Finite Group Databasebeta

← #29662856

Subgroups — #29662856

Order 15616.

728 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456213C211yesmaximal, characteristic, complemented
2#21455819C211yesmaximal, characteristic
3C2×#16839967C211yesmaximal, characteristic, complemented
4#21445380C212yesmaximal, complemented
5#21445366C212yesmaximal, complemented
6#21445256C212yesmaximal, complemented
7#21445242C212yesmaximal, complemented
8#21444571C211yesmaximal, characteristic, complemented
9#21444505C211yesmaximal, characteristic, complemented
10#21444497C211yesmaximal, characteristic, complemented
11D4×M16×C61C211yesmaximal, characteristic, complemented
12C22×#11758200—21no—
13#16840312C2211yescharacteristic
14C2×#14519777C2211yescharacteristic
15#16840122—22no—
16#16840113C2212yescomplemented
17#16840109C2212yescomplemented
18#16840075—21no—
19#16840042C2211yescharacteristic
20#16840042C2211yescharacteristic
21#16839967—21no—
22#16839967—21no—
23#16839967C2212yes—
24#16839967C2212yes—
25#16839957C2211yescharacteristic
26#16839957C2211yescharacteristic
27#16839896—21no—
28C4×#11758199—21no—
29C4×#11758199—21no—
30#16839815C2211yescharacteristic
31C2×C4⋊#92418C2211yescharacteristic
32#16839764—21no—
33#16839764—21no—
34C2×#14519700—21no—
35#16839515C2212yescomplemented
36#16839513C2212yescomplemented
37#16839509C2212yescomplemented
38#16839507C2212yescomplemented
39#16839507C2212yescomplemented
40#16839505C2212yescomplemented
41C22×M16×C61—21no—
42C2×C4×D4×C61C2211yescharacteristic
43C61×#436C2212yes—
44C61×#435—22no—
45C8×D4×C61C2212yes—
46C61×#423C2211yescharacteristic
47C61×#407—21no—
48C4×M16×C61C2211yescharacteristic
49C2×#11758222—21no—
50C22×Dic122—21no—
51C2×#11758200—22no—
52C2×#11758200—22no—
53C2×#11758200—44no—
54C2×#11758199—21no—
55C2×#11758199—21no—
56C22×#92418—21no—
57#14519777—21no—
58#14519777—21no—
59#14519777—21no—
60#14519777C2312yes—
61#14519777C2311yescharacteristic
62#14519774C2311yescharacteristic
63#14519765—21no—
64#14519751—21no—
65#14519751—21no—
66C4×D244—21no—
67C4×D244—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70#14519710—21no—
71#14519710—21no—
72C4⋊#92418—21no—
73C4⋊#92418C2311yescharacteristic
74C4⋊#92418C2311yescharacteristic
75C4⋊#92418C2312yes—
76#14519702—22no—
77#14519702—22no—
78#14519700—22no—
79#14519700—22no—
80#14519699—22no—
81#14519699—22no—
82#14519698—21no—
83#14519698—21no—
84#14519697—21no—
85#14519697—21no—
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—
112C2×D4×C61—22no—
113C2×D4×C61—21no—
114C2×D4×C61—41no—
115C2×D4×C61C2×D411yescharacteristic
116C2×D4×C61C2×D411yescharacteristic
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—42no—
123C22×C4×C61—42no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61#4111yescharacteristic
127C2×C22⋊D61—41no—
128C22×Dic61—41no—
129#11758222—42no—
130#11758222—42no—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244—21no—
134C2×C4×D61—41no—
135C2×Dic122—22no—
136C2×Dic122—21no—
137C2×Dic122—42no—
138C2×Dic122—42no—
139C2×Dic122—41no—
140M16×C61—22no—
141M16×C61—22no—
142M16×C61—44no—
143M16×C61—42no—
144C2×C8×C61—22no—
145C2×C8×C61—44no—
146C2×C8×C61C2×D412yes—
147C2×C8×C61C2×D412yes—
148C4⋊C4×C61—21no—
149C4⋊C4×C61C2×D411yescharacteristic
150C4⋊C4×C61C2×D411yescharacteristic
151C22⋊C4×C61—21no—
152C22⋊C4×C61—21no—
153C22⋊C4×C61—21no—
154C22⋊C4×C61—21no—
155C42×C61C2×D411yescharacteristic
156C42×C61C2×D411yescharacteristic
157C42×C61#4111yescharacteristic
158C42×C61C2411yescharacteristic
159#11758200—44no—
160#11758200—44no—
161#11758199—42no—
162#11758199—42no—
163#11758198—41no—
164#11758198—41no—
165C4⋊Dic61—21no—
166C4⋊Dic61—22no—
167C4⋊Dic61—22no—
168C4⋊Dic61—21no—
169C4⋊Dic61—41no—
