Finite Group Databasebeta

← #29658849

Subgroups — #29658849

Order 15616.

1170 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840417C211yesmaximal, characteristic
2#21456967C211yesmaximal, characteristic, complemented
3#21455823C211yesmaximal, characteristic, complemented
4#21445053C214yesmaximal, complemented
5#21444969C214yesmaximal, complemented
6#21444505C211yesmaximal, characteristic, complemented
7#21444243C211yesmaximal, characteristic, complemented
8#21444210C211yesmaximal, characteristic, complemented
9C61×#3077C211yesmaximal, characteristic, complemented
10C22×#11758222—21no—
11#16840417—22no—
12#16840417—22no—
13#16840417—22no—
14#16840417C2214yes—
15C2×#14519777C2211yescharacteristic
16C2×#14519759—22no—
17C2×#14519756—21no—
18C2×#14519748—21no—
19C2×C4×D244C2211yescharacteristic
20C2×C4⋊C4×D61—21no—
21#16840165C2212yescomplemented
22#16840112C2212yescomplemented
23#16840095—22no—
24#16840090—21no—
25#16840057—22no—
26#16840049C2212yescomplemented
27#16839964C2212yescomplemented
28#16839912—21no—
29C4×#11758201—22no—
30#16839815C2211yescharacteristic
31#16839815C2211yescharacteristic
32#16839507C2214yescomplemented
33D4⋊#92418C2214yescomplemented
34#16839489C2214yescomplemented
35C61×#533C2211yescharacteristic
36C2×C4×D4×C61C2211yescharacteristic
37C61×#478—22no—
38C61×#476C2212yescomplemented
39C61×#474C2212yescomplemented
40C61×#428—21no—
41C61×#423C2211yescharacteristic
42C61×#418—21no—
43C2×#11758222—24no—
44C2×#11758222—24no—
45C2×#11758222—48no—
46C22×Dic122—21no—
47C22×D244D411yescharacteristic
48C22×C22⋊D61—22no—
49C22×C4×D61—22no—
50C22×C4×D61—21no—
51C2×#11758201—22no—
52#14519777—21no—
53#14519777—21no—
54#14519777D412yes—
55#14519777C2312yes—
56#14519759—28no—
57#14519759—28no—
58#14519756—24no—
59#14519756—24no—
60#14519748—24no—
61#14519748—24no—
62C4×D244—21no—
63C4×D244—21no—
64C4×D244D412yes—
65C4×D244C2312yes—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61D411yescharacteristic
68C2×C4⋊Dic61D412yes—
69C2×#11758196—22no—
70C4⋊C4×D61—24no—
71C4⋊C4×D61—24no—
72C2×C22⋊Dic61—22no—
73C2×#11758198—21no—
74#14519710—24no—
75#14519708—22no—
76C4⋊#92418—21no—
77C4⋊#92418C2312yes—
78#14519701—22no—
79#14519701—24no—
80#14519697—22no—
81C22⋊#92418—21no—
82C22⋊#92418—21no—
83C4×#92418—21no—
84C4×#92418C2312yes—
85C22×D4×C61D411yescharacteristic
86C23×C4×C61—21no—
87C2×M16×C61—21no—
88C4⋊Q8×C61C2311yescharacteristic
89C4⋊Q8×C61C2311yescharacteristic
90#125×C61—21no—
91C22⋊Q8×C61—21no—
92C22⋊Q8×C61—21no—
93C4×Q8×C61—21no—
94C4×Q8×C61—21no—
95C4×D4×C61—21no—
96C4×D4×C61—21no—
97C4×D4×C61D412yes—
98C4×D4×C61C2312yes—
99#117×C61—21no—
100C2×C4⋊C4×C61D411yescharacteristic
101C2×C22⋊C4×C61—21no—
102C2×C42×C61C2311yescharacteristic
103#107×C61—22no—
104#106×C61—22no—
105C4⋊C8×C61—21no—
106C4⋊C8×C61C2312yes—
107#102×C61—22no—
108#102×C61—22no—
109C24×C61—21no—
