Finite Group Databasebeta

← #29684386

Subgroups — #29684386

Order 15616.

814 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839863C211yesmaximal, characteristic
2#21448464C211yesmaximal, characteristic, complemented
3#21448456C211yesmaximal, characteristic
4#21446675C211yesmaximal, characteristic
5#21446557C211yesmaximal, characteristic, complemented
6#21446550C211yesmaximal, characteristic, complemented
7#21446513C211yesmaximal, characteristic
8#21446506C211yesmaximal, characteristic, complemented
9#21446505C211yesmaximal, characteristic
10#21446498C211yesmaximal, characteristic, complemented
11#21445718C211yesmaximal, characteristic, complemented
12#103×Dic61C211yesmaximal, characteristic, complemented
13#102×Dic61C211yesmaximal, characteristic
14#21445507C211yesmaximal, characteristic, complemented
15C61×#1973C211yesmaximal, characteristic
16C2×Q8×Dic61C2211yescharacteristic
17C2×D4×Dic61C2211yescharacteristic
18#16839863C418yescomplemented
19C2×#14519711C2211yescharacteristic
20C2×#14519709C2211yescharacteristic
21C2×#14519701C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519687C2211yescharacteristic
26C2×#14519687C2211yescharacteristic
27C2×C4×#92418C2211yescharacteristic
28C2×C8×Dic61C2211yescharacteristic
29C2×C8×Dic61C2211yescharacteristic
30#16839669C2211yescharacteristic
31#16839665C2211yescharacteristic
32#16839660C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839658C2211yescharacteristic
35#16839658C2211yescharacteristic
36#16839656—21no—
37#16839650—21no—
38#16839647—21no—
39#16839645—21no—
40#16839629—21no—
41#16839626—21no—
42#16839623—21no—
43#16839620—21no—
44C4⋊C4×Dic61C2211yescharacteristic
45C4⋊C4×Dic61C2211yescharacteristic
46#16839608C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53C22×SD16×C61C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×#97×C61C2211yescharacteristic
58C61×#391C2211yescharacteristic
59C61×#390C2211yescharacteristic
60C23×Dic61—21no—
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63Q8×Dic61—22no—
64Q8×Dic61C2×C414yes—
65D4×Dic61—22no—
66D4×Dic61C2×C414yes—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61C2311yescharacteristic
83C2×C4×Dic61C2311yescharacteristic
84#14519711C2×C414yes—
85#14519709—22no—
86#14519701—22no—
87#14519701C2×C414yes—
88#14519698—22no—
89#14519698C2×C414yes—
90#14519687—22no—
91#14519687C2×C414yes—
92C4×#92418—22no—
93C8×Dic61—22no—
94C8×Dic61—22no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684D411yescharacteristic
104#14519684D411yescharacteristic
105#14519684D411yescharacteristic
106#14519684D411yescharacteristic
107C22×Q8×C61C2311yescharacteristic
108C22×D4×C61C2311yescharacteristic
109C2×SD16×C61—22no—
110C2×SD16×C61—22no—
111C2×SD16×C61C2×C414yes—
112C22×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C22⋊C4×C61—21no—
117C2×C22⋊C4×C61—21no—
118C2×C42×C61C2311yescharacteristic
119#107×C61—22no—
120#103×C61—22no—
121#102×C61—22no—
122#97×C61—22no—
123#95×C61—21no—
124#95×C61—21no—
125C24×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61C22×C412yes—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61C22×C412yes—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2411yescharacteristic
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—24no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—81no—
150C22×Dic61—81no—
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155SD16×C61—28no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61C22×C412yes—
159C2×C8×C61#4112yes—
160C2×C8×C61#4112yes—
161C2×C8×C61#4112yes—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C22⋊C4×C61—81no—
167C22⋊C4×C61—81no—
168C42×C61—21no—
169C42×C61—21no—
170C22⋊Dic61—24no—
171C22⋊Dic61—81no—
172C22⋊Dic61—81no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—24no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61C22×C412yes—
