Finite Group Databasebeta

← #29662825

Subgroups — #29662825

Order 15616.

758 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456194C211yesmaximal, characteristic, complemented
2#21455827C211yesmaximal, characteristic, complemented
3C2×#16839957C211yesmaximal, characteristic
4#21445380C212yesmaximal, complemented
5#21445352C212yesmaximal, complemented
6#21445256C212yesmaximal, complemented
7#21445228C212yesmaximal, complemented
8#21444571C211yesmaximal, characteristic, complemented
9#21444510C211yesmaximal, characteristic, complemented
10#21444488C211yesmaximal, characteristic, complemented
11D4×M16×C61C211yesmaximal, characteristic, complemented
12C22×#11758199—21no—
13#16840319C2211yescharacteristic
14C2×#14519774C2211yescharacteristic
15#16840127—22no—
16#16840109C2212yescomplemented
17#16840109C2212yescomplemented
18#16840092—21no—
19#16840052C2211yescharacteristic, complemented
20#16840052C2211yescharacteristic, complemented
21#16839967C2211yescharacteristic, complemented
22#16839967C2211yescharacteristic, complemented
23#16839957—21no—
24#16839957—21no—
25#16839957C2212yes—
26#16839957C2212yes—
27#16839913—21no—
28C4×#11758200—21no—
29C4×#11758200—21no—
30#16839815C2211yescharacteristic
31C2×C4⋊#92418C2211yescharacteristic
32#16839768—21no—
33#16839768—21no—
34C2×#14519697—21no—
35#16839515C2212yescomplemented
36#16839513C2212yescomplemented
37#16839510C2212yescomplemented
38#16839507C2212yescomplemented
39#16839505C2212yescomplemented
40#16839505C2212yescomplemented
41C22×M16×C61—21no—
42C2×C4×D4×C61C2211yescharacteristic
43C61×#436C2212yescomplemented
44C61×#435—22no—
45C8×D4×C61C2212yescomplemented
46C61×#423C2211yescharacteristic
47C61×#407—21no—
48C4×M16×C61C2211yescharacteristic
49C22×D244—21no—
50C22×D244—21no—
51C22×D244—21no—
52C2×#11758200—21no—
53C2×#11758200—21no—
54C2×#11758199—22no—
55C2×#11758199—22no—
56C2×#11758199—44no—
57C22×#92418—21no—
58#14519777C2311yescharacteristic
59#14519777C2311yescharacteristic
60#14519774—21no—
61#14519774—21no—
62#14519774—21no—
63#14519774C2312yes—
64#14519769—21no—
65#14519756—21no—
66#14519756—21no—
67C4×Dic122—21no—
68C4×Dic122—21no—
69#14519736—21no—
70#14519708—21no—
71#14519708—21no—
72C4⋊#92418—21no—
73C4⋊#92418C2312yes—
74C4⋊#92418C2311yescharacteristic
75C4⋊#92418C2311yescharacteristic
76#14519700—21no—
77#14519700—21no—
78#14519699—22no—
79#14519699—22no—
80#14519699—22no—
81#14519699—22no—
82#14519698—21no—
83#14519698—21no—
84#14519697—22no—
85#14519697—22no—
86C22⋊#92418—21no—
87C4×#92418—21no—
88C22×D4×C61D411yescharacteristic
89C23×C4×C61—21no—
90C2×M16×C61—21no—
91C2×M16×C61—22no—
92C2×M16×C61—44no—
93C2×M16×C61D411yescharacteristic
94C2×M16×C61D411yescharacteristic
95C22×C8×C61—22no—
96C4×D4×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61D411yescharacteristic
99C4×D4×C61D411yescharacteristic
100C4×D4×C61C2311yescharacteristic
101C4×D4×C61C2311yescharacteristic
102C2×C4⋊C4×C61D411yescharacteristic
103C2×C22⋊C4×C61—21no—
104C2×C42×C61C2311yescharacteristic
105C4⋊C8×C61—21no—
106C4⋊C8×C61C2312yes—
