Finite Group Databasebeta

← #29687382

Subgroups — #29687382

Order 15616.

844 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840102C211yesmaximal, characteristic
2#21448504C211yesmaximal, characteristic
3#21448173C211yesmaximal, characteristic
4#21446450C211yesmaximal, characteristic, complemented
5#21446412C211yesmaximal, characteristic, complemented
6#21446397C211yesmaximal, characteristic, complemented
7#21446123C211yesmaximal, characteristic
8#21446010C211yesmaximal, characteristic
9#21445912C211yesmaximal, characteristic
10#21445719C211yesmaximal, characteristic, complemented
11#21445707C211yesmaximal, characteristic, complemented
12#21445651C211yesmaximal, characteristic, complemented
13#103×Dic61C211yesmaximal, characteristic, complemented
14#21445584C211yesmaximal, characteristic
15C61×#1927C211yesmaximal, characteristic, complemented
16C2×#14519753C2211yescharacteristic
17C2×Q8×Dic61C2211yescharacteristic
18C2×#14519735C2211yescharacteristic
19C2×#14519734C2211yescharacteristic
20C2×C4⋊C4×D61C2211yescharacteristic
21#16840102C418yescomplemented
22C2×#14519711C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519689C2211yescharacteristic
28C2×#14519689C2211yescharacteristic
29C2×#14519687C2211yescharacteristic
30C2×C8×Dic61C2211yescharacteristic
31#16839660C2211yescharacteristic
32#16839659—21no—
33#16839654—21no—
34#16839651—21no—
35#16839644—21no—
36#16839619C2211yescharacteristic
37C4⋊C4×Dic61C2211yescharacteristic
38C4⋊C4×Dic61C2211yescharacteristic
39C4⋊C4×Dic61C2211yescharacteristic
40C4×#11758196—21no—
41C4×C4⋊Dic61C2211yescharacteristic
42#16839613—21no—
43C4×#11758198—21no—
44#16839609—21no—
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839517C2211yescharacteristic
53C2×#106×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C61×#391C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C22×C4×D61—21no—
61C22×C4×D61C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64#14519753—22no—
65#14519753C2×C414yes—
66Q8×Dic61—22no—
67Q8×Dic61—22no—
68#14519735—22no—
69#14519735C2×C414yes—
70#14519734—22no—
71#14519734—22no—
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×#11758196D411yescharacteristic
81C2×#11758196D411yescharacteristic
82C4⋊C4×D61—22no—
83C4⋊C4×D61—22no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198D411yescharacteristic
88C2×#11758198D411yescharacteristic
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61C2311yescharacteristic
96#14519711C2×C414yescomplemented
97#14519708—22no—
98#14519708C2×C414yescomplemented
99#14519701—22no—
100#14519701C2×C414yescomplemented
101#14519689—22no—
102#14519689C2×C414yes—
103#14519687—22no—
104C8×Dic61—22no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111C22×Q8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C42×C61—21no—
118C2×C42×C61C2311yescharacteristic
119#106×C61—22no—
120#103×C61—22no—
121#103×C61C2×C414yescomplemented
122#95×C61—21no—
123#95×C61—21no—
124#95×C61—21no—
125C23×D61C2×D411yescharacteristic
126C2×Q8×C61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61C22×C412yes—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2411yescharacteristic
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61C2×D411yescharacteristic
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C2×C4×D61—21no—
147C2×C4×D61—21no—
148C2×C4×D61—21no—
149C2×C4×D61—21no—
150C2×C4×D61—24no—
151C2×C4×D61—81no—
152C2×C4×D61—81no—
153C2×C4×D61C22×C412yes—
154C2×C4×D61#4112yes—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61C22×C412yes—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61C22×C412yes—
166C4⋊C4×C61#4112yes—
167C42×C61—21no—
168C42×C61—21no—
169C42×C61—81no—
170#11758198—24no—
171#11758198—24no—
172#11758198—22no—
173#11758198—22no—
174#11758198—81no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—42no—
181C4⋊Dic61C22×C412yes—
182C4⋊Dic61C22×C412yes—
183C4⋊Dic61#4112yes—
184#11758196—24no—
185#11758196—24no—
186#11758196—22no—
187#11758196—22no—
188#11758196—81no—
189C4×Dic61—24no—
