Finite Group Databasebeta

← #29606665

Subgroups — #29606665

Order 15616.

1934 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519763C211yesmaximal, characteristic
2C2×#16839629C211yesmaximal, characteristic, complemented
3C2×C4×#11758196C211yesmaximal, characteristic, complemented
4#21444089C212yesmaximal, complemented
5#21444063C212yesmaximal, complemented
6#21444062C212yesmaximal, complemented
7#21444058C212yesmaximal
8#21444057C211yesmaximal, characteristic, complemented
9#21444056C211yesmaximal, characteristic, complemented
10#21444055C211yesmaximal, characteristic, complemented
11C61×#1437C211yesmaximal, characteristic, complemented
12C23×Dic122C2211yescharacteristic
13C23×C22⋊D61C2211yescharacteristic
14C2×#14519763—24no—
15C2×#14519763—24no—
16C2×#14519763C418yescomplemented
17C22×#11758196C2212yes—
18C22×C22⋊Dic61C2211yescharacteristic
19C22×C22⋊Dic61C2212yes—
20C22×#11758198C2211yescharacteristic
21C22×#11758198C2211yescharacteristic
22C22×#11758198C2212yes—
23C22×#11758198C2212yes—
24C22×C4×Dic61C2211yescharacteristic
25C22×C4×Dic61C2211yescharacteristic
26#16839629—24no—
27C4×#11758196—24no—
28C2×#14519684C2211yescharacteristic
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2212yes—
32C2×#14519684C2212yes—
33C2×#14519684C2212yes—
34C2×#14519684C2212yes—
35C22×C22⋊C4×C61C2211yescharacteristic
36C22×C22⋊C4×C61C2212yes—
37C22×C42×C61C2211yescharacteristic
38C2×#95×C61C2211yescharacteristic
39C2×#95×C61C2212yes—
40C24×D61C2311yescharacteristic
41C22×Dic122—22no—
42C22×Dic122—22no—
43C22×Dic122—42no—
44C22×Dic122—42no—
45C22×Dic122C2×C414yes—
46C22×C22⋊D61—22no—
47C22×C22⋊D61—22no—
48C22×C22⋊D61—42no—
49C22×C22⋊D61—42no—
50C22×C22⋊D61C2×C414yes—
51C23×Dic61C2312yes—
52C23×Dic61C2311yescharacteristic
53C23×Dic61C2312yes—
54C23×Dic61C2311yescharacteristic
55C23×Dic61C2311yescharacteristic
56#14519763—232no—
57C2×#11758196—24no—
58C2×#11758196—24no—
59C2×#11758196—24no—
60C2×C22⋊Dic61—22no—
61C2×C22⋊Dic61—22no—
62C2×C22⋊Dic61—24no—
63C2×C22⋊Dic61—42no—
64C2×C22⋊Dic61—42no—
65C2×C22⋊Dic61C2×C414yes—
66C2×#11758198—22no—
67C2×#11758198—22no—
68C2×#11758198—22no—
69C2×#11758198—22no—
70C2×#11758198—24no—
71C2×#11758198—24no—
72C2×#11758198—42no—
73C2×#11758198—42no—
74C2×#11758198—42no—
75C2×#11758198—42no—
76C2×#11758198C2×C414yes—
77C2×#11758198C2×C414yes—
78C2×C4×Dic61—22no—
79C2×C4×Dic61—42no—
80C2×C4×Dic61C2×C414yes—
81C2×C4×Dic61D414yes—
82C2×C4×Dic61D414yes—
83#14519684—22no—
84#14519684—24no—
85#14519684—24no—
86#14519684—24no—
87#14519684—42no—
88#14519684—42no—
89#14519684—42no—
90C25×C61C2311yescharacteristic
91C23×C4×C61C2312yes—
92C23×C4×C61C2311yescharacteristic
93C23×C4×C61C2312yes—
94C23×C4×C61C2311yescharacteristic
95C2×C22⋊C4×C61—22no—
96C2×C22⋊C4×C61—22no—
97C2×C22⋊C4×C61—24no—
98C2×C22⋊C4×C61—42no—
99C2×C22⋊C4×C61—42no—
100C2×C22⋊C4×C61C2×C414yes—
101C2×C42×C61—22no—
102C2×C42×C61—42no—
103#95×C61—22no—
104#95×C61—42no—
