Finite Group Databasebeta

← #29602706

Subgroups — #29602706

Order 15616.

1862 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519735C211yesmaximal, characteristic
2C2×#16839627C211yesmaximal, characteristic, complemented
3C2×#16839607C211yesmaximal, characteristic, complemented
4#21444072C211yesmaximal, characteristic, complemented
5#21444065C211yesmaximal, characteristic
6#21444061C211yesmaximal, characteristic
7#21444055C212yesmaximal
8#21444053C212yesmaximal, complemented
9#21444045C212yesmaximal, complemented
10#21444040C212yesmaximal
11C61×#1441C211yesmaximal, characteristic, complemented
12C23×C4×D61C2211yescharacteristic
13C2×#14519735—24no—
14C2×#14519735—24no—
15C2×#14519735C418yescomplemented
16C22×C4⋊Dic61C2211yescharacteristic
17C22×C4⋊Dic61C2211yescharacteristic
18C22×#11758198C2212yes—
19C22×#11758198C2212yes—
20C22×#11758198C2211yescharacteristic
21C22×#11758198C2211yescharacteristic
22C22×C4×Dic61C2212yes—
23C22×C4×Dic61C2211yescharacteristic
24C22×C4×Dic61C2211yescharacteristic
25#16839627—24no—
26#16839607—24no—
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2212yes—
32C2×#14519684C2212yes—
33C2×#14519684C2211yescharacteristic
34C2×#14519684C2212yes—
35C22×C4⋊C4×C61C2211yescharacteristic
36C22×C4⋊C4×C61C2211yescharacteristic
37C2×#95×C61C2212yes—
38C2×#95×C61C2211yescharacteristic
39C2×#95×C61C2212yes—
40C24×D61C2311yescharacteristic
41C23×Dic61C2312yes—
42C23×Dic61C2311yescharacteristic
43C23×Dic61C2312yes—
44C23×Dic61C2311yescharacteristic
45C23×Dic61C2311yescharacteristic
46C22×C4×D61—22no—
47C22×C4×D61—22no—
48C22×C4×D61—22no—
49C22×C4×D61—22no—
50C22×C4×D61—22no—
51C22×C4×D61—22no—
52C22×C4×D61C2×C414yes—
53#14519735—232no—
54C2×C4⋊Dic61—22no—
55C2×C4⋊Dic61—22no—
56C2×C4⋊Dic61—22no—
57C2×C4⋊Dic61—42no—
58C2×C4⋊Dic61C2×C414yes—
59C2×#11758198—22no—
60C2×#11758198—22no—
61C2×#11758198—22no—
62C2×#11758198—22no—
63C2×#11758198—24no—
64C2×#11758198—24no—
65C2×#11758198—24no—
66C2×#11758198—24no—
67C2×#11758198—42no—
68C2×#11758198—42no—
69C2×#11758198C2×C414yes—
70C2×#11758198C2×C414yes—
71C2×C4×Dic61—24no—
72C2×C4×Dic61—22no—
73C2×C4×Dic61—22no—
74C2×C4×Dic61—24no—
75C2×C4×Dic61—24no—
76C2×C4×Dic61C2×C414yes—
77C2×C4×Dic61C2×C414yes—
78C2×C4×Dic61D414yes—
79C2×C4×Dic61D414yes—
80#14519684—22no—
81#14519684—24no—
82#14519684—24no—
83#14519684—24no—
84#14519684—24no—
85#14519684—24no—
86#14519684—24no—
87#14519684—42no—
88C23×C4×C61C2312yes—
89C23×C4×C61C2311yescharacteristic
90C23×C4×C61C2312yes—
91C23×C4×C61C2311yescharacteristic
92C23×C4×C61C2311yescharacteristic
93C2×C4⋊C4×C61—22no—
94C2×C4⋊C4×C61—22no—
95C2×C4⋊C4×C61—22no—
96C2×C4⋊C4×C61—42no—
97C2×C4⋊C4×C61C2×C414yes—
98#95×C61—22no—
99#95×C61—24no—
100#95×C61—42no—
101C24×C61C2411yescharacteristic
