Finite Group Databasebeta

← #29699370

Subgroups — #29699370

Order 15616.

866 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×#16839862C211yesmaximal, characteristic
2#21448774C211yesmaximal, characteristic, complemented
3#21446508C211yesmaximal, characteristic, complemented
4#21446494C211yesmaximal, characteristic, complemented
5#21446408C211yesmaximal, characteristic, complemented
6#21446387C211yesmaximal, characteristic, complemented
7#21446153C211yesmaximal, characteristic
8#21446153C211yesmaximal, characteristic
9#21445952C211yesmaximal, characteristic, complemented
10#21445831C211yesmaximal, characteristic, complemented
11#21445819C211yesmaximal, characteristic
12#21445694C211yesmaximal, characteristic, complemented
13#21445644C211yesmaximal, characteristic
14C4×#14519698C211yesmaximal, characteristic
15C61×#2349C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18#16839862C418yescomplemented
19C2×#14519710C2211yescharacteristic
20C2×C4⋊#92418C2211yescharacteristic
21C2×#14519700C2211yescharacteristic
22C2×#14519700C2211yescharacteristic
23C2×#14519700C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519688C2211yescharacteristic
28C2×#14519688C2211yescharacteristic
29C2×C4×#92418C2211yescharacteristic
30#16839668C2211yescharacteristic
31#16839668C2211yescharacteristic
32#16839649—21no—
33#16839649—21no—
34#16839632C2211yescharacteristic
35#16839627—21no—
36C4⋊C4×Dic61—21no—
37C4×C4⋊Dic61C2211yescharacteristic
38C4×C4⋊Dic61C2211yescharacteristic
39#16839524C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839517C2211yescharacteristic
48#16839517C2211yescharacteristic
49C2×C4×D4×C61C2211yescharacteristic
50C2×C4×D4×C61C2211yescharacteristic
51C2×C4×D4×C61C2211yescharacteristic
52C61×#389C2211yescharacteristic
53C61×#389C2211yescharacteristic
54C61×#385—21no—
55C61×#385—21no—
56C61×#382C2211yescharacteristic
57C61×#380—21no—
58C4×C4⋊C4×C61C2211yescharacteristic
59C2×#95×C61—21no—
60C22×Dic122C2311yescharacteristic
61C2×#11758200—22no—
62C2×#11758200—22no—
63C2×#11758200C2×C414yes—
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68C4×Dic122—22no—
69C4×Dic122C2×C414yescomplemented
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×C4×Dic61—21no—
77C2×C4×Dic61—21no—
78C2×C4×Dic61—21no—
79#14519710C2×C414yescomplemented
80C4⋊#92418—22no—
81#14519700—41no—
82#14519700—41no—
83#14519700C2×C414yescomplemented
84#14519698—22no—
85#14519698—22no—
86#14519698C2×C414yes—
87#14519688—41no—
88#14519688C2×C414yescomplemented
89C4×#92418—41no—
90#14519684—21no—
91#14519684—21no—
92#14519684—21no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96C22×D4×C61C2311yescharacteristic
97C23×C4×C61—21no—
98C23×C4×C61—21no—
99C23×C4×C61—21no—
100C4×D4×C61—22no—
101C4×D4×C61—22no—
102C4×D4×C61—22no—
103C4×D4×C61—41no—
104C4×D4×C61—41no—
105C4×D4×C61C2×C414yes—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61D411yescharacteristic
109C2×C4⋊C4×C61D411yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61—21no—
115C2×C22⋊C4×C61—21no—
116C2×C42×C61C2311yescharacteristic
117C2×C42×C61C2311yescharacteristic
118C2×C42×C61C2311yescharacteristic
119#95×C61—21no—
120#95×C61—21no—
121#95×C61—21no—
122#95×C61—21no—
123#95×C61D411yescharacteristic
124#95×C61D411yescharacteristic
125C24×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61C22×C412yes—
130C2×D4×C61#4112yes—
131C2×D4×C61#4112yes—
132C22×C4×C61—21no—
133C22×C4×C61—24no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—42no—
137C22×C4×C61—42no—
138C22×C4×C61—42no—
139C22×C4×C61—42no—
140C22×C4×C61—81no—
141C22×C4×C61—81no—
142C22×C4×C61—81no—
143C22×C4×C61—81no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122C22×C412yes—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—41no—
168C4⋊C4×C61—81no—
169C4⋊C4×C61—81no—
170C4⋊C4×C61C22×C412yes—
171C22⋊C4×C61—24no—
172C22⋊C4×C61—42no—
173C22⋊C4×C61—42no—
174C42×C61—21no—
175C42×C61—21no—
176C42×C61—21no—
177C42×C61—21no—
178C42×C61C22×C412yes—
179C42×C61#4112yes—
