Finite Group Databasebeta

← #29748141

Subgroups — #29748141

Order 15616.

764 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450196C211yesmaximal, characteristic
2#21448826C211yesmaximal, characteristic
3#21447142C211yesmaximal, characteristic, complemented
4#21447025C211yesmaximal, characteristic
5#21446957C211yesmaximal, characteristic, complemented
6#21446830C211yesmaximal, characteristic
7#21446496C211yesmaximal, characteristic, complemented
8#21446460C211yesmaximal, characteristic, complemented
9#21446424C211yesmaximal, characteristic
10#21446414C211yesmaximal, characteristic, complemented
11#21446161C211yesmaximal, characteristic, complemented
12#21446125C211yesmaximal, characteristic
13#21445908C211yesmaximal, characteristic
14#21445638C211yesmaximal, characteristic, complemented
15C61×#1987C211yesmaximal, characteristic, complemented
16C22×Dic244C2211yescharacteristic
17C2×#14519757C2211yescharacteristic
18C2×#14519757C2211yescharacteristic
19C2×#14519742C2211yescharacteristic
20C2×C4⋊C4×D61C2211yescharacteristic
21C2×#14519710C2211yescharacteristic
22C2×#14519702C2211yescharacteristic
23C2×#14519702C2211yescharacteristic
24C2×#14519700C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×#14519700C2211yescharacteristic
28C2×#14519689C2211yescharacteristic
29C2×#14519689C2211yescharacteristic
30C2×C8×Dic61C2211yescharacteristic
31#16839701—21no—
32#16839673—21no—
33#16839669—21no—
34#16839668C2211yescharacteristic
35#16839668C2211yescharacteristic
36#16839667C2211yescharacteristic
37#16839661C2211yescharacteristic
38#16839660—21no—
39#16839652—21no—
40#16839647—21no—
41#16839636—21no—
42#16839619C2211yescharacteristic
43C4⋊C4×Dic61C2211yescharacteristic
44C4×#11758198—21no—
45C4×#11758198C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839517C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×C4⋊Q8×C61C2211yescharacteristic
53C2×C4⋊C8×C61C2211yescharacteristic
54C4×C4⋊C4×C61C2211yescharacteristic
55C61×#340C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×Dic122C2311yescharacteristic
60C22×Dic122C2311yescharacteristic
61C22×C4×D61—21no—
62C22×C4×D61C2311yescharacteristic
63C2×Dic244—22no—
64C2×Dic244—41no—
65C2×Dic244—41no—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519757—22no—
69#14519757—22no—
70#14519757—22no—
71#14519757—41no—
72#14519742—22no—
73#14519742—41no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C4⋊C4×D61—22no—
82C4⋊C4×D61—41no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198D411yescharacteristic
87C2×#11758198D411yescharacteristic
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61C2311yescharacteristic
91#14519710—22no—
92#14519702—22no—
93#14519702—41no—
94#14519700—22no—
95#14519700—22no—
96#14519700—22no—
97#14519700—41no—
98#14519689—41no—
99#14519689—41no—
100C8×Dic61—41no—
101#14519684—21no—
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×C61—21no—
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C4⋊Q8×C61—22no—
116C4⋊Q8×C61—41no—
117C2×C4⋊C4×C61—21no—
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C42×C61—21no—
123C2×C42×C61C2311yescharacteristic
124C4⋊C8×C61—41no—
125#95×C61—21no—
126C23×D61C2×D411yescharacteristic
127C2×Q8×C61—42no—
128C2×Q8×C61—42no—
129C2×Q8×C61—81no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2411yescharacteristic
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61C2×D411yescharacteristic
142C22×Dic61C2×D411yescharacteristic
143C22×Dic61C2×D411yescharacteristic
144C2×C4×D61—21no—
145C2×C4×D61—21no—
146C2×C4×D61—21no—
147C2×C4×D61—21no—
148C2×C4×D61—21no—
149C2×C4×D61—21no—
150C2×C4×D61—42no—
151C2×C4×D61—42no—
152C2×C4×D61—81no—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122#4112yes—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—42no—
177C4⋊C4×C61#4112yes—
178C4⋊C4×C61#4112yes—
179C4⋊C4×C61#4112yes—
180C42×C61—21no—
181C42×C61—21no—
182C42×C61—81no—
183#11758198—24no—
184#11758198—41no—
185#11758198—42no—
186#11758198—81no—
187#11758198Q1614yescomplemented
