Finite Group Databasebeta

← #29571946

Subgroups — #29571946

Order 15616.

772 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459730C211yesmaximal, characteristic
2#21446742C211yesmaximal, characteristic, complemented
3#21444501C211yesmaximal, characteristic, complemented
4#21444497C211yesmaximal, characteristic, complemented
5#21444313C211yesmaximal, characteristic, complemented
6#21444206C211yesmaximal, characteristic, complemented
7C61×#1806C211yesmaximal, characteristic, complemented
8#16840485—21no—
9#16840477C412yescomplemented
10#16840472C412yescomplemented
11#16840458—22no—
12#16840421—22no—
13#16840417—21no—
14#16840383—22no—
15#16840340—21no—
16#16840319—21no—
17#16840312—21no—
18#16840312C2211yescharacteristic
19C2×#14519777C2211yescharacteristic
20#16840242—21no—
21#16839815C2211yescharacteristic
22C2×C4⋊#92418C2211yescharacteristic
23C2×#14519700—22no—
24#16839510—22no—
25#16839509—21no—
26#16839505—21no—
27#16839495—22no—
28Q8⋊#92418—21no—
29D4⋊#92418—21no—
30C4×#41×C61C2211yescharacteristic
31C61×#423C2211yescharacteristic
32C2×#104×C61—21no—
33C4×C4⋊C4×C61C2211yescharacteristic
34C61×#339—21no—
35C2×#11758222—21no—
36C2×#11758222—22no—
37C22×Dic122D412yes—
38C22×#92418—21no—
39#14519777—21no—
40#14519777—21no—
41#14519777C2×C412yes—
42#14519777D412yes—
43#14519777D411yescharacteristic
44#14519774D411yescharacteristic
45#14519772—24no—
46#14519769—22no—
47#14519765—22no—
48#14519765—21no—
49#14519763—24no—
50#14519759—24no—
51#14519756—22no—
52#14519756—22no—
53#14519755—24no—
54#14519753—22no—
55#14519751—22no—
56#14519751—21no—
57#14519751—22no—
58#14519751D412yes—
59#14519749—24no—
60#14519748—22no—
61C4×Dic122—22no—
62C4×Dic122—22no—
63C4×D244—22no—
64C4×D244—22no—
65C4×D244—21no—
66C4×D244C2×C412yes—
67#14519737—24no—
68#14519736—22no—
69#14519735—22no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C4⋊C4×D61—22no—
73C4⋊#92418—21no—
74C4⋊#92418—21no—
75C4⋊#92418—21no—
76#14519700—44no—
77C22⋊#92418—21no—
78C4×#92418—21no—
79C2×#41×C61D411yescharacteristic
80C2×M16×C61—21no—
81C4×Q8×C61C2×C411yescharacteristic
82C4×Q8×C61D411yescharacteristic
83C4×D4×C61—21no—
84C4×D4×C61—21no—
85C4×D4×C61C2×C411yescharacteristic
86C4×D4×C61D411yescharacteristic
87#117×C61—21no—
88#117×C61D411yescharacteristic
89C2×C4⋊C4×C61—21no—
90C2×C42×C61—21no—
91C2×C42×C61—21no—
92C2×C42×C61C2311yescharacteristic
93C4⋊C8×C61—21no—
94C4⋊C8×C61—21no—
95#104×C61—42no—
96#95×C61—21no—
97#41×C61—21no—
98#41×C61—21no—
99#41×C61—21no—
100#41×C61—21no—
101C2×Q8×C61C22⋊C411yescharacteristic
102C2×D4×C61—21no—
103C2×D4×C61C22⋊C411yescharacteristic
104C22×C4×C61—21no—
105C22×C4×C61—21no—
106C22×C4×C61—21no—
107C22×C4×C61—41no—
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C2×C22⋊D61—22no—
112C2×C22⋊D61—24no—
113C22×Dic61—22no—
114#11758222—41no—
