Finite Group Databasebeta

← #29684271

Subgroups — #29684271

Order 15616.

814 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839853C211yesmaximal, characteristic
2#21448466C211yesmaximal, characteristic, complemented
3#21448456C211yesmaximal, characteristic
4#21446674C211yesmaximal, characteristic
5#21446672C211yesmaximal, characteristic
6#21446546C211yesmaximal, characteristic, complemented
7#21446544C211yesmaximal, characteristic, complemented
8#21446518C211yesmaximal, characteristic
9#21446509C211yesmaximal, characteristic, complemented
10#21446507C211yesmaximal, characteristic, complemented
11#21445707C211yesmaximal, characteristic, complemented
12#21445648C211yesmaximal, characteristic, complemented
13#21445646C211yesmaximal, characteristic
14#21445507C211yesmaximal, characteristic, complemented
15C61×#1936C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×#14519742C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19#16839853C418yescomplemented
20C2×#14519708C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×#14519702C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519687C2211yescharacteristic
28C2×#14519687C2211yescharacteristic
29C2×C4×#92418C2211yescharacteristic
30C2×C8×Dic61C2211yescharacteristic
31C2×C8×Dic61C2211yescharacteristic
32#16839668C2211yescharacteristic
33#16839667C2211yescharacteristic
34#16839665C2211yescharacteristic
35#16839661C2211yescharacteristic
36#16839660C2211yescharacteristic
37#16839658C2211yescharacteristic
38#16839656—21no—
39#16839650—21no—
40#16839649—21no—
41#16839643—21no—
42#16839629—21no—
43#16839627—21no—
44#16839623—21no—
45#16839620—21no—
46C4⋊C4×Dic61C2211yescharacteristic
47C4⋊C4×Dic61C2211yescharacteristic
48#16839608C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53C2×#102×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×#97×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—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61C2311yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86#14519708—22no—
87#14519708—22no—
88#14519708C2×C414yes—
89#14519702—22no—
90#14519702C2×C414yes—
91#14519700—22no—
92#14519698—22no—
93#14519687—22no—
94#14519687—22no—
95C4×#92418—22no—
96C8×Dic61—22no—
97C8×Dic61C2×C414yes—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684D411yescharacteristic
108#14519684D411yescharacteristic
109#14519684D411yescharacteristic
110#14519684D411yescharacteristic
111C22×D4×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C22⋊C4×C61—21no—
119C2×C22⋊C4×C61—21no—
120C2×C42×C61C2311yescharacteristic
121#102×C61—22no—
122#102×C61C2×C414yes—
123#97×C61—22no—
124#95×C61—21no—
125C24×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61C22×C412yes—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2411yescharacteristic
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—24no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—81no—
144C22×Dic61—81no—
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C2×Dic122—21no—
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122C22×C412yes—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61C22×C412yes—
160C2×C8×C61#4112yes—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—81no—
167C4⋊C4×C61C22×C412yes—
168C22⋊C4×C61—81no—
169C22⋊C4×C61—81no—
170C42×C61—21no—
171C42×C61—21no—
172C22⋊Dic61—24no—
173C22⋊Dic61—81no—
174C22⋊Dic61—81no—
175#11758200—28no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—81no—
182C4⋊Dic61C22×C412yes—
183#11758196—24no—
184#11758196—81no—
185#11758196—81no—
186C4×Dic61—21no—
187C4×Dic61—21no—
188C4×Dic61—24no—
189C4×Dic61—81no—
190C4×Dic61—81no—
191C4×Dic61C22×C412yes—
