Finite Group Databasebeta

← #29853203

Subgroups — #29853203

Order 15616.

2042 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454125C214yesmaximal, complemented
2#21451791C214yesmaximal, complemented
3#21451045C214yesmaximal, complemented
4#21451040C214yesmaximal, complemented
5#21449542C214yesmaximal, complemented
6#21449540C214yesmaximal, complemented
7#21448433C211yesmaximal, characteristic
8#21448431C211yesmaximal, characteristic
9#21448431C211yesmaximal, characteristic
10C2×#16839620C211yesmaximal, characteristic
11C2×#16839620C211yesmaximal, characteristic
12C2×C22⋊C4×Dic61C211yesmaximal, characteristic
13C61×#2315C211yesmaximal, characteristic
14C24×Dic61C2211yescharacteristic
15C2×#14519767C2214yescomplemented
16C22×#11758196C2212yes—
17C22×#11758196C2212yes—
18C22×C22⋊Dic61C2211yescharacteristic
19C22×C22⋊Dic61C2211yescharacteristic
20C22×C22⋊Dic61C2211yescharacteristic
21C22×C4×Dic61C2212yes—
22#16839703C2214yescomplemented
23#16839703C2214yescomplemented
24#16839702C2214yescomplemented
25#16839700C2214yescomplemented
26#16839696C2214yescomplemented
27#16839695C2214yescomplemented
28#16839693C2214yescomplemented
29#16839692C2214yescomplemented
30#16839659C2214yescomplemented
31#16839657C2214yescomplemented
32#16839648C2214yescomplemented
33#16839647C2214yescomplemented
34#16839644C2214yescomplemented
35#16839643C2214yescomplemented
36#16839637C2214yes—
37#16839637C2214yes—
38#16839635C2214yes—
39#16839635C2214yes—
40#16839634C2214yes—
41#16839634C2214yes—
42#16839626C2214yescomplemented
43#16839622C2214yes—
44#16839620C2212yes—
45#16839620C2212yes—
46#16839620C2212yes—
47#16839620C2214yes—
48#16839620C2212yes—
49#16839620C2214yes—
50#16839620C2212yes—
51#16839620C2212yes—
52#16839620C2212yes—
53#16839620C2212yes—
54C22⋊C4×Dic61C2212yes—
55C22⋊C4×Dic61C2212yes—
56C22⋊C4×Dic61C2212yes—
57C22⋊C4×Dic61C2212yes—
58C2×#14519684C2212yes—
59C2×#14519684C2212yes—
60C2×#14519684C2212yes—
61C22×C22⋊C4×C61C2211yescharacteristic
62C22×C22⋊C4×C61C2211yescharacteristic
63C22×C22⋊C4×C61C2211yescharacteristic
64C61×#385C2214yes—
65C61×#385C2214yes—
66C61×#380C2214yes—
67C23×Dic61—44no—
68C23×Dic61Q818yes—
69C23×Dic61C2311yescharacteristic
70C23×Dic61C2311yescharacteristic
71C23×Dic61C2311yescharacteristic
72C23×Dic61C2312yes—
73C23×Dic61C2312yes—
74C23×Dic61C2312yes—
75C23×Dic61C2311yescharacteristic
76C23×Dic61C2311yescharacteristic
77C23×Dic61C2311yescharacteristic
78#14519767—48no—
79C2×C4⋊Dic61C2314yes—
80C2×C4⋊Dic61C2314yes—
81C2×#11758196—44no—
82C2×#11758196—44no—
83C2×#11758196C2314yes—
84C2×#11758196C2314yes—
85C2×#11758196C2314yes—
86C2×#11758196C2314yes—
87C2×#11758196C2314yes—
88C2×#11758196C2314yes—
89C2×#11758196C2314yes—
90C2×#11758196C2314yes—
91C2×C22⋊Dic61—24no—