170C4⋊Dic61—41no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418—22no—
174C2×#92418—42no—
175C23×C61—21no—
176C23×C61—21no—
177C23×C61—42no—
178C23×C61—42no—
179C23×C61—81no—
180C23×C61#12111yescharacteristic
181C22×D61—41no—
182D4×C61—21no—
183D4×C61—21no—
184D4×C61—21no—
185D4×C61—42no—
186D4×C61AGL(1,ℤ/8)12yes—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—41no—
192C2×C4×C61—42no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61#12111yescharacteristic
200C2×C4×C61#12111yescharacteristic
201C2×C4×C61#12111yescharacteristic
202C2×C4×C61#13512yes—
203C2×C4×C61AGL(1,ℤ/8)12yes—
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C2×#4111yescharacteristic
207C22⋊D61—82no—
208C2×Dic61—42no—
209C2×Dic61—41no—
210C2×Dic61—82no—
211D244—41no—
212D244—41no—
213D244—41no—
214D244—41no—
215D244—81no—
216C4×D61—82no—
217Dic122—41no—
218Dic122—42no—
219Dic122—41no—
220Dic122—42no—
221Dic122—82no—
222C8×C61—22no—
223C8×C61—22no—
224C8×C61—42no—
225#92418—42no—
226#92418—41no—
227#92418—41no—
228#19169—611nomaximal, Sylow
229C22×C61—21no—
230C22×C61—21no—
231C22×C61—22no—
232C22×C61—41no—
233C22×C61—41no—
234C22×C61—41no—
235C22×C61—41no—
236C22×C61—41no—
237C22×C61—84no—
238C22×C61#46312yes—
239C22×C61C2×#12111yescharacteristic
240C2×D61—81no—
241C2×D61—81no—
242C2×D61—81no—
243C4×C61—21no—
244C4×C61—21no—
245C4×C61—21no—
246C4×C61—41no—
247C4×C61—41no—
248C4×C61#46312yes—
249C4×C61C2×#13511yescharacteristic
250C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
251C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
252C4×C61#57511yescharacteristic
253Dic61—81no—
254Dic61—82no—
255#3141—611no—
256#3100—611no—
257C2×#461—611no—
258D4×M16D6111yescharacteristic, complemented
259#1701—612no—
260#1699—612no—
261#1681—612no—
262#1679—612no—
263#1565—611no—
264#1549—611no—
265#1547—611no—
266C2×C61—22no—
267C2×C61—42no—
268C2×C61—81no—
269C2×C61C2×#46311yescharacteristic
270C2×C61#305511yescharacteristic
271C2×C61#306911yescharacteristic
272D61—161no—
273C22×SD16—1221no—
274C22×M16—21no—
275#532—611no—
276C2×C4⋊Q8—611no—
277C2×C4×D4C2×D6111yescharacteristic
278#489—1222no—
279#487—612no—
280#486—612no—
281#480—1221no—
282#474—611no—
283#474—611no—
284#461—612no—
285#461—612no—
286#461—1221no—
287#461—1221no—
288#459—611no—
289#459—611no—
290#447—1221no—
291C4×D8—1221no—
292C4×D8—1221no—
293#436C2×D6112yes—
294#435—22no—
295C8×D4C2×D6112yes—
296#431—611no—
297#423C2×D6111yescharacteristic
298C2×C4⋊C8—611no—
299#417—1221no—
300#417—1221no—
301C2×#103—1221no—
302#407—21no—
303C4×M16C2×D6111yescharacteristic
304#335—612no—
305#334—612no—
306#332—612no—
307#332—612no—
308#332—612no—
309#331—612no—
310C61#1916911yescharacteristic, complemented, Sylow
311C2×#41—1221no—
312C22×Q8—1221no—
313C22×D4D24411yescharacteristic
314C23×C4—21no—
315C2×SD16—1222no—
316C2×SD16—1222no—
317C2×SD16—2444no—
318C2×D8—1221no—
319C2×D8—1221no—
320C2×M16—21no—
321C2×M16—22no—
322C2×M16—44no—
323C2×M16C22⋊D6111yescharacteristic
324C2×M16C22⋊D6111yescharacteristic
325C22×C8—22no—
326C22×C8—1221no—
327C4⋊Q8—612no—
328C4⋊Q8—611no—
329C4⋊Q8—1221no—
330C4⋊Q8—1221no—
331C4⋊Q8—1221no—
332#127—611no—
333#124—1221no—
334#121—1221no—
335#121—1221no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4—1221no—
339C4×D4—1221no—
340C4×D4D24411yescharacteristic
341C4×D4D24411yescharacteristic
342C4×D4C22×D6111yescharacteristic
343C4×D4C22×D6111yescharacteristic
344C2×C4⋊C4—1221no—
345C2×C4⋊C4—1221no—
346C2×C4⋊C4D24411yescharacteristic
347C2×C22⋊C4—21no—
348C2×C42C22×D6111yescharacteristic
349#107—1221no—
350#107—1221no—
351C4⋊C8—21no—
352C4⋊C8—612no—
353C4⋊C8—611no—
354C4⋊C8—611no—
355C4⋊C8—1221no—
356C4⋊C8C22×D6112yes—
357#103—1222no—
358#103—1222no—
359#103—1222no—
360#103—1222no—
361#102—1221no—
362#102—1221no—
