Finite Group Databasebeta

← #29609023

Subgroups — #29609023

Order 15616.

1918 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519764C211yesmaximal, characteristic
2C2×#16839666C212yesmaximal, complemented
3#21444062C212yesmaximal, complemented
4#21444062C212yesmaximal, complemented
5#21444060C211yesmaximal, characteristic
6#21444056C212yesmaximal, complemented
7#21444056C212yesmaximal, complemented
8#21444043C212yesmaximal, complemented
9C61×#1438C211yesmaximal, characteristic
10C23×Dic122C2211yescharacteristic
11C2×#14519764—28no—
12C2×#14519764C418yescomplemented
13C22×#11758196C2212yes—
14C22×C22⋊Dic61C2211yescharacteristic
15C22×C22⋊Dic61C2212yes—
16C22×C22⋊Dic61C2212yes—
17C22×C22⋊Dic61C2212yes—
18C22×C22⋊Dic61C2211yescharacteristic
19C22×C4×Dic61C2211yescharacteristic
20C22×C4×Dic61C2212yes—
21#16839666—28no—
22C2×#14519684C2212yes—
23C2×#14519684C2212yes—
24C2×#14519684C2212yes—
25C2×#14519684C2212yes—
26C2×#14519684C2212yes—
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C23×D4×C61C2211yescharacteristic
30C22×C22⋊C4×C61C2211yescharacteristic
31C22×C22⋊C4×C61C2212yes—
32C22×C22⋊C4×C61C2211yescharacteristic
33C2×#95×C61C2212yes—
34C22×Dic122—41no—
35C22×Dic122—42no—
36C22×Dic122—41no—
37C22×Dic122C2×C414yes—
38C22×Dic122D418yes—
39C23×Dic61C2312yes—
40C23×Dic61C2312yes—
41C23×Dic61C2312yes—
42C23×Dic61C2312yes—
43#14519764—232no—
44C2×#11758196—24no—
45C2×#11758196—42no—
46C2×#11758196—42no—
47C2×C22⋊Dic61—24no—
48C2×C22⋊Dic61—24no—
49C2×C22⋊Dic61—24no—
50C2×C22⋊Dic61—24no—
51C2×C22⋊Dic61—24no—
52C2×C22⋊Dic61—24no—
53C2×C22⋊Dic61—41no—
54C2×C22⋊Dic61—42no—
55C2×C22⋊Dic61—42no—
56C2×C22⋊Dic61—42no—
57C2×C22⋊Dic61—41no—
58C2×C22⋊Dic61C2×C418yes—
59C2×C22⋊Dic61C2×C418yes—
60C2×C4×Dic61—24no—
61C2×C4×Dic61—24no—
62C2×C4×Dic61—42no—
63C2×C4×Dic61—42no—
64C2×C4×Dic61C2×C414yes—
65#14519684—24no—
66#14519684—24no—
67#14519684—24no—
68#14519684—24no—
69#14519684—24no—
70#14519684—42no—
71#14519684—42no—
72C25×C61C2311yescharacteristic
73C25×C61C2311yescharacteristic
74C22×D4×C61—24no—
75C22×D4×C61—42no—
76C22×D4×C61—41no—
77C22×D4×C61—41no—
78C22×D4×C61C2×C414yes—
79C23×C4×C61C2312yes—
80C23×C4×C61C2312yes—
81C23×C4×C61C2311yescharacteristic
82C2×C22⋊C4×C61—24no—
83C2×C22⋊C4×C61—24no—
84C2×C22⋊C4×C61—41no—
85C2×C22⋊C4×C61—42no—
86C2×C22⋊C4×C61—42no—
87C2×C22⋊C4×C61—42no—
88C2×C22⋊C4×C61—41no—
89#95×C61—24no—
90C24×C61—22no—
91C24×C61—22no—
92C24×C61—22no—
93C24×C61—22no—
94C24×C61—22no—
95C24×C61—22no—
96C24×C61—22no—
97C24×C61—21no—
98C24×C61—21no—
99C24×C61—22no—
100C24×C61—22no—
101C24×C61—22no—
102C24×C61—21no—
103C24×C61—21no—
104C24×C61—22no—
105C24×C61—22no—
106C24×C61C22×C412yes—
107C24×C61C22×C412yes—
108C24×C61C2411yescharacteristic
109C2×D4×C61—28no—
110C2×D4×C61—44no—
111C2×D4×C61—44no—
112C2×D4×C61—48no—
113C2×D4×C61—48no—
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×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61C22×C412yes—
131C22×C4×C61C2×D414yes—
132C22×C4×C61#4114yes—
