Finite Group Databasebeta

← #29658747

Subgroups — #29658747

Order 15616.

986 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840477C211yesmaximal, characteristic
2#21456764C211yesmaximal, characteristic, complemented
3#21455823C211yesmaximal, characteristic, complemented
4#21445046C214yesmaximal, complemented
5#21444963C214yesmaximal, complemented
6#21444501C211yesmaximal, characteristic, complemented
7#21444243C211yesmaximal, characteristic, complemented
8#21444210C211yesmaximal, characteristic, complemented
9C61×#3077C211yesmaximal, characteristic, complemented
10#16840477—22no—
11#16840477—22no—
12#16840477—22no—
13#16840477C2214yes—
14C2×#14519777C2211yescharacteristic
15C2×#14519763—22no—
16C2×#14519751—21no—
17C2×#14519748—21no—
18C2×C4×D244C2211yescharacteristic
19C2×#14519736—21no—
20C2×#14519735—21no—
21#16840141C2212yescomplemented
22#16840114C2212yescomplemented
23#16840090—21no—
24#16840049C2212yescomplemented
25#16839972—22no—
26#16839964C2212yescomplemented
27#16839917—22no—
28#16839912—21no—
29C4×#11758201—22no—
30#16839815C2211yescharacteristic
31#16839815C2211yescharacteristic
32#16839510C2214yescomplemented
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—
43C22×Dic122D412yes—
44C22×D244D411yescharacteristic
45C22×C22⋊D61—22no—
46C22×C4×D61—21no—
47C2×#11758202—24no—
48C2×#11758201—22no—
49#14519777—21no—
50#14519777—21no—
51#14519777D412yes—
52#14519777C2312yes—
53#14519763—28no—
54#14519763—28no—
55#14519751—24no—
56#14519751—24no—
57#14519748—24no—
58#14519748—24no—
59C4×D244—21no—
60C4×D244—21no—
61C4×D244D412yes—
62C4×D244C2312yes—
63#14519736—24no—
64#14519736—24no—
65#14519735—24no—
66#14519735—24no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61D411yescharacteristic
70C2×C22⋊Dic61—22no—
71C2×#11758198—22no—
72C2×#11758198—21no—
73C2×C4×Dic61—22no—
74#14519708—22no—
75C4⋊#92418—21no—
76C4⋊#92418C2312yes—
77#14519701—22no—
78#14519700—24no—
79#14519697—22no—
80C22⋊#92418—21no—
81C22⋊#92418—21no—
82C4×#92418—21no—
83C4×#92418C2312yes—
84C22×D4×C61D411yescharacteristic
85C23×C4×C61—21no—
86C2×M16×C61—21no—
87C4⋊Q8×C61C2311yescharacteristic
88C4⋊Q8×C61C2311yescharacteristic
89#125×C61—21no—
90C22⋊Q8×C61—21no—
91C22⋊Q8×C61—21no—
92C4×Q8×C61—21no—
93C4×Q8×C61—21no—
94C4×D4×C61—21no—
95C4×D4×C61—21no—
96C4×D4×C61D412yes—
97C4×D4×C61C2312yes—
98#117×C61—21no—
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C22⋊C4×C61—21no—
101C2×C42×C61C2311yescharacteristic
102#107×C61—22no—
103#106×C61—22no—
104C4⋊C8×C61—21no—
105C4⋊C8×C61C2312yes—
106#102×C61—22no—
107#102×C61—22no—
108C24×C61—21no—
109C23×D61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61C2×D412yes—
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—44no—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C2×C22⋊D61—28no—
124C2×C22⋊D61—84no—
125C22×Dic61—21no—
126C22×Dic61—22no—
127C2×D244—21no—
128C2×D244—42no—
129C2×D244C2×D412yes—
130C2×C4×D61—22no—
131C2×C4×D61—22no—
132C2×C4×D61—42no—
133C2×C4×D61—42no—
134C2×Dic122—22no—
135C2×Dic122—82no—
136C2×Dic122C2×D414yes—
137M16×C61—42no—
138C2×C8×C61—22no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—41no—
145C4⋊C4×C61—41no—
146C4⋊C4×C61C2×D412yes—
147C22⋊C4×C61—22no—
148C22⋊C4×C61—22no—
149C22⋊C4×C61—41no—
150C42×C61—41no—
151C42×C61C2×D411yescharacteristic
152C42×C61C2×D411yescharacteristic
153C42×C61C2×D411yescharacteristic
154C42×C61C2411yescharacteristic
155C22⋊Dic61—28no—
156#11758202—84no—
157#11758201—44no—
158#11758198—28no—
159#11758198—22no—
160#11758198—22no—
161C4⋊Dic61—22no—
162C4⋊Dic61—22no—
