Finite Group Databasebeta

← #29602127

Subgroups — #29602127

Order 15616.

1602 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839629C211yesmaximal, characteristic, complemented
2C2×C4×#11758198C211yesmaximal, characteristic
3C2×#16839609C211yesmaximal, characteristic
4#21444067C212yesmaximal
5#21444061C212yesmaximal
6#21444051C212yesmaximal, complemented
7#21444046C211yesmaximal, characteristic, complemented
8#21444045C212yesmaximal, complemented
9#21444043C211yesmaximal, characteristic, complemented
10#21444040C211yesmaximal, characteristic
11C61×C4⋊#95C211yesmaximal, characteristic, complemented
12C23×C4×D61C2211yescharacteristic
13C22×#11758198C2212yes—
14C22×#11758198C2211yescharacteristic
15C22×#11758198C2211yescharacteristic
16C22×#11758198C2212yes—
17C22×C4×Dic61C2212yes—
18C22×C4×Dic61C2211yescharacteristic
19#16839629—24no—
20C4×#11758198C418yescomplemented
21#16839609—24no—
22C2×#14519684C2212yes—
23C2×#14519684C2212yes—
24C2×#14519684C2211yescharacteristic
25C2×#14519684C2212yes—
26C2×#14519684C2212yes—
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2212yes—
30C2×#14519684C2211yescharacteristic
31C2×#14519684C2211yescharacteristic
32C2×#14519684C2211yescharacteristic
33C22×C4⋊C4×C61C2212yes—
34C22×C42×C61C2211yescharacteristic
35C2×#95×C61C2211yescharacteristic
36C2×#95×C61C2211yescharacteristic
37C2×#95×C61C2212yes—
38C24×D61C2311yescharacteristic
39C23×Dic61C2312yes—
40C23×Dic61C2311yescharacteristic
41C23×Dic61C2311yescharacteristic
42C23×Dic61C2312yes—
43C23×Dic61C2311yescharacteristic
44C22×C4×D61—22no—
45C22×C4×D61—22no—
46C22×C4×D61—24no—
47C22×C4×D61—42no—
48C22×C4×D61C2×C414yes—
49C2×#11758198—24no—
50C2×#11758198—24no—
51C2×#11758198—24no—
52C2×#11758198—22no—
53C2×#11758198—22no—
54C2×#11758198—24no—
55C2×#11758198—24no—
56C2×#11758198—42no—
57C2×#11758198—42no—
58C2×#11758198C2×C414yes—
59C2×#11758198C2×C414yes—
60C2×C4×Dic61—22no—
61C2×C4×Dic61—22no—
62C2×C4×Dic61—24no—
63C2×C4×Dic61—24no—
64C2×C4×Dic61—42no—
65C2×C4×Dic61C2×C414yes—
66#14519684—24no—
67#14519684—24no—
68#14519684—22no—
69#14519684—24no—
70#14519684—24no—
71#14519684—24no—
72#14519684—42no—
73#14519684—42no—
74#14519684C2×C414yes—
75#14519684C2×C414yes—
76#14519684D414yes—
77C23×C4×C61C2312yes—
78C23×C4×C61C2311yescharacteristic
79C23×C4×C61C2312yes—
80C23×C4×C61C2311yescharacteristic
81C23×C4×C61C2311yescharacteristic
82C2×C4⋊C4×C61—24no—
83C2×C4⋊C4×C61—24no—
84C2×C4⋊C4×C61—42no—
85C2×C42×C61—42no—
86C2×C42×C61C2×C414yes—
87#95×C61—22no—
88#95×C61—22no—
89#95×C61—42no—
90C24×C61C2411yescharacteristic
91C23×D61—21no—
92C23×D61—22no—
93C23×D61—21no—
94C23×D61—21no—
95C23×D61—21no—
96C23×D61—22no—
97C23×D61—22no—
98C23×D61—21no—
99C23×D61—21no—
100C23×D61C22×C412yes—
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—21no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—21no—
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—22no—
123C22×C4×C61C22×C412yes—
124C22×C4×C61C22×C412yes—
125C22×C4×C61C22×C412yes—
126C22×C4×C61C2×D412yes—
127C22×C4×C61#4114yes—
128C22×C4×C61#4112yes—
129C22×C4×C61#4112yes—
130C22×C4×C61#4112yes—
