Finite Group Databasebeta

← #29748130

Subgroups — #29748130

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#21448822C211yesmaximal, characteristic
2#21447141C211yesmaximal, characteristic, complemented
3#21447022C211yesmaximal, characteristic
4#21446957C211yesmaximal, characteristic, complemented
5#21446827C211yesmaximal, characteristic
6#21446768C211yesmaximal, characteristic, complemented
7#21446498C211yesmaximal, characteristic, complemented
8#21446457C211yesmaximal, characteristic, complemented
9#21446427C211yesmaximal, characteristic
10#21446411C211yesmaximal, characteristic, complemented
11#21446163C211yesmaximal, characteristic, complemented
12#21446127C211yesmaximal, characteristic
13#21445907C211yesmaximal, characteristic
14C4×#14519700C211yesmaximal, characteristic, complemented
15C61×#2492C211yesmaximal, characteristic
16C22×#11758202C2211yescharacteristic
17C2×#14519756C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19C22×C4⋊Dic61C2211yescharacteristic
20C2×#14519710C2211yescharacteristic
21C2×C4⋊#92418C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×#14519700C2211yescharacteristic
28C2×C22⋊#92418C2211yescharacteristic
29C2×C22⋊#92418C2211yescharacteristic
30C2×C4×#92418C2211yescharacteristic
31#16839668C2211yescharacteristic
32#16839660C2211yescharacteristic
33#16839659—21no—
34#16839649—21no—
35#16839634—21no—
36C4×C4⋊Dic61C2211yescharacteristic
37C22⋊C4×Dic61—21no—
38#16839524C2211yescharacteristic
39#16839524C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48C2×C4⋊Q8×C61C2211yescharacteristic
49C2×C22⋊Q8×C61C2211yescharacteristic
50C61×#400—21no—
51C61×#393—21no—
52C61×#391—21no—
53C61×#391C2211yescharacteristic
54C61×#391C2211yescharacteristic
55C61×#388—21no—
56C61×#380C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C4×C22⋊C4×C61C2211yescharacteristic
59C22×Dic122C2311yescharacteristic
60C23×Dic61—21no—
61C2×#11758202—22no—
62C2×#11758202—41no—
63C2×#11758202—41no—
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519756—22no—
69#14519756—22no—
70C4×Dic122—22no—
71C4×Dic122—41no—
72C2×C4⋊Dic61—22no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—41no—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×C22⋊Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84#14519710—22no—
85C4⋊#92418—41no—
86#14519701—22no—
87#14519701—22no—
88#14519701—41no—
89#14519700—22no—
90#14519700—22no—
91#14519700—41no—
92C22⋊#92418—41no—
93C22⋊#92418—41no—
94C4×#92418—41no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99C22×Q8×C61—21no—
100C22×Q8×C61C2311yescharacteristic
101C23×C4×C61C2311yescharacteristic
102C4⋊Q8×C61—22no—
103C4⋊Q8×C61—41no—
104C22⋊Q8×C61—22no—
105C22⋊Q8×C61—41no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61D411yescharacteristic
114C2×C22⋊C4×C61D411yescharacteristic
115C2×C42×C61—21no—
116C2×C42×C61C2311yescharacteristic
117C2×C42×C61C2311yescharacteristic
118#95×C61—21no—
119#95×C61—21no—
120#95×C61—21no—
121#95×C61—21no—
122#95×C61—21no—
123#95×C61—21no—
124#95×C61D411yescharacteristic
125#95×C61D411yescharacteristic
126C24×C61C2×D411yescharacteristic
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—42no—
134C2×Q8×C61—42no—
135C2×Q8×C61—81no—
136C22×C4×C61—21no—
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×C61—21no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—42no—
157C22×Dic61—42no—
158C22×Dic61—81no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122#4112yes—
165C4⋊C4×C61—21no—
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#4112yes—
173C22⋊C4×C61—41no—
174C22⋊C4×C61—42no—
175C22⋊C4×C61—81no—
176C22⋊C4×C61Q1614yescomplemented
177C42×C61—21no—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61—21no—
181C42×C61—81no—
182C22⋊Dic61—24no—
