Finite Group Databasebeta

← #29647776

Subgroups — #29647776

Order 15616.

1270 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459583C211yesmaximal, characteristic
2#21448759C211yesmaximal, characteristic, complemented
3#21444388C212yesmaximal, complemented
4#21444367C212yesmaximal, complemented
5#21444349C212yesmaximal, complemented
6#21444336C212yesmaximal, complemented
7#21444328C212yesmaximal, complemented
8C2×#16839497C212yesmaximal, complemented
9C61×#2973C211yesmaximal, characteristic, complemented
10C2×#14519822—21no—
11#16840416C414yescomplemented
12#16840407—24no—
13D4×D244C414yescomplemented
14#16840343—21no—
15#16840324—21no—
16#16840324—21no—
17#16840312C2212yes—
18C2×#14519751C2212yes—
19C2×#14519751C2212yes—
20#16839655—21no—
21#16839648—21no—
22#16839642—21no—
23#16839630C2211yescharacteristic, complemented
24#16839625C2211yescharacteristic, complemented
25#16839622C2211yescharacteristic, complemented
26C22⋊C4×Dic61—21no—
27C4×C22⋊Dic61C2211yescharacteristic, complemented
28#16839608C2211yescharacteristic, complemented
29C2×#14519684C2211yescharacteristic, complemented
30#16839497C2214yescomplemented
31#16839497C2214yescomplemented
32#16839497C2214yescomplemented
33#16839497C2214yescomplemented
34C2×C4×D4×C61C2211yescharacteristic
35C61×#435—24no—
36C61×#432C2212yescomplemented
37C61×#407C2212yescomplemented
38C2×C22⋊C8×C61C2212yescomplemented
39#14519822—44no—
40#14519822—44no—
41C2×#11758222—22no—
42C2×#11758224—24no—
43C22×D244C2×C412yes—
44C22×D244C2×C412yescomplemented
45C22×C22⋊D61—22no—
46C22×C22⋊D61—22no—
47C2×D4×D61—24no—
48C23×Dic61—21no—
49#14519777C2×C412yescomplemented
50#14519774C2×C412yescomplemented
51#14519765—22no—
52#14519761—24no—
53#14519761—24no—
54#14519757—24no—
55#14519752—24no—
56#14519751—22no—
57#14519751—22no—
58#14519751—22no—
59#14519751—22no—
60#14519751—22no—
61#14519751C2×C414yes—
62#14519751C2×C414yescomplemented
63#14519751C2314yes—
64#14519747—24no—
65#14519747—24no—
66C4×D244—22no—
67C4×D244C2×C414yescomplemented
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61C2×C412yes—
70C2×C4⋊Dic61C2×C412yescomplemented
71C2×C22⋊Dic61—21no—
72C2×C22⋊Dic61—21no—
73C2×C22⋊Dic61—21no—
74C2×#11758198—22no—
75C2×#11758198—22no—
76C2×C4×Dic61—21no—
77C2×C4×Dic61—21no—
78#14519684—21no—
79#14519684—21no—
80#14519684—21no—
81#14519684—21no—
82#14519684—21no—
83#14519684C2311yescharacteristic
84#14519684C2311yescharacteristic
85#14519684C2311yescharacteristic
86#14519684C2311yescharacteristic
87C22×D4×C61D411yescharacteristic
88C23×C4×C61C2311yescharacteristic
89C23×C4×C61C2311yescharacteristic
90C2×M16×C61—22no—
91C22×C8×C61—22no—
92C4×D4×C61C2×C414yes—
93C4×D4×C61D414yes—
94C2×C4⋊C4×C61D411yescharacteristic
95C2×C22⋊C4×C61D411yescharacteristic
96C2×C22⋊C4×C61D411yescharacteristic
97C2×C42×C61C2311yescharacteristic
98C4⋊C8×C61—22no—
99C22⋊C8×C61—22no—
100C22⋊C8×C61—22no—
101C22⋊C8×C61C2314yes—
102#97×C61—22no—
103C24×C61C2×D411yescharacteristic
104C24×C61C2×D411yescharacteristic
105C23×D61—22no—
106C23×D61—22no—
107C2×D4×C61—22no—
108C2×D4×C61—22no—
109C2×D4×C61C22⋊C414yes—
110C22×C4×C61—81no—
111C22×C4×C61—81no—