92C2×C22⋊Dic61—24no—
93C2×C22⋊Dic61—24no—
94C2×C22⋊Dic61C2312yes—
95C2×C22⋊Dic61C2312yes—
96C2×C22⋊Dic61C2312yes—
97C2×C22⋊Dic61C2312yes—
98C2×C22⋊Dic61C2312yes—
99C2×C22⋊Dic61C2312yes—
100C2×C4×Dic61—44no—
101C2×C4×Dic61C2314yes—
102C2×C4×Dic61C2314yes—
103#14519684C2314yes—
104#14519684C2314yes—
105#14519684C2314yes—
106#14519684C2314yes—
107#14519684C2314yes—
108#14519684C2314yes—
109#14519684C2314yes—
110#14519684C2314yes—
111#14519684C2314yes—
112#14519684C2314yes—
113#14519684C2314yes—
114#14519684C2314yes—
115C25×C61C2311yescharacteristic
116C23×C4×C61C2312yes—
117C23×C4×C61C2312yes—
118C23×C4×C61C2312yes—
119C2×C4⋊C4×C61C2314yes—
120C2×C4⋊C4×C61C2314yes—
121C2×C22⋊C4×C61—24no—
122C2×C22⋊C4×C61—24no—
123C2×C22⋊C4×C61—24no—
124C2×C22⋊C4×C61C2312yes—
125C2×C22⋊C4×C61C2312yes—
126C2×C22⋊C4×C61C2312yes—
127C2×C22⋊C4×C61C2312yes—
128C2×C22⋊C4×C61C2312yes—
129C2×C22⋊C4×C61C2312yes—
130#95×C61C2314yes—
131#95×C61C2314yes—
132C24×C61—42no—
133C24×C61—42no—
134C24×C61C2×Q814yes—
135C24×C61#4114yes—
136C24×C61#4114yes—
137C24×C61C2411yescharacteristic
138C24×C61C2411yescharacteristic
139C24×C61C2411yescharacteristic
140C22×C4×C61—24no—
141C22×C4×C61—24no—
142C22×C4×C61—24no—
143C22×C4×C61—42no—
144C22×C4×C61—42no—
145C22×C4×C61—42no—
146C22×C4×C61—42no—
147C22×C4×C61—42no—
148C22×C4×C61—42no—
149C22×C4×C61C2414yes—
150C22×C4×C61C2414yes—
151C22×C4×C61C2414yes—
152C22×Dic61—24no—
153C22×Dic61—24no—
154C22×Dic61—24no—
155C22×Dic61—24no—
156C22×Dic61—24no—
157C22×Dic61—24no—
158C22×Dic61—416no—
159C22×Dic61—42no—
160C22×Dic61—42no—
161C22×Dic61—42no—
162C22×Dic61—42no—
163C22×Dic61—42no—
164C22×Dic61—42no—
165C22×Dic61—42no—
166C22×Dic61—42no—
167C22×Dic61—42no—
168C22×Dic61C2×Q814yes—
169C22×Dic61C2×Q814yes—
170C22×Dic61C2×Q814yes—
171C22×Dic61C2×Q814yes—
172C22×Dic61C2×Q814yes—
173C22×Dic61C2×Q814yes—
174C22×Dic61C2414yes—
175C22×Dic61C2414yes—
176C22×Dic61C2414yes—
177C22×Dic61C2411yescharacteristic
178C22×Dic61C2411yescharacteristic
179C22×Dic61C2411yescharacteristic
180C22×Dic61C2411yescharacteristic
181C4⋊C4×C61—44no—
182C4⋊C4×C61—44no—
183C22⋊C4×C61—24no—
184C22⋊C4×C61—24no—
185C22⋊C4×C61—24no—
186C22⋊C4×C61—24no—
187C22⋊C4×C61—24no—
188C22⋊C4×C61—24no—
189C22⋊Dic61—24no—
190C22⋊Dic61—24no—
191C22⋊Dic61—24no—
192C22⋊Dic61—24no—
193C22⋊Dic61—24no—
194C22⋊Dic61—24no—
195C4⋊Dic61—44no—
196C4⋊Dic61—44no—
197#11758196—28no—
198#11758196—28no—
199#11758196—44no—
200#11758196—44no—
201#11758196—44no—
202#11758196—44no—
203#11758196—44no—
204#11758196—44no—
205C4×Dic61—28no—
206C4×Dic61—44no—
207C23×C61—22no—
208C23×C61—22no—
209C23×C61—22no—
210C23×C61—22no—
211C23×C61—22no—
212C23×C61—22no—
213C23×C61—44no—
214C23×C61—44no—
215C23×C61—48no—
216C23×C61—48no—
217C23×C61#125116yescomplemented
