Finite Group Databasebeta

← #37527253

Subgroups — #37527253

Order 22976.

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

normal subgroups only

This group has 258 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#26040062C211yesmaximal, characteristic, complemented
2#26040060C211yesmaximal, characteristic, complemented
3#26040048C211yesmaximal, characteristic, complemented
4#26040044C212yesmaximal, complemented
5#26040039C212yesmaximal, complemented
6#26040035C211yesmaximal, characteristic, complemented
7#26040034C212yesmaximal, complemented
8C2×#19153329C211yesmaximal, characteristic, complemented
9SD16×D359C212yesmaximal, complemented
10#26040023C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C359C211yesmaximal, characteristic, complemented
12#19153341—21no—
13#19153341C2211yescharacteristic
14#19153341C2211yescharacteristic
15#19153340—21no—
16#19153340—22no—
17#19153340C2212yes—
18#19153340C2212yescomplemented
19C2×Dic718—21no—
20C2×Dic718C2211yescharacteristic, complemented
21Q8×D359C2212yescomplemented
22C2×C22⋊D359—22no—
23D4×D359—21no—
24D4×D359C2212yes—
25Dic1436C2212yescomplemented
26#19153331C2211yescharacteristic, complemented
27#19153330C2212yescomplemented
28#19153329C2212yescomplemented
29#19153329C2212yescomplemented
30#19153329C2211yescharacteristic, complemented
31#19153328C2211yescharacteristic, complemented
32#19153326C2211yescharacteristic, complemented
33#19153324C2212yescomplemented
34C2×C359⋊C8C2211yescharacteristic, complemented
35C8×D359C2212yescomplemented
36#41×C359C2211yescharacteristic, complemented
37C2×D4×C359C2211yescharacteristic
38SD16×C359C2212yescomplemented
39D8×C359C2212yescomplemented
40M16×C359C2211yescharacteristic, complemented
41C22×D359—22no—
42Dic718—21no—
43Dic718—22no—
44Dic718D411yescharacteristic
45Dic718C2312yes—
46Dic718C2311yescharacteristic
47D1436D411yescharacteristic
48D1436C2311yescharacteristic
49C22⋊D359—21no—
50C22⋊D359—22no—
51C22⋊D359—22no—
52C22⋊D359—22no—
53C22⋊D359D412yes—
54C2×Dic359—21no—
55C2×Dic359—22no—
56C2×Dic359—22no—
57C4×D359—22no—
58C4×D359—21no—
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—82no—
77C2×D359C2×D412yes—
78C4×C359—21no—
79C4×C359C2×D411yescharacteristic
80C4×C359C2411yescharacteristic
81Dic359—21no—
82Dic359—22no—
83Dic359C2×D412yes—
84C2×C359—21no—
85C2×C359—42no—
86C2×C359—41no—
87C2×C359C22×D411yescharacteristic
88D359—22no—
89D359—41no—
90C359#57711yescharacteristic, complemented, Sylow
91#577—3591nomaximal, Sylow
922−1+4—3591no—
93H(5,𝔽2)—3591no—
94#137—3592no—
95#137—3591no—
96AGL(1,ℤ/8)—3592no—
97AGL(1,ℤ/8)D35911yescharacteristic, complemented
98#135—3591no—
99#135—3592no—
100C2×SD16—3591no—
101C2×SD16—3592no—
102#131—3591no—
103#41—3591no—
104#41—3592no—
105#41—3592no—
106#41—3591no—
107#41—7181no—
108#41—7182no—
109#41—7181no—
110#41C2×D35911yescharacteristic, complemented
111C2×Q8—3592no—
112C2×Q8—3591no—
113C2×Q8—7181no—
114C2×D4—3592no—
115C2×D4—7181no—
116C2×D4—7182no—
117C2×D4C2×D35911yescharacteristic
118Q16—3591no—
119Q16—3592no—
120SD16—3591no—
121SD16—3592no—
122SD16—3592no—
123SD16—3591no—
124SD16—3592no—
125SD16C2×D35912yescomplemented
126D8—3591no—
127D8C2×D35912yescomplemented
128M16—3592no—
129M16C2×D35911yescharacteristic, complemented
130C2×C8—3591no—
131C2×C8—3592no—
132C23—21no—
133C23—7182no—
134Q8—3591no—
135Q8—3592no—
136Q8—3591no—
137Q8—7182no—
138Q8—7181no—
139Q8C22×D35911yescharacteristic
140D4—21no—
141D4—21no—
142D4—3592no—
143D4—3591no—
144D4—3591no—
145D4—7182no—
146D4—7182no—
147D4—7182no—
148D4—7181no—
149D4C22×D35911yescharacteristic
150D4C22×D35912yes—
151C2×C4—21no—
152C2×C4—3592no—
153C2×C4—7182no—
154C2×C4—7182no—
155C2×C4—7182no—
156C2×C4—7181no—
157C2×C4—7181no—
158C2×C4C22×D35911yescharacteristic
159C8—3591no—
160C8—3591no—
161C8C22×D35912yes—
162C22—21no—
163C22—22no—
164C22—81no—
165C22—28722no—
166C22—3592no—
167C22—7181no—
168C22D4×D35911yescharacteristic
169C4—21no—
170C4—3592no—
171C4—7181no—
172C4—7182no—
173C4D4×D35911yescharacteristic
174C4C23×D35911yescharacteristic
175C2—21no—
176C2—42no—
177C2—41no—
178C2—14361no—
179C2—7182no—
180C2C2×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#577
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. #26040062C2semidirect
2. #26040060C2semidirect
3. #26040048C2semidirect
4. #26040044C2semidirect
5. #26040039C2semidirect
6. #26040035C2semidirect
7. #26040034C2semidirect
8. C2×#19153329C2semidirect
9. SD16×D359C2semidirect
10. #26040023C2semidirect
11. AGL(1,ℤ/8)×C359C2semidirect
18. #19153340C22semidirect
20. C2×Dic718C22semidirect
21. Q8×D359C22semidirect
25. Dic1436C22semidirect
26. #19153331C22semidirect
27. #19153330C22semidirect
28. #19153329C22semidirect
29. #19153329C22semidirect
30. #19153329C22semidirect
31. #19153328C22semidirect
32. #19153326C22semidirect
33. #19153324C22semidirect
34. C2×C359⋊C8C22semidirect
35. C8×D359C22semidirect
36. #41×C359C22semidirect
38. SD16×C359C22semidirect
39. D8×C359C22semidirect
40. M16×C359C22semidirect
90. C359#577semidirect
97. AGL(1,ℤ/8)D359semidirect
110. #41C2×D359semidirect
125. SD16C2×D359semidirect
127. D8C2×D359semidirect
129. M16C2×D359semidirect

Maximal subgroups

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

subgroupindex
#260400622
#260400602
#260400482
#260400442
#260400392
#260400352
#260400342
C2×#191533292
SD16×D3592
#260400232
AGL(1,ℤ/8)×C3592
#577359