363#102—1222no—
364#102—1222no—
365#102—1221no—
366#102—1221no—
367C22⋊C8—22no—
368C22⋊C8—22no—
369C22⋊C8—1221no—
370#97C22⋊D6112yes—
371C4×C8—1221no—
372C4×C8C22×D6112yes—
373C24—21no—
374#41—2442no—
375#41—2442no—
376C2×Q8—1221no—
377C2×Q8—1222no—
378C2×Q8—2442no—
379C2×Q8—2442no—
380C2×Q8—2441no—
381C2×D4—22no—
382C2×D4—21no—
383C2×D4—41no—
384C2×D4—1221no—
385C2×D4—1221no—
386C2×D4—1221no—
387C2×D4—2441no—
388C2×D4C2×D24411yescharacteristic
389C2×D4C2×D24411yescharacteristic
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—2441no—
398C22×C4—2441no—
399C22×C4C2×D24411yescharacteristic
400C22×C4#1175822211yescharacteristic
401C22×C4C2×C22⋊D6111yescharacteristic
402SD16—2444no—
403SD16—2444no—
404D8—2442no—
405D8—2442no—
406M16—22no—
407M16—22no—
408M16—44no—
409M16—42no—
410C2×C8—22no—
411C2×C8—44no—
412C2×C8—1221no—
413C2×C8—1221no—
414C2×C8—1222no—
415C2×C8—2442no—
416C2×C8C2×C22⋊D6112yes—
417C2×C8C2×C22⋊D6112yes—
418C4⋊C4—21no—
419C4⋊C4—1222no—
420C4⋊C4—1221no—
421C4⋊C4—1222no—
422C4⋊C4—1221no—
423C4⋊C4—2441no—
424C4⋊C4—2441no—
425C4⋊C4C2×D24411yescharacteristic
426C4⋊C4C2×D24411yescharacteristic
427C22⋊C4—21no—
428C22⋊C4—21no—
429C22⋊C4—21no—
430C22⋊C4—21no—
431C22⋊C4—2441no—
432C22⋊C4—2441no—
433C42C2×D24411yescharacteristic
434C42#1175822211yescharacteristic
435C42C2×C22⋊D6111yescharacteristic
436C42C23×D6111yescharacteristic
437C23—21no—
438C23—21no—
439C23—42no—
440C23—42no—
441C23—81no—
442C23—2441no—
443C23#1451975111yescharacteristic
444Q8—4882no—
445Q8—2441no—
446Q8—2441no—
447Q8—2442no—
448Q8—2442no—
449D4—21no—
450D4—21no—
451D4—21no—
452D4—42no—
453D4—4881no—
454D4—4882no—
455D4—2441no—
456D4—2441no—
457D4—2441no—
458D4—2441no—
459D4#1451980512yes—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—4882no—
473C2×C4—4882no—
474C2×C4—2441no—
475C2×C4—2442no—
476C2×C4#1451975111yescharacteristic
477C2×C4#1451975111yescharacteristic
478C2×C4#1451975111yescharacteristic
479C2×C4#1451979712yes—
480C2×C4#1451980312yes—
481C2×C4C22×C22⋊D6111yescharacteristic
482C2×C4C22×D24411yescharacteristic
483C2×C4C2×#1175822211yescharacteristic
484C8—22no—
485C8—22no—
486C8—42no—
487C8—2442no—
488C8—2441no—
489C8—2441no—
490C22—21no—
491C22—21no—
492C22—22no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—84no—
499C22—4881no—
500C22—4881no—
501C22—4881no—
502C22#1683997712yes—
503C22C2×#1451975111yescharacteristic
504C4—21no—
505C4—21no—
506C4—21no—
507C4—41no—
508C4—41no—
509C4—4882no—
510C4—4881no—
511C4#1683997712yes—
512C4C2×#1451979711yescharacteristic
513C4C2×#1451980311yescharacteristic
514C4C2×#1451980511yescharacteristic
515C4#1684056211yescharacteristic
516C2—22no—
517C2—42no—
518C2—81no—
519C2—9761no—
520C2C2×#1683997711yescharacteristic
521C2#2145543811yescharacteristic
522C2#2145555611yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingD4×M16×C61C2
socleC22×C61C2×#121
prime radical#21455819C2
hypercenterD4×M16D61
hypocenterC61#19169
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. #21456213C2semidirect
3. C2×#16839967C2semidirect
4. #21445380C2semidirect
5. #21445366C2semidirect
6. #21445256C2semidirect
7. #21445242C2semidirect
8. #21444571C2semidirect
9. #21444505C2semidirect
10. #21444497C2semidirect
11. D4×M16×C61C2semidirect
16. #16840113C22semidirect
17. #16840109C22semidirect
35. #16839515C22semidirect
36. #16839513C22semidirect
37. #16839509C22semidirect
38. #16839507C22semidirect
39. #16839507C22semidirect
40. #16839505C22semidirect
258. D4×M16D61semidirect
310. C61#19169semidirect

Maximal subgroups

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

subgroupindex
#214562132
#214558192
C2×#168399672
#214453802
#214453662
#214452562
#214452422
#214445712
#214445052
#214444972
D4×M16×C612
#1916961