Finite Group Databasebeta

← #29658499

Subgroups — #29658499

Order 15616.

1020 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840472C211yesmaximal, characteristic, complemented
2#21456741C211yesmaximal, characteristic, complemented
3#21455279C211yesmaximal, characteristic
4#21445025C214yesmaximal, complemented
5#21444890C214yesmaximal, complemented
6#21444499C211yesmaximal, characteristic, complemented
7#21444248C211yesmaximal, characteristic, complemented
8#21444209C211yesmaximal, characteristic, complemented
9C61×#3103C211yesmaximal, characteristic, complemented
10C2×#14519802—21no—
11#16840472—22no—
12#16840472—22no—
13#16840472—22no—
14#16840472C2214yes—
15C2×#14519777C2211yescharacteristic
16C2×#14519769—21no—
17C2×#14519755—22no—
18C2×#14519753—21no—
19C2×C4×D244—21no—
20C2×C4×D244C2211yescharacteristic
21C2×C4⋊C4×D61—21no—
22#16840134C2212yescomplemented
23#16840114C2212yescomplemented
24#16839974—22no—
25#16839967—22no—
26#16839964C2212yes—
27#16839954C2212yes—
28#16839900—22no—
29#16839815C2211yescharacteristic
30#16839786C2211yescharacteristic
31#16839763—21no—
32#16839510C2214yescomplemented
33Q8⋊#92418C2214yescomplemented
34#16839489C2214yes—
35C61×#532C2211yescharacteristic
36C2×C4×Q8×C61C2211yescharacteristic
37C61×#481—21no—
38C61×#475C2212yescomplemented
39C61×#460C2212yescomplemented
40C61×#448—21no—
41C4×SD16×C61—22no—
42C61×#431C2211yescharacteristic
43C22×Dic122D412yes—
44C22×D244D411yescharacteristic
45C22×C4×D61—21no—
46C22×C4×D61—22no—
47#14519802—44no—
48C2×#11758200—24no—
49C2×#11758201—22no—
50C2×#11758199—22no—
51C2×#11758194—21no—
52#14519777—21no—
53#14519777—21no—
54#14519777D412yes—
55#14519777C2312yes—
56#14519769—24no—
57#14519769—24no—
58#14519755—28no—
59#14519755—28no—
60#14519753—24no—
61#14519753—24no—
62C4×D244—24no—
63C4×D244—24no—
64C4×D244—21no—
65C4×D244—21no—
66C4×D244D412yes—
67C4×D244C2312yes—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61D411yescharacteristic
71C2×#11758196—22no—
72C2×#11758196—22no—
73C4⋊C4×D61—24no—
74C4⋊C4×D61—24no—
75C2×#11758198—22no—
76C2×#11758198—21no—
77C4⋊#92418—21no—
78C4⋊#92418—21no—
79C4⋊#92418C2312yes—
80#14519701—22no—
81#14519700—24no—
82#14519698—22no—
83C22⋊#92418—21no—
84C4×#92418C2312yes—
85C2×#41×C61—21no—
86C22×Q8×C61D411yescharacteristic
87C2×SD16×C61—22no—
88C4⋊Q8×C61C2311yescharacteristic
89#127×C61C2311yescharacteristic
90#124×C61—21no—
91#121×C61—21no—
92#121×C61—21no—
93C4×Q8×C61—21no—
94C4×Q8×C61—21no—
95C4×Q8×C61D412yes—
96C4×Q8×C61C2312yes—
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61D411yescharacteristic
101C2×C42×C61—21no—
102C2×C42×C61C2311yescharacteristic
103#106×C61—22no—
104C4⋊C8×C61C2312yes—
105#103×C61—22no—
106#102×C61—22no—
107C22⋊C8×C61—21no—
108C4×C8×C61—21no—
109C23×D61—21no—
110#41×C61—42no—
