Finite Group Databasebeta

← #13157215

Subgroups — #13157215

Order 1728.

272 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 37 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
1C3×#64128C211yesmaximal, characteristic
2#11723096C211yesmaximal, characteristic, complemented
3#11723089C211yesmaximal, characteristic, complemented
4#10590198—31nomaximal
5C3×#3832—21no—
6#88989—21no—
7C33×Q16C411yescharacteristic
8C3×#3755—21no—
9C3×#3745C411yescharacteristic
10C3×C8×C3⋊D3C2211yescharacteristic
11C3⋊#3708—21no—
12#64128D311yescharacteristic
13C3×#919—62no—
14#63771—31no—
15#63764—31no—
16C33×Q8—21no—
17C3×C4⋊(C3⋊D3)—21no—
18C3×C4⋊(C3⋊D3)—41no—
19C3×C4×C3⋊D3—41no—
20C3×C4×C3⋊D3D411yescharacteristic
21C3⋊#622—41no—
22C33×C8C2×C411yescharacteristic
23C3×C3⋊(C3⋊C8)D411yescharacteristic
24#4567—91nomaximal
25#3832—21no—
26C3×#239—64no—
27#3788—61no—
28C32×Q16—22no—
29C32×Q16—42no—
30C32×Q16Dic311yescharacteristic
31#3755—21no—
32#3745Dic311yescharacteristic
33C8×C3⋊D3C2×D311yescharacteristic
34C3×#215—64no—
35C3×D24—62no—
36C3×D3×C8—62no—
37#3708—61no—
38C2×C3×C3⋊D3—41no—
39C2×C3×C3⋊D3SD1611yescharacteristic
40C33×C4—41no—
41C33×C4C22⋊C411yescharacteristic
42C3×C3⋊Dic3D811yescharacteristic
43C3×#135—91no—
44#919—62no—
45#819—91no—
46C2×C3⋊C16—91no—
47C32×Q8—21no—
48C32×Q8—24no—
49C32×Q8—44no—
50C4⋊(C3⋊D3)—21no—
51C4⋊(C3⋊D3)—41no—
52C4×C3⋊D3—41no—
53C4×C3⋊D3#6711yescharacteristic
54C3×D12—64no—
55C3×D12—124no—
56C3×C4×D3—62no—
57C3×C4×D3—124no—
58#622—121no—
59C32×C8—22no—
60C32×C8—42no—
61C32×C8C2×Dic311yescharacteristic
62C3⋊(C3⋊C8)#6711yescharacteristic
63C3×C3⋊C8—62no—
64#359—271noSylow
65C2×C33#10211yescharacteristic
66C3×C3⋊D3—21no—
67C3×C3⋊D3—81no—
68C3×#41—181no—
69#239—64no—
70C3×Q16—22no—
71C3×Q16—42no—
72C3×Q16#15711yescharacteristic
73C3×SD16—181no—
74C3×D8—91no—
75C2×C3×C8—91no—
76#215—64no—
77#208—181no—
78D24—62no—
79D3×C8—62no—
80C3⋊C16—181no—
81C2×C3⋊D3—41no—
82C2×C3⋊D3#21411yescharacteristic
83C2×C3×D3—62no—
84C2×C3×D3—124no—
85C32×C4—22no—
86C32×C4—42no—
87C32×C4—41no—
88C32×C4—44no—
89C32×C4—84no—
90C32×C4#21711yescharacteristic
91C3⋊Dic3#21311yescharacteristic
92C3×Dic3—62no—
93#135—91no—
94C2×C16—271no—
95#108—271no—
96C33#35911yescharacteristic, complemented, Sylow
97C3×Q8—24no—
98C3×Q8—21no—
99C3×Q8—44no—
100C3×D4—181no—
101C3×D4—361no—
102C2×C3×C4—91no—
103C2×C3×C4—361no—
104D12—64no—
105D12—124no—
106C4×D3—62no—
107C4×D3—124no—
108C3×C8—22no—
109C3×C8—42no—
110C3×C8—91no—
111C3×C8C2×#15711yescharacteristic
112C3⋊C8—62no—
113C3⋊C8—361no—
114C2×C32—22no—
115C2×C32—42no—
116C2×C32#83211yescharacteristic
117C3⋊D3—21no—
118C3⋊D3—81no—
119C3×D3—122no—
120C3×D3—244no—
121#41—181no—
122Q16C3⋊#15711yescharacteristic
123SD16—181no—
124D8—91no—
125M16—541no—
126C2×C8—91no—
127C16—541no—
128C22×C3—91no—
129C22×C3—361no—
130C2×D3—62no—
131C2×D3—124no—
132C3×C4—22no—
133C3×C4—44no—
134C3×C4—41no—
135C3×C4—42no—
136C3×C4—84no—
137C3×C4—91no—
138C3×C4C22⋊#15711yescharacteristic
139Dic3—62no—
140C32—22no—
141C32—42no—
142C32#456711yescharacteristic, complemented
143Q8—21no—
144D4—181no—
145D4—361no—
146C2×C4—91no—
147C2×C4—361no—
148C8—91no—
149C8—1081no—
150C8C2×C3⋊#15711yescharacteristic
151C2×C3—22no—
152C2×C3—42no—
153C2×C3—181no—
154C2×C3—721no—
155C2×C3#6378211yescharacteristic
156D3—122no—
157D3—244no—
158C22—91no—
159C22—361no—
160C4—41no—
161C4—91no—
162C4#8899411yescharacteristic
163C3—22no—
164C3—42no—
165C3#1059019811yescharacteristic, complemented
166C2—181no—
167C2—721no—
168C2#1172310711yescharacteristic
subgroupG/H
centerC2#11723107
commutatorC33×C8C2×C4
FrattiniC8C2×C3⋊#157
FittingC33×Q16C4
socleC2×C33#102
prime radicalC3×#64128C2
cosocleC3×C8×C3⋊D3C22
hypercenterQ16C3⋊#157
hypocenterC33#359
LCSG ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ Q16

Complements

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

NH ≅ G/Ntype
2. #11723096C2semidirect
3. #11723089C2semidirect
96. C33#359semidirect
142. C32#4567semidirect
165. C3#10590198semidirect

Maximal subgroups

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

subgroupindex
C3×#641282
#117230962
#117230892
#105901983
#45679