Finite Group Databasebeta

← #37565410

Subgroups — #37565410

Order 24256.

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#27131377C211yesmaximal, characteristic, complemented
2#27131357C211yesmaximal, characteristic, complemented
3#27131327C211yesmaximal, characteristic, complemented
4#27131258C211yesmaximal, characteristic, complemented
5#27131257C211yesmaximal, characteristic, complemented
6#27131247C211yesmaximal, characteristic, complemented
7AGL(1,ℤ/8)×C379C211yesmaximal, characteristic, complemented
8#20258607—21no—
9#20258606—21no—
10#20258606C2211yescharacteristic
11C2×D1516—21no—
12C2×D1516C2211yescharacteristic
13C2×C22⋊D379—22no—
14D4×D379—22no—
15D4×D379—21no—
16#20258594—21no—
17#20258591—21no—
18#20258590C2211yescharacteristic, complemented
19C4×Dic379C2211yescharacteristic, complemented
20#41×C379C2211yescharacteristic
21C2×D4×C379C2211yescharacteristic, complemented
22SD16×C379—21no—
23D8×C379—21no—
24M16×C379C2211yescharacteristic, complemented
25C22×D379—22no—
26C22×D379D412yes—
27Dic758D411yescharacteristic, complemented
28D1516—22no—
29D1516—21no—
30D1516—21no—
31D1516D411yescharacteristic, complemented
32C22⋊D379—22no—
33C22⋊D379—21no—
34C22⋊D379—44no—
35C2×Dic379—21no—
36C4×D379—22no—
37C4×D379—21no—
38C379⋊C8—21no—
39C23×C379—21no—
40Q8×C379D411yescharacteristic, complemented
41D4×C379—21no—
42D4×C379—21no—
43D4×C379—21no—
44D4×C379D411yescharacteristic, complemented
45C2×C4×C379—21no—
46C2×C4×C379C2311yescharacteristic
47C8×C379—21no—
48C22×C379—21no—
49C22×C379—42no—
50C22×C379—41no—
51C22×C379C2×D411yescharacteristic
52C2×D379—22no—
53C2×D379—21no—
54C2×D379—42no—
55C2×D379—82no—
56C4×C379—21no—
57C4×C379C2×D411yescharacteristic
58C4×C379C2×D411yescharacteristic
59Dic379—22no—
60Dic379—21no—
61C2×C379—21no—
62C2×C379—41no—
63C2×C379—81no—
64C2×C379#12011yescharacteristic
65D379—42no—
66D379—41no—
67C379#45311yescharacteristic, complemented, Sylow
68#453—3791nomaximal, Sylow
69H(5,𝔽2)—3791no—
70AGL(1,ℤ/8)—3791no—
71AGL(1,ℤ/8)D37911yescharacteristic, complemented
72#127—3791no—
73#104—3791no—
74#104—3791no—
75#100—3791no—
76#41—3791no—
77#41—7581no—
78#41—7581no—
79#41C2×D37911yescharacteristic
80C2×D4—3791no—
81C2×D4—7581no—
82C2×D4—7582no—
83C2×D4—7582no—
84C2×D4—7581no—
85C2×D4C2×D37911yescharacteristic, complemented
86SD16—21no—
87SD16—7581no—
88D8—21no—
89D8—7581no—
90M16—3791no—
91M16C2×D37911yescharacteristic, complemented
92C42—3791no—
93C23—21no—
94C23—3792no—
95C23—7582no—
96Q8—3791no—
97Q8C22⋊D37911yescharacteristic, complemented
98D4—21no—
99D4—21no—
100D4—21no—
101D4—3791no—
102D4—15164no—
103D4—7582no—
104D4—7582no—
105D4—7581no—
106D4—7581no—
107D4—7581no—
108D4C22⋊D37911yescharacteristic, complemented
109C2×C4—21no—
110C2×C4—7582no—
111C2×C4—7581no—
112C2×C4—7581no—
113C2×C4C22×D37911yescharacteristic
114C8—21no—
115C8—7581no—
116C22—21no—
117C22—42no—
118C22—41no—
119C22—30322no—
120C22—15162no—
121C22—7581no—
122C22—7582no—
123C22D4×D37911yescharacteristic
124C4—21no—
125C4—7582no—
126C4—7581no—
127C4D4×D37911yescharacteristic
128C4C2×C22⋊D37911yescharacteristic
129C2—21no—
130C2—41no—
131C2—81no—
132C2—15161no—
133C2—15162no—
134C2#2713129911yescharacteristic
subgroupG/H
centerC2#27131299
commutator, cosocleC2×C4×C379C23
FrattiniC2×C4C22×D379
FittingAGL(1,ℤ/8)×C379C2
socleC2×C379#120
hypercenterAGL(1,ℤ/8)D379
hypocenterC379#453
LCSG ▷ C2×C4×C379 ▷ C2×C379 ▷ C379
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #27131377C2semidirect
2. #27131357C2semidirect
3. #27131327C2semidirect
4. #27131258C2semidirect
5. #27131257C2semidirect
6. #27131247C2semidirect
7. AGL(1,ℤ/8)×C379C2semidirect
18. #20258590C22semidirect
19. C4×Dic379C22semidirect
21. C2×D4×C379C22semidirect
24. M16×C379C22semidirect
27. Dic758D4semidirect
31. D1516D4semidirect
40. Q8×C379D4semidirect
44. D4×C379D4semidirect
67. C379#453semidirect
71. AGL(1,ℤ/8)D379semidirect
85. C2×D4C2×D379semidirect
91. M16C2×D379semidirect
97. Q8C22⋊D379semidirect
108. D4C22⋊D379semidirect

Maximal subgroups

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

subgroupindex
#271313772
#271313572
#271313272
#271312582
#271312572
#271312472
AGL(1,ℤ/8)×C3792
#453379