Finite Group Databasebeta

← #27153503

Subgroups — #27153503

Order 12416.

464 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261547C211yesmaximal, characteristic, complemented
2#20261542C211yesmaximal, characteristic, complemented
3#20261540C211yesmaximal, characteristic, complemented
4#20261537C211yesmaximal, characteristic, complemented
5#20261124C211yesmaximal, characteristic, complemented
6#20261006C211yesmaximal, characteristic, complemented
7#20261005C211yesmaximal, characteristic, complemented
8#20260999C211yesmaximal, characteristic, complemented
9#20260998C211yesmaximal, characteristic, complemented
10#20260914C211yesmaximal, characteristic, complemented
11#20260909C211yesmaximal, characteristic, complemented
12#20260904C211yesmaximal, characteristic, complemented
13#20260904C211yesmaximal, characteristic, complemented
14#20260858C211yesmaximal, characteristic, complemented
15#444×C97C211yesmaximal, characteristic, complemented
16#15690857C2211yescharacteristic, complemented
17#15690856C2211yescharacteristic
18#15690856C2211yescharacteristic, complemented
19#15690856C2211yescharacteristic, complemented
20#15690843—21no—
21#15690843C2211yescharacteristic, complemented
22#15690842—21no—
23#15690840C2211yescharacteristic, complemented
24#15690838—21no—
25#15690838—21no—
26#15690838—21no—
27#15690838C2211yescharacteristic
28#15690838C2211yescharacteristic, complemented
29#15690838C2211yescharacteristic, complemented
30#15690837—21no—
31#15690837—21no—
32#15690835C2211yescharacteristic
33#15690835C2211yescharacteristic, complemented
34#15690835C2211yescharacteristic, complemented
35#15690834—21no—
36#15690833—21no—
37#15690832—21no—
38#15690832—21no—
39#15690832C2211yescharacteristic
40#15690831—21no—
41C2×#13095500—21no—
42C2×#13095500C2211yescharacteristic, complemented
43C2×#13095500C2211yescharacteristic, complemented
44SD16×D97—21no—
45C2×D776—21no—
46C2×D776C2211yescharacteristic, complemented
47C2×D776C2211yescharacteristic, complemented
48D8×D97—21no—
49#15690816C2211yescharacteristic, complemented
50#15690816C2211yescharacteristic, complemented
51#15690807C2211yescharacteristic, complemented
52#15690798C2211yescharacteristic, complemented
53#15690746C2211yescharacteristic, complemented
54#15690738C2211yescharacteristic, complemented
55#15690738C2211yescharacteristic, complemented
56#15690728C2211yescharacteristic, complemented
57#15690726C2211yescharacteristic, complemented
58#15690726C2211yescharacteristic, complemented
59#15690726C2211yescharacteristic, complemented
60#135×C97C2211yescharacteristic
61#131×C97C2211yescharacteristic, complemented
62#131×C97C2211yescharacteristic, complemented
63#108×C97C2211yescharacteristic, complemented
64#104×C97C2211yescharacteristic, complemented
65#104×C97C2211yescharacteristic, complemented
66#97×C97C2211yescharacteristic, complemented
67#13095512—21no—
68#13095512—21no—
69#13095512—21no—
70#13095511—21no—
71#13095511—21no—
72#13095511—21no—
73#13095511—21no—
74#13095511C2311yescharacteristic
75#13095511C2311yescharacteristic
76#13095510—21no—
77#13095510—41no—
78C2×Dic194—21no—
79C2×Dic194C2311yescharacteristic
80Q8×D97—21no—
81C2×D388—21no—
82C2×D388—21no—
83C2×D388—21no—
84C2×D388—41no—
85C2×D388C2311yescharacteristic
86C2×D388C2311yescharacteristic
87C2×D388C2311yescharacteristic
88D4×D97—21no—
89D4×D97—21no—
90D4×D97—21no—
91D4×D97—41no—
92D4×D97—41no—
93D4×D97—41no—
94Dic388—21no—
95Dic388D411yescharacteristic, complemented
96#13095501—21no—
97#13095500—21no—
98#13095500—21no—
99#13095500—21no—
100#13095500—21no—
101#13095500—21no—
102#13095500—21no—
103#13095500—21no—
