Finite Group Databasebeta

← #29687264

Subgroups — #29687264

Order 15616.

900 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840078C211yesmaximal, characteristic
2#21448175C211yesmaximal, characteristic, complemented
3#21446443C211yesmaximal, characteristic, complemented
4#21446436C211yesmaximal, characteristic, complemented
5#21446401C211yesmaximal, characteristic, complemented
6#21446394C211yesmaximal, characteristic, complemented
7#21446142C211yesmaximal, characteristic
8#21446016C211yesmaximal, characteristic, complemented
9#21445903C211yesmaximal, characteristic, complemented
10#21445717C211yesmaximal, characteristic, complemented
11C4×#14519708C211yesmaximal, characteristic, complemented
12#21445644C211yesmaximal, characteristic, complemented
13C4×#14519698C211yesmaximal, characteristic, complemented
14#21445585C211yesmaximal, characteristic, complemented
15C61×#2324C211yesmaximal, characteristic
16C2×#14519736C2211yescharacteristic
17C22×C4⋊Dic61C2211yescharacteristic
18#16840078C418yescomplemented
19C2×#14519710C2211yescharacteristic
20C2×#14519708C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×C22⋊#92418C2211yescharacteristic
28C2×C22⋊#92418C2211yescharacteristic
29C2×#14519688C2211yescharacteristic
30C2×C4×#92418C2211yescharacteristic
31C4⋊C4×Dic61—21no—
32C4⋊C4×Dic61—21no—
33C4×C4⋊Dic61C2211yescharacteristic
34C4×C4⋊Dic61C2211yescharacteristic
35C4×C4⋊Dic61C2211yescharacteristic
36C4×C4⋊Dic61C2211yescharacteristic
37C22⋊C4×Dic61—21no—
38C22⋊C4×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×C61C2211yescharacteristic
51C2×#117×C61C2211yescharacteristic
52C61×#390—21no—
53C61×#390C2211yescharacteristic
54C61×#389C2211yescharacteristic
55C61×#388—21no—
56C61×#386—21no—
57C61×#385—21no—
58C61×#380C2211yescharacteristic
59C4×C4⋊C4×C61C2211yescharacteristic
60C23×Dic61—21no—
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65#14519736—22no—
66#14519736—22no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—22no—
69C2×C4⋊Dic61C2×C414yescomplemented
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C22⋊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#14519708—22no—
82#14519708C2×C414yescomplemented
83#14519708C2×C414yescomplemented
84#14519698—22no—
85#14519698—22no—
86#14519698C2×C414yes—
87#14519698C2×C414yes—
88C22⋊#92418—22no—
89C22⋊#92418C2×C414yescomplemented
90#14519688—22no—
91C4×#92418—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×C61C2311yescharacteristic
100#121×C61—22no—
101#121×C61C2×C414yes—
102C4×D4×C61—22no—
103C4×D4×C61—22no—
104#117×C61—22no—
105#117×C61—22no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61D411yescharacteristic
108C2×C4⋊C4×C61D411yescharacteristic
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61—21no—
115C2×C22⋊C4×C61—21no—
116C2×C22⋊C4×C61—21no—
117C2×C22⋊C4×C61D411yescharacteristic
118C2×C22⋊C4×C61D411yescharacteristic
119C2×C42×C61—21no—
120C2×C42×C61C2311yescharacteristic
121C2×C42×C61C2311yescharacteristic
122#95×C61—21no—
123#95×C61—21no—
124#95×C61—21no—
125C24×C61—21no—
126C24×C61C2×D411yescharacteristic
127C2×D4×C61—24no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—81no—
134C2×D4×C61—81no—
135C2×D4×C61C22×C412yes—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—42no—
144C22×C4×C61—81no—
145C22×C4×C61—81no—
146C22×C4×C61C22×C412yes—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61#4112yes—
153C22×C4×C61C2411yescharacteristic
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—24no—
159C22×Dic61—81no—
160C22×Dic61—81no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—81no—
169C4⋊C4×C61C22×C412yes—
170C22⋊C4×C61—24no—
171C22⋊C4×C61—22no—
172C22⋊C4×C61—22no—
173C22⋊C4×C61—42no—
174C22⋊C4×C61—81no—
175C22⋊C4×C61—81no—
176C22⋊C4×C61—81no—
177C42×C61—21no—
178C42×C61—21no—
179C42×C61—81no—
180C42×C61#4112yes—
181C42×C61#4112yes—
182C22⋊Dic61—24no—
183C4⋊Dic61—24no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61C22×C412yes—
189C4⋊Dic61C22×C412yes—
190C4⋊Dic61#4112yes—
191C4⋊Dic61#4112yes—
192C4×Dic61—24no—
193C4×Dic61—81no—
194C4×Dic61—81no—
195C4×Dic61—81no—
