Finite Group Databasebeta

← #29606616

Subgroups — #29606616

Order 15616.

1502 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519767C211yesmaximal, characteristic
2C2×#16839631C211yesmaximal, characteristic
3C2×#16839607C211yesmaximal, characteristic
4#21444073C211yesmaximal, characteristic
5#21444072C211yesmaximal, characteristic
6#21444071C212yesmaximal
7#21444070C212yesmaximal
8#21444069C211yesmaximal, characteristic
9#21444047C212yesmaximal
10#21444047C212yesmaximal
11C61×#1441C211yesmaximal, characteristic
12C2×#14519767—24no—
13C2×#14519767—24no—
14C2×#14519767C418yescomplemented
15C22×C4⋊Dic61C2211yescharacteristic
16C22×C4⋊Dic61C2211yescharacteristic
17C22×#11758196C2212yes—
18C22×#11758196C2211yescharacteristic
19C22×#11758196C2211yescharacteristic
20C22×#11758196C2211yescharacteristic
21C22×#11758196C2211yescharacteristic
22C22×#11758196C2212yes—
23C22×C4×Dic61C2212yes—
24C22×C4×Dic61C2211yescharacteristic
25#16839631—24no—
26#16839607—24no—
27C2×#14519684C2211yescharacteristic
28C2×#14519684C2212yes—
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2212yes—
32C2×#14519684C2212yes—
33C2×#14519684C2212yes—
34C2×#14519684C2212yes—
35C22×C4⋊C4×C61C2211yescharacteristic
36C22×C4⋊C4×C61C2211yescharacteristic
37C2×#95×C61C2211yescharacteristic
38C2×#95×C61C2212yes—
39C2×#95×C61C2212yes—
40C23×Dic61C2312yes—
41C23×Dic61C2311yescharacteristic
42C23×Dic61C2311yescharacteristic
43C23×Dic61C2312yes—
44C23×Dic61C2311yescharacteristic
45C23×Dic61C2311yescharacteristic
46#14519767—232no—
47C2×C4⋊Dic61—22no—
48C2×C4⋊Dic61—22no—
49C2×C4⋊Dic61—22no—
50C2×C4⋊Dic61—42no—
51C2×C4⋊Dic61C2×C414yes—
52C2×#11758196—22no—
53C2×#11758196—22no—
54C2×#11758196—22no—
55C2×#11758196—22no—
56C2×#11758196—22no—
57C2×#11758196—22no—
58C2×#11758196—22no—
59C2×#11758196—22no—
60C2×#11758196—24no—
61C2×#11758196—24no—
62C2×#11758196—24no—
63C2×#11758196—24no—
64C2×#11758196—42no—
65C2×#11758196—42no—
66C2×#11758196C2×C414yes—
67C2×#11758196C2×C414yes—
68C2×#11758196C2×C414yes—
69C2×#11758196C2×C414yes—
70C2×C4×Dic61—24no—
71C2×C4×Dic61—24no—
72C2×C4×Dic61C2×C414yes—
73C2×C4×Dic61D414yes—
74C2×C4×Dic61D414yes—
75C2×C4×Dic61Q818yes—
76#14519684—22no—
77#14519684—24no—
78#14519684—42no—
79#14519684—42no—
80#14519684—42no—
81#14519684—42no—
82#14519684—42no—
83#14519684—42no—
84C23×C4×C61C2312yes—
85C23×C4×C61C2311yescharacteristic
86C23×C4×C61C2311yescharacteristic
87C23×C4×C61C2312yes—
88C23×C4×C61C2311yescharacteristic
89C2×C4⋊C4×C61—22no—
90C2×C4⋊C4×C61—22no—
91C2×C4⋊C4×C61—22no—
92C2×C4⋊C4×C61—42no—
93C2×C4⋊C4×C61C2×C414yes—
94#95×C61—22no—
95#95×C61—42no—
96#95×C61—42no—
97C24×C61C2411yescharacteristic
98C22×C4×C61—22no—
99C22×C4×C61—22no—
100C22×C4×C61—22no—
101C22×C4×C61—21no—
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61—21no—
105C22×C4×C61—22no—
106C22×C4×C61—21no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—21no—
111C22×C4×C61—21no—
