Finite Group Databasebeta

← #29687244

Subgroups — #29687244

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×#16840088C211yesmaximal, characteristic
2C4×#14519752C211yesmaximal, characteristic
3C4×#14519735C211yesmaximal, characteristic, complemented
4#21446444C211yesmaximal, characteristic, complemented
5#21446438C211yesmaximal, characteristic, complemented
6#21446399C211yesmaximal, characteristic, complemented
7#21446393C211yesmaximal, characteristic, complemented
8#21446140C211yesmaximal, characteristic
9#21446015C211yesmaximal, characteristic, complemented
10#21445912C211yesmaximal, characteristic, complemented
11#21445717C211yesmaximal, characteristic, complemented
12#21445647C211yesmaximal, characteristic, complemented
13#21445645C211yesmaximal, characteristic, complemented
14#21445582C211yesmaximal, characteristic, complemented
15C61×#1776C211yesmaximal, characteristic, complemented
16C2×#14519752C2211yescharacteristic
17C2×C4×D244—21no—
18C2×C4×D244C2211yescharacteristic
19C2×#14519735C2211yescharacteristic
20C2×#14519735C2211yescharacteristic
21C2×C42×D61C2211yescharacteristic
22#16840088C418yescomplemented
23C2×#14519710C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×#14519689C2211yescharacteristic
30C2×#14519689C2211yescharacteristic
31C2×#14519688C2211yescharacteristic
32#16839663C2211yescharacteristic
33#16839661C2211yescharacteristic
34#16839654—21no—
35#16839640—21no—
36#16839619C2211yescharacteristic
37C4⋊C4×Dic61C2211yescharacteristic
38C4⋊C4×Dic61C2211yescharacteristic
39C4×C4⋊Dic61C2211yescharacteristic
40C4×#11758198—21no—
41C4×#11758198—21no—
42#16839609—21no—
43#16839607—21no—
44C42×Dic61—21no—
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#107×C61C2211yescharacteristic
54C2×#106×C61C2211yescharacteristic
55C2×#97×C61C2211yescharacteristic
56C4×C4⋊C4×C61C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C22×D244—21no—
61C22×D244C2311yescharacteristic
62C22×C4×D61—21no—
63C22×C4×D61D411yescharacteristic
64C22×C4×D61D411yescharacteristic
65C22×C4×D61C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519752—22no—
69#14519752C2×C414yes—
70C4×D244—22no—
71C4×D244—22no—
72C4×D244—44no—
73#14519735—22no—
74#14519735—22no—
75#14519735—22no—
76#14519735C2×C414yescomplemented
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61D411yescharacteristic
94C2×C4×Dic61D411yescharacteristic
95C2×C4×Dic61C2311yescharacteristic
96C42×D61—22no—
97C42×D61—22no—
98#14519710C2×C414yescomplemented
99#14519708—22no—
100#14519699—22no—
101#14519699C2×C414yescomplemented
102#14519697—22no—
103#14519697C2×C414yescomplemented
104#14519689—22no—
105#14519689C2×C414yescomplemented
106#14519688—22no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112C22×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C42×C61—21no—
119C2×C42×C61—21no—
120C2×C42×C61C2311yescharacteristic
121#107×C61—22no—
122#106×C61C2×C414yescomplemented
123#97×C61—22no—
124#95×C61—21no—
125#95×C61—21no—
126C23×D61—21no—
127C23×D61C2×D411yescharacteristic
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2411yescharacteristic
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C22×Dic61C2×D411yescharacteristic