104#13095500—21no—
105#13095500D411yescharacteristic, complemented
106#13095500D411yescharacteristic, complemented
107#13095499—21no—
108#13095498—21no—
109#13095498—21no—
110#13095498—21no—
111D776—21no—
112D776—21no—
113D776—21no—
114D776—21no—
115D776—21no—
116D776—21no—
117D776—21no—
118D776D411yescharacteristic, complemented
119#13095496—21no—
120#13095496—21no—
121#13095496—21no—
122#13095495C2311yescharacteristic
123#13095495C2311yescharacteristic
124#13095494—21no—
125#13095494—21no—
126C8×D97—21no—
127C8×D97—21no—
128#13095488—41no—
129#41×C97C2311yescharacteristic
130#41×C97C2311yescharacteristic
131Q16×C97D411yescharacteristic, complemented
132SD16×C97D411yescharacteristic, complemented
133SD16×C97D411yescharacteristic, complemented
134D8×C97D411yescharacteristic, complemented
135M16×C97—21no—
136M16×C97—21no—
137M16×C97C2311yescharacteristic
138M16×C97C2311yescharacteristic
139C2×C8×C97—21no—
140C2×C8×C97—21no—
141C2×C8×C97C2311yescharacteristic
142C2×C8×C97C2311yescharacteristic
143C42×C97C2311yescharacteristic
144C22×D97—21no—
145C22×D97—21no—
146C22×D97—21no—
147C22×D97—41no—
148C22×D97—41no—
149C22×D97—41no—
150Q8×C97C2×D411yescharacteristic
151Q8×C97C2×D411yescharacteristic
152D4×C97—21no—
153D4×C97—21no—
154D4×C97C2×D411yescharacteristic
155D4×C97C2×D411yescharacteristic
156C2×C4×C97—21no—
157C2×C4×C97—21no—
158C2×C4×C97—21no—
159C2×C4×C97C2411yescharacteristic
160C22⋊D97—21no—
161C22⋊D97—21no—
162C22⋊D97—41no—
163C22⋊D97—41no—
164C22⋊D97—41no—
165C22⋊D97—41no—
166C2×Dic97—21no—
167C2×Dic97—41no—
168D388—21no—
169D388—21no—
170D388—21no—
171D388—21no—
172D388—21no—
173D388—21no—
174D388—41no—
175D388—41no—
176D388—41no—
177D388—82no—
178D388C2×D411yescharacteristic
179D388C2×D411yescharacteristic
180C4×D97—21no—
181C4×D97—21no—
182C4×D97—41no—
183C4×D97—41no—
184C4×D97—41no—
185C4×D97—41no—
186Dic194—21no—
187Dic194—21no—
188Dic194—41no—
189Dic194C2×D411yescharacteristic
190Dic194C2×D411yescharacteristic
191C8×C97—21no—
192C8×C97—21no—
193C8×C97C2×D411yescharacteristic
194C8×C97C2×D411yescharacteristic
195C8×C97C2×D411yescharacteristic
196C8×C97C2×D411yescharacteristic
197#11715895—21no—
198#11715895—21no—
199C22×C97—21no—
200C22×C97—21no—
201C22×C97H(5,𝔽2)11yescharacteristic
202C2×D97—21no—
203C2×D97—21no—
204C2×D97—41no—
205C2×D97—41no—
206C2×D97—41no—
207C2×D97—81no—
208C2×D97—81no—
209C2×D97—81no—
210C2×D97—161no—
211C2×D97—161no—
212C2×D97—161no—
213C4×C97—21no—
214C4×C97—21no—
215C4×C97—41no—
216C4×C97C22×D411yescharacteristic
217C4×C97C22×D411yescharacteristic
218Dic97—21no—
219Dic97—21no—
220Dic97—41no—
221C2×C97—21no—
222C2×C97—41no—
223C2×C97—41no—
224C2×C97D4211yescharacteristic
225D97—41no—
226D97—41no—
227D97—81no—
228D97—81no—
229D97—81no—
230#3288—971nomaximal, Sylow
231C97#328811yescharacteristic, complemented, Sylow
232#577—971no—
233#577—971no—
234#576—971no—
235#576—971no—
236#496—971no—
237#472—971no—
238#472—971no—
239#471—971no—
240#471—971no—
241#455—971no—
242#454—971no—
243#453—971no—
244#453—971no—
245#444—971no—
246#444D9711yescharacteristic, complemented
2472−1+4—971no—
248H(5,𝔽2)—971no—
249H(5,𝔽2)—971no—
250H(5,𝔽2)—971no—
251#137—971no—
252#137—971no—
253#137—1941no—
254#137—1941no—
255AGL(1,ℤ/8)—971no—
256AGL(1,ℤ/8)—971no—
257AGL(1,ℤ/8)—971no—
258AGL(1,ℤ/8)—971no—
259AGL(1,ℤ/8)—971no—
260AGL(1,ℤ/8)—971no—
261AGL(1,ℤ/8)—1941no—
262AGL(1,ℤ/8)—1941no—
263AGL(1,ℤ/8)—1941no—
264AGL(1,ℤ/8)—1941no—
265AGL(1,ℤ/8)—1941no—
266AGL(1,ℤ/8)—1941no—
267#135—971no—
268#135—1941no—
269#135—1941no—
270#135—1941no—
271#135—1941no—
272#135C2×D9711yescharacteristic
273C2×SD16—971no—
274C2×SD16—971no—
275C2×SD16—1941no—
