Finite Group Databasebeta

← #29847173

Subgroups — #29847173

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#21451853C214yesmaximal, complemented
2#21451808C214yesmaximal, complemented
3#21451798C211yesmaximal, characteristic, complemented
4#21451780C212yesmaximal, complemented
5#21451764C211yesmaximal, characteristic, complemented
6#21451649C211yesmaximal, characteristic, complemented
7#21451609C212yesmaximal, complemented
8#21449075C214yesmaximal, complemented
9#21449054C214yesmaximal, complemented
10#127×Dic61C211yesmaximal, characteristic, complemented
11#21448374C211yesmaximal, characteristic, complemented
12#121×Dic61C212yesmaximal, complemented
13#21448281C212yesmaximal, complemented
14#117×Dic61C211yesmaximal, characteristic, complemented
15C61×#2619C211yesmaximal, characteristic
16C2×#14519769C2211yescharacteristic
17C2×D4×Dic61C2212yes—
18C2×D4×Dic61C2214yes—
19C2×D4×Dic61C2214yes—
20C2×#14519736C2211yescharacteristic
21C2×#14519733C2214yes—
22C22×C4×Dic61C2211yescharacteristic
23#16839704C2214yes—
24#16839698C2214yes—
25#16839695C2214yes—
26#16839694C2214yes—
27#16839690C2214yes—
28#16839690C2214yes—
29#16839687C2214yes—
30#16839687C2214yes—
31#16839685C2214yes—
32#16839685C2214yes—
33#16839676C2214yes—
34#16839676C2214yes—
35#16839676C2214yes—
36#16839676C2214yes—
37#16839665C2212yes—
38#16839665C2212yes—
39#16839665C2214yes—
40#16839665C2212yes—
41#16839661C2212yes—
42#16839660C2212yes—
43#16839658C2212yes—
44#16839658C2212yes—
45#16839656C2214yes—
46#16839655C2214yes—
47#16839650C2214yes—
48#16839644C2214yes—
49#16839642C2214yes—
50#16839632C2211yescharacteristic
51#16839622C2212yes—
52C4⋊C4×Dic61C2212yes—
53C4⋊C4×Dic61C2212yes—
54C4⋊C4×Dic61C2212yes—
55#16839613C2214yes—
56C22⋊C4×Dic61C2212yes—
57C22⋊C4×Dic61C2214yes—
58C22⋊C4×Dic61C2212yes—
59C22⋊C4×Dic61C2214yes—
60#16839608C2211yescharacteristic
61#16839608C2211yescharacteristic
62C42×Dic61C2211yescharacteristic
63C42×Dic61C2211yescharacteristic
64C2×#127×C61C2211yescharacteristic
65C2×#121×C61C2212yes—
66C2×#117×C61C2211yescharacteristic
67C61×#394C2214yes—
68C61×#392C2214yes—
69C61×#390C2212yes—
70C61×#382C2211yescharacteristic
71C23×Dic61C2311yescharacteristic
72C23×Dic61C2314yes—
73C23×Dic61C2311yescharacteristic
74#14519769—24no—
75D4×Dic61—28no—
76D4×Dic61—48no—
77D4×Dic61—48no—
78#14519736—24no—
79#14519733—48no—
80C2×C4⋊Dic61C2312yes—
81C2×C4⋊Dic61C2312yes—
82C2×C4⋊Dic61C2312yes—
83C2×C4⋊Dic61C2314yes—
84C2×C4⋊Dic61C2314yes—
85C2×#11758196C2314yes—
86C2×#11758196C2314yes—
87C2×#11758196C2314yes—
88C2×#11758196C2314yes—
89C2×C22⋊Dic61C2312yes—
90C2×C22⋊Dic61C2314yes—
91C2×C22⋊Dic61C2314yes—
92C2×C22⋊Dic61C2314yes—
93C2×C22⋊Dic61C2314yes—
94C2×C22⋊Dic61C2314yes—
95C2×C22⋊Dic61C2314yes—
96C2×C22⋊Dic61C2314yes—
97C2×C22⋊Dic61C2314yes—
98C2×C4×Dic61—24no—
99C2×C4×Dic61C2311yescharacteristic
100C2×C4×Dic61C2311yescharacteristic
101C2×C4×Dic61C2311yescharacteristic
102C2×C4×Dic61C2312yes—
