Finite Group Databasebeta

← #29718363

Subgroups — #29718363

Order 15616.

1130 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840093C211yesmaximal, characteristic, complemented
2C22×#14519699C211yesmaximal, characteristic
3#21449470C211yesmaximal, characteristic
4#21448686C211yesmaximal, characteristic, complemented
5#21447192C212yesmaximal, complemented
6#21447061C212yesmaximal, complemented
7#21446749C212yesmaximal, complemented
8#21446447C212yesmaximal, complemented
9#21446242C211yesmaximal, characteristic, complemented
10#21445751C211yesmaximal, characteristic
11C61×#1850C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C2×#14519765C2212yes—
14C2×#14519756C2211yescharacteristic
15C2×#14519733—21no—
16C22×C4⋊Dic61C2211yescharacteristic
17#16840093—24no—
18C2×#14519709C2211yescharacteristic, complemented
19C2×#14519709C2211yescharacteristic, complemented
20C2×#14519702C2211yescharacteristic, complemented
21C2×#14519702C2211yescharacteristic, complemented
22C2×#14519702C2212yescomplemented
23C2×#14519699—24no—
24C2×#14519699C2212yes—
25C2×#14519699C2212yes—
26C2×#14519699C2212yes—
27C2×#14519699C2212yes—
28#16839680—22no—
29#16839664C2212yes—
30#16839660C2212yescomplemented
31#16839647—22no—
32#16839623—21no—
33#16839622C2211yescharacteristic
34#16839617—21no—
35#16839617—21no—
36C4×C4⋊Dic61C2212yescomplemented
37#16839525C2212yescomplemented
38C23×C8×C61C2211yescharacteristic
39C2×C4⋊C8×C61C2212yescomplemented
40C2×C22⋊C8×C61C2211yescharacteristic
41C4×C22⋊C4×C61C2211yescharacteristic
42C61×#336C2212yescomplemented
43C24×D61—21no—
44C22×Dic122C2311yescharacteristic
45C22×D244—22no—
46C22×D244—24no—
47C22×D244—22no—
48C22×D244—21no—
49C22×D244—42no—
50C22×D244C2312yes—
51C23×Dic61—21no—
52#14519765—24no—
53#14519765—42no—
54#14519756—22no—
55#14519756—22no—
56#14519733—44no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61—22no—
59C2×C4⋊Dic61D414yes—
60C2×C4⋊Dic61C2311yescharacteristic
61C2×C4⋊Dic61C2312yes—
62C2×C4⋊Dic61C2311yescharacteristic
63C2×C4⋊Dic61C2311yescharacteristic
64C2×#11758196—21no—
65C2×#11758196—22no—
66C2×C22⋊Dic61—21no—
67C2×C22⋊Dic61—21no—
68C2×#11758198—22no—
69C2×#11758198—22no—
70C2×#11758198—22no—
71C2×#11758198—22no—
72C2×C4×Dic61—22no—
73#14519709—22no—
74#14519709—22no—
75#14519702—22no—
76#14519702—22no—
77#14519702—42no—
78#14519699—24no—
79#14519699—24no—
80#14519699—24no—
81#14519699—42no—
82#14519684—22no—
83#14519684—21no—
84#14519684—22no—
85C23×C4×C61C2311yescharacteristic
86C22×C8×C61—22no—
87C22×C8×C61—42no—
88C22×C8×C61C2312yes—
89C22×C8×C61C2311yescharacteristic
90C22×C8×C61C2311yescharacteristic
91C2×C22⋊C4×C61D411yescharacteristic
92C2×C22⋊C4×C61D411yescharacteristic
93C2×C42×C61C2312yes—
94C4⋊C8×C61—42no—
95C22⋊C8×C61—22no—
96#95×C61D412yes—
97C24×C61C2×D411yescharacteristic
98C23×D61—22no—
99C23×D61—44no—
100C23×D61—42no—
101C23×D61—42no—
102C23×D61—42no—
103C23×D61—82no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61C2×D412yes—
