Finite Group Databasebeta

← #29654316

Subgroups — #29654316

Order 15616.

1156 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459152C211yesmaximal, characteristic
2#21457009C212yesmaximal, complemented
3#21444855C212yesmaximal, complemented
4#21444838C212yesmaximal, complemented
5#21444821C212yesmaximal, complemented
6#21444804C212yesmaximal, complemented
7#21444326C212yesmaximal, complemented
8#21444153C211yesmaximal, characteristic, complemented
9C61×#2611C211yesmaximal, 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#16840172—22no—
21#16840128—22no—
22#16840080C2214yescomplemented
23#16840075C2214yescomplemented
24#16840059—24no—
25#16839819C2212yescomplemented
26#16839495C2214yescomplemented
27#16839495C2214yescomplemented
28#16839488C2212yescomplemented
29#16839486C2212yes—
30C2×#125×C61C2211yescharacteristic, complemented
31C2×#117×C61C2211yescharacteristic
32C61×#400C2211yescharacteristic, complemented
33C61×#399C2211yescharacteristic
34C61×#398C2211yescharacteristic, complemented
35C61×#397C2211yescharacteristic
36C61×#389—21no—
37C61×#388—21no—
38C61×#387—21no—
39C61×#385—21no—
40C61×#376C2211yescharacteristic, 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—
51#14519777—21no—
52#14519777D412yes—
53#14519774—21no—
54#14519774D412yes—
55#14519765—22no—
56#14519765—22no—
57#14519761—24no—
58#14519761—24no—
59#14519757—24no—
60#14519757—24no—
61#14519752—24no—
62#14519752—24no—
63#14519751—22no—
64#14519751—22no—
65#14519751—22no—
66#14519751C2314yes—
67#14519747—24no—
68#14519747—24no—
69C4×D244—22no—
70C4×D244—22no—
71C4×D244—22no—
72C4×D244D414yes—
73#14519710—24no—
74#14519708—24no—
75C4⋊#92418—22no—
76#14519697—24no—
77#14519697—24no—
78C22⋊#92418C2314yes—
79#14519688—22no—
80C23×C4×C61D411yescharacteristic
81#125×C61—22no—
82#125×C61—22no—
83#117×C61—22no—
84#117×C61—22no—
85C2×C4⋊C4×C61—21no—
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×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×C61D411yescharacteristic
99C2×C42×C61C2311yescharacteristic
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—
109C22×C4×C61—22no—
110C22×C4×C61—21no—
111C22×C4×C61—21no—
112C22×C4×C61—22no—
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2411yescharacteristic
123C2×C22⋊D61—24no—
124C2×C22⋊D61—24no—
125C2×C22⋊D61—44no—
126C2×C22⋊D61—44no—
127C22×Dic61—22no—
128#11758222—48no—
129#11758222—48no—
130#11758222—44no—
131#11758222—44no—
132#11758222—44no—
133#11758222—44no—
134C2×D244—22no—
135C2×D244—22no—
136C2×D244—22no—
137C2×D244—22no—
138C2×D244—22no—
139C2×D244—44no—
140C2×D244C2×D414yes—
141C2×C4×D61—24no—
142C2×C4×D61—24no—
143C2×C4×D61—48no—
144C2×Dic122—22no—
145C2×Dic122—22no—
146C2×Dic122—44no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—42no—
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#11758198—24no—
166#11758198—24no—
167#11758198—24no—
168#11758198—24no—
169C4⋊Dic61—22no—
170C4⋊Dic61—22no—
171C4⋊Dic61—22no—
172C4⋊Dic61C2×D414yes—
173C2×#92418—24no—
174C23×C61—21no—
175C23×C61—21no—
176C23×C61—41no—
177C23×C61—41no—
178C23×C61AGL(1,ℤ/8)12yes—
179C23×C61C22×D411yescharacteristic
180C22×D61—24no—
181C22×D61—44no—
182C22×D61—82no—
183C22×D61—82no—
184C2×C4×C61—24no—
185C2×C4×C61—24no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—21no—
190C2×C4×C61—22no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
