Finite Group Databasebeta

← #37527247

Subgroups — #37527247

Order 22976.

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#26040061C211yesmaximal, characteristic, complemented
2#26040060C211yesmaximal, characteristic, complemented
3#26040043C211yesmaximal, characteristic, complemented
4#26040042C212yesmaximal, complemented
5#26040039C212yesmaximal, complemented
6#26040036C212yesmaximal, complemented
7#26040035C211yesmaximal, characteristic, complemented
8C2×#19153326C211yesmaximal, characteristic, complemented
9D8×D359C212yesmaximal, complemented
10#26040023C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C359C211yesmaximal, characteristic, complemented
12#19153342C2212yescomplemented
13#19153341—21no—
14#19153341C2211yescharacteristic
15#19153341C2211yescharacteristic, complemented
16#19153340—21no—
17#19153340C2212yescomplemented
18C2×D1436—21no—
19C2×D1436C2211yescharacteristic, complemented
20C2×C22⋊D359—22no—
21D4×D359—21no—
22D4×D359—22no—
23D4×D359C2212yescomplemented
24D4×D359C2212yescomplemented
25#19153331C2211yescharacteristic, complemented
26#19153330C2212yescomplemented
27#19153329C2211yescharacteristic, complemented
28#19153328C2211yescharacteristic, complemented
29D2872C2212yescomplemented
30#19153326C2211yescharacteristic, complemented
31#19153326C2212yescomplemented
32#19153326C2212yescomplemented
33#19153324C2212yescomplemented
34C2×C359⋊C8C2211yescharacteristic, complemented
35C8×D359C2212yescomplemented
36#41×C359C2211yescharacteristic, complemented
37C2×D4×C359C2211yescharacteristic, complemented
38SD16×C359C2212yescomplemented
39D8×C359C2212yescomplemented
40M16×C359C2211yescharacteristic, complemented
41C22×D359—21no—
42C22×D359—22no—
43C22×D359—22no—
44Dic718D411yescharacteristic
45Dic718C2311yescharacteristic
46D1436—22no—
47D1436—21no—
48D1436D411yescharacteristic
49D1436C2312yes—
50D1436C2311yescharacteristic
51C22⋊D359—21no—
52C22⋊D359—22no—
53C22⋊D359—22no—
54C22⋊D359—22no—
55C22⋊D359D412yes—
56C2×Dic359—22no—
57C4×D359—21no—
58C4×D359—22no—
59C4×D359C2312yes—
60C359⋊C8C2311yescharacteristic
61C359⋊C8C2311yescharacteristic
62C23×C359—21no—
63Q8×C359C2311yescharacteristic
64D4×C359—21no—
65D4×C359—21no—
66D4×C359C2311yescharacteristic
67D4×C359C2312yes—
68C2×C4×C359—21no—
69C2×C4×C359C2311yescharacteristic
70C8×C359C2312yes—
71C22×C359—21no—
72C22×C359—22no—
73C22×C359—81no—
74C22×C359C2×D411yescharacteristic
75C2×D359—21no—
76C2×D359—22no—
77C2×D359—82no—
78C2×D359—82no—
79C2×D359—81no—
80C2×D359C2×D412yes—
81C4×C359—21no—
82C4×C359C2×D411yescharacteristic
83C4×C359C2411yescharacteristic
84Dic359—21no—
85Dic359C2×D412yes—
86C2×C359—21no—
87C2×C359—41no—
88C2×C359—42no—
89C2×C359C22×D411yescharacteristic
90D359—22no—
91D359—41no—
92D359—42no—
93C359#57611yescharacteristic, complemented, Sylow
94#576—3591nomaximal, Sylow
95H(5,𝔽2)—3591no—
96H(5,𝔽2)—3591no—
97AGL(1,ℤ/8)—3592no—
98AGL(1,ℤ/8)—3592no—
99AGL(1,ℤ/8)—3591no—
100AGL(1,ℤ/8)D35911yescharacteristic, complemented
101#135—3591no—
102#135—3592no—
103C2×D8—3591no—
