Finite Group Databasebeta

← #29601404

Subgroups — #29601404

Order 15616.

1958 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×C4×#11758198C211yesmaximal, characteristic
2C2×#16839609C211yesmaximal, characteristic
3C2×#95×D61C211yesmaximal, characteristic
4#21444046C211yesmaximal, characteristic, complemented
5#21444045C212yesmaximal, complemented
6#21444044C211yesmaximal, characteristic, complemented
7#21444044C211yesmaximal, characteristic, complemented
8#21444043C212yesmaximal, complemented
9#21444040C212yesmaximal
10#21444040C212yesmaximal
11C61×#1434C211yesmaximal, characteristic, complemented
12C23×C4×D61C2212yes—
13C23×C4×D61C2211yescharacteristic
14C22×#11758198C2212yes—
15C22×#11758198C2211yescharacteristic
16C22×#11758198C2211yescharacteristic
17C22×C4×Dic61C2212yes—
18C22×C4×Dic61C2211yescharacteristic
19C4×#11758198C418yescomplemented
20#16839609C418yescomplemented
21C2×#14519684C2212yes—
22C2×#14519684C2212yes—
23C2×#14519684C2212yes—
24C2×#14519684C2212yes—
25C2×#14519684C2212yes—
26C2×#14519684C2212yes—
27C2×#14519684C2212yes—
28C2×#14519684C2211yescharacteristic
29C2×#14519684C2211yescharacteristic
30C2×#14519684C2211yescharacteristic
31C2×#14519684C2211yescharacteristic
32#95×D61C418yescomplemented
33C22×C42×C61C2211yescharacteristic
34C2×#95×C61C2212yes—
35C2×#95×C61C2212yes—
36C2×#95×C61C2211yescharacteristic
37C2×#95×C61C2211yescharacteristic
38C24×D61C2311yescharacteristic
39C23×Dic61C2312yes—
40C23×Dic61C2311yescharacteristic
41C23×Dic61C2311yescharacteristic
42C23×Dic61C2312yes—
43C23×Dic61C2311yescharacteristic
44C22×C4×D61—24no—
45C22×C4×D61—24no—
46C22×C4×D61—42no—
47C22×C4×D61—42no—
48C22×C4×D61—42no—
49C22×C4×D61—42no—
50C22×C4×D61—42no—
51C22×C4×D61—42no—
52C22×C4×D61C2×C418yes—
53C22×C4×D61C2×C414yes—
54C22×C4×D61C2×C414yes—
55C22×C4×D61C2×C414yes—
56C2×#11758198—24no—
57C2×#11758198—24no—
58C2×#11758198—24no—
59C2×#11758198—24no—
60C2×#11758198C2×C418yes—
61C2×#11758198C2×C414yes—
62C2×#11758198C2×C414yes—
63C2×C4×Dic61—42no—
64C2×C4×Dic61—42no—
65C2×C4×Dic61—42no—
66C2×C4×Dic61C2×C418yescomplemented
67C2×C4×Dic61C2×C414yescomplemented
68#14519684—42no—
69#14519684—42no—
70#14519684—42no—
71#14519684—42no—
72#14519684—42no—
73#14519684—42no—
74#14519684C2×C414yes—
75#14519684C2×C418yes—
76#14519684C2×C414yescomplemented
77#14519684C2×C414yescomplemented
78#14519684C2×C414yescomplemented
79C23×C4×C61C2312yes—
80C23×C4×C61C2311yescharacteristic
81C23×C4×C61C2311yescharacteristic
82C23×C4×C61C2312yes—
83C23×C4×C61C2311yescharacteristic
84C2×C42×C61—42no—
85C2×C42×C61C2×C414yescomplemented
86#95×C61—42no—
87#95×C61—42no—
88#95×C61C2×C414yes—
89#95×C61C2×C414yescomplemented
90C24×C61C2411yescharacteristic
91C23×D61—22no—
92C23×D61—22no—
93C23×D61—21no—
94C23×D61—21no—
95C23×D61—21no—
96C23×D61—21no—
97C23×D61C22×C412yes—
98C23×D61C22×C412yes—
99C23×D61C22×C412yes—
100C23×D61#4114yes—
101C23×D61#4114yes—
102C22×C4×C61—22no—
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—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—21no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—21no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61C22×C414yes—
125C22×C4×C61C22×C414yes—
126C22×C4×C61C22×C412yes—
127C22×C4×C61C22×C412yes—
128C22×C4×C61C22×C412yes—