196C4×Dic61—81no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418C22×C412yes—
208C2×#92418C22×C412yes—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61—21no—
212C23×C61—21no—
213C23×C61—21no—
214C23×C61—21no—
215C23×C61—22no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61C4×D412yes—
223C23×C61C22×D411yescharacteristic
224D4×C61—22no—
225D4×C61—22no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—84no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—84no—
255C2×C4×C61C4×D412yes—
256C2×C4×C61#12312yes—
257C2×C4×C61#12312yes—
258C2×C4×C61C23×C411yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×Dic61—22no—
265C2×Dic61—22no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—84no—
277#92418—22no—
278#92418—22no—
279#92418—41no—
280#92418—41no—
281#21584—611nomaximal, Sylow
282C22×C61—22no—
283C22×C61—22no—
284C22×C61—22no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61—42no—
290C22×C61—42no—
291C22×C61—42no—
292C22×C61—81no—
293C22×C61—81no—
294C22×C61—81no—
295C22×C61—81no—
296C22×C61C2×C4×D411yescharacteristic
297C22×C61C2×#12311yescharacteristic
298C22×C61#52511yescharacteristic
299C22×C61#57611yescharacteristic
300C22×C61#57711yescharacteristic
301C22×C61#57711yescharacteristic
302C22×C61#57711yescharacteristic
303C4×C61—22no—
304C4×C61—22no—
305C4×C61—22no—
306C4×C61—41no—
307C4×C61—41no—
308C4×C61—42no—
309C4×C61—81no—
310C4×C61—81no—
311C4×C61C4×#4111yescharacteristic
312C4×C61C4×#4111yescharacteristic
313C4×C61#52911yescharacteristic
314C4×C61#52911yescharacteristic
315Dic61—42no—
316Dic61—42no—
317Dic61—42no—
318Dic61—42no—
319C2×#481—611no—
320#2324D6111yescharacteristic
321#2274—611no—
322#1926—611no—
323#1925—611no—
324#1920—611no—
325#1919—611no—
326#1875—611no—
327#1854—611no—
328#1823—611no—
329#1776—611no—
330C4×#106—611no—
331#1762—611no—
332C4×#102—611no—
333#1750—611no—
334C2×C61—42no—
335C2×C61—82no—
336C2×C61C4×#12311yescharacteristic
337C2×C61#294611yescharacteristic
338C2×C61#294911yescharacteristic
339C2×C61#309611yescharacteristic
340C2×C61#309911yescharacteristic
341C2×C61#311911yescharacteristic
342C2×C61#312511yescharacteristic
343C2×#121C2×D6111yescharacteristic
344C2×C4×D4C2×D6111yescharacteristic
345C2×#117—611no—
346C2×#117C2×D6111yescharacteristic
347C22×C4⋊C4—611no—
348#481—618no—
349C2×#107—611no—
350C2×#106—611no—
351C2×#106—611no—
352C2×#106—611no—
353C2×#102—611no—
354C2×#102—611no—
355C2×#102—611no—
356C2×#102—611no—
357C2×C22⋊C8—611no—
358C2×C22⋊C8—611no—
359C2×#97—611no—
360C2×C4×C8—611no—
361#390—21no—
362#390C2×D6111yescharacteristic
363#389C2×D6111yescharacteristic
364#388—21no—
365#386—21no—
366#385—21no—
367#380C2×D6111yescharacteristic
368C4×C4⋊C4—611no—
369C4×C4⋊C4—611no—
370C4×C4⋊C4—611no—
371C4×C4⋊C4—611no—
372C4×C4⋊C4—1221no—
373C4×C4⋊C4—1221no—
374C4×C4⋊C4C2×D6111yescharacteristic
375C4×C22⋊C4—1221no—
376C4×C22⋊C4—1221no—
377#340—611no—
378#340—611no—
379#340—611no—
380#340—611no—
381#340—611no—
382#340—611no—
383#340—611no—
384#340—611no—
385#336—611no—
386#336—611no—
387C61#2158411yescharacteristic, complemented, Sylow
388C22×D4—21no—
389C22×D4C22×D6111yescharacteristic
390C23×C4—21no—
391C23×C4—1221no—
392C23×C4C22×D6111yescharacteristic
393C22×C8—611no—
394C22×C8—611no—
395C22×C8—611no—
396C22×C8—611no—
397#121—22no—
398#121C4×D6114yes—
399C4×D4—22no—
400C4×D4—22no—
401#117—22no—
402#117—22no—
403#117—1222no—
404#117—1222no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—614no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—1222no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4C22⋊D6111yescharacteristic
414C2×C4⋊C4C22⋊D6111yescharacteristic
415C2×C4⋊C4C22×D6111yescharacteristic
416C2×C4⋊C4C22×D6111yescharacteristic
417C2×C4⋊C4C22×D6111yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4C22⋊D6111yescharacteristic
425C2×C22⋊C4C22⋊D6111yescharacteristic
426C2×C42—21no—
427C2×C42—1221no—
428C2×C42—1221no—
429C2×C42—1221no—
430C2×C42—1221no—
431C2×C42—1221no—
432C2×C42C22×D6111yescharacteristic
433C2×C42C22×D6111yescharacteristic
434#107—1222no—
435#106—614no—
436#106—614no—
437#106—1222no—
438#102—614no—
439#102—614no—
440#102—1222no—
441#102—1222no—
