Finite Group Databasebeta

← #29652548

Subgroups — #29652548

Order 15616.

1184 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459152C211yesmaximal, characteristic
2#21455941C212yesmaximal, complemented
3#21444821C212yesmaximal, complemented
4#21444804C212yesmaximal, complemented
5#21444787C212yesmaximal, complemented
6#21444770C212yesmaximal, complemented
7#21444365C212yesmaximal, complemented
8#21444153C211yesmaximal, characteristic, complemented
9C61×#2599C211yesmaximal, 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#16840075C2214yescomplemented
21#16840057—22no—
22#16839962—22no—
23#16839901C2214yescomplemented
24#16839860—24no—
25#16839819C2212yescomplemented
26#16839495C2214yescomplemented
27#16839495C2214yescomplemented
28#16839488C2212yescomplemented
29#16839486C2212yes—
30C2×C4×Q8×C61C2211yescharacteristic, complemented
31C2×#117×C61C2211yescharacteristic
32C61×#398C2211yescharacteristic, complemented
33C61×#397C2211yescharacteristic
34C61×#395C2211yescharacteristic, complemented
35C61×#393C2211yescharacteristic
36C61×#388—21no—
37C61×#387—21no—
38C61×#387—21no—
39C61×#385—21no—
40C61×#383C2211yescharacteristic, complemented
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#14519710—24no—
75C4⋊#92418—22no—
76#14519701—24no—
77#14519697—24no—
78C22⋊#92418C2314yes—
79#14519688—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×C61D411yescharacteristic
90C2×C4⋊C4×C61C2311yescharacteristic
91C2×C4⋊C4×C61C2311yescharacteristic
92C2×C22⋊C4×C61—21no—
93C2×C22⋊C4×C61—21no—
94C2×C22⋊C4×C61D411yescharacteristic
95C2×C22⋊C4×C61C2311yescharacteristic
96C2×C42×C61—21no—
97C2×C42×C61D411yescharacteristic
98C2×C42×C61C2311yescharacteristic
99#95×C61—21no—
100#95×C61—21no—
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—22no—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
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—48no—
131#11758222—48no—
132#11758222—44no—
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—41no—
154C4⋊C4×C61—41no—
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—42no—
163C42×C61C2×D412yes—
164C42×C61C2×D412yes—
165#11758201—48no—
166#11758198—24no—
167#11758198—24no—
168#11758198—24no—
169#11758198—24no—
170C4⋊Dic61—22no—
171C4⋊Dic61—22no—
172C4⋊Dic61—22no—
173C4⋊Dic61C2×D414yes—
174C2×#92418—24no—
175C23×C61—21no—
176C23×C61—21no—
177C23×C61—41no—
178C23×C61—41no—
179C23×C61AGL(1,ℤ/8)12yes—
180C23×C61C22×D411yescharacteristic
181C22×D61—24no—
182C22×D61—44no—
183C22×D61—82no—
184C22×D61—82no—
185Q8×C61—24no—
186Q8×C61—42no—
187C2×C4×C61—24no—
188C2×C4×C61—24no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—22no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—41no—
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×C61AGL(1,ℤ/8)12yes—
215C2×C4×C61C22×D412yes—
216C22⋊D61—48no—
217C22⋊D61—44no—
218C22⋊D61—44no—
219C2×Dic61—24no—
220C2×Dic61—44no—
221D244—44no—
222D244—44no—
223D244—44no—
224D244—82no—
225D244—82no—
226D244—82no—
227C4×D61—48no—
228C4×D61—48no—
229Dic122—44no—
230Dic122—44no—
231#92418—44no—
232#18153—611nomaximal, Sylow
233C22×C61—21no—
234C22×C61—21no—
235C22×C61—22no—
236C22×C61—42no—