102C23×D61—21no—
103C23×D61—21no—
104C23×D61—21no—
105C23×D61—21no—
106C23×D61—22no—
107C23×D61—22no—
108C23×D61—22no—
109C23×D61—22no—
110C23×D61C22×C412yes—
111C23×D61#4112yes—
112C23×D61#4112yes—
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
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×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61C22×C412yes—
137C22×C4×C61C22×C412yes—
138C22×C4×C61C22×C412yes—
139C22×C4×C61C2×D412yes—
140C22×C4×C61C2×D412yes—
141C22×C4×C61#4114yes—
142C22×C4×C61#4112yes—
143C22×C4×C61#4112yes—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61—22no—
156C22×Dic61—22no—
157C22×Dic61—22no—
158C22×Dic61—22no—
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×Dic61C22×C412yes—
168C22×Dic61C22×C412yes—
169C22×Dic61C22×C412yes—
170C22×Dic61C2×D412yes—
171C22×Dic61C2×D412yes—
172C22×Dic61C2×D412yes—
173C22×Dic61C2×D412yes—
174C22×Dic61#4114yes—
175C2×C4×D61—28no—
176C2×C4×D61—28no—
177C2×C4×D61—28no—
178C2×C4×D61—28no—
179C2×C4×D61—28no—
180C2×C4×D61—28no—
181C2×C4×D61—28no—
182C4⋊C4×C61—28no—
183C4⋊C4×C61—44no—
184#11758198—28no—
185#11758198—28no—
186#11758198—48no—
187#11758198—48no—
188C4⋊Dic61—28no—
189C4⋊Dic61—44no—
190C4×Dic61—216no—
191C4×Dic61—28no—
192C4×Dic61C22⋊C4116yescomplemented
193C23×C61C23×C411yescharacteristic
194C23×C61C22×D411yescharacteristic
195C23×C61C22×D411yescharacteristic
196C23×C61C2×#4111yescharacteristic
197C23×C61C2×#4111yescharacteristic
198C23×C61C2×#4111yescharacteristic
199C23×C61C2×#4111yescharacteristic
200C23×C61C2×#4112yes—
201C23×C61C2×#4112yes—
202C23×C61H(5,𝔽2)12yes—
203C23×C612−1+412yes—
204C22×D61—22no—
205C22×D61—22no—
206C22×D61—24no—
207C22×D61—24no—
208C22×D61—22no—
209C22×D61—22no—
210C22×D61—22no—
211C22×D61—22no—
212C22×D61—24no—
213C22×D61—24no—
214C22×D61—24no—
215C22×D61—24no—
216C22×D61—42no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—42no—
222C22×D61—42no—
223C22×D61—42no—
224C22×D61—42no—
225C22×D61—42no—
226C22×D61#11714yes—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—24no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—24no—
237C2×C4×C61—22no—
238C2×C4×C61—24no—
239C2×C4×C61—24no—
240C2×C4×C61—24no—
241C2×C4×C61—24no—
242C2×C4×C61—42no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61—42no—
251C2×C4×C61—42no—
252C2×C4×C61—42no—
253C2×C4×C61—42no—
254C2×C4×C61—42no—
255C2×C4×C61—42no—
256C2×C4×C61C2×C22⋊C414yes—
257C2×C4×C61#11714yes—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—24no—
261C2×Dic61—24no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—22no—
265C2×Dic61—22no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—24no—
270C2×Dic61—24no—
271C2×Dic61—24no—
272C2×Dic61—24no—
273C2×Dic61—42no—