104C2×D8—3592no—
105#131—3591no—
106#41—3591no—
107#41—3592no—
108#41—3592no—
109#41—3591no—
110#41—7181no—
111#41—7181no—
112#41C2×D35911yescharacteristic, complemented
113C2×D4—3592no—
114C2×D4—3592no—
115C2×D4—3591no—
116C2×D4—7181no—
117C2×D4—7182no—
118C2×D4—7181no—
119C2×D4—7182no—
120C2×D4C2×D35911yescharacteristic, complemented
121Q16—3591no—
122SD16—3591no—
123SD16—3591no—
124SD16—3592no—
125SD16C2×D35912yescomplemented
126D8—3592no—
127D8—3591no—
128D8—3592no—
129D8—3592no—
130D8C2×D35912yescomplemented
131M16—3592no—
132M16C2×D35911yescharacteristic, complemented
133C2×C8—3591no—
134C2×C8—3592no—
135C23—21no—
136C23—7182no—
137C23—7182no—
138C23—7181no—
139Q8—3591no—
140Q8—3591no—
141Q8C22×D35911yescharacteristic
142D4—21no—
143D4—21no—
144D4—3592no—
145D4—3591no—
146D4—3592no—
147D4—3591no—
148D4—7182no—
149D4—7182no—
150D4—7182no—
151D4—7182no—
152D4—7181no—
153D4—7181no—
154D4C22×D35912yes—
155D4C22×D35911yescharacteristic
156C2×C4—21no—
157C2×C4—3592no—
158C2×C4—7182no—
159C2×C4—7182no—
160C2×C4—7181no—
161C2×C4C22×D35911yescharacteristic
162C8—3591no—
163C8—3591no—
164C8C22×D35912yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—28721no—
169C22—28722no—
170C22—28722no—
171C22—3592no—
172C22—7182no—
173C22—7181no—
174C22D4×D35911yescharacteristic
175C4—21no—
176C4—3592no—
177C4—7181no—
178C4D4×D35911yescharacteristic
179C4C23×D35911yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—14362no—
184C2—14361no—
185C2—7182no—
186C2C2×D4×D35911yescharacteristic
subgroupG/H
centerC2C2×D4×D359
commutator, cosocleC4×C359C24
FrattiniC4C23×D359
FittingAGL(1,ℤ/8)×C359C2
socleC2×C359C22×D4
hypercenterAGL(1,ℤ/8)D359
hypocenterC359#576
LCSG ▷ C4×C359 ▷ C2×C359 ▷ C359
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #26040061C2semidirect
2. #26040060C2semidirect
3. #26040043C2semidirect
4. #26040042C2semidirect
5. #26040039C2semidirect
6. #26040036C2semidirect
7. #26040035C2semidirect
8. C2×#19153326C2semidirect
9. D8×D359C2semidirect
10. #26040023C2semidirect
11. AGL(1,ℤ/8)×C359C2semidirect
12. #19153342C22semidirect
15. #19153341C22semidirect
17. #19153340C22semidirect
19. C2×D1436C22semidirect
23. D4×D359C22semidirect
24. D4×D359C22semidirect
25. #19153331C22semidirect
26. #19153330C22semidirect
27. #19153329C22semidirect
28. #19153328C22semidirect
29. D2872C22semidirect
30. #19153326C22semidirect
31. #19153326C22semidirect
32. #19153326C22semidirect
33. #19153324C22semidirect
34. C2×C359⋊C8C22semidirect
35. C8×D359C22semidirect
36. #41×C359C22semidirect
37. C2×D4×C359C22semidirect
38. SD16×C359C22semidirect
39. D8×C359C22semidirect
40. M16×C359C22semidirect
93. C359#576semidirect
100. AGL(1,ℤ/8)D359semidirect
112. #41C2×D359semidirect
120. C2×D4C2×D359semidirect
125. SD16C2×D359semidirect
130. D8C2×D359semidirect
132. M16C2×D359semidirect

Maximal subgroups

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

subgroupindex
#260400612
#260400602
#260400432
#260400422
#260400392
#260400362
#260400352
C2×#191533262
D8×D3592
#260400232
AGL(1,ℤ/8)×C3592
#576359