Finite Group Databasebeta

← #29718385

Subgroups — #29718385

Order 15616.

1102 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840098C211yesmaximal, characteristic, complemented
2C22×#14519699C211yesmaximal, characteristic
3#21449470C211yesmaximal, characteristic
4#21448694C211yesmaximal, characteristic, complemented
5#21447206C212yesmaximal, complemented
6#21447061C212yesmaximal, complemented
7#21446749C212yesmaximal, complemented
8#21446440C212yesmaximal, complemented
9#21446242C211yesmaximal, characteristic, complemented
10#21445751C211yesmaximal, characteristic
11C61×#1850C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C2×#14519765C2212yes—
14C2×#14519756C2211yescharacteristic
15C22×C4⋊Dic61C2211yescharacteristic
16#16840098—24no—
17C2×#14519711C2211yescharacteristic, complemented
18C2×#14519711C2211yescharacteristic, complemented
19C2×#14519702C2211yescharacteristic, complemented
20C2×#14519702C2211yescharacteristic, complemented
21C2×#14519702C2212yescomplemented
22C2×#14519699—24no—
23C2×#14519699C2212yes—
24C2×#14519699C2212yes—
25C2×#14519699C2212yes—
26C2×#14519699C2212yes—
27#16839680—22no—
28#16839664C2212yes—
29#16839660C2212yescomplemented
30#16839645—22no—
31#16839622C2211yescharacteristic
32#16839620—21no—
33#16839617—21no—
34#16839617—21no—
35C4×C4⋊Dic61C2212yescomplemented
36C22⋊C4×Dic61—21no—
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—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—22no—
58C2×C4⋊Dic61D414yes—
59C2×C4⋊Dic61C2311yescharacteristic
60C2×C4⋊Dic61C2312yes—
61C2×C4⋊Dic61C2311yescharacteristic
62C2×C4⋊Dic61C2311yescharacteristic
63C2×#11758196—21no—
64C2×#11758196—22no—
65C2×C22⋊Dic61—21no—
66C2×C22⋊Dic61—21no—
67C2×#11758198—22no—
68C2×#11758198—22no—
69C2×#11758198—22no—
70C2×#11758198—22no—
71C2×C4×Dic61—22no—
72#14519711—22no—
73#14519711—22no—
74#14519702—22no—
75#14519702—22no—
76#14519702—42no—
77#14519699—24no—
78#14519699—24no—
79#14519699—24no—
80#14519699—42no—
81#14519684—22no—
82#14519684—22no—
83#14519684—21no—
84C23×C4×C61C2311yescharacteristic
85C22×C8×C61—22no—
86C22×C8×C61—42no—
87C22×C8×C61C2312yes—
88C22×C8×C61C2311yescharacteristic
89C22×C8×C61C2311yescharacteristic
90C2×C22⋊C4×C61D411yescharacteristic
91C2×C22⋊C4×C61D411yescharacteristic
92C2×C42×C61C2312yes—
93C4⋊C8×C61—42no—
94C22⋊C8×C61—22no—
95#95×C61D412yes—
96C24×C61C2×D411yescharacteristic
97C23×D61—22no—
98C23×D61—44no—
99C23×D61—42no—
100C23×D61—42no—
101C23×D61—42no—
102C23×D61—82no—
103C22×C4×C61—22no—
104C22×C4×C61—22no—
105C22×C4×C61C2×D412yes—
106C22×C4×C61C2×D412yes—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D412yes—
109C22×C4×C61#4112yes—
110C22×C4×C61C2411yescharacteristic
111C22×Dic61—21no—
112C22×Dic61—21no—
113C22×Dic61—24no—
114C22×Dic61—21no—
115C22×Dic61—22no—
