Finite Group Databasebeta

← #29835208

Subgroups — #29835208

Order 15616.

546 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454696C211yesmaximal, characteristic, complemented
2#21448014C211yesmaximal, characteristic, complemented
3#21448002C211yesmaximal, characteristic, complemented
4#21447963C211yesmaximal, characteristic, complemented
5#21447958C211yesmaximal, characteristic, complemented
6#21447878C211yesmaximal, characteristic, complemented
7#21447871C211yesmaximal, characteristic, complemented
8#21447757C211yesmaximal, characteristic, complemented
9#21447740C211yesmaximal, characteristic
10#21447693C211yesmaximal, characteristic, complemented
11#21447688C211yesmaximal, characteristic, complemented
12C4×#14519720C211yesmaximal, characteristic, complemented
13C4×#14519721C211yesmaximal, characteristic, complemented
14#21447626C211yesmaximal, characteristic, complemented
15C61×#3175C211yesmaximal, characteristic, complemented
16C2×#14519799C2211yescharacteristic
17#16840242—21no—
18C2×#14519723C2211yescharacteristic
19C2×#14519721C2211yescharacteristic
20C2×#14519720C2211yescharacteristic
21C2×#14519718C2211yescharacteristic
22#16839864—21no—
23#16839864—21no—
24C4×Dic244C2211yescharacteristic
25C4×Dic244C2211yescharacteristic
26C4×D488C2211yescharacteristic
27C4×D488C2211yescharacteristic
28C2×#14519711C2211yescharacteristic
29#16839764—21no—
30#16839589C2211yescharacteristic
31#16839589C2211yescharacteristic
32#16839589C2211yescharacteristic
33#16839589C2211yescharacteristic
34C4⋊#11758185C2211yescharacteristic
35C4⋊#11758185C2211yescharacteristic
36#16839566C2211yescharacteristic
37#16839566C2211yescharacteristic
38#16839565C2211yescharacteristic
39#16839565C2211yescharacteristic
40#16839563C2211yescharacteristic
41#16839563C2211yescharacteristic
42#16839562C2211yescharacteristic
43#16839562C2211yescharacteristic
44C22⋊#11758185C2211yescharacteristic
45C22⋊#11758185C2211yescharacteristic
46#16839538C2211yescharacteristic
47C4×#11758185C2211yescharacteristic
48C61×#542—21no—
49C61×#500C2211yescharacteristic
50C61×#495C2211yescharacteristic
51C61×#489—21no—
52C61×#468—21no—
53C61×#467C2211yescharacteristic
54C61×#466C2211yescharacteristic
55C61×#440—21no—
56C4×Q16×C61C2211yescharacteristic
57C4×D8×C61C2211yescharacteristic
58C61×#405C2211yescharacteristic
59C2×#11758222—21no—
60#14519799—22no—
61#14519799—22no—
62C2×Dic244C2311yescharacteristic
63C2×#11758190—21no—
64C2×D488C2311yescharacteristic
65C4×Dic122—21no—
66C4×Dic122—21no—
67C4×D244—21no—
68C4×D244—21no—
69#14519737—41no—
70#14519737—41no—
71#14519735—41no—
72C2×C4⋊Dic61—21no—
73C4⋊C4×D61—41no—
74#14519723—22no—
75#14519721—22no—
76#14519720—22no—
77#14519718—22no—
78C2×#11758185C2311yescharacteristic
79C2×#11758185C2311yescharacteristic
80C2×#11758185C2311yescharacteristic
81C2×#11758185C2311yescharacteristic
82#14519711—21no—
83#14519711C2311yescharacteristic
84#14519711C2311yescharacteristic
85#14519702—21no—
86#14519702—21no—
87#14519699—21no—
