Finite Group Databasebeta

← #29662865

Subgroups — #29662865

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#21456269C211yesmaximal, characteristic
2#21455850C211yesmaximal, characteristic, complemented
3Q8×#11758194C211yesmaximal, characteristic, complemented
4#21445301C212yesmaximal, complemented
5#21445289C212yesmaximal, complemented
6#21445177C212yesmaximal, complemented
7#21445165C212yesmaximal, complemented
8#21444569C211yesmaximal, characteristic, complemented
9#21444494C211yesmaximal, characteristic, complemented
10#21444487C211yesmaximal, characteristic, complemented
11C2×C61×#460C211yesmaximal, characteristic, complemented
12C2×#14519803—21no—
13C2×#14519803—21no—
14#16840312C2211yescharacteristic
15#16840149—22no—
16#16840138C2212yes—
17#16840134C2212yes—
18#16840080—21no—
19#16840047C2211yescharacteristic, complemented
20#16840047C2211yescharacteristic, complemented
21#16839962C2211yescharacteristic, complemented
22#16839962C2211yescharacteristic, complemented
23#16839901—21no—
24#16839893—22no—
25#16839893C2212yescomplemented
26Q8×#92418C2212yescomplemented
27C4×#11758201—21no—
28C4×#11758201—21no—
29#16839815C2211yescharacteristic
30#16839786—21no—
31#16839786C2211yescharacteristic
32C4×#11758194—21no—
33C4×#11758194C2211yescharacteristic
34#16839514C2212yescomplemented
35#16839512C2212yescomplemented
36#16839507C2212yescomplemented
37#16839506C2212yes—
38#16839506C2212yes—
39#16839505C2212yescomplemented
40C22×SD16×C61—21no—
41C2×#127×C61C2211yescharacteristic
42C2×C4×Q8×C61C2211yescharacteristic
43C61×#460—21no—
44C61×#460—21no—
45C61×#460C2212yes—
46C61×#460C2212yes—
47C2×C4⋊C8×C61C2211yescharacteristic
48C2×#102×C61—21no—
49C2×#11758222—21no—
50#14519803—44no—
51#14519803—44no—
52C2×#11758200—22no—
53C2×#11758200—22no—
54C2×#11758201—21no—
55C2×#11758201—21no—
56C2×#11758199—22no—
57C2×#11758199—22no—
58C2×#11758194—21no—
59C2×#11758194D411yescharacteristic
60C2×#11758194D411yescharacteristic
61#14519777C2311yescharacteristic
62#14519774C2311yescharacteristic
63#14519765—21no—
64#14519751—21no—
65#14519751—21no—
66C4×D244—21no—
67C4×D244—21no—
68#14519708—21no—
69#14519708—21no—
70C4⋊#92418—21no—
71C4⋊#92418—22no—
72C4⋊#92418—22no—
73C4⋊#92418C2311yescharacteristic
74C4⋊#92418C2311yescharacteristic
75C4⋊#92418C2312yes—
76#14519701—21no—
77#14519701—21no—
78#14519697—21no—
79#14519697—21no—
80C22⋊#92418—21no—
81#14519688—22no—
82#14519688D412yes—
83C4×#92418—21no—
84C4×#92418—22no—
85C4×#92418C2312yes—
86C22×Q8×C61D411yescharacteristic
87C22×D4×C61—21no—
88C22×D4×C61—21no—
89C22×D4×C61—21no—
90C2×SD16×C61—22no—
91C2×SD16×C61—22no—
92C2×SD16×C61—44no—
93C22×C8×C61—21no—
94#127×C61—21no—
95#127×C61—21no—
96#127×C61—21no—
97#127×C61C2312yes—
98C4×Q8×C61—21no—
99C4×Q8×C61—21no—
100C4×Q8×C61D411yescharacteristic
101C4×Q8×C61D411yescharacteristic
102C4×Q8×C61C2311yescharacteristic
