Finite Group Databasebeta

← #30615846

Subgroups — #30615846

Order 15616.

926 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462279C211yesmaximal, characteristic
2#117×#6923C211yesmaximal, characteristic
3#21461883C212yesmaximal, complemented
4#21461882C212yesmaximal, complemented
5#21461881C212yesmaximal, complemented
6#21461880C212yesmaximal, complemented
7#21461505C211yesmaximal, characteristic
8#21461504C211yesmaximal, characteristic
9#21461504C211yesmaximal, characteristic
10C2×(C2×#6923)⋊C8C211yesmaximal, characteristic
11#427×D61C211yesmaximal, characteristic
12C22×D61⋊C8C2211yescharacteristic
13C2×#14519856C2211yescharacteristic
14C2×#14519856C2211yescharacteristic
15C22×C4×#6923C2211yescharacteristic
16#16840713C2212yes—
17#16840713C2212yes—
18#16840709—22no—
19C4⋊C4×#6923C2212yes—
20C4⋊C4×#6923C2212yes—
21C4×C4⋊#6923—21no—
22C4×C4⋊#6923—21no—
23C22⋊C4×#6923—22no—
24#16840698—21no—
25C42×#6923—21no—
26#16840636C2212yes—
27#16840636C2212yes—
28#16840635C2212yes—
29#16840635C2212yes—
30#16840635C2212yes—
31#16840635C2212yes—
32(C2×#6923)⋊C8C2212yes—
33(C2×#6923)⋊C8C2212yes—
34C22×C8×D61C2211yescharacteristic
35#117×D61C2211yescharacteristic
36#117×D61C2211yescharacteristic
37#16839800C411yescharacteristic, complemented
38#16839799C411yescharacteristic, complemented
39#16839799C411yescharacteristic, complemented
40#16839798C411yescharacteristic, complemented
41#16839798C411yescharacteristic, complemented
42#16839797C411yescharacteristic, complemented
43#16839797C411yescharacteristic, complemented
44#107×D61C2211yescharacteristic
45#107×D61C2211yescharacteristic
46#106×D61C2211yescharacteristic
47#106×D61C2211yescharacteristic
48C61×#427C411yescharacteristic, complemented
49C23×#6923D412yes—
50C2×D61⋊C8—22no—
51C2×D61⋊C8—22no—
52C2×D61⋊C8—21no—
53C2×D61⋊C8C2312yes—
54C22×#92420—21no—
55#14519856—22no—
56#14519856—22no—
57#14519856—22no—
58#14519856Q814yes—
59C2×C4⋊#6923—22no—
60C2×C4⋊#6923—22no—
61C2×C4⋊#6923C2312yes—
62C2×C4⋊#6923C2312yes—
63C2×C22⋊#6923—21no—
64C2×C22⋊#6923—21no—
65C2×C4×#6923—21no—
66C2×C4×#6923—22no—
67C2×C4×#6923—22no—
68C2×C4×#6923—22no—
69C2×C4×#6923—22no—
70C2×C4×#6923—21no—
71C2×C4×#6923D411yescharacteristic
72C2×C4×#6923D411yescharacteristic
73C2×C4×#6923C2312yes—
74(C2×#6923)⋊C4—22no—
75(C2×#6923)⋊C4—22no—
76(C2×#6923)⋊C4—22no—
77(C2×#6923)⋊C4—22no—
78C22×C4×D61C2311yescharacteristic
79C22×#92418C2×C411yescharacteristic
80C2×C8×D61—22no—
81C2×C8×D61—22no—
82C2×C8×D61C2×C411yescharacteristic
83C2×C8×D61C2×C411yescharacteristic
84C2×C8×D61C2311yescharacteristic
85C2×C8×D61C2311yescharacteristic
86#14519736C2×C411yescharacteristic
87#14519736C2×C411yescharacteristic
88#14519735C2×C411yescharacteristic
89#14519735C2×C411yescharacteristic
90#14519735C2×C411yescharacteristic
91#14519735C2×C411yescharacteristic
92#14519734—21no—
93#14519734—21no—
94#14519733—21no—
95#14519733—21no—
96C4⋊C4×D61C2311yescharacteristic