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61—22no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—21no—
147C22×Dic61—22no—
148C22×Dic61—21no—
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×Dic61C22×C412yes—
158C22×Dic61C22×C412yes—
159C22×Dic61C22×C412yes—
160C22×Dic61C2×D414yes—
161C22×Dic61#4114yes—
162C2×C4×D61—28no—
163C2×C4×D61—28no—
164C2×C4×D61—44no—
165C2×C4×D61—48no—
166C2×C4×D61—48no—
167C4⋊C4×C61—48no—
168C42×C61—44no—
169#11758198—216no—
170#11758198—28no—
171#11758198—28no—
172#11758198—48no—
173C4×Dic61—28no—
174C4×Dic61—48no—
175C23×C61C23×C411yescharacteristic
176C23×C61C22×D411yescharacteristic
177C23×C61C2×#4112yes—
178C23×C61C2×#4112yes—
179C23×C61C2×#4111yescharacteristic
180C23×C61C2×#4111yescharacteristic
181C23×C61C2×#4111yescharacteristic
182C23×C61C2×#4112yes—
183C23×C61H(5,𝔽2)12yes—
184C23×C61H(5,𝔽2)11yescharacteristic
185C23×C612−1+411yescharacteristic
186C22×D61—22no—
187C22×D61—22no—
188C22×D61—22no—
189C22×D61—22no—
190C22×D61—22no—
191C22×D61—22no—
192C22×D61—24no—
193C22×D61—24no—
194C22×D61—24no—
195C22×D61—24no—
196C22×D61—24no—
197C22×D61—24no—
198C22×D61—42no—
199C22×D61—41no—
200C22×D61—41no—
201C22×D61—42no—
202C22×D61—42no—
203C22×D61—42no—
204C22×D61—42no—
205C22×D61—41no—
206C22×D61—42no—
207C22×D61—41no—
208C22×D61#11714yes—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—24no—
214C2×C4×C61—24no—
215C2×C4×C61—24no—
216C2×C4×C61—24no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—24no—
221C2×C4×C61—24no—
222C2×C4×C61—24no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
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#11714yes—
237C2×C4×C61C4×D414yes—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—24no—
241C2×Dic61—24no—
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—24no—
249C2×Dic61—24no—
250C2×Dic61—24no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—42no—
258C2×Dic61—41no—
259C2×Dic61—41no—
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#11714yes—
269C4×D61—416no—
270#13892—611nomaximal, Sylow
271C22×C61C2×#11711yescharacteristic
272C22×C61C2×#11711yescharacteristic
273C22×C61C2×#11711yescharacteristic
274C22×C61C2×C4×D411yescharacteristic
275C22×C61C4×#4111yescharacteristic
276C22×C61#51811yescharacteristic
277C22×C61#52011yescharacteristic
278C22×C61#52512yes—
279C22×C61#52912yes—
280C22×C61#52912yes—
281C22×C61#52912yes—
282C22×C61#53812yes—
283C22×C61#53912yes—
284C22×C61#54612yes—
285C22×C61#54711yescharacteristic
286C22×C61#55011yescharacteristic
287C22×C61#55112yes—
288C22×C61#55112yes—
289C22×C61#55312yes—
290C22×C61#55312yes—
291C22×C61#55511yescharacteristic
292C22×C61#55612yes—
293C22×C61#55611yescharacteristic
294C2×D61—24no—
295C2×D61—24no—
296C2×D61—24no—
297C2×D61—42no—
298C2×D61—42no—
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—44no—
302C2×D61—44no—
303C2×D61—44no—
304C2×D61—44no—
305C4×C61—24no—
306C4×C61—24no—
307C4×C61—42no—
308C4×C61—44no—
309C4×C61—82no—
310Dic61—24no—
311Dic61—42no—
312Dic61—44no—
313Dic61—42no—
314Dic61—82no—
315C2×#383—611no—
316C2×C4×C22⋊C4—611no—
317C2×C4×C22⋊C4—611no—
318#1440—612no—
319#1439—612no—
320#1436—612no—
321C4⋊#95D6111yescharacteristic, complemented
