Finite Group Databasebeta

← #29647428

Subgroups — #29647428

Order 15616.

1070 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840296C211yesmaximal, characteristic
2C2×#16839819C211yesmaximal, characteristic, complemented
3#21444390C212yesmaximal, complemented
4#21444386C212yesmaximal
5#21444355C214yesmaximal, complemented
6#21444342C214yesmaximal, complemented
7C61×#2285C211yesmaximal, characteristic, complemented
8#16840296—21no—
9#16840296—21no—
10#16840296—21no—
11#16840296—21no—
12#16840296C412yescomplemented
13#16840296C412yescomplemented
14#16840296C2212yes—
15#16840296C2212yes—
16C2×#14519771—21no—
17C2×#14519769C411yescharacteristic, complemented
18C2×#14519765C411yescharacteristic, complemented
19C2×#14519760—21no—
20C2×#14519756C2211yescharacteristic
21C2×#14519751C2211yescharacteristic
22C2×C4×Dic122C411yescharacteristic, complemented
23C2×C4×D244C411yescharacteristic, complemented
24C2×C4×C22⋊D61—21no—
25C2×#14519734—21no—
26#16839819—21no—
27#16839819—21no—
28#16839819C2212yescomplemented
29#16839819C2212yescomplemented
30C2×C4⋊#92418—21no—
31C2×C22⋊#92418C2212yes—
32C2×#14519688—21no—
33#16839498C2218yescomplemented
34#16839495C2218yes—
35C2×#117×C61—21no—
36C22×C42×C61C2211yescharacteristic
37C61×#384—21no—
38C61×#382C2212yescomplemented
39C61×#380—21no—
40C61×#380C2212yes—
41C4×C4⋊C4×C61—21no—
42C4×C4⋊C4×C61C2212yescomplemented
43C22×Dic122C2×C411yescharacteristic
44C22×D244C2×C411yescharacteristic
45C22×C22⋊D61—21no—
46C22×C4×D61—21no—
47C22×#92418—22no—
48#14519771—22no—
49#14519771—22no—
50#14519771—22no—
51#14519771—22no—
52#14519769—21no—
53#14519769—21no—
54#14519769C2×C412yes—
55#14519769C2×C412yes—
56#14519765—21no—
57#14519765—21no—
58#14519765C2×C412yes—
59#14519765C2×C412yes—
60#14519760—22no—
61#14519760—22no—
62#14519760—22no—
63#14519760—22no—
64#14519756—21no—
65#14519756—21no—
66#14519756C2×C412yes—
67#14519756C2312yes—
68#14519751—21no—
69#14519751—21no—
70#14519751C2×C412yes—
71#14519751C2312yes—
72C4×Dic122—21no—
73C4×Dic122—21no—
74C4×Dic122C2×C412yes—
75C4×Dic122C2×C412yes—
76C4×D244—21no—
77C4×D244—21no—
78C4×D244C2×C412yes—
79C4×D244C2×C412yes—
80C4×C22⋊D61—22no—
81C4×C22⋊D61—22no—
82C4×C22⋊D61—22no—
83C4×C22⋊D61—22no—
84#14519734—22no—
85#14519734—22no—
86#14519734—22no—
87#14519734—22no—
88C2×C4⋊Dic61C2×C411yescharacteristic
89C2×C4⋊Dic61C2×C411yescharacteristic
90C2×#11758196—21no—
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×C22⋊Dic61—21no—
95C2×#11758198—21no—
96C2×#11758198—21no—
97C2×#11758198—21no—
98C2×#11758198—21no—
99C2×C4×Dic61—21no—
100C4⋊#92418—22no—
101C4⋊#92418—22no—
102C22⋊#92418—22no—
103C22⋊#92418C2314yes—
104#14519688—22no—
105#14519688—22no—
106C23×C4×C61D411yescharacteristic
107C23×C4×C61D411yescharacteristic
108C23×C4×C61C2311yescharacteristic
109#117×C61—44no—
110C2×C4⋊C4×C61—22no—
111C2×C4⋊C4×C61—22no—
112C2×C4⋊C4×C61—22no—
113C2×C22⋊C4×C61—22no—
114C2×C42×C61—22no—