103C4×Q8×C61C2311yescharacteristic
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61D411yescharacteristic
106C2×C42×C61—21no—
107C2×C42×C61C2311yescharacteristic
108C4⋊C8×C61—21no—
109C4⋊C8×C61C2312yes—
110#102×C61—22no—
111#102×C61—22no—
112C24×C61—41no—
113C2×Q8×C61—21no—
114C2×Q8×C61—22no—
115C2×Q8×C61—41no—
116C2×Q8×C61C2×D411yescharacteristic
117C2×Q8×C61C2×D411yescharacteristic
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C2×D4×C61—22no—
121C2×D4×C61—42no—
122C2×D4×C61—42no—
123C2×D4×C61—41no—
124C2×D4×C61—41no—
125C2×D4×C61—41no—
126C2×D4×C61—42no—
127C2×D4×C61—42no—
128C2×D4×C61—82no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61#4111yescharacteristic
134C2×C22⋊D61—41no—
135#11758222—42no—
136#11758222—42no—
137C2×D244—21no—
138C2×D244—21no—
139C2×D244—21no—
140C2×C4×D61—41no—
141C2×Dic122—21no—
142SD16×C61—44no—
143SD16×C61—44no—
144C2×C8×C61—22no—
145C2×C8×C61—42no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61C2×D411yescharacteristic
152C4⋊C4×C61C2×D411yescharacteristic
153C42×C61—21no—
154C42×C61—21no—
155C42×C61—21no—
156C42×C61—21no—
157C42×C61C2×D411yescharacteristic
158C42×C61C2×D411yescharacteristic
159C42×C61#4111yescharacteristic
160C42×C61C2411yescharacteristic
161#11758201—42no—
162#11758201—42no—
163#11758200—44no—
164#11758200—44no—
165#11758199—44no—
166#11758199—44no—
167#11758198—41no—
168#11758198—41no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171#11758194—22no—
172#11758194—42no—
173#11758194SD1614yescomplemented
174C2×#92418—22no—
175C2×#92418—21no—
176C2×#92418—21no—
177C2×#92418C2×D412yes—
178C2×#92418C2×D412yes—
179C23×C61—42no—
180C23×C61—82no—
181C23×C61—82no—
182C23×C61—82no—
183C23×C61#12111yescharacteristic
184C22×D61—41no—
185Q8×C61—21no—
186Q8×C61—21no—
187Q8×C61—21no—
188Q8×C61—42no—
189Q8×C61AGL(1,ℤ/8)12yescomplemented
190D4×C61—42no—
191D4×C61—42no—
192D4×C61—42no—
193D4×C61—42no—
194D4×C61—82no—
195D4×C61—82no—
196D4×C61—82no—
197D4×C61—82no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—41no—
205C2×C4×C61—42no—
206C2×C4×C61—81no—
207C2×C4×C61#12111yescharacteristic
208C2×C4×C61#12111yescharacteristic
209C2×C4×C61#12111yescharacteristic
210C2×C4×C61C2×SD1612yes—
211C2×C4×C61AGL(1,ℤ/8)12yes—
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C2×#4111yescharacteristic
215C22⋊D61—82no—
216C2×Dic61—41no—
217D244—41no—
218D244—41no—
219D244—41no—
220D244—41no—
221D244—81no—
222C4×D61—82no—
223Dic122—41no—
224Dic122—41no—
225C8×C61—42no—
226#92418—22no—
227#92418—42no—
228#92418—41no—
229#92418—41no—
230#92418C2×SD1614yescomplemented
231#19170—611nomaximal, Sylow
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—82no—
235C22×C61—82no—
236C22×C61—82no—
237C22×C61—164no—
238C22×C61#46012yes—
239C22×C61C2×#12111yescharacteristic
240C2×D61—81no—
241C2×D61—81no—
242C2×D61—81no—
