Finite Group Databasebeta

← #29652921

Subgroups — #29652921

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
1#21459583C211yesmaximal, characteristic, complemented
2C2×#16839896C212yesmaximal, complemented
3#21444753C214yesmaximal, complemented
4#21444736C214yesmaximal, complemented
5#21444386C212yesmaximal, complemented
6#21444137C211yesmaximal, characteristic, complemented
7C61×#2408C211yesmaximal, characteristic, complemented
8C2×#14519822—21no—
9C22×#11758199—22no—
10#16840416—22no—
11#16840407—24no—
12D4×D244—22no—
13#16840343C2212yescomplemented
14#16840324—21no—
15#16840324C2212yescomplemented
16#16840312—21no—
17C2×#14519751—21no—
18C2×#14519751C2212yes—
19#16839896—24no—
20#16839896C2218yescomplemented
21C2×#14519698—22no—
22C2×C22⋊#92418C2212yescomplemented
23#16839495C2218yescomplemented
24#16839488C2212yescomplemented
25#16839486C2212yescomplemented
26C23×D4×C61C2211yescharacteristic, complemented
27C2×#121×C61—21no—
28C2×#120×C61—21no—
29C2×C4×D4×C61C2211yescharacteristic
30C61×#394C2212yescomplemented
31C61×#392C2212yescomplemented
32C61×#390—21no—
33C61×#380C2211yescharacteristic, complemented
34C61×#379—21no—
35#14519822—44no—
36#14519822—44no—
37C2×#11758222—22no—
38C2×#11758224—24no—
39C22×D244—21no—
40C22×D244D412yescomplemented
41C22×C22⋊D61—22no—
42C22×C22⋊D61—22no—
43C2×D4×D61—24no—
44C2×#11758199—28no—
45C2×#11758199—48no—
46C22×#92418—22no—
47#14519777—21no—
48#14519774—21no—
49#14519765—22no—
50#14519761—24no—
51#14519761—24no—
52#14519757—24no—
53#14519752—24no—
54#14519751—22no—
55#14519751—22no—
56#14519751—22no—
57#14519751—22no—
58#14519751—22no—
59#14519751—22no—
60#14519751D414yescomplemented
61#14519751C2314yes—
62#14519747—24no—
63#14519747—24no—
64C4×D244—22no—
65C4×D244—22no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61D412yescomplemented
68C2×#11758198—22no—
69C2×#11758198—22no—
70#14519698—28no—
71C22⋊#92418—22no—
72C22⋊#92418C2314yes—
73C25×C61—21no—
74C22×D4×C61—22no—
75C22×D4×C61—22no—
76C22×D4×C61—21no—
77C22×D4×C61—22no—
78C22×D4×C61—44no—
79C22×D4×C61D411yescharacteristic
80C22×D4×C61C2312yes—
81C23×C4×C61D411yescharacteristic
82C23×C4×C61C2311yescharacteristic
83#121×C61—44no—
84#120×C61—44no—
85C4×D4×C61—22no—
86C4×D4×C61—22no—
87C2×C4⋊C4×C61—21no—
88C2×C4⋊C4×C61C2311yescharacteristic
89C2×C22⋊C4×C61—21no—
90C2×C22⋊C4×C61—22no—
91C2×C22⋊C4×C61—22no—
92C2×C22⋊C4×C61—22no—
93C2×C22⋊C4×C61D411yescharacteristic
94C2×C22⋊C4×C61C2311yescharacteristic
95C2×C42×C61D411yescharacteristic
96#95×C61C2312yes—
97C24×C61—22no—
98C24×C61—44no—
99C24×C61—42no—
100C24×C61—42no—
101C24×C61—42no—
102C24×C61—82no—
103C24×C61C2×D411yescharacteristic
104C24×C61C2×D411yescharacteristic
105C23×D61—22no—
106C23×D61—22no—
107C2×D4×C61—22no—
108C2×D4×C61—22no—
109C2×D4×C61—28no—
110C2×D4×C61—24no—
111C2×D4×C61—24no—
112C2×D4×C61—44no—
113C2×D4×C61—42no—
114C2×D4×C61—44no—
115C2×D4×C61—48no—
116C2×D4×C61—44no—
117C2×D4×C61—44no—
118C2×D4×C61—41no—
119C2×D4×C61—41no—
120C2×D4×C61—82no—
121C2×D4×C61D814yescomplemented
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—22no—
