Finite Group Databasebeta

← #29687234

Subgroups — #29687234

Order 15616.

960 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×#16840083C211yesmaximal, characteristic
2C4×#14519736C211yesmaximal, characteristic, complemented
3#21446443C211yesmaximal, characteristic, complemented
4#21446436C211yesmaximal, characteristic, complemented
5#21446394C211yesmaximal, characteristic, complemented
6#21446387C211yesmaximal, characteristic, complemented
7#21446142C211yesmaximal, characteristic
8#21446016C211yesmaximal, characteristic, complemented
9#21445911C211yesmaximal, characteristic, complemented
10#21445717C211yesmaximal, characteristic, complemented
11#21445717C211yesmaximal, characteristic, complemented
12#21445644C211yesmaximal, characteristic, complemented
13#21445644C211yesmaximal, characteristic, complemented
14#21445581C211yesmaximal, characteristic, complemented
15C4×#121×C61C211yesmaximal, characteristic
16C2×#14519736C2211yescharacteristic
17C2×#14519736C2211yescharacteristic
18#16840083C418yescomplemented
19C2×#14519710C2211yescharacteristic
20C2×#14519710C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×C22⋊#92418C2211yescharacteristic
28C2×C22⋊#92418C2211yescharacteristic
29C2×#14519688C2211yescharacteristic
30C2×#14519688C2211yescharacteristic
31C4×C4⋊Dic61C2211yescharacteristic
32C4×C4⋊Dic61C2211yescharacteristic
33C4×C4⋊Dic61C2211yescharacteristic
34C4×C4⋊Dic61C2211yescharacteristic
35C4×C22⋊Dic61—21no—
36C4×C22⋊Dic61—21no—
37#16839608—21no—
38C42×Dic61—21no—
39#16839524C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839517C2211yescharacteristic
48#16839517C2211yescharacteristic
49C2×#121×C61C2211yescharacteristic
50C2×C4×D4×C61—21no—
51C2×C4×D4×C61C2211yescharacteristic
52C22×C42×C61C2211yescharacteristic
53C61×#390C2211yescharacteristic
54C61×#389C2211yescharacteristic
55C61×#388—21no—
56C61×#386—21no—
57C61×#380C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C4×C22⋊C4×C61—21no—
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64#14519736—22no—
65#14519736—22no—
66#14519736—22no—
67#14519736C2×C414yescomplemented
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C22⋊Dic61—21no—
73C2×C22⋊Dic61—21no—
74C2×C4×Dic61—21no—
75C2×C4×Dic61—21no—
76C2×C4×Dic61—21no—
77C2×C4×Dic61—21no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61—21no—
80#14519710—22no—
81#14519710C2×C414yescomplemented
82#14519708—22no—
83#14519708C2×C414yescomplemented
84#14519698—22no—
85#14519698—22no—
86#14519698C2×C414yescomplemented
87#14519698C2×C414yescomplemented
88C22⋊#92418—22no—
89C22⋊#92418C2×C414yescomplemented
90#14519688—22no—
91#14519688—22no—
92#14519684—21no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96C22×D4×C61—21no—
97C22×D4×C61C2311yescharacteristic
98C23×C4×C61—21no—
99C23×C4×C61D411yescharacteristic
100C23×C4×C61D411yescharacteristic
101C23×C4×C61C2311yescharacteristic
102#121×C61—22no—
103#121×C61C2×C414yes—
104C4×D4×C61—22no—
105C4×D4×C61—22no—
106C4×D4×C61—44no—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61—21no—
115C2×C22⋊C4×C61—21no—
116C2×C42×C61—22no—
117C2×C42×C61—22no—
118C2×C42×C61—21no—
119C2×C42×C61D411yescharacteristic
120C2×C42×C61D411yescharacteristic
121C2×C42×C61C2311yescharacteristic
122C2×C42×C61C2311yescharacteristic
123#95×C61—21no—
124#95×C61—21no—
125#95×C61—21no—
126C24×C61—21no—
127C24×C61C2×D411yescharacteristic
128C2×D4×C61—24no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—42no—
135C2×D4×C61—42no—
136C2×D4×C61C22×C412yes—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—22no—
143C22×C4×C61—22no—
144C22×C4×C61—22no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—41no—
148C22×C4×C61—41no—
149C22×C4×C61—41no—
150C22×C4×C61—42no—
151C22×C4×C61—42no—
152C22×C4×C61—42no—
153C22×C4×C61C22×C412yes—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61#4112yes—
160C22×C4×C61C2411yescharacteristic
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—42no—
171C4⋊C4×C61C22×C412yes—
172C22⋊C4×C61—24no—
173C22⋊C4×C61—42no—
174C22⋊C4×C61—42no—
175C22⋊C4×C61—42no—
176C22⋊C4×C61—42no—
177C42×C61—22no—
178C42×C61—22no—
179C42×C61—21no—
180C42×C61—21no—
181C42×C61—42no—
182C42×C61#4112yes—