188C4⋊Dic61—24no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—42no—
194C4⋊Dic61—81no—
195#11758196—42no—
196#11758196—42no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C4×Dic61—42no—
200C4×Dic61—81no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207Dic244—44no—
208C23×C61C22×D411yescharacteristic
209C22×D61—21no—
210C22×D61—21no—
211C22×D61—21no—
212C22×D61—21no—
213C22×D61—21no—
214C22×D61—21no—
215C22×D61C2×Q1612yes—
216Q8×C61—84no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
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×C61C2×Q1612yes—
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61H(5,𝔽2)11yescharacteristic
240C2×C4×C61H(5,𝔽2)11yescharacteristic
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—22no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
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×Dic61C2×Q1612yes—
262C4×D61—41no—
263C4×D61—41no—
264C4×D61—41no—
265C4×D61—41no—
266C4×D61—84no—
267Dic122—22no—
268Dic122—22no—
269Dic122—41no—
270Dic122—41no—
271Dic122—41no—
272Dic122—41no—
273Dic122—41no—
274Dic122—41no—
275C8×C61—41no—
276C8×C61—41no—
277#92418—41no—
278#92418—41no—
279#26726—611nomaximal, Sylow
280C22×C61#52511yescharacteristic
281C22×C61#54611yescharacteristic
282C22×C61#54611yescharacteristic
283C22×C61C22×Q1611yescharacteristic
284C22×C61#57511yescharacteristic
285C22×C61#57711yescharacteristic
286C22×C61#57711yescharacteristic
287C2×D61—22no—
288C2×D61—22no—
289C2×D61—22no—
290C2×D61—41no—
291C2×D61—41no—
292C2×D61—41no—
293C2×D61—41no—
294C4×C61—22no—
295C4×C61—42no—
296C4×C61—42no—
297C4×C61—41no—
298C4×C61—42no—
299C4×C61—81no—
300C4×C61#52911yescharacteristic
301C4×C61#53811yescharacteristic
302C4×C61#55511yescharacteristic
303C4×C61#55911yescharacteristic
304Dic61—22no—
305Dic61—42no—
306Dic61—41no—
307Dic61—41no—
308Dic61—81no—
309Dic61—81no—
310Dic61—81no—
311#2492—611no—
312#2360—611no—
313#2056—611no—
314#2036—611no—
315#2024—611no—
316#2001—611no—
317#1987D6111yescharacteristic, complemented
318#1933—611no—
319#1928—611no—
320#1923—611no—
321#1921—611no—
322#1878—611no—
323#1872—611no—
324#1824—611no—
325C4×#103—611no—
326C2×C61#245911yescharacteristic
327C2×C61#310411yescharacteristic
328C2×C61#312511yescharacteristic
329C2×C61#329911yescharacteristic
330C2×C61#329911yescharacteristic
331C2×C61#331111yescharacteristic
332C2×C61#332311yescharacteristic
333D61—42no—
334C22×Q16—611no—
335C2×C4⋊Q8C2×D6111yescharacteristic
336C2×C22⋊Q8—611no—
337C2×C22⋊Q8—611no—
338C2×C4×Q8—611no—
339C22×C4⋊C4—611no—
340C2×#107—611no—
341C2×C4⋊C8C2×D6111yescharacteristic
342C2×#103—611no—
343C2×#103—611no—
344C2×#103—611no—
345C2×#103—611no—
346C2×#103—611no—
347C2×#103—611no—
348C2×C22⋊C8—611no—
349C2×C22⋊C8—611no—
350C2×C4×C8—611no—
351#400—1221no—
352#393—1221no—
353#391—611no—
354#391—611no—
355#391—611no—
356#391—1221no—
357#389—611no—
358#389—1221no—
359#388—1221no—
360#387—1221no—
361#385—1221no—
362#380—611no—
363C4×C4⋊C4—611no—
364C4×C4⋊C4C2×D6111yescharacteristic
365C4×C22⋊C4—611no—
366C4×C22⋊C4—1221no—
367#340—611no—
368#340—611no—
369#340—611no—
370#340—611no—
371#340C2×D6111yescharacteristic
372#340C2×D6111yescharacteristic
373#340C2×D6111yescharacteristic
374#340C2×D6111yescharacteristic
375#336—611no—
376#336—611no—
377C61#2672611yescharacteristic, complemented, Sylow
378C22×Q8—21no—
379C22×Q8—611no—
380C22×Q8—611no—
381C23×C4—611no—
382C23×C4—1221no—
383C2×Q16—1222no—
384C2×Q16—2441no—
385C2×Q16—2441no—
386C22×C8—611no—
387C22×C8—611no—
388C22×C8C22×D6111yescharacteristic
389C22×C8C22×D6111yescharacteristic
390C4⋊Q8—22no—
391C4⋊Q8—41no—
392C22⋊Q8—1222no—
393C22⋊Q8—1222no—
394C22⋊Q8—1222no—
395C22⋊Q8—2441no—
396C4×Q8—1222no—
397C4×Q8—2441no—
398C2×C4⋊C4—21no—
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—2441no—
408C2×C4⋊C4C22×D6111yescharacteristic
409C2×C4⋊C4C22×D6111yescharacteristic
410C2×C4⋊C4C22×D6111yescharacteristic
411C2×C4⋊C4C22×D6111yescharacteristic
412C2×C22⋊C4—611no—
413C2×C22⋊C4—611no—
414C2×C22⋊C4—1221no—
