Finite Group Databasebeta

← #42119863

Subgroups — #42119863

Order 27584.

268 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 99 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#28352235C211yesmaximal, characteristic, complemented
2#28352234C211yesmaximal, characteristic, complemented
3#28352217C211yesmaximal, characteristic, complemented
4#28352216C212yesmaximal, complemented
5#28352213C212yesmaximal, complemented
6#28352210C212yesmaximal, complemented
7#28352209C211yesmaximal, characteristic, complemented
8C2×#20293785C211yesmaximal, characteristic, complemented
9D8×D431C212yesmaximal, complemented
10#28352197C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C431C211yesmaximal, characteristic, complemented
12#20293801C2212yescomplemented
13#20293800—21no—
14#20293800C2211yescharacteristic
15#20293800C2211yescharacteristic, complemented
16#20293799—21no—
17#20293799C2212yescomplemented
18C2×D1724—21no—
19C2×D1724C2211yescharacteristic, complemented
20C2×C22⋊D431—22no—
21D4×D431—21no—
22D4×D431—22no—
23D4×D431C2212yescomplemented
24D4×D431C2212yescomplemented
25#20293790C2211yescharacteristic, complemented
26#20293789C2212yescomplemented
27#20293788C2211yescharacteristic, complemented
28#20293787C2211yescharacteristic, complemented
29D3448C2212yescomplemented
30#20293785C2211yescharacteristic, complemented
31#20293785C2212yescomplemented
32#20293785C2212yescomplemented
33#20293783C2212yescomplemented
34C2×C431⋊C8C2211yescharacteristic, complemented
35C8×D431C2212yescomplemented
36#41×C431C2211yescharacteristic, complemented
37C2×D4×C431C2211yescharacteristic, complemented
38SD16×C431C2212yescomplemented
39D8×C431C2212yescomplemented
40M16×C431C2211yescharacteristic, complemented
41C22×D431—21no—
42C22×D431—22no—
43C22×D431—22no—
44Dic862D411yescharacteristic
45Dic862C2311yescharacteristic
46D1724—22no—
47D1724—21no—
48D1724D411yescharacteristic
49D1724C2312yes—
50D1724C2311yescharacteristic
51C22⋊D431—21no—
52C22⋊D431—22no—
53C22⋊D431—22no—
54C22⋊D431—22no—
55C22⋊D431D412yes—
56C2×Dic431—22no—
57C4×D431—21no—
58C4×D431—22no—
59C4×D431C2312yes—
60C431⋊C8C2311yescharacteristic
61C431⋊C8C2311yescharacteristic
62C23×C431—21no—
63Q8×C431C2311yescharacteristic
64D4×C431—21no—
65D4×C431—21no—
66D4×C431C2311yescharacteristic
67D4×C431C2312yes—
68C2×C4×C431—21no—
69C2×C4×C431C2311yescharacteristic
70C8×C431C2312yes—
71C22×C431—21no—
72C22×C431—22no—
73C22×C431—81no—
74C22×C431C2×D411yescharacteristic
75C2×D431—21no—
76C2×D431—22no—
77C2×D431—82no—
78C2×D431—82no—
79C2×D431—81no—
80C2×D431C2×D412yes—
81C4×C431—21no—
82C4×C431C2×D411yescharacteristic
83C4×C431C2411yescharacteristic
84Dic431—21no—
85Dic431C2×D412yes—
86C2×C431—21no—
87C2×C431—41no—
88C2×C431—42no—
89C2×C431C22×D411yescharacteristic
90D431—22no—
91D431—41no—
92D431—42no—
93C431#57611yescharacteristic, complemented, Sylow
94#576—4311nomaximal, Sylow
95H(5,𝔽2)—4311no—
96H(5,𝔽2)—4311no—
97AGL(1,ℤ/8)—4312no—
98AGL(1,ℤ/8)—4312no—
99AGL(1,ℤ/8)—4311no—
100AGL(1,ℤ/8)D43111yescharacteristic, complemented
101#135—4311no—
102#135—4312no—
