Finite Group Databasebeta

← #13136945

Subgroups — #13136945

Order 1728.

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

normal subgroups only

This group has 345 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#11721850C211yesmaximal, characteristic
2#11720780C212yesmaximal, complemented
3C2×D4×H(3,𝔽3)C411yescharacteristic, complemented
4C2×#6235—22no—
5C2×C4⋊#272C2211yescharacteristic
6C22⋊#1088—22no—
7C4×#1084C411yescharacteristic, complemented
8C3×#940—62no—
9C23×H(3,𝔽3)—21no—
10C22×#272D412yes—
11C2×#1088—42no—
12D4×H(3,𝔽3)—41no—
13C2×C4×H(3,𝔽3)C2×C411yescharacteristic
14#6235—44no—
15C2×#1084—21no—
16C4⋊#272—41no—
17C3×#353—91nomaximal
18C2×C32×D4—22no—
19C2×C3×#67—68no—
20C2×C3×D12—62no—
21C3×C4×Dic3—62no—
22C22×H(3,𝔽3)—42no—
23C22×H(3,𝔽3)—41no—
24C22×H(3,𝔽3)C22⋊C411yescharacteristic
25C2×#272—22no—
26C2×#272—42no—
27#1088—82no—
28C4×H(3,𝔽3)—21no—
29#1084—22no—
30C3×#127—91no—
31#940—62no—
32C3×#99—92no—
33C23×C32—24no—
34C22×C3×D3—64no—
35C32×D4—44no—
36C2×C32×C4—22no—
37C3×#67—1216no—
38C2×C3×Dic3—64no—
39C3×D12—124no—
40#353—91noSylow
41C2×H(3,𝔽3)—21no—
42C2×H(3,𝔽3)—81no—
43C2×H(3,𝔽3)#9911yescharacteristic
44#272—42no—
45C2×C3×D4—62no—
46C2×C3×D4—91no—
47C2×C3×D4—182no—
48C2×C3×D4#15711yescharacteristic
49C2×#67—68no—
50C2×D12—62no—
51C3×C22⋊C4—182no—
52C3×C42—91no—
53C4×Dic3—62no—
54C22×C32—22no—
55C22×C32—48no—
56C22×C32—44no—
57C2×C3×D3—124no—
58C2×C3×D3—128no—
59C32×C4—24no—
60C3×Dic3—68no—
61#127—91no—
62#99—92no—
63H(3,𝔽3)#35311yescharacteristic, complemented, Sylow
64C23×C3—21no—
65C23×C3—64no—
66C23×C3—92no—
67C22×D3—64no—
68C3×D4—41no—
69C3×D4—124no—
70C3×D4—361no—
71C3×D4—364no—
72C2×C3×C4—62no—
73C2×C3×C4—181no—
74C2×C3×C4—362no—
75C2×C3×C4C2×#15711yescharacteristic
76#67—1216no—
77C2×Dic3—64no—
78D12—124no—
79C2×C32—24no—
80C2×C32—22no—
81C2×C32—84no—
82C3×D3—244no—
83C2×D4—91no—
84C2×D4—182no—
85C2×D4#108811yescharacteristic, complemented
86C22⋊C4—182no—
87C42—91no—
88C22×C3—42no—
89C22×C3—41no—
90C22×C3—62no—
91C22×C3—128no—
92C22×C3—124no—
93C22×C3—182no—
94C22×C3—362no—
95C22×C3C22⋊#15711yescharacteristic
96C2×D3—124no—
97C2×D3—128no—
98C3×C4—21no—
99C3×C4—64no—
100C3×C4—182no—
101C3×C4—722no—
102Dic3—68no—
103C32—22no—
104C23—21no—
105C23—92no—
106D4—41no—
107D4—361no—
108D4—364no—
109C2×C4—181no—
110C2×C4—362no—
111C2×C4C2×#108811yescharacteristic
112C2×C3—21no—
113C2×C3—64no—
114C2×C3—62no—
115C2×C3—81no—
116C2×C3—244no—
117C2×C3—362no—
118C2×C3#6378611yescharacteristic
119D3—244no—
120C22—42no—
121C22—41no—
122C22—182no—
123C22—362no—
124C22C22⋊#108811yescharacteristic
125C4—21no—
126C4—182no—
127C4—722no—
128C3—62no—
129C3#1059021111yescharacteristic
130C2—21no—
131C2—81no—
132C2—362no—
133C2#1172079211yescharacteristic
subgroupG/H
center, socleC2×C3#63786
commutatorC2×C4×H(3,𝔽3)C2×C4
FrattiniC2×C3×C4C2×#157
FittingC2×D4×H(3,𝔽3)C4
prime radical#11721850C2
cosocleC2×C4⋊#272C22
hypercenterC2×C3×D4#157
hypocenterH(3,𝔽3)#353
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×D4

Complements

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

NH ≅ G/Ntype
2. #11720780C2semidirect
3. C2×D4×H(3,𝔽3)C4semidirect
7. C4×#1084C4semidirect
63. H(3,𝔽3)#353semidirect
85. C2×D4#1088semidirect

Maximal subgroups

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

subgroupindex
#117218502
#117207802
C3×#3539