105C24×C61—21no—
106C24×C61—21no—
107C24×C61—22no—
108C24×C61—22no—
109C24×C61—22no—
110C24×C61—22no—
111C24×C61—22no—
112C24×C61—22no—
113C24×C61C22×C412yes—
114C24×C61C2411yescharacteristic
115C23×D61—21no—
116C23×D61—21no—
117C23×D61—22no—
118C23×D61—22no—
119C23×D61—22no—
120C23×D61—22no—
121C23×D61—22no—
122C23×D61—22no—
123C23×D61C22×C412yes—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61C22×C412yes—
144C22×C4×C61C22×C412yes—
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2×D412yes—
147C22×C4×C61#4114yes—
148C2×C22⋊D61—28no—
149C2×C22⋊D61—48no—
150C2×C22⋊D61—44no—
151C2×C22⋊D61—48no—
152C2×C22⋊D61—44no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61—22no—
156C22×Dic61—22no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—22no—
160C22×Dic61—22no—
161C22×Dic61—22no—
162C22×Dic61—22no—
163C22×Dic61—22no—
164C22×Dic61—22no—
165C22×Dic61—22no—
166C22×Dic61—22no—
167C22×Dic61—22no—
168C22×Dic61—22no—
169C22×Dic61—22no—
170C22×Dic61—22no—
171C22×Dic61—22no—
172C22×Dic61—22no—
173C22×Dic61—22no—
174C22×Dic61C22×C412yes—
175C22×Dic61C22×C412yes—
176C22×Dic61C22×C412yes—
177C22×Dic61C2×D412yes—
178C22×Dic61C2×D412yes—
179C22×Dic61C2×D412yes—
180C22×Dic61C2×D412yes—
181C22×Dic61#4114yes—
182C2×Dic122—28no—
183C2×Dic122—48no—
184C2×Dic122—44no—
185C2×Dic122—48no—
186C2×Dic122—44no—
187C22⋊C4×C61—28no—
188C22⋊C4×C61—48no—
189C42×C61—44no—
190C22⋊Dic61—28no—
191C22⋊Dic61—48no—
192#11758198—28no—
193#11758198—28no—
194#11758198—48no—
195#11758198—48no—
196#11758196—216no—
197C4×Dic61—44no—
198C4×Dic61C22⋊C4116yescomplemented
199C23×C61—22no—
200C23×C61—22no—
201C23×C61—22no—
202C23×C61—24no—
203C23×C61—24no—
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—42no—
214C23×C61—42no—
215C23×C61—42no—
216C23×C61—42no—
217C23×C61—42no—
218C23×C61—42no—
219C23×C61—42no—
220C23×C61C23×C411yescharacteristic
221C23×C61C22×D411yescharacteristic
222C23×C61C22×D411yescharacteristic
223C23×C61C2×#4112yes—
224C23×C61C2×#4112yes—
225C23×C61H(5,𝔽2)12yes—
226C23×C61H(5,𝔽2)12yes—
227C23×C61H(5,𝔽2)12yes—
228C23×C612−1+412yes—
229C22×D61—22no—
230C22×D61—22no—
231C22×D61—22no—
232C22×D61—24no—
233C22×D61—24no—
234C22×D61—42no—
235C22×D61—42no—
236C22×D61—42no—
237C22×D61—41no—
238C22×D61—41no—
239C22×D61—41no—
240C22×D61—41no—
241C22×D61—42no—
242C22×D61—42no—
243C22×D61—42no—
244C22×D61—42no—
245C22×D61—42no—
246C22×D61—42no—
247C22×D61—42no—
248C22×D61—42no—
249C22×D61—42no—
250C2×C4×C61—24no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—24no—
254C2×C4×C61—22no—
255C2×C4×C61—24no—
256C2×C4×C61—24no—
257C2×C4×C61—24no—
258C2×C4×C61—24no—
259C2×C4×C61—42no—
260C2×C4×C61—41no—
261C2×C4×C61—42no—
262C2×C4×C61—41no—
263C2×C4×C61—42no—
264C2×C4×C61—42no—
265C2×C4×C61—42no—
266C2×C4×C61—42no—
267C2×C4×C61—42no—
268C2×C4×C61—42no—
269C2×C4×C61—42no—
270C2×C4×C61—42no—
271C2×C4×C61—42no—
272C2×C4×C61—42no—
273C2×C4×C61C2×C22⋊C414yes—
274C22⋊D61—416no—
275C2×Dic61—22no—
