Finite Group Databasebeta

← #29647772

Subgroups — #29647772

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#21454848C211yesmaximal, characteristic, complemented
2#21448759C211yesmaximal, characteristic, complemented
3#21444387C212yesmaximal, complemented
4#21444366C212yesmaximal, complemented
5#21444348C212yesmaximal, complemented
6#21444335C212yesmaximal, complemented
7#21444327C212yesmaximal, complemented
8C2×#16839496C212yesmaximal, complemented
9C61×#3496C211yesmaximal, characteristic
10#16839834—24no—
11#16839819C2212yescomplemented
12#16839726C2212yescomplemented
13C2×C22⋊#92418C2212yescomplemented
14#16839655—21no—
15#16839648—21no—
16#16839642—21no—
17#16839630C2211yescharacteristic, complemented
18#16839625C2211yescharacteristic, complemented
19#16839622C2211yescharacteristic, complemented
20C22⋊C4×Dic61—21no—
21C4×C22⋊Dic61C2211yescharacteristic, complemented
22#16839608C2211yescharacteristic, complemented
23C2×#14519684C2211yescharacteristic, complemented
24#16839496C2214yescomplemented
25#16839496C2214yescomplemented
26#16839496C2214yescomplemented
27#16839496C2214yescomplemented
28C2×H(5,𝔽2)×C61—21no—
29C61×#547C414yescomplemented
30C61×#546—24no—
31D42×C61C414yescomplemented
32C61×#537—21no—
33C61×#534—21no—
34C61×#534—21no—
35C61×#532C2212yes—
36C2×#121×C61C2212yes—
37C2×#121×C61C2212yes—
38C2×C4×D4×C61C2211yescharacteristic
39C23×Dic61—21no—
40C2×#11758194—22no—
41C22×#92418—22no—
42C2×C4⋊Dic61—21no—
43C2×C22⋊Dic61—21no—
44C2×C22⋊Dic61—21no—
45C2×C22⋊Dic61—21no—
46C2×C4×Dic61—21no—
47C2×C4×Dic61—21no—
48C4⋊#92418—22no—
49C22⋊#92418—22no—
50C22⋊#92418—22no—
51C22⋊#92418C2314yes—
52#14519688—22no—
53#14519684—21no—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57#14519684—21no—
58#14519684C2311yescharacteristic
59#14519684C2311yescharacteristic
60#14519684C2311yescharacteristic
61#14519684C2311yescharacteristic
62H(5,𝔽2)×C61—44no—
63H(5,𝔽2)×C61—44no—
64C2×#41×C61—24no—
65C2×#41×C61—22no—
66C22×D4×C61—22no—
67C22×D4×C61—22no—
68C22×D4×C61—24no—
69C22×D4×C61C2×C412yes—
70C22×D4×C61C2×C412yescomplemented
71C22×D4×C61D411yescharacteristic
72C23×C4×C61C2311yescharacteristic
73C23×C4×C61C2311yescharacteristic
74C4⋊Q8×C61C2×C412yescomplemented
75#127×C61C2×C412yescomplemented
76#124×C61—22no—
77#123×C61—24no—
78#123×C61—24no—
79C22⋊Q8×C61—24no—
80#121×C61—22no—
81#121×C61—22no—
82#121×C61—22no—
83#121×C61—22no—
84#121×C61—22no—
85#121×C61—24no—
86#121×C61C2×C414yes—
87#121×C61C2×C414yescomplemented
88#121×C61C2314yes—
89#120×C61—24no—
90#120×C61—24no—
91C4×D4×C61—22no—
92C4×D4×C61C2×C414yes—
93C4×D4×C61C2×C414yescomplemented
94C4×D4×C61D414yes—
95C2×C4⋊C4×C61C2×C412yes—
96C2×C4⋊C4×C61C2×C412yescomplemented
97C2×C4⋊C4×C61D411yescharacteristic
98C2×C22⋊C4×C61—22no—
99C2×C22⋊C4×C61—22no—
100C2×C22⋊C4×C61D411yescharacteristic
101C2×C22⋊C4×C61D411yescharacteristic
102C2×C42×C61C2311yescharacteristic
103C24×C61—22no—
104C24×C61—22no—
105C24×C61C2×D411yescharacteristic
