Finite Group Databasebeta

← #29847178

Subgroups — #29847178

Order 15616.

1918 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#21451855C214yesmaximal, complemented
2#21451810C214yesmaximal, complemented
3#21451799C211yesmaximal, characteristic, complemented
4#21451782C212yesmaximal, complemented
5#21451765C211yesmaximal, characteristic, complemented
6#21451610C212yesmaximal, complemented
7#21451569C211yesmaximal, characteristic
8#21449083C214yesmaximal, complemented
9#21449064C214yesmaximal, complemented
10#21448381C211yesmaximal, characteristic, complemented
11#125×Dic61C211yesmaximal, characteristic, complemented
12#21448325C212yesmaximal, complemented
13#21448277C212yesmaximal, complemented
14#117×Dic61C211yesmaximal, characteristic, complemented
15C61×#2629C211yesmaximal, characteristic, complemented
16C2×#14519774C2211yescharacteristic
17C2×#14519751C2212yes—
18C2×#14519739C2212yes—
19C2×#14519739C2214yes—
20C2×#14519737C2214yes—
21C2×#14519736C2211yescharacteristic
22C2×#14519735C2214yes—
23C22×C4×Dic61C2211yescharacteristic
24#16839701C2214yes—
25#16839699C2214yes—
26#16839695C2214yes—
27#16839691C2214yes—
28#16839689C2214yes—
29#16839689C2214yes—
30#16839688C2214yes—
31#16839684C2214yes—
32#16839679C2214yes—
33#16839679C2214yes—
34#16839679C2214yes—
35#16839678C2214yes—
36#16839678C2214yes—
37#16839671C2214yes—
38#16839664C2212yes—
39#16839663C2212yes—
40#16839663C2212yes—
41#16839663C2214yes—
42#16839663C2212yes—
43#16839661C2212yes—
44#16839661C2212yes—
45#16839658C2212yes—
46#16839654C2214yes—
47#16839652C2214yes—
48#16839651C2214yes—
49#16839648C2214yes—
50#16839639C2214yes—
51#16839628C2211yescharacteristic
52C4⋊C4×Dic61C2212yes—
53C4⋊C4×Dic61C2212yes—
54C4⋊C4×Dic61C2214yes—
55C4⋊C4×Dic61C2212yes—
56C22⋊C4×Dic61C2212yes—
57C22⋊C4×Dic61C2212yes—
58#16839609C2214yes—
59#16839609C2214yes—
60#16839608C2211yescharacteristic
61#16839608C2211yescharacteristic
62C42×Dic61C2211yescharacteristic
63C42×Dic61C2211yescharacteristic
64C2×#125×C61C2211yescharacteristic
65C2×#117×C61C2211yescharacteristic
66C61×#398C2214yes—
67C61×#396C2214yes—
68C61×#389C2212yes—
69C61×#384C2211yescharacteristic
70C61×#380C2212yes—
71C22×D244C2312yes—
72C22×D244C2314yes—
73C22×C22⋊D61C2314yes—
74C23×Dic61C2311yescharacteristic
75C23×Dic61C2311yescharacteristic
76C22×C4×D61C2314yes—
77#14519774—24no—
78#14519751—28no—
79#14519739—28no—
80#14519739—48no—
81#14519737—48no—
82#14519736—24no—
83#14519735—48no—
84C2×C4⋊Dic61C2312yes—
85C2×C4⋊Dic61C2312yes—
86C2×C4⋊Dic61C2314yes—
87C2×C4⋊Dic61C2314yes—
88C2×#11758196C2314yes—
89C2×C22⋊Dic61C2312yes—
90C2×#11758198C2314yes—
91C2×#11758198C2314yes—
92C2×#11758198C2314yes—
