Finite Group Databasebeta

← #29662869

Subgroups — #29662869

Order 15616.

766 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#21456271C211yesmaximal, characteristic
2#21455850C211yesmaximal, characteristic, complemented
3C2×#16839962C211yesmaximal, characteristic, complemented
4#21445303C212yesmaximal, complemented
5#21445291C212yesmaximal, complemented
6#21445179C212yesmaximal, complemented
7#21445167C212yesmaximal, complemented
8#21444571C211yesmaximal, characteristic, complemented
9#21444497C211yesmaximal, characteristic, complemented
10#21444488C211yesmaximal, characteristic, complemented
11Q8×M16×C61C211yesmaximal, characteristic, complemented
12C2×#14519805—21no—
13C2×#14519805—21no—
14C22×#11758201—21no—
15#16840312C2211yescharacteristic
16C2×#14519774C2211yescharacteristic
17#16840151—22no—
18#16840139C2212yes—
19#16840135C2212yes—
20#16840080—21no—
21#16840047C2211yescharacteristic, complemented
22#16840047C2211yescharacteristic, complemented
23#16839962—21no—
24#16839962—21no—
25#16839962C2212yes—
26#16839962C2212yes—
27#16839962C2211yescharacteristic, complemented
28#16839962C2211yescharacteristic, complemented
29#16839901—21no—
30C4×#11758201—21no—
31C4×#11758201—21no—
32#16839815C2211yescharacteristic
33C2×C4⋊#92418C2211yescharacteristic
34C2×#14519697—21no—
35#16839515C2212yescomplemented
36#16839513C2212yescomplemented
37#16839509C2212yescomplemented
38#16839505C2212yes—
39#16839505C2212yes—
40#16839505C2212yescomplemented
41C2×C4×Q8×C61C2211yescharacteristic
42C61×#446—22no—
43C61×#446C2212yescomplemented
44C8×Q8×C61C2212yescomplemented
45C61×#423—21no—
46C61×#423C2211yescharacteristic
47C4×M16×C61—21no—
48C4×M16×C61C2211yescharacteristic
49C2×#11758222—21no—
50C22×D244—21no—
51C22×D244—21no—
52C22×D244—21no—
53#14519805—44no—
54#14519805—44no—
55C2×#11758190—22no—
56C2×#11758190—22no—
57C2×#11758201—22no—
58C2×#11758201—22no—
59C2×#11758201—21no—
60C2×#11758201—21no—
61C2×#11758201—44no—
62C2×D488—22no—
63C2×D488—22no—
64C22×#92418—21no—
65#14519777C2311yescharacteristic
66#14519774—21no—
67#14519774—21no—
68#14519774—21no—
69#14519774C2312yes—
70#14519774C2311yescharacteristic
71#14519765—21no—
72#14519751—21no—
73#14519751—21no—
74C4×D244—21no—
75C4×D244—21no—
76#14519708—21no—
77#14519708—21no—
78C4⋊#92418—21no—
79C4⋊#92418C2312yes—
80C4⋊#92418C2311yescharacteristic
81C4⋊#92418C2311yescharacteristic
82#14519701—21no—
83#14519701—21no—
84#14519697—22no—
85#14519697—22no—
86#14519697—21no—
87#14519697—21no—
88C22⋊#92418—21no—
89C4×#92418—21no—
90C22×Q8×C61D411yescharacteristic
91C2×M16×C61—21no—
92C2×M16×C61D411yescharacteristic
93C2×M16×C61D411yescharacteristic
94C4×Q8×C61—21no—
95C4×Q8×C61—21no—
96C4×Q8×C61D411yescharacteristic
97C4×Q8×C61D411yescharacteristic
98C4×Q8×C61C2311yescharacteristic
99C4×Q8×C61C2311yescharacteristic
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C42×C61—21no—
103C2×C42×C61C2311yescharacteristic
104C4⋊C8×C61—22no—
105C4⋊C8×C61—22no—
106C4⋊C8×C61—21no—
107C4⋊C8×C61C2312yes—
108#97×C61—22no—
109#97×C61D412yes—
