Finite Group Databasebeta

← #29843605

Subgroups — #29843605

Order 15616.

2390 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840367C211yesmaximal, characteristic
2#21448972C214yesmaximal, complemented
3#21448915C212yesmaximal, complemented
4#21448883C212yesmaximal, complemented
5#21448832C214yesmaximal, complemented
6#21448794C212yesmaximal, complemented
7#21448719C212yesmaximal, complemented
8#21448608C214yesmaximal, complemented
9#21448508C212yesmaximal, complemented
10#21448492C214yesmaximal, complemented
11#21448325C212yesmaximal, complemented
12#21448303C211yesmaximal, characteristic, complemented
13C61×#2314C211yesmaximal, characteristic, complemented
14#16840367C4116yescomplemented
15C2×#14519774C2211yescharacteristic
16C2×#14519769C2211yescharacteristic
17C2×#14519762C2214yes—
18C2×#14519751C2212yes—
19C2×#14519750C2214yes—
20C2×C4×D244C2214yes—
21C2×C4×D244C2212yes—
22C2×#14519739C2212yes—
23C2×#14519739C2212yes—
24C2×#14519737C2214yes—
25C2×#14519736C2211yescharacteristic
26C2×C4⋊C4×D61C2214yes—
27C22×C4×Dic61C2211yescharacteristic
28#16839664C2214yes—
29#16839663C2212yes—
30#16839663C2212yes—
31#16839662C2214yes—
32#16839661C2212yes—
33#16839661C2212yes—
34#16839659C2214yes—
35#16839658C2212yes—
36#16839658C2212yes—
37#16839654C2214yes—
38#16839654C2214yes—
39#16839654C2214yes—
40#16839651C2214yes—
41#16839649C2214yes—
42#16839646C2214yes—
43#16839640C2214yes—
44#16839640C2214yes—
45#16839639C2214yes—
46#16839638C2214yes—
47#16839637C2212yes—
48#16839637C2212yes—
49#16839636C2214yes—
50#16839633C2214yes—
51#16839633C2214yes—
52#16839624C2214yes—
53#16839619C2214yes—
54C4⋊C4×Dic61C2212yes—
55C4⋊C4×Dic61C2212yes—
56C4×#11758196C2214yes—
57C4×C4⋊Dic61C2212yes—
58C4×C4⋊Dic61C2212yes—
59#16839609C2214yes—
60#16839609C2214yes—
61C2×#117×C61C2211yescharacteristic
62C61×#388C2214yes—
63C61×#387C2214yes—
64C61×#384C2211yescharacteristic
65C61×#382C2211yescharacteristic
66C61×#380C2212yes—
67C4×C4⋊C4×C61C2212yes—
68C22×D244C2312yes—
69C22×D244C2314yes—
70C22×C22⋊D61C2314yes—
71C23×Dic61C2311yescharacteristic
72C23×Dic61C2311yescharacteristic
73C22×C4×D61C2314yes—
74C22×C4×D61C2314yes—
75C22×C4×D61C2314yes—
76#14519774C2×C418yescomplemented
77#14519769C2×C418yescomplemented
78#14519762C2×C4132yescomplemented
79#14519751C2×C4116yescomplemented
80#14519750C2×C4132yescomplemented
81C4×D244—48no—
82C4×D244C2×C4116yescomplemented
83#14519739—44no—
84#14519739—44no—
85#14519737—48no—
86#14519736—42no—
87C2×C4⋊Dic61C2312yes—
88C2×C4⋊Dic61C2312yes—
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×C4⋊Dic61C2312yes—
92C2×C4⋊Dic61C2312yes—
93C2×C4⋊Dic61C2312yes—
94C2×#11758196C2314yes—