93C2×#11758198C2314yes—
94C2×#11758198C2314yes—
95C2×#11758198C2314yes—
96C2×#11758198C2314yes—
97C2×#11758198C2314yes—
98C2×#11758198C2314yes—
99C2×#11758198C2314yes—
100C2×C4×Dic61—24no—
101C2×C4×Dic61C2311yescharacteristic
102C2×C4×Dic61C2311yescharacteristic
103C2×C4×Dic61C2311yescharacteristic
104C2×C4×Dic61C2312yes—
105C2×C4×Dic61C2314yes—
106C2×C4×Dic61C2311yescharacteristic
107C2×C4×Dic61C2311yescharacteristic
108C2×C4×Dic61C2312yes—
109C2×C4×Dic61C2312yes—
110C2×C4×Dic61C2312yes—
111C2×C4×Dic61C2311yescharacteristic
112#14519684C2312yes—
113#14519684C2312yes—
114#14519684C2314yes—
115#14519684C2314yes—
116#14519684C2314yes—
117#14519684C2312yes—
118#14519684C2314yes—
119#14519684C2312yes—
120#14519684C2314yes—
121C23×C4×C61C2311yescharacteristic
122#125×C61—24no—
123#117×C61—24no—
124C2×C4⋊C4×C61C2312yes—
125C2×C4⋊C4×C61C2312yes—
126C2×C4⋊C4×C61C2314yes—
127C2×C4⋊C4×C61C2312yes—
128C2×C4⋊C4×C61C2314yes—
129C2×C4⋊C4×C61C2314yes—
130C2×C22⋊C4×C61C2312yes—
131C2×C22⋊C4×C61C2314yes—
132C2×C42×C61C2311yescharacteristic
133C2×C42×C61C2311yescharacteristic
134#95×C61C2314yes—
135#95×C61C2314yes—
136C24×C61C2411yescharacteristic
137C23×D61C2414yes—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61#4114yes—
141C22×C4×C61C2414yes—
142C22×C4×C61C2412yes—
143C22×C4×C61C2414yes—
144C22×C4×C61C2412yes—
145C22×C4×C61C2411yescharacteristic
146C22×C4×C61C2411yescharacteristic
147C2×C22⋊D61—28no—
148C2×C22⋊D61—44no—
149C2×C22⋊D61—44no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—24no—
153C22×Dic61—42no—
154C22×Dic61C2411yescharacteristic
155C22×Dic61C2411yescharacteristic
156C22×Dic61C2414yes—
157C22×Dic61C2412yes—
158C22×Dic61C2412yes—
159C22×Dic61C2411yescharacteristic
160C22×Dic61C2411yescharacteristic
161C2×D244—24no—
162C2×D244—24no—
163C2×D244—28no—
164C2×D244—24no—
165C2×D244—44no—
166C2×D244—44no—
167C2×C4×D61—28no—
168C2×C4×D61—28no—
169C2×C4×D61—28no—
170C4⋊C4×C61—24no—
171C4⋊C4×C61—28no—
172C4⋊C4×C61—24no—
173C4⋊C4×C61—24no—
174C4⋊C4×C61—44no—
175C4⋊C4×C61—44no—
176C22⋊C4×C61—24no—
177C22⋊C4×C61—44no—
178C42×C61—22no—
179C42×C61#4114yes—
180C22⋊Dic61—24no—
181#11758198—28no—
182#11758198—28no—
183#11758198—44no—
184#11758198—44no—
185#11758198—44no—
186#11758198—44no—
187#11758198—44no—
188#11758198—44no—
189#11758198—44no—
190#11758198—44no—
191C4⋊Dic61—24no—
192C4⋊Dic61—24no—
193C4⋊Dic61—44no—
194C4⋊Dic61—44no—
195#11758196—44no—
196C4×Dic61—22no—
197C4×Dic61—22no—
198C4×Dic61—22no—
199C4×Dic61—22no—
200C4×Dic61—22no—
201C4×Dic61—24no—
202C4×Dic61—28no—
203C4×Dic61—24no—
204C4×Dic61—42no—
205C4×Dic61—42no—
206C4×Dic61#4114yes—
207C23×C61—21no—
208C23×C61—21no—
209C23×C61—22no—
