Finite Group Databasebeta

← #29687247

Subgroups — #29687247

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×#16840085C211yesmaximal, characteristic
2#21448335C211yesmaximal, characteristic
3C4×#14519735C211yesmaximal, characteristic, complemented
4#21446443C211yesmaximal, characteristic, complemented
5#21446398C211yesmaximal, characteristic, complemented
6#21446390C211yesmaximal, characteristic, complemented
7#21446138C211yesmaximal, characteristic
8#21446015C211yesmaximal, characteristic, complemented
9#21445913C211yesmaximal, characteristic, complemented
10#21445718C211yesmaximal, characteristic, complemented
11#21445715C211yesmaximal, characteristic, complemented
12#21445646C211yesmaximal, characteristic, complemented
13#21445646C211yesmaximal, characteristic, complemented
14#21445580C211yesmaximal, characteristic, complemented
15C61×#1925C211yesmaximal, characteristic, complemented
16C2×#14519748C2211yescharacteristic
17C2×D4×Dic61C2211yescharacteristic
18C2×#14519737—21no—
19C2×#14519735C2211yescharacteristic
20C2×#14519735C2211yescharacteristic
21C2×C42×D61C2211yescharacteristic
22#16840085C418yescomplemented
23C2×#14519710C2211yescharacteristic
24C2×#14519709C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519689C2211yescharacteristic
29C2×#14519689C2211yescharacteristic
30C2×#14519687C2211yescharacteristic
31C2×#14519687C2211yescharacteristic
32#16839660C2211yescharacteristic
33#16839656—21no—
34#16839638—21no—
35#16839619C2211yescharacteristic
36C4⋊C4×Dic61C2211yescharacteristic
37C4⋊C4×Dic61C2211yescharacteristic
38C4⋊C4×Dic61C2211yescharacteristic
39C4×C4⋊Dic61C2211yescharacteristic
40C22⋊C4×Dic61—21no—
41C4×#11758198—21no—
42#16839609—21no—
43#16839607—21no—
44C42×Dic61—21no—
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839517C2211yescharacteristic
53C2×#107×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C61×#390C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C22×C22⋊D61—21no—
61C23×Dic61—21no—
62C22×C4×D61D411yescharacteristic
63C22×C4×D61D411yescharacteristic
64C22×C4×D61C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67#14519748—22no—
68#14519748C2×C414yes—
69D4×Dic61—22no—
70D4×Dic61—22no—
71#14519737—44no—
72#14519735—22no—
73#14519735—22no—
74#14519735—22no—
75#14519735C2×C414yescomplemented
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61D411yescharacteristic
93C2×C4×Dic61D411yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95C42×D61—22no—
96C42×D61—22no—
97#14519710C2×C414yescomplemented
98#14519709C2×C414yescomplemented
99#14519708—22no—
100#14519698—22no—
101#14519698C2×C414yescomplemented
102#14519689—22no—
103#14519689C2×C414yescomplemented
104#14519687—22no—
105#14519687—22no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112C22×D4×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C22⋊C4×C61—21no—
119C2×C22⋊C4×C61—21no—
120C2×C42×C61—21no—
121C2×C42×C61C2311yescharacteristic
122#107×C61—22no—
123#102×C61—22no—
124#102×C61C2×C414yescomplemented
125#95×C61—21no—
126C24×C61—21no—
127C23×D61C2×D411yescharacteristic
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61C22×C412yes—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2411yescharacteristic
140C2×C22⋊D61—24no—
141C2×C22⋊D61—42no—
142C2×C22⋊D61—42no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—42no—
148C22×Dic61—42no—
149C22×Dic61—42no—
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C2×C4×D61—21no—
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×C4×D61—21no—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—22no—
160C2×C4×D61—41no—
161C2×C4×D61—41no—
162C2×C4×D61—41no—
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×C61C22×C412yes—
176C4⋊C4×C61#4112yes—
177C22⋊C4×C61—42no—
178C22⋊C4×C61—42no—
179C42×C61—21no—
180C42×C61—21no—
181C42×C61—42no—
182C22⋊Dic61—24no—
183C22⋊Dic61—42no—
184#11758198—24no—
185#11758198—24no—
