Finite Group Databasebeta

← #29601976

Subgroups — #29601976

Order 15616.

2222 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519735C211yesmaximal, characteristic
2C2×#16839624C211yesmaximal, characteristic
3C2×#16839619C212yesmaximal
4C2×#95×D61C211yesmaximal, characteristic
5#21444084C211yesmaximal, characteristic, complemented
6#21444078C211yesmaximal, characteristic, complemented
7#21444075C211yesmaximal, characteristic, complemented
8#21444053C212yesmaximal, complemented
9#21444051C212yesmaximal, complemented
10#21444040C212yesmaximal
11C61×#1444C211yesmaximal, characteristic, complemented
12C23×C4×D61C2212yes—
13C23×C4×D61C2211yescharacteristic
14C2×#14519735—28no—
15C2×#14519735—24no—
16C22×C4⋊Dic61C2211yescharacteristic
17C22×C4⋊Dic61C2212yes—
18C22×#11758198C2211yescharacteristic
19C22×#11758198C2211yescharacteristic
20C22×#11758198C2212yes—
21C22×C4×Dic61C2212yes—
22#16839624—24no—
23#16839619—28no—
24C2×#14519684C2211yescharacteristic
25C2×#14519684C2211yescharacteristic
26C2×#14519684C2211yescharacteristic
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2212yes—
32C2×#14519684C2212yes—
33C2×#14519684C2211yescharacteristic
34#95×D61C418yescomplemented
35C22×C4⋊C4×C61C2211yescharacteristic
36C22×C4⋊C4×C61C2212yes—
37C2×#95×C61C2211yescharacteristic
38C2×#95×C61C2212yes—
39C2×#95×C61C2211yescharacteristic
40C24×D61C2311yescharacteristic
41C23×Dic61C2311yescharacteristic
42C23×Dic61C2311yescharacteristic
43C23×Dic61C2312yes—
44C23×Dic61C2312yes—
45C23×Dic61C2311yescharacteristic
46C22×C4×D61—24no—
47C22×C4×D61—24no—
48C22×C4×D61—24no—
49C22×C4×D61—24no—
50C22×C4×D61—24no—
51C22×C4×D61—24no—
52C22×C4×D61—42no—
53C22×C4×D61—42no—
54C22×C4×D61C2×C418yes—
55C22×C4×D61C2×C414yes—
56C22×C4×D61D414yes—
57C22×C4×D61Q814yes—
58#14519735—232no—
59C2×C4⋊Dic61—22no—
60C2×C4⋊Dic61—24no—
61C2×C4⋊Dic61—24no—
62C2×C4⋊Dic61—24no—
63C2×C4⋊Dic61—42no—
64C2×#11758198—24no—
65C2×#11758198—24no—
66C2×#11758198—24no—
67C2×#11758198—22no—
68C2×#11758198—22no—
69C2×#11758198—24no—
70C2×#11758198D418yes—
71C2×C4×Dic61—24no—
72C2×C4×Dic61—24no—
73C2×C4×Dic61—24no—
74#14519684—24no—
75#14519684—22no—
76#14519684—24no—
77#14519684—24no—
78#14519684—22no—
79#14519684—22no—
80#14519684—42no—
81#14519684—42no—
82#14519684C2×C414yes—
83#14519684C2×C418yes—
84C23×C4×C61C2311yescharacteristic
85C23×C4×C61C2311yescharacteristic
86C23×C4×C61C2312yes—
87C23×C4×C61C2311yescharacteristic
88C23×C4×C61C2312yes—
89C2×C4⋊C4×C61—24no—
90C2×C4⋊C4×C61—22no—
91C2×C4⋊C4×C61—24no—
92C2×C4⋊C4×C61—24no—
93C2×C4⋊C4×C61—42no—
94#95×C61—22no—
95#95×C61—24no—
96#95×C61C2×C414yes—
97C24×C61C2411yescharacteristic
98C23×D61—22no—
99C23×D61—22no—
100C23×D61—21no—
101C23×D61—21no—
102C23×D61—21no—
103C23×D61—21no—
104C23×D61C22×C412yes—
105C23×D61C2×D412yes—
106C23×D61C2×D414yes—
107C23×D61C2×Q812yes—
108C23×D61#4114yes—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
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—21no—
122C22×C4×C61—21no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61C22×C412yes—
132C22×C4×C61C22×C414yes—
133C22×C4×C61C2×D414yes—
134C22×C4×C61C2×D412yes—
135C22×C4×C61C2×Q812yes—
136C22×C4×C61#4114yes—
137C22×C4×C61#4114yes—
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—21no—