163C4⋊Dic61—21no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61C2×D412yes—
167C4×Dic61—28no—
168C2×#92418—22no—
169C2×#92418—22no—
170C23×C61—22no—
171C23×C61—42no—
172C23×C61—42no—
173C23×C61—42no—
174C23×C61#12011yescharacteristic
175C22×D61—22no—
176C22×D61—42no—
177C22×D61—42no—
178C22×D61—42no—
179Q8×C61—42no—
180Q8×C61—42no—
181D4×C61—22no—
182D4×C61—22no—
183D4×C61—42no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—41no—
187C2×C4×C61—44no—
188C2×C4×C61—42no—
189C2×C4×C61—44no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61#12011yescharacteristic
193C2×C4×C61#12011yescharacteristic
194C2×C4×C61#12011yescharacteristic
195C2×C4×C61#13512yes—
196C2×C4×C61#13712yes—
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C22⋊D61—88no—
201C2×Dic61—22no—
202C2×Dic61—24no—
203C2×Dic61—44no—
204C2×Dic61—42no—
205D244—22no—
206D244—22no—
207D244—42no—
208C4×D61—42no—
209C4×D61—42no—
210C4×D61—44no—
211Dic122—44no—
212Dic122—82no—
213C8×C61—42no—
214#92418—42no—
215#92418—42no—
216#18773—611nomaximal, Sylow
217C22×C61—21no—
218C22×C61—21no—
219C22×C61—42no—
220C22×C61—42no—
221C22×C61—42no—
222C22×C61—84no—
223C22×C61#45212yes—
224C22×C61C2×#12011yescharacteristic
225C2×D61—42no—
226C2×D61—42no—
227C2×D61—42no—
228C2×D61—84no—
229C4×C61—21no—
230C4×C61—21no—
231C4×C61—21no—
232C4×C61—42no—
233C4×C61—81no—
234C4×C61—81no—
235C4×C61#45212yes—
236C4×C61C2×#13511yescharacteristic
237C4×C61C2×#13711yescharacteristic
238C4×C61#57512yes—
239Dic61—42no—
240Dic61—44no—
241C2×#555—611no—
242#3209—611no—
243#3101—611no—
244#3077D6111yescharacteristic, complemented
245#1646—614no—
246#1634—614no—
247#1548—611no—
248#1486—611no—
249#1478—611no—
250C2×C61—22no—
251C2×C61—82no—
252C2×C61C2×#45211yescharacteristic
253C2×C61#300511yescharacteristic
254C2×C61#301911yescharacteristic
255D61—82no—
256#555—614no—
257#555—1222no—
258#555—1222no—
259#555—1222no—
260#533C2×D6111yescharacteristic
261C2×C4⋊Q8—611no—
262C2×#124—1222no—
263C2×#121—1221no—
264C2×#121—1221no—
265C2×C4×D4—611no—
266C2×C4×D4C2×D6111yescharacteristic
267C2×#117—1221no—
268C2×#117—1221no—
269#494—612no—
270#487—612no—
271#483—1221no—
272#478—22no—
273#476—612no—
274#476C2×D6112yescomplemented
275#474C2×D6112yescomplemented
276#462—1222no—
277#461—612no—
278#451—1222no—
279#450—1221no—
280C4×SD16—1222no—
281#431—611no—
282#431—611no—
283#428—21no—
284#423C2×D6111yescharacteristic
285#418—21no—
286#333—614no—
287D4⋊C8—614no—
288D4⋊C8—614no—
289C61#1877311yescharacteristic, complemented, Sylow
290C22×Q8—612no—
291C22×D4—611no—
292C22×D4—1222no—
293C22×D4C22⋊D6111yescharacteristic
294C23×C4—21no—
295C23×C4—1221no—
296C2×Q16—1224no—
297C2×SD16—1222no—
298C2×M16—21no—
299C4⋊Q8—612no—
300C4⋊Q8—612no—
301C4⋊Q8—1221no—
302C4⋊Q8—1221no—
303C4⋊Q8C22×D6111yescharacteristic
304C4⋊Q8C22×D6111yescharacteristic
305#125—21no—
306#124—1228no—
307#124—1228no—
308C22⋊Q8—21no—
309C22⋊Q8—21no—
310#121—1224no—
311#121—1224no—
312#121—1224no—
313#121—1224no—
314C4×Q8—21no—
315C4×Q8—21no—
316C4×D4—21no—
317C4×D4—21no—
318C4×D4—612no—
319C4×D4—612no—
320C4×D4—1221no—
321C4×D4—1221no—
322C4×D4C22⋊D6112yes—
323C4×D4C22×D6112yes—
324#117—21no—
325#117—1224no—
326#117—1224no—
327#117—1224no—
328#117—1224no—
329C2×C4⋊C4—611no—
330C2×C4⋊C4—1221no—
331C2×C4⋊C4—1221no—
332C2×C4⋊C4C22⋊D6111yescharacteristic
333C2×C22⋊C4—21no—
334C2×C22⋊C4—1222no—
335C2×C22⋊C4—1222no—
336C2×C22⋊C4—1221no—
337C2×C42—1222no—
338C2×C42C22×D6111yescharacteristic
339#107—22no—
340#106—22no—
341#106—1222no—
342C4⋊C8—21no—
343C4⋊C8—612no—
344C4⋊C8—1221no—
345C4⋊C8C22×D6112yes—