107C22×C4×C61C2×D412yes—
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D412yes—
110C22×C4×C61#4112yes—
111C22×C4×C61C2411yescharacteristic
112C22×Dic61—24no—
113C22×Dic61—21no—
114C22×Dic61—21no—
115C22×Dic61—21no—
116C22×Dic61—22no—
117C22×Dic61—21no—
118C22×Dic61—42no—
119C22×Dic61—42no—
120C2×D244—28no—
121C2×D244—22no—
122C2×D244—22no—
123C2×D244—22no—
124C2×D244—22no—
125C2×D244—22no—
126C2×D244—48no—
127C2×D244—44no—
128C2×D244—44no—
129C2×D244—42no—
130C2×D244#4114yes—
131C2×Dic122—21no—
132C2×Dic122—21no—
133C2×Dic122—22no—
134C2×Dic122—22no—
135C2×C8×C61—21no—
136C2×C8×C61—21no—
137C2×C8×C61—22no—
138C2×C8×C61—22no—
139C2×C8×C61—22no—
140C2×C8×C61—22no—
141C2×C8×C61—22no—
142C2×C8×C61—44no—
143C22⋊C4×C61—22no—
144C22⋊C4×C61—22no—
145C42×C61—22no—
146C42×C61—22no—
147C22⋊Dic61—42no—
148C22⋊Dic61—42no—
149#11758198—44no—
150#11758198—44no—
151#11758198—82no—
152#11758198—82no—
153C4⋊Dic61—24no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—22no—
158C4⋊Dic61—21no—
159C4⋊Dic61—22no—
160C4⋊Dic61C2×D412yes—
161C4⋊Dic61C2×D412yes—
162#11758196—44no—
163#11758196—42no—
164C4×Dic61—44no—
165C23×C61—22no—
166C23×C61—22no—
167C23×C61#12112yes—
168C23×C61#13512yes—
169C23×C61AGL(1,ℤ/8)12yes—
170C23×C61C22×D411yescharacteristic
171C22×D61—24no—
172C22×D61—48no—
173C22×D61—42no—
174C22×D61—42no—
175C22×D61—42no—
176C22×D61—42no—
177C22×D61—42no—
178C22×D61—42no—
179C22×D61—88no—
180C22×D61—88no—
181C22×D61—84no—
182C22×D61—84no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—24no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61#12112yes—
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C2×#4112yes—
196C2×C4×C61C2×#4111yescharacteristic
197C2×C4×C61H(5,𝔽2)12yes—
198C2×Dic61—22no—
199C2×Dic61—22no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—48no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—42no—
208C2×Dic61—42no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211C2×Dic61—42no—
212D244—24no—
213D244—24no—
214D244—48no—
215D244—42no—
216D244—42no—
217Dic122—41no—
218Dic122—41no—
219Dic122—42no—
220C8×C61—41no—
221C8×C61—41no—
222C8×C61—42no—
223#24477—611nomaximal, Sylow
224C22×C61—22no—
225C22×C61—24no—
226C22×C61—22no—
227C22×C61—24no—
228C22×C61#46314yes—
229C22×C61C2×#12111yescharacteristic
230C22×C61#54612yes—
231C22×C61C2×#13511yescharacteristic
232C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
233C22×C61#57712yes—
234C2×D61—44no—
235C2×D61—44no—
236C2×D61—44no—
237C2×D61—816no—
238C2×D61—82no—
239C2×D61—82no—
240C2×D61—82no—
241C2×D61—82no—
242C4×C61—22no—
243C4×C61—42no—
244C4×C61—42no—
245C4×C61#52511yescharacteristic
246C4×C61#53511yescharacteristic
247C4×C61#55112yes—
248Dic61—42no—
249Dic61—42no—
250Dic61—81no—
251Dic61—81no—
252Dic61—82no—
253C2×#483—611no—
254C22×#102—611no—
255#2435—611no—
256#2344—611no—
257#2065—612no—
258#2042—612no—
259#1983—612no—
260#1926—612no—
261#1892—611no—