97C4⋊C4×D61C2311yescharacteristic
98C4⋊C4×D61C2311yescharacteristic
99C4⋊C4×D61C2311yescharacteristic
100C22⋊C4×D61—21no—
101C22⋊C4×D61—21no—
102C42×D61—21no—
103C42×D61—21no—
104#14519711C2×C411yescharacteristic, complemented
105#14519711C2×C411yescharacteristic, complemented
106#14519710C2×C411yescharacteristic, complemented
107#14519710C2×C411yescharacteristic, complemented
108#14519710C2×C411yescharacteristic, complemented
109#14519710C2×C411yescharacteristic, complemented
110#14519709C2×C411yescharacteristic, complemented
111#14519709C2×C411yescharacteristic, complemented
112#14519708C2×C411yescharacteristic, complemented
113#14519708C2×C411yescharacteristic, complemented
114#14519708C2×C411yescharacteristic, complemented
115#14519708C2×C411yescharacteristic, complemented
116C22×C8×C61C2×C411yescharacteristic
117#117×C61C2×C411yescharacteristic
118#117×C61C2×C411yescharacteristic
119#107×C61C2×C411yescharacteristic, complemented
120#107×C61C2×C411yescharacteristic, complemented
121#106×C61C2×C411yescharacteristic, complemented
122#106×C61C2×C411yescharacteristic, complemented
123C23×D61C2×D411yescharacteristic
124C22×#6923—24no—
125C22×#6923—22no—
126C22×#6923—24no—
127C22×#6923—22no—
128C22×#6923—42no—
129C22×#6923C2×D412yes—
130C22×#6923C2×D412yes—
131C22×C4×C61C22×C411yescharacteristic
132C22×Dic61#4111yescharacteristic
133C2×C4×D61—21no—
134C2×C4×D61—21no—
135C2×C4×D61—21no—
136C2×C4×D61—21no—
137C2×C4×D61C22×C411yescharacteristic
138C2×C4×D61C2×D411yescharacteristic
139C2×C4×D61C2×Q812yes—
140C2×C4×D61#4112yes—
141C2×C4×D61#4112yes—
142C2×C4×D61#4112yes—
143C2×C4×D61#4111yescharacteristic
144C2×C4×D61C2411yescharacteristic
145C22⋊#6923—24no—
146C22⋊#6923—42no—
147C2×#92420—22no—
148C2×#92420—42no—
149C4⋊#6923—22no—
150C4⋊#6923—22no—
151C4⋊#6923—22no—
152C4⋊#6923—44no—
153C4⋊#6923—44no—
154C4⋊#6923C2×Q814yes—
155C4×#6923—24no—
156C4×#6923—22no—
157C4×#6923—22no—
158C4×#6923—22no—
159C4×#6923—22no—
160C4×#6923—44no—
161C4×#6923—44no—
162C4×#6923—42no—
163D61⋊C8—22no—
164D61⋊C8—22no—
165D61⋊C8—42no—
166C2×C8×C61—22no—
167C2×C8×C61C22×C411yescharacteristic
168C2×C8×C61C22×C411yescharacteristic
169C4⋊C4×C61C22×C411yescharacteristic
170C4⋊C4×C61C22×C411yescharacteristic
171C4⋊C4×C61C22×C411yescharacteristic
172C4⋊C4×C61C22×C411yescharacteristic
173C22⋊C4×C61—21no—
174C22⋊C4×C61—21no—
175C42×C61—21no—
176C42×C61—21no—
177C22⋊Dic61—21no—
178C22⋊Dic61—21no—
179#11758198—21no—
180#11758198—21no—
181#11758198—21no—
182#11758198—21no—
183C4⋊Dic61C22×C411yescharacteristic
184C4⋊Dic61C22×C411yescharacteristic
185C4⋊Dic61C22×C411yescharacteristic
186C4⋊Dic61C22×C411yescharacteristic
187#11758196—21no—
188#11758196—21no—
189#11758196—21no—
190#11758196—21no—
191C4×Dic61—21no—
192C4×Dic61—21no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C2×#92418—22no—
198C2×#92418C22×C411yescharacteristic
199C2×#92418C22×C411yescharacteristic
200C8×D61—22no—
201C8×D61—22no—
202C8×D61—21no—