110C4×C8×C61—22no—
111C4×C8×C61C2312yes—
112C23×D61—41no—
113C2×Q8×C61—21no—
114C2×Q8×C61—22no—
115C2×Q8×C61—41no—
116C2×Q8×C61C2×D411yescharacteristic
117C2×Q8×C61C2×D411yescharacteristic
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61#4111yescharacteristic
123C2×C22⋊D61—41no—
124#11758222—42no—
125#11758222—42no—
126C2×D244—22no—
127C2×D244—22no—
128C2×D244—21no—
129C2×D244—21no—
130C2×D244—22no—
131C2×D244—21no—
132C2×D244—42no—
133C2×D244—42no—
134C2×D244—42no—
135C2×D244—42no—
136C2×D244—41no—
137C2×D244—41no—
138C2×D244—41no—
139C2×D244—82no—
140C2×C4×D61—41no—
141C2×Dic122—21no—
142M16×C61—22no—
143M16×C61—42no—
144M16×C61SD1614yescomplemented
145C2×C8×C61—22no—
146C2×C8×C61C2×D412yes—
147C2×C8×C61C2×D412yes—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61C2×D411yescharacteristic
154C4⋊C4×C61C2×D411yescharacteristic
155C42×C61—21no—
156C42×C61—21no—
157C42×C61—21no—
158C42×C61—21no—
159C42×C61C2×D411yescharacteristic
160C42×C61C2×D411yescharacteristic
161C42×C61#4111yescharacteristic
162C42×C61C2411yescharacteristic
163#11758201—42no—
164#11758201—44no—
165#11758201—42no—
166#11758201—44no—
167#11758198—41no—
168#11758198—41no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418—22no—
174C2×#92418—42no—
175D488—44no—
176D488—44no—
177#11758190—44no—
178#11758190—44no—
179C23×C61#12111yescharacteristic
180C22×D61—41no—
181C22×D61—42no—
182C22×D61—82no—
183C22×D61—82no—
184C22×D61—82no—
185Q8×C61—21no—
186Q8×C61—21no—
187Q8×C61—21no—
188Q8×C61—42no—
189Q8×C61AGL(1,ℤ/8)12yescomplemented
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—41no—
197C2×C4×C61—42no—
198C2×C4×C61—81no—
199C2×C4×C61#12111yescharacteristic
200C2×C4×C61#12111yescharacteristic
201C2×C4×C61#12111yescharacteristic
202C2×C4×C61C2×SD1612yes—
203C2×C4×C61AGL(1,ℤ/8)12yes—
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C2×#4111yescharacteristic
207C22⋊D61—82no—
208C2×Dic61—41no—
209D244—42no—
210D244—41no—
211D244—42no—
212D244—41no—
213D244—42no—
214D244—41no—
215D244—41no—
216D244—42no—
217D244—81no—
218D244—82no—
219D244—82no—
220D244—82no—
221D244—82no—
222C4×D61—82no—
223Dic122—41no—
224Dic122—41no—
225C8×C61—22no—
226C8×C61—42no—
227C8×C61C2×SD1614yescomplemented
228#92418—42no—
229#92418—41no—
230#92418—41no—
231#19170—611nomaximal, Sylow
232C22×C61—21no—
233C22×C61—21no—
234C22×C61#46012yes—
235C22×C61C2×#12111yescharacteristic
236C2×D61—81no—
237C2×D61—82no—
238C2×D61—82no—
239C2×D61—81no—
240C2×D61—82no—
241C2×D61—81no—
242C2×D61—164no—
243C4×C61—21no—
244C4×C61—22no—
245C4×C61—21no—
246C4×C61—21no—
247C4×C61—41no—
248C4×C61—41no—
249C4×C61—41no—
250C4×C61#46012yes—
251C4×C61C22×SD1611yescharacteristic
252C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
253C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
254C4×C61#57511yescharacteristic
255Dic61—81no—
256#3151—611no—
257#3103—611no—
258C2×#460—611no—
259Q8×M16D6111yescharacteristic, complemented
260#1688—612no—
261#1686—612no—
262#1668—612no—
263#1666—612no—
264#1565—611no—
265#1547—611no—
266#1545—611no—
267C2×C61—22no—
