Finite Group Databasebeta

← #29683077

Subgroups — #29683077

Order 15616.

922 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839859C211yesmaximal, characteristic
2#21448281C211yesmaximal, characteristic
3C4×#14519742C211yesmaximal, characteristic, complemented
4#21446448C211yesmaximal, characteristic
5#21446409C211yesmaximal, characteristic, complemented
6#21446390C211yesmaximal, characteristic, complemented
7#21446177C211yesmaximal, characteristic
8#21446171C211yesmaximal, characteristic
9#21446090C211yesmaximal, characteristic, complemented
10#21446070C211yesmaximal, characteristic, complemented
11#21445718C211yesmaximal, characteristic, complemented
12#21445648C211yesmaximal, characteristic, complemented
13#21445646C211yesmaximal, characteristic
14#21445566C211yesmaximal, characteristic, complemented
15C61×#1925C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18C2×#14519742C2211yescharacteristic
19C2×#14519742C2211yescharacteristic
20C2×D4×Dic61C2211yescharacteristic
21C2×#14519733—21no—
22#16839859C418yescomplemented
23C2×#14519710C2211yescharacteristic
24C2×#14519709C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519700C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519687C2211yescharacteristic
30C2×#14519687C2211yescharacteristic
31#16839665C2211yescharacteristic
32#16839660C2211yescharacteristic
33#16839658C2211yescharacteristic
34#16839656—21no—
35#16839650—21no—
36C4⋊C4×Dic61C2211yescharacteristic
37C4⋊C4×Dic61C2211yescharacteristic
38C4⋊C4×Dic61C2211yescharacteristic
39C4×#11758196—21no—
40#16839613—21no—
41#16839613—21no—
42C22⋊C4×Dic61—21no—
43C42×Dic61—21no—
44C42×Dic61C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839517C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#107×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C61×#390C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C22×Dic122C2311yescharacteristic
61C23×Dic61—21no—
62C2×#11758200—22no—
63C2×#11758200—22no—
64C2×#11758200C2×C414yes—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C4×Dic122—22no—
68C4×Dic122—22no—
69#14519742—22no—
70#14519742—22no—
71#14519742—22no—
72#14519742C2×C414yes—
73D4×Dic61—22no—
74D4×Dic61C2×C414yes—
75#14519733—44no—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×C4×Dic61—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×Dic61D411yescharacteristic
96C2×C4×Dic61D411yescharacteristic
97C2×C4×Dic61C2311yescharacteristic
98C2×C4×Dic61C2311yescharacteristic
99#14519710C2×C414yes—
100#14519709—22no—
101#14519702—22no—
102#14519702C2×C414yes—
103#14519700—22no—
104#14519698—22no—
105#14519687—22no—
106#14519687C2×C414yes—
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×C414yes—
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—21no—
142C22×Dic61—24no—
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×Dic122C22×C412yes—
154C2×Dic122#4112yes—
155C2×Dic122#4112yes—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61C22×C412yes—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61C22×C412yes—
167C4⋊C4×C61#4112yes—
168C22⋊C4×C61—42no—
169C22⋊C4×C61—42no—
170C42×C61—21no—
171C42×C61—21no—
172C42×C61—42no—
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⋊Dic61C22×C412yes—
182C4⋊Dic61#4112yes—
183#11758196—24no—
184#11758196—24no—
185#11758196—24no—
186#11758196—42no—
187#11758196—42no—
188C4×Dic61—24no—
189C4×Dic61—24no—
190C4×Dic61—22no—
191C4×Dic61—21no—
192C4×Dic61—21no—
193C4×Dic61—24no—
194C4×Dic61—22no—
195C4×Dic61—41no—
196C4×Dic61—41no—
197C4×Dic61—42no—
198C4×Dic61—42no—
199C4×Dic61—42no—
200C4×Dic61C22×C412yes—
201C4×Dic61#4112yes—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418C22×C412yes—
208C23×C61—22no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—41no—
