Finite Group Databasebeta

← #29684355

Subgroups — #29684355

Order 15616.

874 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839859C211yesmaximal, characteristic
2#21448290C211yesmaximal, characteristic, complemented
3#21448281C211yesmaximal, characteristic
4#21446701C211yesmaximal, characteristic
5#21446672C211yesmaximal, characteristic
6#21446556C211yesmaximal, characteristic, complemented
7#21446549C211yesmaximal, characteristic, complemented
8#21446518C211yesmaximal, characteristic
9#21446507C211yesmaximal, characteristic, complemented
10#21446495C211yesmaximal, characteristic, complemented
11#21445715C211yesmaximal, characteristic, complemented
12#21445648C211yesmaximal, characteristic, complemented
13#21445646C211yesmaximal, characteristic
14C4×#14519687C211yesmaximal, characteristic, complemented
15C61×#1934C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×#14519742C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19C2×#14519733—21no—
20#16839859C418yescomplemented
21C2×#14519710C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519687C2211yescharacteristic
29C2×#14519687C2211yescharacteristic
30C2×#14519687C2211yescharacteristic
31C2×#14519687C2211yescharacteristic
32C2×C4×#92418C2211yescharacteristic
33#16839668C2211yescharacteristic
34#16839667C2211yescharacteristic
35#16839665C2211yescharacteristic
36#16839661C2211yescharacteristic
37#16839660C2211yescharacteristic
38#16839658C2211yescharacteristic
39#16839656—21no—
40#16839650—21no—
41#16839649—21no—
42#16839643—21no—
43C4⋊C4×Dic61—21no—
44C4⋊C4×Dic61C2211yescharacteristic
45C4⋊C4×Dic61C2211yescharacteristic
46C22⋊C4×Dic61—21no—
47#16839607—21no—
48C42×Dic61C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53C2×#102×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×C4×C8×C61C2211yescharacteristic
56C61×#390C2211yescharacteristic
57C61×#389C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C22×Dic122C2311yescharacteristic
61C23×Dic61—21no—
62C2×#11758200—22no—
63C2×#11758200—22no—
64C2×#11758200C2×C414yes—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67#14519742—22no—
68#14519742C2×C414yes—
69D4×Dic61—22no—
70D4×Dic61C2×C414yes—
71#14519733—44no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61D411yescharacteristic
88C2×C4×Dic61D411yescharacteristic
89C2×C4×Dic61C2311yescharacteristic
90C2×C4×Dic61C2311yescharacteristic
91#14519710—22no—
92#14519710C2×C414yes—
93#14519708—22no—
94#14519702—22no—
95#14519702C2×C414yes—
96#14519700—22no—
97#14519698—22no—
98#14519687—22no—
99#14519687—22no—
100#14519687—22no—
101#14519687C2×C414yes—
102C4×#92418—22no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
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×C61—21no—
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C22⋊C4×C61—21no—
120C2×C22⋊C4×C61—21no—
121C2×C42×C61C2311yescharacteristic
122#102×C61—22no—
123#102×C61C2×C414yes—
124C4×C8×C61—22no—
125#95×C61—21no—
126C24×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61C22×C412yes—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2411yescharacteristic
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—24no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—42no—
145C22×Dic61—42no—
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122C22×C412yes—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61C22×C412yes—
161C2×C8×C61#4112yes—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—42no—
168C4⋊C4×C61C22×C412yes—
169C22⋊C4×C61—42no—
170C22⋊C4×C61—42no—
171C42×C61—21no—
172C42×C61—21no—
173C22⋊Dic61—24no—
174C22⋊Dic61—42no—
175C22⋊Dic61—42no—
176#11758200—28no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—42no—
183C4⋊Dic61C22×C412yes—
184#11758196—24no—
185#11758196—42no—
186#11758196—42no—
187C4×Dic61—22no—
188C4×Dic61—22no—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61—24no—
