Finite Group Databasebeta

← #29653847

Subgroups — #29653847

Order 15616.

1486 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456567C212yesmaximal, complemented
2#21451553C211yesmaximal, characteristic, complemented
3#21444805C214yesmaximal, complemented
4#21444737C214yesmaximal, complemented
5#21444387C212yesmaximal, complemented
6C2×#16839486C211yesmaximal, characteristic, complemented
7C2×C61×#534C211yesmaximal, characteristic, complemented
8C2×#14519751C2211yescharacteristic, complemented
9C2×#14519746—21no—
10#16840075—22no—
11#16840073C2214yescomplemented
12#16839996—24no—
13#16839896—22no—
14#16839895C2214yescomplemented
15C2×C22⋊#92418C2212yescomplemented
16#16839687C2212yescomplemented
17#16839678—22no—
18#16839671C2212yescomplemented
19#16839641—21no—
20#16839622C2211yescharacteristic, complemented
21#16839496C2218yescomplemented
22#16839486C2214yescomplemented
23C23×D4×C61—21no—
24C61×#534—24no—
25C61×#534—22no—
26C61×#534C2214yes—
27C2×#123×C61—22no—
28C2×#121×C61—21no—
29C2×#121×C61C2212yes—
30C2×#120×C61—22no—
31C22×C22⋊C4×C61C2211yescharacteristic
32C22×D244C2311yescharacteristic, complemented
33C22×C22⋊D61—21no—
34C22×C22⋊D61—22no—
35C2×#11758199—24no—
36C22×#92418—22no—
37#14519751—22no—
38#14519751—22no—
39#14519746—44no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61C2311yescharacteristic, complemented
42C2×C22⋊Dic61—22no—
43C2×C22⋊Dic61—21no—
44C2×#11758198—21no—
45C2×#11758198—22no—
46#14519710—24no—
47#14519698—24no—
48#14519697—24no—
49C22⋊#92418—22no—
50C22⋊#92418C2314yescomplemented
51#14519684—22no—
52#14519684C2312yescomplemented
53C25×C61—21no—
54C25×C61D411yescharacteristic
55C22×D4×C61—24no—
56C22×D4×C61—22no—
57C22×D4×C61—24no—
58C22×D4×C61—22no—
59C22×D4×C61—21no—
60C22×D4×C61—44no—
61C22×D4×C61—44no—
62C22×D4×C61D412yescomplemented
63C23×C4×C61—21no—
64C23×C4×C61D411yescharacteristic
65C23×C4×C61C2311yescharacteristic
66#123×C61—28no—
67#123×C61—28no—
68#121×C61—24no—
69#121×C61—22no—
70#121×C61—22no—
71#121×C61—24no—
72#121×C61D414yescomplemented
73#121×C61C2314yes—
74#120×C61—28no—
75#120×C61—28no—
76C2×C4⋊C4×C61—21no—
77C2×C4⋊C4×C61D412yescomplemented
78C2×C22⋊C4×C61—22no—
79C2×C22⋊C4×C61—22no—
80C2×C22⋊C4×C61—22no—
81C2×C22⋊C4×C61—22no—
82C2×C22⋊C4×C61D412yes—
83C2×C22⋊C4×C61C2312yes—
84C24×C61—21no—
85C24×C61—21no—
86C24×C61—22no—
87C24×C61—24no—
88C24×C61—22no—
89C24×C61—44no—
90C24×C61—81no—
91C24×C61—81no—
92C24×C61—82no—
93C24×C61—82no—
94C24×C61D814yescomplemented
95C24×C61C2×D412yes—
96C24×C61C2×D411yescharacteristic
97C23×D61—21no—
98C2×D4×C61—28no—
99C2×D4×C61—24no—
100C2×D4×C61—24no—
101C2×D4×C61—24no—
102C2×D4×C61—22no—
103C2×D4×C61—48no—
104C2×D4×C61—416no—
105C2×D4×C61—48no—
106C2×D4×C61—48no—
107C2×D4×C61—44no—
108C2×D4×C61—44no—
109C2×D4×C61—82no—
110C2×D4×C61—82no—
111C2×D4×C61C2×D414yes—
112C22×C4×C61—24no—
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61—21no—
116C22×C4×C61—44no—
117C22×C4×C61—81no—
118C22×C4×C61—81no—
119C22×C4×C61C2×D412yes—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D412yes—
122C22×C4×C61C2×D412yes—
123C22×C4×C61C2411yescharacteristic
124C2×C22⋊D61—41no—
125C2×C22⋊D61—41no—
126C2×C22⋊D61—44no—
127C2×C22⋊D61—42no—
128C2×C22⋊D61—42no—
129C2×C22⋊D61—82no—
130C2×C22⋊D61—82no—
131C22×Dic61—21no—
132C22×Dic61—22no—
133C2×D244—22no—
134C2×D244—81no—
135C4⋊C4×C61—24no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61C2×D414yes—
138C22⋊C4×C61—28no—
139C22⋊C4×C61—24no—
140C22⋊C4×C61—24no—
141C22⋊C4×C61—24no—
142C22⋊C4×C61—24no—
