Finite Group Databasebeta

← #29683091

Subgroups — #29683091

Order 15616.

890 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839864C211yesmaximal, characteristic
2#21448471C211yesmaximal, characteristic, complemented
3#21448181C211yesmaximal, characteristic
4#21446461C211yesmaximal, characteristic, complemented
5#21446454C211yesmaximal, characteristic, complemented
6#21446412C211yesmaximal, characteristic, complemented
7#21446391C211yesmaximal, characteristic, complemented
8#21446152C211yesmaximal, characteristic
9#21446152C211yesmaximal, characteristic
10#21446083C211yesmaximal, characteristic
11#21446069C211yesmaximal, characteristic
12#21445716C211yesmaximal, characteristic, complemented
13C4×#14519702C211yesmaximal, characteristic, complemented
14C4×#14519699C211yesmaximal, characteristic
15C61×#1746C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19C2×C4×D244C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21#16839864C418yescomplemented
22C2×#14519711C2211yescharacteristic
23C2×#14519711C2211yescharacteristic
24C2×#14519709C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519699C2211yescharacteristic
29C2×#14519699C2211yescharacteristic
30C2×#14519699C2211yescharacteristic
31#16839668C2211yescharacteristic
32#16839668C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839660C2211yescharacteristic
35#16839644—21no—
36#16839644—21no—
37#16839629—21no—
38#16839626—21no—
39C4×C4⋊Dic61C2211yescharacteristic
40C4×C4⋊Dic61C2211yescharacteristic
41C4×C4⋊Dic61C2211yescharacteristic
42C4×C4⋊Dic61C2211yescharacteristic
43#16839609—21no—
44#16839609—21no—
45#16839609—21no—
46#95×D61—21no—
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53C2×#97×C61C2211yescharacteristic
54C2×C4×C8×C61C2211yescharacteristic
55C61×#376C2211yescharacteristic
56C61×#336C2211yescharacteristic
57C61×#336C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C22×Dic122C2311yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C4×D61—21no—
63C22×C4×D61—21no—
64C22×C4×D61—21no—
65C2×#11758190—22no—
66C2×#11758190—22no—
67C2×#11758190C2×C414yes—
68C4×Dic122—22no—
69C4×Dic122C2×C414yescomplemented
70C4×D244—22no—
71C4×D244—22no—
72C4×D244—22no—
73C4×D244—22no—
74C4×D244—22no—
75C4×D244C2×C414yes—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93#14519711—22no—
94#14519711C2×C414yescomplemented
95#14519709—22no—
96#14519702—22no—
97#14519702—22no—
98#14519702C2×C414yescomplemented
99#14519699—22no—
100#14519699—22no—
101#14519699C2×C414yes—
102#14519684—21no—
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×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116C2×C42×C61C2311yescharacteristic
117C2×C42×C61C2311yescharacteristic
118C2×C42×C61C2311yescharacteristic
119#97×C61C2×C414yescomplemented
120C4×C8×C61—22no—
121#95×C61D411yescharacteristic
122#95×C61D411yescharacteristic
123#95×C61D411yescharacteristic
124#95×C61D411yescharacteristic
125C23×D61—21no—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2411yescharacteristic
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244C22×C412yes—
144C2×D244#4112yes—
145C2×D244#4112yes—
146C2×C4×D61—24no—
147C2×C4×D61—24no—
148C2×C4×D61—24no—
149C2×C4×D61—81no—
150C2×C4×D61—81no—
151C2×C4×D61—81no—
152C2×C4×D61—81no—
153C2×C4×D61—81no—
154C2×C4×D61—81no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122C22×C412yes—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61C22×C412yes—
172C2×C8×C61C22×C412yes—
173C42×C61—21no—
174C42×C61—21no—
175C42×C61—21no—
176C42×C61—21no—
177C42×C61C22×C412yes—
178C42×C61#4112yes—
179#11758198—24no—
180#11758198—24no—
181#11758198—24no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61C22×C412yes—
191C4⋊Dic61C22×C412yes—
192C4⋊Dic61#4112yes—
193C4⋊Dic61#4112yes—
194#11758196—24no—
195#11758196—81no—
196#11758196—81no—
197C4×Dic61—24no—
198C4×Dic61—81no—
199C4×Dic61—81no—
200C4×Dic61—81no—
201C4×Dic61—81no—
202#11758190—28no—
203C23×C61C22×D411yescharacteristic
204C22×D61—22no—