276C2×SD16—1941no—
277C2×D8—971no—
278C2×D8—971no—
279C2×D8—1941no—
280C2×D8—1941no—
281#131—971no—
282#131—971no—
283#131C2×D9711yescharacteristic, complemented
284#131C2×D9711yescharacteristic, complemented
285#127—971no—
286#124—971no—
287#108—971no—
288#108C2×D9711yescharacteristic, complemented
289#104—971no—
290#104—971no—
291#104C2×D9711yescharacteristic, complemented
292#104C2×D9711yescharacteristic, complemented
293#101—971no—
294#100—971no—
295#100—971no—
296#100—971no—
297#97C2×D9711yescharacteristic, complemented
298#41—971no—
299#41—971no—
300#41—3881no—
301#41—1941no—
302#41—1941no—
303#41—1941no—
304#41—1941no—
305#41—1941no—
306#41—1941no—
307#41—1941no—
308#41—1941no—
309#41C22×D9711yescharacteristic
310#41C22×D9711yescharacteristic
311C2×Q8—971no—
312C2×Q8—1941no—
313C2×Q8—1941no—
314C2×D4—971no—
315C2×D4—971no—
316C2×D4—971no—
317C2×D4—3881no—
318C2×D4—3881no—
319C2×D4—3881no—
320C2×D4—3881no—
321C2×D4—1941no—
322C2×D4—1941no—
323C2×D4—1941no—
324C2×D4—1941no—
325C2×D4—1941no—
326C2×D4—1941no—
327Q16—971no—
328Q16—1941no—
329Q16—1941no—
330Q16D38811yescharacteristic, complemented
331SD16—971no—
332SD16—971no—
333SD16—1941no—
334SD16—1941no—
335SD16—1941no—
336SD16—1941no—
337SD16—1941no—
338SD16—1941no—
339SD16—1941no—
340SD16—1941no—
341SD16—1941no—
342SD16—1941no—
343SD16—1941no—
344SD16—1941no—
345SD16D38811yescharacteristic, complemented
346SD16D38811yescharacteristic, complemented
347D8—971no—
348D8—1941no—
349D8—1941no—
350D8—1941no—
351D8—1941no—
352D8—1941no—
353D8—1941no—
354D8—1941no—
355D8—1941no—
356D8—1941no—
357D8—1941no—
358D8D38811yescharacteristic, complemented
359M16—21no—
360M16—21no—
361M16—971no—
362M16—971no—
363M16—1941no—
364M16—1941no—
365M16C22×D9711yescharacteristic
366M16C22×D9711yescharacteristic
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—1941no—
370C2×C8—1941no—
371C2×C8C22×D9711yescharacteristic
372C2×C8C22×D9711yescharacteristic
373C22⋊C4—3881no—
374C42C22×D9711yescharacteristic
375C23—3881no—
376C23—3881no—
377C23—3881no—
378C23—1941no—
379C23—1941no—
380C23—1941no—
381Q8—971no—
382Q8—971no—
383Q8—3881no—
384Q8—1941no—
385Q8—1941no—
386Q8C2×D38811yescharacteristic
387Q8C2×D38811yescharacteristic
388D4—21no—
389D4—21no—
390D4—7762no—
391D4—971no—
392D4—971no—
393D4—3881no—
394D4—3881no—
395D4—3881no—
396D4—3881no—
397D4—3881no—
398D4—3881no—
399D4—3881no—
400D4—1941no—
401D4—1941no—
402D4—1941no—
403D4—1941no—
404D4—1941no—
405D4—1941no—
406D4—1941no—
407D4—1941no—
408D4C2×D38811yescharacteristic
409D4C2×D38811yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—3881no—
414C2×C4—3881no—
415C2×C4—3881no—
416C2×C4—3881no—
417C2×C4—3881no—
418C2×C4—1941no—
419C2×C4—1941no—
420C2×C4—1941no—
421C2×C4C23×D9711yescharacteristic
422C8—21no—
423C8—21no—
424C8—1941no—
425C8—1941no—
426C8D4×D9711yescharacteristic
427C8D4×D9711yescharacteristic
428C8C2×D38811yescharacteristic
429C8C2×D38811yescharacteristic
430C22—21no—
431C22—21no—
432C22—7761no—
433C22—7761no—
434C22—7761no—
435C22—15521no—
436C22—15521no—
437C22—15521no—
438C22—3881no—
439C22—3881no—
440C22—3881no—
441C22—1941no—
442C22—1941no—
443C22#1569085611yescharacteristic
444C4—21no—
445C4—21no—
446C4—41no—
447C4—3881no—
448C4—1941no—
449C4—1941no—
450C4C2×D4×D9711yescharacteristic
451C4C22×D38811yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—7761no—
456C2—7761no—
457C2—7761no—
458C2—3881no—
459C2—3881no—
460C2D4×D38811yescharacteristic
subgroupG/H
centerC2D4×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#444×C97C2
socleC2×C97D42
hypercenter#444D97
hypocenterC97#3288
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ #444

