Finite Group Databasebeta

← #29839121

Subgroups — #29839121

Order 15616.

2518 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840368C211yesmaximal, characteristic
2#21448612C214yesmaximal, complemented
3#21448583C214yesmaximal, complemented
4#21448544C211yesmaximal, characteristic, complemented
5#21448513C212yesmaximal, complemented
6#21448458C212yesmaximal, complemented
7#21448427C211yesmaximal, characteristic, complemented
8#21448381C211yesmaximal, characteristic, complemented
9#125×Dic61C211yesmaximal, characteristic, complemented
10#21448340C214yesmaximal, complemented
11#21448327C212yesmaximal, complemented
12#21448327C214yesmaximal, complemented
13C4×#14519739C212yesmaximal, complemented
14#21448158C211yesmaximal, characteristic, complemented
15C61×#2330C211yesmaximal, characteristic, complemented
16#16840368C4116yescomplemented
17C2×#14519774C2211yescharacteristic
18C2×#14519762C2214yes—
19C2×#14519752C2212yes—
20C2×#14519752C2214yes—
21C2×C4×D244C2212yes—
22C2×#14519739C2212yes—
23C2×#14519739C2214yes—
24C2×#14519739C2214yes—
25C2×#14519739C2212yes—
26C2×#14519734C2211yescharacteristic
27C2×#14519734C2211yescharacteristic
28C2×C42×D61C2211yescharacteristic
29C2×C42×D61C2214yes—
30#16839664C2212yes—
31#16839664C2214yes—
32#16839663C2212yes—
33#16839663C2214yes—
34#16839661C2212yes—
35#16839654C2214yes—
36#16839653C2214yes—
37#16839652C2214yes—
38#16839652C2214yes—
39#16839652C2214yes—
40#16839648C2214yes—
41#16839643C2214yes—
42#16839640C2214yes—
43#16839640C2214yes—
44#16839640C2214yes—
45#16839640C2214yes—
46#16839636C2214yes—
47#16839628C2211yescharacteristic
48#16839624C2214yes—
49#16839619C2214yes—
50#16839619C2212yes—
51C4⋊C4×Dic61C2212yes—
52C4⋊C4×Dic61C2214yes—
53C4⋊C4×Dic61C2212yes—
54#16839613C2212yes—
55#16839613C2212yes—
56C4×#11758198C2214yes—
57#16839609C2212yes—
58#16839609C2214yes—
59#16839609C2214yes—
60#16839609C2212yes—
61#16839609C2214yes—
62#16839608C2211yescharacteristic
63C42×Dic61C2211yescharacteristic
64C42×Dic61C2211yescharacteristic
65C2×#125×C61C2211yescharacteristic
66C61×#389C2212yes—
67C61×#389C2214yes—
68C61×#387C2214yes—
69C61×#384C2211yescharacteristic
70C4×C4⋊C4×C61C2212yes—
71C61×#376C2211yescharacteristic
72C22×D244C2312yes—
73C22×D244C2314yes—
74C22×D244C2314yes—
75C22×C4×D61C2311yescharacteristic
76C22×C4×D61C2314yes—
77C22×C4×D61C2311yescharacteristic
78C22×C4×D61C2314yes—
79C22×C4×D61C2311yescharacteristic
80C22×C4×D61C2314yes—
81#14519774C2×C418yescomplemented
82#14519762C2×C4132yescomplemented
83#14519752C2×C4132yescomplemented
84#14519752C2×C4116yescomplemented
85C4×D244—28no—
86#14519739—28no—
87#14519739—48no—
88#14519739—48no—
89#14519739C2×C4116yescomplemented
90#14519734—24no—
91#14519734C2×C418yescomplemented
92C2×C4⋊Dic61C2312yes—
93C2×C4⋊Dic61C2314yes—
94C2×#11758196C2312yes—
95C2×#11758196C2312yes—
96C2×#11758198C2312yes—