218C23×C61C22×Q812yes—
219C23×C61C22×Q812yes—
220C23×C61C22×Q812yes—
221C23×C61C2×#4112yes—
222C23×C61C2×#4112yes—
223C23×C61C2×#4112yes—
224C23×C61C2×#4112yes—
225C23×C61C2×#4112yes—
226C23×C61C2×#4112yes—
227C23×C61C2511yescharacteristic
228C2×C4×C61—24no—
229C2×C4×C61—24no—
230C2×C4×C61—24no—
231C2×C4×C61—24no—
232C2×C4×C61—24no—
233C2×C4×C61—24no—
234C2×C4×C61—24no—
235C2×C4×C61—24no—
236C2×C4×C61—24no—
237C2×C4×C61—48no—
238C2×C4×C61—84no—
239C2×C4×C61—84no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—28no—
244C2×Dic61—28no—
245C2×Dic61—28no—
246C2×Dic61—24no—
247C2×Dic61—24no—
248C2×Dic61—24no—
249C2×Dic61—24no—
250C2×Dic61—24no—
251C2×Dic61—24no—
252C2×Dic61—24no—
253C2×Dic61—24no—
254C2×Dic61—24no—
255C2×Dic61—44no—
256C2×Dic61—44no—
257C2×Dic61—44no—
258C2×Dic61—48no—
259C2×Dic61—84no—
260C2×Dic61—84no—
261C2×Dic61C22×Q812yes—
262C2×Dic61C22×Q812yes—
263C2×Dic61C22×Q812yes—
264C2×Dic61C22×Q812yes—
265C2×Dic61H(5,𝔽2)14yes—
266#35020—611nomaximal, Sylow
267C22×C61—22no—
268C22×C61—22no—
269C22×C61—22no—
270C22×C61—22no—
271C22×C61—22no—
272C22×C61—22no—
273C22×C61—24no—
274C22×C61—24no—
275C22×C61—24no—
276C22×C61—24no—
277C22×C61—24no—
278C22×C61—24no—
279C22×C61—416no—
280C22×C61C2×#12514yes—
281C22×C61C2×#12514yes—
282C22×C61C2×#12514yes—
283C22×C61C23×Q811yescharacteristic
284C22×C61C22×#4111yescharacteristic
285C22×C61C22×#4111yescharacteristic
286C22×C61C2×H(5,𝔽2)12yes—
287C22×C61#58512yes—
288C4×C61—44no—
289C4×C61—44no—
290C4×C61—44no—
291Dic61—22no—
292Dic61—22no—
293Dic61—22no—
294Dic61—44no—
295Dic61—44no—
296Dic61—44no—
297Dic61#54314yes—
298#2851—614no—
299#2646—614no—
300#2569—614no—
301#2569—614no—
302#2442—614no—
303#2442—614no—
304#2315—611no—
305#2315—611no—
306#2315—611no—
307#2315D6111yescharacteristic
308C2×#380—611no—
309C2×#380—611no—
310C2×C4×C22⋊C4—611no—
311C2×C61—24no—
312C2×C61—24no—
313C2×C61—24no—
314C2×C61C22×#12511yescharacteristic
315C2×C61C2×#54311yescharacteristic
316C2×C61#345811yescharacteristic
317C2×C61#350512yes—
318C2×C61#350512yes—
319C24×C4—611no—
320C2×#125—614no—
321C22×C4⋊C4—612no—
322C22×C4⋊C4—612no—
323C22×C22⋊C4—611no—
324C22×C22⋊C4—611no—
325C22×C22⋊C4—611no—
326C22×C22⋊C4C2×D6111yescharacteristic
327C22×C22⋊C4C2×D6111yescharacteristic
328C22×C22⋊C4C2×D6111yescharacteristic
329C22×C42—612no—
330#400—614no—
331#400—614no—
332#400—614no—
333#400—614no—
334#398—614no—
335#398—614no—
336#398—614no—
337#398—614no—
338#389—614no—
339#389—614no—
340#387—614no—
341#387—614no—
342#387—614no—
343#387—614no—
344#385—614no—
345#385—614no—
346#385—614no—
347#385—614no—
348#385—614no—
349#385—614no—
350#385C2×D6114yes—
351#385C2×D6114yes—
352#382—614no—
353#380—614no—
354#380—612no—
355#380—612no—
356#380—614no—
357#380—614no—