112C22×C4×C61—22no—
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
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×C61C2×D412yes—
128C22×C4×C61C2×Q814yes—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61—22no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—21no—
146C22×Dic61—21no—
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×C412yes—
157C22×Dic61C22×C412yes—
158C22×Dic61C22×C412yes—
159C22×Dic61C22×C412yes—
160C22×Dic61C2×D412yes—
161C22×Dic61C2×D412yes—
162C22×Dic61C2×D412yes—
163C22×Dic61C2×D412yes—
164C22×Dic61C2×Q814yes—
165C22×Dic61C2×Q814yes—
166C22×Dic61#4114yes—
167C4⋊C4×C61—28no—
168C4⋊C4×C61—44no—
169C4⋊Dic61—28no—
170C4⋊Dic61—44no—
171#11758196—28no—
172#11758196—28no—
173#11758196—28no—
174#11758196—28no—
175#11758196—48no—
176#11758196—48no—
177C4×Dic61—216no—
178C4×Dic61C22⋊C4116yes—
179C23×C61C23×C411yescharacteristic
180C23×C61C22×D411yescharacteristic
181C23×C61C22×D411yescharacteristic
182C23×C61C22×Q812yes—
183C23×C61C2×#4112yes—
184C23×C61H(5,𝔽2)12yes—
185C23×C61H(5,𝔽2)11yescharacteristic
186C23×C61H(5,𝔽2)11yescharacteristic
187C23×C612−1+412yes—
188C23×C612−1+411yescharacteristic
189C23×C612−1+411yescharacteristic
190C2×C4×C61—24no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—24no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—24no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—24no—
200C2×C4×C61—24no—
201C2×C4×C61—24no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—42no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61C2×C22⋊C414yes—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—24no—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—24no—
233C2×Dic61—24no—
234C2×Dic61—24no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—42no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61C2×C22⋊C414yes—
254C2×Dic61C2×C22⋊C414yes—
255C2×Dic61C4×Q818yes—
256C2×Dic61C22⋊Q818yes—
257C2×Dic61#12414yes—
258C2×Dic61C4⋊Q814yes—
259#14283—611nomaximal, Sylow
260C22×C61C22×C22⋊C411yescharacteristic
261C22×C61C2×C4×Q812yes—
262C22×C61#52012yes—
263C22×C61#52011yescharacteristic
264C22×C61#52011yescharacteristic
265C22×C61C2×C22⋊Q812yes—
266C22×C61C2×#12411yescharacteristic
267C22×C61C2×C4⋊Q811yescharacteristic
268C22×C61#53511yescharacteristic
269C22×C61#53511yescharacteristic
270C22×C61#53611yescharacteristic
271C22×C61#53611yescharacteristic
272C22×C61#53712yes—
273C22×C61#53711yescharacteristic
274C22×C61#53711yescharacteristic
275C22×C61#54012yes—
276C22×C61#54012yes—
277C22×C61#54112yes—
278C22×C61#54112yes—
279C22×C61#54412yes—
280C22×C61#54412yes—
281C22×C61#54412yes—
282C22×C61#54412yes—
283C4×C61—24no—
284C4×C61—42no—
285C4×C61—42no—
286C4×C61—44no—
287C4×C61—82no—
288Dic61—24no—
289Dic61—42no—
290Dic61—42no—
291Dic61—44no—
292Dic61—82no—
293Dic61#39118yes—
294C22×#125—611no—
295C2×#384—611no—
296C2×#376—611no—
297#1441—612no—
298#1441—612no—