107C22⋊C8×C61—22no—
108C22⋊C8×C61—22no—
109#97×C61D412yes—
110C4×C8×C61C2312yes—
111C24×C61—21no—
112C23×D61—41no—
113C2×D4×C61—22no—
114C2×D4×C61—21no—
115C2×D4×C61—41no—
116C2×D4×C61C2×D411yescharacteristic
117C2×D4×C61C2×D411yescharacteristic
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—42no—
124C22×C4×C61—42no—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61#4111yescharacteristic
128C2×D244—22no—
129C2×D244—22no—
130C2×D244—22no—
131C2×D244—42no—
132C2×D244—42no—
133C2×D244—42no—
134C2×D244—42no—
135C2×D244—41no—
136C2×D244—41no—
137C2×D244—41no—
138C2×D244—82no—
139C2×Dic122—21no—
140C2×Dic122—21no—
141M16×C61—22no—
142M16×C61—22no—
143M16×C61—44no—
144M16×C61—42no—
145C2×C8×C61—22no—
146C2×C8×C61—44no—
147C2×C8×C61C2×D412yes—
148C2×C8×C61C2×D412yes—
149C4⋊C4×C61—21no—
150C4⋊C4×C61C2×D411yescharacteristic
151C4⋊C4×C61C2×D411yescharacteristic
152C22⋊C4×C61—21no—
153C22⋊C4×C61—21no—
154C22⋊C4×C61—21no—
155C22⋊C4×C61—21no—
156C42×C61C2×D411yescharacteristic
157C42×C61C2×D411yescharacteristic
158C42×C61#4111yescharacteristic
159C42×C61C2411yescharacteristic
160C22⋊Dic61—41no—
161#11758200—42no—
162#11758200—42no—
163#11758199—44no—
164#11758199—44no—
165C4⋊Dic61—21no—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169#11758196—41no—
170#11758196—41no—
171C4×Dic61—41no—
172C2×#92418—21no—
173C2×#92418—21no—
174C2×#92418—22no—
175C2×#92418—42no—
176C23×C61—21no—
177C23×C61—21no—
178C23×C61—42no—
179C23×C61—42no—
180C23×C61—81no—
181C23×C61#12111yescharacteristic
182C22×D61—42no—
183C22×D61—82no—
184C22×D61—82no—
185C22×D61—82no—
186D4×C61—21no—
187D4×C61—21no—
188D4×C61—21no—
189D4×C61—42no—
190D4×C61AGL(1,ℤ/8)12yes—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—41no—
196C2×C4×C61—42no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61#12111yescharacteristic
204C2×C4×C61#12111yescharacteristic
205C2×C4×C61#12111yescharacteristic
206C2×C4×C61#13512yes—
207C2×C4×C61AGL(1,ℤ/8)12yes—
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61C22×D411yescharacteristic
210C2×C4×C61C2×#4111yescharacteristic
211C2×Dic61—41no—
212C2×Dic61—41no—
213D244—42no—
214D244—42no—
215D244—42no—
216D244—42no—
217D244—82no—
218D244—82no—
219D244—82no—
220D244—82no—
221Dic122—41no—
222Dic122—41no—
223Dic122—81no—
224C8×C61—22no—
225C8×C61—22no—
226C8×C61—42no—
227#92418—42no—
228#92418—41no—
229#92418—41no—
230#19166—611nomaximal, Sylow
231C22×C61—21no—
232C22×C61—21no—
233C22×C61—22no—
234C22×C61—41no—
235C22×C61—41no—
236C22×C61—41no—
237C22×C61—41no—
238C22×C61—41no—
239C22×C61—84no—
240C22×C61#46312yes—
241C22×C61C2×#12111yescharacteristic
242C2×D61—82no—
243C2×D61—82no—
244C2×D61—82no—
245C2×D61—164no—
246C4×C61—21no—
247C4×C61—21no—
248C4×C61—21no—
249C4×C61—41no—
250C4×C61—41no—
251C4×C61#46312yes—
252C4×C61C2×#13511yescharacteristic
253C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
254C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
255C4×C61#57511yescharacteristic
256Dic61—81no—
257Dic61—81no—
258#3138—611no—
259#3101—611no—
260C2×#459—611no—
261D4×M16D6111yescharacteristic, complemented