106C24×C61C2×D411yescharacteristic
107#41×C61—44no—
108#41×C61—44no—
109#41×C61—416no—
110C2×Q8×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61—22no—
113C2×D4×C61—24no—
114C2×D4×C61—24no—
115C2×D4×C61—24no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C2×D4×C61—416no—
121C2×D4×C61—44no—
122C2×D4×C61—44no—
123C2×D4×C61—44no—
124C2×D4×C61—44no—
125C2×D4×C61—82no—
126C2×D4×C61—82no—
127C2×D4×C61C22⋊C414yes—
128C2×D4×C61C22×C414yes—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—24no—
132C22×C4×C61—81no—
133C22×C4×C61—81no—
134C22×C4×C61C22×C412yes—
135C22×C4×C61C22×C412yes—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D412yes—
141C22×C4×C61C2×D412yes—
142C22×C4×C61C2411yescharacteristic
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—42no—
148C22×Dic61—42no—
149C22×Dic61—81no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61C22⋊C414yes—
154C4⋊C4×C61C22×C414yes—
155C22⋊C4×C61—24no—
156C22⋊C4×C61—24no—
157C22⋊C4×C61—24no—
158C22⋊C4×C61—24no—
159C22⋊C4×C61C22⋊C414yes—
160C22⋊C4×C61C22⋊C414yes—
161C42×C61C22×C412yes—
162C42×C61C2×D412yes—
163C22⋊Dic61—42no—
164C22⋊Dic61—42no—
165C22⋊Dic61—81no—
166C4⋊Dic61—81no—
167C4×Dic61—42no—
168C4×Dic61—81no—
169#11758194—82no—
170C2×#92418—24no—
171C2×#92418—82no—
172C23×C61—21no—
173C23×C61—21no—
174C23×C61—24no—
175C23×C61—41no—
176C23×C61—41no—
177C23×C61—41no—
178C23×C61—41no—
179C23×C61—44no—
180C23×C61—44no—
181C23×C61—82no—
182C23×C61—82no—
183C23×C61—82no—
184C23×C61—82no—
185C23×C61C2×C22⋊C412yes—
186C23×C61C2×C22⋊C412yes—
187C23×C61AGL(1,ℤ/8)12yes—
188C23×C61AGL(1,ℤ/8)12yes—
189C23×C61C22×D411yescharacteristic
190Q8×C61—44no—
191D4×C61—28no—
192D4×C61—44no—
193D4×C61—44no—
194D4×C61—48no—
195D4×C61—44no—
196D4×C61—44no—
197D4×C61—48no—
198D4×C61—82no—
199D4×C61—82no—
200D4×C61—82no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—24no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—48no—
213C2×C4×C61—44no—
214C2×C4×C61—44no—
215C2×C4×C61C2×C22⋊C412yes—
216C2×C4×C61C2×C22⋊C412yes—
217C2×C4×C61C2×C22⋊C412yes—
218C2×C4×C61C2×C22⋊C412yes—
219C2×C4×C61C23×C411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×Dic61—22no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—84no—
234#92418—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—44no—
246C22×C61—44no—
247C22×C61—44no—
248C22×C61—81no—
249C22×C61—81no—
250C22×C61—164no—
251C22×C61—164no—
252C22×C61#41814yes—
253C22×C61C22×C22⋊C411yescharacteristic
254C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
255C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
256C4×C61—21no—
257C4×C61—21no—
258C4×C61—22no—
259C4×C61—22no—
260C4×C61—22no—
261C4×C61—22no—
262C4×C61—44no—
263Dic61—42no—
264Dic61—81no—
265Dic61—81no—
266Dic61—81no—
267#3496D6111yescharacteristic
268#2973—611no—
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—84no—
281C2×C61#288311yescharacteristic
282C2×C61C2×#41811yescharacteristic
283C2×C61#290111yescharacteristic
284C2×H(5,𝔽2)—21no—