299#1441—611no—
300#1441—611no—
301#1441—611no—
302#1441D6111yescharacteristic
303C4⋊#95—612no—
304C4⋊#95—612no—
305C2×C61C2×#39111yescharacteristic
306C2×C61#231111yescharacteristic
307C2×C61#231111yescharacteristic
308C2×C61#233012yes—
309C2×C61#233012yes—
310C2×C61#262912yes—
311C2×C61#262912yes—
312C2×C61#264111yescharacteristic
313C2×C61#264211yescharacteristic
314C2×C61#266011yescharacteristic
315C2×C61#266011yescharacteristic
316C2×#125—618no—
317C2×#125—1224no—
318C2×#125—1224no—
319C22×C4⋊C4—611no—
320C22×C4⋊C4—611no—
321C22×C4⋊C4—611no—
322C22×C4⋊C4—611no—
323C22×C4⋊C4—611no—
324C22×C4⋊C4—612no—
325C22×C4⋊C4—612no—
326C22×C4⋊C4—611no—
327C22×C4⋊C4C2×D6111yescharacteristic
328C22×C4⋊C4C2×D6111yescharacteristic
329C22×C42—611no—
330C22×C42—612no—
331#384—1224no—
332#376—1224no—
333C2×#95—612no—
334C2×#95—611no—
335C2×#95—612no—
336C2×#95—612no—
337C2×#95—612no—
338C2×#95—612no—
339C2×#95—612no—
340C2×#95—612no—
341C2×#95C2×D6111yescharacteristic
342C2×#95C2×D6112yes—
343C2×#95C2×D6112yes—
344C61#1428311yescharacteristic, complemented, Sylow
345C23×C4—611no—
346C23×C4—611no—
347C23×C4—611no—
348C23×C4—611no—
349C23×C4—612no—
350C23×C4—612no—
351C23×C4C22×D6112yes—
352C23×C4C22×D6111yescharacteristic
353C23×C4C22×D6111yescharacteristic
354C23×C4C22×D6112yes—
355C23×C4C22×D6111yescharacteristic
356#125—12232no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—42no—
361C2×C4⋊C4—614no—
362C2×C4⋊C4—614no—
363C2×C4⋊C4—614no—
364C2×C4⋊C4—614no—
365C2×C4⋊C4—614no—
366C2×C4⋊C4—1222no—
367C2×C4⋊C4—1222no—
368C2×C4⋊C4—1222no—
369C2×C4⋊C4—1222no—
370C2×C4⋊C4—1222no—
371C2×C4⋊C4—1222no—
372C2×C4⋊C4—1222no—
373C2×C4⋊C4—1222no—
374C2×C4⋊C4—1222no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4—1224no—
377C2×C4⋊C4—1224no—
378C2×C4⋊C4—1224no—
379C2×C4⋊C4—1224no—
380C2×C4⋊C4—1222no—
381C2×C4⋊C4—2442no—
382C2×C4⋊C4—2442no—
383C2×C4⋊C4—2442no—
384C2×C4⋊C4C4×D6114yes—
385C2×C42—614no—
386C2×C42—614no—
387C2×C42—614no—
388C2×C42—618no—
389C2×C42—1224no—
390C2×C42—1224no—
391#95—22no—
392#95—42no—
393#95—42no—
394#95—1224no—
395#95—1222no—
396#95—2442no—
397#95—2442no—
398#95—2442no—
399#95—2442no—
400#95—2442no—
401#95—2442no—
402C24C23×D6111yescharacteristic
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—612no—
429C22×C4—612no—
430C22×C4—612no—
431C22×C4—614no—
432C22×C4—614no—
433C22×C4—612no—
434C22×C4—612no—
435C22×C4—612no—
436C22×C4—612no—
437C22×C4—612no—
438C22×C4—614no—
439C22×C4—1221no—
440C22×C4—1221no—
441C22×C4—1221no—
442C22×C4—1221no—
443C22×C4—1222no—
444C22×C4—1221no—
445C22×C4—1221no—
446C22×C4—1221no—
447C22×C4—1221no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1221no—
451C22×C4—1222no—
452C22×C4—1221no—
453C22×C4—1222no—
454C22×C4—1222no—
455C22×C4—1221no—
456C22×C4—1222no—
457C22×C4—1221no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—1222no—
462C22×C4—1222no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4C2×C4×D6112yes—
