Finite Group Databasebeta

← #29697160

Subgroups — #29697160

Order 15616.

1002 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#103×D61C211yesmaximal, characteristic
2#21448684C211yesmaximal, characteristic
3#21448653C211yesmaximal, characteristic
4#21446498C211yesmaximal, characteristic, complemented
5#21446492C211yesmaximal, characteristic, complemented
6#21446455C211yesmaximal, characteristic, complemented
7#21446090C211yesmaximal, characteristic, complemented
8#21446084C211yesmaximal, characteristic, complemented
9#21445904C211yesmaximal, characteristic
10#21445841C211yesmaximal, characteristic
11#21445752C211yesmaximal, characteristic
12C4×#14519700C211yesmaximal, characteristic, complemented
13#21445638C211yesmaximal, characteristic, complemented
14#21445589C211yesmaximal, characteristic
15C61×#1927C211yesmaximal, characteristic, complemented
16C22×Q8×D61C2211yescharacteristic
17C22×C8×D61C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19C2×#14519742C2211yescharacteristic
20C2×#14519734—21no—
21C2×C4⋊C4×D61C2211yescharacteristic
22C2×#14519709C2211yescharacteristic
23C2×#14519702C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519700C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29#103×D61C418yescomplemented
30C2×#14519689C2211yescharacteristic
31C2×C4×#92418C2211yescharacteristic
32C2×C8×Dic61C2211yescharacteristic
33#16839669C2211yescharacteristic
34#16839666—21no—
35#16839666—21no—
36#16839660C2211yescharacteristic
37#16839658C2211yescharacteristic
38#16839646—21no—
39#16839644—21no—
40#16839624—21no—
41#16839621—21no—
42#16839619C2211yescharacteristic
43#16839619C2211yescharacteristic
44C4⋊C4×Dic61C2211yescharacteristic
45C4×#11758198C2211yescharacteristic
46#16839609—21no—
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#106×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C61×#391C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C22×Dic122—21no—
61C22×Dic122C2311yescharacteristic
62C2×Q8×D61—22no—
63C2×Q8×D61—22no—
64C2×Q8×D61—22no—
65C2×Q8×D61—41no—
66C2×Q8×D61—41no—
67C2×Q8×D61C2×C414yes—
68C22×C4×D61—21no—
69C22×C4×D61—21no—
70C22×C4×D61C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73C2×C8×D61—41no—
74C2×C8×D61—41no—
75C2×C8×D61C2×C414yes—
76C4×Dic122—22no—
77C4×Dic122—41no—
78#14519742—22no—
79#14519742—41no—
80#14519734—44no—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C4⋊C4×D61—22no—
90C4⋊C4×D61C2×C414yes—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198D411yescharacteristic
94C2×#11758198D411yescharacteristic
95C2×C4×Dic61—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61C2311yescharacteristic
99#14519709—41no—
100#14519702—22no—
101#14519702C2×C414yes—
102#14519701—41no—
103#14519701C2×C414yes—
104#14519700—22no—
105#14519700C2×C414yes—
106#14519689—22no—
107C4×#92418—41no—
108C8×Dic61—41no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684D411yescharacteristic
117#14519684D411yescharacteristic
118C22×Q8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C42×C61—21no—
125C2×C42×C61C2311yescharacteristic
126#106×C61—41no—
127#103×C61—41no—
128#103×C61C2×C414yes—
129#95×C61—21no—
130#95×C61—21no—
131#95×C61—21no—
132C23×D61C2×D411yescharacteristic
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61C22×C412yes—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2411yescharacteristic
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153Q8×D61—28no—
154Q8×D61—44no—
155Q8×D61—44no—
156Q8×D61—44no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—24no—
162C2×C4×D61—24no—
163C2×C4×D61—42no—
164C2×C4×D61—42no—
165C2×C4×D61—42no—
166C2×C4×D61—42no—
167C2×C4×D61C22×C412yes—
168C2×C4×D61#4112yes—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—24no—
173C2×Dic122—42no—
174C2×Dic122—42no—
175C2×Dic122C22×C412yes—
176C2×Dic122#4112yes—
177C2×Dic122#4112yes—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61C22×C412yes—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61C22×C412yes—
190C42×C61—21no—
191C42×C61—21no—
192C42×C61—42no—
193#11758198—22no—
194#11758198—42no—
195#11758198—42no—
196#11758198SD1614yes—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61C22×C412yes—
203#11758196—24no—
204#11758196—42no—
205#11758196—42no—
206#11758196—42no—
