Finite Group Databasebeta

← #29742955

Subgroups — #29742955

Order 15616.

804 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840067C211yesmaximal, characteristic, complemented
2C2×#16839804C211yesmaximal, characteristic
3#21449544C211yesmaximal, characteristic
4#21448850C211yesmaximal, characteristic
5#21447365C211yesmaximal, characteristic, complemented
6#21447314C211yesmaximal, characteristic, complemented
7#21447212C211yesmaximal, characteristic, complemented
8#21447160C211yesmaximal, characteristic, complemented
9#21446586C211yesmaximal, characteristic, complemented
10#21446396C211yesmaximal, characteristic, complemented
11#21446127C211yesmaximal, characteristic
12#21446089C211yesmaximal, characteristic, complemented
13#21446016C211yesmaximal, characteristic
14#21446002C211yesmaximal, characteristic
15C61×#1894C211yesmaximal, characteristic
16C2×#14519768C2211yescharacteristic
17C2×Q8×Dic61C2211yescharacteristic
18C2×#14519736C2211yescharacteristic
19C22×C4⋊Dic61C2211yescharacteristic
20#16840067—24no—
21#16839804—24no—
22C2×#14519711C2211yescharacteristic
23C2×#14519711C2211yescharacteristic
24C2×#14519710C2211yescharacteristic
25C2×#14519709C2211yescharacteristic
26C2×#14519709C2211yescharacteristic
27C2×#14519708C2211yescharacteristic
28C2×#14519706C2211yescharacteristic
29C2×#14519706C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519701C2211yescharacteristic
32C2×C22⋊#92418C2211yescharacteristic
33#16839704—21no—
34#16839694—21no—
35#16839669C2211yescharacteristic
36#16839661C2211yescharacteristic
37#16839661C2211yescharacteristic
38#16839658C2211yescharacteristic
39#16839655—21no—
40#16839655—21no—
41#16839648—21no—
42#16839637—21no—
43#16839627C2211yescharacteristic
44#16839620—21no—
45C4⋊C4×Dic61—21no—
46C4⋊C4×Dic61C2211yescharacteristic
47C4×C22⋊Dic61C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C22×M16×C61C2211yescharacteristic
55C2×C22⋊Q8×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C61×#380C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C23×Dic61—21no—
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64#14519768—22no—
65#14519768—41no—
66Q8×Dic61—22no—
67Q8×Dic61—41no—
68#14519736—22no—
69#14519736—41no—
70C2×C4⋊Dic61—22no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—41no—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61D411yescharacteristic
85C2×C22⋊Dic61D411yescharacteristic
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61C2311yescharacteristic
90C2×C4×Dic61C2311yescharacteristic
91#14519711—22no—
92#14519711—22no—
93#14519710—22no—
94#14519709—22no—
95#14519709D414yes—
96#14519708—41no—
97#14519706—22no—
98#14519706—22no—
99#14519701—22no—
100#14519701—41no—
101C22⋊#92418—22no—
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#14519684D411yescharacteristic
111#14519684D411yescharacteristic
112C22×Q8×C61C2311yescharacteristic
113C23×C4×C61C2311yescharacteristic
114C2×M16×C61—22no—
115C2×M16×C61—22no—
116C2×M16×C61—22no—
117C22×C8×C61C2311yescharacteristic
118C22×C8×C61C2311yescharacteristic
119C22⋊Q8×C61—22no—
120C22⋊Q8×C61Q814yes—
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C22⋊C4×C61—21no—
126C2×C22⋊C4×C61—21no—
127#103×C61—22no—
128#103×C61—22no—
129#95×C61—21no—
130C24×C61C2×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×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×Q812yes—
147C22×C4×C61#4112yes—
148C22×C4×C61C2411yescharacteristic
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—42no—
154C22×Dic61—42no—
155C22×Dic61—81no—
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160M16×C61—28no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61C2×D412yes—
166C2×C8×C61#4112yes—
167C4⋊C4×C61—24no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61C2×Q812yes—
174C22⋊C4×C61—24no—
175C22⋊C4×C61—42no—