115#11758222—41no—
116#11758222—41no—
117#11758222—41no—
118#11758222—82no—
119#11758222—82no—
120C2×D244—21no—
121C2×D244—21no—
122C2×D244C22⋊C412yes—
123C2×C4×D61—22no—
124C2×C4×D61—24no—
125C2×Dic122—22no—
126C2×Dic122—21no—
127C2×Dic122—42no—
128C2×Dic122—42no—
129C2×Dic122C22⋊C414yes—
130M16×C61—41no—
131M16×C61—41no—
132C2×C8×C61—41no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—41no—
137C4⋊C4×C61C22⋊C411yescharacteristic
138C4⋊C4×C61C22⋊C411yescharacteristic
139C22⋊C4×C61—21no—
140C22⋊C4×C61—21no—
141C22⋊C4×C61—21no—
142C42×C61—21no—
143C42×C61—21no—
144C42×C61—21no—
145C42×C61—41no—
146C42×C61—41no—
147C42×C61C22×C411yescharacteristic
148C42×C61C2×D411yescharacteristic
149C42×C61C2×D411yescharacteristic
150C42×C61C2×D411yescharacteristic
151C22⋊Dic61—24no—
152#11758198—24no—
153#11758198—24no—
154#11758198—22no—
155#11758198—22no—
156C4⋊Dic61—22no—
157C4⋊Dic61—22no—
158C4⋊Dic61—22no—
159C4⋊Dic61—22no—
160C4⋊Dic61—21no—
161C4⋊Dic61C22⋊C412yes—
162#11758196—24no—
163#11758196—24no—
164C4×Dic61—24no—
165C2×#92418—42no—
166C2×#92418—41no—
167C2×#92418—41no—
168C23×C61—21no—
169C23×C61#12011yescharacteristic
170C22×D61—22no—
171Q8×C61—21no—
172Q8×C61—21no—
173D4×C61—21no—
174D4×C61—21no—
175D4×C61—41no—
176D4×C61—41no—
177C2×C4×C61—21no—
178C2×C4×C61—21no—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—21no—
182C2×C4×C61—41no—
183C2×C4×C61—42no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61—42no—
189C2×C4×C61—81no—
190C2×C4×C61—81no—
191C2×C4×C61C2×C22⋊C411yescharacteristic
192C2×C4×C61C2×C22⋊C411yescharacteristic
193C2×C4×C61C2×C22⋊C411yescharacteristic
194C2×C4×C61#12011yescharacteristic
195C2×C4×C61#12011yescharacteristic
196C2×C4×C61#12011yescharacteristic
197C2×C4×C61#13511yescharacteristic
198C2×C4×C61#13511yescharacteristic
199C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
200C2×C4×C61#13711yescharacteristic
201C22⋊D61—42no—
202C22⋊D61—42no—
203C22⋊D61—84no—
204C2×Dic61—22no—
205C2×Dic61—24no—
206C2×Dic61—82no—
207D244—42no—
208D244—41no—
209D244—41no—
210D244—41no—
211D244—41no—
212C4×D61—42no—
213C4×D61—42no—
214C4×D61—84no—
215Dic122—41no—
216Dic122—41no—
217Dic122—42no—
218Dic122—42no—
219Dic122—44no—
220C8×C61—81no—
221#92418—81no—
222#92418—81no—
223#8423—611nomaximal, Sylow
224C22×C61—21no—
225C22×C61—21no—
226C22×C61—41no—
227C22×C61—41no—
228C22×C61—41no—
229C22×C61#37911yescharacteristic
230C22×C61#44911yescharacteristic
231C22×C61#45211yescharacteristic
232C2×D61—42no—
233C2×D61—42no—
234C2×D61—42no—
235C4×C61—21no—
236C4×C61—21no—
237C4×C61—21no—
238C4×C61—41no—
239C4×C61—41no—
240C4×C61—41no—
241C4×C61—82no—
242C4×C61#41611yescharacteristic
243C4×C61#41711yescharacteristic
244C4×C61#44911yescharacteristic
245C4×C61#45211yescharacteristic
246C4×C61#45311yescharacteristic
247C4×C61#45511yescharacteristic