112C22×C4×C61C22×C412yes—
113C22×C4×C61C22×C412yes—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D412yes—
119C22×C4×C61C2×D412yes—
120C22×C4×C61C2411yescharacteristic
121C2×C22⋊D61—24no—
122C2×C22⋊D61—24no—
123C2×C22⋊D61—24no—
124C2×C22⋊D61—44no—
125C2×C22⋊D61—44no—
126C2×C22⋊D61—44no—
127C2×C22⋊D61—44no—
128C22×Dic61—22no—
129C22×Dic61—22no—
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61—21no—
134C22×Dic61—42no—
135C22×Dic61—42no—
136C22×Dic61—81no—
137#11758224—416no—
138D4×D61—416no—
139#11758222—44no—
140#11758222—44no—
141C2×D244—22no—
142C2×D244—22no—
143C2×D244—22no—
144C2×D244—22no—
145C2×D244—82no—
146C2×D244—82no—
147C2×D244C22×C414yes—
148C2×C4×D61—24no—
149C2×Dic122—22no—
150M16×C61—82no—
151C2×C8×C61—24no—
152C2×C8×C61—82no—
153C4⋊C4×C61C22⋊C414yes—
154C22⋊C4×C61C22⋊C414yes—
155C22⋊C4×C61C22⋊C414yes—
156C42×C61C22×C412yes—
157C42×C61C2×D412yes—
158C22⋊Dic61—42no—
159C22⋊Dic61—42no—
160C22⋊Dic61—81no—
161#11758198—24no—
162#11758198—24no—
163#11758198—24no—
164#11758198—24no—
165C4⋊Dic61—22no—
166C4⋊Dic61—22no—
167C4⋊Dic61—22no—
168C4⋊Dic61—81no—
169C4⋊Dic61C22×C414yes—
170C4×Dic61—42no—
171C4×Dic61—81no—
172C23×C61—21no—
173C23×C61—21no—
174C23×C61—41no—
175C23×C61—41no—
176C23×C61—41no—
177C23×C61—41no—
178C23×C61C2×C22⋊C412yes—
179C23×C61C2×C22⋊C412yes—
180C23×C61AGL(1,ℤ/8)12yes—
181C23×C61AGL(1,ℤ/8)12yes—
182C23×C61C22×D411yescharacteristic
183C22×D61—24no—
184C22×D61—44no—
185C22×D61—44no—
186C22×D61—82no—
187C22×D61—82no—
188C22×D61—82no—
189C22×D61—82no—
190D4×C61—28no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61—41no—
201C2×C4×C61C2×C22⋊C412yes—
202C2×C4×C61C2×C22⋊C412yes—
203C2×C4×C61C2×C22⋊C412yes—
204C2×C4×C61C2×C22⋊C412yes—
205C2×C4×C61C23×C411yescharacteristic
206C2×C4×C61C22×D411yescharacteristic
207C22⋊D61—48no—
208C22⋊D61—44no—
209C22⋊D61—44no—
210C22⋊D61—48no—
211C2×Dic61—22no—
212C2×Dic61—24no—
213C2×Dic61—41no—
214C2×Dic61—44no—
215C2×Dic61—41no—
216C2×Dic61—44no—
217C2×Dic61—41no—
218C2×Dic61—41no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—84no—
227D244—44no—
228D244—44no—
229D244—82no—
230D244—82no—
231D244—82no—
232C4×D61—48no—
233Dic122—44no—
234C8×C61—44no—
235#17620—611nomaximal, Sylow
236C22×C61—21no—
237C22×C61—22no—
238C22×C61—21no—
239C22×C61—22no—
240C22×C61—22no—
241C22×C61—22no—
242C22×C61—22no—
243C22×C61—42no—
244C22×C61—42no—
245C22×C61—81no—
246C22×C61—81no—
247C22×C61#41814yes—
248C22×C61C22×C22⋊C411yescharacteristic
249C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
250C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
251C2×D61—44no—
252C2×D61—44no—
253C2×D61—44no—
254C2×D61—164no—
255C2×D61—164no—
256C4×C61—21no—
257C4×C61—21no—
258C4×C61—22no—
259C4×C61—22no—
260C4×C61—22no—
261C4×C61—22no—
262Dic61—42no—
263Dic61—44no—
264Dic61—81no—
265Dic61—81no—
266Dic61—81no—
267#3496—611no—
268#2973D6111yescharacteristic, complemented
269#2353—611no—
270#1519—612no—
271#1514—612no—
272#1510—612no—
273#1507—612no—
274#1505—612no—
275C2×#327—612no—
276C2×C61—21no—
277C2×C61—21no—
278C2×C61—24no—
279C2×C61—42no—
280C2×C61#288311yescharacteristic