176C22⋊Dic61—41no—
177C22⋊Dic61—41no—
178C22⋊Dic61—42no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—42no—
186C4⋊Dic61—42no—
187C4⋊Dic61—42no—
188C4⋊Dic61—42no—
189C4⋊Dic61—81no—
190C4⋊Dic61C2×D412yes—
191C4⋊Dic61C2×D412yes—
192#11758196—42no—
193#11758196—81no—
194C4×Dic61—21no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C4×Dic61—42no—
199C4×Dic61—42no—
200C4×Dic61—42no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C23×C61—21no—
208C23×C61—21no—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61—21no—
212C23×C61—21no—
213C23×C61C22⋊Q812yes—
214C23×C61C22×D411yescharacteristic
215Q8×C61—22no—
216Q8×C61—22no—
217Q8×C61—41no—
218Q8×C61—41no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
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×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61C22⋊Q812yes—
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×Q811yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61H(5,𝔽2)11yescharacteristic
243C2×C4×C612−1+411yescharacteristic
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—84no—
269C8×C61—22no—
270C8×C61—22no—
271#92418—41no—
272#92418—41no—
273#26367—611nomaximal, Sylow
274C22×C61—22no—
275C22×C61—22no—
276C22×C61—22no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61C2×C22⋊Q811yescharacteristic
282C22×C61D4211yescharacteristic
283C22×C61#54711yescharacteristic
284C22×C61#57511yescharacteristic
285C22×C61#57511yescharacteristic
286C22×C61#57711yescharacteristic
287C22×C61#57811yescharacteristic
288C4×C61—22no—
289C4×C61—42no—
290C4×C61—42no—
291C4×C61—81no—
292C4×C61—81no—
293C4×C61#52511yescharacteristic
294C4×C61#53911yescharacteristic
295C4×C61D4×Q811yescharacteristic
296C4×C61#55411yescharacteristic
297Dic61—41no—
298Dic61—41no—
299Dic61—41no—
300Dic61—42no—
301Dic61—41no—
302Dic61—42no—
303Dic61—81no—
304Dic61—81no—
305C2×#479—611no—
306C2×#428—611no—
307#2442—611no—
308#2362—611no—
309#2097—611no—
310#2087—611no—
311#2068—611no—
312#2059—611no—
313#1949—611no—
314#1919—611no—
315#1894D6111yescharacteristic
316#1872—611no—
317#1865—611no—
318#1854—611no—
319#1851—611no—
320C2×C61—42no—
321C2×C61#244911yescharacteristic
322C2×C61#308011yescharacteristic
323C2×C61#308211yescharacteristic
324C2×C61#326911yescharacteristic
325C2×C61#327511yescharacteristic
326C2×C61#334911yescharacteristic
327C2×C61#335111yescharacteristic
328C22×M16C2×D6111yescharacteristic
329C2×#125—611no—
330C2×C22⋊Q8C2×D6111yescharacteristic
331C2×C4×Q8—611no—
332C2×#117—611no—
333C22×C4⋊C4—611no—
334#479—1224no—
335#428—1224no—
336C2×#107—611no—
337C2×#107—611no—
338C2×#107—611no—
339C2×#106—611no—
340C2×#106—611no—
341C2×#106—611no—
342C2×C4⋊C8—611no—
343C2×C4⋊C8—611no—
344C2×#103—611no—
345C2×#103—611no—
346C2×#103C2×D6111yescharacteristic
347C2×#103C2×D6111yescharacteristic
348C2×C22⋊C8—611no—
349#400—1221no—
350#398—1221no—
351#391—611no—
352#389—611no—
353#389—611no—
354#389—611no—
355#388—1221no—
356#388—1221no—
357#387—1221no—
358#385—1221no—
359#382—611no—
360#380—1221no—
361#380C2×D6111yescharacteristic
362C4×C4⋊C4—611no—
363C4×C4⋊C4—1221no—
364C4×C22⋊C4—611no—
365#340—611no—
366#340—611no—
367#340—611no—
368#340—611no—
369#340C2×D6111yescharacteristic
370#340C2×D6111yescharacteristic
371#336—611no—
372#336—611no—
373C61#2636711yescharacteristic, complemented, Sylow
374C22×Q8C22×D6111yescharacteristic
375C23×C4—1221no—
376C23×C4C22×D6111yescharacteristic
377C2×M16—22no—
378C2×M16—22no—
379C2×M16—22no—
380C22×C8—611no—
381C22×C8—611no—
382C22×C8C22×D6111yescharacteristic
383C22×C8C22×D6111yescharacteristic
384#125—1222no—
385#125—2441no—
386C22⋊Q8—22no—
387C22⋊Q8Dic12214yes—
388C4×Q8—1222no—
389C4×Q8—2441no—
390#117—1222no—
391#117—2441no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—1222no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—2441no—
