Finite Group Databasebeta

← #29684352

Subgroups — #29684352

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×#16839864C211yesmaximal, characteristic
2#21448296C211yesmaximal, characteristic, complemented
3#21448283C211yesmaximal, characteristic
4#21446700C211yesmaximal, characteristic
5#21446673C211yesmaximal, characteristic
6#21446557C211yesmaximal, characteristic, complemented
7#21446552C211yesmaximal, characteristic, complemented
8#21446516C211yesmaximal, characteristic
9#21446509C211yesmaximal, characteristic, complemented
10#21446502C211yesmaximal, characteristic
11#21446495C211yesmaximal, characteristic, complemented
12#21445716C211yesmaximal, characteristic, complemented
13#21445650C211yesmaximal, characteristic, complemented
14#21445645C211yesmaximal, characteristic
15C4×#97×C61C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19C2×#14519734—21no—
20#16839864C418yescomplemented
21C2×#14519711C2211yescharacteristic
22C2×#14519711C2211yescharacteristic
23C2×#14519709C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28C2×#14519699C2211yescharacteristic
29C2×#14519699C2211yescharacteristic
30#16839668C2211yescharacteristic
31#16839668C2211yescharacteristic
32#16839664C2211yescharacteristic
33#16839664C2211yescharacteristic
34#16839660C2211yescharacteristic
35#16839660C2211yescharacteristic
36#16839660C2211yescharacteristic
37#16839660C2211yescharacteristic
38#16839653—21no—
39#16839653—21no—
40#16839646—21no—
41#16839646—21no—
42C4×#11758196—21no—
43C4×C4⋊Dic61C2211yescharacteristic
44C4×C4⋊Dic61C2211yescharacteristic
45C4×#11758198—21no—
46#16839608—21no—
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53C2×#97×C61C2211yescharacteristic
54C2×#97×C61C2211yescharacteristic
55C2×#97×C61C2211yescharacteristic
56C2×#97×C61C2211yescharacteristic
57C2×C4×C8×C61C2211yescharacteristic
58C2×C4×C8×C61C2211yescharacteristic
59C43×C61C2211yescharacteristic
60C22×Dic122C2311yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C4×D61—21no—
63C2×#11758190—22no—
64C2×#11758190—22no—
65C2×#11758190C2×C414yes—
66C4×Dic122—22no—
67C4×Dic122C2×C414yes—
68C4×D244—22no—
69C4×D244C2×C414yes—
70#14519734—44no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90#14519711—22no—
91#14519711C2×C414yes—
92#14519709—22no—
93#14519702—22no—
94#14519702—22no—
95#14519702C2×C414yes—
96#14519699—22no—
97#14519699—22no—
98#14519699C2×C414yes—
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#14519684—21no—
108#14519684—21no—
109C22×C8×C61C2311yescharacteristic
110C22×C8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C2×C42×C61D411yescharacteristic
114C2×C42×C61D411yescharacteristic
115C2×C42×C61D411yescharacteristic
116C2×C42×C61D411yescharacteristic
117C2×C42×C61C2311yescharacteristic
118C2×C42×C61C2311yescharacteristic
119C2×C42×C61C2311yescharacteristic
120#97×C61—22no—
121#97×C61—22no—
122#97×C61—22no—
123#97×C61C2×C414yes—
124C4×C8×C61—22no—
125C4×C8×C61—22no—
126C23×D61—21no—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2411yescharacteristic
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244C22×C412yes—
147C2×C4×D61—24no—
148C2×C4×D61—42no—
149C2×C4×D61—42no—
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×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61C22×C412yes—
163C2×C8×C61C22×C412yes—
164C2×C8×C61#4112yes—
165C2×C8×C61#4112yes—
166C2×C8×C61#4112yes—
167C2×C8×C61#4112yes—
168C42×C61—22no—
169C42×C61—22no—
170C42×C61—21no—
171C42×C61—21no—
172C42×C61—21no—
173C42×C61—21no—
174C42×C61—41no—
175C42×C61—41no—
176C42×C61C22×C412yes—
177C42×C61#4112yes—
178#11758198—24no—
179#11758198—42no—
180#11758198—42no—
181#11758198—42no—
182#11758198—42no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61C22×C412yes—
194C4⋊Dic61C22×C412yes—