186#11758198—42no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61C22×C412yes—
193C4⋊Dic61C22×C412yes—
194C4⋊Dic61#4112yes—
195#11758196—42no—
196C4×Dic61—24no—
197C4×Dic61—24no—
198C4×Dic61—22no—
199C4×Dic61—22no—
200C4×Dic61—42no—
201C4×Dic61—42no—
202C4×Dic61—42no—
203C4×Dic61—42no—
204C4×Dic61#4112yes—
205C4×Dic61#4112yes—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418C22×C412yes—
212C23×C61—22no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61C22×D411yescharacteristic
220C22×D61—21no—
221C22×D61—21no—
222C22×D61—21no—
223C22×D61—21no—
224C22×D61—21no—
225C22×D61—21no—
226C22×D61C4×D412yes—
227D4×C61—22no—
228D4×C61—22no—
229D4×C61—41no—
230D4×C61—41no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—21no—
235C2×C4×C61—22no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
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—41no—
245C2×C4×C61AGL(1,ℤ/8)12yes—
246C2×C4×C61C23×C411yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C22⋊D61—48no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—84no—
272C2×Dic61C4×D412yes—
273C2×Dic61#12312yes—
274C2×Dic61#12312yes—
275C4×D61—22no—
276C4×D61—22no—
277C4×D61—22no—
278C4×D61—22no—
279C4×D61—44no—
280C4×D61—44no—
281C8×C61—22no—
282C8×C61—41no—
283#92418—22no—
284#92418—41no—
285#21582—611nomaximal, Sylow
286C22×C61—42no—
287C22×C61—42no—
288C22×C61—42no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61—81no—
293C22×C61C2×C4×D411yescharacteristic
294C22×C61C2×#12311yescharacteristic
295C22×C61#52511yescharacteristic
296C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
297C22×C61#57511yescharacteristic
298C22×C61#57511yescharacteristic
299C22×C61#57511yescharacteristic
300C2×D61—22no—
301C2×D61—22no—
302C2×D61—22no—
303C2×D61—41no—
304C2×D61—41no—
305C2×D61—41no—
306C2×D61—41no—
307C4×C61—22no—
308C4×C61—41no—
309C4×C61—42no—
310C4×C61—42no—
311C4×C61—81no—
312C4×C61C4×#4111yescharacteristic
313C4×C61C4×#4111yescharacteristic
314C4×C61#52911yescharacteristic
315C4×C61#52911yescharacteristic
316Dic61—22no—
317Dic61—22no—
318Dic61—22no—
319Dic61—42no—
320Dic61—42no—
321Dic61—42no—
322Dic61—81no—
323C2×#482—611no—
324C4×#121—611no—
325C4×#117—611no—
326#1926—611no—
327#1925D6111yescharacteristic, complemented
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—82no—
339C2×C61C4×#12311yescharacteristic
340C2×C61#294311yescharacteristic
341C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
342C2×C61#309311yescharacteristic
343C2×C61#309411yescharacteristic
344C2×C61#311111yescharacteristic
345C2×C61#311511yescharacteristic
346D61—42no—
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×#106—611no—
358C2×#102—611no—
359C2×#102—611no—
360C2×#102C2×D6111yescharacteristic
361C2×#102C2×D6111yescharacteristic
362C2×C22⋊C8—611no—
363C2×C22⋊C8—611no—
364C2×#97—611no—
365C2×#97—611no—
366#390C2×D6111yescharacteristic
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⋊C4—611no—
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#340—611no—
388#340C2×D6111yescharacteristic
389#336—611no—
390#336C2×D6111yescharacteristic
391C61#2158211yescharacteristic, complemented, Sylow
392C22×D4—1221no—
393C22×D4C22×D6111yescharacteristic
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—611no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4C22×D6111yescharacteristic
417C2×C4⋊C4C22×D6111yescharacteristic
418C2×C4⋊C4C22×D6111yescharacteristic
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
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—611no—
428C2×C42—611no—
429C2×C42—611no—
430C2×C42—1222no—
431C2×C42—1222no—
432C2×C42—1221no—
433C2×C42—1221no—
434C2×C42—1221no—
435C2×C42—1221no—
436C2×C42—1221no—
437C2×C42—1221no—
438C2×C42C22×D6111yescharacteristic
439#107—22no—
440#107—614no—
441#106—614no—
442#106—1222no—
443#102—22no—
444#102—614no—
445#102—1222no—
446#102C4×D6114yescomplemented
447C22⋊C8—614no—
448C22⋊C8—1222no—
449#97—1222no—
450#97—1222no—