203C8×D61—21no—
204C8×D61—21no—
205C8×D61—21no—
206C23×C61C4×D411yescharacteristic
207C22×D61—22no—
208C22×D61—22no—
209C22×D61C4×D411yescharacteristic
210C22×D61C22⋊Q812yes—
211C22×D61#12312yes—
212C22×D61C22×D411yescharacteristic
213C2×#6923—22no—
214C2×#6923—22no—
215C2×#6923—24no—
216C2×#6923—22no—
217C2×#6923—22no—
218C2×#6923—22no—
219C2×#6923—48no—
220C2×#6923—42no—
221C2×#6923—42no—
222C2×#6923—42no—
223C2×#6923—42no—
224C2×#6923—42no—
225C2×#6923#13514yes—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61#11712yes—
231C2×C4×C61C4×D411yescharacteristic
232C2×C4×C61C4×Q812yes—
233C2×C4×C61C23×C411yescharacteristic
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61C4×D411yescharacteristic
239C2×Dic61#12512yes—
240C2×Dic61#12612yes—
241C2×Dic61C2×#4111yescharacteristic
242C4×D61—42no—
243C4×D61—42no—
244C4×D61—42no—
245C4×D61—42no—
246C4×D61#11712yes—
247C4×D61C4×Q812yes—
248C4×D61C22⋊Q812yes—
249C4×D61#12312yes—
250C4×D61#12512yes—
251C4×D61#12612yes—
252C4×D61C22×Q811yescharacteristic
253C4×D61C2×#4111yescharacteristic
254C4×D61C2×#4111yescharacteristic
255C4×D61C2×#4111yescharacteristic
256#92420—42no—
257C8×C61—21no—
258C8×C61—21no—
259#92418—21no—
260#92418—21no—
261#21032—611nomaximal, Sylow
262C22×C61—21no—
263C22×C61—21no—
264C22×C61#38712yes—
265C22×C61C2×C4×D411yescharacteristic
266C2×D61—21no—
267C2×D61—21no—
268C2×D61—24no—
269C2×D61—24no—
270C2×D61#38712yes—
271C2×D61C2×C22⋊Q811yescharacteristic
272C2×D61C2×#12311yescharacteristic
273C2×D61C2×#13512yes—
274C2×D61#57512yes—
275#6923—24no—
276#6923—44no—
277#6923—42no—
278#6923—82no—
279C4×C61—41no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61#38712yes—
284C4×C61C2×#11711yescharacteristic
285C4×C61C2×C4×Q811yescharacteristic
286Dic61—41no—
287Dic61—41no—
288Dic61—41no—
289Dic61—41no—
290Dic61#38712yes—
291Dic61C2×#12511yescharacteristic
292Dic61C2×#12611yescharacteristic
293C2×#427—611no—
294#2296—611no—
295C4×#117—611no—
296#1921—612no—
297#1920—612no—
298#1919—612no—
299#1918—612no—
300#1735—611no—
301#1735—611no—
302#1735—611no—
303C2×#336—611no—
304C2×C61—22no—
305C2×C61C2×#38711yescharacteristic
306C2×C61C4×#13511yescharacteristic
307C2×C61#294311yescharacteristic
308D61—22no—
309D61#307612yes—
310D61#309112yes—
311C23×C8—611no—
312C23×C8—611no—
313C2×#117—611no—
314C2×#117—611no—
315C2×#117—611no—
316C2×#117—611no—
317C22×C42—611no—
318#427—611no—
319#427—611no—
320#427—611no—
321#427—611no—
322#427—611no—
323#427—611no—
324#427—611no—
325#427#692311yescharacteristic, complemented
326C2×#107—611no—
327C2×#107—611no—
328C2×#106—611no—
329C2×#106—611no—
330#389—612no—
331#389—612no—
332#385—1222no—
333C4×C4⋊C4—612no—
334C4×C4⋊C4—612no—
335C4×C4⋊C4—1221no—
336C4×C4⋊C4—1221no—
337C4×C22⋊C4—1222no—
338#376—1221no—
339C43—1221no—
340#340—612no—
341#340—612no—
342#340—612no—
343#340—612no—
344#340—612no—
345#340—612no—
346#336—612no—
347#336—612no—
