Finite Group Databasebeta

← #29839111

Subgroups — #29839111

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×#16840367C211yesmaximal, characteristic
2#21448615C214yesmaximal, complemented
3#21448582C214yesmaximal, complemented
4#21448558C211yesmaximal, characteristic, complemented
5#21448542C211yesmaximal, characteristic, complemented
6#21448508C212yesmaximal, complemented
7#21448457C212yesmaximal, complemented
8#21448381C211yesmaximal, characteristic, complemented
9#21448374C211yesmaximal, characteristic, complemented
10#21448340C214yesmaximal, complemented
11#21448331C214yesmaximal, complemented
12#21448325C212yesmaximal, complemented
13C4×#14519739C212yesmaximal, complemented
14#117×Dic61C211yesmaximal, characteristic, complemented
15C61×#2314C211yesmaximal, characteristic, complemented
16#16840367C4116yescomplemented
17C2×#14519774C2211yescharacteristic
18C2×#14519769C2211yescharacteristic
19C2×#14519762C2214yes—
20C2×#14519751C2212yes—
21C2×#14519750C2214yes—
22C2×C4×D244C2212yes—
23C2×#14519739C2212yes—
24C2×#14519739C2212yes—
25C2×#14519739C2214yes—
26C2×#14519737C2214yes—
27C2×#14519736C2211yescharacteristic
28C22×C4×Dic61C2211yescharacteristic
29C2×C42×D61C2214yes—
30#16839663C2212yes—
31#16839663C2212yes—
32#16839663C2214yes—
33#16839662C2214yes—
34#16839658C2212yes—
35#16839658C2212yes—
36#16839657C2214yes—
37#16839654C2214yes—
38#16839653C2214yes—
39#16839653C2214yes—
40#16839651C2214yes—
41#16839644C2214yes—
42#16839643C2214yes—
43#16839640C2214yes—
44#16839640C2214yes—
45#16839639C2214yes—
46#16839638C2214yes—
47#16839636C2214yes—
48#16839624C2214yes—
49#16839621C2214yes—
50C4⋊C4×Dic61C2212yes—
51C4⋊C4×Dic61C2212yes—
52C4⋊C4×Dic61C2212yes—
53C4⋊C4×Dic61C2212yes—
54#16839613C2214yes—
55C22⋊C4×Dic61C2212yes—
56C22⋊C4×Dic61C2212yes—
57C4×#11758198C2214yes—
58#16839609C2214yes—
59#16839609C2214yes—
60#16839609C2214yes—
61#16839608C2211yescharacteristic
62#16839608C2211yescharacteristic
63C42×Dic61C2211yescharacteristic
64C42×Dic61C2211yescharacteristic
65C2×#117×C61C2211yescharacteristic
66C61×#388C2214yes—
67C61×#387C2214yes—
68C61×#384C2211yescharacteristic
69C61×#382C2211yescharacteristic
70C61×#380C2212yes—
71C4×C4⋊C4×C61C2212yes—
72C22×D244C2312yes—
73C22×D244C2314yes—
74C22×C22⋊D61C2314yes—
75C23×Dic61C2311yescharacteristic
76C23×Dic61C2311yescharacteristic
77C22×C4×D61C2314yes—
78C22×C4×D61C2314yes—
79C22×C4×D61C2314yes—
80#14519774C2×C418yescomplemented
81#14519769C2×C418yescomplemented
82#14519762C2×C4132yescomplemented
83#14519751C2×C4116yescomplemented
84#14519750C2×C4132yescomplemented
85C4×D244C2×C4116yescomplemented
86#14519739—28no—
87#14519739—28no—
88#14519739—48no—
89#14519737—48no—
90#14519736—24no—
