427#95C4×D6118yescomplemented
428#95D24414yes—
429C24—614no—
430C24—612no—
431C24—1221no—
432C24—1222no—
433C24—1221no—
434C24—1222no—
435C24—1221no—
436C24—1221no—
437C24—1222no—
438C24—1221no—
439C24—1221no—
440C24C23×D6111yescharacteristic
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—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—612no—
459C22×C4—612no—
460C22×C4—614no—
461C22×C4—614no—
462C22×C4—612no—
463C22×C4—614no—
464C22×C4—614no—
465C22×C4—1228no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1222no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4—1221no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1221no—
478C22×C4—1222no—
479C22×C4—1221no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1221no—
484C22×C4—1222no—
485C22×C4—1222no—
486C22×C4—1221no—
487C22×C4—1222no—
488C22×C4—1222no—
489C22×C4—1221no—
490C22×C4—1228no—
491C22×C4—2444no—
492C22×C4—2448no—
493C22×C4—2448no—
494C22×C4C2×C4×D6114yes—
495C22×C4C2×C4×D6112yes—
496C22×C4C2×C4×D6114yes—
497C22×C4C2×C4×D6112yes—
498C22×C4C2×C4×D6114yes—
499C22×C4C2×C4×D6112yes—
500C22×C4C2×D24412yes—
501C22×C4C2×D24414yes—
502C22×C4#1175822212yes—
503C22×C4#1175822214yes—
504C22×C4#1175822412yes—
505C22×C4#1175822612yes—
506C4⋊C4—2448no—
507C22⋊C4—6116no—
508C22⋊C4—1228no—
509C22⋊C4—12216no—
510C22⋊C4—2448no—
511C42—44no—
512C42—48no—
513C42—12216no—
514C42—1228no—
515C42—2448no—
516C23—614no—
517C23—1222no—
518C23—1222no—
519C23—1224no—
520C23—1222no—
521C23—1224no—
522C23—1222no—
523C23—1224no—
524C23—1224no—
525C23—1224no—
526C23—1222no—
527C23—1224no—
528C23—1222no—
529C23—2442no—
530C23—2442no—
531C23—2441no—
532C23—2441no—
533C23—2442no—
534C23—2441no—
535C23—2441no—
536C23—2442no—
537C23—2442no—
538C23—2442no—
539C23C22×C4×D6111yescharacteristic
540C23C22×C4×D6112yes—
541C23C22×D24411yescharacteristic
542C23#41×D6112yes—
543C23C2×#1175822411yescharacteristic
544C23C2×#1175822212yes—
545C23C2×#1175822211yescharacteristic
546C23C2×#1175822611yescharacteristic
547C23#1451982212yes—
548C23#1451982311yescharacteristic
549C23#1451982411yescharacteristic
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—614no—
576C2×C4—614no—
577C2×C4—1222no—
578C2×C4—1224no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—1224no—
582C2×C4—1222no—
583C2×C4—1224no—
584C2×C4—1224no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—1224no—
588C2×C4—1224no—
589C2×C4—1224no—
590C2×C4—1224no—
591C2×C4—1224no—
592C2×C4—1224no—
593C2×C4—1222no—
594C2×C4—24416no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4—2442no—
598C2×C4—2442no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2442no—
602C2×C4—2442no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2442no—
606C2×C4—2442no—
607C2×C4C42×D6114yes—
608C2×C4#1451973514yes—
609C2×C4C4×D24418yes—
610C2×C4C4×D24414yes—
611C22—1224no—
612C22—1224no—
613C22—1224no—
614C22—2442no—
615C22—2442no—
616C22—2442no—
617C22—2444no—
618C22—2444no—
619C22—2442no—
620C22—2444no—
621C22—2444no—
622C22C2×C42×D6111yescharacteristic
623C22#117×D6111yescharacteristic
624C22C2×#1451973511yescharacteristic
625C22C2×C4×D24411yescharacteristic
626C22C2×C4×D24412yes—
627C22C4×#1175822412yes—
628C22C4×#1175822211yescharacteristic
629C22C4×#1175822212yes—
630C22C4×#1175822612yes—
631C22#1684023412yes—
632C22#1684023511yescharacteristic
633C22#1684024112yes—
634C22#1684024211yescharacteristic
635C22#1684040712yes—
636C22#1684041611yescharacteristic
637C22#1684043411yescharacteristic
638C22#1684043812yes—
639C22#1684044112yes—
640C22#1684044512yes—
641C22#1684046012yes—
642C22#1684047512yes—
643C22#1684047711yescharacteristic
644C22#1684048511yescharacteristic
645C4—24no—
646C4—28no—
647C4—24no—
648C4—24no—
649C4—44no—
650C4—1224no—
651C4—1224no—
652C4—2442no—
653C4—2444no—
654C4—2444no—
655C2—2444no—
656C2C4×#1451973511yescharacteristic
657C2C42×D24411yescharacteristic
658C2#2144817811yescharacteristic
659C2#2144867111yescharacteristic
660C2#2144870511yescharacteristic
661C2#2144875511yescharacteristic
662C2#2144876611yescharacteristic
663C2#2144881312yes—
664C2#2144885912yes—
665C2#2144892712yes—
666C2#2144893312yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC22×C61C22×C42
FittingC4×#95×C61C2
socle, cosocleC24×C61C24
prime radicalC24×D61C23
hypercenterC4×#95D61
hypocenterC61#13816
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C24 ◁ C4×#95

Complements

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

NH ≅ G/Ntype
4. C2×#16839607C2semidirect
6. #21444053C2semidirect
7. #21444049C2semidirect
8. #21444043C2semidirect
10. C4×#14519684C2semidirect
11. C4×#95×C61C2semidirect
21. #16839654C4semidirect
22. C4×#11758198C4semidirect
52. C2×C4⋊Dic61C2×C4semidirect
72. C2×C4×Dic61C2×C4semidirect
81. #14519684C2×C4semidirect
93. #95×C61C2×C4semidirect
177. #11758198C42semidirect
332. C4×#95D61semidirect
372. C61#13816semidirect
427. #95C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168396542
C2×C4×#117581982
C2×#168396092
C2×#168396072
#214440572
#214440532
#214440492
#214440432
#214440402
C4×#145196842
C4×#95×C612
#1381661