116C22×Dic61—21no—
117C22×Dic61—82no—
118C2×D244—28no—
119C2×D244—22no—
120C2×D244—22no—
121C2×D244—22no—
122C2×D244—22no—
123C2×D244—22no—
124C2×D244—48no—
125C2×D244—44no—
126C2×D244—44no—
127C2×D244—42no—
128C2×D244#4114yes—
129C2×Dic122—21no—
130C2×Dic122—21no—
131C2×Dic122—22no—
132C2×Dic122—22no—
133C2×C8×C61—21no—
134C2×C8×C61—21no—
135C2×C8×C61—22no—
136C2×C8×C61—22no—
137C2×C8×C61—22no—
138C2×C8×C61—22no—
139C2×C8×C61—22no—
140C2×C8×C61—44no—
141C22⋊C4×C61—22no—
142C22⋊C4×C61—22no—
143C42×C61—22no—
144C42×C61—22no—
145C22⋊Dic61—42no—
146C22⋊Dic61—42no—
147#11758198—44no—
148#11758198—44no—
149#11758198—82no—
150#11758198—82no—
151C4⋊Dic61—24no—
152C4⋊Dic61—21no—
153C4⋊Dic61—21no—
154C4⋊Dic61—21no—
155C4⋊Dic61—22no—
156C4⋊Dic61—21no—
157C4⋊Dic61—22no—
158C4⋊Dic61C2×D412yes—
159C4⋊Dic61C2×D412yes—
160#11758196—42no—
161#11758196—82no—
162C4×Dic61—82no—
163C23×C61—22no—
164C23×C61—22no—
165C23×C61#12112yes—
166C23×C61#13512yes—
167C23×C61#13712yes—
168C23×C61C22×D411yescharacteristic
169C22×D61—24no—
170C22×D61—48no—
171C22×D61—42no—
172C22×D61—42no—
173C22×D61—42no—
174C22×D61—42no—
175C22×D61—42no—
176C22×D61—42no—
177C22×D61—88no—
178C22×D61—88no—
179C22×D61—84no—
180C22×D61—84no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—24no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61#12112yes—
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C2×#4112yes—
194C2×C4×C61C2×#4111yescharacteristic
195C2×C4×C61H(5,𝔽2)12yes—
196C2×Dic61—22no—
197C2×Dic61—22no—
198C2×Dic61—42no—
199C2×Dic61—42no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61—42no—
203C2×Dic61—42no—
204C2×Dic61—42no—
205C2×Dic61—42no—
206C2×Dic61—42no—
207C2×Dic61—84no—
208D244—24no—
209D244—24no—
210D244—48no—
211D244—42no—
212D244—42no—
213Dic122—41no—
214Dic122—41no—
215Dic122—42no—
216C8×C61—41no—
217C8×C61—41no—
218C8×C61—42no—
219#24479—611nomaximal, Sylow
220C22×C61—22no—
221C22×C61—22no—
222C22×C61—24no—
223C22×C61—24no—
224C22×C61#46414yes—
225C22×C61C2×#12111yescharacteristic
226C22×C61#54612yes—
227C22×C61C2×#13511yescharacteristic
228C22×C61C2×#13711yescharacteristic
229C22×C61#57612yes—
230C2×D61—44no—
231C2×D61—44no—
232C2×D61—44no—
233C2×D61—816no—
234C2×D61—82no—
235C2×D61—82no—
236C2×D61—82no—
237C2×D61—82no—
238C4×C61—22no—
239C4×C61—42no—
240C4×C61—42no—
241C4×C61#52511yescharacteristic
242C4×C61#53511yescharacteristic
243C4×C61#55112yes—
244Dic61—42no—
245Dic61—42no—
246Dic61—81no—
247Dic61—81no—
248Dic61—82no—
249C2×#484—611no—
250C22×#102—611no—
251#2435—611no—
252#2345—611no—
253#2067—612no—
254#2042—612no—
255#1983—612no—
256#1925—612no—
257#1892—611no—
258#1850D6111yescharacteristic, complemented
259#1786—611no—
260C2×C61—24no—
261C2×C61#241111yescharacteristic
262C2×C61C2×#46411yescharacteristic