95C2×#11758196C2314yes—
96C2×#11758196C2314yes—
97C4⋊C4×D61—48no—
98C2×C22⋊Dic61C2312yes—
99C2×#11758198C2314yes—
100C2×#11758198C2314yes—
101C2×#11758198C2314yes—
102C2×#11758198C2314yes—
103C2×#11758198C2314yes—
104C2×#11758198C2314yes—
105C2×#11758198C2314yes—
106C2×#11758198C2314yes—
107C2×C4×Dic61—42no—
108C2×C4×Dic61C2312yes—
109C2×C4×Dic61C2312yes—
110C2×C4×Dic61C2312yes—
111C2×C4×Dic61C2314yes—
112C2×C4×Dic61C2314yes—
113#14519684C2312yes—
114#14519684C2314yes—
115#14519684C2312yes—
116#14519684C2312yes—
117#14519684C2312yes—
118#14519684C2314yes—
119#14519684C2314yes—
120#14519684C2314yes—
121C23×C4×C61C2311yescharacteristic
122#117×C61C2×C418yescomplemented
123C2×C4⋊C4×C61C2312yes—
124C2×C4⋊C4×C61C2312yes—
125C2×C4⋊C4×C61C2314yes—
126C2×C4⋊C4×C61C2312yes—
127C2×C22⋊C4×C61C2314yes—
128C2×C22⋊C4×C61C2312yes—
129C2×C42×C61C2314yes—
130C2×C42×C61C2311yescharacteristic
131C2×C42×C61C2311yescharacteristic
132#95×C61C2314yes—
133#95×C61C2314yes—
134#95×C61C2314yes—
135C24×C61C2411yescharacteristic
136C23×D61C2414yes—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61C22×C414yes—
140C22×C4×C61C2414yes—
141C22×C4×C61C2414yes—
142C22×C4×C61C2412yes—
143C22×C4×C61C2412yes—
144C22×C4×C61C2411yescharacteristic
145C22×C4×C61C2411yescharacteristic
146C2×C22⋊D61—44no—
147C2×C22⋊D61—44no—
148C2×C22⋊D61C22×C4116yes—
149C22×Dic61—24no—
150C22×Dic61—42no—
151C22×Dic61—41no—
152C22×Dic61—41no—
153C22×Dic61C2412yes—
154C22×Dic61C2412yes—
155C22×Dic61C2412yes—
156C22×Dic61C2414yes—
157C22×Dic61C2412yes—
158C2×D244—24no—
159C2×D244—24no—
160C2×D244—44no—
161C2×D244—44no—
162C2×D244C22×C418yes—
163C2×D244C22×C4116yes—
164C2×C4×D61—28no—
165C2×C4×D61—44no—
166C2×C4×D61—44no—
167C2×C4×D61—44no—
168C2×C4×D61—44no—
169C2×C4×D61—44no—
170C2×C4×D61—44no—
171C2×C4×D61—44no—
172C2×C4×D61C22×C4116yes—
173C4⋊C4×C61—24no—
174C4⋊C4×C61—24no—
175C4⋊C4×C61—44no—
176C4⋊C4×C61C22×C418yes—
177C22⋊C4×C61—44no—
178C22⋊C4×C61C22×C418yes—
179C42×C61—44no—
180C42×C61C22×C414yes—
181C42×C61C22×C414yes—
182C22⋊Dic61—42no—
183#11758198—28no—
184#11758198—44no—
185#11758198—44no—
186#11758198—44no—
187#11758198—44no—
188#11758198—44no—
189#11758198C22×C4116yes—
190#11758198C22×C4116yes—
191C4⋊Dic61—24no—
192C4⋊Dic61—24no—
193C4⋊Dic61—42no—
194C4⋊Dic61—42no—
195C4⋊Dic61—41no—
196C4⋊Dic61—41no—
197C4⋊Dic61C22×C418yes—
198#11758196—28no—
199#11758196—44no—
200#11758196C22×C4116yes—
201C4×Dic61—24no—
202C4×Dic61—24no—
203C4×Dic61—42no—
204C4×Dic61—44no—
205C4×Dic61—44no—
206C23×C61—21no—
207C23×C61—22no—
208C23×C61—21no—
209C23×C61—22no—
210C23×C61C23×C412yes—
211C23×C61C2511yescharacteristic
212C22×D61—24no—