103C2×C4×Dic61C2314yes—
104C2×C4×Dic61C2311yescharacteristic
105C2×C4×Dic61C2311yescharacteristic
106C2×C4×Dic61C2312yes—
107C2×C4×Dic61C2312yes—
108C2×C4×Dic61C2312yes—
109C2×C4×Dic61C2311yescharacteristic
110#14519684C2312yes—
111#14519684C2314yes—
112#14519684C2314yes—
113#14519684C2314yes—
114#14519684C2314yes—
115#14519684C2312yes—
116#14519684C2314yes—
117#14519684C2312yes—
118#14519684C2312yes—
119#14519684C2314yes—
120C22×D4×C61C2312yes—
121C22×D4×C61C2314yes—
122C22×D4×C61C2314yes—
123C23×C4×C61C2311yescharacteristic
124#127×C61—24no—
125#121×C61—28no—
126#117×C61—24no—
127C2×C4⋊C4×C61C2312yes—
128C2×C4⋊C4×C61C2312yes—
129C2×C22⋊C4×C61C2312yes—
130C2×C22⋊C4×C61C2314yes—
131C2×C22⋊C4×C61C2314yes—
132C2×C22⋊C4×C61C2314yes—
133C2×C42×C61C2311yescharacteristic
134C2×C42×C61C2311yescharacteristic
135#95×C61C2314yes—
136C24×C61C2414yes—
137C24×C61C2411yescharacteristic
138C2×D4×C61—28no—
139C2×D4×C61—24no—
140C2×D4×C61—28no—
141C2×D4×C61—24no—
142C2×D4×C61—24no—
143C2×D4×C61—44no—
144C2×D4×C61—44no—
145C2×D4×C61—44no—
146C2×D4×C61—44no—
147C22×C4×C61—22no—
148C22×C4×C61—22no—
149C22×C4×C61#4114yes—
150C22×C4×C61C2414yes—
151C22×C4×C61C2412yes—
152C22×C4×C61C2412yes—
153C22×C4×C61C2411yescharacteristic
154C22×C4×C61C2411yescharacteristic
155C22×Dic61—22no—
156C22×Dic61—22no—
157C22×Dic61—28no—
158C22×Dic61—28no—
159C22×Dic61—24no—
160C22×Dic61—28no—
161C22×Dic61—42no—
162C22×Dic61C2411yescharacteristic
163C22×Dic61C2411yescharacteristic
164C22×Dic61C2414yes—
165C22×Dic61C2412yes—
166C22×Dic61C2412yes—
167C22×Dic61C2414yes—
168C22×Dic61C2411yescharacteristic
169C22×Dic61C2411yescharacteristic
170C4⋊C4×C61—24no—
171C4⋊C4×C61—24no—
172C22⋊C4×C61—24no—
173C22⋊C4×C61—28no—
174C22⋊C4×C61—44no—
175C22⋊C4×C61—44no—
176C42×C61—22no—
177C42×C61#4114yes—
178C22⋊Dic61—24no—
179C22⋊Dic61—28no—
180C22⋊Dic61—44no—
181C22⋊Dic61—44no—
182C22⋊Dic61—44no—
183C22⋊Dic61—44no—
184C22⋊Dic61—44no—
185C22⋊Dic61—44no—
186C22⋊Dic61—44no—
187C4⋊Dic61—24no—
188C4⋊Dic61—24no—
189C4⋊Dic61—24no—
190C4⋊Dic61—44no—
191C4⋊Dic61—44no—
192#11758196—28no—
193#11758196—44no—
194#11758196—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×C61—24no—
212C23×C61—24no—
213C23×C61—24no—
214C23×C61—24no—
215C23×C61—24no—
216C23×C61—24no—
217C23×C61—24no—
218C23×C61C2×#4112yes—
219C23×C61C2511yescharacteristic
220D4×C61—416no—
221D4×C61—416no—
222D4×C61—48no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—28no—
228C2×C4×C61—22no—
229C2×C4×C61—24no—
230C2×C4×C61—22no—
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—
239C2×Dic61—22no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—24no—
245C2×Dic61—24no—
246C2×Dic61—22no—
247C2×Dic61—24no—
248C2×Dic61—22no—
249C2×Dic61—24no—
250C2×Dic61—22no—
251C2×Dic61—24no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—24no—
255C2×Dic61—22no—
256C2×Dic61—44no—
257C2×Dic61—44no—
258C2×Dic61—416no—
259C2×Dic61C2×#4112yes—
260C2×Dic61C2×#4112yes—
261C2×Dic61H(5,𝔽2)14yes—
