Finite Group Databasebeta

← #13136927

Subgroups — #13136927

Order 1728.

251 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 251 conjugacy classes of subgroups — too many to draw. Only its 27 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
1D8×#272C211yesmaximal, characteristic
2#11720775C211yesmaximal, characteristic, complemented
3#11720770C211yesmaximal, characteristic, complemented
4D4×#272—21no—
5C4⋊#1088—21no—
6D8×H(3,𝔽3)C411yescharacteristic, complemented
7#88545—21no—
8#88529C411yescharacteristic, complemented
9C8×#272C2211yescharacteristic
10#88410—21no—
11C3×D3×D8—62no—
12C22×#272—41no—
13C2×#1088—41no—
14D4×H(3,𝔽3)—21no—
15#6235—41no—
16C4⋊#272—21no—
17C4×#272D411yescharacteristic
18C8×H(3,𝔽3)C2×C411yescharacteristic
19#6180D411yescharacteristic
20C3×#357—91nomaximal
21C3×D3×D4—64no—
22C32×D8—22no—
23C3×#213—64no—
24C3×D24—62no—
25C3×D3×C8—62no—
26C22×H(3,𝔽3)—41no—
27C2×#272—41no—
28C2×#272—81no—
29C2×#272—81no—
30C2×#272D811yescharacteristic
31#1088—81no—
32C4×H(3,𝔽3)C22⋊C411yescharacteristic
33#1084SD1611yescharacteristic
34C2×C3×D8—91no—
35D3×D8—62no—
36C2×C3×C16—91no—
37C3×#107—91no—
38C22×C3×D3—124no—
39C32×D4—24no—
40C3×#67—124no—
41C3×D12—64no—
42C3×C4×D3—62no—
43C32×C8—22no—
44C3×C3⋊C8—62no—
45#357—91noSylow
46C2×H(3,𝔽3)—81no—
47C2×H(3,𝔽3)#10211yescharacteristic
48#272—81no—
49#272D1611yescharacteristic
50#272SD3211yescharacteristic
51C2×C3×D4—181no—
52D3×D4—64no—
53C3×D8—62no—
54C3×D8—91no—
55C3×D8—181no—
56C3×D8#15711yescharacteristic
57C2×C3×C8—91no—
58C3×C4⋊C4—181no—
59#213—64no—
60D24—62no—
61D3×C8—62no—
62C3×C16—181no—
63C22×C32—44no—
64C2×C3×D3—62no—
65C2×C3×D3—124no—
66C2×C3×D3—244no—
67C2×C3×D3—244no—
68C32×C4—22no—
69C3×Dic3—62no—
70C2×D8—91no—
71C2×C16—91no—
72#107—91no—
73H(3,𝔽3)#35711yescharacteristic, complemented, Sylow
74C23×C3—361no—
75C22×D3—124no—
76C3×D4—21no—
77C3×D4—64no—
78C3×D4—181no—
79C3×D4—361no—
80C2×C3×C4—91no—
81C2×C3×C4—361no—
82#67—124no—
83D12—64no—
84C4×D3—62no—
85C3×C8—62no—
86C3×C8—91no—
87C3×C8C2×#15711yescharacteristic
88C3⋊C8—62no—
89C2×C32—22no—
90C2×C32—84no—
91C3×D3—62no—
92C3×D3—62no—
93C3×D3—244no—
94C2×D4—181no—
95D8—91no—
96D8—181no—
97D8#108811yescharacteristic, complemented
98C2×C8—91no—
99C4⋊C4—181no—
100C16—181no—
101C22×C3—41no—
102C22×C3—91no—
103C22×C3—124no—
104C22×C3—361no—
105C22×C3—721no—
106C22×C3—721no—
107C2×D3—62no—
108C2×D3—124no—
109C2×D3—244no—
110C2×D3—244no—
111C3×C4—62no—
112C3×C4—91no—
113C3×C4—721no—
114C3×C4C22⋊#15711yescharacteristic
115Dic3—62no—
116C32—22no—
117C23—361no—
118D4—21no—
119D4—181no—
120D4—361no—
121C2×C4—91no—
122C2×C4—361no—
123C8—91no—
124C8C2×#108811yescharacteristic
125C2×C3—62no—
126C2×C3—81no—
127C2×C3—91no—
128C2×C3—91no—
129C2×C3—244no—
130C2×C3—721no—
131C2×C3#6378211yescharacteristic
132D3—62no—
133D3—62no—
134D3—244no—
135C22—41no—
136C22—91no—
137C22—361no—
138C22—721no—
139C22—721no—
140C4—91no—
141C4—721no—
142C4C22⋊#108811yescharacteristic
143C3—62no—
144C3#1059019311yescharacteristic
145C2—81no—
146C2—91no—
147C2—91no—
148C2—721no—
149C2#1172078811yescharacteristic
subgroupG/H
center, socleC2×C3#63782
commutatorC8×H(3,𝔽3)C2×C4
FrattiniC3×C8C2×#157
FittingD8×H(3,𝔽3)C4
prime radicalD8×#272C2
cosocleC8×#272C22
hypercenterC3×D8#157
hypocenterH(3,𝔽3)#357
LCSG ▷ C8×H(3,𝔽3) ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8 ◁ C3×D8

Complements

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

NH ≅ G/Ntype
2. #11720775C2semidirect
3. #11720770C2semidirect
6. D8×H(3,𝔽3)C4semidirect
8. #88529C4semidirect
73. H(3,𝔽3)#357semidirect
97. D8#1088semidirect

Maximal subgroups

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

subgroupindex
D8×#2722
#117207752
#117207702
C3×#3579