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—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×Dic61C22×C414yes—
157C22×Dic61C22×C414yes—
158C22×Dic61C2×D414yes—
159C22×Dic61C2×D414yes—
160C22×Dic61#4114yes—
161C2×Dic122—44no—
162C2×Dic122—44no—
163C2×Dic122—48no—
164C2×Dic122—48no—
165C2×Dic122C22⋊C4116yescomplemented
166C22⋊C4×C61—44no—
167C22⋊C4×C61—48no—
168C22⋊C4×C61—44no—
169C22⋊Dic61—216no—
170C22⋊Dic61—216no—
171C22⋊Dic61—44no—
172C22⋊Dic61—48no—
173C22⋊Dic61—44no—
174#11758196—48no—
175C4×Dic61—28no—
176C4×Dic61—48no—
177C23×C61—22no—
178C23×C61—22no—
179C23×C61—24no—
180C23×C61—24no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—24no—
186C23×C61—24no—
187C23×C61—42no—
188C23×C61—42no—
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×C61—42no—
198C23×C61—42no—
199C23×C61—42no—
200C23×C61—42no—
201C23×C61—42no—
202C23×C61—42no—
203C23×C61—42no—
204C23×C61—42no—
205C23×C61—42no—
206C23×C61—42no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—42no—
212C23×C61—42no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61—42no—
216C23×C61—42no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61C23×C411yescharacteristic
220C23×C61C22×D412yes—
221C23×C61C2×#4112yes—
222C23×C61C2×#4112yes—
223C23×C61H(5,𝔽2)12yes—
224C23×C61H(5,𝔽2)11yescharacteristic
225C23×C61H(5,𝔽2)11yescharacteristic
226C23×C61H(5,𝔽2)12yes—
227C23×C612−1+412yes—
228D4×C61—416no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—24no—
232C2×C4×C61—24no—
233C2×C4×C61—24no—
234C2×C4×C61—24no—
235C2×C4×C61—24no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61C2×C22⋊C414yes—
248C2×Dic61—24no—
249C2×Dic61—24no—
250C2×Dic61—24no—
251C2×Dic61—24no—
252C2×Dic61—24no—
253C2×Dic61—24no—
254C2×Dic61—24no—
255C2×Dic61—24no—
256C2×Dic61—24no—
257C2×Dic61—24no—
258C2×Dic61—24no—
259C2×Dic61—24no—
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×Dic61—42no—
270C2×Dic61—42no—
271C2×Dic61—42no—
272C2×Dic61—42no—
273C2×Dic61—42no—
274C2×Dic61—42no—
275C2×Dic61C2×C22⋊C418yes—
276Dic122—416no—
277#14492—611nomaximal, Sylow
278C22×C61—42no—
279C22×C61—42no—
280C22×C61—44no—
281C22×C61—44no—
282C22×C61—44no—
283C22×C61—44no—
284C22×C61—42no—
285C22×C61—42no—
286C22×C61—42no—
287C22×C61—42no—
288C22×C61—82no—
289C22×C61—82no—
290C22×C61—82no—
291C22×C61—82no—
292C22×C61—82no—
293C22×C61—82no—
294C22×C61—82no—
295C22×C61—82no—
296C22×C61C22×C22⋊C411yescharacteristic
297C22×C61C4×#4112yes—
298C22×C61#51811yescharacteristic
299C22×C61#51811yescharacteristic
300C22×C61#52012yes—
301C22×C61#53412yes—
302C22×C61#53412yes—
303C22×C61#53812yes—
304C22×C61#53812yes—
305C22×C61#54212yes—
306C22×C61#54212yes—
307C22×C61#54812yes—
308C22×C61#54812yes—
309C22×C61#55012yes—
310C22×C61#55012yes—
311C22×C61#55111yescharacteristic
312C22×C61#55111yescharacteristic
313C22×C61#55512yes—
314C22×C61H(3,𝔽4)11yescharacteristic
315C22×C61#56212yes—
316C22×C61#56311yescharacteristic
317C4×C61—24no—
318C4×C61—44no—
319C4×C61—82no—
320Dic61—28no—
321Dic61—44no—
322Dic61—44no—
