Finite Group Databasebeta

← #30677758

Subgroups — #30677758

Order 16768.

476 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22554115C212yesmaximal, complemented
2#22554114C211yesmaximal, characteristic, complemented
3#22553687C211yesmaximal, characteristic, complemented
4#22553582C211yesmaximal, characteristic, complemented
5#22553580C212yesmaximal, complemented
6#22553579C211yesmaximal, characteristic, complemented
7#22553480C212yesmaximal, complemented
8#22553478C212yesmaximal, complemented
9#22553429C211yesmaximal, characteristic, complemented
10#22553427C211yesmaximal, characteristic, complemented
11#576×C131C211yesmaximal, characteristic, complemented
12#16862292C2212yescomplemented
13#16862273—22no—
14#16862272—22no—
15#16862272C2212yescomplemented
16#16862271—22no—
17#16862271C2212yescomplemented
18#16862271C2212yescomplemented
19#16862269—21no—
20#16862266—22no—
21#16862266C2211yescharacteristic
22C2×D1048C2211yescharacteristic, complemented
23C2×#14523498—22no—
24C2×#14523498C2212yescomplemented
25D8×D131—21no—
26#16862253C2212yescomplemented
27#16862252C2211yescharacteristic, complemented
28#16862242C2212yescomplemented
29#16862182C2211yescharacteristic, complemented
30#16862181C2211yescharacteristic, complemented
31#16862173C2211yescharacteristic, complemented
32#16862173C2212yescomplemented
33#16862172C2211yescharacteristic, complemented
34#16862163C2212yescomplemented
35#16862162C2212yescomplemented
36C8×Dic131C2211yescharacteristic, complemented
37H(5,𝔽2)×C131C2212yescomplemented
38AGL(1,ℤ/8)×C131—22no—
39AGL(1,ℤ/8)×C131C2212yescomplemented
40#135×C131—21no—
41#135×C131C2211yescharacteristic
42C2×D8×C131—21no—
43C2×D8×C131C2211yescharacteristic, complemented
44#131×C131C2211yescharacteristic, complemented
45#14523514—22no—
46#14523513—22no—
47#14523513C2311yescharacteristic
48C2×D524—22no—
49C2×D524C2312yes—
50C2×C22⋊D131—42no—
51D4×D131—22no—
52D4×D131—42no—
53#14523503D411yescharacteristic, complemented
54#14523501—22no—
55#14523501D411yescharacteristic, complemented
56#14523500—22no—
57#14523500—22no—
58#14523500D411yescharacteristic, complemented
59D1048—21no—
60D1048—21no—
61#14523498—22no—
62#14523498—22no—
63#14523498—22no—
64#14523498—22no—
65#14523498—22no—
66#14523498D411yescharacteristic, complemented
67#14523497—22no—
68#14523497C2311yescharacteristic
69#14523497C2312yes—
70#14523496—21no—
71C2×C131⋊C8—22no—
72C2×C131⋊C8C2311yescharacteristic
73C8×D131—21no—
74C4×Dic131C2311yescharacteristic
75#41×C131—22no—
76#41×C131—22no—
77#41×C131C2312yes—
78#41×C131C2311yescharacteristic
79C2×D4×C131—22no—
80C2×D4×C131—22no—
81C2×D4×C131—42no—
82C2×D4×C131C2312yes—
83Q16×C131D411yescharacteristic, complemented
84SD16×C131—22no—
85SD16×C131D412yescomplemented
86D8×C131—22no—
87D8×C131—21no—
88D8×C131—21no—
89D8×C131D411yescharacteristic, complemented
90M16×C131—21no—
91M16×C131C2311yescharacteristic
92C2×C8×C131—21no—
93C2×C8×C131C2311yescharacteristic