205C22×D61—22no—
206C22×D61—22no—
207C22×D61—41no—
208C22×D61—41no—
209C22×D61—41no—
210C22×D61—41no—
211C2×C4×C61—21no—
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—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61C4×D412yes—
228C2×C4×C61C4×D412yes—
229C2×C4×C61C23×C411yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
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—41no—
255C2×Dic61—41no—
256D244—22no—
257D244—22no—
258D244—22no—
259D244#11714yes—
260C4×D61—84no—
261C4×D61—84no—
262C4×D61—84no—
263Dic122—22no—
264Dic122—22no—
265Dic122—41no—
266Dic122—41no—
267C8×C61—22no—
268C8×C61—22no—
269C8×C61—41no—
270C8×C61—41no—
271#21125—611nomaximal, Sylow
272C22×C61C2×C4×D411yescharacteristic
273C22×C61#52511yescharacteristic
274C22×C61#52511yescharacteristic
275C22×C61#57611yescharacteristic
276C22×C61#57711yescharacteristic
277C22×C61#57711yescharacteristic
278C22×C61#57811yescharacteristic
279C2×D61—24no—
280C2×D61—42no—
281C2×D61—42no—
282C2×D61—42no—
283C2×D61—42no—
284C2×D61—42no—
285C2×D61—42no—
286C4×C61—22no—
287C4×C61—22no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61—41no—
291C4×C61—41no—
292C4×C61C2×#11711yescharacteristic
293C4×C61C4×#4111yescharacteristic
294C4×C61#52911yescharacteristic
295C4×C61#52911yescharacteristic
296Dic61—42no—
297Dic61—42no—
298Dic61—42no—
299Dic61—42no—
300Dic61—42no—
301Dic61—81no—
302Dic61—81no—
303C2×#440—611no—
304#2320—611no—
305#2275—611no—
306#1928—611no—
307#1927—611no—
308#1921—611no—
309#1918—611no—
310#1877—611no—
311#1877—611no—
312#1864—611no—
313#1862—611no—
314#1776—611no—
315C4×#103—611no—
316C4×#102—611no—
317#1746D6111yescharacteristic, complemented
318C2×C61#229011yescharacteristic
319C2×C61#294611yescharacteristic
320C2×C61#294711yescharacteristic
321C2×C61#311811yescharacteristic
322C2×C61#312011yescharacteristic
323C2×C61#312311yescharacteristic
324C2×C61#312411yescharacteristic
325D61—44no—
326C22×SD16—611no—
327C2×C4×Q8—611no—
328C2×C4×D4—611no—
329C2×C4×D4—611no—
330C2×C4×D4—611no—
331#440—618no—
332C2×#107—611no—
333C2×#107—611no—
334C2×#106—611no—
335C2×#103—611no—
336C2×#103—611no—
337C2×#103—611no—
338C2×#102—611no—
339C2×#102—611no—
340C2×#102—611no—
341C2×#97C2×D6111yescharacteristic
342C2×C4×C8C2×D6111yescharacteristic
343#391—611no—
344#391—611no—
345#389—611no—
346#389—611no—
347#387—1221no—
348#387—1221no—
349#383—1221no—
350#382—1221no—
351C4×C4⋊C4—611no—
352C4×C4⋊C4—611no—
353C4×C4⋊C4—611no—
354C4×C4⋊C4—611no—
355C4×C22⋊C4—1221no—
356C4×C22⋊C4—1221no—
357C4×C22⋊C4—1221no—
358#376C2×D6111yescharacteristic
359C2×#95—1221no—
360#340—611no—
361#340—611no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#336C2×D6111yescharacteristic
367#336C2×D6111yescharacteristic
368#336C2×D6111yescharacteristic
369#336C2×D6111yescharacteristic
370C61#2112511yescharacteristic, complemented, Sylow
371C22×Q8—611no—
372C22×D4—611no—
373C23×C4—1221no—
374C23×C4—1221no—
375C23×C4—1221no—
376C2×SD16—614no—
377C2×SD16—1222no—
378C2×SD16—1222no—
379C22×C8C22×D6111yescharacteristic
380C22×C8C22×D6111yescharacteristic
381C22×C8C22×D6111yescharacteristic
382C22×C8C22×D6111yescharacteristic
383C4×Q8—614no—
384C4×Q8—1222no—
385C4×D4—614no—
386C4×D4—1222no—
387C4×D4—1222no—
388C4×D4—1222no—
389C4×D4—1222no—
390C4×D4—1222no—
391C2×C4⋊C4—611no—
392C2×C4⋊C4—611no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C22⋊C4—1221no—
402C2×C22⋊C4—1221no—
403C2×C42—1221no—
404C2×C42—1221no—
405C2×C42—1221no—
406C2×C42—1221no—
407C2×C42—1221no—
408C2×C42C22×D6111yescharacteristic
409C2×C42C22×D6111yescharacteristic
410C2×C42C22×D6111yescharacteristic
411#107—614no—
412#107—1222no—
413#106—1222no—
414#103—614no—
415#103—1222no—
416#103—1222no—
417#102—614no—
418#102—1222no—
419#102—1222no—
420#97C4×D6114yescomplemented
421C4×C8—22no—
422#95—1221no—
423#95—1221no—
424#95—1221no—
425#95—1221no—
426#95—1221no—
427#95—1221no—
428#95—1221no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432#95D24411yescharacteristic