110C23×D61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×D4×C61—21no—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61C2×D412yes—
117C22×C4×C61—21no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—44no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C2×C22⋊D61—28no—
125C2×C22⋊D61—48no—
126C22×Dic61—22no—
127C22×Dic61—21no—
128#11758222—416no—
129#11758222—48no—
130#11758222—48no—
131C2×D244—21no—
132C2×D244—24no—
133C2×D244C2×D412yes—
134C2×C4×D61—28no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—48no—
138C2×C4×D61—42no—
139C2×C4×D61—42no—
140C2×Dic122—22no—
141C2×Dic122—22no—
142C2×Dic122—44no—
143M16×C61—42no—
144C2×C8×C61—22no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—41no—
151C4⋊C4×C61—41no—
152C4⋊C4×C61C2×D412yes—
153C22⋊C4×C61—22no—
154C22⋊C4×C61—22no—
155C22⋊C4×C61—41no—
156C42×C61—41no—
157C42×C61C2×D411yescharacteristic
158C42×C61C2×D411yescharacteristic
159C42×C61C2×D411yescharacteristic
160C42×C61C2411yescharacteristic
161C22⋊Dic61—28no—
162#11758201—44no—
163#11758198—22no—
164#11758198—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61—21no—
167C4⋊Dic61—22no—
168C4⋊Dic61—22no—
169C4⋊Dic61C2×D412yes—
170C4⋊Dic61C2×D414yes—
171#11758196—28no—
172C2×#92418—22no—
173C2×#92418—22no—
174C23×C61—22no—
175C23×C61—42no—
176C23×C61—42no—
177C23×C61—42no—
178C23×C61#12011yescharacteristic
179C22×D61—24no—
180C22×D61—22no—
181C22×D61—42no—
182C22×D61—42no—
183Q8×C61—42no—
184Q8×C61—42no—
185D4×C61—22no—
186D4×C61—22no—
187D4×C61—42no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—41no—
191C2×C4×C61—44no—
192C2×C4×C61—42no—
193C2×C4×C61—44no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61#12011yescharacteristic
197C2×C4×C61#12011yescharacteristic
198C2×C4×C61#12011yescharacteristic
199C2×C4×C61#13512yes—
200C2×C4×C61#13712yes—
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C22×D411yescharacteristic
203C2×C4×C61C22×D411yescharacteristic
204C22⋊D61—416no—
205C2×Dic61—22no—
206C2×Dic61—24no—
207C2×Dic61—44no—
208C2×Dic61—42no—
209D244—24no—
210D244—22no—
211D244AGL(1,ℤ/8)14yes—
212C4×D61—416no—
213C4×D61—42no—
214C4×D61—42no—
215C4×D61—44no—
216Dic122—44no—
217Dic122—44no—
218C8×C61—42no—
219#92418—42no—
220#92418—42no—
221#18783—611nomaximal, Sylow
222C22×C61—21no—
223C22×C61—21no—
224C22×C61—42no—
225C22×C61—42no—
226C22×C61—42no—
227C22×C61—84no—
228C22×C61#45212yes—
229C22×C61C2×#12011yescharacteristic
230C2×D61—24no—
231C2×D61—48no—
232C2×D61—42no—
233C2×D61—42no—
234C4×C61—21no—
235C4×C61—21no—
236C4×C61—21no—
237C4×C61—42no—
238C4×C61—81no—
239C4×C61—81no—
240C4×C61#45212yes—
241C4×C61C2×#13511yescharacteristic
242C4×C61C2×AGL(1,ℤ/8)12yes—
243C4×C61C2×#13711yescharacteristic
244Dic61—42no—
245Dic61—44no—
246C2×#547—611no—
247#3230—611no—
248#3101—611no—
249#3077D6111yescharacteristic, complemented
250#1647—614no—
251#1635—614no—