262#1701—612no—
263#1697—612no—
264#1681—612no—
265#1677—612no—
266#1565—611no—
267#1550—611no—
268#1545—611no—
269C2×C61—22no—
270C2×C61—42no—
271C2×C61—81no—
272C2×C61C2×#46311yescharacteristic
273C2×C61#305511yescharacteristic
274C2×C61#306811yescharacteristic
275D61—162no—
276C22×D8—1221no—
277C22×M16—21no—
278#533—611no—
279C2×#127—611no—
280C2×C4×D4C2×D6111yescharacteristic
281#490—1222no—
282#486—612no—
283#486—612no—
284#483—1221no—
285#476—611no—
286#476—611no—
287#461—611no—
288#461—611no—
289#459—612no—
290#459—612no—
291#459—1221no—
292#459—1221no—
293#450—1221no—
294C4×SD16—1221no—
295C4×SD16—1221no—
296#436C2×D6112yescomplemented
297#435—22no—
298C8×D4C2×D6112yescomplemented
299#431—611no—
300#423C2×D6111yescharacteristic
301C2×C4⋊C8—611no—
302#418—1221no—
303#418—1221no—
304C2×#102—1221no—
305#407—21no—
306C4×M16C2×D6111yescharacteristic
307#335—612no—
308#334—612no—
309#333—612no—
310#332—612no—
311#331—612no—
312#331—612no—
313C61#1916611yescharacteristic, complemented, Sylow
314C22×D4—1221no—
315C22×D4—1221no—
316C22×D4—1221no—
317C22×D4D24411yescharacteristic
318C23×C4—21no—
319C2×SD16—1221no—
320C2×SD16—1221no—
321C2×D8—1222no—
322C2×D8—1222no—
323C2×D8—2444no—
324C2×M16—21no—
325C2×M16—22no—
326C2×M16—44no—
327C2×M16C22⋊D6111yescharacteristic
328C2×M16C22⋊D6111yescharacteristic
329C22×C8—22no—
330C22×C8—1221no—
331C4⋊Q8—611no—
332C4⋊Q8—611no—
333#127—612no—
334#127—1221no—
335#127—1221no—
336#127—1221no—
337#125—1221no—
338C22⋊Q8—1221no—
339C22⋊Q8—1221no—
340C4×Q8—1221no—
341C4×Q8—1221no—
342C4×D4—21no—
343C4×D4—21no—
344C4×D4D24411yescharacteristic
345C4×D4D24411yescharacteristic
346C4×D4C22×D6111yescharacteristic
347C4×D4C22×D6111yescharacteristic
348#117—1221no—
349C2×C4⋊C4D24411yescharacteristic
350C2×C22⋊C4—21no—
351C2×C42C22×D6111yescharacteristic
352#106—1221no—
353#106—1221no—
354C4⋊C8—21no—
355C4⋊C8—611no—
356C4⋊C8—611no—
357C4⋊C8—612no—
358C4⋊C8—1221no—
359C4⋊C8C22×D6112yes—
360#103—1221no—
361#103—1221no—
362#102—1221no—
363#102—1221no—
364#102—1222no—
365#102—1222no—
366#102—1222no—
367#102—1222no—
368#102—1222no—
369#102—1222no—
370C22⋊C8—22no—
371C22⋊C8—22no—
372C22⋊C8—1221no—
373#97C22⋊D6112yes—
374C4×C8—1221no—
375C4×C8C22×D6112yes—
376C24—21no—
377C24—2441no—
378C2×Q8—1221no—
379C2×Q8—1221no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—41no—
383C2×D4—1222no—
384C2×D4—1222no—
385C2×D4—1222no—
386C2×D4—4882no—
387C2×D4—2442no—
388C2×D4—2442no—
389C2×D4—2441no—
390C2×D4—2441no—
391C2×D4—2441no—
392C2×D4—2442no—
393C2×D4—2442no—
394C2×D4C2×D24411yescharacteristic
395C2×D4C2×D24411yescharacteristic
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4C2×D24411yescharacteristic
404C22×C4#1175822211yescharacteristic
405C22×C4C2×C22⋊D6111yescharacteristic
406SD16—2442no—
407SD16—2442no—
408D8—2444no—
409D8—2444no—
410M16—22no—
411M16—22no—
412M16—44no—
413M16—42no—
414C2×C8—22no—
415C2×C8—44no—
416C2×C8—1221no—
417C2×C8—1221no—
418C2×C8—1222no—
419C2×C8—2442no—
420C2×C8C2×C22⋊D6112yes—
421C2×C8C2×C22⋊D6112yes—
422C4⋊C4—21no—