153C22×Dic61—21no—
154C22×Dic61—22no—
155C22×Dic61—22no—
156C22×Dic61—22no—
157C22×Dic61—22no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61C22×C414yes—
163C22×Dic61C22×C412yes—
164C22×Dic61C2×D414yes—
165C22×Dic61C2×D412yes—
166C22×Dic61C2×Q812yes—
167C2×C4×D61—216no—
168C2×C4×D61—28no—
169C2×C4×D61—216no—
170C2×C4×D61—28no—
171C2×C4×D61—216no—
172C2×C4×D61—216no—
173C2×C4×D61—48no—
174C2×C4×D61—48no—
175C2×C4×D61—48no—
176C2×C4×D61—48no—
177C2×C4×D61—48no—
178C2×C4×D61C4⋊C4116yescomplemented
179C4⋊C4×C61—28no—
180C4⋊C4×C61—48no—
181#11758198—216no—
182#11758198—44no—
183#11758198—44no—
184C4⋊Dic61—28no—
185C4⋊Dic61—48no—
186C4×Dic61—216no—
187C23×C61C23×C411yescharacteristic
188C23×C61C22×D411yescharacteristic
189C23×C61C22×D412yes—
190C23×C61C22×Q811yescharacteristic
191C23×C61C2×#4112yes—
192C23×C61C2×#4112yes—
193C23×C61C2×#4112yes—
194C23×C61H(5,𝔽2)11yescharacteristic
195C23×C61H(5,𝔽2)11yescharacteristic
196C23×C612−1+411yescharacteristic
197C23×C612−1+411yescharacteristic
198C22×D61—24no—
199C22×D61—22no—
200C22×D61—22no—
201C22×D61—22no—
202C22×D61—22no—
203C22×D61—24no—
204C22×D61—22no—
205C22×D61—22no—
206C22×D61—22no—
207C22×D61—22no—
208C22×D61—24no—
209C22×D61—24no—
210C22×D61—24no—
211C22×D61—24no—
212C22×D61—24no—
213C22×D61—24no—
214C22×D61—24no—
215C22×D61—24no—
216C22×D61C2×C4⋊C414yes—
217C22×D61C4×D418yes—
218C22×D61#12014yes—
219C22×D61C22⋊Q818yes—
220C22×D61#12314yes—
221C2×C4×C61—24no—
222C2×C4×C61—24no—
223C2×C4×C61—24no—
224C2×C4×C61—22no—
225C2×C4×C61—24no—
226C2×C4×C61—22no—
227C2×C4×C61—24no—
228C2×C4×C61—24no—
229C2×C4×C61—24no—
230C2×C4×C61—24no—
231C2×C4×C61—24no—
232C2×C4×C61—24no—
233C2×C4×C61—24no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61C2×C4⋊C414yes—
249C2×C4×C61#11718yescomplemented
250C2×Dic61—24no—
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—42no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263C2×Dic61—42no—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—42no—
275C2×Dic61—42no—
276C2×Dic61—42no—
277C2×Dic61—42no—
278C2×Dic61C2×C4⋊C414yes—
279C4×D61—232no—
280C4×D61—432no—
281#13878—611nomaximal, Sylow
282C22×C61C22×C4⋊C411yescharacteristic
283C22×C61C2×#11712yes—
284C22×C61C2×C4×D412yes—
285C22×C61#51811yescharacteristic
286C22×C61#51911yescharacteristic
287C22×C61C2×#12011yescharacteristic
288C22×C61C2×C22⋊Q812yes—
289C22×C61C2×#12311yescharacteristic
290C22×C61#53211yescharacteristic
291C22×C61#53211yescharacteristic
292C22×C61#53312yes—
293C22×C61#53711yescharacteristic
294C22×C61#53711yescharacteristic
295C22×C61#54411yescharacteristic
296C22×C61#54411yescharacteristic
297C22×C61#54612yes—
298C22×C61#54612yes—
299C22×C61#54712yes—
300C22×C61#54712yes—
301C22×C61#55112yes—
302C22×C61#55112yes—
303C22×C61#55212yes—
304C22×C61#55212yes—
305C2×D61—24no—
306C2×D61—28no—
307C2×D61—28no—
308C2×D61—24no—
309C2×D61—24no—
310C2×D61—28no—
311C2×D61—28no—
312C2×D61—24no—
313C2×D61—24no—
314C2×D61—24no—
315C2×D61#38518yes—
316C4×C61—28no—
317C4×C61—24no—
318C4×C61—44no—
319C4×C61—81no—
320C4×C61—81no—
321Dic61—24no—
322Dic61—44no—
323Dic61—44no—
324Dic61—81no—
325Dic61—81no—
326C22×#117—611no—
327C2×#381—611no—
328C2×#380—612no—
329C22×#95—611no—
330#1444—611no—
331#1444D6111yescharacteristic, complemented
332#1442—611no—