91C2×C4⋊Dic61C2312yes—
92C2×C4⋊Dic61C2312yes—
93C2×#11758196C2314yes—
94C2×#11758196C2314yes—
95C2×#11758196C2314yes—
96C2×C22⋊Dic61C2312yes—
97C2×#11758198C2314yes—
98C2×#11758198C2314yes—
99C2×#11758198C2314yes—
100C2×#11758198C2314yes—
101C2×#11758198C2314yes—
102C2×#11758198C2314yes—
103C2×#11758198C2314yes—
104C2×#11758198C2314yes—
105C2×C4×Dic61—24no—
106C2×C4×Dic61C2312yes—
107C2×C4×Dic61C2314yes—
108C2×C4×Dic61C2311yescharacteristic
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—48no—
119#14519684C2312yes—
120#14519684C2312yes—
121#14519684C2312yes—
122#14519684C2312yes—
123#14519684C2314yes—
124#14519684C2314yes—
125#14519684C2314yes—
126#14519684C2314yes—
127C23×C4×C61C2311yescharacteristic
128#117×C61C2×C418yescomplemented
129C2×C4⋊C4×C61C2312yes—
130C2×C4⋊C4×C61C2312yes—
131C2×C4⋊C4×C61C2314yes—
132C2×C4⋊C4×C61C2312yes—
133C2×C22⋊C4×C61C2314yes—
134C2×C22⋊C4×C61C2312yes—
135C2×C42×C61C2314yes—
136C2×C42×C61C2311yescharacteristic
137C2×C42×C61C2311yescharacteristic
138#95×C61C2314yes—
139#95×C61C2314yes—
140#95×C61C2314yes—
141C24×C61C2411yescharacteristic
142C23×D61C2414yes—
143C22×C4×C61—22no—
144C22×C4×C61—22no—
145C22×C4×C61C22×C414yes—
146C22×C4×C61C2414yes—
147C22×C4×C61C2414yes—
148C22×C4×C61C2412yes—
149C22×C4×C61C2412yes—
150C22×C4×C61C2411yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C2×C22⋊D61—44no—
153C2×C22⋊D61—44no—
154C2×C22⋊D61C22×C4116yes—
155C22×Dic61—22no—
156C22×Dic61—22no—
157C22×Dic61—24no—
158C22×Dic61—42no—
159C22×Dic61C2411yescharacteristic
160C22×Dic61C2411yescharacteristic
161C22×Dic61C2412yes—
162C22×Dic61C2414yes—
163C22×Dic61C2412yes—
164C22×Dic61C2411yescharacteristic
165C22×Dic61C2411yescharacteristic
166C2×D244—24no—
167C2×D244—24no—
168C2×D244—44no—
169C2×D244—44no—
170C2×D244C22×C418yes—
171C2×D244C22×C4116yes—
172C2×C4×D61—28no—
173C2×C4×D61—28no—
174C2×C4×D61—28no—
175C2×C4×D61—28no—
176C2×C4×D61—44no—
177C2×C4×D61—44no—
178C2×C4×D61—44no—
179C2×C4×D61—44no—
180C2×C4×D61C22×C4116yes—
181C4⋊C4×C61—24no—
182C4⋊C4×C61—24no—
183C4⋊C4×C61—44no—
184C4⋊C4×C61C22×C418yes—
185C22⋊C4×C61—44no—
186C22⋊C4×C61C22×C418yes—
187C42×C61—44no—
188C42×C61C22×C414yes—
189C42×C61C22×C414yes—
190C22⋊Dic61—24no—
191#11758198—28no—
192#11758198—28no—
193#11758198—44no—
194#11758198—44no—
195#11758198—44no—
196#11758198—44no—
197#11758198C22×C4116yes—
198#11758198C22×C4116yes—
199C4⋊Dic61—24no—
200C4⋊Dic61C22×C418yes—
201#11758196—44no—
202#11758196—44no—
203#11758196C22×C4116yes—
204C4×Dic61—22no—
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×Dic61#4114yes—
215C4×Dic61#4114yes—
216C23×C61—21no—
217C23×C61—21no—
218C23×C61—22no—
219C23×C61—22no—
220C23×C61C23×C412yes—
221C23×C61C2511yescharacteristic