262#1850D6111yescharacteristic, complemented
263#1786—611no—
264C2×C61—24no—
265C2×C61#241111yescharacteristic
266C2×C61C2×#46311yescharacteristic
267C2×C61#304311yescharacteristic
268C2×C61#329612yes—
269C2×C61#330612yes—
270D61—84no—
271C23×D4—611no—
272C23×C8C2×D6111yescharacteristic
273C2×#124—612no—
274C2×C22⋊Q8—611no—
275C2×#117—1221no—
276C22×C4⋊C4—611no—
277#483—1224no—
278C2×#106—611no—
279C2×#106—611no—
280C2×C4⋊C8C2×D6112yescomplemented
281C2×#103—611no—
282C2×#103—611no—
283C2×#103—612no—
284C2×#102—612no—
285C2×#102—612no—
286C2×#102—612no—
287C2×#102—612no—
288C2×#102—1224no—
289C2×C22⋊C8C2×D6111yescharacteristic
290#394—1222no—
291#390—612no—
292#389—612no—
293#387—1222no—
294#381—1221no—
295#380—611no—
296#379—1221no—
297#379—1221no—
298C4×C4⋊C4—612no—
299C4×C22⋊C4C2×D6111yescharacteristic
300#340—612no—
301#336C2×D6112yescomplemented
302C61#2447711yescharacteristic, complemented, Sylow
303C25—1221no—
304C22×Q8—611no—
305C22×D4—612no—
306C22×D4—1224no—
307C22×D4—1221no—
308C22×D4—1222no—
309C22×D4—1222no—
310C22×D4—2442no—
311C23×C4—1221no—
312C23×C4C22×D6111yescharacteristic
313C22×C8—22no—
314C22×C8—42no—
315C22×C8C22×D6112yes—
316C22×C8C22×D6111yescharacteristic
317C22×C8C22×D6111yescharacteristic
318#124—1224no—
319#124—2442no—
320C22⋊Q8—1222no—
321C22⋊Q8—1222no—
322#117—2444no—
323C2×C4⋊C4—614no—
324C2×C4⋊C4—611no—
325C2×C4⋊C4—611no—
326C2×C4⋊C4—611no—
327C2×C4⋊C4—612no—
328C2×C4⋊C4—1221no—
329C2×C4⋊C4—1222no—
330C2×C4⋊C4—1222no—
331C2×C4⋊C4—1221no—
332C2×C22⋊C4—1222no—
333C2×C22⋊C4—1222no—
334C2×C22⋊C4—1221no—
335C2×C22⋊C4—1221no—
336C2×C22⋊C4—1222no—
337C2×C22⋊C4—1222no—
338C2×C22⋊C4D24411yescharacteristic
339C2×C22⋊C4D24411yescharacteristic
340C2×C42—1222no—
341C2×C42C22×D6112yes—
342#106—1222no—
343#106—1222no—
344C4⋊C8—42no—
345#103—1222no—
346#103—1222no—
347#103—2442no—
348#102—1224no—
349#102—1224no—
350#102—1224no—
351#102—2442no—
352C22⋊C8—22no—
353#95—1222no—
354#95—1221no—
355#95—1222no—
356#95D24412yes—
357C24—1222no—
358C24—4882no—
359C24—2444no—
360C24—2442no—
361C24—2442no—
362C24—2442no—
363C24C2×D24411yescharacteristic
364C2×Q8—1221no—
365C2×Q8—1222no—
366C2×Q8—1221no—
367C2×Q8—1222no—
368C2×D4—614no—
369C2×D4—1228no—
370C2×D4—1222no—
371C2×D4—1222no—
372C2×D4—1222no—
373C2×D4—1222no—
374C2×D4—1222no—
375C2×D4—2448no—
376C2×D4—2444no—
377C2×D4—2444no—
378C2×D4—2442no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—1221no—
382C22×C4—1221no—
383C22×C4—1224no—
384C22×C4—1222no—
385C22×C4—1221no—
386C22×C4—1221no—
387C22×C4—2442no—
388C22×C4—2442no—
389C22×C4C2×D24412yes—
390C22×C4C2×D24412yes—
391C22×C4C2×D24411yescharacteristic
392C22×C4D4×D6112yes—
393C22×C4#1175822612yes—
394C22×C4C23×D6111yescharacteristic
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—44no—
403C4⋊C4—612no—
404C4⋊C4—612no—
405C4⋊C4—1221no—
406C4⋊C4—1221no—
407C4⋊C4—1224no—
408C4⋊C4—1221no—
409C4⋊C4—1222no—
410C4⋊C4—1221no—
411C4⋊C4—1222no—
412C4⋊C4—2444no—