323Dic61—82no—
324C22×#124—611no—
325C2×#391—612no—
326#1439—612no—
327#1439—612no—
328#1438—611no—
329#1438D6111yescharacteristic
330#1437—612no—
331#1437—612no—
332#1434—612no—
333C2×C61—82no—
334C2×C61—82no—
335C2×C61#230811yescharacteristic
336C2×C61#233712yes—
337C2×C61#234112yes—
338C2×C61#237211yescharacteristic
339C2×C61#237911yescharacteristic
340C2×C61#273012yes—
341C2×C61#273012yes—
342C2×C61#273112yes—
343C2×C61#273312yes—
344C23×Q8—611no—
345C23×D4C2×D6111yescharacteristic
346C2×#124—618no—
347C2×#124—1228no—
348C22×C4⋊C4—612no—
349C22×C22⋊C4—612no—
350C22×C22⋊C4—612no—
351C22×C22⋊C4—611no—
352C22×C22⋊C4—612no—
353C22×C22⋊C4—611no—
354C22×C22⋊C4C2×D6111yescharacteristic
355C22×C22⋊C4C2×D6112yes—
356C22×C22⋊C4C2×D6111yescharacteristic
357C22×C42—611no—
358C22×C42—612no—
359#391—1228no—
360C2×#95—612no—
361C2×#95—612no—
362C2×#95—612no—
363C2×#95—612no—
364C2×#95—612no—
365C2×#95—612no—
366C2×#95—612no—
367C2×#95C2×D6112yes—
368C61#1449211yescharacteristic, complemented, Sylow
369C25C22×D6111yescharacteristic
370C25C22×D6111yescharacteristic
371C22×Q8—614no—
372C22×Q8—618no—
373C22×Q8—2441no—
374C22×Q8—2442no—
375C22×Q8—2441no—
376C22×D4—24no—
377C22×D4—41no—
378C22×D4—41no—
379C22×D4—42no—
380C22×D4C4×D6114yes—
381C23×C4—612no—
382C23×C4—612no—
383C23×C4—612no—
384C23×C4—612no—
385C23×C4C22×D6112yes—
386C23×C4C22×D6112yes—
387C23×C4C22×D6111yescharacteristic
388#124—12232no—
389C2×C4⋊C4—1224no—
390C2×C4⋊C4—2442no—
391C2×C4⋊C4—2442no—
392C2×C22⋊C4—24no—
393C2×C22⋊C4—24no—
394C2×C22⋊C4—41no—
395C2×C22⋊C4—42no—
396C2×C22⋊C4—42no—
397C2×C22⋊C4—42no—
398C2×C22⋊C4—41no—
399C2×C22⋊C4—618no—
400C2×C22⋊C4—618no—
401C2×C22⋊C4—1224no—
402C2×C22⋊C4—1224no—
403C2×C22⋊C4—1224no—
404C2×C22⋊C4—1224no—
405C2×C22⋊C4—1224no—
406C2×C22⋊C4—1224no—
407C2×C22⋊C4—2441no—
408C2×C22⋊C4—2442no—
409C2×C22⋊C4—2442no—
410C2×C22⋊C4—2442no—
411C2×C22⋊C4—2441no—
412C2×C42—614no—
413C2×C42—1224no—
414C2×C42—1224no—
415C2×C42—2442no—
416C2×C42—2442no—
417#95—24no—
418#95—1224no—
419#95—1224no—
420#95—1224no—
421#95—1224no—
422#95—1224no—
423#95—2442no—
424#95—2442no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24—21no—
434C24—21no—
435C24—22no—
436C24—21no—
437C24—21no—
438C24—22no—
439C24—22no—
440C24—22no—
441C24C2×C4×D6112yes—
442C24C2×C4×D6112yes—
443C24C23×D6111yescharacteristic
444C2×Q8—6116no—
445C2×Q8—2444no—
446C2×Q8—2444no—
447C2×Q8—2448no—
448C2×Q8—2448no—
449C2×D4—28no—
450C2×D4—44no—
451C2×D4—44no—
452C2×D4—48no—
453C2×D4—48no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—614no—
471C22×C4—614no—
472C22×C4—614no—
473C22×C4—614no—
474C22×C4—614no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—1222no—
488C22×C4—1222no—
489C22×C4—1222no—
490C22×C4—1222no—
491C22×C4—1222no—
492C22×C4—1222no—
493C22×C4—1222no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—1222no—
498C22×C4C2×C4×D6112yes—
499C22×C4D4×D6114yes—
500C22×C4#1175822414yes—
501C4⋊C4—2448no—
502C22⋊C4—44no—
503C22⋊C4—48no—
504C22⋊C4—44no—
505C22⋊C4—12216no—
