Finite Group Databasebeta

← #38734265

Subgroups — #38734265

Order 24896.

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#27159430C211yesmaximal, characteristic, complemented
2#27159410C211yesmaximal, characteristic, complemented
3#27159380C211yesmaximal, characteristic, complemented
4#27159311C211yesmaximal, characteristic, complemented
5#27159310C211yesmaximal, characteristic, complemented
6#27159300C211yesmaximal, characteristic, complemented
7AGL(1,ℤ/8)×C389C211yesmaximal, characteristic, complemented
8#20261967—21no—
9#20261966—21no—
10#20261966C2211yescharacteristic
11C2×D1556—21no—
12C2×D1556C2211yescharacteristic
13C2×C22⋊D389—22no—
14D4×D389—21no—
15D4×D389—22no—
16#20261954—21no—
17#20261951—21no—
18#20261950C2211yescharacteristic, complemented
19C4×Dic389C2211yescharacteristic, complemented
20#41×C389C2211yescharacteristic
21C2×D4×C389C2211yescharacteristic, complemented
22SD16×C389—21no—
23D8×C389—21no—
24M16×C389C2211yescharacteristic, complemented
25C22×D389—22no—
26C22×D389D412yes—
27Dic778D411yescharacteristic, complemented
28D1556—22no—
29D1556—21no—
30D1556—21no—
31D1556D411yescharacteristic, complemented
32C22⋊D389—22no—
33C22⋊D389—21no—
34C22⋊D389—44no—
35C2×Dic389—21no—
36C4×D389—22no—
37C4×D389—21no—
38#15691005—21no—
39C23×C389—21no—
40Q8×C389D411yescharacteristic, complemented
41D4×C389—21no—
42D4×C389—21no—
43D4×C389—21no—
44D4×C389D411yescharacteristic, complemented
45C2×C4×C389—21no—
46C2×C4×C389C2311yescharacteristic
47C8×C389—21no—
48C22×C389—21no—
49C22×C389—42no—
50C22×C389—41no—
51C22×C389C2×D411yescharacteristic
52C2×D389—22no—
53C2×D389—21no—
54C2×D389—42no—
55C2×D389—82no—
56C4×C389—21no—
57C4×C389C2×D411yescharacteristic
58C4×C389C2×D411yescharacteristic
59Dic389—22no—
60Dic389—21no—
61C2×C389—21no—
62C2×C389—41no—
63C2×C389—81no—
64C2×C389#12011yescharacteristic
65D389—42no—
66D389—41no—
67C389#45311yescharacteristic, complemented, Sylow
68#453—3891nomaximal, Sylow
69H(5,𝔽2)—3891no—
70AGL(1,ℤ/8)—3891no—
71AGL(1,ℤ/8)D38911yescharacteristic, complemented
72#127—3891no—
73#104—3891no—
74#104—3891no—
75#100—3891no—
76#41—7781no—
77#41—7781no—
78#41—3891no—
79#41C2×D38911yescharacteristic
80C2×D4—7781no—
81C2×D4—7782no—
82C2×D4—7781no—
83C2×D4—7782no—
84C2×D4—3891no—
85C2×D4C2×D38911yescharacteristic, complemented
86SD16—21no—
87SD16—7781no—
88D8—21no—
89D8—7781no—
90M16—3891no—
91M16C2×D38911yescharacteristic, complemented
92C42—3891no—
93C23—21no—
94C23—7782no—
95C23—3892no—
96Q8—3891no—
97Q8C22⋊D38911yescharacteristic, complemented
98D4—21no—
99D4—21no—
100D4—21no—
101D4—7782no—
102D4—7782no—
103D4—7781no—
104D4—7781no—
105D4—7781no—
106D4—15564no—
107D4—3891no—
108D4C22⋊D38911yescharacteristic, complemented
109C2×C4—21no—
110C2×C4—7782no—
111C2×C4—7781no—
112C2×C4—7781no—
113C2×C4C22×D38911yescharacteristic
114C8—21no—
115C8—7781no—
116C22—21no—
117C22—42no—
118C22—41no—
119C22—7782no—
120C22—7781no—
121C22—15562no—
122C22—31122no—
123C22D4×D38911yescharacteristic
124C4—21no—
125C4—7782no—
126C4—7781no—
127C4D4×D38911yescharacteristic
128C4C2×C22⋊D38911yescharacteristic
129C2—21no—
130C2—41no—
131C2—81no—
132C2—15562no—
133C2—15561no—
134C2#2715935211yescharacteristic
subgroupG/H
centerC2#27159352
commutator, cosocleC2×C4×C389C23
FrattiniC2×C4C22×D389
FittingAGL(1,ℤ/8)×C389C2
socleC2×C389#120
hypercenterAGL(1,ℤ/8)D389
hypocenterC389#453
LCSG ▷ C2×C4×C389 ▷ C2×C389 ▷ C389
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #27159430C2semidirect
2. #27159410C2semidirect
3. #27159380C2semidirect
4. #27159311C2semidirect
5. #27159310C2semidirect
6. #27159300C2semidirect
7. AGL(1,ℤ/8)×C389C2semidirect
18. #20261950C22semidirect
19. C4×Dic389C22semidirect
21. C2×D4×C389C22semidirect
24. M16×C389C22semidirect
27. Dic778D4semidirect
31. D1556D4semidirect
40. Q8×C389D4semidirect
44. D4×C389D4semidirect
67. C389#453semidirect
71. AGL(1,ℤ/8)D389semidirect
85. C2×D4C2×D389semidirect
91. M16C2×D389semidirect
97. Q8C22⋊D389semidirect
108. D4C22⋊D389semidirect

Maximal subgroups

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

subgroupindex
#271594302
#271594102
#271593802
#271593112
#271593102
#271593002
AGL(1,ℤ/8)×C3892
#453389