97C2×#11758198C2314yes—
98C2×#11758198C2314yes—
99C2×#11758198C2314yes—
100C2×#11758198C2314yes—
101C2×#11758198C2314yes—
102C2×#11758198C2314yes—
103C2×#11758198C2314yes—
104C2×#11758198C2312yes—
105C2×#11758198C2314yes—
106C2×#11758198C2314yes—
107C2×C4×Dic61C2312yes—
108C2×C4×Dic61C2314yes—
109C2×C4×Dic61C2311yescharacteristic
110C2×C4×Dic61C2311yescharacteristic
111C2×C4×Dic61C2311yescharacteristic
112C2×C4×Dic61C2312yes—
113C2×C4×Dic61C2314yes—
114C2×C4×Dic61C2312yes—
115C2×C4×Dic61C2312yes—
116C2×C4×Dic61C2311yescharacteristic
117C2×C4×Dic61C2311yescharacteristic
118C42×D61—24no—
119C42×D61—48no—
120#14519684C2312yes—
121#14519684C2312yes—
122#14519684C2314yes—
123#14519684C2314yes—
124#14519684C2314yes—
125#14519684C2314yes—
126#14519684C2314yes—
127#125×C61C2×C418yescomplemented
128C2×C4⋊C4×C61C2312yes—
129C2×C4⋊C4×C61C2314yes—
130C2×C4⋊C4×C61C2314yes—
131C2×C4⋊C4×C61C2312yes—
132C2×C4⋊C4×C61C2314yes—
133C2×C42×C61C2311yescharacteristic
134C2×C42×C61C2314yes—
135C2×C42×C61C2311yescharacteristic
136C2×C42×C61C2311yescharacteristic
137#95×C61C2312yes—
138#95×C61C2312yes—
139#95×C61C2314yes—
140#95×C61C2314yes—
141C23×D61C2411yescharacteristic
142C23×D61C2414yes—
143C22×C4×C61C2411yescharacteristic
144C22×C4×C61C2411yescharacteristic
145C22×C4×C61C2414yes—
146C22×C4×C61C2414yes—
147C22×C4×C61C2412yes—
148C22×C4×C61C2412yes—
149C22×C4×C61C2411yescharacteristic
150C22×Dic61C2414yes—
151C22×Dic61C2412yes—
152C22×Dic61C2412yes—
153C22×Dic61C2411yescharacteristic
154C22×Dic61C2411yescharacteristic
155C22×Dic61C2411yescharacteristic
156C2×D244—24no—
157C2×D244—24no—
158C2×D244—44no—
159C2×D244—44no—
160C2×D244—44no—
161C2×D244—44no—
162C2×D244C22×C418yes—
163C2×D244C22×C4116yes—
164C2×D244C22×C4116yes—
165C2×C4×D61—22no—
166C2×C4×D61—22no—
167C2×C4×D61—28no—
168C2×C4×D61—22no—
169C2×C4×D61—22no—
170C2×C4×D61—28no—
171C2×C4×D61—28no—
172C2×C4×D61—24no—
173C2×C4×D61—28no—
174C2×C4×D61—44no—
175C2×C4×D61—42no—
176C2×C4×D61—44no—
177C2×C4×D61—44no—
178C2×C4×D61—44no—
179C2×C4×D61C22×C414yes—
180C2×C4×D61C22×C4116yes—
181C4⋊C4×C61—24no—
182C4⋊C4×C61—44no—
183C4⋊C4×C61—44no—
184C4⋊C4×C61C22×C418yes—
185C4⋊C4×C61C22×C4116yes—
186C42×C61—22no—
187C42×C61—44no—
188C42×C61C22×C414yes—
189C42×C61#4114yes—
190#11758198—24no—
191#11758198—28no—
192#11758198—28no—
193#11758198—44no—
194#11758198—44no—
195#11758198—44no—
196#11758198—44no—
197#11758198—44no—
198#11758198C22×C418yes—
199#11758198C22×C4116yes—
200#11758198C22×C4116yes—
201C4⋊Dic61—24no—
202C4⋊Dic61—44no—
203#11758196—24no—
204#11758196C22×C418yes—
205C4×Dic61—22no—
206C4×Dic61—22no—
207C4×Dic61—22no—
208C4×Dic61—24no—
209C4×Dic61—28no—
210C4×Dic61—24no—
211C4×Dic61—44no—
212C4×Dic61—42no—
213C4×Dic61—42no—
214C4×Dic61C22×C414yes—
215C4×Dic61#4114yes—
216C23×C61C2511yescharacteristic
217C22×D61—21no—
218C22×D61—24no—
219C22×D61—24no—
