Finite Group Databasebeta

← #30620677

Subgroups — #30620677

Order 15616.

708 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#21461878C211yesmaximal, characteristic, complemented
2#21461875C211yesmaximal, characteristic, complemented
3#21461874C211yesmaximal, characteristic, complemented
4#21461872C211yesmaximal, characteristic, complemented
5#21461766C211yesmaximal, characteristic
6#21461763C211yesmaximal, characteristic
7#21461748C211yesmaximal, characteristic, complemented
8#21461735C211yesmaximal, characteristic, complemented
9#21461712C211yesmaximal, characteristic
10#21461607C211yesmaximal, characteristic, complemented
11#21461561C211yesmaximal, characteristic
12C22⋊C4×#92420C211yesmaximal, characteristic
13#21461534C211yesmaximal, characteristic, complemented
14#21461485C211yesmaximal, characteristic
15#21448313C211yesmaximal, characteristic
16C22×#11758217C2211yescharacteristic
17C23×#92420C2211yescharacteristic
18C2×#14519844C2211yescharacteristic
19C2×#14519844C2211yescharacteristic
20C2×#14519844C2211yescharacteristic
21C2×C4⋊#92420C2211yescharacteristic
22C2×C22⋊#92420C2211yescharacteristic
23C2×C22⋊#92420C2211yescharacteristic
24C2×#14519832C2211yescharacteristic
25C2×C4×#92420C2211yescharacteristic
26C2×C4×#92420C2211yescharacteristic
27C2×C4×#92420C2211yescharacteristic
28#16840628C2211yescharacteristic
29#16840628C2211yescharacteristic
30#16840628C2211yescharacteristic
31#16840628C2211yescharacteristic
32#16840628C2211yescharacteristic
33#16840628C2211yescharacteristic
34#16840628C2211yescharacteristic
35#16840628C2211yescharacteristic
36#16840628C2211yescharacteristic
37#16840628C2211yescharacteristic
38#16840628C2211yescharacteristic
39#16840628C2211yescharacteristic
40#16840628C2211yescharacteristic
41#16840628C2211yescharacteristic
42#16840628C2211yescharacteristic
43#16840628C2211yescharacteristic
44C2×#14519733C2211yescharacteristic
45#16839656C411yescharacteristic
46#16839655C411yescharacteristic
47#16839650C411yescharacteristic
48#16839649C411yescharacteristic
49#16839646C411yescharacteristic
50#16839644C411yescharacteristic
51#16839643C411yescharacteristic
52C4⋊C4×Dic61C2211yescharacteristic
53#16839613C2211yescharacteristic
54C22⋊C4×Dic61C2211yescharacteristic
55C22⋊C4×Dic61C2211yescharacteristic
56#16839607C2211yescharacteristic
57C42×Dic61C2211yescharacteristic
58C61×#388C411yescharacteristic
59C2×#11758217—22no—
60C2×#11758217—22no—
61C2×#11758217—41no—
62C22×#92420—22no—
63C22×#92420—22no—
64C22×#92420—41no—
65C22×#92420C2311yescharacteristic
66C22×#92420C2311yescharacteristic
67C22×#92420C2311yescharacteristic
68C22×#92420C2311yescharacteristic
69C22×#92420C2311yescharacteristic
70C22×#92420C2311yescharacteristic
71C22×#92420C2311yescharacteristic
72C22×#92420C2311yescharacteristic
73#14519844—22no—
74#14519844—41no—
75#14519844—41no—
76C4⋊#92420—22no—
77C22⋊#92420—22no—
78C22⋊#92420—22no—
79#14519832—41no—
80C4×#92420—22no—
81C4×#92420—22no—
82C4×#92420—41no—
83C23×Dic61C2311yescharacteristic
84#14519733—22no—
85#14519733—22no—
86C2×C4⋊Dic61C2×C411yescharacteristic
87C2×#11758196C2×C411yescharacteristic
88C2×#11758196C2×C411yescharacteristic
89C2×#11758196C2×C411yescharacteristic
90C2×#11758196C2×C411yescharacteristic
91C2×C22⋊Dic61C2×C411yescharacteristic
92C2×C22⋊Dic61C2×C411yescharacteristic