192C4×Dic61—41no—
193C4×Dic61—41no—
194C4×Dic61—42no—
195C4×Dic61—42no—
196C4×Dic61C22×C412yes—
197C4×Dic61#4112yes—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418C22×C412yes—
201C2×#92418#4112yes—
202C2×#92418#4112yes—
203C2×#92418#4112yes—
204C23×C61—22no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61C22×D411yescharacteristic
212D4×C61—22no—
213D4×C61—22no—
214D4×C61—41no—
215D4×C61—41no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—22no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61AGL(1,ℤ/8)12yes—
230C2×C4×C61#13712yes—
231C2×C4×C61C23×C411yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—48no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61C4×D412yes—
261C2×Dic61C4×D412yes—
262Dic122—22no—
263Dic122—22no—
264Dic122—41no—
265Dic122—41no—
266C8×C61—22no—
267C8×C61—22no—
268#92418—22no—
269#92418#11714yescomplemented
270#21288—611nomaximal, Sylow
271C22×C61—42no—
272C22×C61—42no—
273C22×C61—42no—
274C22×C61—81no—
275C22×C61—81no—
276C22×C61—81no—
277C22×C61—81no—
278C22×C61C2×C4×D411yescharacteristic
279C22×C61#53211yescharacteristic
280C22×C61#53211yescharacteristic
281C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C61C2×#13711yescharacteristic
283C22×C61#57511yescharacteristic
284C22×C61#57511yescharacteristic
285C4×C61—22no—
286C4×C61—22no—
287C4×C61—42no—
288C4×C61—81no—
289C4×C61—81no—
290C4×C61C2×#11711yescharacteristic
291C4×C61C4×#4111yescharacteristic
292C4×C61#52911yescharacteristic
293C4×C61#52911yescharacteristic
294Dic61—22no—
295Dic61—22no—
296Dic61—41no—
297Dic61—41no—
298Dic61—42no—
299Dic61—42no—
300Dic61—81no—
301Dic61—81no—
302C2×#440—611no—
303#2295—611no—
304#2293—611no—
305#1973—611no—
306#1968—611no—
307#1943—611no—
308#1942—611no—
309#1936—611no—
310#1935—611no—
311#1934D6111yescharacteristic
312#1933—611no—
313#1776—611no—
314#1763—611no—
315#1762—611no—
316C4×#97—611no—
317C2×C61—82no—
318C2×C61#229311yescharacteristic
319C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
320C2×C61C4×#13711yescharacteristic
321C2×C61#318011yescharacteristic
322C2×C61#318111yescharacteristic
323C2×C61#318211yescharacteristic
324C2×C61#318311yescharacteristic
325C22×SD16—611no—
326C2×C4×Q8—611no—
327C2×C4×D4—611no—
328C2×#117—1221no—
329#440—618no—
330C2×#107—611no—
331C2×#107—611no—
332C2×#106—611no—
333C2×#103—611no—
334C2×#103—611no—
335C2×#103—611no—
336C2×#102—611no—
337C2×#102C2×D6111yescharacteristic
338C2×#102C2×D6111yescharacteristic
339C2×#97—611no—
340C2×#97—611no—
341C2×#97—611no—
342C2×#97—611no—
343C2×C4×C8—611no—
344C2×C4×C8C2×D6111yescharacteristic
345#391—611no—
346#391—611no—
347#390—611no—
348#390C2×D6111yescharacteristic
349#389—611no—
350#389—611no—
351#389—611no—
352#389C2×D6111yescharacteristic
353#388—1221no—
354#388—1221no—
355#387—1221no—
356#387—1221no—
357C4×C4⋊C4—611no—
358C4×C4⋊C4—611no—
359C4×C4⋊C4—1221no—
360C4×C22⋊C4—1221no—
361#376—1221no—
362C43—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367#340C2×D6111yescharacteristic
368#340C2×D6111yescharacteristic
369C61#2128811yescharacteristic, complemented, Sylow
370C22×Q8—611no—
371C22×D4C22×D6111yescharacteristic
372C23×C4—1221no—
373C2×SD16—614no—
374C2×SD16—1222no—
375C2×SD16—1222no—
376C22×C8—611no—
377C22×C8—611no—
378C22×C8C22×D6111yescharacteristic
379C22×C8C22×D6111yescharacteristic
380C4×Q8—614no—
381C4×Q8—1222no—
382C4×D4—614no—
383C4×D4—1222no—
384#117—2444no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4C22×D6111yescharacteristic
394C2×C4⋊C4C22×D6111yescharacteristic
395C2×C4⋊C4C22×D6111yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C42—611no—
402C2×C42—611no—
403C2×C42—611no—
404C2×C42—611no—
405C2×C42—611no—
406C2×C42—611no—
407C2×C42—1221no—
408C2×C42—1221no—
409C2×C42—1221no—
410C2×C42C22×D6111yescharacteristic
411#107—614no—
412#107—1222no—
413#106—1222no—
414#103—614no—
415#103—1222no—
