Finite Group Databasebeta

← #29653984

Subgroups — #29653984

Order 15616.

1312 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456734C212yesmaximal, complemented
2#21451569C211yesmaximal, characteristic, complemented
3#21444754C214yesmaximal, complemented
4#21444737C214yesmaximal, complemented
5#21444379C212yesmaximal, complemented
6#21444133C211yesmaximal, characteristic, complemented
7C61×#3449C211yesmaximal, characteristic, complemented
8C2×#14519774C2211yescharacteristic, complemented
9C2×#14519751—21no—
10#16840138—22no—
11#16840110—22no—
12#16839957—24no—
13#16839895C2218yescomplemented
14#16839815C2212yescomplemented
15#16839679—21no—
16#16839679C2212yescomplemented
17#16839671—21no—
18#16839671C2212yes—
19#16839664—21no—
20#16839628C2211yescharacteristic, complemented
21#16839496C2218yescomplemented
22#16839488C2212yescomplemented
23#16839486C2212yes—
24C22×#41×C61—21no—
25C61×#537—21no—
26C61×#537C2212yes—
27C61×#536—22no—
28C61×#535—22no—
29C61×#534—21no—
30C61×#534C2212yes—
31C61×#532—21no—
32C61×#532C2212yes—
33C61×#525—24no—
34C2×#117×C61C2211yescharacteristic
35C22×D244—21no—
36C22×D244—21no—
37C22×D244C2312yes—
38C22×C22⋊D61—22no—
39C2×#11758199—28no—
40#14519774—22no—
41#14519774—22no—
42#14519751—44no—
43C2×C4⋊Dic61—21no—
44C2×#11758198—22no—
45C2×#11758198—22no—
46C2×#11758198—22no—
47C4⋊#92418—22no—
48#14519697—28no—
49C22⋊#92418C2314yes—
50C4×#92418—22no—
51#14519684—21no—
52#14519684C2312yes—
53C2×#41×C61—22no—
54C2×#41×C61—22no—
55C2×#41×C61—22no—
56C2×#41×C61—22no—
57C2×#41×C61—44no—
58C2×#41×C61—44no—
59C22×Q8×C61—21no—
60C22×D4×C61—21no—
61C22×D4×C61—22no—
62C23×C4×C61—22no—
63C23×C4×C61D411yescharacteristic
64C4⋊Q8×C61—21no—
65C4⋊Q8×C61D412yescomplemented
66#127×C61—21no—
67#127×C61D412yescomplemented
68#124×C61—22no—
69#124×C61—22no—
70#123×C61—24no—
71#123×C61—24no—
72C22⋊Q8×C61—24no—
73C22⋊Q8×C61—24no—
74#121×C61—24no—
75#121×C61—24no—
76#121×C61—22no—
77#121×C61—22no—
78#121×C61—22no—
79#121×C61C2314yes—
80#120×C61—24no—
81#120×C61—24no—
82C4×D4×C61—22no—
83C4×D4×C61—22no—
84C4×D4×C61—22no—
85C4×D4×C61D414yescomplemented
86#117×C61—22no—
87#117×C61—22no—
88C2×C4⋊C4×C61D411yescharacteristic
89C2×C4⋊C4×C61C2311yescharacteristic
90C2×C22⋊C4×C61D411yescharacteristic
91C2×C22⋊C4×C61C2311yescharacteristic
92C2×C42×C61D411yescharacteristic
93C2×C42×C61C2311yescharacteristic
94C24×C61—22no—
95C24×C61C2×D411yescharacteristic
96C23×D61—22no—
97#41×C61—44no—
98#41×C61—48no—
99#41×C61—48no—
100#41×C61—44no—
101#41×C61—44no—
102#41×C61—44no—
103C2×Q8×C61—22no—
104C2×Q8×C61—22no—
105C2×Q8×C61—44no—
106C2×D4×C61—24no—
107C2×D4×C61—22no—
108C2×D4×C61—24no—
109C2×D4×C61—22no—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61—22no—
113C2×D4×C61—44no—
114C2×D4×C61—44no—
115C2×D4×C61—44no—