125C22×C4×C61—42no—
126C22×C4×C61—42no—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D412yes—
133C22×C4×C61C2411yescharacteristic
134C2×C22⋊D61—24no—
135C2×C22⋊D61—24no—
136C2×C22⋊D61—24no—
137C2×C22⋊D61—44no—
138C2×C22⋊D61—44no—
139C2×C22⋊D61—44no—
140C2×C22⋊D61—44no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143#11758224—416no—
144D4×D61—416no—
145#11758222—44no—
146#11758222—44no—
147C2×D244—22no—
148C2×D244—22no—
149C2×D244—22no—
150C2×D244—22no—
151C2×D244—44no—
152C2×D244—82no—
153C2×D244C2×D414yes—
154C2×C4×D61—24no—
155C2×Dic122—22no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—42no—
158C22⋊C4×C61—22no—
159C22⋊C4×C61—22no—
160C22⋊C4×C61—44no—
161C22⋊C4×C61—42no—
162C22⋊C4×C61—44no—
163C22⋊C4×C61—82no—
164C42×C61—21no—
165C42×C61—21no—
166#11758199—416no—
167#11758198—24no—
168#11758198—24no—
169#11758198—24no—
170#11758198—24no—
171C4⋊Dic61—22no—
172C4⋊Dic61—22no—
173C4⋊Dic61—22no—
174C4⋊Dic61C2×D414yes—
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—42no—
185C23×C61—48no—
186C23×C61—42no—
187C23×C61—42no—
188C23×C61—42no—
189C23×C61—42no—
190C23×C61—42no—
191C23×C61—84no—
192C23×C61—88no—
193C23×C61—88no—
194C23×C61—84no—
195C23×C61#12012yes—
196C23×C61C2×D812yes—
197C23×C61C2×D812yescomplemented
198C23×C61AGL(1,ℤ/8)12yes—
199C23×C61C22×D411yescharacteristic
200C22×D61—24no—
201C22×D61—44no—
202C22×D61—44no—
203C22×D61—82no—
204C22×D61—82no—
205C22×D61—82no—
206C22×D61—82no—
207D4×C61—28no—
208D4×C61—28no—
209D4×C61—44no—
210D4×C61—44no—
211D4×C61—44no—
212D4×C61—88no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—24no—
216C2×C4×C61—22no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—42no—
220C2×C4×C61—41no—
221C2×C4×C61—42no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—81no—
228C2×C4×C61#12011yescharacteristic
229C2×C4×C61#12011yescharacteristic
230C2×C4×C61C2×D812yescomplemented
231C2×C4×C61C22×D412yes—
232C22⋊D61—48no—
233C22⋊D61—44no—
234C22⋊D61—44no—
235C22⋊D61—48no—
236C2×Dic61—24no—
237C2×Dic61—44no—
238C2×Dic61—44no—
239D244—44no—
240D244—44no—
241D244—44no—
242D244—82no—
243D244—82no—
244C4×D61—48no—
245Dic122—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—44no—
257C22×C61—44no—
258C22×C61—41no—
259C22×C61—41no—
260C22×C61—44no—
261C22×C61—81no—
262C22×C61—82no—
263C22×C61—82no—
264C22×C61—816no—
265C22×C61—82no—
266C22×C61—82no—
267C22×C61#44714yes—
268C22×C61C2×#12011yescharacteristic
269C22×C61C22×D811yescharacteristic
270C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
271C2×D61—44no—
272C2×D61—44no—
273C2×D61—44no—
274C2×D61—164no—
275C2×D61—164no—
276C4×C61—22no—
277C4×C61—22no—
278C4×C61—21no—
279C4×C61—41no—
280C4×C61—41no—
281C4×C61—82no—
282C4×C61B(4,𝔽2)12yes—
283Dic61—44no—
284#3496—611no—
285C2×#447—612no—
286#2408D6111yescharacteristic, complemented
287#1598—614no—
288#1595—614no—
289#1519—612no—
290#1459—611no—
291C2×C61—21no—
292C2×C61—24no—
293C2×C61—21no—
294C2×C61—42no—
295C2×C61—84no—
296C2×C61C2×#44711yescharacteristic
297C2×C61#302111yescharacteristic