115C2×C42×C61C2×C411yescharacteristic
116C2×C42×C61C2×C411yescharacteristic
117C2×C42×C61C2×C411yescharacteristic
118C2×C42×C61C2×C411yescharacteristic
119C2×C42×C61D411yescharacteristic
120C2×C42×C61D411yescharacteristic
121C2×C42×C61D411yescharacteristic
122C2×C42×C61D411yescharacteristic
123C2×C42×C61D411yescharacteristic
124C2×C42×C61D411yescharacteristic
125C2×C42×C61C2311yescharacteristic
126C2×C42×C61C2311yescharacteristic
127#95×C61—22no—
128#95×C61C2314yes—
129C24×C61C2×D411yescharacteristic
130C23×D61—21no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—24no—
134C22×C4×C61—42no—
135C22×C4×C61—42no—
136C22×C4×C61C22⋊C411yescharacteristic
137C22×C4×C61C22⋊C411yescharacteristic
138C22×C4×C61C22⋊C411yescharacteristic
139C22×C4×C61C22⋊C411yescharacteristic
140C22×C4×C61C22⋊C411yescharacteristic
141C22×C4×C61C22⋊C411yescharacteristic
142C22×C4×C61C22⋊C411yescharacteristic
143C22×C4×C61C22⋊C411yescharacteristic
144C22×C4×C61C22×C411yescharacteristic
145C22×C4×C61C22×C411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2411yescharacteristic
154C2×C22⋊D61—22no—
155C2×C22⋊D61—22no—
156C2×C22⋊D61—42no—
157C2×C22⋊D61—42no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C2×D244—21no—
162C2×D244—42no—
163C2×D244C22×C412yes—
164C2×C4×D61—22no—
165C2×C4×D61—22no—
166C2×C4×D61—44no—
167C2×Dic122—21no—
168C2×Dic122—42no—
169C2×Dic122C22×C412yes—
170C4⋊C4×C61—44no—
171C4⋊C4×C61—44no—
172C4⋊C4×C61—44no—
173C22⋊C4×C61—44no—
174C42×C61—44no—
175C42×C61C22⋊C411yescharacteristic
176C42×C61C22⋊C411yescharacteristic
177C42×C61C22⋊C411yescharacteristic
178C42×C61C22⋊C411yescharacteristic
179C42×C61C22⋊C411yescharacteristic
180C42×C61C22⋊C411yescharacteristic
181C42×C61C22⋊C411yescharacteristic
182C42×C61C22⋊C411yescharacteristic
183C42×C61C22×C411yescharacteristic
184C42×C61C22×C411yescharacteristic
185C42×C61C22×C411yescharacteristic
186C42×C61C22×C411yescharacteristic
187C42×C61C2×D411yescharacteristic
188C42×C61C2×D411yescharacteristic
189C42×C61C2×D411yescharacteristic
190C42×C61C2×D411yescharacteristic
191C22⋊Dic61—22no—
192C22⋊Dic61—22no—
193#11758198—22no—
194#11758198—22no—
195#11758198—22no—
196#11758198—22no—
197#11758198—22no—
198#11758198—22no—
199#11758198—22no—
200#11758198—22no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61C22×C412yes—
204C4⋊Dic61C22×C412yes—
205#11758196—22no—
206#11758196—22no—
207#11758196—22no—
208#11758196—22no—
209#11758196—22no—
210#11758196—22no—
211#11758196—22no—
212#11758196—22no—
213C4×Dic61—22no—
214C4×Dic61—22no—
215C2×#92418—24no—
216C2×#92418—44no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61C2×C22⋊C411yescharacteristic
220C23×C61C2×C22⋊C411yescharacteristic
221C23×C61AGL(1,ℤ/8)12yes—
222C23×C61#13712yes—
223C23×C61C22×D411yescharacteristic
224C22×D61—22no—
225C22×D61—42no—
226C22×D61—42no—
227C22×D61—42no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
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—44no—