116C2×D4×C61C2×D414yes—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—22no—
120C22×C4×C61—24no—
121C22×C4×C61—22no—
122C22×C4×C61—24no—
123C22×C4×C61—48no—
124C22×C4×C61D814yescomplemented
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2411yescharacteristic
131C2×C22⋊D61—44no—
132C2×C22⋊D61—82no—
133C2×C22⋊D61—82no—
134C22×Dic61—22no—
135C2×D244—24no—
136C2×D244—42no—
137C2×D244—41no—
138C2×D244—41no—
139C2×D244—42no—
140C2×D244—42no—
141C2×D244—82no—
142C2×D244—82no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61C2×D414yes—
149C22⋊C4×C61—24no—
150C22⋊C4×C61—24no—
151C22⋊C4×C61—22no—
152C22⋊C4×C61—24no—
153C22⋊C4×C61—22no—
154C22⋊C4×C61—24no—
155C42×C61—21no—
156C42×C61—21no—
157C42×C61C2×D412yes—
158C42×C61C2×D412yes—
159#11758199—88no—
160#11758198—44no—
161#11758198—82no—
162#11758198—82no—
163C4⋊Dic61—42no—
164C2×#92418—24no—
165C23×C61—21no—
166C23×C61—21no—
167C23×C61—24no—
168C23×C61—41no—
169C23×C61—41no—
170C23×C61—44no—
171C23×C61—82no—
172C23×C61—82no—
173C23×C61C2×D812yes—
174C23×C61C22×D411yescharacteristic
175C22×D61—42no—
176C22×D61—42no—
177C22×D61—42no—
178C22×D61—42no—
179C22×D61—42no—
180C22×D61—42no—
181C22×D61—42no—
182Q8×C61—44no—
183Q8×C61—44no—
184D4×C61—44no—
185D4×C61—44no—
186D4×C61—48no—
187D4×C61—44no—
188D4×C61—44no—
189D4×C61—44no—
190D4×C61—82no—
191D4×C61—82no—
192D4×C61—82no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—24no—
196C2×C4×C61—24no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—24no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—48no—
203C2×C4×C61—44no—
204C2×C4×C61—48no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61#12012yes—
208C2×C4×C61#12011yescharacteristic
209C2×C4×C61#12011yescharacteristic
210C2×C4×C61C2×D812yes—
211C2×C4×C61C2×D812yes—
212C2×C4×C61AGL(1,ℤ/8)12yes—
213C2×C4×C61C22×D412yes—
214C22⋊D61—88no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217C2×Dic61—42no—
218D244—48no—
219D244—84no—
220D244—82no—
221D244—82no—
222#92418—44no—
223#18300—611nomaximal, Sylow
224C22×C61—21no—
225C22×C61—21no—
226C22×C61—22no—
227C22×C61—42no—
228C22×C61—41no—
229C22×C61—41no—
230C22×C61—44no—
231C22×C61—44no—
232C22×C61—44no—
233C22×C61—81no—
234C22×C61—88no—
235C22×C61C2×#12011yescharacteristic
236C22×C61C22×D811yescharacteristic
237C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
238C2×D61—82no—
239C2×D61—82no—
240C2×D61—82no—
241C2×D61—82no—
242C2×D61—82no—
243C2×D61—82no—
244C2×D61—82no—
245C4×C61—22no—
246C4×C61—22no—
247C4×C61—22no—
248C4×C61—44no—
249C4×C61—41no—
250C4×C61—41no—
251C4×C61#44712yes—
252C4×C61#44712yes—
253Dic61—82no—
254#3449D6111yescharacteristic, complemented
255#3205—612no—
256#2619—611no—
257#1598—614no—
258#1595—614no—
259#1517—612no—
260#1458—611no—
261C2×C61—21no—
262C2×C61—21no—
263C2×C61—42no—
264C2×C61—84no—
265C2×C61C2×#44711yescharacteristic