93C2×C22⋊Dic61C2×C411yescharacteristic
94C2×C4×Dic61C2×C411yescharacteristic
95C2×C4×Dic61C2×C411yescharacteristic
96C2×C4×Dic61C2×C411yescharacteristic
97C2×C4×Dic61C2311yescharacteristic
98C2×C4×Dic61C2311yescharacteristic
99C2×C4×Dic61C2311yescharacteristic
100C2×C4×Dic61C2311yescharacteristic
101C2×C4×Dic61C2311yescharacteristic
102C2×C4×Dic61C2311yescharacteristic
103#14519684C2×C411yescharacteristic
104#14519684C2×C411yescharacteristic
105#14519684C2×C411yescharacteristic
106#14519684C2×C411yescharacteristic
107#14519684C2×C411yescharacteristic
108#14519684C2×C411yescharacteristic
109#14519684C2×C411yescharacteristic
110#14519684C2×C411yescharacteristic
111#14519684C2×C411yescharacteristic
112#14519684C2×C411yescharacteristic
113C2×C4⋊C4×C61C2×C411yescharacteristic
114C2×C22⋊C4×C61C2×C411yescharacteristic
115C2×C22⋊C4×C61C2×C411yescharacteristic
116C2×C22⋊C4×C61C2×C411yescharacteristic
117C2×C42×C61C2×C411yescharacteristic
118#95×C61C2×C411yescharacteristic
119#95×C61C2×C411yescharacteristic
120C24×C61C22×C411yescharacteristic
121C22×C4×C61C22×C411yescharacteristic
122C22×C4×C61C22×C411yescharacteristic
123C22×C4×C61C22×C411yescharacteristic
124C22×C4×C61C22×C411yescharacteristic
125C22×C4×C61C22×C411yescharacteristic
126C22×C4×C61C22×C411yescharacteristic
127C22×Dic61—21no—
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61C22×C411yescharacteristic
132C22×Dic61C22×C411yescharacteristic
133C22×Dic61C22×C411yescharacteristic
134C22×Dic61C22×C411yescharacteristic
135C22×Dic61C22×C411yescharacteristic
136C22×Dic61C22×C411yescharacteristic
137C22×Dic61C22×C411yescharacteristic
138C22×Dic61#4112yes—
139C22×Dic61#4112yes—
140C22×Dic61C2411yescharacteristic
141#11758217—44no—
142C2×#92420—21no—
143C2×#92420—21no—
144C2×#92420—21no—
145C2×#92420—21no—
146C2×#92420—21no—
147C2×#92420—21no—
148C2×#92420—21no—
149C2×#92420—21no—
150C2×#92420—21no—
151C2×#92420—21no—
152C2×#92420—21no—
153C2×#92420—21no—
154C2×#92420—21no—
155C2×#92420—21no—
156C2×#92420—21no—
157C2×#92420—21no—
158C2×#92420—21no—
159C2×#92420—21no—
160C2×#92420—21no—
161C2×#92420—21no—
162C2×#92420—21no—
163C2×#92420—21no—
164C2×#92420—44no—
165C2×#92420#4112yes—
166C2×#92420#4112yes—
167C4⋊C4×C61—22no—
168C22⋊C4×C61—22no—
169C22⋊C4×C61—22no—
170C22⋊C4×C61—41no—
171C42×C61—41no—
172C22⋊Dic61—22no—
173C22⋊Dic61—22no—
174C22⋊Dic61—22no—
175C4⋊Dic61—22no—
176#11758196—22no—
177#11758196—22no—
178#11758196—22no—
179#11758196—41no—
180C4×Dic61—22no—
181C4×Dic61—22no—
182C4×Dic61—22no—
183C4×Dic61—21no—
184C4×Dic61—21no—
185C4×Dic61—21no—
186C4×Dic61—21no—
187C4×Dic61—21no—
188C4×Dic61—21no—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61—21no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C23×C61—21no—
196C23×C61—21no—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61—21no—
200C23×C61—21no—
201C23×C61#11712yes—
202C23×C61C23×C411yescharacteristic
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61#13112yes—
221C2×Dic61—21no—
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61#11712yes—
244C2×Dic61#13112yes—
245C2×Dic61#13112yes—
246C2×Dic61C2×#4111yescharacteristic
247C2×Dic61C2×#4111yescharacteristic
248C2×Dic61C2×#4111yescharacteristic
249C2×Dic61C2×#4111yescharacteristic
250C2×Dic61H(5,𝔽2)11yescharacteristic
251C2×Dic612−1+411yescharacteristic
252#92420—22no—
253#92420—22no—