181C4⋊Dic61C22×C412yes—
182#11758196—81no—
183#11758196—81no—
184#11758196—81no—
185#11758196—81no—
186C4×Dic61—21no—
187C4×Dic61—21no—
188C4×Dic61—24no—
189C4×Dic61—81no—
190C4×Dic61—81no—
191C4×Dic61C22×C412yes—
192C4×Dic61#4112yes—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418C22×C412yes—
198C2×#92418#4112yes—
199C23×C61—22no—
200C23×C61—41no—
201C23×C61—41no—
202C23×C61—41no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61C22×D411yescharacteristic
207Q8×C61—22no—
208Q8×C61—22no—
209Q8×C61—41no—
210Q8×C61—41no—
211D4×C61—22no—
212D4×C61—22no—
213D4×C61—41no—
214D4×C61—41no—
215C2×C4×C61—22no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
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×C61C23×C411yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
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×Dic61—41no—
258C2×Dic61—84no—
259C2×Dic61C4×D412yes—
260C2×Dic61C4×D412yes—
261C8×C61—22no—
262C8×C61#11714yes—
263#92418—22no—
264#92418—22no—
265#21291—611nomaximal, Sylow
266C22×C61—42no—
267C22×C61—42no—
268C22×C61—42no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61—81no—
273C22×C61C2×C4×D411yescharacteristic
274C22×C61#53211yescharacteristic
275C22×C61#53211yescharacteristic
276C22×C61#57611yescharacteristic
277C22×C61#57711yescharacteristic
278C22×C61#57711yescharacteristic
279C22×C61#57811yescharacteristic
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—42no—
283C4×C61—81no—
284C4×C61—81no—
285C4×C61C2×#11711yescharacteristic
286C4×C61C4×#4111yescharacteristic
287C4×C61#52911yescharacteristic
288C4×C61#52911yescharacteristic
289Dic61—22no—
290Dic61—22no—
291Dic61—41no—
292Dic61—41no—
293Dic61—42no—
294Dic61—42no—
295Dic61—81no—
296Dic61—81no—
297C2×#440—611no—
298#2320—611no—
299#2319—611no—
300#1973D6111yescharacteristic
301#1968—611no—
302#1943—611no—
303#1942—611no—
304#1936—611no—
305#1935—611no—
306#1934—611no—
307#1933—611no—
308#1776—611no—
309C4×#103—611no—
310C4×#102—611no—
311#1726—611no—
312C2×C61—82no—
313C2×C61#229311yescharacteristic
314C2×C61#294611yescharacteristic
315C2×C61#294711yescharacteristic
316C2×C61#317411yescharacteristic
317C2×C61#317511yescharacteristic
318C2×C61#317811yescharacteristic
319C2×C61#317911yescharacteristic
320C22×SD16C2×D6111yescharacteristic
321C2×C4×Q8—611no—
322C2×C4×D4—611no—
323#440—618no—
324C2×#107—611no—
325C2×#107C2×D6111yescharacteristic
326C2×#106—611no—
327C2×#103—611no—
328C2×#103—611no—
329C2×#103C2×D6111yescharacteristic
330C2×#102—611no—
331C2×#102—611no—
332C2×#102C2×D6111yescharacteristic
333C2×#97—611no—
334C2×#97—611no—
335C2×#97C2×D6111yescharacteristic
336C2×C4×C8—611no—
337C2×C4×C8—611no—
338C2×C4×C8—611no—
339#391—611no—
340#391C2×D6111yescharacteristic
341#390—611no—
342#390C2×D6111yescharacteristic
343#389—611no—
344#389—611no—
345#389—611no—
346#389—611no—
347#388—1221no—
348#388—1221no—
349#387—1221no—
350#387—1221no—
351#383—1221no—
352#382—1221no—
353#381—1221no—
354#380—1221no—
355C4×C4⋊C4—611no—
356C4×C4⋊C4—611no—
357#376—611no—
358#340—611no—
359#340—611no—
360#340—611no—
361#340—611no—
362#340—611no—
363#340—611no—
364C61#2129111yescharacteristic, complemented, Sylow
365C22×Q8C22×D6111yescharacteristic
366C22×D4C22×D6111yescharacteristic
367C23×C4—1221no—
368C2×SD16—22no—
369C2×SD16—22no—
370C2×SD16C4×D6114yes—
371C22×C8—611no—
372C22×C8—611no—
373C22×C8C22×D6111yescharacteristic
374C22×C8C22×D6111yescharacteristic
375C4×Q8—614no—
376C4×Q8—1222no—
377C4×D4—614no—
378C4×D4—1222no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4C22×D6111yescharacteristic
389C2×C4⋊C4C22×D6111yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—1221no—
393C2×C22⋊C4—1221no—
394C2×C22⋊C4—1221no—
395C2×C42—611no—
396C2×C42—611no—
397C2×C42—1221no—
398C2×C42—1221no—
399C2×C42—1221no—
400C2×C42C22×D6111yescharacteristic
401#107—22no—
402#107—614no—