88#14519699—21no—
89C2×Q16×C61C2311yescharacteristic
90C2×SD16×C61—21no—
91C2×D8×C61C2311yescharacteristic
92C2×M16×C61D411yescharacteristic
93C22×C8×C61C2311yescharacteristic
94C4⋊Q8×C61—21no—
95#126×C61—41no—
96#124×C61—21no—
97#124×C61—41no—
98#123×C61—41no—
99C22⋊Q8×C61—21no—
100C22⋊Q8×C61—41no—
101#121×C61—21no—
102C4×Q8×C61—21no—
103C4×D4×C61—21no—
104#117×C61D411yescharacteristic
105#107×C61—21no—
106#107×C61C2311yescharacteristic
107#107×C61C2311yescharacteristic
108#103×C61—21no—
109#103×C61—21no—
110#102×C61—21no—
111#102×C61—21no—
112#97×C61D411yescharacteristic
113#97×C61D411yescharacteristic
114C4×C8×C61C2311yescharacteristic
115C4×C8×C61C2311yescharacteristic
116C2×Q8×C61—21no—
117C2×Q8×C61—41no—
118C2×D4×C61—21no—
119C2×D4×C61—41no—
120C22×C4×C61—41no—
121C22×C4×C61C2×D411yescharacteristic
122C2×C22⋊D61—41no—
123C22×Dic61—41no—
124#11758222—24no—
125#11758222—42no—
126C2×D244—21no—
127C2×C4×D61—41no—
128C2×C4×D61—41no—
129C2×C4×D61—41no—
130C2×Dic122—21no—
131Q16×C61—21no—
132Q16×C61—21no—
133SD16×C61—42no—
134D8×C61—21no—
135D8×C61—21no—
136M16×C61—41no—
137C2×C8×C61—21no—
138C2×C8×C61—21no—
139C2×C8×C61C2×D411yescharacteristic
140C2×C8×C61C2×D411yescharacteristic
141C2×C8×C61C2×D411yescharacteristic
142C2×C8×C61C2411yescharacteristic
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—41no—
146C4⋊C4×C61—41no—
147C4⋊C4×C61—41no—
148C4⋊C4×C61D811yescharacteristic
149C4⋊C4×C61D811yescharacteristic
150C22⋊C4×C61—41no—
151C22⋊C4×C61—41no—
152C22⋊C4×C61—41no—
153C22⋊C4×C61—41no—
154C22⋊C4×C61—41no—
155C22⋊C4×C61D811yescharacteristic
156C22⋊C4×C61D811yescharacteristic
157C42×C61—41no—
158C42×C61C2×D411yescharacteristic
159C42×C61C2×D411yescharacteristic
160#11758198—41no—
161#11758198—41no—
162C4⋊Dic61—21no—
163C4⋊Dic61—21no—
164C4⋊Dic61—41no—
165#11758196—41no—
166#11758196—41no—
167C4×Dic61—41no—
168C4×Dic61—41no—
169Dic244—21no—
170Dic244#4112yes—
171D488—21no—
172D488#4112yes—
173#11758190—24no—
174#11758185—21no—
175#11758185—21no—
176#11758185—21no—
177#11758185—21no—
178C23×C61—41no—
179C23×C61C2×D811yescharacteristic
180C22×D61—41no—
181Q8×C61—41no—
182Q8×C61—41no—
183Q8×C61—161no—
184D4×C61—41no—
185D4×C61—41no—
186D4×C61—161no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61—161no—
193C2×C4×C61C2×D811yescharacteristic
194C2×C4×C61C2×D811yescharacteristic
195C2×C4×C61C2×D811yescharacteristic
196C2×C4×C61C2×D811yescharacteristic
197C2×C4×C61C2×D811yescharacteristic
198C2×C4×C61C22×D411yescharacteristic
199C22⋊D61—44no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—161no—
204D244—22no—
205D244—41no—
206C4×D61—44no—
207C4×D61—161no—
208C4×D61—161no—
209Dic122—22no—
210Dic122—41no—
211C8×C61—21no—
212C8×C61—21no—
213C8×C61—21no—
214C8×C61C2×#4111yescharacteristic
215C8×C61C2×#4111yescharacteristic
216#33685—611nomaximal, Sylow
217C22×C61—21no—
218C22×C61—21no—
219C22×C61—21no—
220C22×C61—81no—
221C22×C61—81no—
222C22×C61—81no—
223C22×C61C22×D811yescharacteristic
224C2×D61—42no—
225C2×D61—81no—
226C2×D61—81no—