263C2×C61#304111yescharacteristic
264C2×C61#329712yes—
265C2×C61#331412yes—
266D61—84no—
267C23×D4—611no—
268C23×C8C2×D6111yescharacteristic
269C2×#124—612no—
270C2×C22⋊Q8—611no—
271C22×C4⋊C4—611no—
272#484—1224no—
273C2×#107—611no—
274C2×#107—611no—
275C2×C4⋊C8C2×D6112yescomplemented
276C2×#103—611no—
277C2×#103—611no—
278C2×#103—612no—
279C2×#102—612no—
280C2×#102—612no—
281C2×#102—612no—
282C2×#102—612no—
283C2×#102—1224no—
284C2×C22⋊C8C2×D6111yescharacteristic
285#394—1222no—
286#390—612no—
287#389—612no—
288#387—1222no—
289#380—611no—
290#380—1221no—
291#379—1221no—
292#379—1221no—
293C4×C4⋊C4—612no—
294C4×C22⋊C4—1221no—
295C4×C22⋊C4C2×D6111yescharacteristic
296#340—612no—
297#336C2×D6112yescomplemented
298C61#2447911yescharacteristic, complemented, Sylow
299C25—1221no—
300C22×Q8—611no—
301C22×D4—612no—
302C22×D4—1224no—
303C22×D4—1221no—
304C22×D4—1222no—
305C22×D4—1222no—
306C22×D4—2442no—
307C23×C4—1221no—
308C23×C4C22×D6111yescharacteristic
309C22×C8—22no—
310C22×C8—42no—
311C22×C8C22×D6112yes—
312C22×C8C22×D6111yescharacteristic
313C22×C8C22×D6111yescharacteristic
314#124—1224no—
315#124—2442no—
316C22⋊Q8—1222no—
317C22⋊Q8—1222no—
318C2×C4⋊C4—614no—
319C2×C4⋊C4—611no—
320C2×C4⋊C4—611no—
321C2×C4⋊C4—611no—
322C2×C4⋊C4—612no—
323C2×C4⋊C4—1221no—
324C2×C4⋊C4—1222no—
325C2×C4⋊C4—1222no—
326C2×C4⋊C4—1221no—
327C2×C22⋊C4—1222no—
328C2×C22⋊C4—1222no—
329C2×C22⋊C4—1221no—
330C2×C22⋊C4—1221no—
331C2×C22⋊C4—1222no—
332C2×C22⋊C4—1222no—
333C2×C22⋊C4D24411yescharacteristic
334C2×C22⋊C4D24411yescharacteristic
335C2×C42—1222no—
336C2×C42C22×D6112yes—
337#107—1222no—
338#107—1222no—
339C4⋊C8—42no—
340#103—1222no—
341#103—1222no—
342#103—2442no—
343#102—1224no—
344#102—1224no—
345#102—1224no—
346#102—2442no—
347C22⋊C8—22no—
348#95—1222no—
349#95—1222no—
350#95—1221no—
351#95D24412yes—
352C24—1222no—
353C24—4882no—
354C24—2444no—
355C24—2442no—
356C24—2442no—
357C24—2442no—
358C24C2×D24411yescharacteristic
359C2×Q8—1221no—
360C2×Q8—1222no—
361C2×Q8—1221no—
362C2×Q8—1222no—
363C2×D4—614no—
364C2×D4—1228no—
365C2×D4—1222no—
366C2×D4—1222no—
367C2×D4—1222no—
368C2×D4—1222no—
369C2×D4—1222no—
370C2×D4—2448no—
371C2×D4—2444no—
372C2×D4—2444no—
373C2×D4—2442no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—1224no—
377C22×C4—1221no—
378C22×C4—1221no—
379C22×C4—1222no—
380C22×C4—1221no—
381C22×C4—1221no—
382C22×C4—4882no—
383C22×C4C2×D24412yes—
384C22×C4C2×D24412yes—
385C22×C4C2×D24411yescharacteristic
386C22×C4D4×D6112yes—
387C22×C4#1175822612yes—
388C22×C4C23×D6111yescharacteristic
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—44no—
397C4⋊C4—612no—
398C4⋊C4—612no—
399C4⋊C4—1221no—
400C4⋊C4—1221no—
401C4⋊C4—1224no—
402C4⋊C4—1221no—
403C4⋊C4—1222no—
404C4⋊C4—1221no—
405C4⋊C4—1222no—
406C4⋊C4—4882no—