506C22⋊C4—12216no—
507C22⋊C4—2444no—
508C22⋊C4—2448no—
509C22⋊C4—2444no—
510C42—1228no—
511C42—2448no—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—24no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—24no—
521C23—24no—
522C23—42no—
523C23—42no—
524C23—41no—
525C23—41no—
526C23—42no—
527C23—42no—
528C23—41no—
529C23—41no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23C22×C4×D6111yescharacteristic
555C23C2×D4×D6112yes—
556C23#41×D6112yes—
557C23C2×#1175822412yes—
558C23#1451982211yescharacteristic
559C23#1451982211yescharacteristic
560C23#1451982212yes—
561C23#1451982212yes—
562C23#1451982412yes—
563Q8—24416no—
564D4—416no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—618no—
584C2×C4—1224no—
585C2×C4—1224no—
586C2×C4—1224no—
587C2×C4—1224no—
588C2×C4—1224no—
589C2×C4—1224no—
590C2×C4—1224no—
591C2×C4—1224no—
592C2×C4—1224no—
593C2×C4—1224no—
594C2×C4—1224no—
595C2×C4—1224no—
596C2×C4—2442no—
597C2×C4—2442no—
598C2×C4—2442no—
599C2×C4—2442no—
600C2×C4—2442no—
601C2×C4—2442no—
602C2×C4—2442no—
603C2×C4—2442no—
604C2×C4—2442no—
605C2×C4—2442no—
606C2×C4—2442no—
607C2×C4—2442no—
608C2×C4—2442no—
609C2×C4—2442no—
610C2×C4—2442no—
611C2×C4C22⋊C4×D6114yes—
612C22—42no—
613C22—42no—
614C22—44no—
615C22—44no—
616C22—42no—
617C22—42no—
618C22—44no—
619C22—44no—
620C22—42no—
621C22—42no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22C2×C22⋊C4×D6111yescharacteristic
631C22C4×#1175822412yes—
632C22#1684023511yescharacteristic
633C22#1684023511yescharacteristic
634C22#1684024112yes—
635C22#1684032212yes—
636C22#1684032212yes—
637C22#1684034412yes—
638C22#1684034712yes—
639C22#1684037912yes—
640C22#1684038012yes—
641C22#1684041812yes—
642C22#1684042012yes—
643C22#1684043012yes—
644C22#1684043212yes—
645C22#1684044011yescharacteristic
646C22#1684044311yescharacteristic
647C22#1684047612yes—
648C22#1684050511yescharacteristic
649C22#1684050712yes—
650C22#1684051511yescharacteristic
651C4—24no—
652C4—44no—
653C4—82no—
654C4—1228no—
655C4—4882no—
656C4—2444no—
657C4—2444no—
658C2—82no—
659C2—82no—
660C2#2144839511yescharacteristic
661C2#2144862812yes—
662C2#2144865812yes—
663C2#2144892811yescharacteristic
664C2#2144898911yescharacteristic
665C2#2145281812yes—
666C2#2145282212yes—
667C2#2145283112yes—
668C2#2145284712yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1438C2
socle, cosocleC24×C61C24
prime radicalC23×D4×C61C22
hypercenter#1438D61
hypocenterC61#14492
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1438

Complements

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

NH ≅ G/Ntype
2. C2×#16839666C2semidirect
3. #21444062C2semidirect
4. #21444062C2semidirect
6. #21444056C2semidirect
7. #21444056C2semidirect
8. #21444043C2semidirect
12. C2×#14519764C4semidirect
165. C2×Dic122C22⋊C4semidirect
368. C61#14492semidirect

Maximal subgroups

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

subgroupindex
C22×#145197642
C2×#168396662
#214440622
#214440602
#214440562
#214440432
C61×#14382
#1449261