252#1549—611no—
253#1486—611no—
254#1478—611no—
255C2×C61—22no—
256C2×C61—82no—
257C2×C61C2×#45211yescharacteristic
258C2×C61#300311yescharacteristic
259C2×C61#302011yescharacteristic
260D61—44no—
261C22×#41—1221no—
262#547—614no—
263#547—1222no—
264#547—1222no—
265#547—1222no—
266#533C2×D6111yescharacteristic
267C2×C4⋊Q8—611no—
268C2×#123—1222no—
269C2×C22⋊Q8—1221no—
270C2×#121—1221no—
271C2×C4×D4—611no—
272C2×C4×D4C2×D6111yescharacteristic
273C22×C4⋊C4—1221no—
274#500—612no—
275#487—612no—
276#484—1222no—
277#483—1221no—
278#478—22no—
279#477—1222no—
280#476—612no—
281#476C2×D6112yescomplemented
282#474C2×D6112yescomplemented
283#461—612no—
284#450—1221no—
285C4×SD16—1222no—
286#431—611no—
287#431—611no—
288#428—21no—
289#423C2×D6111yescharacteristic
290#418—21no—
291#332—614no—
292D4⋊C8—614no—
293D4⋊C8—614no—
294C61#1878311yescharacteristic, complemented, Sylow
295C2×#41—1224no—
296C2×#41—1224no—
297C2×#41—2448no—
298C22×Q8—1221no—
299C22×D4—611no—
300C22×D4—1222no—
301C22×D4C22⋊D6111yescharacteristic
302C23×C4—21no—
303C23×C4—1222no—
304C23×C4—1221no—
305C2×SD16—1222no—
306C2×M16—21no—
307C4⋊Q8—612no—
308C4⋊Q8—612no—
309C4⋊Q8—1221no—
310C4⋊Q8—1221no—
311C4⋊Q8C22×D6111yescharacteristic
312C4⋊Q8C22×D6111yescharacteristic
313#125—21no—
314#123—1228no—
315#123—1228no—
316C22⋊Q8—21no—
317C22⋊Q8—21no—
318C22⋊Q8—1224no—
319C22⋊Q8—1224no—
320#121—1224no—
321#121—1224no—
322C4×Q8—21no—
323C4×Q8—21no—
324C4×D4—21no—
325C4×D4—21no—
326C4×D4—612no—
327C4×D4—612no—
328C4×D4—1221no—
329C4×D4—1221no—
330C4×D4C22⋊D6112yes—
331C4×D4C22×D6112yes—
332#117—21no—
333C2×C4⋊C4—611no—
334C2×C4⋊C4—612no—
335C2×C4⋊C4—1224no—
336C2×C4⋊C4—1222no—
337C2×C4⋊C4—1221no—
338C2×C4⋊C4—1224no—
339C2×C4⋊C4C22⋊D6111yescharacteristic
340C2×C22⋊C4—21no—
341C2×C22⋊C4—1222no—
342C2×C22⋊C4—1221no—
343C2×C42C22×D6111yescharacteristic
344#107—22no—
345#107—1224no—
346#106—22no—
347#106—1222no—
348C4⋊C8—21no—
349C4⋊C8—612no—
350C4⋊C8—1221no—
351C4⋊C8C22×D6112yes—
352#103—1222no—
353#103—1224no—
354#102—22no—
355#102—22no—
356#102—1222no—
357C22⋊C8—1221no—
358C22⋊C8—1221no—
359C4×C8—612no—
360C4×C8—1221no—
361C24—21no—
362C24—1221no—
363#41—2448no—
364#41—24416no—
365#41—2448no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—1222no—
369C2×Q8—1222no—
370C2×Q8—2444no—
371C2×D4—22no—
372C2×D4—21no—
373C2×D4—22no—
374C2×D4—612no—
375C2×D4—1224no—
376C2×D4—1228no—
377C2×D4—1221no—
378C2×D4—2448no—
379C2×D4C2×C22⋊D6112yes—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—44no—
384C22×C4—1228no—
385C22×C4—1221no—
386C22×C4—1222no—
387C22×C4—1222no—
388C22×C4—1222no—
389C22×C4—2448no—
390C22×C4—2442no—
391C22×C4—2442no—
392C22×C4D4×D6111yescharacteristic
393C22×C4D4×D6111yescharacteristic
394C22×C4C2×C22⋊D6111yescharacteristic
395SD16—2444no—
396M16—42no—
397C2×C8—22no—
398C2×C8—1222no—
399C2×C8—1222no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—612no—