237C22×C61—41no—
238C22×C61—41no—
239C22×C61—81no—
240C22×C61C2×#12011yescharacteristic
241C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
242C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
243C2×D61—44no—
244C2×D61—44no—
245C2×D61—44no—
246C2×D61—88no—
247C4×C61—21no—
248C4×C61—22no—
249C4×C61—21no—
250C4×C61—22no—
251C4×C61—22no—
252C4×C61—41no—
253C4×C61—41no—
254C4×C61—42no—
255C4×C61—81no—
256C4×C61—81no—
257C4×C61—81no—
258Dic61—44no—
259#3449—611no—
260#3112—612no—
261#2599D6111yescharacteristic, complemented
262#1610—612no—
263#1607—612no—
264#1604—612no—
265#1601—612no—
266#1514—612no—
267#1464—611no—
268C2×C61—21no—
269C2×C61—21no—
270C2×C61—42no—
271C2×C61#300311yescharacteristic
272C2×C61#302211yescharacteristic
273C2×C61#302511yescharacteristic
274D61—84no—
275C22×#41—1221no—
276#537—612no—
277#537—1221no—
278#536—1222no—
279#535—1222no—
280#534—612no—
281#534—1221no—
282#532—612no—
283#532—1221no—
284#525—1224no—
285C2×C4×Q8C2×D6111yescharacteristic, complemented
286C2×#117C2×D6111yescharacteristic
287#480—614no—
288#477—1222no—
289#460—1222no—
290#448—614no—
291#440—1224no—
292#432—612no—
293#398C2×D6111yescharacteristic, complemented
294#397C2×D6111yescharacteristic
295#395C2×D6111yescharacteristic, complemented
296#393C2×D6111yescharacteristic
297#388—21no—
298#387—21no—
299#387—21no—
300#385—21no—
301#383C2×D6111yescharacteristic, complemented
302#327—614no—
303#327—614no—
304#324—612no—
305#323—612no—
306C61#1815311yescharacteristic, complemented, Sylow
307C2×#41—1222no—
308C2×#41—1222no—
309C2×#41—1222no—
310C2×#41—1222no—
311C2×#41—2444no—
312C2×#41—2444no—
313C22×Q8—1221no—
314C22×Q8C22×D6111yescharacteristic
315C22×D4—1221no—
316C22×D4—1222no—
317C23×C4—1222no—
318C23×C4C22⋊D6111yescharacteristic
319C2×SD16—1224no—
320C4⋊Q8—612no—
321C4⋊Q8—1221no—
322#127—612no—
323#127—1221no—
324#124—1222no—
325#124—1222no—
326#123—1224no—
327#123—1224no—
328C22⋊Q8—1224no—
329C22⋊Q8—1224no—
330#121—614no—
331#121—1224no—
332#121—1224no—
333#121—1222no—
334#121—1222no—
335#121—1222no—
336#120—1224no—
337#120—1224no—
338C4×Q8—22no—
339C4×Q8—22no—
340C4×D4—614no—
341C4×D4—1222no—
342C4×D4—1222no—
343C4×D4—1222no—
344#117—22no—
345#117—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4C22⋊D6111yescharacteristic
350C2×C4⋊C4C22×D6111yescharacteristic
351C2×C4⋊C4C22×D6111yescharacteristic
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4C22⋊D6111yescharacteristic
355C2×C22⋊C4C22×D6111yescharacteristic
356C2×C42—21no—
357C2×C42C22⋊D6111yescharacteristic
358C2×C42C22×D6111yescharacteristic
359#107—1224no—
360C4⋊C8—1222no—
361#103—1224no—
362#102—1224no—
363C22⋊C8—614no—
364#97—1222no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95C22×D6111yescharacteristic
372#95C22×D6111yescharacteristic
373C24—1222no—
374C24C2×C22⋊D6111yescharacteristic
375#41—2448no—
376#41—2448no—
377#41—2444no—
378#41—2444no—
379#41—2444no—
380#41—2444no—
381C2×Q8—22no—
382C2×Q8—42no—
383C2×Q8—1222no—
384C2×Q8—1222no—
385C2×Q8—2444no—
386C2×D4—614no—
387C2×D4—1222no—
388C2×D4—1222no—
389C2×D4—1222no—
390C2×D4—1224no—
391C2×D4—1224no—
392C2×D4—1222no—
393C2×D4—1222no—
