Finite Group Databasebeta

← #29652922

Subgroups — #29652922

Order 15616.

1650 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839895C212yesmaximal, complemented
2#21449175C211yesmaximal, characteristic, complemented
3#21444754C214yesmaximal, complemented
4#21444737C214yesmaximal, complemented
5#21444387C212yesmaximal, complemented
6#21444137C211yesmaximal, characteristic, complemented
7C61×#3496C211yesmaximal, characteristic, complemented
8C23×D244C2211yescharacteristic, complemented
9C22×#11758199—22no—
10C2×#14519751—21no—
11C2×#14519745—21no—
12#16839895—24no—
13#16839895C2218yescomplemented
14C2×#14519697—22no—
15C2×C22⋊#92418C2212yescomplemented
16#16839679C2212yescomplemented
17#16839671C2212yescomplemented
18#16839664—21no—
19#16839622C2211yescharacteristic, complemented
20#16839617—21no—
21#16839496C2218yescomplemented
22#16839488C2212yescomplemented
23#16839486C2212yescomplemented
24C2×H(5,𝔽2)×C61—21no—
25C61×#547—22no—
26C61×#546—24no—
27D42×C61—22no—
28C61×#537C2212yescomplemented
29C61×#534—21no—
30C61×#534C2212yescomplemented
31C61×#532—21no—
32C2×#121×C61—21no—
33C2×#121×C61C2212yes—
34C2×C4×D4×C61C2211yescharacteristic
35C24×D61—21no—
36C22×D244—22no—
37C22×D244—21no—
38C22×D244—22no—
39C22×D244—44no—
40C22×D244C2312yes—
41C22×C22⋊D61—22no—
42C2×#11758199—28no—
43C2×#11758199—48no—
44C22×#92418—22no—
45#14519751—44no—
46#14519745—44no—
47C2×C4⋊Dic61—21no—
48C2×C22⋊Dic61—21no—
49C2×#11758198—22no—
50C2×#11758198—22no—
51C2×#11758198—22no—
52#14519697—28no—
53C22⋊#92418—22no—
54C22⋊#92418C2314yes—
55#14519684C2312yes—
56H(5,𝔽2)×C61—44no—
57H(5,𝔽2)×C61—44no—
58C2×#41×C61—24no—
59C2×#41×C61—22no—
60C22×D4×C61—24no—
61C22×D4×C61—22no—
62C22×D4×C61—22no—
63C22×D4×C61—21no—
64C22×D4×C61D411yescharacteristic
65C22×D4×C61D412yescomplemented
66C23×C4×C61D411yescharacteristic
67C23×C4×C61C2311yescharacteristic
68C4⋊Q8×C61—21no—
69#127×C61—21no—
70#124×C61—22no—
71#123×C61—24no—
72#123×C61—24no—
73C22⋊Q8×C61—24no—
74#121×C61—22no—
75#121×C61—22no—
76#121×C61—24no—
77#121×C61—22no—
78#121×C61—22no—
79#121×C61—22no—
80#121×C61—22no—
81#121×C61D414yescomplemented
82#121×C61C2314yes—
83#120×C61—24no—
84#120×C61—24no—
85C4×D4×C61—22no—
86C4×D4×C61—22no—
87C4×D4×C61—22no—
88C4×D4×C61—22no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61D412yescomplemented
91C2×C4⋊C4×C61C2311yescharacteristic
92C2×C22⋊C4×C61—22no—
93C2×C22⋊C4×C61—22no—
94C2×C22⋊C4×C61D411yescharacteristic
95C2×C22⋊C4×C61C2311yescharacteristic
96C2×C42×C61D411yescharacteristic
97C24×C61—22no—
98C24×C61—22no—
99C24×C61C2×D411yescharacteristic
100C24×C61C2×D411yescharacteristic
101C23×D61—22no—
102C23×D61—44no—
103C23×D61—42no—
104C23×D61—42no—
105C23×D61—42no—
106C23×D61—82no—
107#41×C61—44no—
108#41×C61—416no—
109#41×C61—44no—
110C2×Q8×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61—22no—
113C2×D4×C61—24no—
114C2×D4×C61—22no—
115C2×D4×C61—24no—
116C2×D4×C61—24no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C2×D4×C61—44no—
121C2×D4×C61—416no—
122C2×D4×C61—44no—
123C2×D4×C61—44no—
124C2×D4×C61—44no—
125C2×D4×C61—44no—
126C2×D4×C61—82no—
127C2×D4×C61D814yescomplemented
128C2×D4×C61C2×D414yes—
129C22×C4×C61—22no—
130C22×C4×C61—24no—
131C22×C4×C61—22no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—42no—
135C22×C4×C61—42no—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D412yes—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D412yes—
142C22×C4×C61C2411yescharacteristic
143C2×C22⋊D61—44no—
144C2×C22⋊D61—44no—
145C2×C22⋊D61—82no—
146C22×Dic61—22no—
147C2×D244—28no—
148C2×D244—24no—
149C2×D244—24no—
150C2×D244—44no—
151C2×D244—48no—
152C2×D244—44no—
153C2×D244—42no—
154C2×D244—41no—
155C2×D244—41no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61C2×D414yes—
161C22⋊C4×C61—24no—
162C22⋊C4×C61—24no—
163C22⋊C4×C61—22no—
164C22⋊C4×C61—24no—
165C22⋊C4×C61—24no—