415C2×C22⋊C4—1221no—
416C2×C22⋊C4—1221no—
417C2×C42—21no—
418C2×C42—611no—
419C2×C42—1221no—
420C2×C42—1221no—
421C2×C42C22×D6111yescharacteristic
422#107—1222no—
423C4⋊C8—41no—
424#103—1222no—
425#103—1222no—
426#103—1222no—
427#103—1222no—
428#103—2441no—
429#103—2441no—
430C22⋊C8—2441no—
431C22⋊C8—2441no—
432C4×C8—2441no—
433#95—21no—
434#95—611no—
435#95—611no—
436#95—1221no—
437#95—1221no—
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—42no—
447C2×Q8—42no—
448C2×Q8—81no—
449C2×Q8—612no—
450C2×Q8—1221no—
451C2×Q8—1221no—
452C2×Q8—1221no—
453C2×Q8—1221no—
454C2×Q8—1221no—
455C2×Q8—1221no—
456C2×Q8—1221no—
457C2×Q8—1221no—
458C2×Q8—1221no—
459C2×Q8—1221no—
460C2×Q8—1221no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—4881no—
479C22×C4—2442no—
480C22×C4—2442no—
481C22×C4D4×D6111yescharacteristic
482C22×C4D4×D6111yescharacteristic
483C22×C4C23×D6111yescharacteristic
484Q16—2444no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8—1221no—
496C2×C8—1221no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—42no—
503C4⋊C4—1224no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—4881no—
509C4⋊C4—2442no—
510C4⋊C4—2442no—
511C4⋊C4—2442no—
512C4⋊C4#1175822212yes—
513C4⋊C4#1175822612yes—
514C4⋊C4#1175822612yes—
515C22⋊C4—614no—
516C22⋊C4—1224no—
517C22⋊C4—4881no—
518C22⋊C4—2441no—
519C22⋊C4—2442no—
520C42—21no—
521C42—21no—
522C42—81no—
523C42—1221no—
524C42—1221no—
525C42—4881no—
526C42—2442no—
527C23—612no—
528C23—1221no—
529C23—1221no—
530C23—1221no—
531C23—1221no—
532C23—1221no—
533C23—1221no—
534C23C2×D4×D6111yescharacteristic
535Q8—84no—
536Q8—1222no—
537Q8—1222no—
538Q8—2441no—
539Q8—2441no—
540Q8—2441no—
541Q8—2441no—
542Q8—2441no—
543Q8—2441no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—612no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1222no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—4884no—
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×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4Q16×D6112yes—
588C2×C4#41×D6111yescharacteristic
589C2×C4C2×#1175822211yescharacteristic
590C2×C4C2×#1175822611yescharacteristic
591C2×C4C2×#1175822611yescharacteristic
592C2×C4#1451982211yescharacteristic
593C2×C4#1451982311yescharacteristic
594C8—41no—
595C8—41no—
596C8—2441no—
597C8—2441no—
598C22—1222no—
599C22—1222no—
600C22—1222no—
601C22—2441no—
602C22—2441no—
603C22—2441no—
604C22—2441no—
605C22#1684027511yescharacteristic
606C22#1684040211yescharacteristic
607C22#1684040311yescharacteristic
608C22C2×Q16×D6111yescharacteristic
609C22#1684055811yescharacteristic
610C22#1684056811yescharacteristic
611C22#1684057211yescharacteristic
612C4—22no—
613C4—41no—
614C4—42no—
615C4—42no—
616C4—42no—
617C4—81no—
618C4—1222no—
619C4—4881no—
620C4—4881no—
621C4—4881no—
622C4—2441no—
623C4—2442no—
624C4—2441no—
625C4#1684029911yescharacteristic
626C4#1684034611yescharacteristic
627C4#1684047811yescharacteristic
628C4#1684049711yescharacteristic
629C2—2442no—
630C2#2144976311yescharacteristic
631C2#2145586011yescharacteristic
632C2#2145610111yescharacteristic
633C2#2145769111yescharacteristic
634C2#2145769211yescharacteristic
635C2#2145785211yescharacteristic
636C2#2145803111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1987C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#1987D61
hypocenterC61#26726
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1987

Complements

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

NH ≅ G/Ntype
3. #21447142C2semidirect
5. #21446957C2semidirect
7. #21446496C2semidirect
8. #21446460C2semidirect
10. #21446414C2semidirect
11. #21446161C2semidirect
14. #21445638C2semidirect
15. C61×#1987C2semidirect
187. #11758198Q16semidirect
317. #1987D61semidirect
377. C61#26726semidirect

Maximal subgroups

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

subgroupindex
#214501962
#214488262
#214471422
#214470252
#214469572
#214468302
#214464962
#214464602
#214464242
#214464142
#214461612
#214461252
#214459082
#214456382
C61×#19872
#2672661