298C2×C61C2×B(4,𝔽2)11yescharacteristic
299D61—84no—
300C2×H(5,𝔽2)—1221no—
301C23×D4C2×D6111yescharacteristic, complemented
302C22×D8—1222no—
303#547—1222no—
304#546—1224no—
305D42—1222no—
306#537—612no—
307#534—612no—
308#534—1221no—
309#532—1221no—
310C2×#121—21no—
311C2×#121—612no—
312C2×#121—1221no—
313C2×#120—21no—
314C2×C4×D4C2×D6111yescharacteristic
315#447—618no—
316#447—1224no—
317C2×#102—1222no—
318C2×C22⋊C8—612no—
319#394C2×D6112yescomplemented
320#392C2×D6112yescomplemented
321#390—21no—
322#380C2×D6111yescharacteristic, complemented
323#379—21no—
324#327—618no—
325#324—612no—
326#323—612no—
327C61#1819411yescharacteristic, complemented, Sylow
328C25—21no—
329H(5,𝔽2)—2444no—
330H(5,𝔽2)—2444no—
331C2×#41—1224no—
332C2×#41—1222no—
333C22×D4—22no—
334C22×D4—22no—
335C22×D4—22no—
336C22×D4—21no—
337C22×D4—44no—
338C22×D4—612no—
339C22×D4—1221no—
340C22×D4—1224no—
341C22×D4—1222no—
342C22×D4—1222no—
343C22×D4C22⋊D6111yescharacteristic
344C22×D4C22×D6112yes—
345C23×C4C22⋊D6111yescharacteristic
346C23×C4C22×D6111yescharacteristic
347C2×D8—1228no—
348C2×D8—2448no—
349C22×C8—1222no—
350C4⋊Q8—1221no—
351#127—1221no—
352#124—1222no—
353#123—1224no—
354#123—1224no—
355C22⋊Q8—1224no—
356#121—44no—
357#121—614no—
358#121—614no—
359#121—1224no—
360#121—1222no—
361#121—1222no—
362#121—1222no—
363#121—1222no—
364#121—1222no—
365#121—1222no—
366#120—44no—
367#120—1224no—
368#120—1224no—
369C4×D4—22no—
370C4×D4—22no—
371C4×D4—1222no—
372C4×D4—1222no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—612no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4C22×D6111yescharacteristic
377C2×C22⋊C4—21no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
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#95C22×D6112yes—
390C24—22no—
391C24—44no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—82no—
396C24—1222no—
397C24—1222no—
398C24C2×C22⋊D6111yescharacteristic
399C24C2×C22⋊D6111yescharacteristic
400#41—2444no—
401#41—24416no—
402#41—2444no—
403C2×Q8—1222no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—28no—
407C2×D4—24no—
408C2×D4—24no—
409C2×D4—42no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—48no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—41no—
416C2×D4—41no—
417C2×D4—82no—
418C2×D4—614no—
419C2×D4—1222no—
420C2×D4—1222no—
421C2×D4—1222no—
422C2×D4—1224no—
423C2×D4—1224no—
424C2×D4—1224no—
425C2×D4—1222no—
426C2×D4—4882no—
427C2×D4—2444no—
428C2×D4—24416no—
429C2×D4—2444no—
430C2×D4—2444no—
431C2×D4—2444no—
432C2×D4—2444no—
433C2×D4#1175819914yescomplemented
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—1224no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4D4×D6112yes—
443C22×C4D4×D6112yes—
444C22×C4C2×C22⋊D6111yescharacteristic
445C22×C4C2×C22⋊D6111yescharacteristic
446C22×C4C2×C22⋊D6111yescharacteristic
447C22×C4C2×C22⋊D6111yescharacteristic
448C22×C4C23×D6111yescharacteristic
449D8—24416no—
450C2×C8—1224no—
451C2×C8—2444no—
452C4⋊C4—22no—
453C4⋊C4—42no—
454C4⋊C4—614no—
455C4⋊C4—1222no—
456C4⋊C4—1222no—
457C4⋊C4—1222no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—44no—