166C22⋊C4×C61—22no—
167C42×C61—21no—
168C42×C61—21no—
169C22⋊Dic61—42no—
170#11758199—416no—
171#11758198—44no—
172#11758198—44no—
173#11758198—82no—
174C4⋊Dic61—42no—
175C2×#92418—24no—
176C2×#92418—44no—
177C23×C61—21no—
178C23×C61—21no—
179C23×C61—24no—
180C23×C61—41no—
181C23×C61—41no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61—44no—
185C23×C61—44no—
186C23×C61—82no—
187C23×C61—82no—
188C23×C61—82no—
189C23×C61—82no—
190C23×C61#12012yes—
191C23×C61C2×D812yes—
192C23×C61C2×D812yescomplemented
193C23×C61AGL(1,ℤ/8)12yes—
194C23×C61C22×D411yescharacteristic
195C22×D61—24no—
196C22×D61—42no—
197C22×D61—48no—
198C22×D61—42no—
199C22×D61—42no—
200C22×D61—42no—
201C22×D61—42no—
202C22×D61—42no—
203C22×D61—88no—
204C22×D61—88no—
205C22×D61—84no—
206C22×D61—84no—
207Q8×C61—44no—
208D4×C61—28no—
209D4×C61—44no—
210D4×C61—44no—
211D4×C61—48no—
212D4×C61—44no—
213D4×C61—44no—
214D4×C61—48no—
215D4×C61—44no—
216D4×C61—82no—
217D4×C61—82no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—24no—
221C2×C4×C61—22no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—24no—
225C2×C4×C61—42no—
226C2×C4×C61—41no—
227C2×C4×C61—48no—
228C2×C4×C61—44no—
229C2×C4×C61—44no—
230C2×C4×C61—42no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—81no—
234C2×C4×C61#12011yescharacteristic
235C2×C4×C61#12011yescharacteristic
236C2×C4×C61C2×D812yescomplemented
237C2×C4×C61C22×D412yes—
238C22⋊D61—88no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242D244—28no—
243D244—44no—
244D244—44no—
245D244—44no—
246#92418—44no—
247#18194—611nomaximal, Sylow
248C22×C61—21no—
249C22×C61—22no—
250C22×C61—21no—
251C22×C61—22no—
252C22×C61—22no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61—41no—
257C22×C61—41no—
258C22×C61—44no—
259C22×C61—44no—
260C22×C61—44no—
261C22×C61—81no—
262C22×C61—164no—
263C22×C61—164no—
264C22×C61#44714yes—
265C22×C61C2×#12011yescharacteristic
266C22×C61C22×D811yescharacteristic
267C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
268C2×D61—44no—
269C2×D61—44no—
270C2×D61—44no—
271C2×D61—82no—
272C2×D61—82no—
273C2×D61—816no—
274C2×D61—82no—
275C2×D61—82no—
276C4×C61—22no—
277C4×C61—22no—
278C4×C61—21no—
279C4×C61—44no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61B(4,𝔽2)12yes—
283Dic61—82no—
284#3496D6111yescharacteristic, complemented
285C2×#447—612no—
286#2408—611no—
287#1598—614no—
288#1595—614no—
289#1519—612no—
290#1459—611no—
291C2×C61—21no—
292C2×C61—21no—
293C2×C61—24no—
294C2×C61—42no—
295C2×C61—84no—
296C2×C61C2×#44711yescharacteristic
297C2×C61#302111yescharacteristic
298C2×C61C2×B(4,𝔽2)11yescharacteristic
299D61—84no—
300C2×H(5,𝔽2)—21no—
301C23×D4—611no—
302C22×D8—1222no—
303#547—22no—
304#546—24no—
305D42—22no—
306#537C2×D6112yescomplemented
307#534—21no—
308#534C2×D6112yescomplemented
309#532—21no—
310C2×#121—21no—
311C2×#121—1221no—
312C2×#121C2×D6112yes—
313C2×#120—1221no—
314C2×C4×D4C2×D6111yescharacteristic
315#447—618no—
316#447—1224no—
317C2×#102—1222no—
318C2×C22⋊C8—612no—
319#394—612no—
320#392—612no—
321#390—1221no—
322#380—611no—
323#379—1221no—
324#327—618no—
325#324—612no—
326#323—612no—
327C61#1819411yescharacteristic, complemented, Sylow
328C25—1221no—
329H(5,𝔽2)—44no—
330H(5,𝔽2)—44no—
331C2×#41—24no—
332C2×#41—22no—
333C22×D4—24no—
334C22×D4—22no—
335C22×D4—22no—
336C22×D4—21no—
337C22×D4—612no—
338C22×D4—1222no—
339C22×D4—1222no—
340C22×D4—1221no—
341C22×D4—1222no—
342C22×D4—2444no—
343C22×D4C22⋊D6111yescharacteristic
344C22×D4C22⋊D6112yescomplemented
345C23×C4C22⋊D6111yescharacteristic
346C23×C4C22×D6111yescharacteristic
347C2×D8—1228no—
348C2×D8—2448no—
349C22×C8—1222no—
350C4⋊Q8—21no—
351#127—21no—
352#124—22no—
353#123—24no—
354#123—24no—
355C22⋊Q8—24no—
356#121—22no—
357#121—22no—
358#121—24no—
359#121—22no—
360#121—22no—
361#121—22no—
362#121—22no—
363#121—2444no—
364#121C22⋊D6114yescomplemented