467C22×C4C2×C4×D6112yes—
468C22×C4C2×C4×D6112yes—
469C22×C4D4×D6112yes—
470C22×C4D4×D6112yes—
471C22×C4Q8×D6114yes—
472C4⋊C4—28no—
473C4⋊C4—44no—
474C4⋊C4—1228no—
475C4⋊C4—1228no—
476C4⋊C4—1228no—
477C4⋊C4—1228no—
478C4⋊C4—1228no—
479C4⋊C4—2444no—
480C4⋊C4—2448no—
481C4⋊C4—2448no—
482C42—6116no—
483C42—12216no—
484C23C22×C4×D6111yescharacteristic
485C23C2×D4×D6111yescharacteristic
486C23C2×D4×D6111yescharacteristic
487C23C2×Q8×D6112yes—
488C23#41×D6112yes—
489C23#1451982212yes—
490C23#1451982211yescharacteristic
491C23#1451982311yescharacteristic
492C23#1451982412yes—
493C23#1451982411yescharacteristic
494C23#1451982511yescharacteristic
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—614no—
526C2×C4—618no—
527C2×C4—618no—
528C2×C4—614no—
529C2×C4—614no—
530C2×C4—614no—
531C2×C4—1222no—
532C2×C4—1222no—
533C2×C4—1222no—
534C2×C4—1222no—
535C2×C4—1222no—
536C2×C4—1222no—
537C2×C4—1222no—
538C2×C4—1222no—
539C2×C4—1224no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—1224no—
543C2×C4—1224no—
544C2×C4—1224no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4—2442no—
551C2×C4—2442no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2442no—
557C2×C4—2442no—
558C2×C4—2441no—
559C2×C4—2442no—
560C2×C4—2441no—
561C2×C4—2442no—
562C2×C4—2442no—
563C2×C4C22⋊C4×D6114yes—
564C22C2×C22⋊C4×D6111yescharacteristic
565C22C4×Q8×D6112yes—
566C22#1684024112yes—
567C22#1684024211yescharacteristic
568C22#1684024311yescharacteristic
569C22C22⋊Q8×D6112yes—
570C22#124×D6111yescharacteristic
571C22C4⋊Q8×D6111yescharacteristic
572C22#1684032711yescharacteristic
573C22#1684032811yescharacteristic
574C22#1684033411yescharacteristic
575C22#1684033511yescharacteristic
576C22#1684033912yes—
577C22#1684034111yescharacteristic
578C22#1684034211yescharacteristic
579C22#1684036212yes—
580C22#1684036412yes—
581C22#1684036912yes—
582C22#1684037112yes—
583C22#1684038912yes—
584C22#1684039112yes—
585C22#1684039112yes—
586C22#1684039212yes—
587C4—24no—
588C4—44no—
589C4—42no—
590C4—42no—
591C4—82no—
592C4—618no—
593C4—1224no—
594C4—4882no—
595C4—2444no—
596C4—2442no—
597C4—2442no—
598C2#391×D6111yescharacteristic
599C2#2144840611yescharacteristic
600C2#2144840811yescharacteristic
601C2#2144855512yes—
602C2#2144855712yes—
603C2#2145164812yes—
604C2#2145165112yes—
605C2#2145176111yescharacteristic
606C2#2145176611yescharacteristic
607C2#2145193711yescharacteristic
608C2#2145194211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1441C2
socle, prime radical, cosocleC24×C61C24
hypercenter#1441D61
hypocenterC61#14283
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
14. C2×#14519767C4semidirect
344. C61#14283semidirect

Maximal subgroups

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

subgroupindex
C22×#145197672
C2×#168396312
C2×#168396072
#214440732
#214440722
#214440712
#214440702
#214440692
#214440472
C61×#14412
#1428361