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#12012yes—
209C2×C4×C61#12011yescharacteristic
210C2×C4×C61#12011yescharacteristic
211C2×C4×C61AGL(1,ℤ/8)12yes—
212C2×C4×C61C22×D412yes—
213C22⋊D61—48no—
214C22⋊D61—44no—
215C22⋊D61—44no—
216C2×Dic61—24no—
217C2×Dic61—44no—
218D244—44no—
219D244—44no—
220D244—44no—
221D244—82no—
222D244—82no—
223D244—82no—
224C4×D61—48no—
225C4×D61—48no—
226Dic122—44no—
227Dic122—44no—
228#92418—44no—
229#18339—611nomaximal, Sylow
230C22×C61—21no—
231C22×C61—21no—
232C22×C61—22no—
233C22×C61—42no—
234C22×C61—41no—
235C22×C61—41no—
236C22×C61—81no—
237C22×C61C2×#12011yescharacteristic
238C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
239C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
240C2×D61—44no—
241C2×D61—44no—
242C2×D61—44no—
243C2×D61—88no—
244C4×C61—21no—
245C4×C61—22no—
246C4×C61—21no—
247C4×C61—22no—
248C4×C61—22no—
249C4×C61—42no—
250C4×C61—41no—
251C4×C61—41no—
252C4×C61—81no—
253C4×C61—81no—
254C4×C61—81no—
255Dic61—44no—
256#3449—611no—
257#3234—612no—
258#2611D6111yescharacteristic, complemented
259#1616—612no—
260#1613—612no—
261#1610—612no—
262#1607—612no—
263#1505—612no—
264#1464—611no—
265C2×C61—21no—
266C2×C61—21no—
267C2×C61—42no—
268C2×C61#300311yescharacteristic
269C2×C61#301811yescharacteristic
270C2×C61#302811yescharacteristic
271D61—84no—
272C22×#41—1221no—
273#537—612no—
274#537—1221no—
275#536—1222no—
276#535—1222no—
277#534—612no—
278#534—1221no—
279#532—612no—
280#532—1221no—
281C2×#125C2×D6111yescharacteristic, complemented
282#525—1224no—
283C2×#117C2×D6111yescharacteristic
284#501—1222no—
285#490—1222no—
286#481—614no—
287#480—614no—
288#478—1224no—
289#432—612no—
290#400C2×D6111yescharacteristic, complemented
291#399C2×D6111yescharacteristic
292#398C2×D6111yescharacteristic, complemented
293#397C2×D6111yescharacteristic
294#389—21no—
295#388—21no—
296#387—21no—
297#385—21no—
298#376C2×D6111yescharacteristic, complemented
299#327—614no—
300#327—614no—
301#324—612no—
302#323—612no—
303C61#1833911yescharacteristic, complemented, Sylow
304C2×#41—1222no—
305C2×#41—1222no—
306C2×#41—1222no—
307C2×#41—1222no—
308C2×#41—2444no—
309C2×#41—2444no—
310C22×Q8—1221no—
311C22×D4—1221no—
312C22×D4—1222no—
313C23×C4—1222no—
314C23×C4C22⋊D6111yescharacteristic
315C4⋊Q8—612no—
316C4⋊Q8—1221no—
317#127—612no—
318#127—1221no—
319#125—22no—
320#125—22no—
321#124—1222no—
322#124—1222no—
323#123—1224no—
324#123—1224no—
325C22⋊Q8—1224no—
326C22⋊Q8—1224no—
327#121—614no—
328#121—1224no—
329#121—1224no—
330#121—1222no—
331#121—1222no—
332#121—1222no—
333#120—1224no—
334#120—1224no—
335C4×D4—614no—
336C4×D4—1222no—
337C4×D4—1222no—
338C4×D4—1222no—
339#117—22no—
340#117—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4C22⋊D6111yescharacteristic
346C2×C4⋊C4C22×D6111yescharacteristic
347C2×C4⋊C4C22×D6111yescharacteristic
348C2×C4⋊C4C22×D6111yescharacteristic
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4C22⋊D6111yescharacteristic
352C2×C22⋊C4C22×D6111yescharacteristic
353C2×C42—21no—
354C2×C42C22⋊D6111yescharacteristic
355C2×C42C22×D6111yescharacteristic
356#107—1224no—
357#106—1224no—
358C4⋊C8—1222no—
359#102—1224no—
360#102—1224no—
361C22⋊C8—614no—
362#97—1222no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95C22×D6111yescharacteristic
369#95C22×D6111yescharacteristic
370C24—1222no—
371C24C2×C22⋊D6111yescharacteristic
372#41—2448no—