268C2×C61C2×#46011yescharacteristic
269C2×C61#305511yescharacteristic
270C2×C61#306811yescharacteristic
271D61—161no—
272D61—162no—
273C2×AGL(1,ℤ/8)—1221no—
274C2×AGL(1,ℤ/8)—1221no—
275C22×SD16—1221no—
276#532—611no—
277C2×#127—611no—
278C2×C4×Q8C2×D6111yescharacteristic
279#496—1222no—
280#493—612no—
281#492—612no—
282#481—1221no—
283#475—611no—
284#475—611no—
285#460—612no—
286#460—612no—
287#460—611no—
288#460—611no—
289#460—1221no—
290#460—1221no—
291#448—1221no—
292#446—22no—
293#446C2×D6112yescomplemented
294C8×Q8C2×D6112yescomplemented
295C4×SD16—1221no—
296C4×SD16—1221no—
297#431—611no—
298#423—21no—
299#423C2×D6111yescharacteristic
300C2×C4⋊C8—611no—
301C2×#102—1221no—
302C4×M16—21no—
303C4×M16C2×D6111yescharacteristic
304#335—612no—
305#334—612no—
306#332—612no—
307#331—612no—
308#331—612no—
309#331—612no—
310C61#1917011yescharacteristic, complemented, Sylow
311C2×#41—1221no—
312C22×Q8D24411yescharacteristic
313C22×D4—1221no—
314C22×D4—1221no—
315C22×D4—1221no—
316AGL(1,ℤ/8)—2444no—
317AGL(1,ℤ/8)—2444no—
318C2×SD16—1222no—
319C2×SD16—1222no—
320C2×SD16—1221no—
321C2×SD16—1221no—
322C2×SD16—1222no—
323C2×SD16—1222no—
324C2×SD16—2444no—
325C2×D8—1222no—
326C2×D8—1222no—
327C2×M16—21no—
328C2×M16C22⋊D6111yescharacteristic
329C2×M16C22⋊D6111yescharacteristic
330C22×C8—1221no—
331C4⋊Q8—611no—
332#127—612no—
333#127—611no—
334#127—1221no—
335#127—1221no—
336#127—1221no—
337#124—1221no—
338#121—1221no—
339#121—1221no—
340C4×Q8—21no—
341C4×Q8—21no—
342C4×Q8D24411yescharacteristic
343C4×Q8D24411yescharacteristic
344C4×Q8C22×D6111yescharacteristic
345C4×Q8C22×D6111yescharacteristic
346C4×D4—1221no—
347C4×D4—1221no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4D24411yescharacteristic
350C2×C42—21no—
351C2×C42C22×D6111yescharacteristic
352#106—1221no—
353#106—1221no—
354C4⋊C8—22no—
355C4⋊C8—22no—
356C4⋊C8—21no—
357C4⋊C8—611no—
358C4⋊C8—611no—
359C4⋊C8—612no—
360C4⋊C8—1221no—
361C4⋊C8C22×D6112yes—
362#103—1221no—
363#103—1221no—
364#102—1222no—
365#102—1222no—
366#102—1221no—
367#102—1221no—
368C22⋊C8—1221no—
369#97—22no—
370#97C22⋊D6112yes—
371C4×C8—22no—
372C4×C8—1221no—
373C4×C8C22×D6112yes—
374C24—2441no—
375#41—2442no—
376#41—2442no—
377C2×Q8—21no—
378C2×Q8—22no—
379C2×Q8—41no—
380C2×Q8—1221no—
381C2×Q8C2×D24411yescharacteristic
382C2×Q8C2×D24411yescharacteristic
383C2×D4—1221no—
384C2×D4—1222no—
385C2×D4—1221no—
386C2×D4—1222no—
387C2×D4—1221no—
388C2×D4—1222no—
389C2×D4—4882no—
390C2×D4—2442no—
391C2×D4—2442no—
392C2×D4—2441no—
393C2×D4—2441no—
394C2×D4—2441no—
395C2×D4—2441no—
396C2×D4—2442no—
397C2×D4—2442no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—2441no—
401C22×C4C2×D24411yescharacteristic
402C22×C4#1175822211yescharacteristic
403C22×C4C2×C22⋊D6111yescharacteristic
404SD16—2444no—
405SD16—2442no—
406SD16—2444no—
407SD16—2442no—
408SD16—2444no—
409SD16—2444no—
410D8—2444no—
411D8—2444no—
412M16—22no—
413M16—42no—
414M16#1175820114yescomplemented
415C2×C8—22no—
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—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—1221no—