274C2×Dic61—42no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—42no—
278C2×Dic61—42no—
279C2×Dic61—42no—
280C2×Dic61—42no—
281C2×Dic61—42no—
282C2×Dic61—42no—
283C2×Dic61—42no—
284C2×Dic61—42no—
285C2×Dic61—42no—
286C2×Dic61—42no—
287C2×Dic61C2×C22⋊C414yes—
288C2×Dic61C2×C22⋊C414yes—
289C4×D61—232no—
290#13957—611nomaximal, Sylow
291C22×C61C22×C22⋊C411yescharacteristic
292C22×C61C2×#11711yescharacteristic
293C22×C61C2×#11711yescharacteristic
294C22×C61C4×#4112yes—
295C22×C61#52012yes—
296C22×C61#52511yescharacteristic
297C22×C61#52511yescharacteristic
298C22×C61#52511yescharacteristic
299C22×C61#52512yes—
300C22×C61#52511yescharacteristic
301C22×C61#53211yescharacteristic
302C22×C61#53211yescharacteristic
303C22×C61#53511yescharacteristic
304C22×C61#53611yescharacteristic
305C22×C61#53712yes—
306C22×C61#55112yes—
307C22×C61#55112yes—
308C22×C61#55212yes—
309C22×C61#55212yes—
310C22×C61#55212yes—
311C22×C61#55212yes—
312C22×C61#55512yes—
313C22×C61#55512yes—
314C2×D61—28no—
315C2×D61—24no—
316C2×D61—42no—
317C2×D61—44no—
318C2×D61—44no—
319C2×D61—44no—
320C2×D61—42no—
321C2×D61—42no—
322C2×D61—42no—
323C2×D61—42no—
324C2×D61—42no—
325C4×C61—24no—
326C4×C61—24no—
327C4×C61—42no—
328C4×C61—44no—
329C4×C61—82no—
330Dic61—24no—
331Dic61—24no—
332Dic61—44no—
333Dic61—42no—
334Dic61—82no—
335C22×#117—611no—
336C2×#382—611no—
337C2×#376—611no—
338#1441—611no—
339#1441D6111yescharacteristic, complemented
340#1439—611no—
341#1439—611no—
342#1437—612no—
343#1436—612no—
344#1434—612no—
345C22⋊#95—612no—
346C2×C61#228011yescharacteristic
347C2×C61#229711yescharacteristic
348C2×C61#231111yescharacteristic
349C2×C61#234312yes—
350C2×C61#234612yes—
351C2×C61#260912yes—
352C2×C61#260912yes—
353C2×C61#261011yescharacteristic
354C2×C61#261011yescharacteristic
355C2×C61#261111yescharacteristic
356C2×C61#261111yescharacteristic
357D61—44no—
358C24×C4—611no—
359C2×#117—618no—
360C2×#117—1224no—
361C2×#117—1224no—
362C22×C4⋊C4—611no—
363C22×C4⋊C4—611no—
364C22×C4⋊C4C2×D6111yescharacteristic
365C22×C4⋊C4C2×D6111yescharacteristic
366C22×C22⋊C4—611no—
367C22×C22⋊C4—611no—
368C22×C22⋊C4—612no—
369C22×C22⋊C4—612no—
370C22×C42—611no—
371C22×C42—611no—
372C22×C42—612no—
373#382—1224no—
374#376—1224no—
375C2×#95—612no—
376C2×#95—612no—
377C2×#95—612no—
378C2×#95—612no—
379C2×#95—612no—
380C2×#95—612no—
381C2×#95—611no—
382C2×#95—612no—
383C2×#95C2×D6112yes—
384C2×#95C2×D6111yescharacteristic
385C2×#95C2×D6112yes—
386C61#1395711yescharacteristic, complemented, Sylow
387C25—611no—
388C23×C4—611no—
389C23×C4—611no—
390C23×C4—611no—
391C23×C4—612no—
392C23×C4—614no—
393C23×C4—612no—
394C23×C4—1222no—
395C23×C4—1222no—
396C23×C4—1222no—
397C23×C4—1222no—
398C23×C4—1222no—
399C23×C4—1222no—
400C23×C4C22×D6112yes—
401C23×C4C22×D6111yescharacteristic