210C23×C61—22no—
211C23×C61C2×#4112yes—
212C23×C61C2511yescharacteristic
213C22×D61—24no—
214C22×D61—24no—
215C22×D61—24no—
216C22×D61—24no—
217C22×D61—24no—
218C22×D61—24no—
219C22×D61—24no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—28no—
225C2×C4×C61—24no—
226C2×C4×C61—22no—
227C2×C4×C61—24no—
228C2×C4×C61—24no—
229C2×C4×C61—22no—
230C2×C4×C61—24no—
231C2×C4×C61—24no—
232C2×C4×C61—22no—
233C2×C4×C61—24no—
234C2×C4×C61C2×#4114yes—
235C2×C4×C61C2×#4112yes—
236C2×C4×C61C2×#4112yes—
237C2×C4×C61C2×#4112yes—
238C2×C4×C61C2×#4112yes—
239C22⋊D61—416no—
240C2×Dic61—22no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—24no—
246C2×Dic61—24no—
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—24no—
253C2×Dic61—22no—
254C2×Dic61—44no—
255C2×Dic61—44no—
256C2×Dic61C2×#4112yes—
257C2×Dic61C2×#4112yes—
258C2×Dic61H(5,𝔽2)14yes—
259D244—416no—
260D244—48no—
261C4×D61—416no—
262#34621—611nomaximal, Sylow
263C22×C61—22no—
264C22×C61—24no—
265C22×C61—41no—
266C22×C61—41no—
267C22×C61—42no—
268C22×C61C22×#4111yescharacteristic
269C22×C61C22×#4111yescharacteristic
270C22×C61C22×#4111yescharacteristic
271C22×C61C2×H(5,𝔽2)12yes—
272C22×C61#58512yes—
273C2×D61—44no—
274C2×D61—44no—
275C2×D61—44no—
276C2×D61—44no—
277C2×D61—44no—
278C2×D61—44no—
279C2×D61—44no—
280C4×C61—22no—
281C4×C61—24no—
282C4×C61—44no—
283C4×C61—42no—
284C4×C61—44no—
285C4×C61#52914yes—
286Dic61—22no—
287Dic61—41no—
288Dic61—41no—
289Dic61—42no—
290Dic61—42no—
291Dic61—44no—
292Dic61#54014yes—
293#2651—614no—
294#2648—614no—
295#2647—611no—
296#2644—612no—
297#2641—611no—
298#2629D6111yescharacteristic, complemented
299#2624—612no—
300#2619—611no—
301#2395—614no—
302#2392—614no—
303C4×#127—611no—
304C4×#125—611no—
305C4×#121—612no—
306#2293—612no—
307C4×#117—611no—
308C2×C61—42no—
309C2×C61C2×#52911yescharacteristic
310C2×C61C2×#54011yescharacteristic
311C2×C61#345511yescharacteristic
312C2×C61#346012yes—
313C2×C61#346012yes—
314D61—84no—
315C2×#127—611no—
316C2×#125C2×D6111yescharacteristic
317C2×#121—612no—
318C2×C4×D4—612no—
319C2×C4×D4—614no—
320C2×C4×D4—614no—
321C2×#117—611no—
322C2×#117—614no—
323C2×#117C2×D6111yescharacteristic
324C22×C42—611no—
325#400—614no—
326#399—614no—
327#398—614no—
328#398C2×D6114yes—
329#397—614no—
330#397—614no—
331#397—614no—
332#397—614no—
333#396—614no—
334#396C2×D6114yes—
335#394—614no—
336#394—614no—
337#394—614no—
338#394—614no—
339#394—614no—
340#392—614no—
341#390—612no—
342#390—614no—
343#390—612no—
344#390—612no—
345#390—612no—
346#389—612no—
347#389—612no—
348#389—612no—
349#389C2×D6112yes—
350#388—614no—
351#388—614no—
352#388—614no—
353#387—614no—
354#386—614no—
355#384C2×D6111yescharacteristic