213C22×D61—24no—
214C22×D61—24no—
215C22×D61—24no—
216C22×D61—24no—
217C22×D61—24no—
218C22×D61C23×C418yes—
219C2×C4×C61—21no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—21no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—28no—
226C2×C4×C61—24no—
227C2×C4×C61—24no—
228C2×C4×C61—24no—
229C2×C4×C61—24no—
230C2×C4×C61—24no—
231C2×C4×C61—24no—
232C2×C4×C61C23×C414yes—
233C2×C4×C61C23×C412yes—
234C2×C4×C61C23×C414yes—
235C2×C4×C61C23×C412yes—
236C2×C4×C61H(5,𝔽2)12yes—
237C2×C4×C612−1+412yes—
238C22⋊D61—416no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—24no—
242C2×Dic61—24no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—44no—
253C2×Dic61—44no—
254C2×Dic61C23×C418yes—
255C2×Dic61H(5,𝔽2)14yes—
256D244—216no—
257D244—416no—
258C4×D61—416no—
259C4×D61—88no—
260C4×D61—88no—
261#34418—611nomaximal, Sylow
262C22×C61—22no—
263C22×C61—24no—
264C22×C61—41no—
265C22×C61—41no—
266C22×C61—42no—
267C22×C61C24×C411yescharacteristic
268C22×C61C2×H(5,𝔽2)12yes—
269C22×C61C2×H(5,𝔽2)11yescharacteristic
270C22×C61C2×2−1+411yescharacteristic
271C22×C61#58512yes—
272C2×D61—28no—
273C2×D61—28no—
274C2×D61—28no—
275C2×D61—44no—
276C2×D61—44no—
277C2×D61—44no—
278C2×D61—44no—
279C4×C61—22no—
280C4×C61—24no—
281C4×C61—24no—
282C4×C61—44no—
283C4×C61—44no—
284C4×C61#52014yes—
285Dic61—28no—
286Dic61—24no—
287Dic61—42no—
288Dic61—42no—
289Dic61—42no—
290C2×#540—611no—
291#2377—614no—
292#2370—612no—
293#2366—612no—
294#2361—614no—
295#2357—612no—
296D4×C4⋊C4—612no—
297#2335—614no—
298#2324—612no—
299#2322—614no—
300#2314D6111yescharacteristic, complemented
301C4×#121—612no—
302#2296—611no—
303C2×C61—42no—
304C2×C61C2×#52011yescharacteristic
305C2×C61C4×H(5,𝔽2)12yes—
306C2×C61#353711yescharacteristic
307C2×C61#353911yescharacteristic
308C2×C61#355412yes—
309D61—48no—
310#540—6116no—
311C2×#127—611no—
312C2×#125—611no—
313C2×#123—614no—
314C2×#121—612no—
315C2×#121—614no—
316C2×C4×D4—612no—
317C2×C4×D4—612no—
318C2×C4×D4—614no—
319C2×C4×D4—612no—
320C2×C4×D4—614no—
321C2×#117—611no—
322C2×#117C2×D6111yescharacteristic
323C22×C4⋊C4—614no—
324C22×C42—611no—
325#390—612no—
326#390—612no—
327#390—614no—
328#390—614no—
329#389—612no—
330#389—612no—
331#389—612no—
332#389—612no—
333#389—614no—
334#388—614no—
335#388—614no—
336#388—614no—
337#388—614no—
338#388C2×D6114yes—
339#387—614no—
340#387—614no—
341#387C2×D6114yes—
342#386—614no—
343#386—614no—
344#386—614no—
345#386—614no—
346#385—614no—
347#385—614no—
348#385—612no—
349#385—612no—
350#385—614no—
351#384C2×D6111yescharacteristic
352#382C2×D6111yescharacteristic
353#381—614no—
354#380—614no—
355#380C2×D6112yes—
356C4×C4⋊C4—612no—