249C2×C4×C61—44no—
250C2×C4×C61—44no—
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61C2×C22⋊C411yescharacteristic
253C2×C4×C61C2×C22⋊C411yescharacteristic
254C2×C4×C61C2×C22⋊C411yescharacteristic
255C2×C4×C61C2×C22⋊C411yescharacteristic
256C2×C4×C61C2×C22⋊C411yescharacteristic
257C2×C4×C61C2×C22⋊C411yescharacteristic
258C2×C4×C61C2×C22⋊C411yescharacteristic
259C2×C4×C61C2×C22⋊C411yescharacteristic
260C2×C4×C61C2×C22⋊C411yescharacteristic
261C2×C4×C61C23×C411yescharacteristic
262C2×C4×C61C22×D411yescharacteristic
263C22⋊D61—42no—
264C22⋊D61—42no—
265C22⋊D61—44no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—42no—
270C2×Dic61—42no—
271C2×Dic61—42no—
272D244—42no—
273D244—42no—
274C4×D61—44no—
275C4×D61—44no—
276Dic122—42no—
277Dic122—42no—
278#92418—44no—
279#17581—611nomaximal, Sylow
280C22×C61—21no—
281C22×C61—21no—
282C22×C61—21no—
283C22×C61—21no—
284C22×C61—21no—
285C22×C61—21no—
286C22×C61—42no—
287C22×C61—42no—
288C22×C61#41712yes—
289C22×C61#41712yes—
290C22×C61C22×C22⋊C411yescharacteristic
291C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
292C22×C61C2×#13711yescharacteristic
293C2×D61—42no—
294C2×D61—42no—
295C2×D61—42no—
296C2×D61—84no—
297C4×C61—21no—
298C4×C61—21no—
299C4×C61—21no—
300C4×C61—21no—
301C4×C61—21no—
302C4×C61—21no—
303C4×C61—21no—
304C4×C61—21no—
305C4×C61—21no—
306C4×C61—21no—
307C4×C61—84no—
308C4×C61#41012yes—
309C4×C61#42112yes—
310Dic61—42no—
311Dic61—42no—
312Dic61—42no—
313C2×#529—611no—
314C2×#432—611no—
315#2285D6111yescharacteristic, complemented
316#1520—612no—
317#1519—612no—
318#1512—614no—
319#1509—614no—
320C2×C61—21no—
321C2×C61—22no—
322C2×C61—21no—
323C2×C61—22no—
324C2×C61C2×#41011yescharacteristic
325C2×C61C2×#41711yescharacteristic
326C2×C61C2×#42111yescharacteristic
327D61—82no—
328#529—612no—
329#529—612no—
330#529—612no—
331#529—612no—
332#529—1221no—
333#529—1221no—
334#529—1221no—
335#529—1221no—
336C2×#126—1221no—
337C2×#125—611no—
338C2×#124—611no—
339C2×#123—1221no—
340C2×C22⋊Q8—611no—
341C2×#121—611no—
342C2×C4×Q8—611no—
343C2×C4×D4—611no—
344C2×C4×D4—1221no—
345C2×#117—21no—
346C2×#117—1221no—
347C22×C42C2×D6111yescharacteristic
348#432—612no—
349#432—612no—
350#432—1221no—
351#432—1221no—
352C2×C4⋊C8—1221no—
353C2×C22⋊C8—612no—
354C2×#97—1221no—
355#384—21no—
356#382C2×D6112yescomplemented
357#380—21no—
358#380C2×D6112yes—
359C4×C4⋊C4—21no—
360C4×C4⋊C4C2×D6112yescomplemented
361#328—618no—
362#327—618no—
363C61#1758111yescharacteristic, complemented, Sylow
364C22×Q8—611no—
365C22×D4—611no—
366C22×D4—1221no—
367C23×C4—1221no—
368C23×C4C22⋊D6111yescharacteristic
369C23×C4C22⋊D6111yescharacteristic
370C23×C4C22×D6111yescharacteristic
371C22×C8—1222no—
372#126—1222no—
373#126—1222no—
374#126—1222no—
375#126—1222no—
376#125—612no—
377#125—612no—
378#125—1221no—
379#125—1221no—
380#124—612no—
381#124—612no—
382#124—1221no—
383#124—1221no—
384#123—1222no—
385#123—1222no—
386#123—1222no—