356#382—611no—
357#380C2×D6112yes—
358C4×C4⋊C4—614no—
359C4×C4⋊C4—612no—
360C4×C4⋊C4—612no—
361C4×C4⋊C4—612no—
362C4×C22⋊C4—612no—
363C4×C22⋊C4—612no—
364C4×C22⋊C4—614no—
365C4×C22⋊C4—614no—
366#376—611no—
367#376—611no—
368C43—611no—
369C43—611no—
370C61#3462111yescharacteristic, complemented, Sylow
371C22×D4—614no—
372C22×D4—612no—
373C22×D4—614no—
374C23×C4—611no—
375C23×C4—611no—
376C23×C4—614no—
377C23×C4C22×D6111yescharacteristic
378#127—1224no—
379#125—24no—
380#121—1228no—
381C4×D4—1228no—
382C4×D4—2448no—
383C4×D4—2448no—
384#117—24no—
385#117—1224no—
386#117—2448no—
387C2×C4⋊C4—612no—
388C2×C4⋊C4—612no—
389C2×C4⋊C4—614no—
390C2×C4⋊C4—614no—
391C2×C4⋊C4—614no—
392C2×C4⋊C4C22×D6112yes—
393C2×C4⋊C4C22×D6112yes—
394C2×C4⋊C4C22×D6112yes—
395C2×C4⋊C4C22×D6114yes—
396C2×C4⋊C4C22×D6114yes—
397C2×C4⋊C4C22×D6114yes—
398C2×C22⋊C4—612no—
399C2×C22⋊C4—614no—
400C2×C22⋊C4—614no—
401C2×C22⋊C4—614no—
402C2×C22⋊C4—614no—
403C2×C22⋊C4—614no—
404C2×C22⋊C4—614no—
405C2×C22⋊C4—614no—
406C2×C22⋊C4—614no—
407C2×C22⋊C4—614no—
408C2×C22⋊C4—614no—
409C2×C22⋊C4C22×D6112yes—
410C2×C22⋊C4C22×D6114yes—
411C2×C42—611no—
412C2×C42—612no—
413C2×C42—614no—
414C2×C42—611no—
415C2×C42—611no—
416C2×C42—612no—
417C2×C42—611no—
418C2×C42—611no—
419C2×C42—611no—
420C2×C42—612no—
421C2×C42—612no—
422C2×C42—1224no—
423C2×C42C22×D6111yescharacteristic
424C2×C42C22×D6111yescharacteristic
425#95—612no—
426#95—612no—
427#95—612no—
428#95—612no—
429#95—614no—
430#95—614no—
431#95—614no—
432#95—614no—
433#95—614no—
434#95C22×D6114yes—
435#95C22×D6114yes—
436C24—614no—
437C24C23×D6111yescharacteristic
438C2×D4—1228no—
439C2×D4—1224no—
440C2×D4—1228no—
441C2×D4—1224no—
442C2×D4—1224no—
443C2×D4—2444no—
444C2×D4—2444no—
445C2×D4—2444no—
446C2×D4—2444no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—614no—
453C22×C4—612no—
454C22×C4—612no—
455C22×C4—611no—
456C22×C4—1228no—
457C22×C4—1228no—
458C22×C4—1228no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—1224no—
462C22×C4—2442no—
463C22×C4#1175822614yes—
464C22×C4C23×D6112yes—
465C22×C4C23×D6114yes—
466C22×C4C23×D6114yes—
467C22×C4C23×D6112yes—
468C22×C4C23×D6111yescharacteristic
469C22×C4C23×D6111yescharacteristic
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—28no—
473C4⋊C4—24no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—1224no—
477C4⋊C4—1224no—
478C4⋊C4—2444no—
479C4⋊C4—2444no—
480C4⋊C4—2444no—
481C22⋊C4—24no—
482C22⋊C4—44no—
483C22⋊C4—1228no—
484C22⋊C4—1224no—
485C22⋊C4—1228no—
486C22⋊C4—2444no—
487C22⋊C4—2444no—
488C22⋊C4—2444no—
489C22⋊C4—2444no—
490C22⋊C4—2444no—
491C22⋊C4—2444no—
492C22⋊C4—2444no—
493C22⋊C4—2444no—
494C42—22no—
495C42—614no—