346#103—1224no—
347#103—1222no—
348#102—22no—
349#102—22no—
350#102—1222no—
351C22⋊C8—1221no—
352C22⋊C8—1221no—
353C4×C8—612no—
354C4×C8—1221no—
355C24—21no—
356C24—1221no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—614no—
360C2×Q8—1222no—
361C2×Q8—4882no—
362C2×D4—22no—
363C2×D4—21no—
364C2×D4—22no—
365C2×D4—612no—
366C2×D4—1228no—
367C2×D4—1221no—
368C2×D4—4884no—
369C2×D4—2442no—
370C2×D4C2×C22⋊D6112yes—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—44no—
375C22×C4—1221no—
376C22×C4—1222no—
377C22×C4—1222no—
378C22×C4—1222no—
379C22×C4—2442no—
380C22×C4—2442no—
381C22×C4D4×D6111yescharacteristic
382C22×C4D4×D6111yescharacteristic
383C22×C4C2×C22⋊D6111yescharacteristic
384Q16—4884no—
385SD16—2444no—
386M16—42no—
387C2×C8—22no—
388C2×C8—1222no—
389C2×C8—1222no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—612no—
398C4⋊C4—1222no—
399C4⋊C4—1222no—
400C4⋊C4—1221no—
401C4⋊C4—1222no—
402C4⋊C4—1222no—
403C4⋊C4C2×C22⋊D6112yes—
404C22⋊C4—22no—
405C22⋊C4—22no—
406C22⋊C4—41no—
407C22⋊C4—1228no—
408C22⋊C4—1228no—
409C22⋊C4—1222no—
410C22⋊C4—1222no—
411C42—41no—
412C42—1228no—
413C42D4×D6111yescharacteristic
414C42D4×D6111yescharacteristic
415C42C2×C22⋊D6111yescharacteristic
416C42C23×D6111yescharacteristic
417C23—22no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—1222no—
422C23—2442no—
423C23—2442no—
424C23—2442no—
425C23#1451974511yescharacteristic
426Q8—42no—
427Q8—42no—
428Q8—4882no—
429Q8—2444no—
430D4—22no—
431D4—22no—
432D4—42no—
433D4—1222no—
434D4—1222no—
435D4—4888no—
436D4—2442no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—41no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—1222no—
446C2×C4—1224no—
447C2×C4—2444no—
448C2×C4—2442no—
449C2×C4—2442no—
450C2×C4—2442no—
451C2×C4—2444no—
452C2×C4#1451974511yescharacteristic
453C2×C4#1451974511yescharacteristic
454C2×C4#1451974511yescharacteristic
455C2×C4#1451979712yes—
456C2×C4#1451980812yes—
457C2×C4C2×D4×D6111yescharacteristic
458C2×C4C2×D4×D6111yescharacteristic
459C2×C4C22×C22⋊D6111yescharacteristic
460C8—42no—
461C8—2442no—
462C8—2442no—
463C22—21no—
464C22—21no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—84no—
469C22—4884no—
470C22—2442no—
471C22—2442no—
472C22—2442no—
473C22#1683992212yes—
474C22C2×#1451974511yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—42no—
479C4—81no—
480C4—81no—
481C4—2444no—
482C4—2442no—
483C4#1683992212yes—
484C4C2×#1451979711yescharacteristic
485C4C2×#1451980811yescharacteristic
486C4#1684055812yes—
487C2—22no—
488C2—82no—
489C2—4882no—
490C2C2×#1683992211yescharacteristic
491C2#2145505811yescharacteristic
492C2#2145517411yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3077C2
socleC22×C61C2×#120
prime radicalC2×#16840477C2
hypercenter#3077D61
hypocenterC61#18773
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. #21456764C2semidirect
3. #21455823C2semidirect
4. #21445046C2semidirect
5. #21444963C2semidirect
6. #21444501C2semidirect
7. #21444243C2semidirect
8. #21444210C2semidirect
9. C61×#3077C2semidirect
21. #16840141C22semidirect
22. #16840114C22semidirect
24. #16840049C22semidirect
26. #16839964C22semidirect
32. #16839510C22semidirect
33. D4⋊#92418C22semidirect
34. #16839489C22semidirect
38. C61×#476C22semidirect
39. C61×#474C22semidirect
244. #3077D61semidirect
274. #476C2×D61semidirect
275. #474C2×D61semidirect
289. C61#18773semidirect

Maximal subgroups

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

subgroupindex
C2×#168404772
#214567642
#214558232
#214450462
#214449632
#214445012
#214442432
#214442102
C61×#30772
#1877361