387#123—1222no—
388C22⋊Q8—612no—
389C22⋊Q8—612no—
390C22⋊Q8—1221no—
391C22⋊Q8—1221no—
392#121—612no—
393#121—612no—
394#121—1221no—
395#121—1221no—
396C4×Q8—612no—
397C4×Q8—612no—
398C4×Q8—1221no—
399C4×Q8—1221no—
400C4×D4—612no—
401C4×D4—612no—
402C4×D4—1222no—
403C4×D4—1222no—
404C4×D4—1222no—
405C4×D4—1222no—
406C4×D4—1221no—
407C4×D4—1221no—
408#117—44no—
409#117—1222no—
410#117—1222no—
411#117—1222no—
412#117—1222no—
413C2×C4⋊C4—22no—
414C2×C4⋊C4—22no—
415C2×C4⋊C4—22no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C22⋊C4—22no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1221no—
426C2×C22⋊C4—1221no—
427C2×C22⋊C4—1221no—
428C2×C42—22no—
429C2×C42—1221no—
430C2×C42C4×D6111yescharacteristic
431C2×C42C4×D6111yescharacteristic
432C2×C42C4×D6111yescharacteristic
433C2×C42C4×D6111yescharacteristic
434C2×C42D24411yescharacteristic
435C2×C42D24411yescharacteristic
436C2×C42D24411yescharacteristic
437C2×C42D24411yescharacteristic
438C2×C42C22⋊D6111yescharacteristic
439C2×C42C22⋊D6111yescharacteristic
440C2×C42C22×D6111yescharacteristic
441C2×C42C22×D6111yescharacteristic
442C4⋊C8—1222no—
443C4⋊C8—1222no—
444C22⋊C8—614no—
445C22⋊C8—1222no—
446#97—1222no—
447#97—1222no—
448#95—22no—
449#95C22×D6114yes—
450C24—1221no—
451C24C2×C22⋊D6111yescharacteristic
452C2×Q8—612no—
453C2×Q8—1221no—
454C2×Q8—2442no—
455C2×D4—612no—
456C2×D4—1222no—
457C2×D4—1222no—
458C2×D4—1221no—
459C2×D4—2442no—
460C2×D4—2442no—
461C2×D4—2442no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—2444no—
473C22×C4#1175819811yescharacteristic
474C22×C4#1175819811yescharacteristic
475C22×C4#1175819811yescharacteristic
476C22×C4#1175819811yescharacteristic
477C22×C4#1175819811yescharacteristic
478C22×C4#1175819811yescharacteristic
479C22×C4#1175819811yescharacteristic
480C22×C4#1175819811yescharacteristic
481C22×C4C2×C4×D6111yescharacteristic
482C22×C4C2×C4×D6111yescharacteristic
483C22×C4C2×D24411yescharacteristic
484C22×C4C2×D24411yescharacteristic
485C22×C4C2×C22⋊D6111yescharacteristic
486C22×C4C2×C22⋊D6111yescharacteristic
487C22×C4C2×C22⋊D6111yescharacteristic
488C22×C4C2×C22⋊D6111yescharacteristic
489C22×C4C2×C22⋊D6111yescharacteristic
490C22×C4C23×D6111yescharacteristic
491C2×C8—1224no—
492C2×C8—2444no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—612no—
497C4⋊C4—612no—
498C4⋊C4—1222no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1222no—
502C4⋊C4—1222no—
503C4⋊C4—1222no—
504C4⋊C4—1222no—
505C4⋊C4—1222no—
506C4⋊C4—1222no—
507C4⋊C4—1222no—
508C22⋊C4—44no—
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—
519C42—44no—
520C42—1222no—
521C42—1222no—
522C42#1175819811yescharacteristic
523C42#1175819811yescharacteristic
524C42#1175819811yescharacteristic
525C42#1175819811yescharacteristic
526C42#1175819811yescharacteristic
527C42#1175819811yescharacteristic
528C42#1175819811yescharacteristic
529C42#1175819811yescharacteristic
530C42C2×C4×D6111yescharacteristic
