Finite Group Databasebeta

← #29742871

Subgroups — #29742871

Order 15616.

812 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839801C211yesmaximal, characteristic
2#21449528C211yesmaximal, characteristic
3#21448820C211yesmaximal, characteristic
4#21447355C211yesmaximal, characteristic, complemented
5#21447307C211yesmaximal, characteristic, complemented
6#21447196C211yesmaximal, characteristic, complemented
7#21447154C211yesmaximal, characteristic, complemented
8#21446574C211yesmaximal, characteristic, complemented
9#21446399C211yesmaximal, characteristic, complemented
10#21446249C211yesmaximal, characteristic
11#21446128C211yesmaximal, characteristic
12#21446073C211yesmaximal, characteristic, complemented
13#21446005C211yesmaximal, characteristic
14#21446000C211yesmaximal, characteristic
15C2×C61×#479C211yesmaximal, characteristic, complemented
16C22×#11758189C2211yescharacteristic
17C2×#14519753C2211yescharacteristic
18C2×#14519735C2211yescharacteristic
19C2×C4⋊C4×D61C2211yescharacteristic
20#16839801—24no—
21C2×#14519711C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519710C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519689C2211yescharacteristic
29#16839697—21no—
30#16839684—21no—
31#16839669C2211yescharacteristic
32#16839661C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839660C2211yescharacteristic
35#16839658—21no—
36#16839657—21no—
37#16839646—21no—
38#16839643—21no—
39#16839633—21no—
40#16839631C2211yescharacteristic
41#16839619C2211yescharacteristic
42C4⋊C4×Dic61C2211yescharacteristic
43C4×#11758198—21no—
44C4×#11758198C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#125×C61C2211yescharacteristic
54C2×C4×Q8×C61C2211yescharacteristic
55C61×#479—24no—
56C2×#107×C61C2211yescharacteristic
57C2×#106×C61C2211yescharacteristic
58C2×C4⋊C8×C61C2211yescharacteristic
59C2×#103×C61C2211yescharacteristic
60C2×#103×C61C2211yescharacteristic
61C22×C4×D61—21no—
62C22×C4×D61C2311yescharacteristic
63C2×#11758189—22no—
64C2×#11758189—22no—
65C2×#11758189—22no—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519753—22no—
69#14519753Q814yes—
70#14519735—22no—
71#14519735—41no—
72C2×C4⋊Dic61—21no—
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×#11758196—21no—
83C4⋊C4×D61—22no—
84C4⋊C4×D61—41no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198D411yescharacteristic
89C2×#11758198D411yescharacteristic
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61C2311yescharacteristic
93#14519711—22no—
94#14519710—22no—
95#14519710—41no—
96#14519708—22no—
97#14519706—22no—
98#14519701—22no—
99#14519701—41no—
100#14519689—22no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684D411yescharacteristic
109#14519684D411yescharacteristic
110C22×Q8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113#125×C61—22no—
114#125×C61—41no—
115C4×Q8×C61—22no—
116C4×Q8×C61—41no—
117C2×C4⋊C4×C61—21no—
118C2×C4⋊C4×C61—21no—
119C2×C4⋊C4×C61—21no—
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C42×C61—21no—
124C2×C42×C61C2311yescharacteristic
125#107×C61—22no—
126#106×C61D414yes—
127C4⋊C8×C61—22no—
128#103×C61—22no—
129#103×C61—22no—
130C23×D61C2×D411yescharacteristic
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61C2×Q812yes—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2411yescharacteristic
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C2×C4×D61—21no—
152C2×C4×D61—21no—
153C2×C4×D61—21no—
154C2×C4×D61—21no—
155C2×C4×D61—42no—
156C2×C4×D61—42no—
157C2×C4×D61—81no—
158C2×C4×D61C2×Q812yes—
159C2×C4×D61#4112yes—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61C2×D412yes—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—42no—
171C4⋊C4×C61—42no—
172C4⋊C4×C61—42no—
173C4⋊C4×C61C2×D412yes—
174C4⋊C4×C61C2×D412yes—
175C42×C61—21no—
176C42×C61—21no—
177C42×C61—42no—
178#11758198—24no—
179#11758198—22no—
180#11758198—41no—
181#11758198—42no—