266C2×C61#302111yescharacteristic
267C2×C61#302711yescharacteristic
268D61—162no—
269C22×#41—21no—
270#537—21no—
271#537C2×D6112yes—
272#536—22no—
273#535—22no—
274#534—21no—
275#534C2×D6112yes—
276#532—21no—
277#532C2×D6112yes—
278C2×#127—611no—
279#525—24no—
280C2×#121—1221no—
281C2×#117C2×D6111yescharacteristic
282#493—1222no—
283#486—1222no—
284#459—1224no—
285#447—618no—
286#431—612no—
287#394—612no—
288#394—1221no—
289#392—612no—
290#392—1221no—
291#390—1221no—
292#382—611no—
293#327—618no—
294#324—612no—
295#323—612no—
296C61#1830011yescharacteristic, complemented, Sylow
297C2×#41—22no—
298C2×#41—22no—
299C2×#41—22no—
300C2×#41—22no—
301C2×#41—44no—
302C2×#41—44no—
303C22×Q8—21no—
304C22×D4—21no—
305C22×D4—22no—
306C22×D4—612no—
307C22×D4—1222no—
308C22×D4—1221no—
309C22×D4—1221no—
310C23×C4—22no—
311C23×C4C22⋊D6111yescharacteristic
312C2×D8—1228no—
313C4⋊Q8—21no—
314C4⋊Q8C22⋊D6112yescomplemented
315#127—21no—
316#127—1222no—
317#127—1222no—
318#127C22⋊D6112yescomplemented
319#124—22no—
320#124—22no—
321#123—24no—
322#123—24no—
323C22⋊Q8—24no—
324C22⋊Q8—24no—
325#121—24no—
326#121—24no—
327#121—22no—
328#121—22no—
329#121—22no—
330#121—2444no—
331#121C22×D6114yes—
332#120—24no—
333#120—24no—
334C4×D4—22no—
335C4×D4—22no—
336C4×D4—22no—
337C4×D4C22⋊D6114yescomplemented
338#117—22no—
339#117—22no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4C22⋊D6111yescharacteristic
342C2×C4⋊C4C22×D6111yescharacteristic
343C2×C22⋊C4—1222no—
344C2×C22⋊C4—1222no—
345C2×C22⋊C4—1222no—
346C2×C22⋊C4C22⋊D6111yescharacteristic
347C2×C22⋊C4C22×D6111yescharacteristic
348C2×C42C22⋊D6111yescharacteristic
349C2×C42C22×D6111yescharacteristic
350C4⋊C8—1222no—
351#102—1228no—
352C22⋊C8—614no—
353C4×C8—1222no—
354#95—612no—
355#95—1221no—
356C24—22no—
357C24—1222no—
358C24C2×C22⋊D6111yescharacteristic
359#41—44no—
360#41—48no—
361#41—48no—
362#41—44no—
363#41—44no—
364#41—44no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—44no—
368C2×D4—24no—
369C2×D4—22no—
370C2×D4—24no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—1224no—
379C2×D4—4882no—
380C2×D4—4882no—
381C2×D4—4882no—
382C2×D4—4882no—
383C2×D4—2444no—
384C2×D4—2442no—
385C2×D4—2441no—
386C2×D4—2441no—
387C2×D4—2442no—
388C2×D4—2442no—
389C2×D4C2×C22⋊D6114yes—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—48no—
397C22×C4—1222no—
398C22×C4#1175819914yescomplemented
399C22×C4C2×C22⋊D6111yescharacteristic
400C22×C4C2×C22⋊D6111yescharacteristic
401C22×C4C2×C22⋊D6111yescharacteristic
402C22×C4C2×C22⋊D6111yescharacteristic
403C22×C4C2×C22⋊D6111yescharacteristic
404C22×C4C23×D6111yescharacteristic
405D8—4888no—
406C2×C8—1224no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—2442no—
413C4⋊C4C2×C22⋊D6114yes—
414C22⋊C4—24no—
415C22⋊C4—24no—
416C22⋊C4—22no—
417C22⋊C4—24no—