357C4×C4⋊C4—612no—
358C4×C4⋊C4—614no—
359C4×C4⋊C4—612no—
360C4×C4⋊C4—612no—
361C4×C4⋊C4C2×D6112yes—
362C4×C22⋊C4—614no—
363C4×C22⋊C4—614no—
364C61#3441811yescharacteristic, complemented, Sylow
365C22×D4—614no—
366C22×D4—612no—
367C22×D4—614no—
368C23×C4—614no—
369C23×C4—611no—
370C23×C4—611no—
371C23×C4—614no—
372C23×C4—614no—
373C23×C4C22×D6111yescharacteristic
374#127—618no—
375#125—618no—
376#123—6132no—
377#121—6116no—
378#121—6132no—
379C4×D4—6116no—
380C4×D4—2448no—
381C4×D4—2444no—
382C4×D4—2444no—
383C4×D4—2448no—
384#117—2442no—
385#117C4×D6118yescomplemented
386C2×C4⋊C4—612no—
387C2×C4⋊C4—614no—
388C2×C4⋊C4—612no—
389C2×C4⋊C4—614no—
390C2×C4⋊C4—612no—
391C2×C4⋊C4—614no—
392C2×C4⋊C4—611no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—612no—
395C2×C4⋊C4—612no—
396C2×C4⋊C4—2448no—
397C2×C4⋊C4C22×D6112yes—
398C2×C4⋊C4C22×D6112yes—
399C2×C4⋊C4C22×D6114yes—
400C2×C4⋊C4C22×D6112yes—
401C2×C22⋊C4—614no—
402C2×C22⋊C4—614no—
403C2×C22⋊C4—614no—
404C2×C22⋊C4—612no—
405C2×C22⋊C4—614no—
406C2×C22⋊C4—614no—
407C2×C22⋊C4—614no—
408C2×C22⋊C4—614no—
409C2×C22⋊C4—614no—
410C2×C22⋊C4C22×D6114yes—
411C2×C22⋊C4C22×D6112yes—
412C2×C42—612no—
413C2×C42—612no—
414C2×C42—614no—
415C2×C42—612no—
416C2×C42—614no—
417C2×C42—2442no—
418C2×C42C22×D6114yes—
419C2×C42C22×D6111yescharacteristic
420C2×C42C22×D6111yescharacteristic
421#95—612no—
422#95—614no—
423#95—612no—
424#95—614no—
425#95—612no—
426#95—614no—
427#95—614no—
428#95—612no—
429#95C22×D6114yes—
430#95C22×D6114yes—
431#95C22×D6114yes—
432C24—614no—
433C24C23×D6111yescharacteristic
434C2×D4—6116no—
435C2×D4—618no—
436C2×D4—6116no—
437C2×D4—1224no—
438C2×D4—1224no—
439C2×D4—2444no—
440C2×D4—2444no—
441C2×D4—2444no—
442C2×D4—2444no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—614no—
446C22×C4—612no—
447C22×C4—6116no—
448C22×C4—612no—
449C22×C4—612no—
450C22×C4—612no—
451C22×C4—1228no—
452C22×C4—1224no—
453C22×C4—2444no—
454C22×C4—2444no—
455C22×C4—2444no—
456C22×C4—2444no—
457C22×C4—2444no—
458C22×C4—2444no—
459C22×C4—2444no—
460C22×C4—2441no—
461C22×C4—2441no—
462C22×C4—2442no—
463C22×C4C2×C4×D6114yes—
464C22×C4C23×D6114yes—
465C22×C4C23×D6114yes—
466C22×C4C23×D6112yes—
467C22×C4C23×D6112yes—
468C22×C4C23×D6111yescharacteristic
469C22×C4C23×D6111yescharacteristic
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—44no—
473C4⋊C4—6116no—
474C4⋊C4—618no—
475C4⋊C4—1224no—
476C4⋊C4—1228no—
477C4⋊C4—1224no—
478C4⋊C4—2444no—
479C4⋊C4—2441no—
480C4⋊C4—2441no—
481C4⋊C4—2442no—
482C4⋊C4—2442no—
483C4⋊C4C2×C4×D6118yes—
484C22⋊C4—44no—
485C22⋊C4—6116no—
486C22⋊C4—6116no—
487C22⋊C4—1228no—
488C22⋊C4—2444no—
489C22⋊C4—2444no—
490C22⋊C4—2444no—