407C4⋊C4—2442no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—4882no—
411C22⋊C4—4882no—
412C22⋊C4—2444no—
413C22⋊C4—2444no—
414C22⋊C4—2442no—
415C22⋊C4—2442no—
416C42—22no—
417C42—22no—
418C42—4882no—
419C23—22no—
420C23—22no—
421C23—1224no—
422C23—4884no—
423C23—4888no—
424C23—4888no—
425C23—4884no—
426C23—2448no—
427C23—2442no—
428C23—2442no—
429C23—2442no—
430C23—2442no—
431C23—2442no—
432C23—2442no—
433C23#1451975212yes—
434C23#1451979912yes—
435C23#1451981012yes—
436C23C22×D24411yescharacteristic
437Q8—2441no—
438Q8—2441no—
439Q8—2442no—
440D4—1224no—
441D4—1224no—
442D4—2448no—
443D4—2442no—
444D4—2442no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—1222no—
456C2×C4—1222no—
457C2×C4—4884no—
458C2×C4—2441no—
459C2×C4—2441no—
460C2×C4—2442no—
461C2×C4—2442no—
462C2×C4—2442no—
463C2×C4—2442no—
464C2×C4—2442no—
465C2×C4—2442no—
466C2×C4—2442no—
467C2×C4#1451975212yes—
468C2×C4C2×D4×D6111yescharacteristic
469C2×C4#41×D6112yes—
470C2×C4C2×#1175822611yescharacteristic
471C2×C4#1451982212yes—
472C8—41no—
473C8—41no—
474C8—42no—
475C22—24no—
476C22—22no—
477C22—24no—
478C22—22no—
479C22—48816no—
480C22—4882no—
481C22—4882no—
482C22—4882no—
483C22—4882no—
484C22—2444no—
485C22—2444no—
486C22—2444no—
487C22#1683998514yes—
488C22C2×#1451975211yescharacteristic
489C22#1684040712yes—
490C22C2×#1451979911yescharacteristic
491C22C2×#1451981011yescharacteristic
492C22#1684056712yes—
493C4—22no—
494C4—42no—
495C4—42no—
496C4—4882no—
497C4—4881no—
498C4—4881no—
499C4—2442no—
500C4—2442no—
501C4#1684027011yescharacteristic
502C4#1684032711yescharacteristic
503C4#1684043912yes—
504C2—24no—
505C2—4884no—
506C2#2144920211yescharacteristic
507C2C2×#1683998511yescharacteristic
508C2#2145532311yescharacteristic
509C2#2145767112yes—
510C2#2145790212yes—
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1850C2
socleC23×C61C22×D4
prime radicalC23×D244C22
hypercenter#1850D61
hypocenterC61#24479
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×#16840098C2semidirect
4. #21448694C2semidirect
5. #21447206C2semidirect
6. #21447061C2semidirect
7. #21446749C2semidirect
8. #21446440C2semidirect
9. #21446242C2semidirect
11. C61×#1850C2semidirect
17. C2×#14519711C22semidirect
18. C2×#14519711C22semidirect
19. C2×#14519702C22semidirect
20. C2×#14519702C22semidirect
21. C2×#14519702C22semidirect
29. #16839660C22semidirect
35. C4×C4⋊Dic61C22semidirect
37. #16839525C22semidirect
39. C2×C4⋊C8×C61C22semidirect
42. C61×#336C22semidirect
258. #1850D61semidirect
275. C2×C4⋊C8C2×D61semidirect
297. #336C2×D61semidirect
298. C61#24479semidirect

Maximal subgroups

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

subgroupindex
C2×#168400982
C22×#145196992
#214494702
#214486942
#214472062
#214470612
#214467492
#214464402
#214462422
#214457512
C61×#18502
#2447961