416#103—1222no—
417#102—22no—
418#102—1222no—
419#102C4×D6114yes—
420#97—614no—
421#97—1222no—
422#97—1222no—
423#97—1222no—
424C4×C8—22no—
425C4×C8—1222no—
426#95—21no—
427#95—1221no—
428#95—1221no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436C24—21no—
437C2×Q8—612no—
438C2×Q8—1221no—
439C2×Q8—1221no—
440C2×Q8—1221no—
441C2×Q8—1221no—
442C2×Q8—1221no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4C2×C4×D6112yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—1224no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—2442no—
462C22×C4—2442no—
463C22×C4D4×D6111yescharacteristic
464C22×C4D4×D6111yescharacteristic
465C22×C4C23×D6111yescharacteristic
466SD16—1228no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—612no—
472C2×C8—612no—
473C2×C8—612no—
474C2×C8—612no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8C2×C4×D6112yes—
478C2×C8#1175822412yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—42no—
485C4⋊C4—612no—
486C4⋊C4—1224no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—2442no—
493C4⋊C4—2442no—
494C4⋊C4—2442no—
495C4⋊C4C2×C4×D6112yes—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—1224no—
499C22⋊C4—2442no—
500C22⋊C4—2442no—
501C42—21no—
502C42—21no—
503C42—612no—
504C42—612no—
505C42—1224no—
506C42—1221no—
507C42—1221no—
508C42—1222no—
509C42—1222no—
510C42—2441no—
511C42—2441no—
512C42—2442no—
513C42—2442no—
514C23—22no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C2×D4×D6111yescharacteristic
522Q8—1222no—
523Q8—1222no—
524Q8—2441no—
525Q8—2441no—
526D4—22no—
527D4—22no—
528D4—41no—
529D4—41no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—612no—
544C2×C4—612no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1222no—
550C2×C4—1222no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—2448no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4#1451980112yes—
569C2×C4#1451980612yes—
570C2×C4C22×C4×D6111yescharacteristic
571C2×C4#41×D6111yescharacteristic
572C2×C4#41×D6111yescharacteristic
573C2×C4#41×D6111yescharacteristic
574C2×C4#41×D6111yescharacteristic
575C2×C4C2×#1175822411yescharacteristic
576C8—22no—
577C8—22no—
578C8—614no—
579C8—1222no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22C4×D4×D6111yescharacteristic
588C22#1684030711yescharacteristic
589C22#1684031111yescharacteristic
590C22C2×#1451980111yescharacteristic
591C22C2×#1451980611yescharacteristic
592C22#1684055911yescharacteristic
593C22#1684055911yescharacteristic
594C4—22no—
595C4—22no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—1222no—
600C4—1222no—
601C4—4881no—
602C4—4881no—
603C4—2442no—
604C4—2442no—
605C4—2441no—
606C4—2441no—
607C4#117×D6111yescharacteristic
608C4C4×#1175822411yescharacteristic
609C4#1684028911yescharacteristic
610C4#1684029511yescharacteristic
611C2—82no—
612C2#2144827911yescharacteristic
613C2C4×#1451980111yescharacteristic
614C2C4×#1451980611yescharacteristic
615C2#2145651211yescharacteristic
616C2#2145652111yescharacteristic
617C2#2145653611yescharacteristic
618C2#2145654711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1934C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenter#1934D61
hypocenterC61#21288
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1934

Complements

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

NH ≅ G/Ntype
2. #21448290C2semidirect
6. #21446556C2semidirect
7. #21446549C2semidirect
9. #21446507C2semidirect
10. #21446495C2semidirect
11. #21445715C2semidirect
12. #21445648C2semidirect
14. C4×#14519687C2semidirect
20. #16839859C4semidirect
269. #92418#117semidirect
369. C61#21288semidirect

Maximal subgroups

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

subgroupindex
C2×#168398592
#214482902
#214482812
#214467012
#214466722
#214465562
#214465492
#214465182
#214465072
#214464952
#214457152
#214456482
#214456462
C4×#145196872
C61×#19342
#2128861