227C4×C61—21no—
228C4×C61—21no—
229C4×C61—21no—
230C4×C61—21no—
231C4×C61—21no—
232C4×C61—81no—
233C4×C61—81no—
234C4×C61—81no—
235C4×C61#52511yescharacteristic
236C4×C61#57511yescharacteristic
237Dic61—42no—
238Dic61—81no—
239Dic61—81no—
240#3175D6111yescharacteristic, complemented
241#2951—611no—
242#2235—611no—
243#2233—611no—
244#2226—611no—
245#2225—611no—
246#2211—611no—
247#2210—611no—
248#2189—611no—
249#2186—611no—
250#2177—611no—
251#2176—611no—
252C4×SD32—611no—
253C4×SD32—611no—
254#2164—611no—
255C2×C61—41no—
256C2×C61—161no—
257C2×C61#310011yescharacteristic
258C2×C61#341611yescharacteristic
259C2×C61#341611yescharacteristic
260D61—82no—
261C2×#135—611no—
262#542—21no—
263#520—1221no—
264C2×Q32—611no—
265C2×SD32—611no—
266C2×SD32—611no—
267C2×D16—611no—
268#500C2×D6111yescharacteristic
269#495C2×D6111yescharacteristic
270#489—21no—
271#468—21no—
272#467C2×D6111yescharacteristic
273#466C2×D6111yescharacteristic
274#440—21no—
275#440—1221no—
276#440—1221no—
277C4×Q16—611no—
278C4×Q16—611no—
279C4×Q16C2×D6111yescharacteristic
280C4×D8—611no—
281C4×D8—611no—
282C4×D8C2×D6111yescharacteristic
283C2×#107—611no—
284#417—1221no—
285#405C2×D6111yescharacteristic
286#367—611no—
287#367—611no—
288#367—611no—
289#367—611no—
290C4⋊C16—611no—
291C4⋊C16—611no—
292#358—611no—
293#358—611no—
294#358—611no—
295#358—611no—
296#357—611no—
297#357—611no—
298#357—611no—
299#357—611no—
300C22⋊C16—611no—
301C22⋊C16—611no—
302#346—611no—
303C4×C16—611no—
304C61#3368511yescharacteristic, complemented, Sylow
305C2×#41—1221no—
306#135—1222no—
307#135—1222no—
308C2×Q16—611no—
309C2×Q16C22×D6111yescharacteristic
310C2×SD16—21no—
311C2×SD16—1221no—
312C2×D8—611no—
313C2×D8C22×D6111yescharacteristic
314C2×M16C22⋊D6111yescharacteristic
315C22×C8C22×D6111yescharacteristic
316C4⋊Q8—21no—
317#126—41no—
318#124—21no—
319#124—41no—
320#123—41no—
321C22⋊Q8—21no—
322C22⋊Q8—41no—
323#121—21no—
324C4×Q8—21no—
325C4×Q8—1221no—
326C4×Q8—1221no—
327C4×D4—21no—
328C4×D4—1221no—
329C4×D4—1221no—
330C4×D4—2441no—
331C4×D4—2441no—
332#117—2441no—
333#117C22⋊D6111yescharacteristic
334C2×C4⋊C4—1221no—
335C2×C4⋊C4—2441no—
336Q32—1222no—
337SD32—1222no—
338SD32—1222no—
339D16—1222no—
340C2×C16—611no—
341C2×C16—611no—
342C2×C16—611no—
343C2×C16—611no—
344#107—21no—
345#107—611no—
346#107—611no—
347#107—1221no—
348#107C22×D6111yescharacteristic
349#107C22×D6111yescharacteristic
350#103—21no—
351#103—21no—
352#103—1221no—
353#103—1221no—
354#102—21no—
355#102—21no—
356#102—1221no—
357#102—1221no—
358#97C22⋊D6111yescharacteristic
359#97C22⋊D6111yescharacteristic
360C4×C8C22×D6111yescharacteristic
361C4×C8C22×D6111yescharacteristic
362#41—1224no—
363#41—2442no—
364C2×Q8—21no—
365C2×Q8—41no—
366C2×Q8—1221no—
367C2×D4—21no—
368C2×D4—41no—
369C2×D4—1221no—
370C2×D4—2441no—
371C22×C4—41no—
372C22×C4—2441no—
373C22×C4—2441no—
374C22×C4—2441no—
375C22×C4—2441no—
376C22×C4C2×C22⋊D6111yescharacteristic
377Q16—21no—
378Q16—21no—
379Q16—612no—
380Q16—1221no—
381SD16—42no—
