Finite Group Databasebeta

← #29731754

Subgroups — #29731754

Order 15616.

764 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840172C211yesmaximal, characteristic, complemented
2C2×#16840043C211yesmaximal, characteristic
3#21451020C211yesmaximal, characteristic
4#21448908C211yesmaximal, characteristic, complemented
5#21447306C211yesmaximal, characteristic, complemented
6#21447294C211yesmaximal, characteristic, complemented
7#21447208C211yesmaximal, characteristic
8#21447194C211yesmaximal, characteristic
9#21446590C211yesmaximal, characteristic, complemented
10#21446576C211yesmaximal, characteristic, complemented
11#21446540C211yesmaximal, characteristic, complemented
12#21446444C211yesmaximal, characteristic
13#21446437C211yesmaximal, characteristic
14#21445647C211yesmaximal, characteristic
15C61×#1989C211yesmaximal, characteristic, complemented
16C2×#14519777C2211yescharacteristic
17C2×C4×D244C2211yescharacteristic
18#16840172—24no—
19#16840043—24no—
20C2×#14519711C2211yescharacteristic
21C2×#14519711C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519710C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×#14519706C2211yescharacteristic
27C2×#14519706C2211yescharacteristic
28C2×#14519699C2211yescharacteristic
29C2×#14519699C2211yescharacteristic
30C2×#14519697C2211yescharacteristic
31C2×#14519697C2211yescharacteristic
32C2×#14519688C2211yescharacteristic
33#16839684—21no—
34#16839684—21no—
35#16839666—21no—
36#16839666—21no—
37#16839663C2211yescharacteristic
38#16839663C2211yescharacteristic
39#16839660—21no—
40#16839658C2211yescharacteristic
41#16839658C2211yescharacteristic
42#16839633—21no—
43#16839633—21no—
44#16839631C2211yescharacteristic
45#16839627C2211yescharacteristic
46C4⋊C4×Dic61C2211yescharacteristic
47C4⋊C4×Dic61C2211yescharacteristic
48C4×C4⋊Dic61—21no—
49C4×C4⋊Dic61C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×#125×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C4×C4⋊C4×C61C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122—21no—
62C22×D244C2311yescharacteristic
63C22×C4×D61—21no—
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66#14519777—22no—
67#14519777D414yes—
68C4×D244—22no—
69C4×D244—22no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61D411yescharacteristic
72C2×C4⋊Dic61D411yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89#14519711—22no—
90#14519711—22no—
91#14519710—22no—
92#14519710—22no—
93#14519708—22no—
94#14519708—22no—
95#14519706—22no—
96#14519706—22no—
97#14519699—22no—
98#14519699—22no—
99#14519697—22no—
100#14519697Q814yescomplemented
101#14519688—22no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684D411yescharacteristic
109#14519684D411yescharacteristic
110C22×C8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112#125×C61—22no—
113#125×C61—22no—
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C42×C61—21no—
121C2×C42×C61C2311yescharacteristic
122C4⋊C8×C61—22no—
123#95×C61—21no—
124C23×D61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2411yescharacteristic
132C22×Dic61—21no—
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61C2×D411yescharacteristic
136C22×Dic61C2×D411yescharacteristic
137C22×Dic61C2×D411yescharacteristic
138C22×Dic61C2×D411yescharacteristic
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244C2×Q812yes—
145C2×C4×D61—42no—
146C2×C4×D61—81no—
147C2×C4×D61—81no—
148C2×Dic122—24no—
149C2×Dic122—81no—
150C2×Dic122—81no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—42no—
165C4⋊C4×C61—42no—
166C4⋊C4×C61C2×Q812yes—
167C42×C61—42no—
168C42×C61C2×D412yes—
169C42×C61#4112yes—
170#11758198—42no—
171#11758198—42no—
172#11758198—42no—
173#11758198—81no—
174#11758198—81no—
175C4⋊Dic61—24no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—22no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—41no—
182C4⋊Dic61C2×D412yes—
183C4⋊Dic61C2×D412yes—
184#11758196—81no—