333#1442—611no—
334#1436—612no—
335#1436—612no—
336C22⋊#95—612no—
337C2×C61#228511yescharacteristic
338C2×C61C2×#38511yescharacteristic
339C2×C61#231611yescharacteristic
340C2×C61#234212yes—
341C2×C61#234412yes—
342C2×C61#243011yescharacteristic
343C2×C61#243011yescharacteristic
344C2×C61#243812yes—
345C2×C61#243812yes—
346C2×C61#249811yescharacteristic
347C2×C61#249811yescharacteristic
348D61—28no—
349C24×C4—611no—
350C24×C4—612no—
351C2×#117—1228no—
352C2×#117—1224no—
353C22×C4⋊C4—611no—
354C22×C4⋊C4—612no—
355C22×C4⋊C4C2×D6111yescharacteristic
356C22×C4⋊C4C2×D6112yes—
357C22×C22⋊C4—612no—
358C22×C22⋊C4—611no—
359C22×C22⋊C4—611no—
360C22×C42—612no—
361#381—1224no—
362#380—1228no—
363C2×#95—612no—
364C2×#95—611no—
365C2×#95—618no—
366C2×#95—612no—
367C2×#95—611no—
368C2×#95—611no—
369C2×#95—612no—
370C2×#95—612no—
371C2×#95—611no—
372C2×#95—612no—
373C2×#95—612no—
374C2×#95C2×D6112yes—
375C2×#95C2×D6111yescharacteristic
376C2×#95C2×D6111yescharacteristic
377C61#1387811yescharacteristic, complemented, Sylow
378C25—611no—
379C23×C4—611no—
380C23×C4—614no—
381C23×C4—614no—
382C23×C4—614no—
383C23×C4—612no—
384C23×C4—612no—
385C23×C4—618no—
386C23×C4—611no—
387C23×C4—611no—
388C23×C4—1224no—
389C23×C4—1224no—
390C23×C4—1224no—
391C23×C4—1224no—
392C23×C4—1224no—
393C23×C4—1224no—
394C23×C4—2442no—
395C23×C4—2442no—
396C23×C4C22×D6111yescharacteristic
397C23×C4C22×D6111yescharacteristic
398C23×C4C22×D6112yes—
399C23×C4C22×D6111yescharacteristic
400C23×C4C22×D6112yes—
401#117—12232no—
402C2×C4⋊C4—24no—
403C2×C4⋊C4—22no—
404C2×C4⋊C4—24no—
405C2×C4⋊C4—24no—
406C2×C4⋊C4—42no—
407C2×C4⋊C4—1224no—
408C2×C4⋊C4—1222no—
409C2×C4⋊C4—1224no—
410C2×C4⋊C4—1224no—
411C2×C4⋊C4—2442no—
412C2×C22⋊C4—618no—
413C2×C22⋊C4—1224no—
414C2×C22⋊C4—1224no—
415C2×C22⋊C4—1224no—
416C2×C22⋊C4—1222no—
417C2×C22⋊C4—1222no—
418C2×C22⋊C4—1224no—
419C2×C42—1224no—
420C2×C42—1224no—
421C2×C42—1224no—
422#95—22no—
423#95—24no—
424#95—618no—
425#95—614no—
426#95—1224no—
427#95—1222no—
428#95—1222no—
429#95—1224no—
430#95—1222no—
431#95—1224no—
432#95—2442no—
433#95—2442no—
434#95C4×D6114yes—
435C24—612no—
436C24—614no—
437C24—612no—
438C24—612no—
439C24—614no—
440C24—1222no—
441C24—1222no—
442C24—1221no—
443C24—1221no—
444C24—1221no—
445C24—1221no—
446C24C23×D6111yescharacteristic
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—612no—
470C22×C4—612no—
471C22×C4—612no—
472C22×C4—6116no—
473C22×C4—614no—
474C22×C4—614no—
475C22×C4—1228no—
476C22×C4—12216no—
477C22×C4—12216no—
478C22×C4—1228no—
479C22×C4—12216no—
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—12216no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1222no—
498C22×C4—1222no—
499C22×C4—1222no—
500C22×C4—1222no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—2448no—
506C22×C4—2448no—
507C22×C4—2448no—
508C22×C4—2448no—
509C22×C4—2448no—
510C22×C4C2×C4×D6112yes—
511C22×C4C2×C4×D6114yes—
512C22×C4D4×D6114yes—
513C22×C4D4×D6112yes—
514C22×C4#1175822414yes—
515C22×C4Q8×D6112yes—
516C22×C4#1175822614yes—
517C4⋊C4—28no—
518C4⋊C4—48no—
519C4⋊C4—1228no—
520C4⋊C4—2448no—
521C22⋊C4—12216no—
522C22⋊C4—2444no—
523C22⋊C4—2444no—
524C42—12216no—
525C23—614no—
526C23—618no—
527C23—614no—
528C23—618no—
529C23—614no—
530C23—1224no—
531C23—1224no—
532C23—1222no—