322#1434—611no—
323#1434—611no—
324#1434—612no—
325C22⋊#95—611no—
326C2×C61#228311yescharacteristic
327C2×C61#229011yescharacteristic
328C2×C61#231811yescharacteristic
329C2×C61#234211yescharacteristic
330C2×C61#234711yescharacteristic
331C2×C61#235311yescharacteristic
332C2×C61#235411yescharacteristic
333C2×C61#246612yes—
334C2×C61#246712yes—
335C2×C61#251812yes—
336C2×C61#252312yes—
337D61—44no—
338C24×C4—611no—
339C22×C4⋊C4C2×D6112yes—
340C22×C22⋊C4—611no—
341C22×C22⋊C4—611no—
342C22×C22⋊C4—612no—
343C22×C22⋊C4—612no—
344C22×C42—611no—
345C22×C42—612no—
346C22×C42C2×D6111yescharacteristic
347#383—1224no—
348C4×C22⋊C4—618no—
349C4×C22⋊C4—1224no—
350C2×#95—611no—
351C2×#95—611no—
352C2×#95—611no—
353C2×#95—611no—
354C2×#95—612no—
355C2×#95—612no—
356C2×#95—612no—
357C2×#95—612no—
358C2×#95—612no—
359C2×#95—612no—
360C2×#95—612no—
361C2×#95C2×D6111yescharacteristic
362C2×#95C2×D6111yescharacteristic
363C2×#95C2×D6112yes—
364C61#1389211yescharacteristic, complemented, Sylow
365C25—611no—
366C23×C4—611no—
367C23×C4—611no—
368C23×C4—611no—
369C23×C4—612no—
370C23×C4—614no—
371C23×C4—612no—
372C23×C4—1224no—
373C23×C4—1222no—
374C23×C4—1222no—
375C23×C4—2442no—
376C23×C4C22×D6112yes—
377C23×C4C22×D6111yescharacteristic
378C23×C4C22×D6112yes—
379C23×C4C22×D6111yescharacteristic
380C23×C4C22×D6111yescharacteristic
381C2×C4⋊C4—24no—
382C2×C4⋊C4—24no—
383C2×C4⋊C4—42no—
384C2×C22⋊C4—614no—
385C2×C22⋊C4—614no—
386C2×C22⋊C4—1224no—
387C2×C22⋊C4—1224no—
388C2×C22⋊C4—1224no—
389C2×C22⋊C4—1224no—
390C2×C22⋊C4—1222no—
391C2×C22⋊C4—1222no—
392C2×C22⋊C4—1224no—
393C2×C22⋊C4—2442no—
394C2×C22⋊C4—2442no—
395C2×C42—42no—
396C2×C42—614no—
397C2×C42—1222no—
398C2×C42—1222no—
399C2×C42—1224no—
400C2×C42—1224no—
401C2×C42—2442no—
402C2×C42C4×D6114yes—
403#95—22no—
404#95—22no—
405#95—42no—
406#95—614no—
407#95—614no—
408#95—614no—
409#95—1224no—
410#95—1224no—
411#95—1224no—
412#95—1224no—
413#95—1224no—
414#95—1222no—
415#95—2442no—
416#95—2442no—
417C24—612no—
418C24—614no—
419C24—1221no—
420C24—1222no—
421C24—1221no—
422C24—1222no—
423C24—1221no—
424C24—1221no—
425C24—1221no—
426C24—1222no—
427C24—1221no—
428C24C23×D6111yescharacteristic
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—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—612no—
451C22×C4—614no—
452C22×C4—612no—
453C22×C4—612no—
454C22×C4—614no—
455C22×C4—1228no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—1222no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—1222no—
462C22×C4—1222no—
463C22×C4—1222no—
464C22×C4—1221no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—1222no—
473C22×C4—1221no—
474C22×C4—1222no—
475C22×C4—1228no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—2444no—
484C22×C4—2448no—
485C22×C4—2448no—
486C22×C4C2×C4×D6112yes—
487C22×C4C2×C4×D6112yes—
488C22×C4C2×C4×D6112yes—
489C22×C4#1175822214yes—
490C22×C4D4×D6112yes—
491C22×C4#1175822412yes—
492C22×C4#1175822612yes—
493C22×C4#1175822612yes—
494C4⋊C4—48no—
495C22⋊C4—1228no—
496C22⋊C4—12216no—
497C22⋊C4—1228no—
498C22⋊C4—2448no—
499C42—44no—
500C42—1228no—
501C42—2448no—
502C23—614no—
503C23—1222no—
504C23—1222no—
505C23—1222no—