182#11758198—81no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—42no—
191C4⋊Dic61C2×Q812yes—
192#11758196—24no—
193#11758196—42no—
194#11758196—42no—
195#11758196—81no—
196#11758196—81no—
197#11758196—81no—
198C4×Dic61—21no—
199C4×Dic61—21no—
200C4×Dic61—42no—
201C4×Dic61—81no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418#4112yes—
208#11758189—28no—
209C23×C61C22×D411yescharacteristic
210C22×D61—21no—
211C22×D61—21no—
212C22×D61—21no—
213C22×D61—21no—
214C22×D61—21no—
215C22×D61—21no—
216C22×D61C22⋊Q812yes—
217Q8×C61—22no—
218Q8×C61—22no—
219Q8×C61—41no—
220Q8×C61—41no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×Q811yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61H(5,𝔽2)11yescharacteristic
244C2×C4×C612−1+411yescharacteristic
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—22no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61C22⋊Q812yes—
265C2×Dic61#13712yes—
266C4×D61—22no—
267C4×D61—22no—
268C4×D61—22no—
269C4×D61—22no—
270C4×D61—84no—
271C8×C61—22no—
272C8×C61—41no—
273#92418—22no—
274#92418—41no—
275#26362—611nomaximal, Sylow
276C22×C61C2×C22⋊Q811yescharacteristic
277C22×C61D4211yescharacteristic
278C22×C61#54711yescharacteristic
279C22×C61C2×#13711yescharacteristic
280C22×C61#57511yescharacteristic
281C22×C61#57611yescharacteristic
282C22×C61#57811yescharacteristic
283C2×D61—22no—
284C2×D61—22no—
285C2×D61—22no—
286C2×D61—41no—
287C2×D61—41no—
288C2×D61—41no—
289C2×D61—41no—
290C4×C61—42no—
291C4×C61—42no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61—42no—
295C4×C61—81no—
296C4×C61#52511yescharacteristic
297C4×C61#53911yescharacteristic
298C4×C61D4×Q811yescharacteristic
299C4×C61#55411yescharacteristic
300Dic61—22no—
301Dic61—22no—
302Dic61—42no—
303Dic61—41no—
304Dic61—81no—
305Dic61—81no—
306Dic61—81no—
307C2×#479D6111yescharacteristic, complemented
308C2×#428—611no—
309#2441—611no—
310#2360—611no—
311#2095—611no—
312#2086—611no—
313#2066—611no—
314#2059—611no—
315#1947—611no—
316#1919—611no—
317#1894—611no—
318#1873—611no—
319#1863—611no—
320#1852—611no—
321#1851—611no—
322C2×C61#244911yescharacteristic
323C2×C61#307811yescharacteristic
324C2×C61#308311yescharacteristic
325C2×C61#326811yescharacteristic
326C2×C61#327611yescharacteristic
327C2×C61#334611yescharacteristic
328C2×C61#335211yescharacteristic
329D61—42no—
330C22×M16—611no—
331C2×#125C2×D6111yescharacteristic
332C2×C22⋊Q8—611no—
333C2×C4×Q8C2×D6111yescharacteristic
334C2×#117—611no—
335C22×C4⋊C4—611no—
336#479—24no—
337#428—1224no—
338C2×#107—611no—
339C2×#107—611no—
340C2×#107—611no—
341C2×#107C2×D6111yescharacteristic
342C2×#106—611no—
343C2×#106C2×D6111yescharacteristic
344C2×C4⋊C8—611no—
345C2×C4⋊C8C2×D6111yescharacteristic
346C2×#103—611no—
347C2×#103—611no—
348C2×#103C2×D6111yescharacteristic
349C2×#103C2×D6111yescharacteristic
350C2×C22⋊C8—611no—
351#399—1221no—
352#396—1221no—
353#391—611no—
354#389—611no—
355#389—611no—
356#389—611no—
357#389—1221no—
358#389—1221no—
359#387—1221no—
360#387—1221no—
361#385—1221no—
362#384—611no—
363#380—611no—
364C4×C4⋊C4—611no—
365C4×C22⋊C4—611no—
366C4×C22⋊C4—1221no—
367#340—611no—
368#340—611no—
369#340—611no—
370#340—611no—
371#340—611no—
372#340—611no—
373#336—611no—
374#336—611no—
375C61#2636211yescharacteristic, complemented, Sylow
376C22×Q8C22×D6111yescharacteristic
377C23×C4—611no—
378C23×C4—1221no—
379C2×M16—1222no—
380C2×M16—1222no—
381C2×M16—1222no—
382C22×C8—611no—
383C22×C8—611no—
384C22×C8C22×D6111yescharacteristic
385C22×C8C22×D6111yescharacteristic
386#125—22no—
387#125—41no—
388C22⋊Q8—614no—
389C22⋊Q8—1222no—
390C4×Q8—22no—
391C4×Q8—41no—
392#117—1222no—
393#117—2441no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1222no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—2441no—
410C2×C4⋊C4C22×D6111yescharacteristic
411C2×C4⋊C4C22×D6111yescharacteristic
412C2×C4⋊C4C22×D6111yescharacteristic