394C2×D4—2444no—
395C2×D4—2444no—
396C2×D4—2444no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—1224no—
406C22×C4—1222no—
407C22×C4—1224no—
408C22×C4—2448no—
409C22×C4C2×C22⋊D6111yescharacteristic
410C22×C4C2×C22⋊D6111yescharacteristic
411C22×C4C2×C22⋊D6111yescharacteristic
412C22×C4C2×C22⋊D6111yescharacteristic
413C22×C4C2×C22⋊D6111yescharacteristic
414C22×C4C23×D6111yescharacteristic
415SD16—2448no—
416C2×C8—1224no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—42no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—42no—
424C4⋊C4—614no—
425C4⋊C4—1222no—
426C4⋊C4—1222no—
427C4⋊C4—1222no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—1224no—
433C22⋊C4—1224no—
434C22⋊C4—1224no—
435C22⋊C4—1224no—
436C42—21no—
437C42—21no—
438C42—42no—
439C42D4×D6112yes—
440C42D4×D6112yes—
441C23—21no—
442C23—21no—
443C23—41no—
444C23—41no—
445C23—1224no—
446C23—4882no—
447C23—4882no—
448C23—2444no—
449C23#1451980212yes—
450C23C22×C22⋊D6111yescharacteristic
451Q8—24no—
452Q8—42no—
453Q8—2444no—
454Q8—2444no—
455D4—4882no—
456D4—4882no—
457D4—4882no—
458D4—2444no—
459D4—2444no—
460D4—2444no—
461D4—2448no—
462D4—2444no—
463D4—2444no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
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—1224no—
489C2×C4—2448no—
490C2×C4—2448no—
491C2×C4—2444no—
492C2×C4#1451974511yescharacteristic
493C2×C4#1451974511yescharacteristic
494C2×C4#1451974512yes—
495C2×C4#1451980212yes—
496C2×C4C2×D4×D6112yes—
497C8—2444no—
498C22—21no—
499C22—21no—
500C22—22no—
501C22—42no—
502C22—41no—
503C22—41no—
504C22—81no—
505C22—4888no—
506C22—2444no—
507C22—2444no—
508C22—2444no—
509C22C2×#1451974511yescharacteristic
510C22C2×#1451980211yescharacteristic
511C22C2×#1451980211yescharacteristic
512C4—22no—
513C4—22no—
514C4—21no—
515C4—22no—
516C4—21no—
517C4—41no—
518C4—42no—
519C4—41no—
520C4—81no—
521C4—81no—
522C4—81no—
523C4—2444no—
524C2—21no—
525C2—21no—
526C2—42no—
527C2—4884no—
528C2#2145504511yescharacteristic
529C2#2145519711yescharacteristic
530C2#2145521011yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2599C2
socleC22×C61C2×#120
prime radical#21459152C2
hypercenter#2599D61
hypocenterC61#18153
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2599

Complements

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

NH ≅ G/Ntype
2. #21455941C2semidirect
3. #21444821C2semidirect
4. #21444804C2semidirect
5. #21444787C2semidirect
6. #21444770C2semidirect
7. #21444365C2semidirect
8. #21444153C2semidirect
9. C61×#2599C2semidirect
20. #16840075C22semidirect
23. #16839901C22semidirect
25. #16839819C22semidirect
26. #16839495C22semidirect
27. #16839495C22semidirect
28. #16839488C22semidirect
30. C2×C4×Q8×C61C22semidirect
32. C61×#398C22semidirect
34. C61×#395C22semidirect
40. C61×#383C22semidirect
261. #2599D61semidirect
285. C2×C4×Q8C2×D61semidirect
293. #398C2×D61semidirect
295. #395C2×D61semidirect
301. #383C2×D61semidirect
306. C61#18153semidirect

Maximal subgroups

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

subgroupindex
#214591522
#214559412
#214448212
#214448042
#214447872
#214447702
#214443652
#214441532
C61×#25992
#1815361