111#41×C61—42no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—42no—
115C2×Q8×C61C2×D412yes—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61—41no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—41no—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×Dic61—22no—
127C22×Dic61—21no—
128C2×D244—21no—
129C2×D244—22no—
130C2×D244SD1614yescomplemented
131C2×D244C2×D412yes—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—28no—
135C2×C4×D61—44no—
136C2×C4×D61—84no—
137C2×Dic122—22no—
138C2×Dic122—82no—
139C2×Dic122C2×D414yes—
140SD16×C61—44no—
141C2×C8×C61—22no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61C2×D412yes—
148C22⋊C4×C61—41no—
149C22⋊C4×C61—41no—
150C42×C61—22no—
151C42×C61—22no—
152C42×C61C2×D411yescharacteristic
153C42×C61C2×D411yescharacteristic
154C42×C61C2×D411yescharacteristic
155C42×C61C2411yescharacteristic
156#11758201—44no—
157#11758200—84no—
158#11758199—44no—
159#11758198—28no—
160#11758198—22no—
161#11758198—22no—
162C4⋊Dic61—22no—
163C4⋊Dic61—22no—
164C4⋊Dic61—21no—
165C4⋊Dic61—22no—
166C4⋊Dic61—22no—
167C4⋊Dic61C2×D412yes—
168#11758196—28no—
169#11758196—28no—
170#11758194—42no—
171C2×#92418—22no—
172C2×#92418—22no—
173C23×C61—41no—
174C23×C61#12011yescharacteristic
175C22×D61—22no—
176C22×D61—24no—
177C22×D61—42no—
178C22×D61—42no—
179Q8×C61—22no—
180Q8×C61—22no—
181Q8×C61—42no—
182Q8×C61—41no—
183Q8×C61—41no—
184D4×C61—42no—
185D4×C61—41no—
186D4×C61—41no—
187D4×C61—42no—
188D4×C61—82no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—41no—
193C2×C4×C61—42no—
194C2×C4×C61—42no—
195C2×C4×C61—41no—
196C2×C4×C61—82no—
197C2×C4×C61#12011yescharacteristic
198C2×C4×C61#12011yescharacteristic
199C2×C4×C61#12011yescharacteristic
200C2×C4×C61C2×SD1612yes—
201C2×C4×C61#13712yes—
202C2×C4×C61C22×D411yescharacteristic
203C2×C4×C61C22×D411yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205C2×Dic61—22no—
206C2×Dic61—24no—
207C2×Dic61—44no—
208C2×Dic61—42no—
209D244—22no—
210D244—24no—
211D244C2×SD1614yes—
212C4×D61—44no—
213C4×D61—44no—
214C4×D61—88no—
215Dic122—44no—
216Dic122—82no—
217C8×C61—42no—
218#92418—42no—
219#92418—42no—
220#18748—611nomaximal, Sylow
221C22×C61—21no—
222C22×C61—21no—
223C22×C61—81no—
224C22×C61—81no—
225C22×C61—81no—
226C22×C61#44812yes—
227C22×C61C2×#12011yescharacteristic
228C2×D61—24no—
229C2×D61—48no—
230C2×D61—42no—
231C2×D61—42no—
232C4×C61—21no—
233C4×C61—21no—
234C4×C61—21no—
235C4×C61—42no—
236C4×C61—42no—
237C4×C61—81no—
238C4×C61#44812yes—
239C4×C61C22×SD1611yescharacteristic
240C4×C61C2×#13711yescharacteristic
241C4×C61#57512yes—
242Dic61—42no—
243Dic61—44no—
244C2×#554—611no—
245#3206—611no—
246#3103D6111yescharacteristic, complemented
247#3036—611no—
248#1643—614no—
249#1622—614no—
250#1548—611no—
251#1487—611no—
252#1478—611no—
