Finite Group Databasebeta

← #13136948

Subgroups — #13136948

Order 1728.

342 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 33 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#11720785C212yesmaximal, complemented
3C2×D4×H(3,𝔽3)C411yescharacteristic
4C2×#6235—22no—
5C2×C4⋊#272C411yescharacteristic
6C2×#6188—22no—
7C4×#1084C2211yescharacteristic
8C3×#940—62no—
9C23×H(3,𝔽3)—21no—
10C22×#272—21no—
11D4×H(3,𝔽3)—41no—
12C2×C4×H(3,𝔽3)C2×C411yescharacteristic
13#6235—42no—
14#6235—42no—
15C2×#1084D412yes—
16C4⋊#272—41no—
17#6188—42no—
18C3×#331—91nomaximal
19C2×C32×D4—22no—
20C2×C3×#67—68no—
21C2×C3×D12—62no—
22C3×C4×Dic3—62no—
23C22×H(3,𝔽3)—41no—
24C22×H(3,𝔽3)—42no—
25C22×H(3,𝔽3)C22⋊C411yescharacteristic
26C2×#272—41no—
27C2×#272—42no—
28C4×H(3,𝔽3)—21no—
29#1084D812yes—
30#1084SD1612yes—
31C3×#127—91no—
32#940—62no—
33C3×C4⋊C8—92no—
34C23×C32—24no—
35C22×C3×D3—64no—
36C32×D4—44no—
37C2×C32×C4—22no—
38C3×#67—128no—
39C3×#67—128no—
40C2×C3×Dic3—64no—
41C3×D12—124no—
42#331—91noSylow
43C2×H(3,𝔽3)—81no—
44C2×H(3,𝔽3)#10011yescharacteristic
45C2×H(3,𝔽3)#10212yes—
46#272—81no—
47C2×C3×D4—62no—
48C2×C3×D4—91no—
49C2×C3×D4—182no—
50C2×C3×D4#15711yescharacteristic
51C2×#67—68no—
52C2×D12—62no—
53C2×C3×C8—182no—
54C3×C42—91no—
55C4×Dic3—62no—
56C22×C32—22no—
57C22×C32—48no—
58C22×C32—44no—
59C2×C3×D3—124no—
60C2×C3×D3—128no—
61C32×C4—24no—
62C3×Dic3—64no—
63C3×Dic3—64no—
64#127—91no—
65C4⋊C8—92no—
66H(3,𝔽3)#33111yescharacteristic, complemented, Sylow
67C23×C3—21no—
68C23×C3—64no—
69C23×C3—181no—
70C22×D3—64no—
71C3×D4—41no—
72C3×D4—124no—
73C3×D4—361no—
74C3×D4—362no—
75C3×D4—362no—
76C2×C3×C4—62no—
77C2×C3×C4—92no—
78C2×C3×C4C2×#15711yescharacteristic
79#67—128no—
80#67—128no—
81C2×Dic3—64no—
82D12—124no—
83C3×C8—362no—
84C2×C32—24no—
85C2×C32—22no—
86C2×C32—84no—
87C3×D3—244no—
88C2×D4—91no—
89C2×D4—182no—
90C2×D4#108811yescharacteristic
91C2×C8—182no—
92C42—91no—
93C22×C3—42no—
94C22×C3—41no—
95C22×C3—62no—
96C22×C3—128no—
97C22×C3—124no—
98C22×C3—362no—
99C22×C3—361no—
100C22×C3C22⋊#15711yescharacteristic
101C2×D3—124no—
102C2×D3—128no—
103C3×C4—21no—
104C3×C4—64no—
105C3×C4—92no—
106C3×C4—92no—
107Dic3—64no—
108Dic3—64no—
109C32—22no—
110C23—21no—
111C23—181no—
112D4—41no—
113D4—361no—
114D4—362no—
115D4—362no—
116C2×C4—92no—
117C2×C4C2×#108811yescharacteristic
118C8—362no—
119C2×C3—64no—
120C2×C3—62no—
121C2×C3—81no—
122C2×C3—244no—
123C2×C3—721no—
124C2×C3#6378212yes—
125C2×C3#6378811yescharacteristic
126D3—244no—
127C22—42no—
128C22—41no—
129C22—362no—
130C22—361no—
131C22C22⋊#108811yescharacteristic
132C4—21no—
133C4—92no—
134C4—92no—
135C3—62no—
136C3#1059021411yescharacteristic
137C2—81no—
138C2—721no—
139C2#1172078812yes—
140C2#1172079411yescharacteristic
subgroupG/H
center, socleC22×C3C22⋊#157
commutatorC2×C4×H(3,𝔽3)C2×C4
FrattiniC2×C3×C4C2×#157
FittingC2×D4×H(3,𝔽3)C4
prime radical#11721850C2
cosocleC4×#1084C22
hypercenterC2×C3×D4#157
hypocenterH(3,𝔽3)#331
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. #11720785C2semidirect
66. H(3,𝔽3)#331semidirect

Maximal subgroups

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

subgroupindex
#117218502
#117207852
C3×#3319