192C4×Dic61#4112yes—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418C22×C412yes—
196C2×#92418#4112yes—
197C2×#92418#4112yes—
198C2×#92418#4112yes—
199C23×C61—22no—
200C23×C61—41no—
201C23×C61—41no—
202C23×C61—41no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61C22×D411yescharacteristic
207D4×C61—22no—
208D4×C61—22no—
209D4×C61—41no—
210D4×C61—41no—
211C2×C4×C61—22no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61C23×C411yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—84no—
255C2×Dic61C4×D412yes—
256C2×Dic61C4×D412yes—
257Dic122—22no—
258Dic122—22no—
259Dic122—41no—
260Dic122—41no—
261C8×C61—22no—
262C8×C61—22no—
263#92418—22no—
264#92418#11714yes—
265#21280—611nomaximal, Sylow
266C22×C61—42no—
267C22×C61—42no—
268C22×C61—42no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61—81no—
273C22×C61C2×C4×D411yescharacteristic
274C22×C61#53211yescharacteristic
275C22×C61#53211yescharacteristic
276C22×C61#57711yescharacteristic
277C22×C61#57711yescharacteristic
278C22×C61#57711yescharacteristic
279C22×C61#57711yescharacteristic
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—42no—
283C4×C61—81no—
284C4×C61—81no—
285C4×C61C2×#11711yescharacteristic
286C4×C61C4×#4111yescharacteristic
287C4×C61#52911yescharacteristic
288C4×C61#52911yescharacteristic
289Dic61—22no—
290Dic61—22no—
291Dic61—41no—
292Dic61—41no—
293Dic61—42no—
294Dic61—42no—
295Dic61—81no—
296Dic61—81no—
297C2×C4×SD16—611no—
298#2320—611no—
299#2319—611no—
300#1968—611no—
301#1968—611no—
302#1941—611no—
303#1941—611no—
304#1936—611no—
305#1936D6111yescharacteristic
306#1935—611no—
307#1935—611no—
308C4×#106—611no—
309#1763—611no—
310#1762—611no—
311#1726—611no—
312C2×C61—82no—
313C2×C61#229311yescharacteristic
314C2×C61#294911yescharacteristic
315C2×C61#295211yescharacteristic
316C2×C61#317311yescharacteristic
317C2×C61#317311yescharacteristic
318C2×C61#317411yescharacteristic
319C2×C61#317411yescharacteristic
320C22×SD16—611no—
321C2×C4×Q8—611no—
322C2×C4×D4—611no—
323C4×SD16—618no—
324C2×#106—611no—
325C2×#106—611no—
326C2×#106—611no—
327C2×#103—611no—
328C2×#103—611no—
329C2×#103—611no—
330C2×#102—611no—
331C2×#102C2×D6111yescharacteristic
332C2×#102C2×D6111yescharacteristic
333C2×#97—611no—
334C2×#97—611no—
335C2×#97C2×D6111yescharacteristic
336C2×C4×C8—611no—
337C2×C4×C8—611no—
338C2×C4×C8—611no—
339#391—611no—
340#391—611no—
341#390—611no—
342#390C2×D6111yescharacteristic
343#389—611no—
344#389—611no—
345#389—611no—
346#389C2×D6111yescharacteristic
347#388—1221no—
348#388—1221no—
349#387—1221no—
350#387—1221no—
351#383—1221no—
352#382—1221no—
353#381—1221no—
354#380—1221no—
355C4×C4⋊C4—611no—
356C4×C4⋊C4—611no—
357#376—611no—
358#340—611no—
359#340—611no—
360#340—611no—
361#340—611no—
362#340C2×D6111yescharacteristic
363#340C2×D6111yescharacteristic
364C61#2128011yescharacteristic, complemented, Sylow
365C22×Q8—611no—
366C22×D4C22×D6111yescharacteristic
367C23×C4—1221no—
368C2×SD16—614no—
369C2×SD16—1222no—
370C2×SD16—1222no—
371C22×C8—611no—
372C22×C8—611no—
373C22×C8C22×D6111yescharacteristic
374C22×C8C22×D6111yescharacteristic
375C4×Q8—614no—
376C4×Q8—1222no—
377C4×D4—614no—
378C4×D4—1222no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4C22×D6111yescharacteristic
388C2×C4⋊C4C22×D6111yescharacteristic
389C2×C4⋊C4C22×D6111yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—1221no—
393C2×C22⋊C4—1221no—
394C2×C22⋊C4—1221no—
395C2×C42—611no—
396C2×C42—611no—
397C2×C42—1221no—
398C2×C42—1221no—
399C2×C42—1221no—
400C2×C42C22×D6111yescharacteristic
401#106—614no—
402#106—1222no—
403#106—1222no—
404#103—614no—
405#103—1222no—
406#103—1222no—
407#102—22no—
408#102—1222no—
409#102C4×D6114yes—
410#97—22no—
411#97—1222no—
412#97—1222no—