129C22×C4×C61C22×C412yes—
130C22×C4×C61C22×C412yes—
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61—22no—
153C22×Dic61C22×C414yes—
154C22×Dic61C22×C414yes—
155C22×Dic61C22×C412yes—
156C22×Dic61C22×C412yes—
157C22×Dic61C22×C412yes—
158C22×Dic61C22×C412yes—
159C22×Dic61C22×C412yes—
160C2×C4×D61—216no—
161C2×C4×D61—48no—
162C2×C4×D61—44no—
163C2×C4×D61—44no—
164C2×C4×D61—48no—
165C2×C4×D61—48no—
166C2×C4×D61—48no—
167C2×C4×D61—48no—
168C2×C4×D61—48no—
169C2×C4×D61—48no—
170C2×C4×D61—48no—
171C2×C4×D61C42116yes—
172C42×C61—44no—
173#11758198—216no—
174#11758198—28no—
175#11758198—28no—
176C4×Dic61—48no—
177C4×Dic61—44no—
178C23×C61C23×C411yescharacteristic
179C23×C61C23×C411yescharacteristic
180C23×C61C23×C411yescharacteristic
181C23×C61C2×#4112yes—
182C23×C61C2×#4112yes—
183C23×C61H(5,𝔽2)12yes—
184C23×C61H(5,𝔽2)12yes—
185C23×C61H(5,𝔽2)11yescharacteristic
186C23×C61H(5,𝔽2)11yescharacteristic
187C23×C612−1+411yescharacteristic
188C23×C612−1+411yescharacteristic
189C22×D61—22no—
190C22×D61—24no—
191C22×D61—24no—
192C22×D61—22no—
193C22×D61—22no—
194C22×D61—22no—
195C22×D61—22no—
196C22×D61—22no—
197C22×D61—22no—
198C22×D61—22no—
199C22×D61—24no—
200C22×D61—24no—
201C22×D61—24no—
202C22×D61—24no—
203C22×D61—24no—
204C22×D61—24no—
205C22×D61—24no—
206C22×D61—24no—
207C22×D61C2×C4214yes—
208C22×D61#11714yes—
209C22×D61#11718yes—
210C22×D61#11714yes—
211C22×D61#11718yes—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—24no—
215C2×C4×C61—24no—
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—22no—
223C2×C4×C61—24no—
224C2×C4×C61—24no—
225C2×C4×C61—24no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
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×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61C2×C4214yes—
241C2×Dic61—24no—
242C2×Dic61—24no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—24no—
247C2×Dic61—22no—
248C2×Dic61—24no—
249C2×Dic61—24no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—24no—
253C2×Dic61—24no—
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—24no—
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×Dic61C2×C4214yes—
270C4×D61—416no—
271C4×D61—432no—
272#13796—611nomaximal, Sylow
273C22×C61C22×C4211yescharacteristic
274C22×C61C2×#11712yes—
275C22×C61C2×#11711yescharacteristic
276C22×C61C2×#11712yes—
277C22×C61C2×#11711yescharacteristic
278C22×C61#51811yescharacteristic
279C22×C61#51812yes—
280C22×C61#51812yes—
281C22×C61#51811yescharacteristic
282C22×C61#51811yescharacteristic
283C22×C61#51911yescharacteristic
284C22×C61#52011yescharacteristic
285C22×C61#52011yescharacteristic
286C22×C61#52011yescharacteristic
287C22×C61#52011yescharacteristic
288C22×C61#53812yes—
289C22×C61#53812yes—
290C22×C61#53812yes—
291C22×C61#53812yes—
292C22×C61#53912yes—
293C22×C61#53912yes—
294C22×C61#53912yes—
295C22×C61#53912yes—
296C2×D61—28no—
297C2×D61—28no—
298C2×D61—24no—
299C2×D61—28no—
300C2×D61—24no—
301C2×D61—28no—
302C2×D61—24no—
303C2×D61—24no—
304C2×D61—24no—
305C2×D61—24no—
306C2×D61#37618yes—
307C4×C61—24no—
308C4×C61—44no—
309C4×C61—42no—
310C4×C61—44no—
311C4×C61—42no—
312Dic61—24no—
313Dic61—44no—
314Dic61—42no—
315Dic61—44no—
316Dic61—42no—
317C2×C4×C22⋊C4—611no—
318C2×C4×C22⋊C4—611no—
319C22×#95—611no—
320#1434—611no—
