Finite Group Databasebeta

← #29652290

Subgroups — #29652290

Order 15616.

1208 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459152C211yesmaximal, characteristic
2#21455850C212yesmaximal, complemented
3#21444855C212yesmaximal, complemented
4#21444838C212yesmaximal, complemented
5#21444787C212yesmaximal, complemented
6#21444770C212yesmaximal, complemented
7#21444313C212yesmaximal, complemented
8#21444133C211yesmaximal, characteristic, complemented
9C61×#2396C211yesmaximal, characteristic, complemented
10C22×#11758222—21no—
11#16840343—21no—
12#16840343C2212yes—
13#16840338—22no—
14#16840331—22no—
15#16840324—21no—
16#16840324C2212yes—
17#16840312—21no—
18#16840312C2212yes—
19#16840274—24no—
20#16840080C2214yescomplemented
21#16840047—22no—
22#16839962—22no—
23#16839901C2214yescomplemented
24C4×#11758201—24no—
25#16839815C2212yescomplemented
26#16839495C2214yescomplemented
27#16839495C2214yescomplemented
28#16839488C2212yescomplemented
29#16839486C2212yes—
30C2×C4×Q8×C61C2211yescharacteristic, complemented
31C2×#117×C61C2211yescharacteristic
32C61×#400C2211yescharacteristic, complemented
33C61×#399C2211yescharacteristic
34C61×#398—21no—
35C61×#396—21no—
36C61×#395C2211yescharacteristic, complemented
37C61×#393C2211yescharacteristic
38C4×C4⋊C4×C61—21no—
39C4×C4⋊C4×C61C2211yescharacteristic, complemented
40C4×C22⋊C4×C61—21no—
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222—44no—
46C2×#11758222—44no—
47C22×Dic122—21no—
48C22×D244—21no—
49C22×C22⋊D61—22no—
50C22×C4×D61—22no—
51C2×#11758201—24no—
52#14519777—21no—
53#14519777D412yes—
54#14519774—21no—
55#14519774D412yes—
56#14519765—22no—
57#14519765—22no—
58#14519761—24no—
59#14519761—24no—
60#14519757—24no—
61#14519757—24no—
62#14519752—24no—
63#14519752—24no—
64#14519751—22no—
65#14519751—22no—
66#14519751—22no—
67#14519751C2314yes—
68#14519747—24no—
69#14519747—24no—
70C4×D244—22no—
71C4×D244—22no—
72C4×D244—22no—
73C4×D244D414yes—
74#14519708—24no—
75C4⋊#92418—22no—
76#14519701—24no—
77#14519697—24no—
78C22⋊#92418C2314yes—
79C4×#92418—22no—
80C22×Q8×C61C2311yescharacteristic
81C23×C4×C61D411yescharacteristic
82C4×Q8×C61—22no—
83C4×Q8×C61—22no—
84#117×C61—22no—
85#117×C61—22no—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61—21no—
88C2×C4⋊C4×C61—21no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61D411yescharacteristic
91C2×C4⋊C4×C61C2311yescharacteristic
92C2×C4⋊C4×C61C2311yescharacteristic
93C2×C22⋊C4×C61—21no—
94C2×C22⋊C4×C61—21no—
95C2×C22⋊C4×C61D411yescharacteristic
96C2×C22⋊C4×C61C2311yescharacteristic
97C2×C42×C61—21no—
98C2×C42×C61—21no—
99C2×C42×C61D411yescharacteristic
100C2×C42×C61C2311yescharacteristic
101#95×C61—21no—
102#95×C61—21no—
103#95×C61—21no—
104#95×C61—21no—
105#95×C61C2311yescharacteristic
106#95×C61C2311yescharacteristic
107C24×C61C2×D411yescharacteristic
108C23×D61—22no—
109C2×Q8×C61—22no—