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×#460D6111yescharacteristic, complemented
259Q8×M16—611no—
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×C61—162no—
269C2×C61C2×#46011yescharacteristic
270C2×C61#305511yescharacteristic
271C2×C61#306811yescharacteristic
272D61—161no—
273C2×AGL(1,ℤ/8)—1221no—
274C2×AGL(1,ℤ/8)—1221no—
275C22×SD16—21no—
276#532—611no—
277C2×#127C2×D6111yescharacteristic
278C2×C4×Q8C2×D6111yescharacteristic
279#496—1222no—
280#493—612no—
281#492—612no—
282#481—1221no—
283#475—611no—
284#475—611no—
285#460—21no—
286#460—21no—
287#460—611no—
288#460—611no—
289#460C2×D6112yes—
290#460C2×D6112yes—
291#448—1221no—
292#446—612no—
293#446—1222no—
294C8×Q8—612no—
295C4×SD16—1221no—
296C4×SD16—1221no—
297#431—611no—
298#423—611no—
299#423—1221no—
300C2×C4⋊C8C2×D6111yescharacteristic
301C2×#102—21no—
302C4×M16—611no—
303C4×M16—1221no—
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×Q8C22⋊D6111yescharacteristic
313C22×D4—21no—
314C22×D4—21no—
315C22×D4—21no—
316AGL(1,ℤ/8)—2444no—
317AGL(1,ℤ/8)—2444no—
318C2×SD16—22no—
319C2×SD16—22no—
320C2×SD16—44no—
321C2×SD16—1222no—
322C2×SD16—1222no—
323C2×SD16—1221no—
324C2×SD16—1221no—
325C2×D8—1222no—
326C2×D8—1222no—
327C2×M16—611no—
328C2×M16—611no—
329C2×M16—1221no—
330C22×C8—21no—
331C4⋊Q8—611no—
332#127—21no—
333#127—21no—
334#127—21no—
335#127—611no—
336#127C22×D6112yes—
337#124—1221no—
338#121—1221no—
339#121—1221no—
340C4×Q8—21no—
341C4×Q8—21no—
342C4×Q8C22⋊D6111yescharacteristic
343C4×Q8C22⋊D6111yescharacteristic
344C4×Q8C22×D6111yescharacteristic
345C4×Q8C22×D6111yescharacteristic
346C4×D4—1221no—
347C4×D4—1221no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4C22⋊D6111yescharacteristic
350C2×C42—21no—
351C2×C42C22×D6111yescharacteristic
352#106—1221no—
353#106—1221no—
354C4⋊C8—21no—
355C4⋊C8—612no—
356C4⋊C8—611no—
357C4⋊C8—611no—
358C4⋊C8—1222no—
359C4⋊C8—1222no—
360C4⋊C8—1221no—
361C4⋊C8C22×D6112yes—
362#103—1221no—
363#103—1221no—
364#102—22no—
365#102—22no—
366#102—1221no—
367#102—1221no—
368C22⋊C8—1221no—
369#97—612no—
370#97—1222no—
371C4×C8—612no—
372C4×C8—1222no—
373C4×C8—1221no—
374C24—41no—
375#41—2442no—
376#41—2442no—
377C2×Q8—21no—
378C2×Q8—22no—
379C2×Q8—41no—
380C2×Q8—1221no—
381C2×Q8C2×C22⋊D6111yescharacteristic
382C2×Q8C2×C22⋊D6111yescharacteristic
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4—82no—
394C2×D4—1221no—
395C2×D4—1221no—
396C2×D4—1221no—
397C2×D4—2441no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—2441no—
401C22×C4D4×D6111yescharacteristic
402C22×C4#1175822411yescharacteristic
403C22×C4C2×C22⋊D6111yescharacteristic
404SD16—44no—
405SD16—44no—
406SD16—2444no—
407SD16—2444no—
408SD16—2442no—
409SD16—2442no—
410D8—2444no—
411D8—2444no—
412M16—614no—
413M16—1222no—
414M16—2442no—
415C2×C8—22no—
416C2×C8—42no—
417C2×C8—612no—
418C2×C8—612no—
419C2×C8—1222no—
420C2×C8—1221no—