373#41—2448no—
374#41—2444no—
375#41—2444no—
376#41—2444no—
377#41—2444no—
378C2×Q8—1222no—
379C2×Q8—1222no—
380C2×Q8—2444no—
381C2×D4—614no—
382C2×D4—1222no—
383C2×D4—1222no—
384C2×D4—1222no—
385C2×D4—1224no—
386C2×D4—1224no—
387C2×D4—1222no—
388C2×D4—1222no—
389C2×D4—2444no—
390C2×D4—2444no—
391C2×D4—2444no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—1224no—
401C22×C4—1222no—
402C22×C4—1224no—
403C22×C4—2448no—
404C22×C4C2×C22⋊D6111yescharacteristic
405C22×C4C2×C22⋊D6111yescharacteristic
406C22×C4C2×C22⋊D6111yescharacteristic
407C22×C4C2×C22⋊D6111yescharacteristic
408C22×C4C2×C22⋊D6111yescharacteristic
409C22×C4C23×D6111yescharacteristic
410C2×C8—1224no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—42no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—614no—
421C4⋊C4—1222no—
422C4⋊C4—1222no—
423C4⋊C4—1222no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C22⋊C4—1224no—
429C22⋊C4—1224no—
430C22⋊C4—1224no—
431C22⋊C4—1224no—
432C42—21no—
433C42—21no—
434C42—42no—
435C42D4×D6112yes—
436C42D4×D6112yes—
437C23—21no—
438C23—21no—
439C23—41no—
440C23—41no—
441C23—1224no—
442C23—4882no—
443C23—4882no—
444C23—2444no—
445C23#1451980212yes—
446C23C22×C22⋊D6111yescharacteristic
447Q8—2444no—
448Q8—2444no—
449D4—4882no—
450D4—4882no—
451D4—4882no—
452D4—2444no—
453D4—2444no—
454D4—2444no—
455D4—2448no—
456D4—2444no—
457D4—2444no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
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—1224no—
483C2×C4—2448no—
484C2×C4—2448no—
485C2×C4—2444no—
486C2×C4#1451974511yescharacteristic
487C2×C4#1451974511yescharacteristic
488C2×C4#1451974512yes—
489C2×C4#1451980212yes—
490C2×C4C2×D4×D6112yes—
491C8—2444no—
492C22—21no—
493C22—21no—
494C22—22no—
495C22—42no—
496C22—41no—
497C22—41no—
498C22—81no—
499C22—4888no—
500C22—2444no—
501C22—2444no—
502C22—2444no—
503C22C2×#1451974511yescharacteristic
504C22C2×#1451980211yescharacteristic
505C22C2×#1451980211yescharacteristic
506C4—22no—
507C4—22no—
508C4—21no—
509C4—22no—
510C4—21no—
511C4—42no—
512C4—41no—
513C4—41no—
514C4—81no—
515C4—81no—
516C4—81no—
517C4—2444no—
518C2—21no—
519C2—21no—
520C2—42no—
521C2—4884no—
522C2#2145504511yescharacteristic
523C2#2145516711yescharacteristic
524C2#2145522411yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2611C2
socleC22×C61C2×#120
prime radical#21459152C2
hypercenter#2611D61
hypocenterC61#18339
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2611

Complements

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

NH ≅ G/Ntype
2. #21457009C2semidirect
3. #21444855C2semidirect
4. #21444838C2semidirect
5. #21444821C2semidirect
6. #21444804C2semidirect
7. #21444326C2semidirect
8. #21444153C2semidirect
9. C61×#2611C2semidirect
22. #16840080C22semidirect
23. #16840075C22semidirect
25. #16839819C22semidirect
26. #16839495C22semidirect
27. #16839495C22semidirect
28. #16839488C22semidirect
30. C2×#125×C61C22semidirect
32. C61×#400C22semidirect
34. C61×#398C22semidirect
40. C61×#376C22semidirect
258. #2611D61semidirect
281. C2×#125C2×D61semidirect
290. #400C2×D61semidirect
292. #398C2×D61semidirect
298. #376C2×D61semidirect
303. C61#18339semidirect

Maximal subgroups

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

subgroupindex
#214591522
#214570092
#214448552
#214448382
#214448212
#214448042
#214443262
#214441532
C61×#26112
#1833961