402C23×C4C22×D6112yes—
403C23×C4C22×D6111yescharacteristic
404C23×C4C22×D6111yescharacteristic
405#117—12232no—
406C2×C4⋊C4—22no—
407C2×C4⋊C4—22no—
408C2×C4⋊C4—22no—
409C2×C4⋊C4—42no—
410C2×C4⋊C4—614no—
411C2×C4⋊C4—1222no—
412C2×C4⋊C4—1222no—
413C2×C4⋊C4—1222no—
414C2×C4⋊C4—2442no—
415C2×C4⋊C4C4×D6114yes—
416C2×C22⋊C4—614no—
417C2×C22⋊C4—614no—
418C2×C22⋊C4—1222no—
419C2×C22⋊C4—1222no—
420C2×C22⋊C4—1222no—
421C2×C22⋊C4—1222no—
422C2×C22⋊C4—1224no—
423C2×C22⋊C4—1224no—
424C2×C22⋊C4—1224no—
425C2×C22⋊C4—1224no—
426C2×C22⋊C4—2442no—
427C2×C22⋊C4—2442no—
428C2×C42—614no—
429C2×C42—614no—
430C2×C42—614no—
431C2×C42—614no—
432C2×C42—1222no—
433C2×C42—1222no—
434C2×C42—1224no—
435C2×C42—1224no—
436C2×C42—1224no—
437#95—22no—
438#95—24no—
439#95—42no—
440#95—1224no—
441#95—1224no—
442#95—1224no—
443#95—1224no—
444#95—1222no—
445#95—1224no—
446#95—1224no—
447#95—2442no—
448C24—612no—
449C24—612no—
450C24—612no—
451C24—1221no—
452C24—1221no—
453C24—1222no—
454C24—1221no—
455C24—1222no—
456C24—1221no—
457C24—1222no—
458C24—1222no—
459C24C23×D6111yescharacteristic
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—612no—
484C22×C4—612no—
485C22×C4—612no—
486C22×C4—612no—
487C22×C4—612no—
488C22×C4—612no—
489C22×C4—612no—
490C22×C4—614no—
491C22×C4—1228no—
492C22×C4—1228no—
493C22×C4—1228no—
494C22×C4—1228no—
495C22×C4—1228no—
496C22×C4—1228no—
497C22×C4—1222no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—1222no—
502C22×C4—1221no—
503C22×C4—1228no—
504C22×C4—1222no—
505C22×C4—1221no—
506C22×C4—1221no—
507C22×C4—1222no—
508C22×C4—1222no—
509C22×C4—1222no—
510C22×C4—1222no—
511C22×C4—1222no—
512C22×C4—1222no—
513C22×C4—1222no—
514C22×C4—1222no—
515C22×C4—1222no—
516C22×C4—1222no—
517C22×C4—1222no—
518C22×C4—1222no—
519C22×C4—1222no—
520C22×C4—1222no—
521C22×C4C2×C4×D6112yes—
522C22×C4C2×C4×D6112yes—
523C22×C4C2×C4×D6112yes—
524C22×C4D4×D6112yes—
525C22×C4D4×D6112yes—
526C22×C4#1175822412yes—
527C22×C4#1175822614yes—
528C22×C4#1175822612yes—
529C4⋊C4—28no—
530C4⋊C4—44no—
531C4⋊C4—1228no—
532C4⋊C4—2444no—
533C22⋊C4—1228no—
534C22⋊C4—1228no—
535C22⋊C4—2448no—
536C22⋊C4—2448no—
537C42—6116no—
538C42—1228no—
539C42—12216no—
540C23—614no—
541C23—1224no—
542C23—1224no—
543C23—1222no—
544C23—1222no—
545C23—1224no—
546C23—1224no—
547C23—1222no—
548C23—1222no—
549C23—1222no—
550C23—1222no—
551C23—1224no—
552C23—1224no—
553C23—2442no—
554C23—2442no—
555C23—2442no—
556C23—2442no—
557C23—2442no—
558C23—2441no—
559C23—2441no—
560C23—2441no—
561C23—2441no—
562C23—2442no—
563C23C22×C4×D6111yescharacteristic
564C23C2×D4×D6111yescharacteristic
565C23C2×D4×D6111yescharacteristic
566C23#41×D6111yescharacteristic
567C23#41×D6112yes—
568C23#41×D6111yescharacteristic
569C23C2×#1175822411yescharacteristic