382SD16—1224no—
383D8—21no—
384D8—21no—
385D8—612no—
386D8—1221no—
387M16—41no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8C2×C22⋊D6111yescharacteristic
391C2×C8C2×C22⋊D6111yescharacteristic
392C2×C8C2×C22⋊D6111yescharacteristic
393C2×C8C23×D6111yescharacteristic
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—1221no—
400C4⋊C4—1221no—
401C4⋊C4—2441no—
402C4⋊C4—2441no—
403C4⋊C4—2441no—
404C4⋊C4#1175819911yescharacteristic
405C4⋊C4#1175819911yescharacteristic
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—2441no—
412C22⋊C4—2441no—
413C22⋊C4#1175819911yescharacteristic
414C22⋊C4#1175819911yescharacteristic
415C42—41no—
416C42—2441no—
417C42—2441no—
418C42C2×C22⋊D6111yescharacteristic
419C42C2×C22⋊D6111yescharacteristic
420C16—1221no—
421C16—1221no—
422C16—1221no—
423C16—1221no—
424C23—41no—
425C23—2441no—
426C23C2×#1175819911yescharacteristic
427Q8—41no—
428Q8—41no—
429Q8—161no—
430Q8—1222no—
431Q8—2441no—
432D4—41no—
433D4—41no—
434D4—161no—
435D4—1222no—
436D4—2444no—
437D4—2441no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—161no—
444C2×C4—9761no—
445C2×C4—9761no—
446C2×C4—9761no—
447C2×C4—2444no—
448C2×C4—2441no—
449C2×C4—2441no—
450C2×C4—2441no—
451C2×C4C2×#1175819911yescharacteristic
452C2×C4C2×#1175819911yescharacteristic
453C2×C4C2×#1175819911yescharacteristic
454C2×C4C2×#1175819911yescharacteristic
455C2×C4C2×#1175819911yescharacteristic
456C2×C4C22×C22⋊D6111yescharacteristic
457C8—21no—
458C8—21no—
459C8—21no—
460C8#41×D6111yescharacteristic
461C8#41×D6111yescharacteristic
462C22—21no—
463C22—21no—
464C22—21no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—4881no—
469C22—4881no—
470C22—2442no—
471C22C22×#1175819911yescharacteristic
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—81no—
478C4—81no—
479C4—81no—
480C4—4881no—
481C4—4881no—
482C4—2442no—
483C4#1684026911yescharacteristic
484C4#1684056011yescharacteristic
485C2—41no—
486C2—161no—
487C2—4882no—
488C2#2145581511yescharacteristic
489C2#2145896311yescharacteristic
490C2#2145896611yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3175C2
socleC22×C61C22×D8
prime radical#21447740C2
hypercenter#3175D61
hypocenterC61#33685
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3175

Complements

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

NH ≅ G/Ntype
1. #21454696C2semidirect
2. #21448014C2semidirect
3. #21448002C2semidirect
4. #21447963C2semidirect
5. #21447958C2semidirect
6. #21447878C2semidirect
7. #21447871C2semidirect
8. #21447757C2semidirect
10. #21447693C2semidirect
11. #21447688C2semidirect
12. C4×#14519720C2semidirect
13. C4×#14519721C2semidirect
14. #21447626C2semidirect
15. C61×#3175C2semidirect
240. #3175D61semidirect
304. C61#33685semidirect

Maximal subgroups

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

subgroupindex
#214546962
#214480142
#214480022
#214479632
#214479582
#214478782
#214478712
#214477572
#214477402
#214476932
#214476882
C4×#145197202
C4×#145197212
#214476262
C61×#31752
#3368561