207#11758196—42no—
208C4×Dic61—21no—
209C4×Dic61—21no—
210C4×Dic61—42no—
211C4×Dic61—42no—
212C4×Dic61—42no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418C22×C412yes—
219C8×D61—44no—
220C23×C61C22×D411yescharacteristic
221C22×D61—21no—
222C22×D61—21no—
223C22×D61—21no—
224C22×D61—21no—
225C22×D61C4×D412yes—
226C22×D61C2×SD1612yes—
227C22×D61#13512yes—
228Q8×C61—22no—
229Q8×C61—22no—
230Q8×C61—41no—
231Q8×C61—41no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61C2×SD1612yes—
250C2×C4×C61C23×C411yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61H(5,𝔽2)11yescharacteristic
255C2×C4×C61H(5,𝔽2)11yescharacteristic
256C2×Dic61—22no—
257C2×Dic61—21no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—22no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61C4×D412yes—
276C2×Dic61C2×SD1612yes—
277C4×D61—22no—
278C4×D61—22no—
279C4×D61—22no—
280C4×D61—22no—
281C4×D61—48no—
282C4×D61—48no—
283Dic122—22no—
284Dic122—22no—
285Dic122—22no—
286Dic122—48no—
287Dic122#11714yescomplemented
288C8×C61—22no—
289C8×C61—41no—
290#92418—22no—
291#92418—41no—
292#22571—611nomaximal, Sylow
293C22×C61C2×C4×D411yescharacteristic
294C22×C61#54611yescharacteristic
295C22×C61#54611yescharacteristic
296C22×C61C22×SD1611yescharacteristic
297C22×C61C2×#13511yescharacteristic
298C22×C61#57811yescharacteristic
299C22×C61#57811yescharacteristic
300C2×D61—22no—
301C2×D61—22no—
302C2×D61—22no—
303C2×D61—22no—
304C2×D61—22no—
305C2×D61—22no—
306C2×D61C4×SD1614yes—
307C4×C61—22no—
308C4×C61—42no—
309C4×C61—42no—
310C4×C61—41no—
311C4×C61—81no—
312C4×C61—81no—
313C4×C61C2×#11711yescharacteristic
314C4×C61#51811yescharacteristic
315C4×C61#55111yescharacteristic
316C4×C61#55111yescharacteristic
317Dic61—24no—
318Dic61—22no—
319Dic61—22no—
320Dic61—42no—
321Dic61—41no—
322Dic61—81no—
323Dic61—81no—
324C22×#103—611no—
325#2344—611no—
326Q8×C22⋊C4—611no—
327#1933—611no—
328#1933—611no—
329#1927—611no—
330#1927D6111yescharacteristic, complemented
331#1865—611no—
332#1865—611no—
333#1823—611no—
334#1807—611no—
335#1787—611no—
336C4×#103—611no—
337C4×#103—611no—
338C8×C22⋊C4—611no—
339C2×C61#234211yescharacteristic
340C2×C61C2×C4×SD1611yescharacteristic
341C2×C61#295011yescharacteristic
342C2×C61#329811yescharacteristic
343C2×C61#329811yescharacteristic
344C2×C61#330011yescharacteristic
345C2×C61#330011yescharacteristic
346D61—24no—
347C23×Q8—611no—
348C23×C8—611no—
349C2×C4×Q8—611no—
350C2×C4×Q8—611no—
351C2×#117—1221no—
352C22×C4⋊C4—611no—
353C2×#106—611no—
354C2×#106C2×D6111yescharacteristic
355C2×#103—611no—
356C2×#103—611no—
357C2×#103—611no—
358C2×#103—618no—
359C2×#103—611no—
360C2×#103—611no—
361C2×#103—611no—
362C2×#103C2×D6111yescharacteristic
363C2×#103C2×D6111yescharacteristic
364C2×C22⋊C8—611no—
365C2×C4×C8—611no—
366C2×C4×C8—611no—
367#391—611no—
368#391—1221no—
369#391—1221no—
370#391C2×D6111yescharacteristic
371#389—611no—
372#389—611no—
373#387—1221no—
374#387—1221no—
375#381—1221no—
376#380—611no—
377#380—611no—
378#380—1221no—
379C4×C4⋊C4—611no—
380C4×C4⋊C4C2×D6111yescharacteristic
381C4×C22⋊C4—611no—
382C4×C22⋊C4—1221no—
383#340—611no—
384#340—611no—
385#340—611no—
386#340—611no—
387#340—611no—
388#340C2×D6111yescharacteristic
389#336—611no—
390#336C2×D6111yescharacteristic
391C61#2257111yescharacteristic, complemented, Sylow
392C22×Q8—614no—
393C22×Q8—611no—
394C22×Q8—1221no—
395C22×Q8—1222no—
396C22×Q8—1222no—
397C22×Q8—1222no—
398C22×Q8—2441no—
399C22×Q8—2441no—
400C22×Q8C22×D6111yescharacteristic
401C23×C4—611no—
402C23×C4—1221no—
403C23×C4—1221no—
404C22×C8—611no—
405C22×C8—614no—
406C22×C8—611no—
407C22×C8—2441no—
408C22×C8—2441no—
409C22×C8C22×D6111yescharacteristic
410C22×C8C22×D6111yescharacteristic
411C4×Q8—1222no—
412C4×Q8—1222no—
413C4×Q8—2441no—
414C4×Q8—2441no—
415#117—2444no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—614no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—1222no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4C22×D6111yescharacteristic
427C2×C4⋊C4C22×D6111yescharacteristic
428C2×C4⋊C4C22×D6111yescharacteristic
429C2×C22⋊C4—611no—
430C2×C22⋊C4—611no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—1221no—
433C2×C42—21no—
434C2×C42—611no—
435C2×C42—1221no—
436C2×C42—1221no—
437C2×C42—1221no—
438C2×C42C22×D6111yescharacteristic
