Finite Group Databasebeta

← #13136950

Subgroups — #13136950

Order 1728.

257 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 257 conjugacy classes of subgroups — too many to draw. Only its 29 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#11720783C212yesmaximal, complemented
3C2×Q8×H(3,𝔽3)C411yescharacteristic
4C2×C4⋊#272C411yescharacteristic
5C2×#6188—22no—
6#88542—22no—
7C4×#1084C2211yescharacteristic
8C3×#947—62no—
9C22×#272—21no—
10Q8×H(3,𝔽3)—41no—
11C2×C4×H(3,𝔽3)—21no—
12C2×C4×H(3,𝔽3)C2×C411yescharacteristic
13C2×#1084D412yes—
14C4⋊#272—41no—
15#6188—24no—
16C3×#329—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—41no—
23C2×#272—42no—
24C4×H(3,𝔽3)—21no—
25C4×H(3,𝔽3)—41no—
26#1084—22no—
27C3×#124—91no—
28#947—62no—
29C3×#97—92no—
30C22×C3×D3—64no—
31C32×Q8—44no—
32C2×C32×C4—24no—
33C2×C32×C4—22no—
34C2×C3×Dic3—64no—
35C3×D12—124no—
36#329—91noSylow
37C2×H(3,𝔽3)#10011yescharacteristic
38C2×H(3,𝔽3)#10412yes—
39#272—81no—
40C2×C3×Q8—62no—
41C2×C3×Q8#15711yescharacteristic
42C2×C3×D4—91no—
43C2×D12—62no—
44C2×C3×C8—182no—
45C3×C22⋊C4—182no—
46C3×C42—91no—
47(C2×D3)⋊C4—68no—
48C4×Dic3—62no—
49C22×C32—22no—
50C2×C3×D3—124no—
51C2×C3×D3—128no—
52C32×C4—24no—
53C32×C4—44no—
54C3×Dic3—124no—
55#124—91no—
56#97—92no—
57H(3,𝔽3)#32911yescharacteristic, complemented, Sylow
58C23×C3—181no—
59C22×D3—64no—
60C3×Q8—41no—
61C3×Q8—124no—
62C3×D4—361no—
63C2×C3×C4—21no—
64C2×C3×C4—64no—
65C2×C3×C4—62no—
66C2×C3×C4—92no—
67C2×C3×C4C2×#15711yescharacteristic
68C2×Dic3—64no—
69D12—124no—
70C3×C8—184no—
71C2×C32—24no—
72C2×C32—22no—
73C3×D3—244no—
74C2×Q8#108811yescharacteristic
75C2×D4—91no—
76C2×C8—182no—
77C22⋊C4—182no—
78C42—91no—
79C22×C3—62no—
80C22×C3—362no—
81C22×C3—361no—
82C22×C3C22⋊#15711yescharacteristic
83C2×D3—124no—
84C2×D3—128no—
85C3×C4—21no—
86C3×C4—41no—
87C3×C4—64no—
88C3×C4—124no—
89C3×C4—182no—
90Dic3—124no—
91C32—22no—
92C23—181no—
93Q8—41no—
94D4—361no—
95C2×C4—21no—
96C2×C4—92no—
97C2×C4C2×#108811yescharacteristic
98C8—184no—
99C2×C3—64no—
100C2×C3—62no—
101C2×C3—721no—
102C2×C3#6378512yes—
103C2×C3#6378811yescharacteristic
104D3—244no—
105C22—362no—
106C22—361no—
107C22C22⋊#108811yescharacteristic
108C4—21no—
109C4—41no—
110C4—182no—
111C3—62no—
112C3#1059021611yescharacteristic
113C2—721no—
114C2#1172079112yes—
115C2#1172079411yescharacteristic
subgroupG/H
center, socleC22×C3C22⋊#157
commutatorC2×C4×H(3,𝔽3)C2×C4
FrattiniC2×C3×C4C2×#157
FittingC2×Q8×H(3,𝔽3)C4
prime radicalC2×C4⋊#272C4
cosocleC4×#1084C22
hypercenterC2×C3×Q8#157
hypocenterH(3,𝔽3)#329
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
2. #11720783C2semidirect
57. H(3,𝔽3)#329semidirect

Maximal subgroups

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

subgroupindex
#117218642
#117207832
C3×#3299