Finite Group Databasebeta

← #43272811

Subgroups — #43272811

Order 28736.

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#28385386C211yesmaximal, characteristic, complemented
2#28385385C211yesmaximal, characteristic, complemented
3#28385368C211yesmaximal, characteristic, complemented
4#28385367C212yesmaximal, complemented
5#28385364C212yesmaximal, complemented
6#28385361C212yesmaximal, complemented
7#28385360C211yesmaximal, characteristic, complemented
8C2×#20297721C211yesmaximal, characteristic, complemented
9D8×D449C212yesmaximal, complemented
10#28385348C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C449C211yesmaximal, characteristic, complemented
12#20297737C2212yescomplemented
13#20297736—21no—
14#20297736C2211yescharacteristic
15#20297736C2211yescharacteristic, complemented
16#20297735—21no—
17#20297735C2212yescomplemented
18C2×D1796—21no—
19C2×D1796C2211yescharacteristic, complemented
20C2×C22⋊D449—22no—
21D4×D449—21no—
22D4×D449—22no—
23D4×D449C2212yescomplemented
24D4×D449C2212yescomplemented
25#20297726C2211yescharacteristic, complemented
26#20297725C2212yescomplemented
27#20297724C2211yescharacteristic, complemented
28#20297723C2211yescharacteristic, complemented
29D3592C2212yescomplemented
30#20297721C2211yescharacteristic, complemented
31#20297721C2212yescomplemented
32#20297721C2212yescomplemented
33#20297719C2212yescomplemented
34C2×#16813311C2211yescharacteristic, complemented
35C8×D449C2212yescomplemented
36#41×C449C2211yescharacteristic, complemented
37C2×D4×C449C2211yescharacteristic, complemented
38SD16×C449C2212yescomplemented
39D8×C449C2212yescomplemented
40M16×C449C2211yescharacteristic, complemented
41C22×D449—21no—
42C22×D449—22no—
43C22×D449—22no—
44Dic898D411yescharacteristic
45Dic898C2311yescharacteristic
46D1796—22no—
47D1796—21no—
48D1796D411yescharacteristic
49D1796C2311yescharacteristic
50D1796C2312yes—
51C22⋊D449—21no—
52C22⋊D449—22no—
53C22⋊D449—22no—
54C22⋊D449—22no—
55C22⋊D449D412yes—
56C2×Dic449—22no—
57C4×D449—21no—
58C4×D449—22no—
59C4×D449C2312yes—
60#16813311C2311yescharacteristic
61#16813311C2311yescharacteristic
62C23×C449—21no—
63Q8×C449C2311yescharacteristic
64D4×C449—21no—
65D4×C449—21no—
66D4×C449C2311yescharacteristic
67D4×C449C2312yes—
68C2×C4×C449—21no—
69C2×C4×C449C2311yescharacteristic
70C8×C449C2312yes—
71C22×C449—22no—
72C22×C449—21no—
73C22×C449—81no—
74C22×C449C2×D411yescharacteristic
75C2×D449—21no—
76C2×D449—22no—
77C2×D449—82no—
78C2×D449—82no—
79C2×D449—81no—
80C2×D449C2×D412yes—
81C4×C449—21no—
82C4×C449C2×D411yescharacteristic
83C4×C449C2411yescharacteristic
84Dic449—21no—
85Dic449C2×D412yes—
86C2×C449—21no—
87C2×C449—41no—
88C2×C449—42no—
89C2×C449C22×D411yescharacteristic
90D449—22no—
91D449—41no—
92D449—42no—
93C449#57611yescharacteristic, complemented, Sylow
94#576—4491nomaximal, Sylow
95H(5,𝔽2)—4491no—
96H(5,𝔽2)—4491no—
97AGL(1,ℤ/8)—4492no—
98AGL(1,ℤ/8)—4492no—
99AGL(1,ℤ/8)—4491no—
100AGL(1,ℤ/8)D44911yescharacteristic, complemented
101#135—4491no—
102#135—4492no—
103C2×D8—4491no—