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244—24no—
146C2×D244—21no—
147C2×D244—42no—
148C2×D244—42no—
149C2×D244C22×C412yes—
150C2×C4×D61—21no—
151C2×C4×D61—21no—
152C2×C4×D61—21no—
153C2×C4×D61—21no—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—41no—
158C2×C4×D61—41no—
159C2×C4×D61—41no—
160C2×C4×D61—42no—
161C2×C4×D61—42no—
162C2×C4×D61—42no—
163C2×C4×D61C22×C412yes—
164C2×C4×D61#4112yes—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61C22×C412yes—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61C22×C412yes—
177C4⋊C4×C61C22×C412yes—
178C4⋊C4×C61#4112yes—
179C42×C61—42no—
180C42×C61—42no—
181C42×C61#4112yes—
182C42×C61#4112yes—
183#11758198—24no—
184#11758198—24no—
185#11758198—24no—
186#11758198—42no—
187#11758198—42no—
188#11758198—42no—
189#11758198—42no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—42no—
195C4⋊Dic61C22×C412yes—
196C4⋊Dic61#4112yes—
197C4×Dic61—24no—
198C4×Dic61—22no—
199C4×Dic61—22no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61—24no—
203C4×Dic61—42no—
204C4×Dic61—42no—
205C4×Dic61—42no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418C22×C412yes—
212C23×C61C22×D411yescharacteristic
213C22×D61—21no—
214C22×D61—21no—
215C22×D61—21no—
216C22×D61—21no—
217C22×D61—21no—
218C22×D61—21no—
219C22×D61—22no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—41no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61C4×D412yes—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61#12312yes—
244C2×C4×C61#12312yes—
245C2×C4×C61C23×C411yescharacteristic
246C2×C4×C61C2×#4111yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—22no—
256C2×Dic61—21no—
257C2×Dic61—22no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61C4×D412yes—
268C2×Dic61AGL(1,ℤ/8)12yes—
269D244—22no—
270D244—22no—
271D244—48no—
272D244—41no—
273D244—41no—
274C4×D61—22no—
275C4×D61—22no—
276C4×D61—22no—
277C4×D61—22no—
278C4×D61—44no—
279C4×D61—44no—
280C4×D61—84no—
281C8×C61—22no—
282C8×C61—41no—
283#92418—22no—
284#92418—41no—
285#21582—611nomaximal, Sylow
286C22×C61C2×C4×D411yescharacteristic
287C22×C61C2×#12311yescharacteristic
288C22×C61#52511yescharacteristic
289C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
290C22×C61#57511yescharacteristic
291C22×C61#57511yescharacteristic
292C22×C61#57511yescharacteristic
293C2×D61—22no—
294C2×D61—22no—
295C2×D61—22no—
296C2×D61—41no—
297C2×D61—41no—
298C2×D61—41no—
299C2×D61—41no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—42no—
303C2×D61—81no—
304C2×D61—81no—
305C2×D61—81no—
306C2×D61—81no—
307C4×C61—22no—
308C4×C61—22no—
309C4×C61—42no—
310C4×C61—42no—
311C4×C61—42no—
312C4×C61—81no—
313C4×C61C4×#4111yescharacteristic
314C4×C61C4×#4111yescharacteristic
315C4×C61#52911yescharacteristic
316C4×C61#52911yescharacteristic
317Dic61—22no—
318Dic61—22no—
319Dic61—41no—
320Dic61—42no—
321Dic61—42no—
322Dic61—81no—
323C2×#482—611no—
324C4×#121—611no—
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#1776D6111yescharacteristic, complemented
335#1762—611no—
336#1762—611no—
337#1749—611no—
338C2×C61C4×#12311yescharacteristic
339C2×C61#294311yescharacteristic
340C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
341C2×C61#309311yescharacteristic