321#1434—612no—
322#1434—611no—
323#1434—611no—
324#1434—612no—
325#1434D6111yescharacteristic, complemented
326C22⋊#95—612no—
327C22⋊#95—612no—
328C2×C61C2×#37611yescharacteristic
329C2×C61#227311yescharacteristic
330C2×C61#227411yescharacteristic
331C2×C61#231212yes—
332C2×C61#231312yes—
333C2×C61#231811yescharacteristic
334C2×C61#231812yes—
335C2×C61#231811yescharacteristic
336C2×C61#231812yes—
337C2×C61#231811yescharacteristic
338C2×C61#231811yescharacteristic
339D61—28no—
340C24×C4—611no—
341C24×C4—612no—
342C22×C22⋊C4—611no—
343C22×C22⋊C4—611no—
344C22×C22⋊C4—612no—
345C22×C42—612no—
346C22×C42—611no—
347C22×C42C2×D6111yescharacteristic
348C4×C22⋊C4—618no—
349C4×C22⋊C4—618no—
350C2×#95—612no—
351C2×#95—611no—
352C2×#95—611no—
353C2×#95—611no—
354C2×#95—611no—
355C2×#95—618no—
356C2×#95—612no—
357C2×#95—612no—
358C2×#95—612no—
359C2×#95—612no—
360C2×#95—612no—
361C2×#95—612no—
362C2×#95C2×D6112yes—
363C2×#95C2×D6112yes—
364C2×#95C2×D6111yescharacteristic
365C2×#95C2×D6111yescharacteristic
366C61#1379611yescharacteristic, complemented, Sylow
367C25—611no—
368C23×C4—611no—
369C23×C4—614no—
370C23×C4—614no—
371C23×C4—614no—
372C23×C4—611no—
373C23×C4—612no—
374C23×C4—611no—
375C23×C4—612no—
376C23×C4—618no—
377C23×C4—1224no—
378C23×C4—1224no—
379C23×C4—2442no—
380C23×C4—2442no—
381C23×C4—2442no—
382C23×C4—2442no—
383C23×C4—2442no—
384C23×C4—2442no—
385C23×C4C22×D6111yescharacteristic
386C23×C4C22×D6111yescharacteristic
387C23×C4C22×D6112yes—
388C23×C4C22×D6112yes—
389C23×C4C22×D6111yescharacteristic
390C2×C22⋊C4—614no—
391C2×C22⋊C4—614no—
392C2×C22⋊C4—618no—
393C2×C22⋊C4—1224no—
394C2×C22⋊C4—1224no—
395C2×C22⋊C4—1224no—
396C2×C22⋊C4—1224no—
397C2×C42—42no—
398C2×C42—618no—
399C2×C42—614no—
400C2×C42—2442no—
401C2×C42—2442no—
402C2×C42—2442no—
403C2×C42C4×D6114yescomplemented
404#95—42no—
405#95—42no—
406#95—618no—
407#95—614no—
408#95—614no—
409#95—614no—
410#95—614no—
411#95—2442no—
412#95—2442no—
413#95—2442no—
414#95—2442no—
415#95—2442no—
416#95—2442no—
417#95C4×D6114yes—
418#95C4×D6114yescomplemented
419C24—612no—
420C24—612no—
421C24—612no—
422C24—614no—
423C24—614no—
424C24—1222no—
425C24—1221no—
426C24—1222no—
427C24—1221no—
428C24—1221no—
429C24—1221no—
430C24C23×D6111yescharacteristic
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—612no—
454C22×C4—612no—
455C22×C4—612no—
456C22×C4—6116no—
457C22×C4—614no—
458C22×C4—614no—
459C22×C4—612no—
460C22×C4—612no—
461C22×C4—12216no—
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—1221no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1221no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1222no—
484C22×C4—2448no—
485C22×C4—2448no—
486C22×C4—2444no—
487C22×C4—2444no—
488C22×C4—2448no—
489C22×C4—2448no—
490C22×C4—2448no—
491C22×C4—2448no—
492C22×C4—2448no—
493C22×C4—2448no—
494C22×C4C2×C4×D6114yes—
495C22×C4C2×C4×D6112yes—
496C22×C4C2×C4×D6112yes—
497C22×C4C2×C4×D6114yes—
498C22×C4C2×C4×D6112yes—
499C22×C4C2×C4×D6112yes—
500C22×C4C2×C4×D6112yes—
501C22⋊C4—1228no—
502C22⋊C4—12216no—
503C22⋊C4—1228no—
504C42—44no—
505C42—2444no—
506C42—2448no—
507C23—614no—
508C23—614no—
509C23—618no—
510C23—614no—
511C23—618no—
512C23—1222no—
513C23—1224no—
514C23—1224no—
515C23—1222no—
516C23—1222no—
517C23—1222no—
518C23—1222no—