110C2×Q8×C61—42no—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—22no—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61SD1614yescomplemented
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2411yescharacteristic
125C2×C22⋊D61—24no—
126C2×C22⋊D61—24no—
127C2×C22⋊D61—44no—
128C2×C22⋊D61—44no—
129C22×Dic61—22no—
130#11758222—44no—
131#11758222—48no—
132#11758222—48no—
133#11758222—44no—
134#11758222—44no—
135#11758222—44no—
136C2×D244—22no—
137C2×D244—22no—
138C2×D244—22no—
139C2×D244—22no—
140C2×D244—22no—
141C2×D244—44no—
142C2×D244C2×D414yes—
143C2×C4×D61—24no—
144C2×C4×D61—24no—
145C2×C4×D61—48no—
146C2×Dic122—22no—
147C2×Dic122—22no—
148C2×Dic122—44no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—42no—
153C4⋊C4×C61—42no—
154C4⋊C4×C61—42no—
155C4⋊C4×C61—42no—
156C22⋊C4×C61—22no—
157C22⋊C4×C61—22no—
158C22⋊C4×C61—42no—
159C22⋊C4×C61—42no—
160C42×C61—21no—
161C42×C61—21no—
162C42×C61—41no—
163C42×C61—41no—
164C42×C61—42no—
165C42×C61C2×D412yes—
166C42×C61C2×D412yes—
167#11758201—48no—
168#11758198—24no—
169#11758198—24no—
170#11758198—24no—
171#11758198—24no—
172C4⋊Dic61—22no—
173C4⋊Dic61—22no—
174C4⋊Dic61—22no—
175C4⋊Dic61C2×D414yes—
176C2×#92418—24no—
177C23×C61—21no—
178C23×C61—21no—
179C23×C61—41no—
180C23×C61—41no—
181C23×C61C2×SD1612yes—
182C23×C61C22×D411yescharacteristic
183C22×D61—24no—
184C22×D61—44no—
185C22×D61—82no—
186C22×D61—82no—
187Q8×C61—24no—
188Q8×C61—42no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—24no—
192C2×C4×C61—24no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61#12012yes—
212C2×C4×C61#12011yescharacteristic
213C2×C4×C61#12011yescharacteristic
214C2×C4×C61C2×SD1612yes—
215C2×C4×C61C2×SD1612yes—
216C2×C4×C61AGL(1,ℤ/8)12yes—
217C2×C4×C61C22×D412yes—
218C22⋊D61—48no—
219C22⋊D61—44no—
220C22⋊D61—44no—
221C2×Dic61—24no—
222C2×Dic61—44no—
223D244—44no—
224D244—44no—
225D244—44no—
226D244—82no—
227D244—82no—
228D244—82no—
229C4×D61—48no—
230C4×D61—48no—
231Dic122—44no—
232Dic122—44no—
233#92418—44no—
234#18123—611nomaximal, Sylow
235C22×C61—21no—
236C22×C61—21no—
237C22×C61—22no—
238C22×C61—42no—
239C22×C61—41no—
240C22×C61—41no—
241C22×C61—81no—
242C22×C61C2×#12011yescharacteristic
243C22×C61C22×SD1611yescharacteristic
244C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
245C2×D61—44no—
246C2×D61—44no—
247C2×D61—44no—
248C2×D61—88no—
249C4×C61—22no—
250C4×C61—22no—
251C4×C61—22no—
252C4×C61—42no—
253C4×C61—41no—
254C4×C61—41no—
255C4×C61—42no—
256C4×C61—81no—
257C4×C61—81no—
258C4×C61#45012yes—
259C4×C61#45012yes—
260Dic61—44no—
261#3449—611no—
262#3103—612no—
263#2396D6111yescharacteristic, complemented
264#1616—612no—
265#1613—612no—
266#1604—612no—
267#1601—612no—
268#1502—612no—
269#1458—611no—
270C2×C61—21no—
271C2×C61—21no—
272C2×C61—42no—
273C2×C61C2×#45011yescharacteristic
274C2×C61#301811yescharacteristic