190C4×Dic61—42no—
191C4×Dic61—81no—
192C4×Dic61—81no—
193C4×Dic61—81no—
194C4×Dic61—81no—
195C4×Dic61#4112yes—
196C4×Dic61#4112yes—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418C22×C412yes—
203C23×C61C22×D411yescharacteristic
204C22×D61—21no—
205C22×D61—21no—
206C22×D61—21no—
207C22×D61—21no—
208C22×D61—21no—
209C22×D61—21no—
210C22×D61C4×D412yes—
211Q8×C61—22no—
212Q8×C61—22no—
213Q8×C61—41no—
214Q8×C61—41no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—22no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—41no—
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×C61C23×C411yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—22no—
246C2×Dic61—22no—
247C2×Dic61—22no—
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—
257C2×Dic61C4×D412yes—
258C2×Dic61#12312yes—
259C2×Dic61#12312yes—
260C4×D61—22no—
261C4×D61—22no—
262C4×D61—22no—
263C4×D61—22no—
264C4×D61—84no—
265C8×C61—22no—
266C8×C61—41no—
267#92418—22no—
268#92418—41no—
269#21591—611nomaximal, Sylow
270C22×C61C2×C4×D411yescharacteristic
271C22×C61C2×#12311yescharacteristic
272C22×C61#52511yescharacteristic
273C22×C61#57711yescharacteristic
274C22×C61#57711yescharacteristic
275C22×C61#57711yescharacteristic
276C22×C61#57811yescharacteristic
277C2×D61—22no—
278C2×D61—22no—
279C2×D61—22no—
280C2×D61—41no—
281C2×D61—41no—
282C2×D61—41no—
283C2×D61—41no—
284C4×C61—41no—
285C4×C61—41no—
286C4×C61—42no—
287C4×C61—42no—
288C4×C61—42no—
289C4×C61—81no—
290C4×C61C4×#4111yescharacteristic
291C4×C61C4×#4111yescharacteristic
292C4×C61#52911yescharacteristic
293C4×C61#52911yescharacteristic
294Dic61—22no—
295Dic61—22no—
296Dic61—22no—
297Dic61—42no—
298Dic61—42no—
299Dic61—42no—
300Dic61—81no—
301C2×#485—611no—
302#2323—611no—
303#2274—611no—
304#1927—611no—
305#1927D6111yescharacteristic, complemented
306#1921—611no—
307#1919—611no—
308#1872—611no—
309#1853—611no—
310#1825—611no—
311#1776—611no—
312C4×#106—611no—
313#1763—611no—
314C4×#103—611no—
315#1750—611no—
316C2×C61C4×#12311yescharacteristic
317C2×C61#294711yescharacteristic
318C2×C61#295211yescharacteristic
319C2×C61#309711yescharacteristic
320C2×C61#309911yescharacteristic
321C2×C61#312111yescharacteristic
322C2×C61#312311yescharacteristic
323D61—42no—
324C2×C22⋊Q8—611no—
325C2×C4×Q8—611no—
326C2×#117—611no—
327C2×#117—611no—
328C22×C4⋊C4—611no—
329#485—618no—
330C2×#107—611no—
331C2×#106—611no—
332C2×#106—611no—
333C2×#106C2×D6111yescharacteristic
334C2×#103—611no—
335C2×#103—611no—
336C2×#103C2×D6111yescharacteristic
337C2×#103C2×D6111yescharacteristic
338C2×C22⋊C8—611no—
339C2×C22⋊C8—611no—
340C2×#97—611no—
341C2×C4×C8—611no—
342#391C2×D6111yescharacteristic
343#389—611no—
344#389—1221no—
345#388—1221no—
346#388—1221no—
347#387—1221no—
348#380—611no—
349C4×C4⋊C4—611no—
350C4×C4⋊C4—611no—
351C4×C4⋊C4—611no—
352C4×C4⋊C4—611no—
353C4×C4⋊C4—1221no—
354C4×C4⋊C4—1221no—
355C4×C4⋊C4C2×D6111yescharacteristic
356C4×C22⋊C4—1221no—
357C4×C22⋊C4—1221no—
358#340—611no—
359#340—611no—
360#340—611no—
361#340—611no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340C2×D6111yescharacteristic
366#336—611no—
367#336C2×D6111yescharacteristic
368C61#2159111yescharacteristic, complemented, Sylow
369C22×Q8C22×D6111yescharacteristic
370C23×C4—611no—
371C23×C4—1221no—
372C22×C8—611no—
373C22×C8—611no—
374C22×C8C22×D6111yescharacteristic
375C22×C8C22×D6111yescharacteristic
376C22⋊Q8—614no—
377C22⋊Q8—1222no—
378C4×Q8—1222no—
379C4×Q8—1222no—
380#117—614no—
381#117—1222no—
382#117—1222no—
383#117—1222no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—1222no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1222no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4C22×D6111yescharacteristic
397C2×C4⋊C4C22×D6111yescharacteristic
398C2×C4⋊C4C22×D6111yescharacteristic
399C2×C22⋊C4—611no—
400C2×C22⋊C4—611no—
401C2×C22⋊C4—1221no—
402C2×C22⋊C4—1221no—
403C2×C22⋊C4—1221no—
404C2×C42—21no—
405C2×C42—611no—
406C2×C42—1221no—
407C2×C42—1221no—
408C2×C42—1221no—
409C2×C42—1221no—
410C2×C42—1221no—