222C22×D61—24no—
223C22×D61—24no—
224C22×D61—24no—
225C22×D61—24no—
226C22×D61—24no—
227C22×D61—24no—
228C22×D61C23×C418yes—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—28no—
236C2×C4×C61—24no—
237C2×C4×C61—24no—
238C2×C4×C61—24no—
239C2×C4×C61—24no—
240C2×C4×C61—24no—
241C2×C4×C61—24no—
242C2×C4×C61C23×C414yes—
243C2×C4×C61C23×C412yes—
244C2×C4×C61C23×C414yes—
245C2×C4×C61C23×C412yes—
246C2×C4×C61C2×#4112yes—
247C2×C4×C61C2×#4112yes—
248C22⋊D61—416no—
249C2×Dic61—22no—
250C2×Dic61—24no—
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—24no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—44no—
262C2×Dic61—44no—
263C2×Dic61C23×C418yes—
264C2×Dic61C2×#4112yes—
265C2×Dic61C2×#4114yes—
266C2×Dic61C2×#4112yes—
267C2×Dic61H(5,𝔽2)14yes—
268D244—216no—
269D244—416no—
270C4×D61—416no—
271C4×D61—416no—
272C4×D61—416no—
273C4×#540—611nomaximal, Sylow
274C22×C61—22no—
275C22×C61—24no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—42no—
279C22×C61C24×C411yescharacteristic
280C22×C61C22×#4111yescharacteristic
281C22×C61C22×#4111yescharacteristic
282C22×C61C2×H(5,𝔽2)12yes—
283C22×C61#58512yes—
284C2×D61—28no—
285C2×D61—28no—
286C2×D61—28no—
287C2×D61—44no—
288C2×D61—44no—
289C2×D61—44no—
290C2×D61—44no—
291C4×C61—22no—
292C4×C61—24no—
293C4×C61—24no—
294C4×C61—44no—
295C4×C61—44no—
296C4×C61C4×#4114yes—
297Dic61—28no—
298Dic61—22no—
299Dic61—24no—
300Dic61—41no—
301Dic61—41no—
302Dic61—42no—
303Dic61#54014yescomplemented
304C2×#540—611no—
305#2335—614no—
306#2333—614no—
307#2330—611no—
308#2328—611no—
309#2324—612no—
310#2319—612no—
311#2314D6111yescharacteristic, complemented
312C4×#127—611no—
313C4×#125—611no—
314C4×#123—614no—
315C4×#121—614no—
316C4×#121—612no—
317C42×D4—612no—
318C4×#117—611no—
319C2×C61—42no—
320C2×C61C2×C4×#4111yescharacteristic
321C2×C61C4×H(5,𝔽2)12yes—
322C2×C61C2×#54011yescharacteristic
323C2×C61#345511yescharacteristic
324C2×C61#346012yes—
325D61—48no—
326#540—6116no—
327C2×#127—611no—
328C2×#125—611no—
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×#117C2×D6111yescharacteristic
339C22×C42—611no—
340C22×C42—614no—
341#390—612no—
342#390—614no—
343#390—612no—
344#390—614no—
345#389—612no—
346#389—612no—
347#389—614no—
348#388—614no—
349#388—614no—
350#388—614no—
351#388—614no—
352#388C2×D6114yes—
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#382C2×D6111yescharacteristic
363#381—614no—
364#380—614no—
365#380C2×D6112yes—
366C4×C4⋊C4—612no—
367C4×C4⋊C4—612no—
368C4×C4⋊C4—612no—
369C4×C4⋊C4—612no—
370C4×C4⋊C4—614no—
371C4×C4⋊C4C2×D6112yes—
372C4×C22⋊C4—612no—
373C4×C22⋊C4—612no—
374C4×C22⋊C4—614no—
375C4×C22⋊C4—614no—
376C4×C22⋊C4—614no—
377C4×C22⋊C4—614no—
378#376—611no—
379#376—611no—