413C4×C8—614no—
414C4×C8—1222no—
415C4×C8—1222no—
416#95—21no—
417#95—611no—
418#95—611no—
419#95—611no—
420#95—611no—
421#95—1221no—
422#95—1221no—
423#95—1221no—
424#95—1221no—
425#95—1221no—
426#95—1221no—
427#95—1221no—
428#95—1221no—
429#95—1221no—
430C24—21no—
431C2×Q8—612no—
432C2×Q8—1221no—
433C2×Q8—1221no—
434C2×Q8—1221no—
435C2×Q8—1221no—
436C2×Q8—1221no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4C2×C4×D6112yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—611no—
447C22×C4—611no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—1224no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—4881no—
456C22×C4—4881no—
457C22×C4D4×D6111yescharacteristic
458C22×C4D4×D6111yescharacteristic
459C22×C4C23×D6111yescharacteristic
460SD16—1228no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—612no—
466C2×C8—612no—
467C2×C8—612no—
468C2×C8—612no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8C2×C4×D6112yes—
472C2×C8#1175822412yes—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—81no—
479C4⋊C4—612no—
480C4⋊C4—1224no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—4881no—
487C4⋊C4—4881no—
488C4⋊C4—4881no—
489C4⋊C4C2×C4×D6112yes—
490C22⋊C4—81no—
491C22⋊C4—81no—
492C22⋊C4—1224no—
493C22⋊C4—4881no—
494C22⋊C4—4881no—
495C42—21no—
496C42—21no—
497C42—612no—
498C42—612no—
499C42—1224no—
500C42—1221no—
501C42—1221no—
502C42—4881no—
503C42—4881no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23C2×D4×D6111yescharacteristic
512Q8—1222no—
513Q8—1222no—
514Q8—2441no—
515Q8—2441no—
516D4—22no—
517D4—22no—
518D4—41no—
519D4—41no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—612no—
536C2×C4—612no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—4884no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4C22×C4×D6111yescharacteristic
561C2×C4#41×D6111yescharacteristic
562C2×C4#41×D6111yescharacteristic
563C2×C4#41×D6111yescharacteristic
564C2×C4#41×D6111yescharacteristic
565C2×C4C2×#1175822411yescharacteristic
566C8—22no—
567C8—22no—
568C8—614no—
569C8—1222no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22C4×D4×D6111yescharacteristic
578C22#1684030711yescharacteristic
579C22#1684031111yescharacteristic
580C22#1684057111yescharacteristic
581C22#1684057211yescharacteristic
582C22#1684057211yescharacteristic
583C22#1684057211yescharacteristic
584C4—41no—
585C4—41no—
586C4—42no—
587C4—81no—
588C4—81no—
589C4—1222no—
590C4—1222no—
591C4—4881no—
592C4—4881no—
593C4—2441no—
594C4—2441no—
595C4—2442no—
596C4—2442no—
597C4#117×D6111yescharacteristic
598C4C4×#1175822411yescharacteristic
599C4#1684028911yescharacteristic
600C4#1684029511yescharacteristic
601C2—82no—
602C2#2144827911yescharacteristic
603C2#2145467511yescharacteristic
604C2#2145470111yescharacteristic
605C2#2145643111yescharacteristic
606C2#2145643511yescharacteristic
607C2#2145644211yescharacteristic
608C2#2145644611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1936C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenter#1936D61
hypocenterC61#21280
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1936

Complements

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

NH ≅ G/Ntype
2. #21448466C2semidirect
6. #21446546C2semidirect
7. #21446544C2semidirect
9. #21446509C2semidirect
10. #21446507C2semidirect
11. #21445707C2semidirect
12. #21445648C2semidirect
14. #21445507C2semidirect
19. #16839853C4semidirect
364. C61#21280semidirect

Maximal subgroups

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

subgroupindex
C2×#168398532
#214484662
#214484562
#214466742
#214466722
#214465462
#214465442
#214465182
#214465092
#214465072
#214457072
#214456482
#214456462
#214455072
C61×#19362
#2128061