Complements

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

NH ≅ G/Ntype
1. #20261547C2semidirect
2. #20261542C2semidirect
3. #20261540C2semidirect
4. #20261537C2semidirect
5. #20261124C2semidirect
6. #20261006C2semidirect
7. #20261005C2semidirect
8. #20260999C2semidirect
9. #20260998C2semidirect
10. #20260914C2semidirect
11. #20260909C2semidirect
12. #20260904C2semidirect
13. #20260904C2semidirect
14. #20260858C2semidirect
15. #444×C97C2semidirect
16. #15690857C22semidirect
18. #15690856C22semidirect
19. #15690856C22semidirect
21. #15690843C22semidirect
23. #15690840C22semidirect
28. #15690838C22semidirect
29. #15690838C22semidirect
33. #15690835C22semidirect
34. #15690835C22semidirect
42. C2×#13095500C22semidirect
43. C2×#13095500C22semidirect
46. C2×D776C22semidirect
47. C2×D776C22semidirect
49. #15690816C22semidirect
50. #15690816C22semidirect
51. #15690807C22semidirect
52. #15690798C22semidirect
53. #15690746C22semidirect
54. #15690738C22semidirect
55. #15690738C22semidirect
56. #15690728C22semidirect
57. #15690726C22semidirect
58. #15690726C22semidirect
59. #15690726C22semidirect
61. #131×C97C22semidirect
62. #131×C97C22semidirect
63. #108×C97C22semidirect
64. #104×C97C22semidirect
65. #104×C97C22semidirect
66. #97×C97C22semidirect
95. Dic388D4semidirect
105. #13095500D4semidirect
106. #13095500D4semidirect
118. D776D4semidirect
131. Q16×C97D4semidirect
132. SD16×C97D4semidirect
133. SD16×C97D4semidirect
134. D8×C97D4semidirect
231. C97#3288semidirect
246. #444D97semidirect
283. #131C2×D97semidirect
284. #131C2×D97semidirect
288. #108C2×D97semidirect
291. #104C2×D97semidirect
292. #104C2×D97semidirect
297. #97C2×D97semidirect
330. Q16D388semidirect
345. SD16D388semidirect
346. SD16D388semidirect
358. D8D388semidirect

Maximal subgroups

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

subgroupindex
#202615472
#202615422
#202615402
#202615372
#202611242
#202610062
#202610052
#202609992
#202609982
#202609142
#202609092
#202609042
#202608582
#444×C972
#328897