212C23×C61—41no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61C22×D411yescharacteristic
216D4×C61—22no—
217D4×C61—22no—
218D4×C61—41no—
219D4×C61—41no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61AGL(1,ℤ/8)12yes—
235C2×C4×C61C23×C411yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×Dic61—22no—
242C2×Dic61—21no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—22no—
251C2×Dic61—21no—
252C2×Dic61—22no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—48no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61C4×D412yes—
264C2×Dic61C4×D412yes—
265C2×Dic61#13712yes—
266Dic122—22no—
267Dic122—22no—
268Dic122—22no—
269Dic122#11714yescomplemented
270C8×C61—22no—
271C8×C61—41no—
272#92418—22no—
273#92418—41no—
274#21124—611nomaximal, Sylow
275C22×C61—42no—
276C22×C61—42no—
277C22×C61—42no—
278C22×C61—81no—
279C22×C61—81no—
280C22×C61—81no—
281C22×C61—81no—
282C22×C61C2×C4×D411yescharacteristic
283C22×C61#52511yescharacteristic
284C22×C61#52511yescharacteristic
285C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C61C2×#13711yescharacteristic
287C22×C61#57511yescharacteristic
288C22×C61#57511yescharacteristic
289C4×C61—22no—
290C4×C61—41no—
291C4×C61—42no—
292C4×C61—42no—
293C4×C61—81no—
294C4×C61C2×#11711yescharacteristic
295C4×C61C4×#4111yescharacteristic
296C4×C61#52911yescharacteristic
297C4×C61#52911yescharacteristic
298Dic61—22no—
299Dic61—22no—
300Dic61—22no—
301Dic61—24no—
302Dic61—42no—
303Dic61—41no—
304Dic61—42no—
305Dic61—81no—
306C2×#440—611no—
307#2293—611no—
308C42×Q8—611no—
309#1926—611no—
310#1925D6111yescharacteristic
311#1921—611no—
312#1918—611no—
313#1880—611no—
314#1880—611no—
315#1865—611no—
316#1863—611no—
317#1776—611no—
318#1763—611no—
319#1762—611no—
320#1745—611no—
321C2×C61—82no—
322C2×C61#229011yescharacteristic
323C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
324C2×C61C4×#13711yescharacteristic
325C2×C61#311111yescharacteristic
326C2×C61#311211yescharacteristic
327C2×C61#311511yescharacteristic
328C2×C61#311711yescharacteristic
329C22×SD16—611no—
330C2×C4×Q8—611no—
331C2×C4×Q8—611no—
332C2×C4×Q8—611no—
333C2×C4×D4—611no—
334C2×#117—1221no—
335#440—618no—
336C2×#107—611no—
337C2×#107C2×D6111yescharacteristic
338C2×#106—611no—
339C2×#103—611no—
340C2×#103—611no—
341C2×#103—611no—
342C2×#102—611no—
343C2×#102C2×D6111yescharacteristic
344C2×#102C2×D6111yescharacteristic
345C2×#97—611no—
346C2×#97—611no—
347#390—611no—
348#390C2×D6111yescharacteristic
349#389—611no—
350#389—611no—
351#388—1221no—
352#388—1221no—
353C4×C4⋊C4—611no—
354C4×C4⋊C4—611no—
355C4×C4⋊C4—611no—
356C4×C4⋊C4—1221no—
357C4×C4⋊C4—1221no—
358C4×C4⋊C4—1221no—
359C4×C4⋊C4C2×D6111yescharacteristic
360C4×C22⋊C4—1221no—
361C43—611no—
362C43—1221no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367#340—611no—
368#340C2×D6111yescharacteristic
369#336—611no—
370#336—611no—
371#336—611no—
372#336C2×D6111yescharacteristic
373C61#2112411yescharacteristic, complemented, Sylow
374C22×Q8—611no—
375C22×D4C22×D6111yescharacteristic
376C23×C4—1221no—
377C2×SD16—614no—
378C2×SD16—1222no—
379C2×SD16—1222no—
380C22×C8—611no—
381C22×C8—611no—
382C22×C8C22×D6111yescharacteristic
383C22×C8C22×D6111yescharacteristic
384C4×Q8—614no—
385C4×Q8—1222no—
386C4×Q8—1222no—
387C4×Q8—1222no—
388C4×Q8—1222no—
389C4×Q8—1222no—
390C4×D4—614no—
391C4×D4—1222no—
392#117—2444no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4C22×D6111yescharacteristic
402C2×C4⋊C4C22×D6111yescharacteristic
403C2×C4⋊C4C22×D6111yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—1221no—
407C2×C22⋊C4—1221no—
408C2×C22⋊C4—1221no—
409C2×C42—21no—
410C2×C42—611no—
411C2×C42—611no—
412C2×C42—611no—
413C2×C42—611no—
414C2×C42—611no—
415C2×C42—611no—
416C2×C42—1221no—
417C2×C42—1221no—
418C2×C42—1221no—
419C2×C42—1221no—
420C2×C42—1221no—
421C2×C42—1221no—