275C2×C61#302511yescharacteristic
276D61—84no—
277C22×#41—1221no—
278#537—612no—
279#537—1221no—
280#536—1222no—
281#535—1222no—
282#534—612no—
283#534—1221no—
284#532—612no—
285#532—1221no—
286#525—1224no—
287C2×C4×Q8C2×D6111yescharacteristic, complemented
288C2×#117C2×D6111yescharacteristic
289#481—614no—
290#475—1222no—
291#460—1222no—
292#448—614no—
293C4×SD16—1224no—
294#431—612no—
295#400C2×D6111yescharacteristic, complemented
296#399C2×D6111yescharacteristic
297#398—21no—
298#396—21no—
299#395C2×D6111yescharacteristic, complemented
300#393C2×D6111yescharacteristic
301C4×C4⋊C4—21no—
302C4×C4⋊C4C2×D6111yescharacteristic, complemented
303C4×C22⋊C4—21no—
304#327—614no—
305#327—614no—
306#324—612no—
307#323—612no—
308C61#1812311yescharacteristic, complemented, Sylow
309C2×#41—1222no—
310C2×#41—1222no—
311C2×#41—1222no—
312C2×#41—1222no—
313C2×#41—2444no—
314C2×#41—2444no—
315C22×Q8—1221no—
316C22×Q8C22×D6111yescharacteristic
317C22×D4—1221no—
318C22×D4—1222no—
319C23×C4—1222no—
320C23×C4C22⋊D6111yescharacteristic
321C2×SD16—1224no—
322C4⋊Q8—612no—
323C4⋊Q8—1221no—
324#127—612no—
325#127—1221no—
326#124—1222no—
327#124—1222no—
328#123—1224no—
329#123—1224no—
330C22⋊Q8—1224no—
331C22⋊Q8—1224no—
332#121—614no—
333#121—1224no—
334#121—1224no—
335#121—1222no—
336#121—1222no—
337#121—1222no—
338#120—1224no—
339#120—1224no—
340C4×Q8—22no—
341C4×Q8—22no—
342C4×D4—614no—
343C4×D4—1222no—
344C4×D4—1222no—
345C4×D4—1222no—
346#117—22no—
347#117—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4C22⋊D6111yescharacteristic
353C2×C4⋊C4C22×D6111yescharacteristic
354C2×C4⋊C4C22×D6111yescharacteristic
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4C22⋊D6111yescharacteristic
358C2×C22⋊C4C22×D6111yescharacteristic
359C2×C42—21no—
360C2×C42—21no—
361C2×C42C22⋊D6111yescharacteristic
362C2×C42C22×D6111yescharacteristic
363#106—1224no—
364C4⋊C8—1222no—
365#103—1224no—
366#102—1224no—
367C22⋊C8—614no—
368C4×C8—1222no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95C22×D6111yescharacteristic
374#95C22×D6111yescharacteristic
375C24—1222no—
376C24C2×C22⋊D6111yescharacteristic
377#41—2444no—
378#41—2448no—
379#41—2448no—
380#41—2444no—
381#41—2444no—
382#41—2444no—
383C2×Q8—22no—
384C2×Q8—42no—
385C2×Q8—1222no—
386C2×Q8—1222no—
387C2×Q8—2444no—
388C2×D4—614no—
389C2×D4—1222no—
390C2×D4—1222no—
391C2×D4—1222no—
392C2×D4—1224no—
393C2×D4—1224no—
394C2×D4—1222no—
395C2×D4—1222no—
396C2×D4—2444no—
397C2×D4—2444no—
398C2×D4—2444no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—1224no—
407C22×C4—1222no—
408C22×C4—1224no—
409C22×C4—2448no—
410C22×C4#1175820114yescomplemented
411C22×C4C2×C22⋊D6111yescharacteristic
412C22×C4C2×C22⋊D6111yescharacteristic
413C22×C4C2×C22⋊D6111yescharacteristic
414C22×C4C2×C22⋊D6111yescharacteristic
415C22×C4C2×C22⋊D6111yescharacteristic
416C22×C4C23×D6111yescharacteristic