433#95D24411yescharacteristic
434#95D24411yescharacteristic
435#95D24411yescharacteristic
436C24—1221no—
437C2×Q8—612no—
438C2×Q8—1221no—
439C2×Q8—1221no—
440C2×Q8—1221no—
441C2×Q8—1221no—
442C2×Q8—1221no—
443C2×D4—612no—
444C2×D4—612no—
445C2×D4—612no—
446C2×D4—1221no—
447C2×D4—1221no—
448C2×D4—1221no—
449C22×C4—1224no—
450C22×C4—1224no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1224no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—4881no—
460C22×C4—4881no—
461C22×C4—4881no—
462C22×C4—4881no—
463C22×C4—4881no—
464C22×C4—4881no—
465C22×C4C2×D24411yescharacteristic
466C22×C4C2×D24411yescharacteristic
467C22×C4C2×D24411yescharacteristic
468C22×C4C2×D24411yescharacteristic
469C22×C4C2×D24411yescharacteristic
470C22×C4C2×D24411yescharacteristic
471C22×C4C23×D6111yescharacteristic
472SD16—1228no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8C2×C4×D6112yes—
484C2×C8C2×C4×D6112yes—
485C4⋊C4—612no—
486C4⋊C4—612no—
487C4⋊C4—612no—
488C4⋊C4—612no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1224no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—4881no—
499C4⋊C4—4881no—
500C22⋊C4—1224no—
501C22⋊C4—1224no—
502C22⋊C4—1224no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—1224no—
508C42—4881no—
509C42—4881no—
510C42—4881no—
511C42—4881no—
512C42C2×C4×D6112yes—
513C42#1175822212yes—
514C23—1222no—
515C23—1222no—
516C23—1222no—
517C23—2441no—
518C23—2441no—
519C23—2441no—
520C23—2441no—
521C23C22×D24411yescharacteristic
522Q8—1222no—
523Q8—1222no—
524Q8—2441no—
525Q8—2441no—
526D4—614no—
527D4—1222no—
528D4—1222no—
529D4—1222no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—1222no—
550C2×C4—1222no—
551C2×C4—4884no—
552C2×C4—4884no—
553C2×C4—4884no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
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—2441no—
569C2×C4—2441no—
570C2×C4C4×D24412yes—
571C2×C4C4×D24412yes—
572C2×C4C22×C4×D6111yescharacteristic
573C2×C4#41×D6111yescharacteristic
574C2×C4#41×D6111yescharacteristic
575C2×C4#41×D6111yescharacteristic
576C2×C4#41×D6111yescharacteristic
577C2×C4C2×#1175822211yescharacteristic
578C8—22no—
579C8—22no—
580C8—41no—
581C8—41no—
582C22—1224no—
583C22—2442no—
584C22—2442no—
585C22—2442no—
586C22—2442no—
587C22—2442no—
588C22—2442no—
589C22C2×C4×D24411yescharacteristic
590C22#1684027411yescharacteristic
591C22#1684027411yescharacteristic
592C22#1684056711yescharacteristic
593C22#1684057411yescharacteristic
594C22#1684057411yescharacteristic
595C22#1684057911yescharacteristic
596C4—22no—
597C4—22no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—4881no—
603C4—4881no—
604C4—2442no—
605C4—2442no—
606C4—2442no—
607C4—2442no—
608C4—2442no—
609C4#117×D6111yescharacteristic
610C4#1684023011yescharacteristic
611C4#1684029011yescharacteristic
612C4#1684029011yescharacteristic
613C2—2444no—
614C2#2144826311yescharacteristic
615C2#2145465311yescharacteristic
616C2#2145466411yescharacteristic
617C2#2145600511yescharacteristic
618C2#2145603511yescharacteristic
619C2#2145607211yescharacteristic
620C2#2145608311yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1746C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1746D61
hypocenterC61#21125
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ #376 ◁ #1746

Complements

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

NH ≅ G/Ntype
2. #21448471C2semidirect
4. #21446461C2semidirect
5. #21446454C2semidirect
6. #21446412C2semidirect
7. #21446391C2semidirect
12. #21445716C2semidirect
13. C4×#14519702C2semidirect
15. C61×#1746C2semidirect
21. #16839864C4semidirect
69. C4×Dic122C2×C4semidirect
94. #14519711C2×C4semidirect
98. #14519702C2×C4semidirect
119. #97×C61C2×C4semidirect
317. #1746D61semidirect
370. C61#21125semidirect
420. #97C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398642
#214484712
#214481812
#214464612
#214464542
#214464122
#214463912
#214461522
#214460832
#214460692
#214457162
C4×#145197022
C4×#145196992
C61×#17462
#2112561