276C2×Dic61—22no—
277C2×Dic61—24no—
278C2×Dic61—24no—
279C2×Dic61—24no—
280C2×Dic61—22no—
281C2×Dic61—24no—
282C2×Dic61—22no—
283C2×Dic61—22no—
284C2×Dic61—24no—
285C2×Dic61—24no—
286C2×Dic61—24no—
287C2×Dic61—24no—
288C2×Dic61—42no—
289C2×Dic61—41no—
290C2×Dic61—42no—
291C2×Dic61—41no—
292C2×Dic61—42no—
293C2×Dic61—42no—
294C2×Dic61—42no—
295C2×Dic61—42no—
296C2×Dic61—42no—
297C2×Dic61—42no—
298C2×Dic61—42no—
299C2×Dic61—42no—
300C2×Dic61—42no—
301C2×Dic61—42no—
302C2×Dic61C2×C22⋊C414yes—
303C2×Dic61C2×C22⋊C414yes—
304Dic122—416no—
305#14288—611nomaximal, Sylow
306C22×C61—42no—
307C22×C61—42no—
308C22×C61—44no—
309C22×C61—44no—
310C22×C61—42no—
311C22×C61—82no—
312C22×C61—82no—
313C22×C61—82no—
314C22×C61—82no—
315C22×C61C22×C22⋊C411yescharacteristic
316C22×C61C4×#4111yescharacteristic
317C22×C61C4×#4111yescharacteristic
318C22×C61#51812yes—
319C22×C61#52012yes—
320C22×C61#52512yes—
321C22×C61#53212yes—
322C22×C61#53511yescharacteristic
323C22×C61#53511yescharacteristic
324C22×C61#53512yes—
325C22×C61#53511yescharacteristic
326C22×C61#53611yescharacteristic
327C22×C61#53712yes—
328C22×C61#53812yes—
329C22×C61#53812yes—
330C22×C61#53812yes—
331C22×C61#53812yes—
332C22×C61#54212yes—
333C22×C61#54212yes—
334C22×C61#54212yes—
335C22×C61#54212yes—
336C2×D61—42no—
337C2×D61—42no—
338C2×D61—44no—
339C2×D61—44no—
340C2×D61—42no—
341C2×D61—82no—
342C2×D61—82no—
343C2×D61—82no—
344C2×D61—82no—
345C4×C61—24no—
346C4×C61—42no—
347C4×C61—44no—
348C4×C61—82no—
349Dic61—24no—
350Dic61—24no—
351Dic61—42no—
352Dic61—44no—
353Dic61—82no—
354C22×#124—611no—
355C2×#383—611no—
356C2×C4×C4⋊C4—611no—
357#1446—612no—
358#1439—612no—
359#1439—612no—
360#1438—612no—
361#1437—611no—
362#1437—611no—
363#1437—611no—
364#1437D6111yescharacteristic, complemented
365C2×C61—82no—
366C2×C61#229711yescharacteristic
367C2×C61#230911yescharacteristic
368C2×C61#231111yescharacteristic
369C2×C61#233712yes—
370C2×C61#233712yes—
371C2×C61#263112yes—
372C2×C61#263212yes—
373C2×C61#266212yes—
374C2×C61#266212yes—
375D61—82no—
376C23×Q8—611no—
377C23×D4—611no—
378C2×#124—618no—
379C2×#124—1224no—
380C2×#124—1224no—
381C22×C4⋊C4—612no—
382C22×C22⋊C4—611no—
383C22×C22⋊C4—611no—
384C22×C22⋊C4—611no—
385C22×C22⋊C4—612no—
386C22×C22⋊C4—612no—
387C22×C22⋊C4—612no—
388C22×C22⋊C4C2×D6111yescharacteristic
389C22×C22⋊C4C2×D6112yes—
390C22×C42—611no—
391C22×C42—611no—
392C22×C42C2×D6111yescharacteristic
393#383—1224no—
394C4×C4⋊C4—1224no—
395C2×#95—611no—
396C2×#95—612no—
397C2×#95—612no—
398C2×#95—612no—
399C2×#95—612no—
400C2×#95—612no—
401C2×#95—612no—
402C2×#95C2×D6111yescharacteristic
403C2×#95C2×D6112yes—
404C61#1428811yescharacteristic, complemented, Sylow
405C25—611no—
406C25C22×D6111yescharacteristic
407C22×Q8—614no—
408C22×Q8—1222no—
409C22×Q8—1222no—
410C22×Q8—2442no—
411C22×Q8—2442no—
412C22×D4—614no—
413C22×D4—1222no—
414C22×D4—1222no—
415C22×D4—2442no—
416C22×D4—2442no—
417C23×C4—611no—
418C23×C4—611no—