380C43—611no—
381C43—611no—
382C61C4×#54011yescharacteristic, complemented, Sylow
383C22×D4—614no—
384C22×D4—612no—
385C22×D4—614no—
386C23×C4—614no—
387C23×C4—611no—
388C23×C4—611no—
389C23×C4—614no—
390C23×C4—614no—
391C23×C4C22×D6111yescharacteristic
392#127—618no—
393#125—618no—
394#123—6132no—
395#121—6116no—
396#121—6132no—
397C4×D4—6116no—
398C4×D4—1228no—
399C4×D4—1228no—
400C4×D4—2448no—
401C4×D4—2448no—
402#117—1224no—
403#117C4×D6118yescomplemented
404C2×C4⋊C4—612no—
405C2×C4⋊C4—614no—
406C2×C4⋊C4—612no—
407C2×C4⋊C4—614no—
408C2×C4⋊C4—614no—
409C2×C4⋊C4C22×D6112yes—
410C2×C4⋊C4C22×D6112yes—
411C2×C4⋊C4C22×D6114yes—
412C2×C4⋊C4C22×D6112yes—
413C2×C22⋊C4—614no—
414C2×C22⋊C4—614no—
415C2×C22⋊C4—612no—
416C2×C22⋊C4—614no—
417C2×C22⋊C4—614no—
418C2×C22⋊C4—614no—
419C2×C22⋊C4—614no—
420C2×C22⋊C4—614no—
421C2×C22⋊C4—614no—
422C2×C22⋊C4C22×D6114yes—
423C2×C22⋊C4C22×D6112yes—
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—611no—
433C2×C42—612no—
434C2×C42—612no—
435C2×C42—614no—
436C2×C42—1224no—
437C2×C42—2448no—
438C2×C42C22×D6114yes—
439C2×C42C22×D6111yescharacteristic
440C2×C42C22×D6111yescharacteristic
441#95—612no—
442#95—612no—
443#95—612no—
444#95—612no—
445#95—614no—
446#95—614no—
447#95—614no—
448#95—614no—
449#95C22×D6114yes—
450#95C22×D6114yes—
451#95C22×D6114yes—
452C24—614no—
453C24C23×D6111yescharacteristic
454C2×D4—6116no—
455C2×D4—618no—
456C2×D4—6116no—
457C2×D4—1224no—
458C2×D4—1224no—
459C2×D4—2444no—
460C2×D4—2444no—
461C2×D4—2444no—
462C2×D4—2444no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—612no—
468C22×C4—614no—
469C22×C4—6116no—
470C22×C4—611no—
471C22×C4—612no—
472C22×C4—611no—
473C22×C4—1228no—
474C22×C4—1228no—
475C22×C4—1228no—
476C22×C4—1228no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1224no—
480C22×C4—2444no—
481C22×C4—2444no—
482C22×C4—2444no—
483C22×C4—2444no—
484C22×C4—2442no—
485C22×C4C2×C4×D6114yes—
486C22×C4C23×D6114yes—
487C22×C4C23×D6114yes—
488C22×C4C23×D6112yes—
489C22×C4C23×D6112yes—
490C22×C4C23×D6111yescharacteristic
491C22×C4C23×D6111yescharacteristic
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—44no—
495C4⋊C4—6116no—
496C4⋊C4—618no—
497C4⋊C4—1224no—
498C4⋊C4—2444no—
499C4⋊C4—2444no—
500C4⋊C4C2×C4×D6118yes—
501C22⋊C4—44no—
502C22⋊C4—6116no—
503C22⋊C4—6116no—
504C22⋊C4—1228no—
505C22⋊C4—1228no—
506C22⋊C4—1224no—
507C22⋊C4—2444no—
508C22⋊C4—2444no—
509C22⋊C4—2444no—
510C22⋊C4—2444no—
511C22⋊C4C2×C4×D6118yes—
512C42—44no—
513C42—614no—
514C42—614no—
515C42—1222no—
516C42—1222no—
517C42—1224no—
518C42—1228no—
519C42—1224no—
520C42—1222no—
521C42—1222no—
522C42—2444no—
523C42—2442no—
524C42—2442no—
525C42C2×C4×D6114yes—
526C42C2×C4×D6114yes—
527C23—21no—
528C23—21no—