183C42×C61#4112yes—
184C22⋊Dic61—24no—
185C22⋊Dic61—24no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61C22×C412yes—
191C4⋊Dic61C22×C412yes—
192C4⋊Dic61#4112yes—
193C4⋊Dic61#4112yes—
194C4×Dic61—24no—
195C4×Dic61—24no—
196C4×Dic61—42no—
197C4×Dic61—42no—
198C4×Dic61—42no—
199C4×Dic61—42no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418C22×C412yes—
211C2×#92418C22×C412yes—
212C23×C61—21no—
213C23×C61—21no—
214C23×C61—21no—
215C23×C61—21no—
216C23×C61—21no—
217C23×C61—21no—
218C23×C61—22no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61C4×D412yes—
226C23×C61C22×D411yescharacteristic
227D4×C61—22no—
228D4×C61—22no—
229D4×C61—48no—
230D4×C61—41no—
231D4×C61—41no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—44no—
249C2×C4×C61—44no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—84no—
259C2×C4×C61C4×D412yes—
260C2×C4×C61#12312yes—
261C2×C4×C61#12312yes—
262C2×C4×C61AGL(1,ℤ/8)12yes—
263C2×C4×C61C23×C411yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C2×C4×C61C2×#4111yescharacteristic
269C2×Dic61—22no—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281#92418—22no—
282#92418—22no—
283#92418—41no—
284#92418—41no—
285#21582—611nomaximal, Sylow
286C22×C61—22no—
287C22×C61—22no—
288C22×C61—22no—
289C22×C61—41no—
290C22×C61—41no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61—42no—
294C22×C61—42no—
295C22×C61—42no—
296C22×C61—81no—
297C22×C61—81no—
298C22×C61—81no—
299C22×C61—81no—
300C22×C61C2×C4×D411yescharacteristic
301C22×C61C2×#12311yescharacteristic
302C22×C61#52511yescharacteristic
303C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
304C22×C61#57511yescharacteristic
305C22×C61#57511yescharacteristic
306C22×C61#57511yescharacteristic
307C4×C61—22no—
308C4×C61—22no—
309C4×C61—22no—
310C4×C61—22no—
311C4×C61—41no—
312C4×C61—42no—
313C4×C61—81no—
314C4×C61—81no—
315C4×C61C4×#4111yescharacteristic
316C4×C61C4×#4111yescharacteristic
317C4×C61#52911yescharacteristic
318C4×C61#52911yescharacteristic
319Dic61—42no—
320Dic61—42no—
321Dic61—42no—
322Dic61—42no—
323C2×#482—611no—
324C4×#121D6111yescharacteristic
325C4×#117—611no—
326#1926—611no—
327#1925—611no—
328#1919—611no—
329#1918—611no—
330#1875—611no—
331#1854—611no—
332#1825—611no—
333#1776—611no—
334#1776—611no—
335#1762—611no—
336#1762—611no—
337#1749—611no—
338C2×C61—42no—
339C2×C61—82no—
340C2×C61C4×#12311yescharacteristic
341C2×C61#294311yescharacteristic
342C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
343C2×C61#309311yescharacteristic
344C2×C61#309411yescharacteristic
345C2×C61#311111yescharacteristic
346C2×C61#311511yescharacteristic
347C2×#121C2×D6111yescharacteristic
348C2×C4×D4—21no—
349C2×C4×D4C2×D6111yescharacteristic
350C2×#117—611no—
351C2×#117—611no—
352C22×C42C2×D6111yescharacteristic
353#482—618no—
354C2×#107—611no—
355C2×#107—611no—
356C2×#106—611no—
357C2×#106—611no—
358C2×#102—611no—
359C2×#102—611no—
360C2×#102—611no—
361C2×#102—611no—
362C2×C22⋊C8—611no—
363C2×C22⋊C8—611no—
364C2×#97—611no—
365C2×#97—611no—
366#390C2×D6111yescharacteristic
367#389C2×D6111yescharacteristic
368#388—21no—
369#386—21no—
370#380C2×D6111yescharacteristic
371C4×C4⋊C4—611no—
372C4×C4⋊C4—611no—
373C4×C4⋊C4—611no—
374C4×C4⋊C4—611no—
375C4×C4⋊C4C2×D6111yescharacteristic
376C4×C22⋊C4—21no—
377C4×C22⋊C4—1221no—
378C4×C22⋊C4—1221no—
379#376—1221no—
380C43—1221no—
381#340—611no—
382#340—611no—
383#340—611no—
384#340—611no—
385#340—611no—
386#340—611no—
387#340—611no—
388#340—611no—
389#336—611no—
390#336—611no—
391C61#2158211yescharacteristic, complemented, Sylow
392C22×D4—21no—
393C22×D4C22×D6111yescharacteristic
394C23×C4—21no—
395C23×C4C22⋊D6111yescharacteristic
396C23×C4C22⋊D6111yescharacteristic
397C23×C4C22×D6111yescharacteristic
398C22×C8—611no—
399C22×C8—611no—
400C22×C8—611no—
401C22×C8—611no—
402#121—22no—
403#121C4×D6114yes—
404C4×D4—22no—
405C4×D4—22no—
406C4×D4—44no—
407#117—614no—
408#117—1222no—
409#117—1222no—
410#117—1222no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4C22×D6111yescharacteristic
417C2×C4⋊C4C22×D6111yescharacteristic
418C2×C4⋊C4C22×D6111yescharacteristic
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1221no—
426C2×C42—22no—