281C2×C61C2×#41811yescharacteristic
282C2×C61#290111yescharacteristic
283D61—84no—
284C2×H(5,𝔽2)—1221no—
285#547—614no—
286#546—1224no—
287D42—614no—
288#537—1221no—
289#534—1221no—
290#534—1221no—
291#532—612no—
292C2×#121—612no—
293C2×#121—612no—
294C2×C4×D4C2×D6111yescharacteristic
295#435—24no—
296#432C2×D6112yescomplemented
297#407C2×D6112yescomplemented
298C2×C22⋊C8C2×D6112yescomplemented
299#388—1221no—
300#387—1221no—
301#386—1221no—
302#383—611no—
303#381—611no—
304#380—611no—
305C4×C22⋊C4—611no—
306C4×C22⋊C4—1221no—
307#376—611no—
308C2×#95—611no—
309#327—614no—
310#327—614no—
311#327—614no—
312#327—614no—
313C61#1762011yescharacteristic, complemented, Sylow
314H(5,𝔽2)—2444no—
315H(5,𝔽2)—2444no—
316C2×#41—1224no—
317C2×#41—1222no—
318C22×D4—612no—
319C22×D4—612no—
320C22×D4—1222no—
321C22×D4—1222no—
322C22×D4—1224no—
323C22×D4D24411yescharacteristic
324C23×C4—1221no—
325C23×C4C22×D6111yescharacteristic
326C23×C4C22×D6111yescharacteristic
327C2×M16—22no—
328C22×C8—22no—
329C4⋊Q8—612no—
330#127—612no—
331#124—1222no—
332#123—1224no—
333#123—1224no—
334C22⋊Q8—1224no—
335#121—614no—
336#121—614no—
337#121—614no—
338#121—1222no—
339#121—1222no—
340#121—1222no—
341#121—1222no—
342#121—1222no—
343#121—1224no—
344#120—1224no—
345#120—1224no—
346C4×D4—614no—
347C4×D4—1222no—
348C4×D4C4×D6114yes—
349C4×D4C22⋊D6114yes—
350C2×C4⋊C4—612no—
351C2×C4⋊C4—612no—
352C2×C4⋊C4—1221no—
353C2×C4⋊C4D24411yescharacteristic
354C2×C22⋊C4—1222no—
355C2×C22⋊C4—1222no—
356C2×C22⋊C4—1221no—
357C2×C22⋊C4—1221no—
358C2×C22⋊C4—1221no—
359C2×C22⋊C4D24411yescharacteristic
360C2×C22⋊C4D24411yescharacteristic
361C2×C42—1221no—
362C2×C42—1221no—
363C2×C42C22×D6111yescharacteristic
364C4⋊C8—22no—
365C22⋊C8—22no—
366C22⋊C8—22no—
367C22⋊C8C22×D6114yes—
368#97—22no—
369#95—611no—
370#95—611no—
371#95—611no—
372#95—611no—
373#95—1221no—
374#95—1221no—
375#95—1221no—
376#95—1221no—
377#95—1221no—
378C24—1222no—
379C24—1222no—
380C24C2×D24411yescharacteristic
381C24C2×D24411yescharacteristic
382#41—2444no—
383#41—2444no—
384#41—24416no—
385C2×Q8—1222no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—614no—
389C2×D4—1224no—
390C2×D4—1224no—
391C2×D4—1224no—
392C2×D4—1222no—
393C2×D4—1222no—
394C2×D4—1222no—
395C2×D4—1222no—
396C2×D4—4882no—
397C2×D4—4882no—
398C2×D4—24416no—
399C2×D4—2444no—
400C2×D4—2444no—
401C2×D4—2444no—
402C2×D4—2444no—
403C2×D4#1175819814yes—
404C22×C4—81no—
405C22×C4—81no—
406C22×C4—1221no—
407C22×C4—1222no—
408C22×C4—1222no—
409C22×C4—1224no—
410C22×C4—1221no—
411C22×C4—1221no—
412C22×C4—1221no—
413C22×C4—4881no—
414C22×C4—2442no—
415C22×C4—2442no—
416C22×C4C2×C4×D6112yes—
417C22×C4C2×C4×D6112yes—
418C22×C4C2×D24411yescharacteristic
419C22×C4C2×D24411yescharacteristic
420C22×C4C2×D24411yescharacteristic
421C22×C4C2×D24411yescharacteristic
422C22×C4C2×C22⋊D6112yes—
423C22×C4C2×C22⋊D6112yes—
424C22×C4C23×D6111yescharacteristic
425M16—82no—
426C2×C8—24no—
427C2×C8—82no—
428C4⋊C4—614no—
429C4⋊C4—1222no—
430C4⋊C4—1222no—
431C4⋊C4—1222no—
432C4⋊C4—4881no—
433C4⋊C4#1175819814yes—
434C22⋊C4—1224no—
435C22⋊C4—1224no—
436C22⋊C4—1224no—
437C22⋊C4—1224no—
438C22⋊C4—4881no—
439C22⋊C4—2442no—