533C23—1222no—
534C23—1222no—
535C23—1222no—
536C23—1224no—
537C23—1222no—
538C23—1222no—
539C23—1222no—
540C23—1222no—
541C23—1224no—
542C23—1224no—
543C23—1224no—
544C23—1224no—
545C23—1224no—
546C23—1224no—
547C23—1224no—
548C23C22×C4×D6111yescharacteristic
549C23C2×D4×D6111yescharacteristic
550C23C2×D4×D6112yes—
551C23C2×Q8×D6111yescharacteristic
552C23#41×D6112yes—
553C23C2×#1175822412yes—
554C23C2×#1175822612yes—
555C23#1451982211yescharacteristic
556C23#1451982311yescharacteristic
557C23#1451982411yescharacteristic
558C23#1451982511yescharacteristic
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
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—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—614no—
587C2×C4—12232no—
588C2×C4—1224no—
589C2×C4—1222no—
590C2×C4—1224no—
591C2×C4—1222no—
592C2×C4—1224no—
593C2×C4—1224no—
594C2×C4—1224no—
595C2×C4—1224no—
596C2×C4—1224no—
597C2×C4—1224no—
598C2×C4—24432no—
599C2×C4—2442no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2442no—
607C2×C4—2442no—
608C2×C4—2442no—
609C2×C4—2442no—
610C2×C4—2442no—
611C2×C4—2442no—
612C2×C4—2442no—
613C2×C4—2442no—
614C2×C4—2442no—
615C2×C4—2442no—
616C2×C4—2442no—
617C2×C4C4⋊C4×D6114yes—
618C2×C4#1451973518yescomplemented
619C22—618no—
620C22—1228no—
621C22—1224no—
622C22—1228no—
623C22—1224no—
624C22—1224no—
625C22—1224no—
626C22—1224no—
627C22—1228no—
628C22—1228no—
629C22—1224no—
630C22C2×C4⋊C4×D6111yescharacteristic
631C22C2×#1451973512yes—
632C22C4×D4×D6112yes—
633C22#1684023411yescharacteristic
634C22#1684023811yescharacteristic
635C22#120×D6111yescharacteristic
636C22C22⋊Q8×D6112yes—
637C22#123×D6111yescharacteristic
638C22#1684031011yescharacteristic
639C22#1684031311yescharacteristic
640C22#1684031712yes—
641C22#1684034111yescharacteristic
642C22#1684034211yescharacteristic
643C22#1684038911yescharacteristic
644C22#1684039211yescharacteristic
645C22#1684039912yes—
646C22#1684040312yes—
647C22#1684041212yes—
648C22#1684041512yes—
649C22#1684043512yes—
650C22#1684043912yes—
651C22#1684044612yes—
652C22#1684045012yes—
653C4—24no—
654C4—28no—
655C4—44no—
656C4—81no—
657C4—81no—
658C4—1224no—
659C4—4881no—
660C4—4881no—
661C4—2444no—
662C4—2444no—
663C2—1228no—
664C2#2144822411yescharacteristic
665C2#385×D6111yescharacteristic
666C2#2144843411yescharacteristic
667C2#2144866612yes—
668C2#2144868712yes—
669C2#2144942211yescharacteristic
670C2#2144943011yescharacteristic
671C2#2144948812yes—
672C2#2144949612yes—
673C2#2145025611yescharacteristic
674C2#2145026411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1444C2
socle, cosocleC24×C61C24
prime radicalC24×D61C23
hypercenter#1444D61
hypocenterC61#13878
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
5. #21444084C2semidirect
6. #21444078C2semidirect
7. #21444075C2semidirect
8. #21444053C2semidirect
9. #21444051C2semidirect
11. C61×#1444C2semidirect
34. #95×D61C4semidirect
178. C2×C4×D61C4⋊C4semidirect
249. C2×C4×C61#117semidirect
331. #1444D61semidirect
377. C61#13878semidirect
618. C2×C4#14519735semidirect

Maximal subgroups

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

subgroupindex
C22×#145197352
C2×#168396242
C2×#168396192
C2×#95×D612
#214440842
#214440782
#214440752
#214440532
#214440512
#214440402
C61×#14442
#1387861