406C2×C4⋊C4C22×D6111yescharacteristic
407C2×C4⋊C4C22×D6111yescharacteristic
408C2×C4⋊C4C22×D6111yescharacteristic
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—611no—
412C2×C22⋊C4—611no—
413C2×C22⋊C4—1221no—
414C2×C42—611no—
415C2×C42—611no—
416C2×C42—1221no—
417C2×C42—1221no—
418C2×C42—1221no—
419#107—1222no—
420#107—1222no—
421#107—1222no—
422#106—614no—
423#106—1222no—
424#106—2441no—
425C4⋊C8—1222no—
426C4⋊C8—1222no—
427#103—22no—
428#103—22no—
429#103—1222no—
430#103—2441no—
431C22⋊C8—1222no—
432#95—21no—
433#95—611no—
434#95—611no—
435#95—1221no—
436#95—1221no—
437#95—1221no—
438#95—1221no—
439#95—1221no—
440#95—1221no—
441#95—1221no—
442#95—1221no—
443C24D4×D6111yescharacteristic
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8C2×Dic12212yes—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—4881no—
467C22×C4—2442no—
468C22×C4—2442no—
469C22×C4C2×Dic12212yes—
470C22×C4D4×D6111yescharacteristic
471C22×C4#1175822412yes—
472C22×C4C23×D6111yescharacteristic
473M16—28no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8D4×D6112yes—
485C2×C8#1175822412yes—
486C4⋊C4—21no—
487C4⋊C4—24no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—612no—
493C4⋊C4—612no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—4881no—
501C4⋊C4—4881no—
502C4⋊C4—2442no—
503C4⋊C4—2442no—
504C4⋊C4—2442no—
505C4⋊C4—2442no—
506C4⋊C4—2442no—
507C4⋊C4C2×Dic12212yes—
508C22⋊C4—24no—
509C22⋊C4—42no—
510C22⋊C4—2441no—
511C22⋊C4—2441no—
512C22⋊C4—2442no—
513C42—1221no—
514C42—1221no—
515C42—1221no—
516C42—1221no—
517C42—2442no—
518C42—2442no—
519C42—2442no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23#1451975412yes—
527C23C2×D4×D6111yescharacteristic
528Q8—22no—
529Q8—22no—
530Q8—41no—
531Q8—41no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—4884no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4#1451975412yes—
576C2×C4C2×D4×D6111yescharacteristic
577C2×C4C22×Dic12211yescharacteristic
578C2×C4C2×#1175822411yescharacteristic
579C2×C4C2×#1175822411yescharacteristic
580C2×C4#1451982311yescharacteristic
581C2×C4#1451982411yescharacteristic
582C8—22no—
583C8—22no—
584C8—2441no—
585C8—2441no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22C2×#1451975411yescharacteristic
594C22#1684039611yescharacteristic
595C22#1684041211yescharacteristic
596C22#1684055811yescharacteristic
597C22#1684056111yescharacteristic
598C22#1684057111yescharacteristic
599C22#1684057811yescharacteristic
600C4—22no—
601C4—42no—
602C4—42no—
603C4—81no—
604C4—81no—
605C4—4881no—
606C4—4881no—
607C4—2441no—
608C4—2442no—
609C4—2441no—
610C4—2441no—
611C4—2441no—
612C4—2442no—
613C4#1684026811yescharacteristic
614C4#1684035611yescharacteristic
615C4D4×Dic12211yescharacteristic
616C4#1684046911yescharacteristic
617C2—42no—
618C2#2144963411yescharacteristic
619C2#2145563911yescharacteristic
620C2#2145566311yescharacteristic
621C2#2145738111yescharacteristic
622C2#2145744711yescharacteristic
623C2#2145834411yescharacteristic
624C2#2145837411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1894C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#1894D61
hypocenterC61#26367
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1894

Complements

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

NH ≅ G/Ntype
1. C2×#16840067C2semidirect
5. #21447365C2semidirect
6. #21447314C2semidirect
7. #21447212C2semidirect
8. #21447160C2semidirect
9. #21446586C2semidirect
10. #21446396C2semidirect
12. #21446089C2semidirect
373. C61#26367semidirect

Maximal subgroups

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

subgroupindex
C2×#168400672
C2×#168398042
#214495442
#214488502
#214473652
#214473142
#214472122
#214471602
#214465862
#214463962
#214461272
#214460892
#214460162
#214460022
C61×#18942
#2636761