348C61#2103211yescharacteristic, complemented, Sylow
349C23×C4—611no—
350C23×C4—612no—
351C22×C8—611no—
352C22×C8—611no—
353C22×C8—611no—
354C22×C8—611no—
355C22×C8—611no—
356C22×C8—612no—
357C22×C8—1222no—
358C22×C8—1222no—
359C22×C8—1221no—
360C22×C8—1222no—
361C22×C8—1222no—
362C22×C8—1221no—
363C22×C8C2×#692311yescharacteristic
364#117—614no—
365#117—611no—
366#117—611no—
367#117—611no—
368#117—611no—
369#117—611no—
370#117—611no—
371#117—1221no—
372#117—1221no—
373#117—1221no—
374#117—1221no—
375#117—1222no—
376#117—1222no—
377#117—1222no—
378#117C2×#692311yescharacteristic
379#117C2×#692311yescharacteristic
380C2×C4⋊C4—612no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—612no—
386C2×C4⋊C4—1222no—
387C2×C4⋊C4—1222no—
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—1221no—
390C2×C22⋊C4—1221no—
391C2×C22⋊C4—1221no—
392C2×C42—611no—
393C2×C42—611no—
394C2×C42—612no—
395C2×C42—1221no—
396C2×C42—1221no—
397C2×C42—1221no—
398C2×C42—1222no—
399C2×C42—1222no—
400C2×C42—1221no—
401C2×C42—1222no—
402C2×C42—1222no—
403#107—611no—
404#107—611no—
405#107—611no—
406#107—611no—
407#107—611no—
408#107—611no—
409#107C2×#692311yescharacteristic, complemented
410#107C2×#692311yescharacteristic, complemented
411#106—611no—
412#106—611no—
413#106—611no—
414#106—611no—
415#106—611no—
416#106—611no—
417#106C2×#692311yescharacteristic, complemented
418#106C2×#692311yescharacteristic, complemented
419#95—1222no—
420#95—1222no—
421#95—1222no—
422#95—1222no—
423C24—611no—
424C22×C4—611no—
425C22×C4—612no—
426C22×C4—612no—
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—612no—
430C22×C4—612no—
431C22×C4—611no—
432C22×C4—611no—
433C22×C4—612no—
434C22×C4—612no—
435C22×C4—1221no—
436C22×C4—1221no—
437C22×C4—1221no—
438C22×C4—1221no—
439C22×C4—1224no—
440C22×C4—1222no—
441C22×C4—1224no—
442C22×C4—1222no—
443C22×C4—2442no—
444C22×C4C22×#692311yescharacteristic
445C2×C8—22no—
446C2×C8—611no—
447C2×C8—611no—
448C2×C8—1222no—
449C2×C8—1222no—
450C2×C8—1221no—
451C2×C8—1221no—
452C2×C8—1221no—
453C2×C8—1221no—
454C2×C8—1222no—
455C2×C8—1222no—
456C2×C8—1222no—
457C2×C8—1222no—
458C2×C8—2442no—
459C2×C8—2442no—
460C2×C8C22×#692311yescharacteristic
461C2×C8C22×#692311yescharacteristic
462C4⋊C4—614no—
463C4⋊C4—611no—
464C4⋊C4—611no—
465C4⋊C4—611no—
466C4⋊C4—611no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1222no—
472C4⋊C4—1222no—
473C4⋊C4—1222no—
474C4⋊C4—2444no—
475C4⋊C4—2444no—
476C4⋊C4C22×#692311yescharacteristic
477C4⋊C4C22×#692311yescharacteristic
478C4⋊C4C22×#692311yescharacteristic
479C4⋊C4C22×#692311yescharacteristic
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—1221no—
483C22⋊C4—1221no—
484C22⋊C4—1221no—
485C22⋊C4—1221no—
486C22⋊C4—1224no—
487C22⋊C4—1221no—
488C22⋊C4—1221no—
489C22⋊C4—2442no—
490C42—21no—
491C42—21no—
492C42—1221no—
493C42—1221no—
494C42—1221no—
495C42—1221no—
496C42—1222no—
497C42—1222no—
498C42—1222no—
499C42—1222no—
500C42—1224no—
501C42—1221no—