417SD16—2448no—
418C2×C8—1224no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—614no—
427C4⋊C4—1222no—
428C4⋊C4—1222no—
429C4⋊C4—1222no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C22⋊C4—1224no—
435C22⋊C4—1224no—
436C22⋊C4—1224no—
437C22⋊C4—1224no—
438C42—21no—
439C42—21no—
440C42—41no—
441C42—41no—
442C42—42no—
443C42D4×D6112yes—
444C42D4×D6112yes—
445C23—21no—
446C23—21no—
447C23—41no—
448C23—41no—
449C23—1224no—
450C23—4882no—
451C23—4882no—
452C23—2444no—
453C23C2×#1175820112yes—
454C23C22×C22⋊D6111yescharacteristic
455Q8—24no—
456Q8—42no—
457Q8—2444no—
458Q8—2444no—
459D4—4882no—
460D4—4882no—
461D4—4882no—
462D4—2444no—
463D4—2444no—
464D4—2448no—
465D4—2444no—
466D4—2444no—
467D4—2444no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—1224no—
491C2×C4—2448no—
492C2×C4—2444no—
493C2×C4—2448no—
494C2×C4#1451974512yes—
495C2×C4#1451974511yescharacteristic
496C2×C4#1451974511yescharacteristic
497C2×C4C2×#1175820112yes—
498C2×C4C2×#1175820112yes—
499C2×C4#1451980212yes—
500C2×C4C2×D4×D6112yes—
501C8—2444no—
502C22—21no—
503C22—21no—
504C22—22no—
505C22—42no—
506C22—41no—
507C22—41no—
508C22—81no—
509C22—4888no—
510C22—2444no—
511C22—2444no—
512C22—2444no—
513C22C2×#1451974511yescharacteristic
514C22C22×#1175820111yescharacteristic
515C22C2×#1451980211yescharacteristic
516C4—22no—
517C4—22no—
518C4—22no—
519C4—42no—
520C4—41no—
521C4—42no—
522C4—41no—
523C4—81no—
524C4—81no—
525C4—2444no—
526C4#1683991212yes—
527C4#1683991212yes—
528C2—21no—
529C2—21no—
530C2—42no—
531C2—4884no—
532C2C2×#1683991211yescharacteristic
533C2#2145516711yescharacteristic
534C2#2145521011yescharacteristic
subgroupG/H
centerC2×C4#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2396C2
socleC22×C61C2×#120
prime radical#21459152C2
hypercenter#2396D61
hypocenterC61#18123
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2396

Complements

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

NH ≅ G/Ntype
2. #21455850C2semidirect
3. #21444855C2semidirect
4. #21444838C2semidirect
5. #21444787C2semidirect
6. #21444770C2semidirect
7. #21444313C2semidirect
8. #21444133C2semidirect
9. C61×#2396C2semidirect
20. #16840080C22semidirect
23. #16839901C22semidirect
25. #16839815C22semidirect
26. #16839495C22semidirect
27. #16839495C22semidirect
28. #16839488C22semidirect
30. C2×C4×Q8×C61C22semidirect
32. C61×#400C22semidirect
36. C61×#395C22semidirect
39. C4×C4⋊C4×C61C22semidirect
118. C22×C4×C61SD16semidirect
263. #2396D61semidirect
287. C2×C4×Q8C2×D61semidirect
295. #400C2×D61semidirect
299. #395C2×D61semidirect
302. C4×C4⋊C4C2×D61semidirect
308. C61#18123semidirect
410. C22×C4#11758201semidirect

Maximal subgroups

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

subgroupindex
#214591522
#214558502
#214448552
#214448382
#214447872
#214447702
#214443132
#214441332
C61×#23962
#1812361