419C23×C4—612no—
420C23×C4—611no—
421C23×C4—612no—
422C23×C4C22×D6112yes—
423C23×C4C22×D6111yescharacteristic
424C23×C4C22×D6112yes—
425C23×C4C22×D6111yescharacteristic
426#124—12232no—
427C2×C4⋊C4—1224no—
428C2×C4⋊C4—1224no—
429C2×C4⋊C4—1224no—
430C2×C22⋊C4—22no—
431C2×C22⋊C4—22no—
432C2×C22⋊C4—24no—
433C2×C22⋊C4—42no—
434C2×C22⋊C4—42no—
435C2×C22⋊C4—614no—
436C2×C22⋊C4—614no—
437C2×C22⋊C4—614no—
438C2×C22⋊C4—1222no—
439C2×C22⋊C4—1222no—
440C2×C22⋊C4—1222no—
441C2×C22⋊C4—1222no—
442C2×C22⋊C4—1222no—
443C2×C22⋊C4—1222no—
444C2×C22⋊C4—1224no—
445C2×C22⋊C4—1224no—
446C2×C22⋊C4—1224no—
447C2×C22⋊C4—2442no—
448C2×C22⋊C4—2442no—
449C2×C22⋊C4—2442no—
450C2×C22⋊C4—2442no—
451C2×C22⋊C4—2442no—
452C2×C22⋊C4—2442no—
453C2×C22⋊C4C4×D6114yes—
454C2×C42—22no—
455C2×C42—42no—
456C2×C42—614no—
457C2×C42—614no—
458C2×C42—614no—
459C2×C42—1222no—
460C2×C42—2442no—
461#95—22no—
462#95—42no—
463#95—1224no—
464#95—1224no—
465#95—1224no—
466#95—1222no—
467#95—2442no—
468#95—2442no—
469#95—2442no—
470C24—21no—
471C24—21no—
472C24—22no—
473C24—22no—
474C24—22no—
475C24—22no—
476C24—22no—
477C24—22no—
478C24—612no—
479C24—1221no—
480C24—1221no—
481C24—1222no—
482C24—1222no—
483C24—1222no—
484C24—1222no—
485C24—1222no—
486C24—1222no—
487C24C2×C4×D6112yes—
488C24C23×D6111yescharacteristic
489C2×Q8—1228no—
490C2×Q8—2448no—
491C2×Q8—2444no—
492C2×Q8—2448no—
493C2×Q8—2444no—
494C2×D4—1228no—
495C2×D4—2448no—
496C2×D4—2444no—
497C2×D4—2448no—
498C2×D4—2444no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4—22no—
513C22×C4—22no—
514C22×C4—22no—
515C22×C4—22no—
516C22×C4—22no—
517C22×C4—22no—
518C22×C4—612no—
519C22×C4—612no—
520C22×C4—612no—
521C22×C4—612no—
522C22×C4—612no—
523C22×C4—612no—
524C22×C4—612no—
525C22×C4—614no—
526C22×C4—1222no—
527C22×C4—1222no—
528C22×C4—1222no—
529C22×C4—1222no—
530C22×C4—1222no—
531C22×C4—1221no—
532C22×C4—1221no—
533C22×C4—1222no—
534C22×C4—1222no—
535C22×C4—1222no—
536C22×C4—1222no—
537C22×C4—1222no—
538C22×C4—1222no—
539C22×C4—1222no—
540C22×C4—1222no—
541C22×C4—1222no—
542C22×C4—1222no—
543C22×C4—1222no—
544C22×C4—1222no—
545C22×C4—1222no—
546C22×C4—1222no—
547C22×C4C2×C4×D6112yes—
548C22×C4C2×C4×D6112yes—
549C22×C4#1175822214yes—
550C22×C4D4×D6112yes—
551C22×C4D4×D6112yes—
552C4⋊C4—12216no—
553C22⋊C4—28no—
554C22⋊C4—48no—
555C22⋊C4—1228no—
556C22⋊C4—1228no—
557C22⋊C4—1228no—
558C22⋊C4—2448no—
559C22⋊C4—2448no—
560C22⋊C4—2448no—
561C42—44no—
562C42—6116no—
563C42—2444no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—24no—
568C23—24no—
569C23—42no—
570C23—42no—
571C23—41no—
572C23—41no—
573C23—42no—
574C23—41no—
575C23—41no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—1222no—
586C23—1222no—
587C23—1222no—
588C23—1224no—
589C23—1224no—
590C23—2442no—
591C23—2442no—
592C23—2441no—
593C23—2441no—
594C23—2442no—
595C23—2441no—
596C23—2441no—
597C23—2442no—
598C23—2442no—
599C23—2442no—
600C23—2442no—
601C23—2442no—
602C23—2442no—