529C23—22no—
530C23—22no—
531C23—618no—
532C23—1224no—
533C23—1224no—
534C23—1224no—
535C23—1224no—
536C23—1224no—
537C23—1224no—
538C23C22×C4×D6112yes—
539C23C24×D6111yescharacteristic
540D4—12216no—
541D4—24416no—
542D4—24416no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—612no—
557C2×C4—614no—
558C2×C4—618no—
559C2×C4—612no—
560C2×C4—614no—
561C2×C4—1222no—
562C2×C4—1224no—
563C2×C4—1222no—
564C2×C4—1222no—
565C2×C4—1224no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1222no—
569C2×C4—1222no—
570C2×C4—1222no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—24416no—
574C2×C4—24416no—
575C2×C4—24416no—
576C2×C4—2444no—
577C2×C4—2444no—
578C2×C4C22×C4×D6114yes—
579C2×C4C22×C4×D6114yes—
580C2×C4C22×C4×D6112yes—
581C2×C4C22×C4×D6112yes—
582C2×C4#41×D6112yes—
583C2×C4#41×D6112yes—
584C22—22no—
585C22—24no—
586C22—41no—
587C22—42no—
588C22—41no—
589C22—1228no—
590C22—1228no—
591C22—1228no—
592C22—2444no—
593C22—2444no—
594C22—2444no—
595C22—2444no—
596C22C23×C4×D6111yescharacteristic
597C22C2×#41×D6111yescharacteristic
598C22C2×#41×D6111yescharacteristic
599C22H(5,𝔽2)×D6112yes—
600C22#1684059912yes—
601C4—22no—
602C4—24no—
603C4—24no—
604C4—44no—
605C4—44no—
606C4—614no—
607C4—1222no—
608C4—1224no—
609C4—1228no—
610C4—2441no—
611C4—2442no—
612C4—2441no—
613C4#1684023014yes—
614C2—42no—
615C2—2448no—
616C2C2×#1684023011yescharacteristic
617C2#2145902512yes—
618C2#540×D6111yescharacteristic
619C2#2145921011yescharacteristic
620C2#2145923912yes—
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingC61×#2314C2
socle, cosocleC23×C61C25
prime radicalC2×#16840367C2
hypercenter#2314D61
hypocenterC61C4×#540
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2314

Complements

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

NH ≅ G/Ntype
2. #21448615C2semidirect
3. #21448582C2semidirect
4. #21448558C2semidirect
5. #21448542C2semidirect
6. #21448508C2semidirect
7. #21448457C2semidirect
8. #21448381C2semidirect
9. #21448374C2semidirect
10. #21448340C2semidirect
11. #21448331C2semidirect
12. #21448325C2semidirect
13. C4×#14519739C2semidirect
14. #117×Dic61C2semidirect
15. C61×#2314C2semidirect
16. #16840367C4semidirect
80. #14519774C2×C4semidirect
81. #14519769C2×C4semidirect
82. #14519762C2×C4semidirect
83. #14519751C2×C4semidirect
84. #14519750C2×C4semidirect
85. C4×D244C2×C4semidirect
128. #117×C61C2×C4semidirect
303. Dic61#540semidirect
311. #2314D61semidirect
382. C61C4×#540semidirect
403. #117C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403672
#214486152
#214485822
#214485582
#214485422
#214485082
#214484572
#214483812
#214483742
#214483402
#214483312
#214483252
C4×#145197392
#117×Dic612
C61×#23142
C4×#54061