220C22×D61—21no—
221C22×D61—24no—
222C22×D61—22no—
223C22×D61—24no—
224C22×D61—24no—
225C22×D61—22no—
226C22×D61—24no—
227C22×D61C23×C412yes—
228C22×D61C23×C418yes—
229C2×C4×C61—22no—
230C2×C4×C61—24no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—24no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—24no—
238C2×C4×C61—24no—
239C2×C4×C61—22no—
240C2×C4×C61—24no—
241C2×C4×C61—22no—
242C2×C4×C61C23×C414yes—
243C2×C4×C61C23×C418yes—
244C2×C4×C61C23×C412yes—
245C2×C4×C61C2×#4114yes—
246C2×C4×C61C2×#4112yes—
247C2×C4×C61C2×#4112yes—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—24no—
255C2×Dic61—24no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—24no—
259C2×Dic61C23×C414yes—
260C2×Dic61C23×C412yes—
261C2×Dic61C2×#4112yes—
262C2×Dic61C2×#4112yes—
263C2×Dic61H(5,𝔽2)14yes—
264D244—216no—
265D244—416no—
266D244—416no—
267C4×D61—28no—
268C4×D61—416no—
269C4×D61—44no—
270C4×D61—416no—
271C4×D61—44no—
272C4×D61—416no—
273C4×#540—611nomaximal, Sylow
274C22×C61C24×C411yescharacteristic
275C22×C61C22×#4111yescharacteristic
276C22×C61C22×#4111yescharacteristic
277C22×C61C2×H(5,𝔽2)12yes—
278C22×C61#58512yes—
279C2×D61—22no—
280C2×D61—28no—
281C2×D61—28no—
282C2×D61—24no—
283C2×D61—28no—
284C2×D61—44no—
285C2×D61—41no—
286C2×D61—41no—
287C2×D61—44no—
288C2×D61—44no—
289C2×D61—42no—
290C2×D61—44no—
291C4×C61—28no—
292C4×C61—24no—
293C4×C61—24no—
294C4×C61—41no—
295C4×C61—44no—
296C4×C61—41no—
297C4×C61C4×#4114yes—
298Dic61—22no—
299Dic61—24no—
300Dic61—22no—
301Dic61—44no—
302Dic61—42no—
303Dic61#54014yescomplemented
304C2×#540—611no—
305#2335—614no—
306#2333—614no—
307#2330D6111yescharacteristic, complemented
308#2328—611no—
309#2324—612no—
310#2319—612no—
311#2314—611no—
312C4×#127—611no—
313C4×#125—611no—
314C4×#123—614no—
315C4×#121—612no—
316C4×#121—614no—
317C42×D4—612no—
318C4×#117—611no—
319C2×C61C2×C4×#4111yescharacteristic
320C2×C61C4×H(5,𝔽2)12yes—
321C2×C61C2×#54011yescharacteristic
322C2×C61#345511yescharacteristic
323C2×C61#346012yes—
324D61—42no—
325D61—48no—
326#540—6116no—
327C2×#127—611no—
328C2×#125C2×D6111yescharacteristic
329C2×#123—614no—
330C2×#121—612no—
331C2×#121—614no—
332C2×C4×D4—612no—
333C2×C4×D4—612no—
334C2×C4×D4—614no—
335C2×C4×D4—614no—
336C2×C4×D4—612no—
337C2×#117—611no—
338C2×#117—611no—
339C22×C42—611no—
340C22×C42—614no—
341#390—612no—
342#390—614no—
343#390—612no—
344#390—614no—
345#389—612no—
346#389C2×D6112yes—
347#389C2×D6114yes—
348#388—614no—
349#388—614no—
350#388—614no—
351#388—614no—
352#388—614no—
353#387—614no—
354#387—614no—
355#387C2×D6114yes—
356#386—614no—
357#386—614no—
358#386—614no—
359#386—614no—
360#385—614no—
361#384C2×D6111yescharacteristic
362#382—611no—
363#381—614no—
364#380—612no—
365#380—614no—
366C4×C4⋊C4—612no—
367C4×C4⋊C4—614no—
368C4×C4⋊C4—612no—
369C4×C4⋊C4—612no—
370C4×C4⋊C4—612no—
371C4×C4⋊C4C2×D6112yes—
372C4×C22⋊C4—614no—