502C42—1221no—
503C42—2442no—
504C42—2444no—
505C42—2444no—
506C23—611no—
507C23—612no—
508C23—612no—
509C23—611no—
510C23—1222no—
511C23—1222no—
512C23D4×#692311yescharacteristic
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—612no—
518C2×C4—612no—
519C2×C4—611no—
520C2×C4—611no—
521C2×C4—612no—
522C2×C4—612no—
523C2×C4—612no—
524C2×C4—612no—
525C2×C4—612no—
526C2×C4—611no—
527C2×C4—611no—
528C2×C4—612no—
529C2×C4—611no—
530C2×C4—611no—
531C2×C4—614no—
532C2×C4—1222no—
533C2×C4—1222no—
534C2×C4—1222no—
535C2×C4—1224no—
536C2×C4—1222no—
537C2×C4—1222no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2448no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4—2442no—
551C2×C4—2442no—
552C2×C4#1451985612yes—
553C2×C4D4×#692311yescharacteristic
554C2×C4Q8×#692312yes—
555C2×C4C23×#692311yescharacteristic
556C8—21no—
557C8—21no—
558C8—1221no—
559C8—1221no—
560C8—2442no—
561C22—21no—
562C22—21no—
563C22—612no—
564C22—612no—
565C22—611no—
566C22—611no—
567C22—612no—
568C22—1224no—
569C22—1224no—
570C22—1221no—
571C22—1221no—
572C22#1684071112yes—
573C22C2×D4×#692311yescharacteristic
574C4—41no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—611no—
579C4—611no—
580C4—612no—
581C4—1224no—
582C4—4882no—
583C4—2442no—
584C4—2444no—
585C4—2441no—
586C4—2441no—
587C4—2441no—
588C4—2441no—
589C4#1684071112yes—
590C4C2×#1451985611yescharacteristic
591C4C2×Q8×#692311yescharacteristic
592C2—22no—
593C2—612no—
594C2—612no—
595C2—1222no—
596C2C2×#1684071111yescharacteristic
597C2#135×#692311yescharacteristic
598C2#2146257311yescharacteristic
subgroupG/H
centerC2×C4D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#427C4
socleC22×C61C2×C4×D4
prime radicalC23×D61C2×D4
cosocleC2×C4×D61C24
hypercenter#427#6923
hypocenterC61#21032
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
3. #21461883C2semidirect
4. #21461882C2semidirect
5. #21461881C2semidirect
6. #21461880C2semidirect
37. #16839800C4semidirect
38. #16839799C4semidirect
39. #16839799C4semidirect
40. #16839798C4semidirect
41. #16839798C4semidirect
42. #16839797C4semidirect
43. #16839797C4semidirect
48. C61×#427C4semidirect
104. #14519711C2×C4semidirect
105. #14519711C2×C4semidirect
106. #14519710C2×C4semidirect
107. #14519710C2×C4semidirect
108. #14519710C2×C4semidirect
109. #14519710C2×C4semidirect
110. #14519709C2×C4semidirect
111. #14519709C2×C4semidirect
112. #14519708C2×C4semidirect
113. #14519708C2×C4semidirect
114. #14519708C2×C4semidirect
115. #14519708C2×C4semidirect
119. #107×C61C2×C4semidirect
120. #107×C61C2×C4semidirect
121. #106×C61C2×C4semidirect
122. #106×C61C2×C4semidirect
325. #427#6923semidirect
348. C61#21032semidirect
409. #107C2×#6923semidirect
410. #107C2×#6923semidirect
417. #106C2×#6923semidirect
418. #106C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214622792
#117×#69232
#214618832
#214618822
#214618812
#214618802
#214615052
#214615042
C2×(C2×#6923)⋊C82
#427×D612
#2103261