Finite Group Databasebeta

← #29601262

Subgroups — #29601262

Order 15616.

2102 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519735C211yesmaximal, characteristic
2C2×#16839621C212yesmaximal
3C2×C4×#11758196C211yesmaximal, characteristic, complemented
4C2×#16839613C211yesmaximal, characteristic, complemented
5#21444057C211yesmaximal, characteristic
6#21444055C211yesmaximal, characteristic
7#21444055C212yesmaximal
8#21444053C212yesmaximal, complemented
9#21444051C211yesmaximal, characteristic, complemented
10#21444047C212yesmaximal, complemented
11C61×#1436C211yesmaximal, characteristic, complemented
12C23×C4×D61C2211yescharacteristic
13C2×#14519735—24no—
14C2×#14519735—24no—
15C2×#14519735C418yescomplemented
16C22×C4⋊Dic61C2211yescharacteristic
17C22×#11758196C2212yes—
18C22×#11758196C2212yes—
19C22×#11758198C2211yescharacteristic
20C22×#11758198C2211yescharacteristic
21C22×#11758198C2212yes—
22C22×#11758198C2212yes—
23C22×C4×Dic61C2211yescharacteristic
24C22×C4×Dic61C2211yescharacteristic
25C22×C4×Dic61C2212yes—
26C22×C4×Dic61C2211yescharacteristic
27#16839621C4116yescomplemented
28C4×#11758196—24no—
29#16839613—24no—
30C2×#14519684C2211yescharacteristic
31C2×#14519684C2212yes—
32C2×#14519684C2212yes—
33C2×#14519684C2212yes—
34C2×#14519684C2212yes—
35C2×#14519684C2212yes—
36C22×C4⋊C4×C61C2211yescharacteristic
37C22×C42×C61C2211yescharacteristic
38C2×#95×C61C2211yescharacteristic
39C2×#95×C61C2212yes—
40C2×#95×C61C2212yes—
41C24×D61C2311yescharacteristic
42C23×Dic61C2312yes—
43C23×Dic61C2312yes—
44C23×Dic61C2311yescharacteristic
45C23×Dic61C2311yescharacteristic
46C23×Dic61C2311yescharacteristic
47C22×C4×D61—22no—
48C22×C4×D61—22no—
49C22×C4×D61—24no—
50C22×C4×D61C2×C414yes—
51C22×C4×D61C2×C418yes—
52#14519735—232no—
53C2×C4⋊Dic61—22no—
54C2×C4⋊Dic61—22no—
55C2×C4⋊Dic61C2×C414yes—
56C2×#11758196—24no—
57C2×#11758196—24no—
58C2×#11758196—24no—
59C2×#11758196—24no—
60C2×#11758196C2×C418yes—
61C2×#11758196C2×C418yes—
62C2×#11758198—22no—
63C2×#11758198—22no—
64C2×#11758198—22no—
65C2×#11758198—22no—
66C2×#11758198—42no—
67C2×#11758198—42no—
68C2×#11758198—42no—
69C2×#11758198—42no—
70C2×#11758198C2×C414yes—
71C2×#11758198C2×C414yes—
72C2×#11758198C2×C418yes—
73C2×#11758198C2×C418yes—
74C2×C4×Dic61—22no—
75C2×C4×Dic61—22no—
76C2×C4×Dic61—22no—
77C2×C4×Dic61—24no—
78C2×C4×Dic61—42no—
79C2×C4×Dic61—42no—
80C2×C4×Dic61—42no—
81C2×C4×Dic61C2×C414yes—
82C2×C4×Dic61C2×C414yes—
83C2×C4×Dic61D414yes—
84C2×C4×Dic61D414yes—
85#14519684—22no—
86#14519684—24no—
87#14519684—42no—
88#14519684—42no—
89#14519684—42no—
90#14519684C2×C418yes—
91C23×C4×C61C2312yes—
92C23×C4×C61C2311yescharacteristic
93C23×C4×C61C2312yes—
94C23×C4×C61C2311yescharacteristic
95C23×C4×C61C2311yescharacteristic
96C2×C4⋊C4×C61—22no—
97C2×C4⋊C4×C61—22no—
98C2×C4⋊C4×C61C2×C414yes—
99C2×C42×C61—22no—
100C2×C42×C61—42no—
101#95×C61—22no—
102#95×C61—42no—
103#95×C61C2×C418yes—
104C24×C61C2411yescharacteristic
105C23×D61—21no—
106C23×D61—21no—
107C23×D61—22no—
108C23×D61—22no—
109C23×D61—22no—
110C23×D61—22no—
111C23×D61—22no—
112C23×D61C22×C412yes—
113C23×D61C22×C414yes—
114C22×C4×C61—21no—
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—21no—
122C22×C4×C61—21no—
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×C61C22×C414yes—