439#106—41no—
440#106—2441no—
441#103—41no—
442#103—614no—
443#103—614no—
444#103—614no—
445#103—1222no—
446#103—1222no—
447#103—2441no—
448#103C4×D6114yes—
449C22⋊C8—1222no—
450C4×C8—2441no—
451C4×C8—2441no—
452#95—21no—
453#95—21no—
454#95—21no—
455#95—611no—
456#95—611no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460#95—1221no—
461#95—1221no—
462#95—1221no—
463#95—1221no—
464C24—611no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—612no—
471C2×Q8—612no—
472C2×Q8—612no—
473C2×Q8—1224no—
474C2×Q8—1221no—
475C2×Q8—1221no—
476C2×Q8—1221no—
477C2×Q8—1228no—
478C2×Q8—2442no—
479C2×Q8—2442no—
480C2×Q8—2444no—
481C2×Q8—2444no—
482C2×Q8—2444no—
483C2×Q8C2×C4×D6112yes—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—611no—
489C22×C4—611no—
490C22×C4—612no—
491C22×C4—612no—
492C22×C4—611no—
493C22×C4—1224no—
494C22×C4—1224no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—2442no—
504C22×C4—2442no—
505C22×C4—2442no—
506C22×C4—2442no—
507C22×C4D4×D6111yescharacteristic
508C22×C4D4×D6111yescharacteristic
509C22×C4C23×D6111yescharacteristic
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—612no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—2444no—
522C2×C8C2×C4×D6112yes—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—612no—
529C4⋊C4—1224no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—2442no—
536C4⋊C4—2442no—
537C4⋊C4—2442no—
538C4⋊C4—2442no—
539C4⋊C4C2×C4×D6112yes—
540C22⋊C4—614no—
541C22⋊C4—1222no—
542C22⋊C4—2442no—
543C22⋊C4—2442no—
544C42—21no—
545C42—21no—
546C42—42no—
547C42—1221no—
548C42—1221no—
549C42—2442no—
550C42—2442no—
551C42—2442no—
552C23—612no—
553C23—612no—
554C23—612no—
555C23—1221no—
556C23—1221no—
557C23—1221no—
558C23—1221no—
559C23C2×D4×D6111yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—41no—
563Q8—41no—
564Q8—614no—
565Q8—1222no—
566Q8—1222no—
567Q8—1222no—
568Q8—2448no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—612no—
587C2×C4—612no—
588C2×C4—1222no—
589C2×C4—1221no—
590C2×C4—1222no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—1222no—
594C2×C4—1221no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—2448no—
604C2×C4—2448no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4SD16×D6112yes—
614C2×C4C22×C4×D6111yescharacteristic
615C2×C4#41×D6111yescharacteristic
616C2×C4#41×D6111yescharacteristic
617C2×C4#41×D6111yescharacteristic
618C2×C4#1451982211yescharacteristic
619C2×C4#1451982311yescharacteristic
620C8—22no—
621C8—41no—
622C8—1222no—
623C8—2441no—
624C22—614no—
625C22—1222no—
626C22—1222no—
627C22—1222no—
628C22—1222no—
629C22—1222no—
630C22—1222no—
631C22C4×D4×D6111yescharacteristic
632C22#1684040411yescharacteristic
633C22#1684040911yescharacteristic
634C22C2×SD16×D6111yescharacteristic
635C22#135×D6111yescharacteristic
636C22#1684057811yescharacteristic
637C22#1684057811yescharacteristic
638C4—22no—
639C4—41no—
640C4—42no—
641C4—42no—
642C4—81no—
643C4—81no—
644C4—1222no—
645C4—1222no—
646C4—1224no—
647C4—4881no—
648C4—4881no—
649C4—2441no—
650C4—2442no—
651C4#117×D6111yescharacteristic
652C4#1684023411yescharacteristic
653C4#1684043711yescharacteristic
654C4#1684044211yescharacteristic
655C2—1224no—
656C2#2144866311yescharacteristic
657C2C4×SD16×D6111yescharacteristic
658C2#2145468411yescharacteristic
659C2#2145768211yescharacteristic
660C2#2145768711yescharacteristic
661C2#2145770911yescharacteristic
662C2#2145771311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1927C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1927D61
hypocenterC61#22571
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1927

Complements

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

NH ≅ G/Ntype
4. #21446498C2semidirect
5. #21446492C2semidirect
6. #21446455C2semidirect
7. #21446090C2semidirect
8. #21446084C2semidirect
12. C4×#14519700C2semidirect
13. #21445638C2semidirect
15. C61×#1927C2semidirect
29. #103×D61C4semidirect
287. Dic122#117semidirect
330. #1927D61semidirect
391. C61#22571semidirect

Maximal subgroups

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

subgroupindex
C2×#103×D612
#214486842
#214486532
#214464982
#214464922
#214464552
#214460902
#214460842
#214459042
#214458412
#214457522
C4×#145197002
#214456382
#214455892
C61×#19272
#2257161