253C2×C61—22no—
254C2×C61—161no—
255C2×C61C2×#44811yescharacteristic
256C2×C61#300511yescharacteristic
257C2×C61#301911yescharacteristic
258D61—44no—
259C2×AGL(1,ℤ/8)—1221no—
260#554—614no—
261#554—1222no—
262#554—1222no—
263#554—1222no—
264#532C2×D6111yescharacteristic
265C2×C4⋊Q8—611no—
266C2×#125—1221no—
267C2×C22⋊Q8—1222no—
268C2×C22⋊Q8—1221no—
269C2×C4×Q8C2×D6111yescharacteristic
270C2×C4×D4—611no—
271C2×C4×D4—1221no—
272C22×C4⋊C4—1221no—
273#492—612no—
274#487—612no—
275#481—21no—
276#475C2×D6112yescomplemented
277#463—1222no—
278#461—612no—
279#461—1222no—
280#460C2×D6112yescomplemented
281#459—612no—
282#448—21no—
283#448—1222no—
284C4×SD16—22no—
285#431—611no—
286#431C2×D6111yescharacteristic
287#423—611no—
288#417—1221no—
289#333—614no—
290Q8⋊C8—614no—
291D4⋊C8—614no—
292C61#1874811yescharacteristic, complemented, Sylow
293C2×#41—21no—
294C22×Q8—612no—
295C22×Q8C22⋊D6111yescharacteristic
296C22×D4—611no—
297C23×C4—1222no—
298C23×C4—1221no—
299AGL(1,ℤ/8)—2444no—
300C2×SD16—22no—
301C2×SD16—1224no—
302C2×SD16—1222no—
303C2×D8—1222no—
304C2×M16—1221no—
305C4⋊Q8—612no—
306C4⋊Q8—612no—
307C4⋊Q8—1221no—
308C4⋊Q8—1221no—
309C4⋊Q8C22×D6111yescharacteristic
310#127C22×D6111yescharacteristic
311#125—1224no—
312#125—1224no—
313#124—21no—
314C22⋊Q8—1228no—
315C22⋊Q8—1228no—
316C22⋊Q8—1224no—
317C22⋊Q8—1224no—
318#121—21no—
319#121—21no—
320C4×Q8—21no—
321C4×Q8—21no—
322C4×Q8C22⋊D6112yes—
323C4×Q8C22×D6112yes—
324C4×D4—21no—
325C4×D4—21no—
326C4×D4—612no—
327C4×D4—612no—
328C4×D4—1224no—
329C4×D4—1224no—
330C4×D4—1221no—
331C4×D4—1221no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—611no—
334C2×C4⋊C4—1224no—
335C2×C4⋊C4—1222no—
336C2×C4⋊C4—1222no—
337C2×C4⋊C4—1221no—
338C2×C4⋊C4—1221no—
339C2×C4⋊C4—1224no—
340C2×C4⋊C4C22⋊D6111yescharacteristic
341C2×C22⋊C4—1222no—
342C2×C22⋊C4—1221no—
343C2×C42—21no—
344C2×C42C22×D6111yescharacteristic
345#106—22no—
346C4⋊C8—612no—
347C4⋊C8—1221no—
348C4⋊C8—1221no—
349C4⋊C8C22×D6112yes—
350#103—22no—
351#103—1224no—
352#103—1222no—
353#102—22no—
354#102—1222no—
355C22⋊C8—21no—
356C22⋊C8—1221no—
357C4×C8—21no—
358C4×C8—612no—
359C24—1221no—
360#41—42no—
361#41—42no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—42no—
365C2×Q8—614no—
366C2×Q8—1222no—
367C2×Q8—4882no—
368C2×Q8C2×C22⋊D6112yes—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—41no—
373C2×D4—612no—
374C2×D4—614no—
375C2×D4—1221no—
376C2×D4—1222no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—41no—
380C22×C4—1228no—
381C22×C4—1221no—
382C22×C4—1222no—
383C22×C4—1222no—
384C22×C4—1222no—
385C22×C4—4884no—
386C22×C4—2444no—
387C22×C4D4×D6111yescharacteristic
388C22×C4D4×D6111yescharacteristic
389C22×C4C2×C22⋊D6111yescharacteristic
390SD16—44no—
391SD16—4884no—
392SD16—2444no—
393D8—2444no—
394M16—2442no—
395C2×C8—22no—