506C23—1222no—
507C23—1222no—
508C23—1222no—
509C23—1224no—
510C23—1224no—
511C23—1224no—
512C23—1224no—
513C23—1224no—
514C23—1224no—
515C23—2442no—
516C23—2442no—
517C23—2441no—
518C23—2442no—
519C23—2441no—
520C23—2442no—
521C23—2441no—
522C23—2441no—
523C23—2442no—
524C23—2442no—
525C23C22×C4×D6111yescharacteristic
526C23C2×D4×D6111yescharacteristic
527C23#41×D6112yes—
528C23#41×D6112yes—
529C23C2×#1175822411yescharacteristic
530C23C2×#1175822212yes—
531C23C2×#1175822611yescharacteristic
532C23C2×#1175822611yescharacteristic
533C23#1451982212yes—
534C23#1451982211yescharacteristic
535C23#1451982511yescharacteristic
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—614no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—1224no—
567C2×C4—1224no—
568C2×C4—1222no—
569C2×C4—1222no—
570C2×C4—1222no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—1224no—
575C2×C4—1224no—
576C2×C4—1224no—
577C2×C4—24416no—
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—2441no—
587C2×C4—2442no—
588C2×C4—2441no—
589C2×C4—2442no—
590C2×C4—2442no—
591C2×C4—2442no—
592C2×C4—2442no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4#1451973514yes—
596C2×C4#1451973914yes—
597C22—1224no—
598C22—1224no—
599C22—1224no—
600C22—2442no—
601C22—2442no—
602C22—2444no—
603C22—2442no—
604C22—2444no—
605C22—2444no—
606C22—2442no—
607C22—2444no—
608C22#117×D6111yescharacteristic
609C22#117×D6111yescharacteristic
610C22C2×#1451973511yescharacteristic
611C22C2×#1451973911yescharacteristic
612C22C4×#1175822211yescharacteristic
613C22#1684023511yescharacteristic
614C22#1684024311yescharacteristic
615C22#1684027112yes—
616C22#1684029112yes—
617C22#1684029712yes—
618C22#1684029912yes—
619C22#1684034412yes—
620C22#1684035812yes—
621C22#1684040112yes—
622C22#1684041511yescharacteristic
623C22#1684043211yescharacteristic
624C22#1684043812yes—
625C22#1684043912yes—
626C22#1684046012yes—
627C22#1684046112yes—
628C22#1684047811yescharacteristic
629C22#1684048412yes—
630C22#1684048611yescharacteristic
631C4—24no—
632C4—24no—
633C4—42no—
634C4—44no—
635C4—82no—
636C4—1224no—
637C4—4882no—
638C4—2442no—
639C4—2442no—
640C4—2444no—
641C2—2444no—
642C2#2144821211yescharacteristic
643C2#2144826011yescharacteristic
644C2#2144845411yescharacteristic
645C2#2144866511yescharacteristic
646C2#2144871311yescharacteristic
647C2#2144875511yescharacteristic
648C2#2144877011yescharacteristic
649C2#2144986112yes—
650C2#2144987612yes—
651C2#2145047612yes—
652C2#2145054912yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×C4⋊#95C2
socle, cosocleC24×C61C24
prime radicalC24×D61C23
hypercenterC4⋊#95D61
hypocenterC61#13892
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
1. C2×#16839629C2semidirect
6. #21444051C2semidirect
7. #21444046C2semidirect
8. #21444045C2semidirect
9. #21444043C2semidirect
11. C61×C4⋊#95C2semidirect
20. C4×#11758198C4semidirect
321. C4⋊#95D61semidirect
364. C61#13892semidirect

Maximal subgroups

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

subgroupindex
C2×#168396292
C2×C4×#117581982
C2×#168396092
#214440672
#214440612
#214440512
#214440462
#214440452
#214440432
#214440402
C61×C4⋊#952
#1389261