180#11758200—28no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—42no—
187C4⋊Dic61—81no—
188C4⋊Dic61C22×C412yes—
189#11758196—24no—
190C4×Dic61—24no—
191C4×Dic61—81no—
192C4×Dic61—81no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418C22×C412yes—
204C2×#92418C22×C412yes—
205C23×C61—22no—
206C23×C61—22no—
207C23×C61—22no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—41no—
212C23×C61C22×D411yescharacteristic
213D4×C61—22no—
214D4×C61—22no—
215D4×C61—22no—
216D4×C61#11714yes—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
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—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
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×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—84no—
242C2×C4×C61—84no—
243C2×C4×C61—84no—
244C2×C4×C61C4×D412yes—
245C2×C4×C61C4×D412yes—
246C2×C4×C61#13712yes—
247C2×C4×C61C23×C411yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61H(5,𝔽2)11yescharacteristic
252C2×C4×C612−1+411yescharacteristic
253C2×Dic61—22no—
254C2×Dic61—22no—
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×Dic61—41no—
265Dic122—22no—
266Dic122—22no—
267Dic122—41no—
268Dic122—41no—
269#92418—22no—
270#92418—22no—
271#92418—41no—
272#92418—41no—
273#22776—611nomaximal, Sylow
274C22×C61—24no—
275C22×C61—42no—
276C22×C61—42no—
277C22×C61—42no—
278C22×C61—42no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61C2×C4×D411yescharacteristic
282C22×C61#54611yescharacteristic
283C22×C61#54811yescharacteristic
284C22×C61C2×#13711yescharacteristic
285C22×C61#57511yescharacteristic
286C22×C61#57611yescharacteristic
287C22×C61#57711yescharacteristic
288C4×C61—22no—
289C4×C61—22no—
290C4×C61—22no—
291C4×C61—42no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61—81no—
296C4×C61—81no—
297C4×C61C2×#11711yescharacteristic
298C4×C61#52011yescharacteristic
299C4×C61#55211yescharacteristic
300C4×C61#55311yescharacteristic
301Dic61—42no—
302Dic61—42no—
303Dic61—81no—
304Dic61—81no—
305C2×#440—611no—
306#2355—611no—
307#2349D6111yescharacteristic
308#1935—611no—
309#1933—611no—
310#1921—611no—
311#1918—611no—
312#1877—611no—
313#1877—611no—
314#1836—611no—
315#1805—611no—
316#1802—611no—
317#1771—611no—
318#1762—611no—
319C4×#102—611no—
320C2×C61—44no—
321C2×C61#235211yescharacteristic
322C2×C61C4×#13711yescharacteristic
323C2×C61#294611yescharacteristic
324C2×C61#331111yescharacteristic
325C2×C61#331411yescharacteristic
326C2×C61#337011yescharacteristic
327C2×C61#337211yescharacteristic
328C22×SD16—611no—
329C2×C4×Q8—611no—
330C2×C4×D4C2×D6111yescharacteristic
331C2×C4×D4C2×D6111yescharacteristic
332C2×C4×D4C2×D6111yescharacteristic
333#440—618no—
334C2×#107—611no—
335C2×C4⋊C8—611no—
336C2×#103—611no—
337C2×#103—611no—
338C2×#103—611no—
339C2×#102—611no—
340C2×#102—611no—
341C2×#102—611no—
342C2×#97—611no—
343C2×#97—611no—
344C2×C4×C8—611no—
345#391—611no—
346#391—611no—
347#389C2×D6111yescharacteristic
348#389C2×D6111yescharacteristic
349#387—1221no—
350#387—1221no—
351#385—21no—
352#385—21no—
353#384—611no—
354#382—1221no—
355#382C2×D6111yescharacteristic
356#380—21no—
357C4×C4⋊C4—611no—
358C4×C4⋊C4—611no—
359C4×C4⋊C4—1221no—
360C4×C4⋊C4C2×D6111yescharacteristic
361C2×#95—21no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367#340—611no—
368#340—611no—
369#340—611no—
370#336—611no—
371#336—611no—
372C61#2277611yescharacteristic, complemented, Sylow
373C22×Q8—611no—
374C22×D4C22×D6111yescharacteristic
375C23×C4—21no—
376C23×C4—21no—
377C23×C4—21no—
378C2×SD16—614no—
379C2×SD16—1222no—
380C2×SD16—1222no—
381C22×C8—611no—
382C22×C8—611no—
383C22×C8—611no—
384C22×C8—611no—
385C4×Q8—614no—
386C4×Q8—1222no—
387C4×D4—22no—
388C4×D4—22no—
389C4×D4—22no—
390C4×D4—41no—
391C4×D4—41no—
392C4×D4C4×D6114yes—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4C22⋊D6111yescharacteristic
402C2×C4⋊C4C22⋊D6111yescharacteristic
403C2×C4⋊C4C22×D6111yescharacteristic
404C2×C4⋊C4C22×D6111yescharacteristic
405C2×C4⋊C4C22×D6111yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C42—1221no—