451#95—21no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457#95—1221no—
458C24—21no—
459C24—611no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—1224no—
466C2×D4—2442no—
467C2×D4—2442no—
468C2×D4C2×C4×D6112yes—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—611no—
473C22×C4—611no—
474C22×C4—611no—
475C22×C4—612no—
476C22×C4—612no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1221no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1221no—
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—612no—
514C4⋊C4—612no—
515C4⋊C4—612no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—2442no—
522C4⋊C4C2×C4×D6112yes—
523C4⋊C4#1175822412yes—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—1224no—
527C22⋊C4—1224no—
528C22⋊C4—1224no—
529C22⋊C4—2442no—
530C22⋊C4—2442no—
531C42—21no—
532C42—21no—
533C42—42no—
534C42—612no—
535C42—612no—
536C42—1222no—
537C42—1222no—
538C42—1224no—
539C42—1224no—
540C42—2442no—
541C42—2442no—
542C42—2442no—
543C42—2442no—
544C23—22no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—612no—
552C23—1221no—
553C23—1221no—
554C23—1221no—
555C23—1221no—
556C23—1221no—
557C23—1221no—
558C23C2×D4×D6111yescharacteristic
559D4—22no—
560D4—22no—
561D4—41no—
562D4—41no—
563D4—2448no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—612no—
579C2×C4—612no—
580C2×C4—612no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—1222no—
591C2×C4—1222no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—4884no—
595C2×C4—2444no—
596C2×C4—2444no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4#1451980112yes—
607C2×C4C22×C4×D6111yescharacteristic
608C2×C4#41×D6111yescharacteristic
609C2×C4#41×D6111yescharacteristic
610C2×C4#41×D6111yescharacteristic
611C2×C4#41×D6111yescharacteristic
612C2×C4C2×#1175822411yescharacteristic
613C8—22no—
614C8—41no—
615C8—1222no—
616C8—2441no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—1222no—
625C22—1222no—
626C22—1222no—
627C22—2441no—
628C22—2441no—
629C22—2441no—
630C22—2441no—
631C22C4×D4×D6111yescharacteristic
632C22#123×D6111yescharacteristic
633C22#1684027211yescharacteristic
634C22C2×#1451980111yescharacteristic
635C22#1684055811yescharacteristic
636C22#1684055911yescharacteristic
637C22#1684056111yescharacteristic
638C4—22no—
639C4—42no—
640C4—42no—
641C4—41no—
642C4—81no—
643C4—1222no—
644C4—1222no—
645C4—1222no—
646C4—4881no—
647C4—2442no—
648C4—2442no—
649C4—2442no—
650C4C4×#1175822411yescharacteristic
651C4#1684023011yescharacteristic
652C4#1684029111yescharacteristic
653C4#1684029511yescharacteristic
654C2—82no—
655C2—2442no—
656C2#2144834611yescharacteristic
657C2#2145461411yescharacteristic
658C2C4×#1451980111yescharacteristic
659C2#2145574711yescharacteristic
660C2#2145575411yescharacteristic
661C2#2145592711yescharacteristic
662C2#2145596711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1925C2
socleC23×C61C22×D4
prime radicalC2×#14519748C22
cosocleC22×C4×C61C24
hypercenter#1925D61
hypocenterC61#21582
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1925

Complements

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

NH ≅ G/Ntype
3. C4×#14519735C2semidirect
4. #21446443C2semidirect
5. #21446398C2semidirect
6. #21446390C2semidirect
8. #21446015C2semidirect
9. #21445913C2semidirect
10. #21445718C2semidirect
11. #21445715C2semidirect
12. #21445646C2semidirect
13. #21445646C2semidirect
14. #21445580C2semidirect
15. C61×#1925C2semidirect
22. #16840085C4semidirect
75. #14519735C2×C4semidirect
97. #14519710C2×C4semidirect
98. #14519709C2×C4semidirect
101. #14519698C2×C4semidirect
103. #14519689C2×C4semidirect
124. #102×C61C2×C4semidirect
327. #1925D61semidirect
391. C61#21582semidirect
446. #102C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400852
#214483352
C4×#145197352
#214464432
#214463982
#214463902
#214461382
#214460152
#214459132
#214457182
#214457152
#214456462
#214455802
C61×#19252
#2158261