423C4⋊C4—1221no—
424C4⋊C4—1221no—
425C4⋊C4—1221no—
426C4⋊C4—1221no—
427C4⋊C4—2441no—
428C4⋊C4—2441no—
429C4⋊C4C2×D24411yescharacteristic
430C4⋊C4C2×D24411yescharacteristic
431C22⋊C4—21no—
432C22⋊C4—21no—
433C22⋊C4—21no—
434C22⋊C4—21no—
435C22⋊C4—2441no—
436C42—2441no—
437C42C2×D24411yescharacteristic
438C42#1175822211yescharacteristic
439C42C2×C22⋊D6111yescharacteristic
440C42C23×D6111yescharacteristic
441C23—21no—
442C23—21no—
443C23—42no—
444C23—42no—
445C23—81no—
446C23—4882no—
447C23—4882no—
448C23—4882no—
449C23—2442no—
450C23#1451975111yescharacteristic
451Q8—4881no—
452Q8—2441no—
453Q8—2441no—
454D4—21no—
455D4—21no—
456D4—21no—
457D4—42no—
458D4—4882no—
459D4—4882no—
460D4—4882no—
461D4—4882no—
462D4—2442no—
463D4—2442no—
464D4—2442no—
465D4—2442no—
466D4#1451980512yes—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4#1451975111yescharacteristic
482C2×C4#1451975111yescharacteristic
483C2×C4#1451975111yescharacteristic
484C2×C4#1451979712yes—
485C2×C4#1451980312yes—
486C2×C4C22×C22⋊D6111yescharacteristic
487C2×C4C22×D24411yescharacteristic
488C2×C4C2×#1175822211yescharacteristic
489C8—22no—
490C8—22no—
491C8—42no—
492C8—2442no—
493C8—2441no—
494C8—2441no—
495C22—21no—
496C22—21no—
497C22—22no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—84no—
504C22—9764no—
505C22—4882no—
506C22—4882no—
507C22—4882no—
508C22#1683997712yes—
509C22C2×#1451975111yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—41no—
514C4—41no—
515C4—4881no—
516C4—4881no—
517C4#1683997712yes—
518C4C2×#1451979711yescharacteristic
519C4C2×#1451980311yescharacteristic
520C4C2×#1451980511yescharacteristic
521C4#1684056211yescharacteristic
522C2—22no—
523C2—42no—
524C2—81no—
525C2—9762no—
526C2C2×#1683997711yescharacteristic
527C2#2145543811yescharacteristic
528C2#2145554711yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingD4×M16×C61C2
socleC22×C61C2×#121
prime radicalC2×#16839957C2
hypercenterD4×M16D61
hypocenterC61#19166
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ D4×M16

Complements

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

NH ≅ G/Ntype
1. #21456194C2semidirect
2. #21455827C2semidirect
4. #21445380C2semidirect
5. #21445352C2semidirect
6. #21445256C2semidirect
7. #21445228C2semidirect
8. #21444571C2semidirect
9. #21444510C2semidirect
10. #21444488C2semidirect
11. D4×M16×C61C2semidirect
16. #16840109C22semidirect
17. #16840109C22semidirect
19. #16840052C22semidirect
20. #16840052C22semidirect
21. #16839967C22semidirect
22. #16839967C22semidirect
35. #16839515C22semidirect
36. #16839513C22semidirect
37. #16839510C22semidirect
38. #16839507C22semidirect
39. #16839505C22semidirect
40. #16839505C22semidirect
43. C61×#436C22semidirect
45. C8×D4×C61C22semidirect
261. D4×M16D61semidirect
296. #436C2×D61semidirect
298. C8×D4C2×D61semidirect
313. C61#19166semidirect

Maximal subgroups

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

subgroupindex
#214561942
#214558272
C2×#168399572
#214453802
#214453522
#214452562
#214452282
#214445712
#214445102
#214444882
D4×M16×C612
#1916661