603C23—2442no—
604C23—2442no—
605C23—2442no—
606C23C22×C4×D6111yescharacteristic
607C23C2×D4×D6111yescharacteristic
608C23C2×D4×D6111yescharacteristic
609C23#41×D6112yes—
610C23C2×#1175822212yes—
611C23#1451982212yes—
612C23#1451982212yes—
613C23#1451982312yes—
614C23#1451982412yes—
615Q8—24416no—
616D4—24416no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—614no—
641C2×C4—614no—
642C2×C4—1222no—
643C2×C4—1222no—
644C2×C4—1224no—
645C2×C4—1224no—
646C2×C4—1224no—
647C2×C4—1222no—
648C2×C4—1224no—
649C2×C4—1222no—
650C2×C4—1222no—
651C2×C4—1224no—
652C2×C4—1224no—
653C2×C4—1224no—
654C2×C4—1224no—
655C2×C4—2442no—
656C2×C4—2442no—
657C2×C4—2442no—
658C2×C4—2442no—
659C2×C4—2442no—
660C2×C4—2442no—
661C2×C4—2442no—
662C2×C4—2441no—
663C2×C4—2442no—
664C2×C4—2442no—
665C2×C4—2442no—
666C2×C4—2441no—
667C2×C4—2442no—
668C2×C4—2442no—
669C2×C4C22⋊C4×D6114yes—
670C22—42no—
671C22—44no—
672C22—44no—
673C22—42no—
674C22—42no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22—4882no—
680C22—4882no—
681C22—4882no—
682C22—4882no—
683C22—2442no—
684C22—2444no—
685C22—2444no—
686C22—2442no—
687C22—2442no—
688C22C2×C22⋊C4×D6111yescharacteristic
689C22#1684023011yescharacteristic
690C22C4×#1175822211yescharacteristic
691C22#1684023412yes—
692C22#1684024112yes—
693C22#1684027112yes—
694C22#1684030912yes—
695C22#1684032511yescharacteristic
696C22#1684032711yescharacteristic
697C22#1684032812yes—
698C22#1684033011yescharacteristic
699C22#1684033411yescharacteristic
700C22#1684033912yes—
701C22#1684034412yes—
702C22#1684034512yes—
703C22#1684034812yes—
704C22#1684035012yes—
705C22#1684037612yes—
706C22#1684037712yes—
707C22#1684037812yes—
708C22#1684038312yes—
709C4—24no—
710C4—44no—
711C4—42no—
712C4—82no—
713C4—1224no—
714C4—1224no—
715C4—4882no—
716C4—2444no—
717C4—2442no—
718C2—82no—
719C2—4882no—
720C2#2144830511yescharacteristic
721C2#2144839811yescharacteristic
722C2#2144840611yescharacteristic
723C2#2144862712yes—
724C2#2144863012yes—
725C2#2145167512yes—
726C2#2145168312yes—
727C2#2145196012yes—
728C2#2145196212yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1437C2
socle, cosocleC24×C61C24
prime radicalC23×C22⋊D61C22
hypercenter#1437D61
hypocenterC61#14288
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1437

Complements

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

NH ≅ G/Ntype
2. C2×#16839629C2semidirect
3. C2×C4×#11758196C2semidirect
4. #21444089C2semidirect
5. #21444063C2semidirect
6. #21444062C2semidirect
8. #21444057C2semidirect
9. #21444056C2semidirect
10. #21444055C2semidirect
11. C61×#1437C2semidirect
16. C2×#14519763C4semidirect
198. C4×Dic61C22⋊C4semidirect
364. #1437D61semidirect
404. C61#14288semidirect

Maximal subgroups

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

subgroupindex
C22×#145197632
C2×#168396292
C2×C4×#117581962
#214440892
#214440632
#214440622
#214440582
#214440572
#214440562
#214440552
C61×#14372
#1428861