408C4⋊C4—614no—
409C4⋊C4—1228no—
410C4⋊C4—1222no—
411C4⋊C4—1221no—
412C4⋊C4—1222no—
413C4⋊C4—1222no—
414C4⋊C4C2×C22⋊D6112yes—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—41no—
418C22⋊C4—1228no—
419C22⋊C4—1222no—
420C22⋊C4—1222no—
421C42—41no—
422C42D4×D6111yescharacteristic
423C42D4×D6111yescharacteristic
424C42C2×C22⋊D6111yescharacteristic
425C42C23×D6111yescharacteristic
426C23—22no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—1224no—
431C23—1222no—
432C23—2442no—
433C23—2442no—
434C23#1451974511yescharacteristic
435Q8—42no—
436Q8—42no—
437Q8—2444no—
438Q8—2444no—
439D4—22no—
440D4—22no—
441D4—42no—
442D4—614no—
443D4—1224no—
444D4—1222no—
445D4—24416no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—41no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—1222no—
455C2×C4—1224no—
456C2×C4—24416no—
457C2×C4—2444no—
458C2×C4—2442no—
459C2×C4—2442no—
460C2×C4—2442no—
461C2×C4—2444no—
462C2×C4#1451974511yescharacteristic
463C2×C4#1451974511yescharacteristic
464C2×C4#1451974511yescharacteristic
465C2×C4#1451979712yes—
466C2×C4#1451980812yes—
467C2×C4C2×D4×D6111yescharacteristic
468C2×C4C2×D4×D6111yescharacteristic
469C2×C4C22×C22⋊D6111yescharacteristic
470C8—42no—
471C8—2442no—
472C8—2442no—
473C22—21no—
474C22—21no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22—84no—
479C22—1224no—
480C22—2448no—
481C22—2442no—
482C22—2442no—
483C22#1683992212yes—
484C22C2×#1451974511yescharacteristic
485C4—21no—
486C4—21no—
487C4—21no—
488C4—42no—
489C4—81no—
490C4—81no—
491C4—2442no—
492C4—2444no—
493C4#1683992212yes—
494C4C2×#1451979711yescharacteristic
495C4AGL(1,ℤ/8)×D6112yes—
496C4C2×#1451980811yescharacteristic
497C2—22no—
498C2—82no—
499C2—2444no—
500C2C2×#1683992211yescharacteristic
501C2#2145504411yescharacteristic
502C2#2145518111yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3077C2
socleC22×C61C2×#120
prime radicalC2×#16840417C2
hypercenter#3077D61
hypocenterC61#18783
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3077

Complements

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

NH ≅ G/Ntype
2. #21456967C2semidirect
3. #21455823C2semidirect
4. #21445053C2semidirect
5. #21444969C2semidirect
6. #21444505C2semidirect
7. #21444243C2semidirect
8. #21444210C2semidirect
9. C61×#3077C2semidirect
21. #16840165C22semidirect
22. #16840112C22semidirect
26. #16840049C22semidirect
27. #16839964C22semidirect
32. #16839507C22semidirect
33. D4⋊#92418C22semidirect
34. #16839489C22semidirect
38. C61×#476C22semidirect
39. C61×#474C22semidirect
249. #3077D61semidirect
281. #476C2×D61semidirect
282. #474C2×D61semidirect
294. C61#18783semidirect

Maximal subgroups

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

subgroupindex
C2×#168404172
#214569672
#214558232
#214450532
#214449692
#214445052
#214442432
#214442102
C61×#30772
#1878361