136C22×C4×C61C22×C414yes—
137C22×C4×C61C22×C412yes—
138C22×C4×C61C22×C414yes—
139C22×C4×C61C22×C412yes—
140C22×C4×C61C22×C412yes—
141C22×C4×C61C2×D412yes—
142C22×C4×C61C2×D412yes—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—21no—
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×Dic61—22no—
158C22×Dic61—22no—
159C22×Dic61—22no—
160C22×Dic61—22no—
161C22×Dic61C22×C414yes—
162C22×Dic61C22×C414yes—
163C22×Dic61C22×C412yes—
164C22×Dic61C22×C412yes—
165C22×Dic61C22×C412yes—
166C22×Dic61C22×C414yes—
167C22×Dic61C2×D412yes—
168C22×Dic61C2×D412yes—
169C22×Dic61C2×D412yes—
170C22×Dic61C2×D412yes—
171C22×Dic61#4114yes—
172C2×C4×D61—28no—
173C2×C4×D61—28no—
174C2×C4×D61—216no—
175C2×C4×D61—216no—
176C2×C4×D61C42116yescomplemented
177C4⋊C4×C61—28no—
178C42×C61—44no—
179#11758198—28no—
180#11758198—28no—
181#11758198—48no—
182#11758198—48no—
183C4⋊Dic61—28no—
184#11758196—216no—
185#11758196—216no—
186C4×Dic61—28no—
187C4×Dic61—48no—
188C4×Dic61—44no—
189C4×Dic61C22⋊C4116yescomplemented
190C23×C61C23×C411yescharacteristic
191C23×C61C23×C412yes—
192C23×C61C22×D411yescharacteristic
193C23×C61C22×D411yescharacteristic
194C23×C61C2×#4112yes—
195C23×C61H(5,𝔽2)12yes—
196C23×C61H(5,𝔽2)12yes—
197C23×C612−1+412yes—
198C23×C612−1+412yes—
199C22×D61—22no—
200C22×D61—22no—
201C22×D61—24no—
202C22×D61—24no—
203C22×D61—24no—
204C22×D61—24no—
205C22×D61—24no—
206C22×D61—24no—
207C22×D61—24no—
208C22×D61—24no—
209C22×D61—41no—
210C22×D61—41no—
211C22×D61—41no—
212C22×D61—41no—
213C22×D61—42no—
214C22×D61—42no—
215C22×D61—42no—
216C22×D61—42no—
217C22×D61—42no—
218C22×D61—42no—
219C22×D61C2×C4214yes—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—24no—
223C2×C4×C61—24no—
224C2×C4×C61—24no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—24no—
228C2×C4×C61—22no—
229C2×C4×C61—24no—
230C2×C4×C61—24no—
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—41no—
239C2×C4×C61—41no—
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×C4214yes—
248C2×C4×C61C2×C22⋊C414yes—
249C2×Dic61—24no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—24no—
253C2×Dic61—22no—
254C2×Dic61—24no—
255C2×Dic61—24no—
256C2×Dic61—24no—
257C2×Dic61—24no—
258C2×Dic61—24no—
259C2×Dic61—24no—
260C2×Dic61—24no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—42no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61—42no—
270C2×Dic61—42no—
271C2×Dic61—42no—
272C2×Dic61C2×C4214yes—
273C2×Dic61C2×C22⋊C414yes—
274C2×Dic61C2×C22⋊C414yes—
275C2×Dic61#11714yes—
276C2×Dic61C4×D418yes—
277C2×Dic61C4×D418yes—
278C4×D61—232no—
279#13777—611nomaximal, Sylow
280C22×C61C22×C4211yescharacteristic
281C22×C61C22×C22⋊C411yescharacteristic
282C22×C61C2×#11711yescharacteristic
283C22×C61C2×C4×D412yes—
284C22×C61C2×C4×D412yes—
285C22×C61#51812yes—
286C22×C61#51912yes—
287C22×C61#52012yes—
288C22×C61#52012yes—
289C22×C61#52012yes—
290C22×C61#52012yes—
291C22×C61#53511yescharacteristic
292C22×C61#53511yescharacteristic
293C22×C61#53611yescharacteristic
294C22×C61#53611yescharacteristic
295C22×C61#53712yes—
296C22×C61#53712yes—
297C22×C61#54012yes—
298C22×C61#54112yes—
299C22×C61#54212yes—
300C22×C61#54212yes—
301C2×D61—24no—
302C2×D61—28no—
303C2×D61—42no—
304C2×D61—42no—
305C2×D61—44no—
306C2×D61—44no—
307C2×D61—44no—
308C2×D61—44no—
309C2×D61—44no—
310C4×C61—24no—