342C2×C61#309411yescharacteristic
343C2×C61#311111yescharacteristic
344C2×C61#311511yescharacteristic
345D61—42no—
346D61—82no—
347C2×#121—611no—
348C2×C4×D4—611no—
349C2×C4×D4—1221no—
350C2×#117—611no—
351C2×#117—611no—
352C22×C42—611no—
353#482—618no—
354C2×#107—611no—
355C2×#107C2×D6111yescharacteristic
356C2×#106—611no—
357C2×#106C2×D6111yescharacteristic
358C2×#102—611no—
359C2×#102—611no—
360C2×#102—611no—
361C2×#102—611no—
362C2×C22⋊C8—611no—
363C2×C22⋊C8—611no—
364C2×#97—611no—
365C2×#97C2×D6111yescharacteristic
366#390—611no—
367#389—611no—
368#388—1221no—
369#386—1221no—
370#380—611no—
371C4×C4⋊C4—611no—
372C4×C4⋊C4—611no—
373C4×C4⋊C4—611no—
374C4×C4⋊C4C2×D6111yescharacteristic
375C4×C4⋊C4C2×D6111yescharacteristic
376C4×C22⋊C4—1221no—
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#340C2×D6111yescharacteristic
388#340C2×D6111yescharacteristic
389#336—611no—
390#336—611no—
391C61#2158211yescharacteristic, complemented, Sylow
392C22×D4—611no—
393C22×D4—1221no—
394C23×C4—611no—
395C23×C4—611no—
396C23×C4—611no—
397C23×C4—1221no—
398C22×C8—611no—
399C22×C8—611no—
400C22×C8C22×D6111yescharacteristic
401C22×C8C22×D6111yescharacteristic
402#121—614no—
403#121—1222no—
404C4×D4—1222no—
405C4×D4—1222no—
406C4×D4—2444no—
407#117—614no—
408#117—1222no—
409#117—1222no—
410#117—1222no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4C22×D6111yescharacteristic
416C2×C4⋊C4C22×D6111yescharacteristic
417C2×C4⋊C4C22×D6111yescharacteristic
418C2×C4⋊C4C22×D6111yescharacteristic
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1221no—
426C2×C42—21no—
427C2×C42—21no—
428C2×C42—611no—
429C2×C42—611no—
430C2×C42—611no—
431C2×C42—1221no—
432C2×C42—1221no—
433C2×C42—1222no—
434C2×C42—1222no—
435C2×C42—1221no—
436C2×C42—1221no—
437C2×C42—1221no—
438C2×C42C22×D6111yescharacteristic
439#107—22no—
440#107—614no—
441#106—1222no—
442#106C4×D6114yescomplemented
443#102—614no—
444#102—614no—
445#102—1222no—
446#102—1222no—
447C22⋊C8—614no—
448C22⋊C8—1222no—
449#97—22no—
450#97—1222no—
451#95—21no—
452#95—21no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457#95—1221no—
458C24—611no—
459C24—1221no—
460C2×D4—612no—
461C2×D4—1224no—
462C2×D4—1221no—
463C2×D4—1221no—
464C2×D4—1221no—
465C2×D4—1221no—
466C2×D4—1221no—
467C2×D4—2442no—
468C2×D4—2442no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—611no—
474C22×C4—611no—
475C22×C4—612no—
476C22×C4—612no—
477C22×C4—611no—
478C22×C4—1222no—
479C22×C4—1221no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—1221no—
488C22×C4—2441no—
489C22×C4—2441no—
490C22×C4—2441no—
491C22×C4—2442no—
492C22×C4—2442no—
493C22×C4—2442no—
494C22×C4D4×D6111yescharacteristic
495C22×C4D4×D6111yescharacteristic
496C22×C4C23×D6111yescharacteristic
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—612no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8C2×C4×D6112yes—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—612no—
515C4⋊C4—612no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—2442no—
521C4⋊C4C2×C4×D6112yes—
522C4⋊C4C2×C4×D6112yes—
523C4⋊C4#1175822212yes—
524C22⋊C4—1224no—
525C22⋊C4—1224no—
526C22⋊C4—1224no—
527C22⋊C4—2442no—
528C22⋊C4—2442no—
529C22⋊C4—2442no—
530C22⋊C4—2442no—
531C42—42no—