262#34621—611nomaximal, Sylow
263C22×C61—22no—
264C22×C61—24no—
265C22×C61—41no—
266C22×C61—41no—
267C22×C61—42no—
268C22×C61—44no—
269C22×C61—44no—
270C22×C61—44no—
271C22×C61—44no—
272C22×C61—44no—
273C22×C61—44no—
274C22×C61—44no—
275C22×C61C22×#4111yescharacteristic
276C22×C61C22×#4111yescharacteristic
277C22×C61C22×#4111yescharacteristic
278C22×C61C2×H(5,𝔽2)12yes—
279C22×C61#58512yes—
280C4×C61—22no—
281C4×C61—24no—
282C4×C61—42no—
283C4×C61—44no—
284C4×C61#52914yes—
285Dic61—22no—
286Dic61—41no—
287Dic61—44no—
288Dic61—41no—
289Dic61—44no—
290Dic61—42no—
291Dic61—42no—
292Dic61#54014yes—
293#2651—614no—
294#2648—614no—
295#2647—611no—
296#2644—612no—
297#2641—611no—
298#2629—611no—
299#2624—612no—
300#2619D6111yescharacteristic
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×C61—84no—
310C2×C61C2×#52911yescharacteristic
311C2×C61C2×#54011yescharacteristic
312C2×C61#345511yescharacteristic
313C2×C61#346012yes—
314C2×C61#346012yes—
315C2×#127C2×D6111yescharacteristic
316C2×#125—611no—
317C2×#121C2×D6112yes—
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#398—614no—
329#397—614no—
330#397—614no—
331#397—614no—
332#397—614no—
333#396—614no—
334#396—614no—
335#394—614no—
336#394—614no—
337#394—614no—
338#394—614no—
339#394C2×D6114yes—
340#392C2×D6114yes—
341#390—612no—
342#390—614no—
343#390—612no—
344#390—612no—
345#390C2×D6112yes—
346#389—612no—
347#389—612no—
348#389—612no—
349#389—612no—
350#388—614no—
351#388—614no—
352#388—614no—
353#387—614no—
354#386—614no—
355#384—611no—
356#382C2×D6111yescharacteristic
357#380—612no—
358C4×C4⋊C4—612no—
359C4×C4⋊C4—612no—
360C4×C4⋊C4—612no—
361C4×C4⋊C4—614no—
362C4×C22⋊C4—612no—
363C4×C22⋊C4—614no—
364C4×C22⋊C4—612no—
365C4×C22⋊C4—614no—
366#376—611no—
367#376—611no—
368C43—611no—
369C43—611no—
370C61#3462111yescharacteristic, complemented, Sylow
371C22×D4C22×D6112yes—
372C22×D4C22×D6114yes—
373C22×D4C22×D6114yes—
374C23×C4—614no—
375C23×C4—611no—
376C23×C4—611no—
377C23×C4C22×D6111yescharacteristic
378#127—24no—
379#125—1224no—
380#121—28no—
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—612no—
390C2×C4⋊C4—614no—
391C2×C4⋊C4—614no—
392C2×C4⋊C4—614no—
393C2×C4⋊C4—614no—
394C2×C4⋊C4—614no—
395C2×C4⋊C4—614no—
396C2×C4⋊C4C22×D6112yes—
397C2×C4⋊C4C22×D6112yes—
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⋊C4C22×D6112yes—
408C2×C22⋊C4C22×D6114yes—
409C2×C22⋊C4C22×D6114yes—
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#95—614no—
435#95C22×D6114yes—
436C24C23×D6114yes—
437C24C23×D6111yescharacteristic
438C2×D4—28no—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—28no—
442C2×D4—24no—
443C2×D4—44no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—44no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—612no—
453C22×C4—612no—
454C22×C4—611no—
455C22×C4—614no—
456C22×C4—614no—
457C22×C4—1222no—
458C22×C4—1222no—
459C22×C4—1228no—
460C22×C4—1228no—
461C22×C4—1224no—
462C22×C4—1228no—