285#547Dic6114yescomplemented
286#546—24no—
287D42Dic6114yescomplemented
288#537—21no—
289#534—21no—
290#534—21no—
291#532C2×D6112yes—
292C2×#121C2×D6112yes—
293C2×#121C2×D6112yes—
294C2×C4×D4C2×D6111yescharacteristic
295#435—1224no—
296#432—612no—
297#407—612no—
298C2×C22⋊C8—612no—
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)—44no—
315H(5,𝔽2)—44no—
316C2×#41—24no—
317C2×#41—22no—
318C22×D4—22no—
319C22×D4—22no—
320C22×D4—24no—
321C22×D4C2×Dic6112yes—
322C22×D4C2×Dic6112yescomplemented
323C22×D4C22⋊D6111yescharacteristic
324C23×C4—1221no—
325C23×C4C22×D6111yescharacteristic
326C23×C4C22×D6111yescharacteristic
327C2×M16—1222no—
328C22×C8—1222no—
329C4⋊Q8C2×Dic6112yescomplemented
330#127C2×Dic6112yescomplemented
331#124—22no—
332#123—24no—
333#123—24no—
334C22⋊Q8—24no—
335#121—22no—
336#121—22no—
337#121—22no—
338#121—22no—
339#121—22no—
340#121—24no—
341#121C2×Dic6114yes—
342#121C2×Dic6114yescomplemented
343#121C22×D6114yes—
344#120—24no—
345#120—24no—
346C4×D4—22no—
347C4×D4C2×Dic6114yes—
348C4×D4C2×Dic6114yescomplemented
349C4×D4C22⋊D6114yes—
350C2×C4⋊C4—1221no—
351C2×C4⋊C4C2×Dic6112yes—
352C2×C4⋊C4C2×Dic6112yescomplemented
353C2×C4⋊C4C22⋊D6111yescharacteristic
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—1221no—
357C2×C22⋊C4—1221no—
358C2×C22⋊C4—1221no—
359C2×C22⋊C4C22⋊D6111yescharacteristic
360C2×C22⋊C4C22⋊D6111yescharacteristic
361C2×C42—1221no—
362C2×C42—1221no—
363C2×C42C22×D6111yescharacteristic
364C4⋊C8—1222no—
365C22⋊C8—614no—
366C22⋊C8—1222no—
367C22⋊C8—1222no—
368#97—1222no—
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—22no—
379C24—22no—
380C24C2×C22⋊D6111yescharacteristic
381C24C2×C22⋊D6111yescharacteristic
382#41—44no—
383#41—44no—
384#41—416no—
385C2×Q8—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—24no—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—416no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—82no—
401C2×D4—82no—
402C2×D4C22⋊Dic6114yes—
403C2×D4C22×Dic6114yes—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—81no—
408C22×C4—81no—
409C22×C4—1221no—
410C22×C4—1221no—
411C22×C4—1221no—
412C22×C4—1221no—
413C22×C4—4881no—
414C22×C4—2442no—
415C22×C4—2442no—
416C22×C4C22×Dic6112yes—
417C22×C4C22×Dic6112yes—
418C22×C4C2×C22⋊D6111yescharacteristic
419C22×C4C2×C22⋊D6112yes—
420C22×C4C2×C22⋊D6112yes—
421C22×C4C2×C22⋊D6111yescharacteristic
422C22×C4C2×C22⋊D6111yescharacteristic
423C22×C4C2×C22⋊D6111yescharacteristic
424C22×C4C23×D6111yescharacteristic
425M16—4882no—
426C2×C8—1224no—
427C2×C8—4882no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—4881no—
432C4⋊C4C22⋊Dic6114yes—
433C4⋊C4C22×Dic6114yes—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—24no—
437C22⋊C4—24no—
438C22⋊C4—4881no—
439C22⋊C4—2442no—
440C22⋊C4—2442no—
441C22⋊C4C22⋊Dic6114yes—
442C22⋊C4C22⋊Dic6114yes—
443C42—4881no—
444C42—2442no—
445C42C22×Dic6112yes—
446C42C2×C22⋊D6112yes—
447C23—21no—
448C23—21no—
449C23—24no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—44no—