183#11758202—44no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—42no—
189C4⋊Dic61#4112yes—
190C4⋊Dic61#4112yes—
191#11758196—24no—
192#11758196—42no—
193#11758196—81no—
194C4×Dic61—42no—
195C4×Dic61—81no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C23×C61—21no—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61—21no—
212C23×C61—21no—
213C23×C61—21no—
214C23×C61C2×Q1612yes—
215C23×C61C22×D411yescharacteristic
216Q8×C61—41no—
217Q8×C61—41no—
218Q8×C61—41no—
219Q8×C61—41no—
220Q8×C61—84no—
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×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—22no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61C2×Q1612yes—
251C2×C4×C61C2×Q1612yes—
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61H(5,𝔽2)11yescharacteristic
257C2×C4×C61H(5,𝔽2)11yescharacteristic
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—84no—
271Dic122—22no—
272Dic122—22no—
273Dic122—41no—
274Dic122—41no—
275#92418—41no—
276#92418—41no—
277#92418—41no—
278#92418—41no—
279#26726—611nomaximal, Sylow
280C22×C61—22no—
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61#52511yescharacteristic
288C22×C61#54611yescharacteristic
289C22×C61#54611yescharacteristic
290C22×C61C22×Q1611yescharacteristic
291C22×C61#57511yescharacteristic
292C22×C61#57711yescharacteristic
293C22×C61#57711yescharacteristic
294C4×C61—22no—
295C4×C61—22no—
296C4×C61—41no—
297C4×C61—41no—
298C4×C61—41no—
299C4×C61—42no—
300C4×C61—81no—
301C4×C61—81no—
302C4×C61—81no—
303C4×C61#52911yescharacteristic
304C4×C61#53811yescharacteristic
305C4×C61#55511yescharacteristic
306C4×C61#55911yescharacteristic
307Dic61—42no—
308Dic61—42no—
309Dic61—42no—
310Dic61—81no—
311#2492D6111yescharacteristic
312#2360—611no—
313#2056—611no—
314#2036—611no—
315#2024—611no—
316#2001—611no—
317#1987—611no—
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—42no—
327C2×C61#245911yescharacteristic
328C2×C61#310411yescharacteristic
329C2×C61#312511yescharacteristic
330C2×C61#329911yescharacteristic
331C2×C61#329911yescharacteristic
332C2×C61#331111yescharacteristic
333C2×C61#332311yescharacteristic
334C22×Q16—611no—
335C2×C4⋊Q8C2×D6111yescharacteristic
336C2×C22⋊Q8—611no—
337C2×C22⋊Q8C2×D6111yescharacteristic
338C2×C4×Q8—611no—
339C22×C4⋊C4—611no—
340C2×#107—611no—
341C2×C4⋊C8—611no—
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—21no—
352#393—21no—
353#391—21no—
354#391—611no—
355#391C2×D6111yescharacteristic
356#391C2×D6111yescharacteristic
357#389—611no—
358#389—1221no—
359#388—21no—
360#387—1221no—
361#385—1221no—
362#380C2×D6111yescharacteristic
363C4×C4⋊C4—611no—
364C4×C4⋊C4C2×D6111yescharacteristic
365C4×C22⋊C4—1221no—
366C4×C22⋊C4C2×D6111yescharacteristic
367#340—611no—
368#340—611no—
369#340—611no—
370#340—611no—
371#340—611no—
372#340—611no—
373#340—611no—
374#340—611no—
375#336—611no—
376#336—611no—
377C61#2672611yescharacteristic, complemented, Sylow
378C22×Q8—21no—
379C22×Q8—611no—
380C22×Q8C22×D6111yescharacteristic
381C23×C4—1221no—
382C23×C4C22×D6111yescharacteristic
383C2×Q16—1222no—
384C2×Q16—2441no—
385C2×Q16—2441no—
386C22×C8—611no—
387C22×C8—611no—
388C22×C8—611no—
389C22×C8—611no—
390C4⋊Q8—22no—
391C4⋊Q8—41no—
392C22⋊Q8—22no—
393C22⋊Q8—41no—
394C22⋊Q8—1222no—
395C22⋊Q8—1222no—
396C4×Q8—1222no—
397C4×Q8—2441no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—1222no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—2441no—
409C2×C4⋊C4C22×D6111yescharacteristic
410C2×C4⋊C4C22×D6111yescharacteristic
411C2×C4⋊C4C22×D6111yescharacteristic
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—1221no—
415C2×C22⋊C4C22⋊D6111yescharacteristic
416C2×C22⋊C4C22⋊D6111yescharacteristic
417C2×C42—21no—
418C2×C42—1221no—
419C2×C42—1221no—