532C42—42no—
533C42—1224no—
534C42—1224no—
535C42—1221no—
536C42—1221no—
537C42—1222no—
538C42—1222no—
539C42—2442no—
540C42—2442no—
541C42—2442no—
542C42#1175822212yes—
543C42#1175822612yes—
544C23—612no—
545C23—1221no—
546C23—1221no—
547C23—1221no—
548C23—1221no—
549C23—1221no—
550C23—1221no—
551C23—1222no—
552C23—2441no—
553C23—2441no—
554C23—2441no—
555C23—2441no—
556C23—2441no—
557C23—2441no—
558C23C2×D4×D6111yescharacteristic
559D4—1222no—
560D4—1222no—
561D4—2448no—
562D4—2441no—
563D4—2441no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—612no—
581C2×C4—612no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—1222no—
591C2×C4—1221no—
592C2×C4—1222no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—4884no—
596C2×C4—2444no—
597C2×C4—2444no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4#1451976212yes—
606C2×C4#1451976212yes—
607C2×C4C22×C4×D6111yescharacteristic
608C2×C4#41×D6111yescharacteristic
609C2×C4#41×D6111yescharacteristic
610C2×C4C2×#1175822211yescharacteristic
611C2×C4C2×#1175822211yescharacteristic
612C2×C4C2×#1175822611yescharacteristic
613C8—22no—
614C8—41no—
615C8—1222no—
616C8—2441no—
617C22—1222no—
618C22—1222no—
619C22—1222no—
620C22—4881no—
621C22—4881no—
622C22—4881no—
623C22—4881no—
624C22—2442no—
625C22—2442no—
626C22—2442no—
627C22—2441no—
628C22—2441no—
629C22—2441no—
630C22—2441no—
631C22C4×D4×D6111yescharacteristic
632C22C2×#1451976211yescharacteristic
633C22#1684027111yescharacteristic
634C22AGL(1,ℤ/8)×D6111yescharacteristic
635C22#1684055811yescharacteristic
636C22#1684056111yescharacteristic
637C22#1684056111yescharacteristic
638C4—22no—
639C4—22no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4—81no—
644C4—1222no—
645C4—1222no—
646C4—4881no—
647C4—2442no—
648C4—2442no—
649C4—2441no—
650C4C4×#1175822211yescharacteristic
651C4C4×#1175822611yescharacteristic
652C4#1684029011yescharacteristic
653C4#1684029911yescharacteristic
654C2—4882no—
655C2—2442no—
656C2C4×#1451976211yescharacteristic
657C2#2145461411yescharacteristic
658C2#2145462411yescharacteristic
659C2#2145575311yescharacteristic
660C2#2145576111yescharacteristic
661C2#2145593411yescharacteristic
662C2#2145596611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1776C2
socleC23×C61C22×D4
prime radicalC2×#14519752C22
cosocleC22×C4×C61C24
hypercenter#1776D61
hypocenterC61#21582
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
3. C4×#14519735C2semidirect
4. #21446444C2semidirect
5. #21446438C2semidirect
6. #21446399C2semidirect
7. #21446393C2semidirect
9. #21446015C2semidirect
10. #21445912C2semidirect
11. #21445717C2semidirect
12. #21445647C2semidirect
13. #21445645C2semidirect
14. #21445582C2semidirect
15. C61×#1776C2semidirect
22. #16840088C4semidirect
76. #14519735C2×C4semidirect
98. #14519710C2×C4semidirect
101. #14519699C2×C4semidirect
103. #14519697C2×C4semidirect
105. #14519689C2×C4semidirect
122. #106×C61C2×C4semidirect
334. #1776D61semidirect
391. C61#21582semidirect
442. #106C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400882
C4×#145197522
C4×#145197352
#214464442
#214464382
#214463992
#214463932
#214461402
#214460152
#214459122
#214457172
#214456472
#214456452
#214455822
C61×#17762
#2158261