103C2×D8—4311no—
104C2×D8—4312no—
105#131—4311no—
106#41—8621no—
107#41—8621no—
108#41—4312no—
109#41—4312no—
110#41—4311no—
111#41—4311no—
112#41C2×D43111yescharacteristic, complemented
113C2×D4—8621no—
114C2×D4—8622no—
115C2×D4—8621no—
116C2×D4—8622no—
117C2×D4—4312no—
118C2×D4—4312no—
119C2×D4—4311no—
120C2×D4C2×D43111yescharacteristic, complemented
121Q16—4311no—
122SD16—4311no—
123SD16—4311no—
124SD16—4312no—
125SD16C2×D43112yescomplemented
126D8—4312no—
127D8—4311no—
128D8—4312no—
129D8—4312no—
130D8C2×D43112yescomplemented
131M16—4312no—
132M16C2×D43111yescharacteristic, complemented
133C2×C8—4311no—
134C2×C8—4312no—
135C23—21no—
136C23—8622no—
137C23—8622no—
138C23—8621no—
139Q8—4311no—
140Q8—4311no—
141Q8C22×D43111yescharacteristic
142D4—21no—
143D4—21no—
144D4—8622no—
145D4—8622no—
146D4—8621no—
147D4—8622no—
148D4—8622no—
149D4—8621no—
150D4—4312no—
151D4—4311no—
152D4—4312no—
153D4—4311no—
154D4C22×D43112yes—
155D4C22×D43111yescharacteristic
156C2×C4—21no—
157C2×C4—8622no—
158C2×C4—8622no—
159C2×C4—8621no—
160C2×C4—4312no—
161C2×C4C22×D43111yescharacteristic
162C8—4311no—
163C8—4311no—
164C8C22×D43112yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—8622no—
169C22—8621no—
170C22—34481no—
171C22—34482no—
172C22—34482no—
173C22—4312no—
174C22D4×D43111yescharacteristic
175C4—21no—
176C4—8621no—
177C4—4312no—
178C4D4×D43111yescharacteristic
179C4C23×D43111yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—8622no—
184C2—17242no—
185C2—17241no—
186C2C2×D4×D43111yescharacteristic
subgroupG/H
centerC2C2×D4×D431
commutator, cosocleC4×C431C24
FrattiniC4C23×D431
FittingAGL(1,ℤ/8)×C431C2
socleC2×C431C22×D4
hypercenterAGL(1,ℤ/8)D431
hypocenterC431#576
LCSG ▷ C4×C431 ▷ C2×C431 ▷ C431
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #28352235C2semidirect
2. #28352234C2semidirect
3. #28352217C2semidirect
4. #28352216C2semidirect
5. #28352213C2semidirect
6. #28352210C2semidirect
7. #28352209C2semidirect
8. C2×#20293785C2semidirect
9. D8×D431C2semidirect
10. #28352197C2semidirect
11. AGL(1,ℤ/8)×C431C2semidirect
12. #20293801C22semidirect
15. #20293800C22semidirect
17. #20293799C22semidirect
19. C2×D1724C22semidirect
23. D4×D431C22semidirect
24. D4×D431C22semidirect
25. #20293790C22semidirect
26. #20293789C22semidirect
27. #20293788C22semidirect
28. #20293787C22semidirect
29. D3448C22semidirect
30. #20293785C22semidirect
31. #20293785C22semidirect
32. #20293785C22semidirect
33. #20293783C22semidirect
34. C2×C431⋊C8C22semidirect
35. C8×D431C22semidirect
36. #41×C431C22semidirect
37. C2×D4×C431C22semidirect
38. SD16×C431C22semidirect
39. D8×C431C22semidirect
40. M16×C431C22semidirect
93. C431#576semidirect
100. AGL(1,ℤ/8)D431semidirect
112. #41C2×D431semidirect
120. C2×D4C2×D431semidirect
125. SD16C2×D431semidirect
130. D8C2×D431semidirect
132. M16C2×D431semidirect

Maximal subgroups

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

subgroupindex
#283522352
#283522342
#283522172
#283522162
#283522132
#283522102
#283522092
C2×#202937852
D8×D4312
#283521972
AGL(1,ℤ/8)×C4312
#576431