185#11758196—81no—
186#11758196—81no—
187#11758196—81no—
188C4×Dic61—21no—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61—81no—
193C4×Dic61—81no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418C2×Q812yes—
199C2×#92418#4112yes—
200C23×C61C22×D411yescharacteristic
201C22×D61—22no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—41no—
206C22×D61—41no—
207C22×D61—41no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—22no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61#12112yes—
225C2×C4×C61#12112yes—
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×Q811yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C612−1+411yescharacteristic
231C2×C4×C612−1+411yescharacteristic
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61#13712yes—
253D244—22no—
254D244—22no—
255D244—41no—
256D244—41no—
257C4×D61—84no—
258Dic122—84no—
259C8×C61—41no—
260C8×C61—41no—
261#92418—22no—
262#92418—22no—
263#25462—611nomaximal, Sylow
264C22×C61C2×#12111yescharacteristic
265C22×C61#54711yescharacteristic
266C22×C61D4×Q811yescharacteristic
267C22×C61C2×#13711yescharacteristic
268C22×C61#57511yescharacteristic
269C22×C61#57611yescharacteristic
270C22×C61#57611yescharacteristic
271C2×D61—42no—
272C2×D61—42no—
273C2×D61—42no—
274C2×D61—81no—
275C2×D61—81no—
276C2×D61—81no—
277C2×D61—81no—
278C4×C61—22no—
279C4×C61—22no—
280C4×C61—42no—
281C4×C61—81no—
282C4×C61—81no—
283C4×C61—81no—
284C4×C61#53311yescharacteristic
285C4×C61#54111yescharacteristic
286C4×C61#54811yescharacteristic
287C4×C61D4×Q811yescharacteristic
288Dic61—22no—
289Dic61—41no—
290Dic61—41no—
291Dic61—41no—
292Dic61—42no—
293Dic61—42no—
294Dic61—81no—
295C2×#501—611no—
296C2×#474—611no—
297#2566—611no—
298#2369—611no—
299#2085—611no—
300#2083—611no—
301#2067—611no—
302#2065—611no—
303#1989D6111yescharacteristic, complemented
304#1949—611no—
305#1947—611no—
306#1940—611no—
307#1926—611no—
308#1925—611no—
309#1762—611no—
310C2×C61#242911yescharacteristic
311C2×C61#303711yescharacteristic
312C2×C61#304111yescharacteristic
313C2×C61#334611yescharacteristic
314C2×C61#334711yescharacteristic
315C2×C61#338511yescharacteristic
316C2×C61#338911yescharacteristic
317D61—82no—
318C2×C4⋊Q8—611no—
319C2×#125C2×D6111yescharacteristic
320C2×C4×D4—611no—
321#501—1224no—
322#474—1224no—
323C2×#107—611no—
324C2×#107—611no—
325C2×#107—611no—
326C2×#107—611no—
327C2×#106—611no—
328C2×#106—611no—
329C2×C4⋊C8—611no—
330C2×C4⋊C8—611no—
331C2×C4⋊C8C2×D6111yescharacteristic
332C2×#102—611no—
333C2×#102—611no—
334C2×#102—611no—
335C2×#102—611no—
336C2×#97—611no—
337#396—1221no—
338#396—1221no—
339#391—1221no—
340#391—1221no—
341#390—611no—
342#390—611no—
343#389—611no—
344#389—611no—
345#389—1221no—
346#385—1221no—
347#385—1221no—
348#384—611no—
349#382—611no—
350C4×C4⋊C4—611no—
351C4×C4⋊C4—611no—
352C4×C4⋊C4—611no—
353C4×C4⋊C4—1221no—
354C4×C4⋊C4C2×D6111yescharacteristic
355#340—611no—
356#340—611no—
357#340C2×D6111yescharacteristic
358#340C2×D6111yescharacteristic
359#340C2×D6111yescharacteristic
360#340C2×D6111yescharacteristic
361#336—611no—
362#336—611no—
363C61#2546211yescharacteristic, complemented, Sylow
364C22×Q8—1221no—
365C22×D4—611no—
366C23×C4—1221no—
367C22×C8—611no—
368C22×C8—611no—
369C22×C8C22×D6111yescharacteristic
370C22×C8C22×D6111yescharacteristic
371C4⋊Q8—614no—
372C4⋊Q8—1222no—
373#125—22no—
374#125—22no—
375C4×D4—1222no—
376C4×D4—1222no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4C22×D6111yescharacteristic
390C2×C4⋊C4C22×D6111yescharacteristic
391C2×C4⋊C4C22×D6111yescharacteristic
392C2×C4⋊C4C22×D6111yescharacteristic
393C2×C22⋊C4—1221no—
394C2×C22⋊C4—1221no—
395C2×C22⋊C4—1221no—
396C2×C22⋊C4—1221no—
397C2×C22⋊C4—1221no—
398C2×C42—21no—
399C2×C42—611no—
400C2×C42—611no—
401C2×C42—1221no—
402C2×C42—1221no—
403C2×C42C22×D6111yescharacteristic
404#107—1222no—
405#107—1222no—
406#107—1222no—
407#107—1222no—
408#106—1222no—