358#380—612no—
359#380—612no—
360#380—612no—
361#380—612no—
362#380—612no—
363#380—612no—
364#380C2×D6114yes—
365C4×C22⋊C4—612no—
366C4×C22⋊C4—612no—
367C4×C22⋊C4—612no—
368C4×C22⋊C4—612no—
369C2×#95—612no—
370C2×#95—612no—
371C2×#95—612no—
372C61#3502011yescharacteristic, complemented, Sylow
373C25C22×D6111yescharacteristic
374C23×C4—611no—
375C23×C4—611no—
376C23×C4—611no—
377C23×C4—618no—
378C23×C4—611no—
379C23×C4—611no—
380C23×C4—611no—
381C23×C4—612no—
382C23×C4—612no—
383C23×C4—612no—
384C23×C4—2444no—
385C23×C4C22×D6112yes—
386C23×C4C22×D6112yes—
387C23×C4C22×D6112yes—
388#125—2448no—
389C2×C4⋊C4—614no—
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⋊C4—614no—
397C2×C4⋊C4—614no—
398C2×C4⋊C4—614no—
399C2×C4⋊C4—2444no—
400C2×C4⋊C4—2444no—
401C2×C4⋊C4C22×D6114yes—
402C2×C4⋊C4C22×D6114yes—
403C2×C22⋊C4—24no—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—24no—
406C2×C22⋊C4—612no—
407C2×C22⋊C4—612no—
408C2×C22⋊C4—612no—
409C2×C22⋊C4—612no—
410C2×C22⋊C4—612no—
411C2×C22⋊C4—612no—
412C2×C22⋊C4—1224no—
413C2×C22⋊C4—1224no—
414C2×C22⋊C4—1224no—
415C2×C22⋊C4C22×D6112yes—
416C2×C22⋊C4C22×D6112yes—
417C2×C22⋊C4C22×D6112yes—
418C2×C22⋊C4C22×D6112yes—
419C2×C22⋊C4C22×D6112yes—
420C2×C22⋊C4C22×D6112yes—
421C2×C42—614no—
422C2×C42—614no—
423C2×C42—2444no—
424#95—614no—
425#95—614no—
426#95—614no—
427#95—614no—
428#95—614no—
429#95—614no—
430#95—614no—
431#95—614no—
432#95—614no—
433#95—614no—
434#95—614no—
435#95—614no—
436#95C22×D6114yes—
437#95C22×D6114yes—
438C24—42no—
439C24—42no—
440C24#1175822214yes—
441C24#1175822414yes—
442C24Q8×D6114yes—
443C24C23×D6111yescharacteristic
444C24C23×D6111yescharacteristic
445C24C23×D6111yescharacteristic
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—611no—
456C22×C4—614no—
457C22×C4—614no—
458C22×C4—614no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—614no—
463C22×C4—614no—
464C22×C4—614no—
465C22×C4—614no—
466C22×C4—614no—
467C22×C4—614no—
468C22×C4—1224no—
469C22×C4—1224no—
470C22×C4—1224no—
471C22×C4—1224no—
472C22×C4—1224no—
473C22×C4—1224no—
474C22×C4—24416no—
475C22×C4—2442no—
476C22×C4—2442no—
477C22×C4—2442no—
478C22×C4—2442no—
479C22×C4—2442no—
480C22×C4—2442no—
481C22×C4—2442no—
482C22×C4—2442no—
483C22×C4—2442no—
484C22×C4C23×D6114yes—
485C22×C4C23×D6114yes—
486C22×C4C23×D6114yes—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—1228no—
490C4⋊C4—1228no—
491C4⋊C4—2444no—
492C4⋊C4—2444no—
493C4⋊C4—2444no—
494C4⋊C4—2444no—
495C4⋊C4—2444no—
496C4⋊C4—2444no—
497C4⋊C4—2444no—
498C4⋊C4—2444no—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—1224no—
506C22⋊C4—1224no—
507C22⋊C4—1224no—
508C22⋊C4—1224no—
509C22⋊C4—1224no—
510C22⋊C4—1224no—
511C42—1228no—
512C42—2444no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—44no—