442C22⋊C8—614no—
443C22⋊C8—1222no—
444#97—1222no—
445C4×C8—1222no—
446#95—21no—
447#95—21no—
448#95—21no—
449#95—1221no—
450#95—1221no—
451#95—1221no—
452#95—1221no—
453C24—21no—
454C24C2×C22⋊D6111yescharacteristic
455C2×D4—24no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—81no—
462C2×D4—81no—
463C2×D4C2×C4×D6112yes—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—42no—
472C22×C4—81no—
473C22×C4—81no—
474C22×C4—1224no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—1221no—
479C22×C4—4881no—
480C22×C4—4881no—
481C22×C4C2×C4×D6112yes—
482C22×C4#1175822212yes—
483C22×C4C2×C22⋊D6111yescharacteristic
484C22×C4C2×C22⋊D6111yescharacteristic
485C22×C4C2×C22⋊D6111yescharacteristic
486C22×C4C2×C22⋊D6111yescharacteristic
487C22×C4C2×C22⋊D6111yescharacteristic
488C22×C4C23×D6111yescharacteristic
489C2×C8—612no—
490C2×C8—612no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8—1221no—
496C2×C8—1221no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—81no—
509C4⋊C4—612no—
510C4⋊C4—612no—
511C4⋊C4—612no—
512C4⋊C4—612no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—1224no—
518C4⋊C4C2×C4×D6112yes—
519C22⋊C4—24no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—42no—
523C22⋊C4—81no—
524C22⋊C4—81no—
525C22⋊C4—81no—
526C22⋊C4—1224no—
527C42—21no—
528C42—21no—
529C42—81no—
530C42—1224no—
531C42—4881no—
532C42—4881no—
533C42—4881no—
534C42—4881no—
535C42#1175822412yes—
536C42#1175822412yes—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—22no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23C4×C22⋊D6112yes—
551C23C22×C22⋊D6111yescharacteristic
552D4—22no—
553D4—22no—
554D4—41no—
555D4—41no—
556D4—84no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—84no—
583C2×C4—1222no—
584C2×C4—1222no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—4884no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4C4×C22⋊D6112yes—
597C2×C4#1451975812yes—
598C2×C4#1451975812yes—
599C2×C4C22×C4×D6111yescharacteristic
600C2×C4#41×D6111yescharacteristic
601C2×C4#41×D6111yescharacteristic
602C2×C4C2×#1175822411yescharacteristic
603C2×C4C2×#1175822411yescharacteristic
604C2×C4C2×#1175822211yescharacteristic
605C8—1222no—
606C8—1222no—
607C8—2441no—
608C8—2441no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22C2×C4×C22⋊D6111yescharacteristic
624C22C2×#1451975811yescharacteristic
625C22#1684026911yescharacteristic
626C22#1684056511yescharacteristic
627C22#1684057011yescharacteristic
628C22#1684057311yescharacteristic
629C22#1684057311yescharacteristic
630C4—22no—
631C4—22no—
632C4—22no—
633C4—42no—
634C4—41no—
635C4—41no—
636C4—81no—
637C4—81no—
638C4—2442no—
639C4—2442no—
640C4—2442no—
641C4—2442no—
642C4C4×#1175822411yescharacteristic
643C4C4×#1175822411yescharacteristic
644C4#1684029211yescharacteristic
645C4#1684029211yescharacteristic
646C2—42no—
647C2—82no—
648C2C4×#1451975811yescharacteristic
649C2#2145464711yescharacteristic
650C2#2145467611yescharacteristic
651C2#2145577111yescharacteristic
652C2#2145580411yescharacteristic
653C2#2145601211yescharacteristic
654C2#2145609011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2324C2
socleC23×C61C22×D4
prime radicalC2×#121×C61C22
cosocleC22×C4×C61C24
hypercenter#2324D61
hypocenterC61#21584
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2324

Complements

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

NH ≅ G/Ntype
2. #21448175C2semidirect
3. #21446443C2semidirect
4. #21446436C2semidirect
5. #21446401C2semidirect
6. #21446394C2semidirect
8. #21446016C2semidirect
9. #21445903C2semidirect
10. #21445717C2semidirect
11. C4×#14519708C2semidirect
12. #21445644C2semidirect
13. C4×#14519698C2semidirect
14. #21445585C2semidirect
18. #16840078C4semidirect
69. C2×C4⋊Dic61C2×C4semidirect
82. #14519708C2×C4semidirect
83. #14519708C2×C4semidirect
89. C22⋊#92418C2×C4semidirect
387. C61#21584semidirect

Maximal subgroups

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

subgroupindex
C2×#168400782
#214481752
#214464432
#214464362
#214464012
#214463942
#214461422
#214460162
#214459032
#214457172
C4×#145197082
#214456442
C4×#145196982
#214455852
C61×#23242
#2158461