94C22×D131—22no—
95C22×D131—42no—
96Dic262C2×D411yescharacteristic
97D524—22no—
98D524—22no—
99D524—42no—
100D524C2×D411yescharacteristic
101C22⋊D131—21no—
102C22⋊D131—42no—
103C22⋊D131—84no—
104C2×Dic131—21no—
105C4×D131—21no—
106C4×D131—42no—
107C131⋊C8—21no—
108C131⋊C8—22no—
109C131⋊C8C2×D411yescharacteristic
110C131⋊C8C2×D411yescharacteristic
111C23×C131—22no—
112C23×C131—42no—
113Q8×C131C2×D411yescharacteristic
114Q8×C131C2×D412yes—
115D4×C131—21no—
116D4×C131—22no—
117D4×C131—22no—
118D4×C131—22no—
119D4×C131—42no—
120D4×C131—42no—
121D4×C131C2×D411yescharacteristic
122D4×C131C2×D412yes—
123C2×C4×C131—21no—
124C2×C4×C131—22no—
125C2×C4×C131—42no—
126C2×C4×C131C2411yescharacteristic
127C8×C131—21no—
128C8×C131C2×D411yescharacteristic
129C8×C131C2×D411yescharacteristic
130C22×C131—21no—
131C22×C131—22no—
132C22×C131—42no—
133C22×C131—82no—
134C22×C131—162no—
135C22×C131H(5,𝔽2)11yescharacteristic
136C2×D131—21no—
137C2×D131—42no—
138C2×D131—82no—
139C2×D131—162no—
140C4×C131—21no—
141C4×C131—22no—
142C4×C131C22×D411yescharacteristic
143C4×C131C22×D411yescharacteristic
144Dic131—22no—
145Dic131—21no—
146C2×C131—21no—
147C2×C131—42no—
148C2×C131—41no—
149C2×C131—82no—
150C2×C131D4211yescharacteristic
151D131—41no—
152D131—82no—
153C131#329211yescharacteristic, complemented, Sylow
154#3292—1311nomaximal, Sylow
155#576—1312no—
156#576—1311no—
157#576D13111yescharacteristic, complemented
158#493—1311no—
159#472—1312no—
160#472—1311no—
161#472—1311no—
162#453—1312no—
163#453—1312no—
164#443—1311no—
165#443—1311no—
166H(5,𝔽2)—1312no—
167H(5,𝔽2)C2×D13112yescomplemented
168AGL(1,ℤ/8)—22no—
169AGL(1,ℤ/8)—2622no—
170AGL(1,ℤ/8)—2622no—
171AGL(1,ℤ/8)—2622no—
172AGL(1,ℤ/8)—1312no—
173AGL(1,ℤ/8)—1312no—
174AGL(1,ℤ/8)—1312no—
175AGL(1,ℤ/8)C2×D13112yescomplemented
176#135—21no—
177#135—2622no—
178#135—2621no—
179#135—1311no—
180#135C2×D13111yescharacteristic
181C2×D8—21no—
182C2×D8—2622no—
183C2×D8—2621no—
184C2×D8—1312no—
185C2×D8—1311no—
186C2×D8C2×D13111yescharacteristic, complemented
187#131—1312no—
188#131—1311no—
189#131C2×D13111yescharacteristic, complemented
190#127—1312no—
191#108—1311no—
192#108—1311no—
193#104—1311no—
194#104—1312no—
195#104—1311no—
196#100—1312no—
197#100—1312no—
198C4×C8—1311no—
199#41—22no—
200#41—22no—
201#41—2622no—
202#41—2622no—
203#41—1311no—
204#41C22×D13111yescharacteristic
205#41C22×D13112yes—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—42no—
209C2×D4—2622no—
210C2×D4—2622no—
211C2×D4—5242no—
212C2×D4—5242no—
213C2×D4—1312no—
214C2×D4C22×D13112yes—
215Q16—1311no—
216Q16C22⋊D13111yescharacteristic, complemented
217SD16—22no—
218SD16—2622no—
219SD16—2622no—
220SD16—2622no—
221SD16—1311no—
222SD16—1311no—
223SD16C22⋊D13112yescomplemented
224D8—22no—
225D8—21no—
226D8—21no—
227D8—2622no—
228D8—2622no—
229D8—2622no—
230D8—2622no—
231D8—2622no—
232D8—2621no—
233D8—2621no—
234D8—1311no—
235D8C22⋊D13111yescharacteristic, complemented
236M16—21no—
237M16—2622no—