409#106—1222no—
410C4⋊C8—22no—
411C4⋊C8—1222no—
412C4⋊C8—1222no—
413#102—614no—
414#102—1222no—
415#102—1222no—
416#102—1222no—
417#97—1222no—
418#95—21no—
419#95—611no—
420#95—611no—
421#95—1221no—
422#95—1221no—
423#95—1221no—
424#95—1221no—
425#95—1221no—
426#95—1221no—
427C24—1221no—
428C2×Q8—1224no—
429C2×Q8—4881no—
430C2×Q8—4881no—
431C2×D4—612no—
432C2×D4—1221no—
433C2×D4—1221no—
434C2×D4—1221no—
435C2×D4—1221no—
436C2×D4—1221no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—1221no—
446C22×C4—1221no—
447C22×C4—1221no—
448C22×C4—4881no—
449C22×C4—4881no—
450C22×C4—2442no—
451C22×C4D4×D6111yescharacteristic
452C22×C4D4×D6111yescharacteristic
453C22×C4C23×D6111yescharacteristic
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—612no—
461C2×C8—612no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1221no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—612no—
476C4⋊C4—612no—
477C4⋊C4—1222no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—1224no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—4881no—
484C4⋊C4—4881no—
485C4⋊C4—4881no—
486C4⋊C4—4881no—
487C4⋊C4—2441no—
488C4⋊C4Q8×D6112yes—
489C22⋊C4—4881no—
490C22⋊C4—4881no—
491C22⋊C4—2442no—
492C22⋊C4—2442no—
493C22⋊C4—2442no—
494C42—42no—
495C42—1221no—
496C42—1221no—
497C42—1221no—
498C42—1221no—
499C42—4881no—
500C42—4881no—
501C42#1175822212yes—
502C42D4×D6112yes—
503C23—1222no—
504C23—2441no—
505C23—2441no—
506C23—2441no—
507C23—2441no—
508C23—2441no—
509C23—2441no—
510C23C2×D4×D6111yescharacteristic
511Q8—4884no—
512D4—1222no—
513D4—1222no—
514D4—2441no—
515D4—2441no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—612no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1222no—
536C2×C4—1222no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—4884no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4#1451975012yes—
555C2×C4#1451975012yes—
556C2×C4C2×D4×D6111yescharacteristic
557C2×C4C2×Q8×D6111yescharacteristic
558C2×C4#41×D6111yescharacteristic
559C2×C4C2×#1175822211yescharacteristic
560C2×C4#1451982411yescharacteristic
561C2×C4#1451982511yescharacteristic
562C8—41no—
563C8—41no—
564C8—1222no—
565C8—1222no—
566C22—4881no—
567C22—4881no—
568C22—4881no—
569C22—4881no—
570C22—2442no—
571C22—2442no—
572C22—2442no—
573C22C2×#1451975011yescharacteristic
574C22#1684041411yescharacteristic
575C22#1684042511yescharacteristic
576C22#137×D6111yescharacteristic
577C22#1684055811yescharacteristic
578C22#1684056411yescharacteristic
579C22#1684056411yescharacteristic
580C4—22no—
581C4—22no—
582C4—42no—
583C4—81no—
584C4—81no—
585C4—81no—
586C4—1222no—
587C4—4881no—
588C4—2441no—
589C4—2441no—
590C4—2441no—
591C4—2442no—
592C4—2442no—
593C4#1684031611yescharacteristic
594C4#1684037011yescharacteristic
595C4#1684042011yescharacteristic
596C4#1684042711yescharacteristic
597C2—4882no—
598C2#2144941911yescharacteristic
599C2#2145529011yescharacteristic
600C2#2145532511yescharacteristic
601C2#2145831111yescharacteristic
602C2#2145831611yescharacteristic
603C2#2145873211yescharacteristic
604C2#2145877611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1989C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1989D61
hypocenterC61#25462
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1989

Complements

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

NH ≅ G/Ntype
1. C2×#16840172C2semidirect
4. #21448908C2semidirect
5. #21447306C2semidirect
6. #21447294C2semidirect
9. #21446590C2semidirect
10. #21446576C2semidirect
11. #21446540C2semidirect
15. C61×#1989C2semidirect
100. #14519697Q8semidirect
303. #1989D61semidirect
363. C61#25462semidirect

Maximal subgroups

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

subgroupindex
C2×#168401722
C2×#168400432
#214510202
#214489082
#214473062
#214472942
#214472082
#214471942
#214465902
#214465762
#214465402
#214464442
#214464372
#214456472
C61×#19892
#2546261