373C4×C22⋊C4—612no—
374C4×C22⋊C4—614no—
375C4×C22⋊C4—612no—
376C4×C22⋊C4—614no—
377C4×C22⋊C4—614no—
378#376—611no—
379#376C2×D6111yescharacteristic
380C43—611no—
381C43—611no—
382C61C4×#54011yescharacteristic, complemented, Sylow
383C22×D4—612no—
384C22×D4—614no—
385C22×D4—614no—
386C23×C4—611no—
387C23×C4—614no—
388C23×C4—611no—
389C23×C4—614no—
390C23×C4—614no—
391C23×C4—611no—
392#127—618no—
393#125C4×D6118yescomplemented
394#123—6132no—
395#121—6132no—
396#121—6116no—
397C4×D4—6116no—
398C4×D4—1228no—
399C4×D4—1228no—
400C4×D4—2448no—
401C4×D4—2448no—
402#117—618no—
403#117—1224no—
404C2×C4⋊C4—612no—
405C2×C4⋊C4—612no—
406C2×C4⋊C4—612no—
407C2×C4⋊C4—614no—
408C2×C4⋊C4C22×D6112yes—
409C2×C4⋊C4C22×D6114yes—
410C2×C4⋊C4C22×D6114yes—
411C2×C4⋊C4C22×D6112yes—
412C2×C4⋊C4C22×D6114yes—
413C2×C22⋊C4—612no—
414C2×C22⋊C4—614no—
415C2×C22⋊C4—614no—
416C2×C22⋊C4—612no—
417C2×C22⋊C4—614no—
418C2×C22⋊C4—614no—
419C2×C22⋊C4—614no—
420C2×C22⋊C4—614no—
421C2×C22⋊C4—614no—
422C2×C22⋊C4—614no—
423C2×C22⋊C4—614no—
424C2×C42—611no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—612no—
428C2×C42—614no—
429C2×C42—612no—
430C2×C42—611no—
431C2×C42—611no—
432C2×C42—612no—
433C2×C42—612no—
434C2×C42—614no—
435C2×C42—1224no—
436C2×C42—2448no—
437C2×C42C22×D6111yescharacteristic
438C2×C42C22×D6114yes—
439C2×C42C22×D6111yescharacteristic
440C2×C42C22×D6111yescharacteristic
441#95—612no—
442#95—612no—
443#95—614no—
444#95—614no—
445#95—614no—
446#95—614no—
447#95—614no—
448#95C22×D6112yes—
449#95C22×D6112yes—
450#95C22×D6114yes—
451#95C22×D6114yes—
452C24—614no—
453C24—611no—
454C2×D4—618no—
455C2×D4—6116no—
456C2×D4—6116no—
457C2×D4—1224no—
458C2×D4—1224no—
459C2×D4—2444no—
460C2×D4—2444no—
461C2×D4—2444no—
462C2×D4—2444no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—612no—
466C22×C4—611no—
467C22×C4—614no—
468C22×C4—6116no—
469C22×C4—614no—
470C22×C4—612no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4—1228no—
474C22×C4—1228no—
475C22×C4—1224no—
476C22×C4—1228no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1228no—
480C22×C4—2444no—
481C22×C4—2444no—
482C22×C4—2444no—
483C22×C4—2444no—
484C22×C4—2442no—
485C22×C4C23×D6111yescharacteristic
486C22×C4C23×D6111yescharacteristic
487C22×C4C23×D6114yes—
488C22×C4C23×D6112yes—
489C22×C4C23×D6114yes—
490C22×C4C23×D6112yes—
491C22×C4C23×D6111yescharacteristic
492C4⋊C4—24no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—618no—
496C4⋊C4—1224no—
497C4⋊C4—1224no—
498C4⋊C4—2444no—
499C4⋊C4C2×C4×D6118yes—
500C4⋊C4C2×C4×D61116yes—
501C22⋊C4—618no—
502C22⋊C4—6116no—
503C22⋊C4—6116no—
504C22⋊C4—1228no—
505C22⋊C4—1224no—
506C22⋊C4—1228no—
507C22⋊C4—2444no—
508C22⋊C4—2444no—
509C22⋊C4—2444no—
510C22⋊C4—2444no—
511C22⋊C4—2444no—
512C42—22no—
513C42—44no—
514C42—614no—
515C42—614no—
516C42—1222no—
517C42—1222no—
518C42—1224no—
519C42—1228no—