248Dic61—44no—
249Dic61—42no—
250#3511—611no—
251#1981—611no—
252#1806D6111yescharacteristic, complemented
253#1548—611no—
254#1547—611no—
255#1502—611no—
256#1477—611no—
257C2×C61—21no—
258C2×C61—21no—
259C2×C61—81no—
260C2×C61#178811yescharacteristic
261C2×C61#179211yescharacteristic
262C2×C61#179611yescharacteristic
263D61—82no—
264#556—1221no—
265#555—612no—
266#554—612no—
267#553—1222no—
268#548—1222no—
269#547—1221no—
270#542—1222no—
271#537—1221no—
272#533—1221no—
273#532—611no—
274#532—1221no—
275C2×C4⋊Q8—611no—
276#520—1221no—
277C4×#41C2×D6111yescharacteristic
278#431—611no—
279#423C2×D6111yescharacteristic
280C2×C4⋊C8—611no—
281C2×#104—21no—
282C2×#103—1222no—
283C4×C4⋊C4C2×D6111yescharacteristic
284#339—21no—
285#333—1222no—
286#332—1221no—
287#331—1221no—
288#327—1222no—
289Q8⋊C8—1221no—
290D4⋊C8—1221no—
291C61#842311yescharacteristic, complemented, Sylow
292C2×#41—1222no—
293C2×#41—1221no—
294C2×#41C22⋊D6111yescharacteristic
295C22×Q8—612no—
296C2×M16—21no—
297C22×C8—1221no—
298C4⋊Q8—612no—
299C4⋊Q8—611no—
300C4⋊Q8—612no—
301C4⋊Q8—1221no—
302C4⋊Q8—1221no—
303#127—611no—
304#126—1224no—
305#125—1222no—
306#124—1222no—
307#124—1224no—
308#124—1221no—
309#123—1224no—
310C22⋊Q8—1224no—
311C22⋊Q8—1222no—
312C22⋊Q8—1222no—
313C22⋊Q8—1222no—
314#121—612no—
315#121—1222no—
316#121—1222no—
317#121—1224no—
318#121—1222no—
319#121—1221no—
320C4×Q8—1222no—
321C4×Q8—1222no—
322C4×Q8C4×D6111yescharacteristic
323C4×Q8D24411yescharacteristic
324C4×D4—21no—
325C4×D4—21no—
326C4×D4—612no—
327C4×D4—1222no—
328C4×D4—1222no—
329C4×D4—1224no—
330C4×D4—1221no—
331C4×D4C4×D6111yescharacteristic
332C4×D4D24411yescharacteristic
333#117—21no—
334#117—1222no—
335#117—1222no—
336#117C22⋊D6111yescharacteristic
337C2×C4⋊C4—21no—
338C2×C4⋊C4—1221no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4—1222no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42C22×D6111yescharacteristic
344C4⋊C8—21no—
345C4⋊C8—21no—
346C4⋊C8—1221no—
347C4⋊C8—1221no—
348C4⋊C8—1221no—
349#104—42no—
350#103—2444no—
351C22⋊C8—1221no—
352C4×C8—1221no—
353#95—21no—
354#41—21no—
355#41—21no—
356#41—21no—
357#41—21no—
358#41—4882no—
359#41—4882no—
360#41—2441no—
361#41—2441no—
362#41—2441no—
363#41—2441no—
364C2×Q8—614no—
365C2×Q8—1221no—
366C2×Q8—1222no—
367C2×Q8—2442no—
368C2×Q8—2442no—
369C2×Q8#1175819811yescharacteristic
370C2×D4—21no—
371C2×D4—612no—
372C2×D4—1224no—
373C2×D4—1222no—
374C2×D4—1221no—
375C2×D4—1221no—
376C2×D4#1175819811yescharacteristic
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—41no—
381C22×C4—1224no—
382C22×C4—1222no—
383C22×C4—1222no—
384C22×C4D4×D6111yescharacteristic
385C22×C4D4×D6111yescharacteristic
386C22×C4C2×C22⋊D6111yescharacteristic
387M16—41no—
388M16—41no—
389C2×C8—41no—
390C2×C8—2442no—
391C2×C8—2441no—
392C2×C8—2441no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—22no—