410C2×C42—1221no—
411C2×C42—1221no—
412C2×C42C22×D6111yescharacteristic
413C2×C42C22×D6111yescharacteristic
414C2×C42C22×D6111yescharacteristic
415#107—614no—
416C4⋊C8—1222no—
417#103—614no—
418#103—2441no—
419#103—2441no—
420#102—614no—
421#102—1222no—
422#102—1222no—
423#97—614no—
424#97—2441no—
425C4×C8—2441no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436#95C22⋊D6111yescharacteristic
437#95C22⋊D6111yescharacteristic
438C24—21no—
439C2×Q8—612no—
440C2×Q8—1221no—
441C2×Q8—1221no—
442C2×Q8—1221no—
443C2×Q8—1221no—
444C2×Q8—1221no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4C2×C4×D6112yes—
449C2×D4#1175822412yes—
450C2×D4#1175822612yes—
451C22×C4—21no—
452C22×C4—24no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—81no—
460C22×C4—81no—
461C22×C4—81no—
462C22×C4—81no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4C2×C22⋊D6111yescharacteristic
468C22×C4C2×C22⋊D6111yescharacteristic
469C22×C4C2×C22⋊D6111yescharacteristic
470C22×C4C2×C22⋊D6111yescharacteristic
471C22×C4C2×C22⋊D6111yescharacteristic
472C22×C4C2×C22⋊D6111yescharacteristic
473C22×C4C23×D6111yescharacteristic
474SD16—1228no—
475C2×C8—612no—
476C2×C8—612no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—41no—
494C4⋊C4—81no—
495C4⋊C4—81no—
496C4⋊C4—612no—
497C4⋊C4—1221no—
498C4⋊C4—1224no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—4881no—
504C4⋊C4—2442no—
505C4⋊C4C2×C4×D6112yes—
506C22⋊C4—24no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—1224no—
514C42—4881no—
515C42—4881no—
516C42C2×C4×D6112yes—
517C42#1175822212yes—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23C22×C22⋊D6111yescharacteristic
526Q8—1222no—
527Q8—1222no—
528Q8—2441no—
529Q8—2441no—
530D4—22no—
531D4—22no—
532D4—22no—
533D4#1451973514yes—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—84no—
559C2×C4—84no—
560C2×C4—84no—
561C2×C4—1222no—
562C2×C4—1222no—
563C2×C4—2441no—
564C2×C4—2441no—
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×C4C4×C22⋊D6112yes—
574C2×C4C4×C22⋊D6112yes—
575C2×C4#1451980712yes—
576C2×C4C22×C4×D6111yescharacteristic
577C2×C4C2×#1175822411yescharacteristic
578C2×C4C2×#1175822211yescharacteristic
579C2×C4C2×#1175822611yescharacteristic
580C2×C4#1451982211yescharacteristic
581C2×C4#1451982511yescharacteristic
582C8—1222no—
583C8—1222no—
584C8—2441no—
585C8—2441no—
586C22—24no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22C2×C4×C22⋊D6111yescharacteristic
594C22#1684040611yescharacteristic
595C22#1684042311yescharacteristic
596C22C2×#1451980711yescharacteristic
597C22#1684056011yescharacteristic
598C22#1684056511yescharacteristic
599C22#1684057311yescharacteristic
600C4—22no—
601C4—22no—
602C4—22no—
603C4—42no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—81no—
608C4—81no—
609C4—4881no—
610C4—4881no—
611C4—2442no—
612C4—2442no—
613C4C2×#1451973511yescharacteristic
614C4#1684024311yescharacteristic
615C4#1684044711yescharacteristic
616C4#1684046011yescharacteristic
617C2—44no—
618C2#2144874411yescharacteristic
619C2C4×#1451980711yescharacteristic
620C2#2145465111yescharacteristic
621C2#2145785511yescharacteristic
622C2#2145790011yescharacteristic
623C2#2145857211yescharacteristic
624C2#2145860011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2349C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenter#2349D61
hypocenterC61#22776
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
2. #21448774C2semidirect
3. #21446508C2semidirect
4. #21446494C2semidirect
5. #21446408C2semidirect
6. #21446387C2semidirect
9. #21445952C2semidirect
10. #21445831C2semidirect
12. #21445694C2semidirect
18. #16839862C4semidirect
69. C4×Dic122C2×C4semidirect
79. #14519710C2×C4semidirect
83. #14519700C2×C4semidirect
88. #14519688C2×C4semidirect
372. C61#22776semidirect

Maximal subgroups

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

subgroupindex
C2×#168398622
#214487742
#214465082
#214464942
#214464082
#214463872
#214461532
#214459522
#214458312
#214458192
#214456942
#214456442
C4×#145196982
C61×#23492
#2277661