195#11758196—24no—
196#11758196—42no—
197#11758196—42no—
198#11758196—42no—
199#11758196—42no—
200C4×Dic61—24no—
201C4×Dic61—42no—
202C4×Dic61—42no—
203#11758190—28no—
204C23×C61C22×D411yescharacteristic
205C22×D61—22no—
206C22×D61—41no—
207C22×D61—41no—
208C22×D61—41no—
209C22×D61—41no—
210C22×D61—41no—
211C22×D61—41no—
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×C61C4×D412yes—
227C2×C4×C61C4×D412yes—
228C2×C4×C61AGL(1,ℤ/8)12yes—
229C2×C4×C61#13712yes—
230C2×C4×C61C23×C411yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—41no—
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—
256C2×Dic61—41no—
257D244—22no—
258D244—22no—
259D244—41no—
260D244—41no—
261C4×D61—48no—
262Dic122—22no—
263Dic122—22no—
264Dic122—41no—
265Dic122—41no—
266C8×C61—22no—
267C8×C61—22no—
268C8×C61—22no—
269C8×C61#11714yescomplemented
270#21288—611nomaximal, Sylow
271C22×C61C2×C4×D411yescharacteristic
272C22×C61#53211yescharacteristic
273C22×C61#53211yescharacteristic
274C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C61C2×#13711yescharacteristic
276C22×C61#57511yescharacteristic
277C22×C61#57511yescharacteristic
278C2×D61—42no—
279C2×D61—42no—
280C2×D61—42no—
281C2×D61—81no—
282C2×D61—81no—
283C2×D61—81no—
284C2×D61—81no—
285C4×C61—22no—
286C4×C61—22no—
287C4×C61—22no—
288C4×C61—22no—
289C4×C61—41no—
290C4×C61—41no—
291C4×C61C2×#11711yescharacteristic
292C4×C61C4×#4111yescharacteristic
293C4×C61#52911yescharacteristic
294C4×C61#52911yescharacteristic
295Dic61—42no—
296Dic61—42no—
297Dic61—42no—
298Dic61—81no—
299Dic61—81no—
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#1934—611no—
312#1933—611no—
313#1776—611no—
314#1763—611no—
315#1762—611no—
316C4×#97D6111yescharacteristic, complemented
317C2×C61#229311yescharacteristic
318C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
319C2×C61C4×#13711yescharacteristic
320C2×C61#318011yescharacteristic
321C2×C61#318111yescharacteristic
322C2×C61#318211yescharacteristic
323C2×C61#318311yescharacteristic
324D61—82no—
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×#102—611no—
338C2×#102—611no—
339C2×#97C2×D6111yescharacteristic
340C2×#97C2×D6111yescharacteristic
341C2×#97C2×D6111yescharacteristic
342C2×#97C2×D6111yescharacteristic
343C2×C4×C8C2×D6111yescharacteristic
344C2×C4×C8C2×D6111yescharacteristic
345#391—611no—
346#391—611no—
347#390—611no—
348#390—611no—
349#389—611no—
350#389—611no—
351#389—611no—
352#389—611no—
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—
362C43C2×D6111yescharacteristic
363#340—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367#340—611no—
368#340—611no—
369C61#2128811yescharacteristic, complemented, Sylow
370C22×Q8—611no—
371C22×D4—611no—
372C23×C4—1221no—
373C2×SD16—614no—
374C2×SD16—1222no—
375C2×SD16—1222no—
376C22×C8C22×D6111yescharacteristic
377C22×C8C22×D6111yescharacteristic
378C22×C8C22×D6111yescharacteristic
379C22×C8C22×D6111yescharacteristic
380C4×Q8—614no—
381C4×Q8—1222no—
382C4×D4—614no—
383C4×D4—1222no—
384#117—2444no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—1221no—
396C2×C22⋊C4—1221no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C42—1221no—
402C2×C42—1221no—
403C2×C42—1221no—
404C2×C42D24411yescharacteristic
405C2×C42D24411yescharacteristic
406C2×C42D24411yescharacteristic
407C2×C42D24411yescharacteristic
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#97—22no—
421#97—22no—
422#97—22no—
423#97C4×D6114yes—
424C4×C8—22no—
425C4×C8—22no—
426#95—1221no—
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—1221no—
437C2×Q8—612no—
438C2×Q8—1221no—
439C2×Q8—1221no—
440C2×Q8—1221no—
441C2×Q8—1221no—
442C2×Q8—1221no—
443C2×D4—612no—
444C2×D4—1221no—
445C2×D4—1221no—
446C2×D4—1221no—
447C2×D4—1221no—