365#121C22×D6114yes—
366#120—24no—
367#120—24no—
368#120—2444no—
369C4×D4—22no—
370C4×D4—22no—
371C4×D4—22no—
372C4×D4—22no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4C22⋊D6112yescomplemented
376C2×C4⋊C4C22×D6111yescharacteristic
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—1222no—
380C2×C22⋊C4—1221no—
381C2×C22⋊C4—1222no—
382C2×C22⋊C4—1222no—
383C2×C22⋊C4C22⋊D6111yescharacteristic
384C2×C22⋊C4C22×D6111yescharacteristic
385C2×C42C22⋊D6111yescharacteristic
386#102—1228no—
387C22⋊C8—614no—
388C22⋊C8—1222no—
389#95—612no—
390C24—22no—
391C24—22no—
392C24—1222no—
393C24—4882no—
394C24—2444no—
395C24—2442no—
396C24—2442no—
397C24—2442no—
398C24C2×C22⋊D6111yescharacteristic
399C24C2×C22⋊D6111yescharacteristic
400#41—44no—
401#41—416no—
402#41—44no—
403C2×Q8—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—44no—
414C2×D4—416no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—82no—
420C2×D4—1228no—
421C2×D4—1224no—
422C2×D4—1224no—
423C2×D4—4882no—
424C2×D4—2442no—
425C2×D4—2444no—
426C2×D4—2444no—
427C2×D4—2448no—
428C2×D4—2444no—
429C2×D4—2444no—
430C2×D4—2441no—
431C2×D4—2441no—
432C2×D4#1175819914yescomplemented
433C2×D4C2×C22⋊D6114yes—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—1222no—
442C22×C4C2×C22⋊D6112yes—
443C22×C4C2×C22⋊D6111yescharacteristic
444C22×C4C2×C22⋊D6111yescharacteristic
445C22×C4C2×C22⋊D6111yescharacteristic
446C22×C4C2×C22⋊D6111yescharacteristic
447C22×C4C2×C22⋊D6112yes—
448C22×C4C23×D6111yescharacteristic
449D8—24416no—
450C2×C8—1224no—
451C2×C8—2444no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—2442no—
457C4⋊C4C2×C22⋊D6114yes—
458C22⋊C4—24no—
459C22⋊C4—24no—
460C22⋊C4—22no—
461C22⋊C4—24no—
462C22⋊C4—22no—
463C22⋊C4—24no—
464C22⋊C4—4882no—
465C22⋊C4—2444no—
466C22⋊C4—2442no—
467C22⋊C4—2444no—
468C42—21no—
469C42—21no—
470C23—21no—
471C23—21no—
472C23—24no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—44no—
478C23—44no—
479C23—82no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23—1224no—
484C23—4884no—
485C23—4888no—
486C23—4888no—
487C23—4884no—
488C23—2442no—
489C23—2448no—
490C23—2442no—
491C23—2442no—
492C23—2442no—
493C23—2442no—
494C23—2442no—
495C23#1451974612yes—
496C23C2×#1175819912yes—
497C23C2×#1175819912yescomplemented
498C23#1451980312yes—
499C23C22×C22⋊D6111yescharacteristic
500Q8—44no—
501D4—28no—
502D4—44no—
503D4—44no—
504D4—48no—
505D4—44no—
506D4—44no—
507D4—48no—
508D4—44no—
509D4—82no—
510D4—82no—
511D4—1228no—
512D4—4888no—
513D4—2444no—
514D4—2444no—
515D4—2444no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—24no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—48no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—81no—
532C2×C4—2442no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4#1451974611yescharacteristic
536C2×C4#1451974611yescharacteristic
537C2×C4C2×#1175819912yescomplemented
538C2×C4C22×C22⋊D6112yes—
539C8—2444no—
540C22—21no—
541C22—22no—
542C22—21no—
543C22—22no—
544C22—22no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—41no—
549C22—41no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—81no—
554C22—164no—
555C22—164no—
556C22—48816no—
557C22—4882no—
558C22—4882no—
559C22—4882no—
560C22—4882no—
561C22—2444no—
562C22—2444no—
563C22—2444no—
564C22#1683989614yes—
565C22C2×#1451974611yescharacteristic
566C22C22×#1175819911yescharacteristic
567C22C2×#1451980311yescharacteristic
568C4—22no—
569C4—21no—
570C4—22no—
571C4—41no—
572C4—41no—
573C4—44no—
574C4—4882no—
575C4#1683994912yes—
576C2—21no—
577C2—21no—
578C2—24no—
579C2—42no—
580C2—84no—
581C2—4884no—
582C2C2×#1683989611yescharacteristic
583C2#2145519111yescharacteristic
584C2C2×#1683994911yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3496C2
socleC22×C61C2×#120
hypercenter#3496D61
hypocenterC61#18194
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3496