520C42—1224no—
521C42—1222no—
522C42—2444no—
523C42—2442no—
524C42—2442no—
525C42C2×C4×D6114yes—
526C42#1175822614yes—
527C23—612no—
528C23—618no—
529C23—1224no—
530C23—1221no—
531C23—1221no—
532C23—1224no—
533C23—1222no—
534C23—1224no—
535C23—1224no—
536C23—1224no—
537C23—1222no—
538C23—1224no—
539C23C24×D6111yescharacteristic
540D4—12216no—
541D4—24416no—
542D4—24416no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—612no—
557C2×C4—614no—
558C2×C4—614no—
559C2×C4—612no—
560C2×C4—612no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1222no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—1228no—
567C2×C4—1222no—
568C2×C4—1224no—
569C2×C4—1224no—
570C2×C4—1222no—
571C2×C4—1222no—
572C2×C4—1224no—
573C2×C4—24416no—
574C2×C4—24416no—
575C2×C4—2444no—
576C2×C4—24416no—
577C2×C4—2444no—
578C2×C4C22×C4×D6114yes—
579C2×C4C22×C4×D6118yes—
580C2×C4C22×C4×D6112yes—
581C2×C4#41×D6112yes—
582C2×C4C2×#1175822614yes—
583C2×C4C2×#1175822612yes—
584C22—1222no—
585C22—1228no—
586C22—1228no—
587C22—1224no—
588C22—1228no—
589C22—2441no—
590C22—2444no—
591C22—2444no—
592C22—2442no—
593C22—2444no—
594C22—2444no—
595C22—2441no—
596C22C23×C4×D6111yescharacteristic
597C22C2×#41×D6111yescharacteristic
598C22C22×#1175822611yescharacteristic
599C22H(5,𝔽2)×D6112yes—
600C22#1684059912yes—
601C4—24no—
602C4—28no—
603C4—24no—
604C4—41no—
605C4—41no—
606C4—44no—
607C4—614no—
608C4—1222no—
609C4—1222no—
610C4—1224no—
611C4—2442no—
612C4—2444no—
613C4C4×#1175822614yes—
614C2—2442no—
615C2—2448no—
616C2C2×C4×#1175822611yescharacteristic
617C2#2145902512yes—
618C2#540×D6111yescharacteristic
619C2#2145921011yescharacteristic
620C2#2145924212yes—
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingC61×#2330C2
socle, cosocleC23×C61C25
prime radicalC2×#16840368C2
hypercenter#2330D61
hypocenterC61C4×#540
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2330

Complements

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

NH ≅ G/Ntype
2. #21448612C2semidirect
3. #21448583C2semidirect
4. #21448544C2semidirect
5. #21448513C2semidirect
6. #21448458C2semidirect
7. #21448427C2semidirect
8. #21448381C2semidirect
9. #125×Dic61C2semidirect
10. #21448340C2semidirect
11. #21448327C2semidirect
12. #21448327C2semidirect
13. C4×#14519739C2semidirect
14. #21448158C2semidirect
15. C61×#2330C2semidirect
16. #16840368C4semidirect
81. #14519774C2×C4semidirect
82. #14519762C2×C4semidirect
83. #14519752C2×C4semidirect
84. #14519752C2×C4semidirect
89. #14519739C2×C4semidirect
91. #14519734C2×C4semidirect
127. #125×C61C2×C4semidirect
303. Dic61#540semidirect
307. #2330D61semidirect
382. C61C4×#540semidirect
393. #125C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403682
#214486122
#214485832
#214485442
#214485132
#214484582
#214484272
#214483812
#125×Dic612
#214483402
#214483272
C4×#145197392
#214481582
C61×#23302
C4×#54061