311C4×C61—24no—
312C4×C61—42no—
313C4×C61—44no—
314C4×C61—44no—
315Dic61—24no—
316Dic61—42no—
317Dic61—44no—
318Dic61—44no—
319Dic61C4×C22⋊C418yes—
320C22×#117—611no—
321C2×#380—612no—
322C2×C4×C4⋊C4—611no—
323C2×C4×C4⋊C4—611no—
324#1437—611no—
325#1437—612no—
326#1437—611no—
327#1436—612no—
328#1436—611no—
329#1436D6111yescharacteristic, complemented
330C4⋊#95—612no—
331C2×C61C2×C4×C22⋊C411yescharacteristic
332C2×C61#227412yes—
333C2×C61#231111yescharacteristic
334C2×C61#231111yescharacteristic
335C2×C61#231412yes—
336C2×C61#232312yes—
337C2×C61#232312yes—
338C2×C61#232412yes—
339C2×C61#232412yes—
340D61—44no—
341C24×C4—611no—
342C2×#117—618no—
343C2×#117—1224no—
344C2×#117—1224no—
345C22×C4⋊C4—612no—
346C22×C4⋊C4—612no—
347C22×C4⋊C4—611no—
348C22×C4⋊C4C2×D6111yescharacteristic
349C22×C22⋊C4—611no—
350C22×C22⋊C4—611no—
351C22×C22⋊C4—612no—
352C22×C22⋊C4—612no—
353C22×C42—611no—
354C22×C42—611no—
355C22×C42—612no—
356C22×C42—611no—
357C22×C42C2×D6111yescharacteristic
358#380—6116no—
359C4×C4⋊C4—1224no—
360C4×C4⋊C4—1224no—
361C2×#95—612no—
362C2×#95—612no—
363C2×#95—611no—
364C2×#95—612no—
365C2×#95—612no—
366C2×#95—612no—
367C2×#95C2×D6111yescharacteristic
368C2×#95C2×D6112yes—
369C2×#95C2×D6112yes—
370C61#1377711yescharacteristic, complemented, Sylow
371C25—611no—
372C23×C4—611no—
373C23×C4—611no—
374C23×C4—614no—
375C23×C4—618no—
376C23×C4—612no—
377C23×C4—612no—
378C23×C4—611no—
379C23×C4—1222no—
380C23×C4—1222no—
381C23×C4—1224no—
382C23×C4C22×D6112yes—
383C23×C4C22×D6112yes—
384C23×C4C22×D6111yescharacteristic
385C23×C4C22×D6111yescharacteristic
386C23×C4C22×D6111yescharacteristic
387#117—12232no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—614no—
391C2×C4⋊C4—618no—
392C2×C4⋊C4—618no—
393C2×C4⋊C4—1222no—
394C2×C4⋊C4—1222no—
395C2×C4⋊C4—1224no—
396C2×C4⋊C4—1224no—
397C2×C4⋊C4—1224no—
398C2×C4⋊C4—1224no—
399C2×C4⋊C4C4×D6114yes—
400C2×C22⋊C4—614no—
401C2×C22⋊C4—614no—
402C2×C22⋊C4—618no—
403C2×C22⋊C4—618no—
404C2×C22⋊C4—1222no—
405C2×C22⋊C4—1222no—
406C2×C22⋊C4—1222no—
407C2×C22⋊C4—1222no—
408C2×C22⋊C4—2442no—
409C2×C22⋊C4—2442no—
410C2×C22⋊C4—2442no—
411C2×C22⋊C4—2442no—
412C2×C42—22no—
413C2×C42—42no—
414C2×C42—614no—
415C2×C42—614no—
416C2×C42—614no—
417C2×C42—614no—
418C2×C42—1222no—
419C2×C42—1222no—
420C2×C42—1224no—
421C2×C42—1222no—
422C2×C42—2442no—
423C2×C42—2442no—
424C2×C42—2442no—
425#95—22no—
426#95—42no—
427#95—618no—
428#95—1224no—
429#95—1222no—
430#95—2442no—
431#95—2442no—
432#95—2442no—
433#95C4×D6118yes—
434C24—612no—
435C24—614no—
436C24—1221no—
437C24—1221no—
438C24—1222no—
439C24—1222no—
440C24—1222no—
441C24—1222no—
442C24—1222no—
443C24C23×D6111yescharacteristic
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
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—612no—
466C22×C4—614no—
467C22×C4—612no—
468C22×C4—612no—
469C22×C4—614no—
470C22×C4—612no—
471C22×C4—612no—
472C22×C4—612no—
473C22×C4—6116no—
474C22×C4—614no—
475C22×C4—614no—
476C22×C4—612no—
477C22×C4—1228no—
478C22×C4—1228no—
479C22×C4—12216no—
480C22×C4—12216no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—1221no—
486C22×C4—1221no—
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×C4—1222no—
499C22×C4C2×C4×D6114yes—
500C22×C4C2×C4×D6114yes—
501C22×C4C2×C4×D6112yes—
502C22×C4C2×C4×D6114yes—