421C2×C8—1221no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—1221no—
428C4⋊C4—1221no—
429C4⋊C4C2×C22⋊D6111yescharacteristic
430C4⋊C4C2×C22⋊D6111yescharacteristic
431C22⋊C4—2441no—
432C22⋊C4—2441no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42D4×D6111yescharacteristic
438C42#1175822411yescharacteristic
439C42C2×C22⋊D6111yescharacteristic
440C42C23×D6111yescharacteristic
441C23—42no—
442C23—82no—
443C23—82no—
444C23—82no—
445C23—2441no—
446C23#1451974911yescharacteristic
447Q8—21no—
448Q8—21no—
449Q8—21no—
450Q8—42no—
451Q8—2441no—
452Q8—2441no—
453Q8#1451980312yescomplemented
454D4—42no—
455D4—42no—
456D4—42no—
457D4—42no—
458D4—82no—
459D4—82no—
460D4—82no—
461D4—82no—
462D4—4881no—
463D4—4882no—
464D4—2441no—
465D4—2441no—
466D4—2441no—
467D4—2441no—
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#1451974911yescharacteristic
480C2×C4#1451974911yescharacteristic
481C2×C4#1451974911yescharacteristic
482C2×C4SD16×D6112yes—
483C2×C4#1451980412yes—
484C2×C4C2×D4×D6111yescharacteristic
485C2×C4C22×C22⋊D6111yescharacteristic
486C2×C4C2×#1175822411yescharacteristic
487C8—42no—
488C8—614no—
489C8—1222no—
490C8—2441no—
491C8—2442no—
492C8—2441no—
493C22—21no—
494C22—21no—
495C22—82no—
496C22—82no—
497C22—82no—
498C22—164no—
499C22—4881no—
500C22—4881no—
501C22—4881no—
502C22#1683996012yes—
503C22C2×#1451974911yescharacteristic
504C4—21no—
505C4—22no—
506C4—21no—
507C4—21no—
508C4—41no—
509C4—41no—
510C4—41no—
511C4—4881no—
512C4#1683996012yes—
513C4C2×SD16×D6111yescharacteristic
514C4C2×#1451980311yescharacteristic
515C4C2×#1451980411yescharacteristic
516C4#1684056011yescharacteristic
517C2—22no—
518C2—162no—
519C2—9761no—
520C2C2×#1683996011yescharacteristic
521C2#2145543411yescharacteristic
522C2#2145554411yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#460C2
socleC22×C61C2×#121
prime radical#21456269C2
hypercenterC2×#460D61
hypocenterC61#19170
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
2. #21455850C2semidirect
3. Q8×#11758194C2semidirect
4. #21445301C2semidirect
5. #21445289C2semidirect
6. #21445177C2semidirect
7. #21445165C2semidirect
8. #21444569C2semidirect
9. #21444494C2semidirect
10. #21444487C2semidirect
11. C2×C61×#460C2semidirect
19. #16840047C22semidirect
20. #16840047C22semidirect
21. #16839962C22semidirect
22. #16839962C22semidirect
25. #16839893C22semidirect
26. Q8×#92418C22semidirect
34. #16839514C22semidirect
35. #16839512C22semidirect
36. #16839507C22semidirect
39. #16839505C22semidirect
173. #11758194SD16semidirect
189. Q8×C61AGL(1,ℤ/8)semidirect
230. #92418C2×SD16semidirect
258. C2×#460D61semidirect
310. C61#19170semidirect
453. Q8#14519803semidirect

Maximal subgroups

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

subgroupindex
#214562692
#214558502
Q8×#117581942
#214453012
#214452892
#214451772
#214451652
#214445692
#214444942
#214444872
C2×C61×#4602
#1917061