254#92420—41no—
255#92420—41no—
256#92420—41no—
257#92420—41no—
258#92420—41no—
259#92420—41no—
260#22593—611nomaximal, Sylow
261C22×C61—22no—
262C22×C61—22no—
263C22×C61—22no—
264C22×C61—41no—
265C22×C61—41no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61C2×#11711yescharacteristic
269C22×C61C4×#4111yescharacteristic
270C22×C61#52011yescharacteristic
271C22×C61C2×#13111yescharacteristic
272C22×C61C2×#13111yescharacteristic
273C22×C61C2×#13111yescharacteristic
274C22×C61#56811yescharacteristic
275C4×C61—22no—
276C4×C61—41no—
277C4×C61—41no—
278C4×C61—41no—
279C4×C61—41no—
280C4×C61—41no—
281Dic61—22no—
282Dic61—22no—
283Dic61—22no—
284Dic61—41no—
285Dic61—41no—
286Dic61—41no—
287Dic61—41no—
288Dic61#55111yescharacteristic
289Dic61#55111yescharacteristic
290Dic61#55211yescharacteristic
291Dic61#55211yescharacteristic
292#2298—611no—
293#1917—611no—
294#1916—611no—
295#1916—611no—
296#1916—611no—
297#1851—611no—
298#1850—611no—
299#1840—611no—
300#1831—611no—
301#1818—611no—
302#1768—611no—
303#1753—611no—
304C8×C22⋊C4—611no—
305#1744—611no—
306#1729—611no—
307C2×C61—42no—
308C2×C61#234311yescharacteristic
309C2×C61#292211yescharacteristic
310C2×C61#292511yescharacteristic
311C2×C61#296611yescharacteristic
312C2×C61#296811yescharacteristic
313C2×C61#299311yescharacteristic
314C2×C61#299411yescharacteristic
315C22×M16—611no—
316C23×C8—611no—
317C2×#117—611no—
318C2×C4⋊C8—611no—
319C2×C4⋊C8—611no—
320C2×C4⋊C8—611no—
321C2×C4⋊C8—611no—
322C2×C22⋊C8—611no—
323C2×C22⋊C8—611no—
324C2×#97—611no—
325C2×C4×C8—611no—
326C2×C4×C8—611no—
327C2×C4×C8—611no—
328#388—611no—
329#388—611no—
330#388—611no—
331#388#692311yescharacteristic
332#387—611no—
333#387—611no—
334#387—611no—
335#387—611no—
336C4×C4⋊C4—611no—
337C4×C4⋊C4—611no—
338C4×C22⋊C4—611no—
339C4×C22⋊C4—611no—
340#376—611no—
341C43—611no—
342#336—611no—
343#336—611no—
344#336—611no—
345#336—611no—
346#336—611no—
347#336—611no—
348#336—611no—
349#336—611no—
350#336—611no—
351#336—611no—
352#336—611no—
353#336—611no—
354#336—611no—
355#336—611no—
356#336—611no—
357#336—611no—
358C61#2259311yescharacteristic, complemented, Sylow
359C23×C4—611no—
360C2×M16—1222no—
361C2×M16—1222no—
362C2×M16—2441no—
363C22×C8—611no—
364C22×C8—611no—
365C22×C8—611no—
366C22×C8—611no—
367C22×C8—611no—
368C22×C8—611no—
369C22×C8—611no—
370C22×C8—611no—
371C22×C8—1222no—
372C22×C8—1222no—
373C22×C8—2441no—
374#117—1222no—
375#117—1222no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4C2×#692311yescharacteristic
382C2×C22⋊C4—611no—
383C2×C22⋊C4—611no—
384C2×C22⋊C4—611no—
385C2×C22⋊C4C2×#692311yescharacteristic
386C2×C22⋊C4C2×#692311yescharacteristic
387C2×C22⋊C4C2×#692311yescharacteristic
388C2×C42—611no—
389C2×C42—611no—
390C2×C42—611no—
391C2×C42—611no—
392C2×C42—611no—
393C2×C42—611no—
394C2×C42—611no—
395C2×C42—611no—
396C2×C42—611no—
397C2×C42C2×#692311yescharacteristic
398C4⋊C8—1222no—
399C4⋊C8—1222no—
400C4⋊C8—2441no—
401C4⋊C8—2441no—
402C22⋊C8—1222no—
403C22⋊C8—1222no—
404#97—2441no—
405C4×C8—1222no—
406C4×C8—1222no—
407C4×C8—2441no—
408#95—611no—
409#95—611no—
410#95—611no—
411#95—611no—
412#95—611no—
413#95—611no—
414#95—611no—
415#95—611no—
416#95—611no—
417#95—611no—
418#95C2×#692311yescharacteristic
419#95C2×#692311yescharacteristic
420C24C22×#692311yescharacteristic
421C22×C4—611no—
422C22×C4—612no—