491C22⋊C4—2444no—
492C22⋊C4—2444no—
493C22⋊C4—2442no—
494C22⋊C4C2×C4×D6118yes—
495C42—44no—
496C42—1224no—
497C42—1224no—
498C42—2442no—
499C42—2444no—
500C42—2444no—
501C42C2×C4×D6114yes—
502C42C2×C4×D6114yes—
503C23—22no—
504C23—21no—
505C23—21no—
506C23—22no—
507C23—618no—
508C23—1224no—
509C23—1224no—
510C23—1224no—
511C23—1224no—
512C23—1224no—
513C23—1224no—
514C23C22×C4×D6112yes—
515C23C24×D6111yescharacteristic
516D4—12216no—
517D4—24416no—
518D4—24416no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—618no—
533C2×C4—614no—
534C2×C4—1222no—
535C2×C4—1224no—
536C2×C4—1222no—
537C2×C4—1224no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—4888no—
548C2×C4—4888no—
549C2×C4—24416no—
550C2×C4—2444no—
551C2×C4—2444no—
552C2×C4C22×C4×D6114yes—
553C2×C4C22×C4×D6114yes—
554C2×C4C22×C4×D6112yes—
555C2×C4C22×C4×D6112yes—
556C2×C4#1451982312yes—
557C2×C4#1451982412yes—
558C22—22no—
559C22—24no—
560C22—42no—
561C22—41no—
562C22—41no—
563C22—1228no—
564C22—1228no—
565C22—1228no—
566C22—2444no—
567C22—2444no—
568C22—2444no—
569C22—2444no—
570C22C23×C4×D6111yescharacteristic
571C22H(5,𝔽2)×D6112yes—
572C22C2×#1451982311yescharacteristic
573C22C2×#1451982411yescharacteristic
574C22#1684059912yes—
575C4—22no—
576C4—24no—
577C4—24no—
578C4—44no—
579C4—44no—
580C4—1228no—
581C4—1224no—
582C4—2442no—
583C4—2442no—
584C4—2442no—
585C4#1684024214yes—
586C2—42no—
587C2—2448no—
588C2C2×#1684024211yescharacteristic
589C2#2145902512yes—
590C2#2146005711yescharacteristic
591C2#2146008111yescharacteristic
592C2#2146028812yes—
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingC61×#2314C2
socle, cosocleC23×C61C25
prime radicalC2×#16840367C2
hypercenter#2314D61
hypocenterC61#34418
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2314

Complements

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

NH ≅ G/Ntype
2. #21448972C2semidirect
3. #21448915C2semidirect
4. #21448883C2semidirect
5. #21448832C2semidirect
6. #21448794C2semidirect
7. #21448719C2semidirect
8. #21448608C2semidirect
9. #21448508C2semidirect
10. #21448492C2semidirect
11. #21448325C2semidirect
12. #21448303C2semidirect
13. C61×#2314C2semidirect
14. #16840367C4semidirect
76. #14519774C2×C4semidirect
77. #14519769C2×C4semidirect
78. #14519762C2×C4semidirect
79. #14519751C2×C4semidirect
80. #14519750C2×C4semidirect
82. C4×D244C2×C4semidirect
122. #117×C61C2×C4semidirect
300. #2314D61semidirect
364. C61#34418semidirect
385. #117C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403672
#214489722
#214489152
#214488832
#214488322
#214487942
#214487192
#214486082
#214485082
#214484922
#214483252
#214483032
C61×#23142
#3441861