104C2×D8—4492no—
105#131—4491no—
106#41—8981no—
107#41—8981no—
108#41—4492no—
109#41—4492no—
110#41—4491no—
111#41—4491no—
112#41C2×D44911yescharacteristic, complemented
113C2×D4—8981no—
114C2×D4—8982no—
115C2×D4—8981no—
116C2×D4—8982no—
117C2×D4—4492no—
118C2×D4—4492no—
119C2×D4—4491no—
120C2×D4C2×D44911yescharacteristic, complemented
121Q16—4491no—
122SD16—4491no—
123SD16—4491no—
124SD16—4492no—
125SD16C2×D44912yescomplemented
126D8—4492no—
127D8—4491no—
128D8—4492no—
129D8—4492no—
130D8C2×D44912yescomplemented
131M16—4492no—
132M16C2×D44911yescharacteristic, complemented
133C2×C8—4491no—
134C2×C8—4492no—
135C23—21no—
136C23—8982no—
137C23—8982no—
138C23—8981no—
139Q8—4491no—
140Q8—4491no—
141Q8C22×D44911yescharacteristic
142D4—21no—
143D4—21no—
144D4—8982no—
145D4—8982no—
146D4—8981no—
147D4—8982no—
148D4—8982no—
149D4—8981no—
150D4—4492no—
151D4—4491no—
152D4—4491no—
153D4—4492no—
154D4C22×D44912yes—
155D4C22×D44911yescharacteristic
156C2×C4—21no—
157C2×C4—8982no—
158C2×C4—8982no—
159C2×C4—8981no—
160C2×C4—4492no—
161C2×C4C22×D44911yescharacteristic
162C8—4491no—
163C8—4491no—
164C8C22×D44912yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—35921no—
169C22—35922no—
170C22—35922no—
171C22—8982no—
172C22—8981no—
173C22—4492no—
174C22D4×D44911yescharacteristic
175C4—21no—
176C4—8981no—
177C4—4492no—
178C4D4×D44911yescharacteristic
179C4C23×D44911yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—17962no—
184C2—17961no—
185C2—8982no—
186C2C2×D4×D44911yescharacteristic
subgroupG/H
centerC2C2×D4×D449
commutator, cosocleC4×C449C24
FrattiniC4C23×D449
FittingAGL(1,ℤ/8)×C449C2
socleC2×C449C22×D4
hypercenterAGL(1,ℤ/8)D449
hypocenterC449#576
LCSG ▷ C4×C449 ▷ C2×C449 ▷ C449
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #28385386C2semidirect
2. #28385385C2semidirect
3. #28385368C2semidirect
4. #28385367C2semidirect
5. #28385364C2semidirect
6. #28385361C2semidirect
7. #28385360C2semidirect
8. C2×#20297721C2semidirect
9. D8×D449C2semidirect
10. #28385348C2semidirect
11. AGL(1,ℤ/8)×C449C2semidirect
12. #20297737C22semidirect
15. #20297736C22semidirect
17. #20297735C22semidirect
19. C2×D1796C22semidirect
23. D4×D449C22semidirect
24. D4×D449C22semidirect
25. #20297726C22semidirect
26. #20297725C22semidirect
27. #20297724C22semidirect
28. #20297723C22semidirect
29. D3592C22semidirect
30. #20297721C22semidirect
31. #20297721C22semidirect
32. #20297721C22semidirect
33. #20297719C22semidirect
34. C2×#16813311C22semidirect
35. C8×D449C22semidirect
36. #41×C449C22semidirect
37. C2×D4×C449C22semidirect
38. SD16×C449C22semidirect
39. D8×C449C22semidirect
40. M16×C449C22semidirect
93. C449#576semidirect
100. AGL(1,ℤ/8)D449semidirect
112. #41C2×D449semidirect
120. C2×D4C2×D449semidirect
125. SD16C2×D449semidirect
130. D8C2×D449semidirect
132. M16C2×D449semidirect

Maximal subgroups

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

subgroupindex
#283853862
#283853852
#283853682
#283853672
#283853642
#283853612
#283853602
C2×#202977212
D8×D4492
#283853482
AGL(1,ℤ/8)×C4492
#576449