423C22×C4—612no—
424C22×C4—611no—
425C22×C4—611no—
426C22×C4—611no—
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—611no—
430C22×C4—611no—
431C22×C4—1221no—
432C22×C4—1221no—
433C22×C4—1221no—
434C22×C4—1221no—
435C22×C4C22×#692311yescharacteristic
436C22×C4C22×#692311yescharacteristic
437C22×C4C22×#692311yescharacteristic
438C22×C4C22×#692311yescharacteristic
439C22×C4C22×#692311yescharacteristic
440C22×C4C22×#692311yescharacteristic
441M16—2444no—
442C2×C8—612no—
443C2×C8—612no—
444C2×C8—1221no—
445C2×C8—1221no—
446C2×C8—1221no—
447C2×C8—1221no—
448C2×C8—1221no—
449C2×C8—1221no—
450C2×C8—1221no—
451C2×C8—1221no—
452C2×C8—1221no—
453C2×C8—1221no—
454C2×C8—1221no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1221no—
466C2×C8—2444no—
467C4⋊C4—22no—
468C4⋊C4—1222no—
469C4⋊C4—1222no—
470C4⋊C4—1222no—
471C4⋊C4—1222no—
472C4⋊C4—2441no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—41no—
476C22⋊C4—1222no—
477C22⋊C4—1222no—
478C22⋊C4—1222no—
479C42—41no—
480C42—1221no—
481C42—1221no—
482C42—1221no—
483C42—1221no—
484C42—1221no—
485C42—1221no—
486C42—1221no—
487C42—1221no—
488C42—1222no—
489C42—1222no—
490C42—1221no—
491C42—1221no—
492C42—1221no—
493C42—1221no—
494C42—1222no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23#1451985612yes—
502C23C23×#692311yescharacteristic
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—611no—
521C2×C4—611no—
522C2×C4—611no—
523C2×C4—611no—
524C2×C4—611no—
525C2×C4—611no—
526C2×C4—612no—
527C2×C4—612no—
528C2×C4—612no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—1222no—
532C2×C4—1222no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1221no—
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—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4#1451986412yes—
552C8—1222no—
553C8—1222no—
554C8—2441no—
555C8—2441no—
556C8—2441no—
557C8—2441no—
558C8—2441no—
559C8—2441no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22C2×#1451985611yescharacteristic
568C22#41×#692311yescharacteristic
569C22#1684083111yescharacteristic
570C22C2×#1451986411yescharacteristic
571C22D61⋊#13111yescharacteristic
572C22D61⋊#13111yescharacteristic
573C22#1684084011yescharacteristic
574C4—22no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—611no—
581C4—611no—
582C4—611no—
583C4—611no—
584C4—1222no—
585C4—1222no—
586C4—1222no—
587C4—2441no—
588C4—2441no—
589C4—2441no—
590C4—2441no—
591C2—42no—
592C2#2146232611yescharacteristic
593C2#2146250211yescharacteristic
594C2#2146251711yescharacteristic
595C2#2146263711yescharacteristic
596C2#2146264411yescharacteristic
597C2#2146276011yescharacteristic
598C2#2146276911yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC61×#388C4
prime radicalC24×C61C22×C4
cosocleC22×Dic61C24
hypercenter#388#6923
hypocenterC61#22593
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. #21461878C2semidirect
2. #21461875C2semidirect
3. #21461874C2semidirect
4. #21461872C2semidirect
7. #21461748C2semidirect
8. #21461735C2semidirect
10. #21461607C2semidirect
13. #21461534C2semidirect
358. C61#22593semidirect

Maximal subgroups

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

subgroupindex
#214618782
#214618752
#214618742
#214618722
#214617662
#214617632
#214617482
#214617352
#214617122
#214616072
#214615612
C22⋊C4×#924202
#214615342
#214614852
#214483132
#2259361