403#106—1222no—
404#103—22no—
405#103—614no—
406#103—1222no—
407#102—22no—
408#102—614no—
409#102—1222no—
410#97—22no—
411#97—614no—
412#97—1222no—
413C4×C8—1222no—
414C4×C8—1222no—
415C4×C8—1222no—
416#95—21no—
417#95—21no—
418#95—611no—
419#95—611no—
420#95—611no—
421#95—611no—
422#95—1221no—
423#95—1221no—
424#95—1221no—
425#95—1221no—
426#95—1221no—
427#95—1221no—
428#95—1221no—
429#95—1221no—
430C24—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8C2×C4×D6112yes—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4C2×C4×D6112yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—611no—
447C22×C4—611no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—1224no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—4881no—
456C22×C4—4881no—
457C22×C4D4×D6111yescharacteristic
458C22×C4D4×D6111yescharacteristic
459C22×C4C23×D6111yescharacteristic
460SD16—28no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—612no—
464C2×C8—612no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—1221no—
469C2×C8C2×C4×D6112yes—
470C2×C8#1175822412yes—
471C2×C8#1175822412yes—
472C2×C8#1175822412yes—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—612no—
478C4⋊C4—612no—
479C4⋊C4—1224no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—4881no—
487C4⋊C4—4881no—
488C4⋊C4—4881no—
489C4⋊C4—4881no—
490C22⋊C4—81no—
491C22⋊C4—81no—
492C22⋊C4—1224no—
493C22⋊C4—4881no—
494C22⋊C4—4881no—
495C42—21no—
496C42—21no—
497C42—612no—
498C42—612no—
499C42—1224no—
500C42—1221no—
501C42—1221no—
502C42—4881no—
503C42—4881no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23C2×D4×D6111yescharacteristic
512Q8—22no—
513Q8—22no—
514Q8—41no—
515Q8—41no—
516D4—22no—
517D4—22no—
518D4—41no—
519D4—41no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—612no—
536C2×C4—612no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—4884no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4C22×C4×D6111yescharacteristic
561C2×C4#41×D6111yescharacteristic
562C2×C4#41×D6111yescharacteristic
563C2×C4C2×#1175822411yescharacteristic
564C2×C4C2×#1175822411yescharacteristic
565C2×C4C2×#1175822411yescharacteristic
566C8—22no—
567C8—1222no—
568C8—1222no—
569C8#1451973314yes—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22C4×D4×D6111yescharacteristic
578C22#1684030711yescharacteristic
579C22#1684031011yescharacteristic
580C22#1684056611yescharacteristic
581C22#1684057111yescharacteristic
582C22#1684057211yescharacteristic
583C22#1684057811yescharacteristic
584C4—41no—
585C4—41no—
586C4—42no—
587C4—81no—
588C4—81no—
589C4—1222no—
590C4—1222no—
591C4—4881no—
592C4—4881no—
593C4—2442no—
594C4—2442no—
595C4—2441no—
596C4—2441no—
597C4C2×#1451973311yescharacteristic
598C4C4×#1175822411yescharacteristic
599C4#1684029111yescharacteristic
600C4#1684029311yescharacteristic
601C2—82no—
602C2#2144828111yescharacteristic
603C2#2145465011yescharacteristic
604C2#2145465911yescharacteristic
605C2#2145644211yescharacteristic
606C2#2145645311yescharacteristic
607C2#2145648411yescharacteristic
608C2#2145649911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1973C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenter#1973D61
hypocenterC61#21291
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1973

Complements

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

NH ≅ G/Ntype
2. #21448464C2semidirect
5. #21446557C2semidirect
6. #21446550C2semidirect
8. #21446506C2semidirect
10. #21446498C2semidirect
11. #21445718C2semidirect
12. #103×Dic61C2semidirect
14. #21445507C2semidirect
18. #16839863C4semidirect
364. C61#21291semidirect

Maximal subgroups

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

subgroupindex
C2×#168398632
#214484642
#214484562
#214466752
#214465572
#214465502
#214465132
#214465062
#214465052
#214464982
#214457182
#103×Dic612
#102×Dic612
#214455072
C61×#19732
#2129161