411C2×C42—1221no—
412C2×C42C22×D6111yescharacteristic
413#107—614no—
414#106—22no—
415#106—614no—
416#106—1222no—
417#103—22no—
418#103—614no—
419#103—1222no—
420#103C4×D6114yescomplemented
421C22⋊C8—614no—
422C22⋊C8—1222no—
423#97—1222no—
424C4×C8—1222no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—1221no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434C24—611no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8C2×C4×D6112yes—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4—611no—
448C22×C4—612no—
449C22×C4—612no—
450C22×C4—1224no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—4881no—
460C22×C4—4881no—
461C22×C4D4×D6111yescharacteristic
462C22×C4D4×D6111yescharacteristic
463C22×C4C23×D6111yescharacteristic
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—612no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8C2×C4×D6112yes—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—612no—
481C4⋊C4—612no—
482C4⋊C4—612no—
483C4⋊C4—1222no—
484C4⋊C4—1222no—
485C4⋊C4—1224no—
486C4⋊C4—1224no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—4881no—
493C4⋊C4—2442no—
494C4⋊C4C2×C4×D6112yes—
495C4⋊C4#1175822412yes—
496C22⋊C4—1222no—
497C22⋊C4—1222no—
498C22⋊C4—1224no—
499C22⋊C4—1224no—
500C22⋊C4—4881no—
501C42—21no—
502C42—21no—
503C42—81no—
504C42—612no—
505C42—612no—
506C42—1224no—
507C42—4881no—
508C42—4881no—
509C42—4881no—
510C42—4881no—
511C42—2442no—
512C23—612no—
513C23—1221no—
514C23—1221no—
515C23—1221no—
516C23—1221no—
517C23—1221no—
518C23—1221no—
519C23C2×D4×D6111yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—41no—
523Q8—41no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—612no—
543C2×C4—612no—
544C2×C4—612no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1222no—
552C2×C4—1222no—
553C2×C4—1222no—
554C2×C4—1222no—
555C2×C4—1222no—
556C2×C4—1222no—
557C2×C4—1222no—
558C2×C4—4884no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4C22×C4×D6111yescharacteristic
569C2×C4#41×D6111yescharacteristic
570C2×C4#41×D6111yescharacteristic
571C2×C4#41×D6111yescharacteristic
572C2×C4#41×D6111yescharacteristic
573C2×C4C2×#1175822411yescharacteristic
574C8—22no—
575C8—41no—
576C8—1222no—
577C8—2441no—
578C22—1222no—
579C22—1222no—
580C22—1222no—
581C22—2441no—
582C22—2441no—
583C22—2441no—
584C22—2441no—
585C22C4×D4×D6111yescharacteristic
586C22#123×D6111yescharacteristic
587C22#1684027211yescharacteristic
588C22#1684056811yescharacteristic
589C22#1684057111yescharacteristic
590C22#1684057111yescharacteristic
591C22#1684057611yescharacteristic
592C4—42no—
593C4—42no—
594C4—41no—
595C4—41no—
596C4—42no—
597C4—81no—
598C4—1222no—
599C4—1222no—
600C4—1222no—
601C4—4881no—
602C4—2442no—
603C4—2442no—
604C4—2442no—
605C4C4×#1175822411yescharacteristic
606C4#1684023011yescharacteristic
607C4#1684029111yescharacteristic
608C4#1684029511yescharacteristic
609C2—2442no—
610C2#2144834611yescharacteristic
611C2#2145465511yescharacteristic
612C2#2145469911yescharacteristic
613C2#2145578311yescharacteristic
614C2#2145580311yescharacteristic
615C2#2145604711yescharacteristic
616C2#2145606911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1927C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1927D61
hypocenterC61#21591
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1927

Complements

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

NH ≅ G/Ntype
4. #21446450C2semidirect
5. #21446412C2semidirect
6. #21446397C2semidirect
10. #21445719C2semidirect
11. #21445707C2semidirect
12. #21445651C2semidirect
13. #103×Dic61C2semidirect
15. C61×#1927C2semidirect
21. #16840102C4semidirect
96. #14519711C2×C4semidirect
98. #14519708C2×C4semidirect
100. #14519701C2×C4semidirect
121. #103×C61C2×C4semidirect
305. #1927D61semidirect
368. C61#21591semidirect
420. #103C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401022
#214485042
#214481732
#214464502
#214464122
#214463972
#214461232
#214460102
#214459122
#214457192
#214457072
#214456512
#103×Dic612
#214455842
C61×#19272
#2159161