143C22⋊Dic61—44no—
144C22⋊Dic61—42no—
145#11758199—84no—
146#11758198—42no—
147#11758198—82no—
148C4⋊Dic61—22no—
149C4⋊Dic61—81no—
150C2×#92418—24no—
151C2×#92418—82no—
152C23×C61—22no—
153C23×C61—24no—
154C23×C61—22no—
155C23×C61—24no—
156C23×C61—24no—
157C23×C61—41no—
158C23×C61—41no—
159C23×C61—42no—
160C23×C61—42no—
161C23×C61—42no—
162C23×C61—42no—
163C23×C61—48no—
164C23×C61—44no—
165C23×C61—44no—
166C23×C61—44no—
167C23×C61—81no—
168C23×C61—81no—
169C23×C61—84no—
170C23×C61—84no—
171C23×C61—84no—
172C23×C61—84no—
173C23×C61—816no—
174C23×C61#12011yescharacteristic
175C23×C61#12011yescharacteristic
176C23×C61C2×D814yes—
177C23×C61C2×D812yes—
178C23×C61AGL(1,ℤ/8)12yes—
179C23×C61C22×D411yescharacteristic
180C22×D61—42no—
181C22×D61—42no—
182C22×D61—41no—
183C22×D61—41no—
184C22×D61—41no—
185D4×C61—44no—
186D4×C61—416no—
187D4×C61—44no—
188D4×C61—44no—
189D4×C61—48no—
190D4×C61—82no—
191D4×C61—84no—
192C2×C4×C61—22no—
193C2×C4×C61—24no—
194C2×C4×C61—42no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—42no—
198C2×C4×C61—48no—
199C2×C4×C61—42no—
200C2×C4×C61—44no—
201C2×C4×C61—42no—
202C2×C4×C61—81no—
203C2×C4×C61#12012yes—
204C2×C4×C61C22×D412yes—
205C22⋊D61—88no—
206C22⋊D61—82no—
207C22⋊D61—82no—
208C2×Dic61—42no—
209C2×Dic61—42no—
210C2×Dic61—41no—
211C2×Dic61—42no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214D244—44no—
215#92418—44no—
216#18284—611nomaximal, Sylow
217C22×C61—21no—
218C22×C61—21no—
219C22×C61—21no—
220C22×C61—21no—
221C22×C61—21no—
222C22×C61—21no—
223C22×C61—24no—
224C22×C61—42no—
225C22×C61—42no—
226C22×C61—44no—
227C22×C61—48no—
228C22×C61—42no—
229C22×C61—42no—
230C22×C61—44no—
231C22×C61—44no—
232C22×C61—44no—
233C22×C61—82no—
234C22×C61—84no—
235C22×C61—88no—
236C22×C61—816no—
237C22×C61#44712yes—
238C22×C61#44712yes—
239C22×C61C2×#12011yescharacteristic
240C22×C61C22×D811yescharacteristic
241C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
242C2×D61—82no—
243C2×D61—82no—
244C2×D61—81no—
245C2×D61—81no—
246C2×D61—81no—
247C4×C61—22no—
248C4×C61—22no—
249C4×C61—44no—
250C4×C61—42no—
251Dic61—82no—
252Dic61—81no—
253C2×#534D6111yescharacteristic, complemented
254#3186—612no—
255#2617—611no—
256#1607—614no—
257#1595—614no—
258#1519—612no—
259C2×#323—611no—
260C2×C61—21no—
261C2×C61—21no—
262C2×C61—22no—
263C2×C61—22no—
264C2×C61—44no—
265C2×C61—84no—
266C2×C61C2×#44711yescharacteristic
267C2×C61#301911yescharacteristic
268C2×C61#302611yescharacteristic
269D61—161no—
270C23×D4—21no—
271#534—24no—
272#534—22no—
273#534C2×D6114yes—
274C2×#123—22no—
275C2×#121—21no—
276C2×#121—611no—
277C2×#121C2×D6112yes—
278C2×#120—22no—
279C2×#120—1221no—
280C22×C22⋊C4C2×D6111yescharacteristic
281#480—614no—
282#480—1222no—
283#466—1224no—
284#447—614no—
285#447—1222no—
286C2×C22⋊C8—612no—
287#397—612no—
288#394—1222no—
289#392—612no—
290#386—1221no—
291#380—611no—
292#327—618no—
293#323—614no—
294C61#1828411yescharacteristic, complemented, Sylow
295C25—21no—
296C25C22⋊D6111yescharacteristic
297C22×D4—22no—
298C22×D4—24no—
299C22×D4—22no—
300C22×D4—24no—
301C22×D4—21no—
302C22×D4—44no—
303C22×D4—44no—
304C22×D4—611no—
305C22×D4—1222no—
306C22×D4—1221no—
307C22×D4C22⋊D6112yescomplemented
308C23×C4—21no—
309C23×C4C22⋊D6111yescharacteristic
310C23×C4C22×D6111yescharacteristic
311C2×D8—1224no—
312C22×C8—1222no—
313#123—28no—
314#123—28no—
315#121—24no—
316#121—24no—
317#121—22no—
318#121—22no—
319#121—1222no—
320#121—1222no—
321#121C22⋊D6114yescomplemented
322#121C22×D6114yes—
323#120—28no—