461C22⋊C4—42no—
462C22⋊C4—44no—
463C22⋊C4—82no—
464C22⋊C4—1224no—
465C22⋊C4—1224no—
466C22⋊C4—1224no—
467C22⋊C4—1224no—
468C42—21no—
469C42—21no—
470C23—21no—
471C23—21no—
472C23—24no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—42no—
478C23—48no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—84no—
485C23—88no—
486C23—88no—
487C23—84no—
488C23—1224no—
489C23—4882no—
490C23—4882no—
491C23—4882no—
492C23—4882no—
493C23—2444no—
494C23—2444no—
495C23#1451974512yes—
496C23C2×#1175819912yes—
497C23C2×#1175819912yescomplemented
498C23#1451980212yes—
499C23C22×C22⋊D6111yescharacteristic
500Q8—2444no—
501D4—28no—
502D4—28no—
503D4—44no—
504D4—44no—
505D4—44no—
506D4—88no—
507D4—4882no—
508D4—4882no—
509D4—2444no—
510D4—2444no—
511D4—2448no—
512D4—2444no—
513D4—2444no—
514D4—2448no—
515D4—2444no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—81no—
531C2×C4—1224no—
532C2×C4—2448no—
533C2×C4—2444no—
534C2×C4—2444no—
535C2×C4#1451974511yescharacteristic
536C2×C4#1451974511yescharacteristic
537C2×C4C2×#1175819912yescomplemented
538C2×C4C2×D4×D6112yes—
539C8—2444no—
540C22—21no—
541C22—22no—
542C22—21no—
543C22—22no—
544C22—22no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—44no—
549C22—44no—
550C22—41no—
551C22—41no—
552C22—44no—
553C22—81no—
554C22—82no—
555C22—82no—
556C22—816no—
557C22—82no—
558C22—82no—
559C22—9764no—
560C22—9764no—
561C22—2444no—
562C22—2444no—
563C22—2444no—
564C22#1683989514yes—
565C22C2×#1451974511yescharacteristic
566C22C22×#1175819911yescharacteristic
567C22C2×#1451980211yescharacteristic
568C4—22no—
569C4—21no—
570C4—22no—
571C4—41no—
572C4—41no—
573C4—82no—
574C4—2444no—
575C4#1683994812yes—
576C2—21no—
577C2—21no—
578C2—24no—
579C2—42no—
580C2—84no—
581C2—4884no—
582C2C2×#1683989511yescharacteristic
583C2#2145519011yescharacteristic
584C2C2×#1683994811yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2408C2
socleC22×C61C2×#120
hypercenter#2408D61
hypocenterC61#18194
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2408

Complements

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

NH ≅ G/Ntype
1. #21459583C2semidirect
2. C2×#16839896C2semidirect
3. #21444753C2semidirect
4. #21444736C2semidirect
5. #21444386C2semidirect
6. #21444137C2semidirect
7. C61×#2408C2semidirect
13. #16840343C22semidirect
15. #16840324C22semidirect
20. #16839896C22semidirect
22. C2×C22⋊#92418C22semidirect
23. #16839495C22semidirect
24. #16839488C22semidirect
25. #16839486C22semidirect
26. C23×D4×C61C22semidirect
30. C61×#394C22semidirect
31. C61×#392C22semidirect
33. C61×#380C22semidirect
40. C22×D244D4semidirect
60. #14519751D4semidirect
67. C2×C4⋊Dic61D4semidirect
121. C2×D4×C61D8semidirect
197. C23×C61C2×D8semidirect
230. C2×C4×C61C2×D8semidirect
286. #2408D61semidirect
301. C23×D4C2×D61semidirect
319. #394C2×D61semidirect
320. #392C2×D61semidirect
322. #380C2×D61semidirect
327. C61#18194semidirect
433. 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
#214595832
C2×#168398962
#214447532
#214447362
#214443862
#214441372
C61×#24082
#1819461