Finite Group Databasebeta

← #13136951

Subgroups — #13136951

Order 1728.

257 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 257 conjugacy classes of subgroups — too many to draw. Only its 23 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#11721864C211yesmaximal, characteristic
2#11720780C212yesmaximal
3C2×Q8×H(3,𝔽3)C411yescharacteristic, complemented
4C2×C4⋊#272C2211yescharacteristic
5C22⋊#1088—22no—
6#88542—22no—
7C4×#1084C411yescharacteristic, complemented
8C3×#947—62no—
9C22×#272D412yes—
10C2×#1088—42no—
11Q8×H(3,𝔽3)—41no—
12C2×C4×H(3,𝔽3)—21no—
13C2×C4×H(3,𝔽3)C2×C411yescharacteristic
14C2×#1084—21no—
15C4⋊#272—41no—
16C3×#354—91nomaximal
17C2×C32×Q8—22no—
18C2×C3×D12—62no—
19C3×(C2×D3)⋊C4—68no—
20C3×C4×Dic3—62no—
21C22×H(3,𝔽3)C22⋊C411yescharacteristic
22C2×#272—22no—
23C2×#272—42no—
24#1088—82no—
25C4×H(3,𝔽3)—21no—
26C4×H(3,𝔽3)—41no—
27#1084—41no—
28C3×#124—91no—
29#947—62no—
30C3×#99—92no—
31C22×C3×D3—64no—
32C32×Q8—44no—
33C2×C32×C4—24no—
34C2×C32×C4—22no—
35C2×C3×Dic3—64no—
36C3×D12—124no—
37#354—91noSylow
38C2×H(3,𝔽3)—21no—
39C2×H(3,𝔽3)#9911yescharacteristic
40#272—42no—
41C2×C3×Q8—62no—
42C2×C3×Q8#15711yescharacteristic
43C2×C3×D4—91no—
44C2×D12—62no—
45C3×C22⋊C4—182no—
46C3×C22⋊C4—182no—
47C3×C42—91no—
48(C2×D3)⋊C4—68no—
49C4×Dic3—62no—
50C22×C32—22no—
51C2×C3×D3—124no—
52C2×C3×D3—128no—
53C32×C4—24no—
54C32×C4—44no—
55C3×Dic3—124no—
56#124—91no—
57#99—92no—
58H(3,𝔽3)#35411yescharacteristic, complemented, Sylow
59C23×C3—92no—
60C22×D3—64no—
61C3×Q8—41no—
62C3×Q8—124no—
63C3×D4—361no—
64C2×C3×C4—21no—
65C2×C3×C4—64no—
66C2×C3×C4—62no—
67C2×C3×C4—181no—
68C2×C3×C4—362no—
69C2×C3×C4C2×#15711yescharacteristic
70C2×Dic3—64no—
71D12—124no—
72C2×C32—24no—
73C2×C32—22no—
74C3×D3—244no—
75C2×Q8#108811yescharacteristic, complemented
76C2×D4—91no—
77C22⋊C4—182no—
78C22⋊C4—182no—
79C42—91no—
80C22×C3—62no—
81C22×C3—182no—
82C22×C3—362no—
83C22×C3C22⋊#15711yescharacteristic
84C2×D3—124no—
85C2×D3—128no—
86C3×C4—21no—
87C3×C4—41no—
88C3×C4—64no—
89C3×C4—124no—
90C3×C4—361no—
91C3×C4—722no—
92Dic3—124no—
93C32—22no—
94C23—92no—
95Q8—41no—
96D4—361no—
97C2×C4—21no—
98C2×C4—181no—
99C2×C4—362no—
100C2×C4C2×#108811yescharacteristic
101C2×C3—21no—
102C2×C3—64no—
103C2×C3—62no—
104C2×C3—362no—
105C2×C3#6378611yescharacteristic
106D3—244no—
107C22—182no—
108C22—362no—
109C22C22⋊#108811yescharacteristic
110C4—21no—
111C4—41no—
112C4—361no—
113C4—722no—
114C3—62no—
115C3#1059021711yescharacteristic
116C2—21no—
117C2—362no—
118C2#1172079211yescharacteristic
subgroupG/H
center, socleC2×C3#63786
commutatorC2×C4×H(3,𝔽3)C2×C4
FrattiniC2×C3×C4C2×#157
FittingC2×Q8×H(3,𝔽3)C4
prime radical, cosocleC2×C4⋊#272C22
hypercenterC2×C3×Q8#157
hypocenterH(3,𝔽3)#354
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
3. C2×Q8×H(3,𝔽3)C4semidirect
7. C4×#1084C4semidirect
58. H(3,𝔽3)#354semidirect
75. C2×Q8#1088semidirect

Maximal subgroups

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

subgroupindex
#117218642
#117207802
C3×#3549