448C2×D4—1221no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—1224no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—2442no—
458C22×C4—2442no—
459C22×C4C2×D24411yescharacteristic
460C22×C4C2×D24411yescharacteristic
461C22×C4C2×D24411yescharacteristic
462C22×C4C2×D24411yescharacteristic
463C22×C4C2×D24411yescharacteristic
464C22×C4C2×D24411yescharacteristic
465C22×C4C23×D6111yescharacteristic
466SD16—1228no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8C2×C4×D6112yes—
474C2×C8C2×C4×D6112yes—
475C2×C8#1175822212yes—
476C2×C8#1175822212yes—
477C2×C8#1175822212yes—
478C2×C8#1175822212yes—
479C4⋊C4—612no—
480C4⋊C4—612no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1224no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
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⋊C4—2442no—
496C22⋊C4—1224no—
497C22⋊C4—2442no—
498C22⋊C4—2442no—
499C22⋊C4—2442no—
500C22⋊C4—2442no—
501C42—22no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—22no—
507C42—41no—
508C42—41no—
509C42—1224no—
510C42—2442no—
511C42—2442no—
512C42C2×C4×D6112yes—
513C42#1175822212yes—
514C23—1222no—
515C23—2441no—
516C23—2441no—
517C23—2441no—
518C23—2441no—
519C23—2441no—
520C23—2441no—
521C23C22×D24411yescharacteristic
522Q8—1222no—
523Q8—1222no—
524Q8—2441no—
525Q8—2441no—
526D4—1222no—
527D4—1222no—
528D4—2441no—
529D4—2441no—
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—1222no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—2448no—
548C2×C4—2441no—
549C2×C4—2441no—
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×C4—2441no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4C4×D24412yes—
567C2×C4C4×D24412yes—
568C2×C4#1451980512yes—
569C2×C4#1451981012yes—
570C2×C4C22×C4×D6111yescharacteristic
571C2×C4C2×#1175822211yescharacteristic
572C2×C4C2×#1175822211yescharacteristic
573C2×C4C2×#1175822211yescharacteristic
574C2×C4C2×#1175822211yescharacteristic
575C2×C4C2×#1175822211yescharacteristic
576C8—22no—
577C8—22no—
578C8—22no—
579C8#1451973414yescomplemented
580C22—4881no—
581C22—4881no—
582C22—4881no—
583C22—4881no—
584C22—2442no—
585C22—2442no—
586C22—2442no—
587C22C2×C4×D24411yescharacteristic
588C22#1684031211yescharacteristic
589C22#1684031211yescharacteristic
590C22C2×#1451980511yescharacteristic
591C22C2×#1451981011yescharacteristic
592C22#1684056211yescharacteristic
593C22#1684056211yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—41no—
599C4—41no—
600C4—4881no—
601C4—4881no—
602C4—4881no—
603C4—4881no—
604C4—2442no—
605C4—2442no—
606C4—2442no—
607C4C2×#1451973411yescharacteristic
608C4C4×#1175822211yescharacteristic
609C4#1684029611yescharacteristic
610C4#1684029611yescharacteristic
611C2—4882no—
612C2#2144828311yescharacteristic
613C2C4×#1451980511yescharacteristic
614C2C4×#1451981011yescharacteristic
615C2#2145651711yescharacteristic
616C2#2145652811yescharacteristic
617C2#2145653711yescharacteristic
618C2#2145654811yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×#97×C61C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenterC4×#97D61
hypocenterC61#21288
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C43 ◁ C4×#97

Complements

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

NH ≅ G/Ntype
2. #21448296C2semidirect
6. #21446557C2semidirect
7. #21446552C2semidirect
9. #21446509C2semidirect
11. #21446495C2semidirect
12. #21445716C2semidirect
13. #21445650C2semidirect
15. C4×#97×C61C2semidirect
20. #16839864C4semidirect
269. C8×C61#117semidirect
316. C4×#97D61semidirect
369. C61#21288semidirect
579. C8#14519734semidirect

Maximal subgroups

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

subgroupindex
C2×#168398642
#214482962
#214482832
#214467002
#214466732
#214465572
#214465522
#214465162
#214465092
#214465022
#214464952
#214457162
#214456502
#214456452
C4×#97×C612
#2128861