503C22×C4C2×C4×D6112yes—
504C22×C4C2×C4×D6112yes—
505C22×C4D4×D6112yes—
506C22×C4D4×D6112yes—
507C4⋊C4—28no—
508C4⋊C4—12216no—
509C4⋊C4—12216no—
510C4⋊C4—1228no—
511C22⋊C4—1228no—
512C22⋊C4—1228no—
513C22⋊C4—2448no—
514C22⋊C4—2448no—
515C42—44no—
516C42—6116no—
517C42—1228no—
518C42—2444no—
519C42—2448no—
520C23—614no—
521C23—1222no—
522C23—1222no—
523C23—1224no—
524C23—1224no—
525C23—1224no—
526C23—1224no—
527C23—1224no—
528C23—1224no—
529C23—1224no—
530C23—1224no—
531C23—2441no—
532C23—2441no—
533C23—2441no—
534C23—2441no—
535C23—2442no—
536C23—2442no—
537C23—2442no—
538C23—2442no—
539C23—2442no—
540C23—2442no—
541C23C22×C4×D6111yescharacteristic
542C23C22×C4×D6112yes—
543C23C2×D4×D6111yescharacteristic
544C23C2×D4×D6111yescharacteristic
545C23#41×D6112yes—
546C23#1451982212yes—
547C23#1451982312yes—
548C23#1451982412yes—
549C23#1451982512yes—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—614no—
578C2×C4—614no—
579C2×C4—614no—
580C2×C4—618no—
581C2×C4—618no—
582C2×C4—614no—
583C2×C4—12232no—
584C2×C4—1224no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—1224no—
588C2×C4—1222no—
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—2441no—
599C2×C4—2441no—
600C2×C4—2442no—
601C2×C4—2442no—
602C2×C4—2442no—
603C2×C4—2442no—
604C2×C4—2442no—
605C2×C4—2442no—
606C2×C4—2442no—
607C2×C4C42×D6114yes—
608C2×C4C22⋊C4×D6114yes—
609C22—1224no—
610C22—1228no—
611C22—2444no—
612C22—2444no—
613C22—2444no—
614C22—2444no—
615C22—2442no—
616C22—2442no—
617C22—2444no—
618C22C2×C42×D6111yescharacteristic
619C22C2×C22⋊C4×D6111yescharacteristic
620C22#117×D6111yescharacteristic
621C22C4×D4×D6112yes—
622C22C4×D4×D6112yes—
623C22#1684023412yes—
624C22#1684023812yes—
625C22#1684024112yes—
626C22#1684024212yes—
627C22#1684024312yes—
628C22#1684024412yes—
629C22#1684032711yescharacteristic
630C22#1684033011yescharacteristic
631C22#1684033411yescharacteristic
632C22#1684033711yescharacteristic
633C22#1684033912yes—
634C22#1684034212yes—
635C22#1684036412yes—
636C22#1684037112yes—
637C22#1684037812yes—
638C22#1684038212yes—
639C4—24no—
640C4—24no—
641C4—44no—
642C4—44no—
643C4—42no—
644C4—618no—
645C4—1224no—
646C4—2444no—
647C4—2444no—
648C4—2442no—
649C2—2444no—
650C2C4×C22⋊C4×D6111yescharacteristic
651C2#2144817312yes—
652C2#2144840611yescharacteristic
653C2#2144841011yescharacteristic
654C2#2144842512yes—
655C2#2144849612yes—
656C2#2144850012yes—
657C2#2144850712yes—
658C2#2144851112yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC22×C61C22×C42
FittingC61×#1436C2
socle, cosocleC24×C61C24
prime radicalC24×D61C23
hypercenter#1436D61
hypocenterC61#13777
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1436

Complements

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

NH ≅ G/Ntype
3. C2×C4×#11758196C2semidirect
4. C2×#16839613C2semidirect
8. #21444053C2semidirect
9. #21444051C2semidirect
10. #21444047C2semidirect
11. C61×#1436C2semidirect
15. C2×#14519735C4semidirect
27. #16839621C4semidirect
176. C2×C4×D61C42semidirect
189. C4×Dic61C22⋊C4semidirect
329. #1436D61semidirect
370. C61#13777semidirect

Maximal subgroups

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

subgroupindex
C22×#145197352
C2×#168396212
C2×C4×#117581962
C2×#168396132
#214440572
#214440552
#214440532
#214440512
#214440472
C61×#14362
#1377761