427C2×C42—22no—
428C2×C42—21no—
429C2×C42—1221no—
430C2×C42—1221no—
431C2×C42—1221no—
432C2×C42—1221no—
433C2×C42—1221no—
434C2×C42—1221no—
435C2×C42C22⋊D6111yescharacteristic
436C2×C42C22⋊D6111yescharacteristic
437C2×C42C22×D6111yescharacteristic
438C2×C42C22×D6111yescharacteristic
439#107—614no—
440#107—1222no—
441#106—614no—
442#106—1222no—
443#102—614no—
444#102—614no—
445#102—1222no—
446#102—1222no—
447C22⋊C8—614no—
448C22⋊C8—1222no—
449#97—1222no—
450#97—1222no—
451#95—21no—
452#95—21no—
453#95—21no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457#95—1221no—
458C24—21no—
459C24C2×C22⋊D6111yescharacteristic
460C2×D4—24no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4C2×C4×D6112yes—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4C2×C4×D6112yes—
490C22×C4#1175822212yes—
491C22×C4C2×C22⋊D6111yescharacteristic
492C22×C4C2×C22⋊D6111yescharacteristic
493C22×C4C2×C22⋊D6111yescharacteristic
494C22×C4C2×C22⋊D6111yescharacteristic
495C22×C4C2×C22⋊D6111yescharacteristic
496C22×C4C23×D6111yescharacteristic
497C2×C8—612no—
498C2×C8—612no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—42no—
515C4⋊C4—612no—
516C4⋊C4—612no—
517C4⋊C4—612no—
518C4⋊C4—612no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4C2×C4×D6112yes—
524C22⋊C4—24no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—1224no—
530C22⋊C4—1224no—
531C42—21no—
532C42—21no—
533C42—22no—
534C42—22no—
535C42—42no—
536C42—1224no—
537C42—1224no—
538C42—2442no—
539C42—2442no—
540C42—2442no—
541C42—2442no—
542C42#1175822412yes—
543C42#1175822412yes—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—22no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23C4×C22⋊D6112yes—
558C23C22×C22⋊D6111yescharacteristic
559D4—22no—
560D4—22no—
561D4—48no—
562D4—41no—
563D4—41no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—84no—
591C2×C4—1222no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4C4×C22⋊D6112yes—
604C2×C4#1451975812yes—
605C2×C4#1451975812yes—
606C2×C4#1451980312yes—
607C2×C4C22×C4×D6111yescharacteristic
608C2×C4#41×D6111yescharacteristic
609C2×C4#41×D6111yescharacteristic
610C2×C4C2×#1175822411yescharacteristic
611C2×C4C2×#1175822411yescharacteristic
612C2×C4C2×#1175822211yescharacteristic
613C8—1222no—
614C8—1222no—
615C8—2441no—
616C8—2441no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22C2×C4×C22⋊D6111yescharacteristic
632C22C2×#1451975811yescharacteristic
633C22#1684026911yescharacteristic
634C22C2×#1451980311yescharacteristic
635C22#1684056011yescharacteristic
636C22#1684056011yescharacteristic
637C22#1684056011yescharacteristic
638C4—22no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—42no—
643C4—41no—
644C4—81no—
645C4—81no—
646C4—2442no—
647C4—2442no—
648C4—2442no—
649C4—2442no—
650C4C4×#1175822411yescharacteristic
651C4C4×#1175822411yescharacteristic
652C4#1684029211yescharacteristic
653C4#1684029211yescharacteristic
654C2—42no—
655C2—82no—
656C2C4×#1451975811yescharacteristic
657C2#2145461711yescharacteristic
658C2C4×#1451980311yescharacteristic
659C2#2145574811yescharacteristic
660C2#2145575511yescharacteristic
661C2#2145592211yescharacteristic
662C2#2145596411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×#121×C61C2
socleC23×C61C22×D4
prime radicalC2×#121×C61C22
cosocleC22×C4×C61C24
hypercenterC4×#121D61
hypocenterC61#21582
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
2. C4×#14519736C2semidirect
3. #21446443C2semidirect
4. #21446436C2semidirect
5. #21446394C2semidirect
6. #21446387C2semidirect
8. #21446016C2semidirect
9. #21445911C2semidirect
10. #21445717C2semidirect
11. #21445717C2semidirect
12. #21445644C2semidirect
13. #21445644C2semidirect
14. #21445581C2semidirect
18. #16840083C4semidirect
67. #14519736C2×C4semidirect
81. #14519710C2×C4semidirect
83. #14519708C2×C4semidirect
86. #14519698C2×C4semidirect
87. #14519698C2×C4semidirect
89. C22⋊#92418C2×C4semidirect
391. C61#21582semidirect

Maximal subgroups

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

subgroupindex
C2×#168400832
C4×#145197362
#214464432
#214464362
#214463942
#214463872
#214461422
#214460162
#214459112
#214457172
#214456442
#214455812
C4×#121×C612
#2158261