519C23—1222no—
520C23—1222no—
521C23—1222no—
522C23—1224no—
523C23—1224no—
524C23—1224no—
525C23—1224no—
526C23—1224no—
527C23—1224no—
528C23—1224no—
529C23—1224no—
530C23C22×C4×D6111yescharacteristic
531C23C22×C4×D6111yescharacteristic
532C23C22×C4×D6111yescharacteristic
533C23#41×D6112yes—
534C23#41×D6112yes—
535C23#1451982212yes—
536C23#1451982211yescharacteristic
537C23#1451982312yes—
538C23#1451982311yescharacteristic
539C23#1451982411yescharacteristic
540C23#1451982511yescharacteristic
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
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—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—614no—
570C2×C4—1224no—
571C2×C4—1224no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—1224no—
576C2×C4—1222no—
577C2×C4—1224no—
578C2×C4—1224no—
579C2×C4—1224no—
580C2×C4—1224no—
581C2×C4—1222no—
582C2×C4—1222no—
583C2×C4—1222no—
584C2×C4—1222no—
585C2×C4—1224no—
586C2×C4—24416no—
587C2×C4—24432no—
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×C4—2442no—
600C2×C4C42×D6114yes—
601C22—618no—
602C22—1224no—
603C22—1228no—
604C22—1224no—
605C22—1228no—
606C22—1224no—
607C22—1224no—
608C22—1228no—
609C22—1228no—
610C22—1224no—
611C22—1224no—
612C22C2×C42×D6111yescharacteristic
613C22#117×D6111yescharacteristic
614C22#117×D6112yes—
615C22#117×D6111yescharacteristic
616C22#117×D6112yes—
617C22#1684023412yes—
618C22#1684023411yescharacteristic
619C22#1684023412yes—
620C22#1684023511yescharacteristic
621C22#1684023611yescharacteristic
622C22#1684023811yescharacteristic
623C22#1684024111yescharacteristic
624C22#1684024211yescharacteristic
625C22#1684024311yescharacteristic
626C22#1684024411yescharacteristic
627C22#1684034412yes—
628C22#1684034612yes—
629C22#1684034812yes—
630C22#1684035112yes—
631C22#1684035312yes—
632C22#1684035512yes—
633C22#1684035812yes—
634C22#1684036012yes—
635C4—24no—
636C4—42no—
637C4—44no—
638C4—42no—
639C4—44no—
640C4—1224no—
641C4—2442no—
642C4—2444no—
643C4—2442no—
644C4—2444no—
645C2—1228no—
646C2#376×D6111yescharacteristic
647C2#2144817111yescharacteristic
648C2#2144817311yescharacteristic
649C2#2144841312yes—
650C2#2144841812yes—
651C2#2144844811yescharacteristic
652C2#2144844912yes—
653C2#2144845011yescharacteristic
654C2#2144845112yes—
655C2#2144845411yescharacteristic
656C2#2144845511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D61
commutatorC22×C61C22×C42
FittingC61×#1434C2
socle, cosocleC24×C61C24
prime radicalC24×D61C23
hypercenter#1434D61
hypocenterC61#13796
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
4. #21444046C2semidirect
5. #21444045C2semidirect
6. #21444044C2semidirect
7. #21444044C2semidirect
8. #21444043C2semidirect
11. C61×#1434C2semidirect
19. C4×#11758198C4semidirect
20. #16839609C4semidirect
32. #95×D61C4semidirect
66. C2×C4×Dic61C2×C4semidirect
67. C2×C4×Dic61C2×C4semidirect
76. #14519684C2×C4semidirect
77. #14519684C2×C4semidirect
78. #14519684C2×C4semidirect
85. C2×C42×C61C2×C4semidirect
89. #95×C61C2×C4semidirect
325. #1434D61semidirect
366. C61#13796semidirect
403. C2×C42C4×D61semidirect
418. #95C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117581982
C2×#168396092
C2×#95×D612
#214440462
#214440452
#214440442
#214440432
#214440402
C61×#14342
#1379661