420C2×C42C22×D6111yescharacteristic
421C2×C42C22×D6111yescharacteristic
422#107—1222no—
423C4⋊C8—2441no—
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—21no—
435#95—21no—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—1221no—
440#95—1221no—
441#95—1221no—
442#95—1221no—
443#95C22⋊D6111yescharacteristic
444#95C22⋊D6111yescharacteristic
445C24C2×C22⋊D6111yescharacteristic
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—81no—
455C2×Q8—612no—
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—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4—1221no—
475C22×C4—4881no—
476C22×C4—2442no—
477C22×C4—2442no—
478C22×C4C2×C22⋊D6111yescharacteristic
479C22×C4C2×C22⋊D6111yescharacteristic
480C22×C4C2×C22⋊D6111yescharacteristic
481C22×C4C2×C22⋊D6111yescharacteristic
482C22×C4C2×C22⋊D6111yescharacteristic
483C22×C4C23×D6111yescharacteristic
484Q16—2444no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—1221no—
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—42no—
504C4⋊C4—612no—
505C4⋊C4—612no—
506C4⋊C4—1224no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—4881no—
512C4⋊C4—2442no—
513C4⋊C4—2442no—
514C4⋊C4#1175822412yes—
515C22⋊C4—41no—
516C22⋊C4—42no—
517C22⋊C4—81no—
518C22⋊C4—1224no—
519C22⋊C4#1175820214yescomplemented
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—81no—
525C42—4881no—
526C42—2442no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23C2×#1175820212yes—
534C23C22×C22⋊D6111yescharacteristic
535Q8—41no—
536Q8—41no—
537Q8—41no—
538Q8—41no—
539Q8—84no—
540Q8—1222no—
541Q8—1222no—
542Q8—2441no—
543Q8—2441no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—1222no—
576C2×C4—4884no—
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×C4C2×#1175820212yes—
587C2×C4C2×#1175820212yes—
588C2×C4#41×D6111yescharacteristic
589C2×C4#41×D6111yescharacteristic
590C2×C4#41×D6111yescharacteristic
591C2×C4C2×#1175822411yescharacteristic
592C2×C4#1451982211yescharacteristic
593C2×C4#1451982311yescharacteristic
594C8—2441no—
595C8—2441no—
596C8—2441no—
597C8—2441no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22#1684026911yescharacteristic
606C22#1684040011yescharacteristic
607C22#1684040611yescharacteristic
608C22C22×#1175820211yescharacteristic
609C22#1684056011yescharacteristic
610C22#1684057011yescharacteristic
611C22#1684057311yescharacteristic
612C4—22no—
613C4—22no—
614C4—41no—
615C4—41no—
616C4—42no—
617C4—41no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4—4881no—
622C4—2442no—
623C4—2442no—
624C4—2442no—
625C4#1684029111yescharacteristic
626C4#1684034611yescharacteristic
627C4#1684047411yescharacteristic
628C4#1684049411yescharacteristic
629C2—42no—
630C2#2144975211yescharacteristic
631C2#2145585411yescharacteristic
632C2#2145609011yescharacteristic
633C2#2145768911yescharacteristic
634C2#2145769511yescharacteristic
635C2#2145784911yescharacteristic
636C2#2145803511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2492C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#2492D61
hypocenterC61#26726
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2492

Complements

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

NH ≅ G/Ntype
2. #21447141C2semidirect
4. #21446957C2semidirect
6. #21446768C2semidirect
7. #21446498C2semidirect
8. #21446457C2semidirect
10. #21446411C2semidirect
11. #21446163C2semidirect
14. C4×#14519700C2semidirect
176. C22⋊C4×C61Q16semidirect
377. C61#26726semidirect
519. C22⋊C4#11758202semidirect

Maximal subgroups

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

subgroupindex
#214488222
#214471412
#214470222
#214469572
#214468272
#214467682
#214464982
#214464572
#214464272
#214464112
#214461632
#214461272
#214459072
C4×#145197002
C61×#24922
#2672661