324#120—28no—
325#120—2444no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—611no—
328C2×C4⋊C4—1221no—
329C2×C4⋊C4C22⋊D6112yescomplemented
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—1222no—
335C2×C22⋊C4—1221no—
336C2×C22⋊C4—1222no—
337C2×C22⋊C4—1221no—
338C2×C22⋊C4C22⋊D6112yes—
339C2×C22⋊C4C22×D6112yes—
340#107—1224no—
341#102—1224no—
342#102—1224no—
343C22⋊C8—614no—
344C22⋊C8—1222no—
345#95—612no—
346#95—1222no—
347C24—21no—
348C24—21no—
349C24—22no—
350C24—24no—
351C24—22no—
352C24—44no—
353C24—81no—
354C24—81no—
355C24—82no—
356C24—82no—
357C24—1221no—
358C24#1175819914yescomplemented
359C24C2×C22⋊D6112yes—
360C24C2×C22⋊D6111yescharacteristic
361C2×D4—28no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—22no—
366C2×D4—48no—
367C2×D4—416no—
368C2×D4—48no—
369C2×D4—48no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—82no—
373C2×D4—82no—
374C2×D4—1222no—
375C2×D4—4881no—
376C2×D4—4882no—
377C2×D4—4882no—
378C2×D4—2441no—
379C2×D4—2441no—
380C2×D4—2444no—
381C2×D4—2442no—
382C2×D4—2442no—
383C2×D4C2×C22⋊D6114yes—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—44no—
389C22×C4—81no—
390C22×C4—81no—
391C22×C4—1222no—
392C22×C4—1221no—
393C22×C4C2×C22⋊D6112yes—
394C22×C4C2×C22⋊D6112yes—
395C22×C4C2×C22⋊D6112yes—
396C22×C4C2×C22⋊D6111yescharacteristic
397C22×C4C23×D6111yescharacteristic
398D8—4884no—
399C2×C8—1224no—
400C2×C8—4882no—
401C4⋊C4—24no—
402C4⋊C4—22no—
403C4⋊C4—1222no—
404C4⋊C4—4881no—
405C4⋊C4C2×C22⋊D6114yes—
406C22⋊C4—28no—
407C22⋊C4—24no—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—4882no—
412C22⋊C4—2444no—
413C22⋊C4—2442no—
414C22⋊C4—2442no—
415C23—22no—
416C23—24no—
417C23—22no—
418C23—24no—
419C23—24no—
420C23—41no—
421C23—41no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—48no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—81no—
431C23—81no—
432C23—84no—
433C23—84no—
434C23—84no—
435C23—84no—
436C23—816no—
437C23—2442no—
438C23—2442no—
439C23—2441no—
440C23—2441no—
441C23—2441no—
442C23#1451974611yescharacteristic
443C23#1451974611yescharacteristic
444C23C2×#1175819914yes—
445C23C2×#1175819912yes—
446C23#1451980312yes—
447C23C22×C22⋊D6111yescharacteristic
448D4—44no—
449D4—416no—
450D4—44no—
451D4—44no—
452D4—48no—
453D4—82no—
454D4—84no—
455D4—4888no—
456D4—4882no—
457D4—4882no—
458D4—2444no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—48no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—81no—
470C2×C4—2442no—
471C2×C4—2442no—
472C2×C4—2441no—
473C2×C4—2442no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4#1451974612yes—
477C2×C4C22×C22⋊D6112yes—
478C8—2444no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—24no—
486C22—42no—
487C22—42no—
488C22—44no—
489C22—48no—
490C22—42no—
491C22—42no—
492C22—44no—
493C22—44no—
494C22—44no—
495C22—82no—
496C22—84no—
497C22—88no—
498C22—816no—
499C22—4882no—
500C22—4882no—
501C22—4881no—
502C22—4881no—
503C22—4881no—
504C22#1683989612yes—
505C22#1683989612yes—
506C22C2×#1451974611yescharacteristic
507C22C22×#1175819911yescharacteristic
508C22C2×#1451980311yescharacteristic
509C4—22no—
510C4—22no—
511C4—42no—
512C4—44no—
513C4—4882no—
514C4—4881no—
515C2—21no—
516C2—22no—
517C2—21no—
518C2—22no—
519C2—44no—
520C2—84no—
521C2—9761no—
522C2C2×#1683989611yescharacteristic
523C2#2145517611yescharacteristic
524C2#2145521611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#534C2
socleC22×C61C2×#120
hypercenterC2×#534D61
hypocenterC61#18284
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#534