531C42C2×C4×D6111yescharacteristic
532C42C2×C4×D6111yescharacteristic
533C42C2×C4×D6111yescharacteristic
534C42C2×D24411yescharacteristic
535C42C2×D24411yescharacteristic
536C42C2×D24411yescharacteristic
537C42C2×D24411yescharacteristic
538C23—41no—
539C23—41no—
540C23—1222no—
541C23—2442no—
542C23—2442no—
543C23—2442no—
544C23C2×#1175819811yescharacteristic
545C23C2×#1175819811yescharacteristic
546C23#1451980312yes—
547C23#1451980812yes—
548C23C22×C22⋊D6111yescharacteristic
549Q8—2442no—
550Q8—2442no—
551D4—2442no—
552D4—2444no—
553D4—2442no—
554D4—2442no—
555D4—2442no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—1222no—
582C2×C4—2444no—
583C2×C4—2442no—
584C2×C4—2442no—
585C2×C4—2442no—
586C2×C4—2444no—
587C2×C4C2×#1175819811yescharacteristic
588C2×C4C2×#1175819811yescharacteristic
589C2×C4C2×#1175819811yescharacteristic
590C2×C4C2×#1175819811yescharacteristic
591C2×C4C2×#1175819811yescharacteristic
592C2×C4C2×#1175819811yescharacteristic
593C2×C4C2×#1175819811yescharacteristic
594C2×C4C2×#1175819811yescharacteristic
595C2×C4C2×#1175819811yescharacteristic
596C2×C4C2×#1175819811yescharacteristic
597C2×C4C22×C4×D6111yescharacteristic
598C2×C4C22×D24411yescharacteristic
599C8—2444no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—42no—
607C22—42no—
608C22—4884no—
609C22—2442no—
610C22—2442no—
611C22—2442no—
612C22#1683976112yes—
613C22#1683976112yes—
614C22C22×#1175819811yescharacteristic
615C22C2×#1451980311yescharacteristic
616C22C2×#1451980811yescharacteristic
617C4—21no—
618C4—21no—
619C4—21no—
620C4—21no—
621C4—21no—
622C4—21no—
623C4—21no—
624C4—21no—
625C4—21no—
626C4—21no—
627C4—84no—
628C4—2442no—
629C4—2442no—
630C4—2442no—
631C4#1683973512yes—
632C4#1683978012yes—
633C2—21no—
634C2—22no—
635C2—22no—
636C2—21no—
637C2—4882no—
638C2C2×#1683973511yescharacteristic
639C2C2×#1683976111yescharacteristic
640C2C2×#1683978011yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2285C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#14519751C22
cosocleC22×C4×C61C24
hypercenter#2285D61
hypocenterC61#17581
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C42 ◁ #2285

Complements

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

NH ≅ G/Ntype
2. C2×#16839819C2semidirect
3. #21444390C2semidirect
5. #21444355C2semidirect
6. #21444342C2semidirect
7. C61×#2285C2semidirect
12. #16840296C4semidirect
13. #16840296C4semidirect
17. C2×#14519769C4semidirect
18. C2×#14519765C4semidirect
22. C2×C4×Dic122C4semidirect
23. C2×C4×D244C4semidirect
28. #16839819C22semidirect
29. #16839819C22semidirect
33. #16839498C22semidirect
38. C61×#382C22semidirect
42. C4×C4⋊C4×C61C22semidirect
315. #2285D61semidirect
356. #382C2×D61semidirect
360. C4×C4⋊C4C2×D61semidirect
363. C61#17581semidirect

Maximal subgroups

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

subgroupindex
C2×#168402962
C2×#168398192
#214443902
#214443862
#214443552
#214443422
C61×#22852
#1758161