418C22⋊C4—22no—
419C22⋊C4—24no—
420C22⋊C4—4882no—
421C22⋊C4—4882no—
422C22⋊C4—2444no—
423C42—21no—
424C42—21no—
425C42C2×C22⋊D6112yes—
426C42C2×C22⋊D6112yes—
427C23—21no—
428C23—21no—
429C23—24no—
430C23—41no—
431C23—41no—
432C23—44no—
433C23—82no—
434C23—82no—
435C23—2442no—
436C23—2442no—
437C23—2442no—
438C23—2442no—
439C23—2442no—
440C23—2442no—
441C23—2442no—
442C23C2×#1175819912yes—
443C23C22×C22⋊D6111yescharacteristic
444Q8—44no—
445Q8—44no—
446D4—44no—
447D4—44no—
448D4—48no—
449D4—44no—
450D4—44no—
451D4—44no—
452D4—82no—
453D4—82no—
454D4—82no—
455D4—4884no—
456D4—4888no—
457D4—4882no—
458D4—4882no—
459D4—2448no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—24no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—48no—
470C2×C4—44no—
471C2×C4—48no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—2442no—
475C2×C4—2442no—
476C2×C4—2442no—
477C2×C4#1451974612yes—
478C2×C4#1451974611yescharacteristic
479C2×C4#1451974611yescharacteristic
480C2×C4C2×#1175819912yes—
481C2×C4C2×#1175819912yes—
482C2×C4#1451980312yes—
483C2×C4C22×C22⋊D6112yes—
484C8—2444no—
485C22—21no—
486C22—21no—
487C22—22no—
488C22—42no—
489C22—41no—
490C22—41no—
491C22—44no—
492C22—44no—
493C22—44no—
494C22—81no—
495C22—88no—
496C22—4882no—
497C22—4882no—
498C22—4882no—
499C22—4882no—
500C22—4882no—
501C22—4882no—
502C22—4882no—
503C22C2×#1451974611yescharacteristic
504C22C22×#1175819911yescharacteristic
505C22C2×#1451980311yescharacteristic
506C4—22no—
507C4—22no—
508C4—22no—
509C4—41no—
510C4—41no—
511C4—44no—
512C4—4882no—
513C4#1683989612yes—
514C4#1683989612yes—
515C2—21no—
516C2—21no—
517C2—42no—
518C2—84no—
519C2—9762no—
520C2C2×#1683989611yescharacteristic
521C2#2145519111yescharacteristic
522C2#2145522111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3449C2
socleC22×C61C2×#120
hypercenter#3449D61
hypocenterC61#18300
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3449

Complements

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

NH ≅ G/Ntype
1. #21456734C2semidirect
2. #21451569C2semidirect
3. #21444754C2semidirect
4. #21444737C2semidirect
5. #21444379C2semidirect
6. #21444133C2semidirect
7. C61×#3449C2semidirect
8. C2×#14519774C22semidirect
13. #16839895C22semidirect
14. #16839815C22semidirect
16. #16839679C22semidirect
20. #16839628C22semidirect
21. #16839496C22semidirect
22. #16839488C22semidirect
65. C4⋊Q8×C61D4semidirect
67. #127×C61D4semidirect
85. C4×D4×C61D4semidirect
124. C22×C4×C61D8semidirect
254. #3449D61semidirect
296. C61#18300semidirect
314. C4⋊Q8C22⋊D61semidirect
318. #127C22⋊D61semidirect
337. C4×D4C22⋊D61semidirect
398. C22×C4#11758199semidirect

Maximal subgroups

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

subgroupindex
#214567342
#214515692
#214447542
#214447372
#214443792
#214441332
C61×#34492
#1830061