413C2×C22⋊C4—611no—
414C2×C22⋊C4—611no—
415C2×C22⋊C4—1221no—
416C2×C22⋊C4—1221no—
417C2×C22⋊C4—1221no—
418C2×C42—21no—
419C2×C42—611no—
420C2×C42—1221no—
421C2×C42—1221no—
422C2×C42C22×D6111yescharacteristic
423#107—22no—
424#107—1222no—
425#107—1222no—
426#107—2441no—
427#106—1222no—
428#106C22⋊D6114yes—
429C4⋊C8—22no—
430C4⋊C8—1222no—
431#103—22no—
432#103—22no—
433#103—1222no—
434#103—2441no—
435C22⋊C8—1222no—
436#95—611no—
437#95—611no—
438#95—1221no—
439#95—1221no—
440#95—1221no—
441#95—1221no—
442#95—1221no—
443#95—1221no—
444#95—1221no—
445C24—611no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8Q8×D6112yes—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—612no—
459C22×C4—612no—
460C22×C4—611no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—4881no—
470C22×C4—2442no—
471C22×C4—2442no—
472C22×C4D4×D6111yescharacteristic
473C22×C4D4×D6111yescharacteristic
474C22×C4C23×D6111yescharacteristic
475M16—1228no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—612no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8C2×C22⋊D6112yes—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—612no—
496C4⋊C4—1224no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—4881no—
505C4⋊C4—4881no—
506C4⋊C4—4881no—
507C4⋊C4—2442no—
508C4⋊C4—2442no—
509C4⋊C4—2442no—
510C4⋊C4C2×C22⋊D6112yes—
511C4⋊C4C2×C22⋊D6112yes—
512C22⋊C4—1222no—
513C22⋊C4—1224no—
514C22⋊C4—4881no—
515C22⋊C4—2441no—
516C22⋊C4—2442no—
517C42—21no—
518C42—21no—
519C42—42no—
520C42—1221no—
521C42—1221no—
522C42—4881no—
523C42—2442no—
524C23—612no—
525C23—1221no—
526C23—1221no—
527C23—1221no—
528C23—1221no—
529C23—1221no—
530C23—1221no—
531C23C2×D4×D6111yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—612no—
555C2×C4—612no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—1222no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—4884no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4C22×C22⋊D6111yescharacteristic
581C2×C4C2×Q8×D6111yescharacteristic
582C2×C4#41×D6111yescharacteristic
583C2×C4#41×D6111yescharacteristic
584C2×C4#1451982211yescharacteristic
585C2×C4#1451982511yescharacteristic
586C8—22no—
587C8—41no—
588C8—1222no—
589C8—2441no—
590C22—1222no—
591C22—1222no—
592C22—1222no—
593C22—2441no—
594C22—2441no—
595C22—2441no—
596C22—2441no—
597C22C22⋊Q8×D6111yescharacteristic
598C22D4×C22⋊D6111yescharacteristic
599C22#1684041411yescharacteristic
600C22#137×D6111yescharacteristic
601C22#1684055911yescharacteristic
602C22#1684056411yescharacteristic
603C22#1684057811yescharacteristic
604C4—41no—
605C4—42no—
606C4—42no—
607C4—41no—
608C4—42no—
609C4—81no—
610C4—1222no—
611C4—1222no—
612C4—4881no—
613C4—4881no—
614C4—4881no—
615C4—2442no—
616C4—2441no—
617C4#1684026911yescharacteristic
618C4#1684035811yescharacteristic
619C4Q8×C22⋊D6111yescharacteristic
620C4#1684046611yescharacteristic
621C2—2442no—
622C2#2144962511yescharacteristic
623C2#2145562211yescharacteristic
624C2#2145567311yescharacteristic
625C2#2145735911yescharacteristic
626C2#2145745911yescharacteristic
627C2#2145831111yescharacteristic
628C2#2145839111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#479C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenterC2×#479D61
hypocenterC61#26362
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#479

Complements

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

NH ≅ G/Ntype
4. #21447355C2semidirect
5. #21447307C2semidirect
6. #21447196C2semidirect
7. #21447154C2semidirect
8. #21446574C2semidirect
9. #21446399C2semidirect
12. #21446073C2semidirect
15. C2×C61×#479C2semidirect
307. C2×#479D61semidirect
375. C61#26362semidirect

Maximal subgroups

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

subgroupindex
C2×#168398012
#214495282
#214488202
#214473552
#214473072
#214471962
#214471542
#214465742
#214463992
#214462492
#214461282
#214460732
#214460052
#214460002
C2×C61×#4792
#2636261