422C2×C42C22×D6111yescharacteristic
423#107—22no—
424#107—614no—
425#106—1222no—
426#103—614no—
427#103—1222no—
428#103—1222no—
429#102—22no—
430#102—1222no—
431#102C4×D6114yes—
432#97—614no—
433#97—1222no—
434#95—21no—
435#95—1221no—
436#95—1221no—
437#95—1221no—
438#95—1221no—
439#95—1221no—
440C24—21no—
441C2×Q8—612no—
442C2×Q8—612no—
443C2×Q8—612no—
444C2×Q8—1221no—
445C2×Q8—1221no—
446C2×Q8—1221no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4C2×C4×D6112yes—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—1224no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—2442no—
466C22×C4—2442no—
467C22×C4D4×D6111yescharacteristic
468C22×C4D4×D6111yescharacteristic
469C22×C4C23×D6111yescharacteristic
470SD16—1228no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—612no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8C2×C4×D6112yes—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—612no—
488C4⋊C4—612no—
489C4⋊C4—1224no—
490C4⋊C4—1224no—
491C4⋊C4—1224no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—2442no—
497C4⋊C4—2442no—
498C4⋊C4C2×C4×D6112yes—
499C4⋊C4#1175822412yes—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—1224no—
503C22⋊C4—2442no—
504C22⋊C4—2442no—
505C42—21no—
506C42—21no—
507C42—42no—
508C42—612no—
509C42—612no—
510C42—1222no—
511C42—1224no—
512C42—1224no—
513C42—1224no—
514C42—1221no—
515C42—1221no—
516C42—1222no—
517C42—2441no—
518C42—2441no—
519C42—2442no—
520C42—2442no—
521C42—2442no—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23C2×D4×D6111yescharacteristic
530Q8—614no—
531Q8—1222no—
532Q8—1222no—
533Q8—1222no—
534D4—22no—
535D4—22no—
536D4—41no—
537D4—41no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—612no—
553C2×C4—612no—
554C2×C4—612no—
555C2×C4—1222no—
556C2×C4—1222no—
557C2×C4—1221no—
558C2×C4—1222no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—2448no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4#1451980112yes—
578C2×C4C22×C4×D6111yescharacteristic
579C2×C4#41×D6111yescharacteristic
580C2×C4#41×D6111yescharacteristic
581C2×C4#41×D6111yescharacteristic
582C2×C4#41×D6111yescharacteristic
583C2×C4C2×#1175822411yescharacteristic
584C8—22no—
585C8—41no—
586C8—1222no—
587C8—2441no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22C4×D4×D6111yescharacteristic
596C22#1684027011yescharacteristic
597C22#1684027211yescharacteristic
598C22C2×#1451980111yescharacteristic
599C22#137×D6111yescharacteristic
600C22#1684055811yescharacteristic
601C22#1684055911yescharacteristic
602C4—22no—
603C4—42no—
604C4—41no—
605C4—42no—
606C4—81no—
607C4—1222no—
608C4—1222no—
609C4—1224no—
610C4—1222no—
611C4—4881no—
612C4—2442no—
613C4—2442no—
614C4—2441no—
615C4#117×D6111yescharacteristic
616C4C4×#1175822411yescharacteristic
617C4#1684029111yescharacteristic
618C4#1684029511yescharacteristic
619C2—82no—
620C2#2144826111yescharacteristic
621C2C4×#1451980111yescharacteristic
622C2#2145464111yescharacteristic
623C2#2145592311yescharacteristic
624C2#2145593811yescharacteristic
625C2#2145596711yescharacteristic
626C2#2145598911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1925C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenter#1925D61
hypocenterC61#21124
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×#14519742C2semidirect
5. #21446409C2semidirect
6. #21446390C2semidirect
9. #21446090C2semidirect
10. #21446070C2semidirect
11. #21445718C2semidirect
12. #21445648C2semidirect
14. #21445566C2semidirect
22. #16839859C4semidirect
269. Dic122#117semidirect
373. C61#21124semidirect

Maximal subgroups

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

subgroupindex
C2×#168398592
#214482812
C4×#145197422
#214464482
#214464092
#214463902
#214461772
#214461712
#214460902
#214460702
#214457182
#214456482
#214456462
#214455662
C61×#19252
#2112461