Complements

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

NH ≅ G/Ntype
1. #21456567C2semidirect
2. #21451553C2semidirect
3. #21444805C2semidirect
4. #21444737C2semidirect
5. #21444387C2semidirect
6. C2×#16839486C2semidirect
7. C2×C61×#534C2semidirect
8. C2×#14519751C22semidirect
11. #16840073C22semidirect
14. #16839895C22semidirect
15. C2×C22⋊#92418C22semidirect
16. #16839687C22semidirect
18. #16839671C22semidirect
20. #16839622C22semidirect
21. #16839496C22semidirect
22. #16839486C22semidirect
32. C22×D244C23semidirect
41. C2×C4⋊Dic61C23semidirect
50. C22⋊#92418C23semidirect
52. #14519684C23semidirect
62. C22×D4×C61D4semidirect
72. #121×C61D4semidirect
77. C2×C4⋊C4×C61D4semidirect
94. C24×C61D8semidirect
253. C2×#534D61semidirect
294. C61#18284semidirect
307. C22×D4C22⋊D61semidirect
321. #121C22⋊D61semidirect
329. C2×C4⋊C4C22⋊D61semidirect
358. C24#11758199semidirect

Maximal subgroups

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

subgroupindex
#214565672
#214515532
#214448052
#214447372
#214443872
C2×#168394862
C2×C61×#5342
#1828461