Finite Group Databasebeta

← #28416601

Subgroups — #28416601

Order 14912.

168 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 39 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#21418581C211yesmaximal, characteristic, complemented
2#21418561C211yesmaximal, characteristic, complemented
3#21418531C211yesmaximal, characteristic, complemented
4#21418462C211yesmaximal, characteristic, complemented
5#21418461C211yesmaximal, characteristic, complemented
6#21418451C211yesmaximal, characteristic, complemented
7AGL(1,ℤ/8)×C233C211yesmaximal, characteristic, complemented
8#16836595—21no—
9#16836594—21no—
10#16836594C2211yescharacteristic
11C2×D932—21no—
12C2×D932C2211yescharacteristic
13C2×C22⋊D233—22no—
14D4×D233—22no—
15D4×D233—21no—
16#16836582—21no—
17#16836579—21no—
18#16836578C2211yescharacteristic, complemented
19C4×Dic233C2211yescharacteristic, complemented
20#41×C233C2211yescharacteristic
21C2×D4×C233C2211yescharacteristic, complemented
22SD16×C233—21no—
23D8×C233—21no—
24M16×C233C2211yescharacteristic, complemented
25C22×D233—22no—
26C22×D233D412yes—
27Dic466D411yescharacteristic, complemented
28D932—22no—
29D932—21no—
30D932—21no—
31D932D411yescharacteristic, complemented
32C22⋊D233—22no—
33C22⋊D233—21no—
34C22⋊D233—44no—
35C2×Dic233—21no—
36C4×D233—22no—
37C4×D233—21no—
38#14272338—21no—
39C23×C233—21no—
40Q8×C233D411yescharacteristic, complemented
41D4×C233—21no—
42D4×C233—21no—
43D4×C233—21no—
44D4×C233D411yescharacteristic, complemented
45C2×C4×C233—21no—
46C2×C4×C233C2311yescharacteristic
47C8×C233—21no—
48C22×C233—21no—
49C22×C233—42no—
50C22×C233—41no—
51C22×C233C2×D411yescharacteristic
52C2×D233—22no—
53C2×D233—21no—
54C2×D233—42no—
55C2×D233—82no—
56C4×C233—21no—
57C4×C233C2×D411yescharacteristic
58C4×C233C2×D411yescharacteristic
59Dic233—22no—
60Dic233—21no—
61C2×C233—21no—
62C2×C233—41no—
63C2×C233—81no—
64C2×C233#12011yescharacteristic
65D233—42no—
66D233—41no—
67C233#45311yescharacteristic, complemented, Sylow
68#453—2331nomaximal, Sylow
69H(5,𝔽2)—2331no—
70AGL(1,ℤ/8)—2331no—
71AGL(1,ℤ/8)D23311yescharacteristic, complemented
72#127—2331no—
73#104—2331no—
74#104—2331no—
75#100—2331no—
76#41—4661no—
77#41—4661no—
78#41—2331no—
79#41C2×D23311yescharacteristic
80C2×D4—4661no—
81C2×D4—4662no—
82C2×D4—4662no—
83C2×D4—4661no—
84C2×D4—2331no—
85C2×D4C2×D23311yescharacteristic, complemented
86SD16—21no—
87SD16—4661no—
88D8—21no—
89D8—4661no—
90M16—2331no—
91M16C2×D23311yescharacteristic, complemented
92C42—2331no—
93C23—21no—
94C23—4662no—
95C23—2332no—
96Q8—2331no—
97Q8C22⋊D23311yescharacteristic, complemented
98D4—21no—
99D4—21no—
100D4—21no—
101D4—9324no—
102D4—4662no—
103D4—4662no—
104D4—4661no—
105D4—4661no—
106D4—4661no—
107D4—2331no—
108D4C22⋊D23311yescharacteristic, complemented
109C2×C4—21no—
110C2×C4—4662no—
111C2×C4—4661no—
112C2×C4—4661no—
113C2×C4C22×D23311yescharacteristic
114C8—21no—
115C8—4661no—
116C22—21no—
117C22—42no—
118C22—41no—
119C22—18642no—
120C22—9322no—
121C22—4662no—
122C22—4661no—
123C22D4×D23311yescharacteristic
124C4—21no—
125C4—4662no—
126C4—4661no—
127C4D4×D23311yescharacteristic
128C4C2×C22⋊D23311yescharacteristic
129C2—21no—
130C2—41no—
131C2—81no—
132C2—9321no—
133C2—9322no—
134C2#2141850311yescharacteristic
subgroupG/H
centerC2#21418503
commutator, cosocleC2×C4×C233C23
FrattiniC2×C4C22×D233
FittingAGL(1,ℤ/8)×C233C2
socleC2×C233#120
hypercenterAGL(1,ℤ/8)D233
hypocenterC233#453
LCSG ▷ C2×C4×C233 ▷ C2×C233 ▷ C233
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #21418581C2semidirect
2. #21418561C2semidirect
3. #21418531C2semidirect
4. #21418462C2semidirect
5. #21418461C2semidirect
6. #21418451C2semidirect
7. AGL(1,ℤ/8)×C233C2semidirect
18. #16836578C22semidirect
19. C4×Dic233C22semidirect
21. C2×D4×C233C22semidirect
24. M16×C233C22semidirect
27. Dic466D4semidirect
31. D932D4semidirect
40. Q8×C233D4semidirect
44. D4×C233D4semidirect
67. C233#453semidirect
71. AGL(1,ℤ/8)D233semidirect
85. C2×D4C2×D233semidirect
91. M16C2×D233semidirect
97. Q8C22⋊D233semidirect
108. D4C22⋊D233semidirect

Maximal subgroups

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

subgroupindex
#214185812
#214185612
#214185312
#214184622
#214184612
#214184512
AGL(1,ℤ/8)×C2332
#453233