428C4⋊C4—1221no—
429C4⋊C4C2×D24411yescharacteristic
430C4⋊C4C2×D24411yescharacteristic
431C22⋊C4—2441no—
432C22⋊C4—2441no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42C2×D24411yescharacteristic
438C42#1175822211yescharacteristic
439C42C2×C22⋊D6111yescharacteristic
440C42C23×D6111yescharacteristic
441C23—4882no—
442C23—4882no—
443C23—4882no—
444C23—2441no—
445C23—2442no—
446C23#1451975111yescharacteristic
447Q8—21no—
448Q8—21no—
449Q8—21no—
450Q8—42no—
451Q8—2441no—
452Q8—2441no—
453Q8#1451980512yescomplemented
454D4—4881no—
455D4—4882no—
456D4—4882no—
457D4—4882no—
458D4—4882no—
459D4—4882no—
460D4—2441no—
461D4—2442no—
462D4—2441no—
463D4—2442no—
464D4—2442no—
465D4—2441no—
466D4—2441no—
467D4—2442no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—81no—
477C2×C4—4882no—
478C2×C4—2441no—
479C2×C4#1451975111yescharacteristic
480C2×C4#1451975111yescharacteristic
481C2×C4#1451975111yescharacteristic
482C2×C4C2×#1175820112yes—
483C2×C4#1451980212yes—
484C2×C4C22×C22⋊D6111yescharacteristic
485C2×C4C22×D24411yescharacteristic
486C2×C4C2×#1175822211yescharacteristic
487C8—22no—
488C8—42no—
489C8—2442no—
490C8—2441no—
491C8—2441no—
492C8C2×#1175820114yescomplemented
493C22—21no—
494C22—21no—
495C22—9764no—
496C22—4882no—
497C22—4881no—
498C22—4881no—
499C22—4882no—
500C22—4882no—
501C22—4881no—
502C22#1683996212yes—
503C22C2×#1451975111yescharacteristic
504C4—21no—
505C4—22no—
506C4—21no—
507C4—21no—
508C4—41no—
509C4—41no—
510C4—41no—
511C4—4881no—
512C4#1683996212yes—
513C4C22×#1175820111yescharacteristic
514C4C2×#1451980211yescharacteristic
515C4C2×#1451980511yescharacteristic
516C4#1684056211yescharacteristic
517C2—22no—
518C2—9762no—
519C2—9761no—
520C2C2×#1683996211yescharacteristic
521C2#2145543811yescharacteristic
522C2#2145554711yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingQ8×M16×C61C2
socleC22×C61C2×#121
prime radical#21456271C2
hypercenterQ8×M16D61
hypocenterC61#19170
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
2. #21455850C2semidirect
3. C2×#16839962C2semidirect
4. #21445303C2semidirect
5. #21445291C2semidirect
6. #21445179C2semidirect
7. #21445167C2semidirect
8. #21444571C2semidirect
9. #21444497C2semidirect
10. #21444488C2semidirect
11. Q8×M16×C61C2semidirect
21. #16840047C22semidirect
22. #16840047C22semidirect
27. #16839962C22semidirect
28. #16839962C22semidirect
35. #16839515C22semidirect
36. #16839513C22semidirect
37. #16839509C22semidirect
40. #16839505C22semidirect
43. C61×#446C22semidirect
44. C8×Q8×C61C22semidirect
144. M16×C61SD16semidirect
189. Q8×C61AGL(1,ℤ/8)semidirect
227. C8×C61C2×SD16semidirect
259. Q8×M16D61semidirect
293. #446C2×D61semidirect
294. C8×Q8C2×D61semidirect
310. C61#19170semidirect
414. M16#11758201semidirect
453. Q8#14519805semidirect
492. C8C2×#11758201semidirect

Maximal subgroups

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

subgroupindex
#214562712
#214558502
C2×#168399622
#214453032
#214452912
#214451792
#214451672
#214445712
#214444972
#214444882
Q8×M16×C612
#1917061