570C23C2×#1175822611yescharacteristic
571C23C2×#1175822612yes—
572C23#1451982212yes—
573C23#1451982412yes—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—614no—
604C2×C4—614no—
605C2×C4—12232no—
606C2×C4—1222no—
607C2×C4—1222no—
608C2×C4—1224no—
609C2×C4—1222no—
610C2×C4—1222no—
611C2×C4—1224no—
612C2×C4—1222no—
613C2×C4—1222no—
614C2×C4—1222no—
615C2×C4—1222no—
616C2×C4—1224no—
617C2×C4—1222no—
618C2×C4—1224no—
619C2×C4—1224no—
620C2×C4—1224no—
621C2×C4—2442no—
622C2×C4—2442no—
623C2×C4—2442no—
624C2×C4—2442no—
625C2×C4—2442no—
626C2×C4—2442no—
627C2×C4—2442no—
628C2×C4—2442no—
629C2×C4—2442no—
630C2×C4—2442no—
631C2×C4—2442no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4—2442no—
635C2×C4C22⋊C4×D6114yes—
636C2×C4#1451973514yes—
637C22—1224no—
638C22—1228no—
639C22—2442no—
640C22—2442no—
641C22—2442no—
642C22—2444no—
643C22—2444no—
644C22—2442no—
645C22—2442no—
646C22—2442no—
647C22—2444no—
648C22C2×C22⋊C4×D6111yescharacteristic
649C22#117×D6111yescharacteristic
650C22C2×#1451973511yescharacteristic
651C22C4×#1175822612yes—
652C22#1684024112yes—
653C22#1684027211yescharacteristic
654C22#1684027211yescharacteristic
655C22#1684027511yescharacteristic
656C22#1684027511yescharacteristic
657C22#1684027512yes—
658C22#1684031111yescharacteristic
659C22#1684031311yescharacteristic
660C22#1684032711yescharacteristic
661C22#1684033411yescharacteristic
662C22#1684033912yes—
663C22#1684043912yes—
664C22#1684044012yes—
665C22#1684044812yes—
666C22#1684045012yes—
667C22#1684045012yes—
668C22#1684045412yes—
669C22#1684047612yes—
670C22#1684047812yes—
671C4—24no—
672C4—24no—
673C4—42no—
674C4—44no—
675C4—82no—
676C4—1224no—
677C4—1224no—
678C4—4882no—
679C4—2442no—
680C4—2444no—
681C2—2444no—
682C2#2144820111yescharacteristic
683C2#2144830811yescharacteristic
684C2#2144840611yescharacteristic
685C2#2144867812yes—
686C2#2144870312yes—
687C2#2145149212yes—
688C2#2145149712yes—
689C2#2145150111yescharacteristic
690C2#2145150911yescharacteristic
691C2#2145151211yescharacteristic
692C2#2145152011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1441C2
socle, cosocleC24×C61C24
prime radicalC24×D61C23
hypercenter#1441D61
hypocenterC61#13957
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
2. C2×#16839627C2semidirect
3. C2×#16839607C2semidirect
4. #21444072C2semidirect
8. #21444053C2semidirect
9. #21444045C2semidirect
11. C61×#1441C2semidirect
15. C2×#14519735C4semidirect
192. C4×Dic61C22⋊C4semidirect
339. #1441D61semidirect
386. C61#13957semidirect

Maximal subgroups

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

subgroupindex
C22×#145197352
C2×#168396272
C2×#168396072
#214440722
#214440652
#214440612
#214440552
#214440532
#214440452
#214440402
C61×#14412
#1395761