463C22×C4—2442no—
464C22×C4#1175822414yes—
465C22×C4C23×D6114yes—
466C22×C4C23×D6112yes—
467C22×C4C23×D6111yescharacteristic
468C22×C4C23×D6112yes—
469C22×C4C23×D6111yescharacteristic
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—1224no—
473C4⋊C4—1228no—
474C4⋊C4—1224no—
475C4⋊C4—1224no—
476C4⋊C4—2444no—
477C4⋊C4—2444no—
478C4⋊C4—2444no—
479C4⋊C4—2444no—
480C4⋊C4—2444no—
481C22⋊C4—24no—
482C22⋊C4—28no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—1228no—
486C22⋊C4—1224no—
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—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23C2×#1175822412yes—
519C23C24×D6111yescharacteristic
520D4—416no—
521D4—416no—
522D4—48no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—28no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—612no—
535C2×C4—614no—
536C2×C4—612no—
537C2×C4—1222no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1224no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1224no—
544C2×C4—1222no—
545C2×C4—1224no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1224no—
549C2×C4—1224no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—1224no—
553C2×C4—1222no—
554C2×C4—2444no—
555C2×C4—24416no—
556C2×C4—2444no—
557C2×C4#41×D6112yes—
558C2×C4C2×#1175822414yes—
559C2×C4C2×#1175822412yes—
560C2×C4C2×#1175822412yes—
561C2×C4C2×#1175822412yes—
562C22—24no—
563C22—22no—
564C22—41no—
565C22—42no—
566C22—41no—
567C22—44no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22C2×#41×D6111yescharacteristic
575C22C22×#1175822411yescharacteristic
576C22C22×#1175822411yescharacteristic
577C22H(5,𝔽2)×D6112yes—
578C22#1684059712yes—
579C4—24no—
580C4—22no—
581C4—42no—
582C4—44no—
583C4—614no—
584C4—1222no—
585C4—2441no—
586C4—2444no—
587C4—2442no—
588C4—2442no—
589C4—2441no—
590C4—2444no—
591C4#1684029314yes—
592C2—42no—
593C2—84no—
594C2C2×#1684029311yescharacteristic
595C2#540×D6111yescharacteristic
596C2#2145920311yescharacteristic
597C2#2145923212yes—
598C2#2145923212yes—
subgroupG/H
center, FrattiniC23C24×D61
commutator, socle, cosocleC23×C61C25
Fitting, prime radicalC61×#2619C2
hypercenter#2619D61
hypocenterC61#34621
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2619

Complements

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

NH ≅ G/Ntype
1. #21451853C2semidirect
2. #21451808C2semidirect
3. #21451798C2semidirect
4. #21451780C2semidirect
5. #21451764C2semidirect
6. #21451649C2semidirect
7. #21451609C2semidirect
8. #21449075C2semidirect
9. #21449054C2semidirect
10. #127×Dic61C2semidirect
11. #21448374C2semidirect
12. #121×Dic61C2semidirect
13. #21448281C2semidirect
14. #117×Dic61C2semidirect
370. C61#34621semidirect

Maximal subgroups

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

subgroupindex
#214518532
#214518082
#214517982
#214517802
#214517642
#214516492
#214516092
#214490752
#214490542
#127×Dic612
#214483742
#121×Dic612
#214482812
#117×Dic612
C61×#26192
#3462161