396C4⋊C4—41no—
397C4⋊C4—612no—
398C4⋊C4—1224no—
399C4⋊C4—1224no—
400C4⋊C4—1222no—
401C4⋊C4—1222no—
402C4⋊C4—1222no—
403C4⋊C4—1222no—
404C4⋊C4—1221no—
405C4⋊C4#1175819811yescharacteristic
406C4⋊C4#1175819811yescharacteristic
407C22⋊C4—21no—
408C22⋊C4—21no—
409C22⋊C4—21no—
410C22⋊C4—1224no—
411C22⋊C4—1224no—
412C22⋊C4—1224no—
413C22⋊C4—1222no—
414C22⋊C4—1222no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—41no—
419C42—41no—
420C42—1224no—
421C42C2×C4×D6111yescharacteristic
422C42C2×D24411yescharacteristic
423C42D4×D6111yescharacteristic
424C42D4×D6111yescharacteristic
425C23—21no—
426C23—1222no—
427C23#1451974511yescharacteristic
428Q8—21no—
429Q8—21no—
430Q8—2444no—
431Q8—2442no—
432Q8—2441no—
433Q8—2441no—
434Q8—2442no—
435D4—21no—
436D4—21no—
437D4—41no—
438D4—41no—
439D4—4884no—
440D4—2442no—
441D4—2442no—
442D4—2442no—
443D4—2441no—
444D4—2441no—
445D4—2441no—
446D4—2441no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—1222no—
462C2×C4—1224no—
463C2×C4—4884no—
464C2×C4—4882no—
465C2×C4—2442no—
466C2×C4—2442no—
467C2×C4C22⋊C4×D6111yescharacteristic
468C2×C4C22⋊C4×D6111yescharacteristic
469C2×C4C2×#1175819811yescharacteristic
470C2×C4#1451974511yescharacteristic
471C2×C4#1451974711yescharacteristic
472C2×C4#1451974711yescharacteristic
473C2×C4#1451979711yescharacteristic
474C2×C4#1451979711yescharacteristic
475C2×C4#1451980311yescharacteristic
476C2×C4#1451980811yescharacteristic
477C8—81no—
478C8—4881no—
479C8—4881no—
480C22—21no—
481C22—21no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—2442no—
486C22—2442no—
487C22—2442no—
488C22#1683961711yescharacteristic
489C22#1683990511yescharacteristic
490C22#1683992211yescharacteristic
491C4—21no—
492C4—21no—
493C4—21no—
494C4—41no—
495C4—41no—
496C4—41no—
497C4—82no—
498C4—2442no—
499C4—2444no—
500C4#1683975611yescharacteristic
501C4#1683976111yescharacteristic
502C4#1683990511yescharacteristic
503C4#1683992211yescharacteristic
504C4#1683993111yescharacteristic
505C4#1683994111yescharacteristic
506C2—21no—
507C2—21no—
508C2—81no—
509C2—4882no—
510C2#2144575711yescharacteristic
511C2#2144577411yescharacteristic
512C2#2144579311yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1806C2
socleC22×C61#379
prime radical#21459730C2
cosocleC2×C42×C61C23
hypercenter#1806D61
hypocenterC61#8423
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
2. #21446742C2semidirect
3. #21444501C2semidirect
4. #21444497C2semidirect
5. #21444313C2semidirect
6. #21444206C2semidirect
7. C61×#1806C2semidirect
9. #16840477C4semidirect
10. #16840472C4semidirect
252. #1806D61semidirect
291. C61#8423semidirect

Maximal subgroups

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

subgroupindex
#214597302
#214467422
#214445012
#214444972
#214443132
#214442062
C61×#18062
#842361