520C23—48no—
521C23—48no—
522C23—44no—
523C23#14519767116yescomplemented
524C23C2×Q8×D6112yes—
525C23C2×Q8×D6112yes—
526C23C2×Q8×D6112yes—
527C23C2×#1175822412yes—
528C23C2×#1175822412yes—
529C23C2×#1175822412yes—
530C23C2×#1175822212yes—
531C23C2×#1175822212yes—
532C23C2×#1175822212yes—
533C23C24×D6111yescharacteristic
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—48no—
544C2×C4—84no—
545C2×C4—84no—
546C2×C4—612no—
547C2×C4—614no—
548C2×C4—612no—
549C2×C4—612no—
550C2×C4—612no—
551C2×C4—1228no—
552C2×C4—1228no—
553C2×C4—1228no—
554C2×C4—1222no—
555C2×C4—1222no—
556C2×C4—1222no—
557C2×C4—1224no—
558C2×C4—1224no—
559C2×C4—1224no—
560C2×C4—1224no—
561C2×C4—1224no—
562C2×C4—1224no—
563C2×C4—1224no—
564C2×C4—1224no—
565C2×C4—1224no—
566C2×C4—4884no—
567C2×C4—4884no—
568C2×C4—2444no—
569C2×C4—2444no—
570C2×C4—2444no—
571C2×C4—2448no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—24no—
579C22—24no—
580C22—24no—
581C22—24no—
582C22—24no—
583C22—24no—
584C22—416no—
585C22C2×#1451976714yes—
586C22C2×#1451976714yes—
587C22C2×#1451976714yes—
588C22C22×Q8×D6111yescharacteristic
589C22C22×#1175822411yescharacteristic
590C22C22×#1175822211yescharacteristic
591C22H(5,𝔽2)×D6112yes—
592C22#1684059712yes—
593C4—44no—
594C4—44no—
595C4—44no—
596C4—614no—
597C4—1222no—
598C4—1222no—
599C4—1222no—
600C4—2444no—
601C4—2444no—
602C4—2444no—
603C2—24no—
604C2—24no—
605C2—24no—
606C2C22×#1451976711yescharacteristic
607C2#543×D6111yescharacteristic
608C2#2145922111yescharacteristic
609C2#2145964912yes—
610C2#2145965612yes—
subgroupG/H
center, FrattiniC23C24×D61
commutator, socle, cosocleC23×C61C25
FittingC61×#2315C2
prime radicalC25×C61C23
hypercenter#2315D61
hypocenterC61#35020
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2315

Complements

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

NH ≅ G/Ntype
1. #21454125C2semidirect
2. #21451791C2semidirect
3. #21451045C2semidirect
4. #21451040C2semidirect
5. #21449542C2semidirect
6. #21449540C2semidirect
15. C2×#14519767C22semidirect
22. #16839703C22semidirect
23. #16839703C22semidirect
24. #16839702C22semidirect
25. #16839700C22semidirect
26. #16839696C22semidirect
27. #16839695C22semidirect
28. #16839693C22semidirect
29. #16839692C22semidirect
30. #16839659C22semidirect
31. #16839657C22semidirect
32. #16839648C22semidirect
33. #16839647C22semidirect
34. #16839644C22semidirect
35. #16839643C22semidirect
42. #16839626C22semidirect
217. C23×C61#125semidirect
372. C61#35020semidirect
523. C23#14519767semidirect

Maximal subgroups

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

subgroupindex
#214541252
#214517912
#214510452
#214510402
#214495422
#214495402
#214484332
#214484312
C2×#168396202
C2×C22⋊C4×Dic612
C61×#23152
#3502061