455C23—44no—
456C23—82no—
457C23—82no—
458C23—82no—
459C23—82no—
460C23C2×C22⋊Dic6112yes—
461C23C2×C22⋊Dic6112yes—
462C23#1451980312yes—
463C23#1451980312yes—
464C23C22×C22⋊D6111yescharacteristic
465Q8—44no—
466D4—28no—
467D4—44no—
468D4—44no—
469D4—48no—
470D4—48no—
471D4—44no—
472D4—44no—
473D4—82no—
474D4—82no—
475D4—82no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—24no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—48no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—1222no—
491C2×C4—4884no—
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—2441no—
501C2×C4—2441no—
502C2×C4—2441no—
503C2×C4C2×C22⋊Dic6112yes—
504C2×C4C2×C22⋊Dic6112yes—
505C2×C4C2×C22⋊Dic6112yes—
506C2×C4C2×C22⋊Dic6112yes—
507C2×C4C23×Dic6111yescharacteristic
508C2×C4C22×C22⋊D6111yescharacteristic
509C8—2444no—
510C22—21no—
511C22—22no—
512C22—21no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—42no—
518C22—42no—
519C22—44no—
520C22—44no—
521C22—44no—
522C22—81no—
523C22—81no—
524C22—164no—
525C22—164no—
526C22#1683976714yes—
527C22C22×C22⋊Dic6111yescharacteristic
528C22C2×#1451980311yescharacteristic
529C22C2×#1451980311yescharacteristic
530C4—22no—
531C4—21no—
532C4—21no—
533C4—22no—
534C4—22no—
535C4—22no—
536C4—44no—
537C4—4881no—
538C4—4881no—
539C4—4881no—
540C4—2442no—
541C2—21no—
542C2—21no—
543C2—24no—
544C2—42no—
545C2—84no—
546C2#2145428011yescharacteristic
547C2C2×#1683976711yescharacteristic
548C2#2145436611yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3496C2
socleC22×C61C22×C22⋊C4
cosocleC22×C4×C61C24
hypercenter#3496D61
hypocenterC61#17620
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #3496

Complements

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

NH ≅ G/Ntype
1. #21454848C2semidirect
2. #21448759C2semidirect
3. #21444387C2semidirect
4. #21444366C2semidirect
5. #21444348C2semidirect
6. #21444335C2semidirect
7. #21444327C2semidirect
8. C2×#16839496C2semidirect
11. #16839819C22semidirect
12. #16839726C22semidirect
13. C2×C22⋊#92418C22semidirect
17. #16839630C22semidirect
18. #16839625C22semidirect
19. #16839622C22semidirect
21. C4×C22⋊Dic61C22semidirect
22. #16839608C22semidirect
23. C2×#14519684C22semidirect
24. #16839496C22semidirect
25. #16839496C22semidirect
26. #16839496C22semidirect
27. #16839496C22semidirect
29. C61×#547C4semidirect
31. D42×C61C4semidirect
70. C22×D4×C61C2×C4semidirect
74. C4⋊Q8×C61C2×C4semidirect
75. #127×C61C2×C4semidirect
87. #121×C61C2×C4semidirect
93. C4×D4×C61C2×C4semidirect
96. C2×C4⋊C4×C61C2×C4semidirect
285. #547Dic61semidirect
287. D42Dic61semidirect
313. C61#17620semidirect
322. C22×D4C2×Dic61semidirect
329. C4⋊Q8C2×Dic61semidirect
330. #127C2×Dic61semidirect
342. #121C2×Dic61semidirect
348. C4×D4C2×Dic61semidirect
352. C2×C4⋊C4C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
#214548482
#214487592
#214443872
#214443662
#214443482
#214443352
#214443272
C2×#168394962
C61×#34962
#1762061