396C2×C8—1222no—
397C2×C8—1222no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—612no—
404C4⋊C4—1228no—
405C4⋊C4—1228no—
406C4⋊C4—1222no—
407C4⋊C4—1222no—
408C4⋊C4—1221no—
409C4⋊C4—1222no—
410C4⋊C4—1222no—
411C4⋊C4C2×C22⋊D6112yes—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—1228no—
415C22⋊C4—1222no—
416C22⋊C4—1222no—
417C42—22no—
418C42—22no—
419C42D4×D6111yescharacteristic
420C42D4×D6111yescharacteristic
421C42C2×C22⋊D6111yescharacteristic
422C42C23×D6111yescharacteristic
423C23—41no—
424C23—1222no—
425C23—1224no—
426C23—2442no—
427C23—2442no—
428C23#1451974511yescharacteristic
429Q8—22no—
430Q8—22no—
431Q8—42no—
432Q8—41no—
433Q8—41no—
434Q8—4882no—
435Q8—2444no—
436D4—42no—
437D4—41no—
438D4—41no—
439D4—42no—
440D4—82no—
441D4—614no—
442D4—1222no—
443D4—1224no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—82no—
452C2×C4—1222no—
453C2×C4—1224no—
454C2×C4—4888no—
455C2×C4—2444no—
456C2×C4—2444no—
457C2×C4—2442no—
458C2×C4—2444no—
459C2×C4#1451974511yescharacteristic
460C2×C4#1451974511yescharacteristic
461C2×C4#1451974511yescharacteristic
462C2×C4SD16×D6112yes—
463C2×C4#1451980612yes—
464C2×C4C2×D4×D6111yescharacteristic
465C2×C4C2×D4×D6111yescharacteristic
466C2×C4C22×C22⋊D6111yescharacteristic
467C8—42no—
468C8—2442no—
469C8—2442no—
470C22—21no—
471C22—21no—
472C22—81no—
473C22—81no—
474C22—81no—
475C22—1224no—
476C22—2448no—
477C22—2442no—
478C22—2442no—
479C22#1683989912yes—
480C22C2×#1451974511yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—42no—
485C4—42no—
486C4—81no—
487C4—2444no—
488C4—2442no—
489C4#1683989912yes—
490C4C2×SD16×D6111yescharacteristic
491C4C2×#1451980611yescharacteristic
492C4#1684056012yes—
493C2—22no—
494C2—161no—
495C2—2444no—
496C2C2×#1683989911yescharacteristic
497C2#2145505911yescharacteristic
498C2#2145517411yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3103C2
socleC22×C61C2×#120
prime radical#21455279C2
hypercenter#3103D61
hypocenterC61#18748
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3103

Complements

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

NH ≅ G/Ntype
1. C2×#16840472C2semidirect
2. #21456741C2semidirect
4. #21445025C2semidirect
5. #21444890C2semidirect
6. #21444499C2semidirect
7. #21444248C2semidirect
8. #21444209C2semidirect
9. C61×#3103C2semidirect
22. #16840134C22semidirect
23. #16840114C22semidirect
32. #16839510C22semidirect
33. Q8⋊#92418C22semidirect
38. C61×#475C22semidirect
39. C61×#460C22semidirect
130. C2×D244SD16semidirect
246. #3103D61semidirect
276. #475C2×D61semidirect
280. #460C2×D61semidirect
292. C61#18748semidirect

Maximal subgroups

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

subgroupindex
C2×#168404722
#214567412
#214552792
#214450252
#214448902
#214444992
#214442482
#214442092
C61×#31032
#1874861