238M16—2621no—
239M16—1312no—
240M16—1311no—
241M16C22×D13111yescharacteristic
242C2×C8—21no—
243C2×C8—2622no—
244C2×C8—2621no—
245C2×C8—1311no—
246C2×C8C22×D13111yescharacteristic
247C42—1311no—
248C23—22no—
249C23—42no—
250C23—2622no—
251C23—5242no—
252Q8—1311no—
253Q8D4×D13111yescharacteristic
254Q8C2×C22⋊D13112yes—
255D4—21no—
256D4—22no—
257D4—22no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—2621no—
262D4—2622no—
263D4—2622no—
264D4—5242no—
265D4—5242no—
266D4—10484no—
267D4—1311no—
268D4D4×D13111yescharacteristic
269D4C2×C22⋊D13112yes—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—2621no—
274C2×C4—2621no—
275C2×C4—5242no—
276C2×C4C23×D13111yescharacteristic
277C8—21no—
278C8—2622no—
279C8—2621no—
280C8—1311no—
281C8—1311no—
282C8C2×C22⋊D13111yescharacteristic
283C8C2×C22⋊D13111yescharacteristic
284C22—21no—
285C22—22no—
286C22—42no—
287C22—2621no—
288C22—82no—
289C22—5242no—
290C22—162no—
291C22—10482no—
292C22—20962no—
293C22#1686229111yescharacteristic
294C4—21no—
295C4—22no—
296C4—2622no—
297C4—2621no—
298C4C2×D4×D13111yescharacteristic
299C4C22×C22⋊D13111yescharacteristic
300C2—21no—
301C2—41no—
302C2—42no—
303C2—82no—
304C2—5241no—
305C2—10482no—
306C2D4×C22⋊D13111yescharacteristic
subgroupG/H
centerC2D4×C22⋊D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#576×C131C2
socleC2×C131D42
hypercenter#576D131
hypocenterC131#3292
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #22554115C2semidirect
2. #22554114C2semidirect
3. #22553687C2semidirect
4. #22553582C2semidirect
5. #22553580C2semidirect
6. #22553579C2semidirect
7. #22553480C2semidirect
8. #22553478C2semidirect
9. #22553429C2semidirect
10. #22553427C2semidirect
11. #576×C131C2semidirect
12. #16862292C22semidirect
15. #16862272C22semidirect
17. #16862271C22semidirect
18. #16862271C22semidirect
22. C2×D1048C22semidirect
24. C2×#14523498C22semidirect
26. #16862253C22semidirect
27. #16862252C22semidirect
28. #16862242C22semidirect
29. #16862182C22semidirect
30. #16862181C22semidirect
31. #16862173C22semidirect
32. #16862173C22semidirect
33. #16862172C22semidirect
34. #16862163C22semidirect
35. #16862162C22semidirect
36. C8×Dic131C22semidirect
37. H(5,𝔽2)×C131C22semidirect
39. AGL(1,ℤ/8)×C131C22semidirect
43. C2×D8×C131C22semidirect
44. #131×C131C22semidirect
53. #14523503D4semidirect
55. #14523501D4semidirect
58. #14523500D4semidirect
66. #14523498D4semidirect
83. Q16×C131D4semidirect
85. SD16×C131D4semidirect
89. D8×C131D4semidirect
153. C131#3292semidirect
157. #576D131semidirect
167. H(5,𝔽2)C2×D131semidirect
175. AGL(1,ℤ/8)C2×D131semidirect
186. C2×D8C2×D131semidirect
189. #131C2×D131semidirect
216. Q16C22⋊D131semidirect
223. SD16C22⋊D131semidirect
235. D8C22⋊D131semidirect

Maximal subgroups

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

subgroupindex
#225541152
#225541142
#225536872
#225535822
#225535802
#225535792
#225534802
#225534782
#225534292
#225534272
#576×C1312
#3292131