496C42—1222no—
497C42—1222no—
498C42—1224no—
499C42—1228no—
500C42—1224no—
501C42—1222no—
502C42—1222no—
503C42—1222no—
504C42—2442no—
505C42—2442no—
506C42#1175822414yes—
507C23—21no—
508C23—21no—
509C23—22no—
510C23—22no—
511C23—1224no—
512C23—1224no—
513C23—1224no—
514C23—1224no—
515C23—1224no—
516C23—1224no—
517C23—1224no—
518C23C2×#1175822612yes—
519C23C24×D6111yescharacteristic
520D4—24416no—
521D4—24416no—
522D4—2448no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—28no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—612no—
538C2×C4—614no—
539C2×C4—612no—
540C2×C4—1222no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—1224no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1224no—
549C2×C4—1222no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—1222no—
553C2×C4—1224no—
554C2×C4—24416no—
555C2×C4—2444no—
556C2×C4—2444no—
557C2×C4#41×D6112yes—
558C2×C4C2×#1175822414yes—
559C2×C4C2×#1175822412yes—
560C2×C4C2×#1175822612yes—
561C2×C4C2×#1175822612yes—
562C22—24no—
563C22—22no—
564C22—41no—
565C22—42no—
566C22—41no—
567C22—2444no—
568C22—2444no—
569C22—2444no—
570C22—2444no—
571C22—2444no—
572C22—2444no—
573C22—2444no—
574C22C2×#41×D6111yescharacteristic
575C22C22×#1175822411yescharacteristic
576C22C22×#1175822611yescharacteristic
577C22H(5,𝔽2)×D6112yes—
578C22#1684059912yes—
579C4—24no—
580C4—22no—
581C4—42no—
582C4—44no—
583C4—44no—
584C4—614no—
585C4—1222no—
586C4—2441no—
587C4—2442no—
588C4—2442no—
589C4—2444no—
590C4—2441no—
591C4#1684029814yes—
592C2—42no—
593C2—4884no—
594C2C2×#1684029811yescharacteristic
595C2#540×D6111yescharacteristic
596C2#2145921011yescharacteristic
597C2#2145923612yes—
598C2#2145924212yes—
subgroupG/H
center, FrattiniC23C24×D61
commutator, socle, cosocleC23×C61C25
FittingC61×#2629C2
prime radical#21451569C2
hypercenter#2629D61
hypocenterC61#34621
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2629

Complements

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

NH ≅ G/Ntype
1. #21451855C2semidirect
2. #21451810C2semidirect
3. #21451799C2semidirect
4. #21451782C2semidirect
5. #21451765C2semidirect
6. #21451610C2semidirect
8. #21449083C2semidirect
9. #21449064C2semidirect
10. #21448381C2semidirect
11. #125×Dic61C2semidirect
12. #21448325C2semidirect
13. #21448277C2semidirect
14. #117×Dic61C2semidirect
15. C61×#2629C2semidirect
298. #2629D61semidirect
370. C61#34621semidirect

Maximal subgroups

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

subgroupindex
#214518552
#214518102
#214517992
#214517822
#214517652
#214516102
#214515692
#214490832
#214490642
#214483812
#125×Dic612
#214483252
#214482772
#117×Dic612
C61×#26292
#3462161