440C22⋊C4—2442no—
441C22⋊C4#1175819814yes—
442C22⋊C4#1175819814yes—
443C42—4881no—
444C42—2442no—
445C42C2×C4×D6112yes—
446C42C2×C22⋊D6112yes—
447C23—21no—
448C23—21no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—1224no—
454C23—4882no—
455C23—4882no—
456C23—4882no—
457C23—4882no—
458C23—2444no—
459C23—2444no—
460C23C2×#1175819812yes—
461C23C2×#1175819812yes—
462C23#1451980512yes—
463C23#1451980512yes—
464C23C22×D24411yescharacteristic
465Q8—2444no—
466D4—28no—
467D4—4882no—
468D4—4882no—
469D4—4882no—
470D4—2444no—
471D4—2444no—
472D4—2448no—
473D4—2448no—
474D4—2444no—
475D4—2444no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—1222no—
487C2×C4—1224no—
488C2×C4—4884no—
489C2×C4—2448no—
490C2×C4—2441no—
491C2×C4—2441no—
492C2×C4—2441no—
493C2×C4—2441no—
494C2×C4—2441no—
495C2×C4—2441no—
496C2×C4—2441no—
497C2×C4—2441no—
498C2×C4—2441no—
499C2×C4—2441no—
500C2×C4—2444no—
501C2×C4—2444no—
502C2×C4—2441no—
503C2×C4C2×#1175819812yes—
504C2×C4C2×#1175819812yes—
505C2×C4C2×#1175819812yes—
506C2×C4C2×#1175819812yes—
507C2×C4C22×C4×D6111yescharacteristic
508C2×C4C22×C22⋊D6111yescharacteristic
509C8—44no—
510C22—21no—
511C22—22no—
512C22—21no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—42no—
518C22—42no—
519C22—81no—
520C22—81no—
521C22—9764no—
522C22—9764no—
523C22—2444no—
524C22—2444no—
525C22—2444no—
526C22#1683976814yes—
527C22C22×#1175819811yescharacteristic
528C22C2×#1451980511yescharacteristic
529C22C2×#1451980511yescharacteristic
530C4—22no—
531C4—21no—
532C4—21no—
533C4—22no—
534C4—22no—
535C4—22no—
536C4—4881no—
537C4—4881no—
538C4—4881no—
539C4—2442no—
540C4—2444no—
541C2—21no—
542C2—21no—
543C2—24no—
544C2—42no—
545C2—4884no—
546C2#2145427911yescharacteristic
547C2C2×#1683976811yescharacteristic
548C2#2145436411yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2973C2
socleC22×C61C22×C22⋊C4
prime radical#21459583C2
cosocleC22×C4×C61C24
hypercenter#2973D61
hypocenterC61#17620
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #21448759C2semidirect
3. #21444388C2semidirect
4. #21444367C2semidirect
5. #21444349C2semidirect
6. #21444336C2semidirect
7. #21444328C2semidirect
8. C2×#16839497C2semidirect
9. C61×#2973C2semidirect
11. #16840416C4semidirect
13. D4×D244C4semidirect
23. #16839630C22semidirect
24. #16839625C22semidirect
25. #16839622C22semidirect
27. C4×C22⋊Dic61C22semidirect
28. #16839608C22semidirect
29. C2×#14519684C22semidirect
30. #16839497C22semidirect
31. #16839497C22semidirect
32. #16839497C22semidirect
33. #16839497C22semidirect
36. C61×#432C22semidirect
37. C61×#407C22semidirect
38. C2×C22⋊C8×C61C22semidirect
44. C22×D244C2×C4semidirect
49. #14519777C2×C4semidirect
50. #14519774C2×C4semidirect
62. #14519751C2×C4semidirect
67. C4×D244C2×C4semidirect
70. C2×C4⋊Dic61C2×C4semidirect
268. #2973D61semidirect
296. #432C2×D61semidirect
297. #407C2×D61semidirect
298. C2×C22⋊C8C2×D61semidirect
313. C61#17620semidirect

Maximal subgroups

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

subgroupindex
#214595832
#214487592
#214443882
#214443672
#214443492
#214443362
#214443282
C2×#168394972
C61×#29732
#1762061