413C4⋊C4—2442no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—4882no—
417C22⋊C4—4882no—
418C22⋊C4—2444no—
419C22⋊C4—2444no—
420C22⋊C4—2442no—
421C22⋊C4—2442no—
422C42—22no—
423C42—22no—
424C42—2444no—
425C23—22no—
426C23—22no—
427C23—1224no—
428C23—4884no—
429C23—4888no—
430C23—4888no—
431C23—4884no—
432C23—2448no—
433C23—2442no—
434C23—2442no—
435C23—2442no—
436C23—2442no—
437C23—2442no—
438C23—2442no—
439C23#1451975212yes—
440C23#1451979912yes—
441C23#1451980512yes—
442C23C22×D24411yescharacteristic
443Q8—2441no—
444Q8—2441no—
445Q8—2442no—
446D4—1224no—
447D4—1224no—
448D4—2448no—
449D4—2442no—
450D4—2442no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—1222no—
462C2×C4—1222no—
463C2×C4—2441no—
464C2×C4—2441no—
465C2×C4—2442no—
466C2×C4—2442no—
467C2×C4—2442no—
468C2×C4—2442no—
469C2×C4—2442no—
470C2×C4—2448no—
471C2×C4—2441no—
472C2×C4—2441no—
473C2×C4—2441no—
474C2×C4—2441no—
475C2×C4#1451975212yes—
476C2×C4C2×D4×D6111yescharacteristic
477C2×C4#41×D6112yes—
478C2×C4C2×#1175822611yescharacteristic
479C2×C4#1451982212yes—
480C8—41no—
481C8—41no—
482C8—42no—
483C22—24no—
484C22—22no—
485C22—22no—
486C22—24no—
487C22—48816no—
488C22—4882no—
489C22—4882no—
490C22—4882no—
491C22—4882no—
492C22—2444no—
493C22—2444no—
494C22—2444no—
495C22#1683997814yes—
496C22C2×#1451975211yescharacteristic
497C22#1684040712yes—
498C22C2×#1451979911yescharacteristic
499C22C2×#1451980511yescharacteristic
500C22#1684057412yes—
501C4—22no—
502C4—42no—
503C4—42no—
504C4—4882no—
505C4—4881no—
506C4—4881no—
507C4—2442no—
508C4—2442no—
509C4#1684027011yescharacteristic
510C4#1684032711yescharacteristic
511C4#1684043912yes—
512C2—24no—
513C2—4884no—
514C2#2144920211yescharacteristic
515C2C2×#1683997811yescharacteristic
516C2#2145534111yescharacteristic
517C2#2145765612yes—
518C2#2145778212yes—
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1850C2
socleC23×C61C22×D4
prime radicalC23×D244C22
hypercenter#1850D61
hypocenterC61#24477
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1850

Complements

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

NH ≅ G/Ntype
1. C2×#16840093C2semidirect
4. #21448686C2semidirect
5. #21447192C2semidirect
6. #21447061C2semidirect
7. #21446749C2semidirect
8. #21446447C2semidirect
9. #21446242C2semidirect
11. C61×#1850C2semidirect
18. C2×#14519709C22semidirect
19. C2×#14519709C22semidirect
20. C2×#14519702C22semidirect
21. C2×#14519702C22semidirect
22. C2×#14519702C22semidirect
30. #16839660C22semidirect
36. C4×C4⋊Dic61C22semidirect
37. #16839525C22semidirect
39. C2×C4⋊C8×C61C22semidirect
42. C61×#336C22semidirect
262. #1850D61semidirect
280. C2×C4⋊C8C2×D61semidirect
301. #336C2×D61semidirect
302. C61#24477semidirect

Maximal subgroups

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

subgroupindex
C2×#168400932
C22×#145196992
#214494702
#214486862
#214471922
#214470612
#214467492
#214464472
#214462422
#214457512
C61×#18502
#2447761