Complements

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

NH ≅ G/Ntype
1. C2×#16839895C2semidirect
2. #21449175C2semidirect
3. #21444754C2semidirect
4. #21444737C2semidirect
5. #21444387C2semidirect
6. #21444137C2semidirect
7. C61×#3496C2semidirect
8. C23×D244C22semidirect
13. #16839895C22semidirect
15. C2×C22⋊#92418C22semidirect
16. #16839679C22semidirect
17. #16839671C22semidirect
19. #16839622C22semidirect
21. #16839496C22semidirect
22. #16839488C22semidirect
23. #16839486C22semidirect
28. C61×#537C22semidirect
30. C61×#534C22semidirect
65. C22×D4×C61D4semidirect
81. #121×C61D4semidirect
90. C2×C4⋊C4×C61D4semidirect
127. C2×D4×C61D8semidirect
192. C23×C61C2×D8semidirect
236. C2×C4×C61C2×D8semidirect
284. #3496D61semidirect
306. #537C2×D61semidirect
308. #534C2×D61semidirect
327. C61#18194semidirect
344. C22×D4C22⋊D61semidirect
364. #121C22⋊D61semidirect
375. C2×C4⋊C4C22⋊D61semidirect
432. C2×D4#11758199semidirect
497. C23C2×#11758199semidirect
537. C2×C4C2×#11758199semidirect

Maximal subgroups

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

subgroupindex
C2×#168398952
#214491752
#214447542
#214447372
#214443872
#214441372
C61×#34962
#1819461