Finite Group Databasebeta

← #13157214

Subgroups — #13157214

Order 1728.

272 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 41 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×Q16×C3⋊D3C211yesmaximal, characteristic
2#11723094C211yesmaximal, characteristic
3#11723089C211yesmaximal, characteristic
4#10590197—31nomaximal
5C3×Q8×C3⋊D3—21no—
6C4⋊(C3⋊#157)—21no—
7C33×Q16C411yescharacteristic, complemented
8C3×#3756—21no—
9C3×#3746C411yescharacteristic, complemented
10C3×C8×C3⋊D3C2211yescharacteristic
11C3⋊#3708—21no—
12Q16×C3⋊D3D311yescharacteristic
13C3×D3×Q16—62no—
14#63769—31no—
15#63764—31no—
16C2×C3⋊#157—41no—
17C33×Q8—21no—
18C3×C4×C3⋊D3—41no—
19C3×C4×C3⋊D3D411yescharacteristic
20C3×#634—21no—
21C3×#634—41no—
22C33×C8C2×C411yescharacteristic
23C3×C3⋊(C3⋊C8)D411yescharacteristic
24#4564—91nomaximal
25Q8×C3⋊D3—21no—
26C3×D3×Q8—64no—
27C4⋊#157—61no—
28C32×Q16—22no—
29C32×Q16—42no—
30C32×Q16Dic311yescharacteristic, complemented
31#3756—21no—
32#3746Dic311yescharacteristic, complemented
33C8×C3⋊D3C2×D311yescharacteristic
34C3×#216—64no—
35C3×Dic12—62no—
36C3×D3×C8—62no—
37#3708—61no—
38C2×C3×C3⋊D3D811yescharacteristic
39C3⋊#157—81no—
40C33×C4—41no—
41C33×C4C22⋊C411yescharacteristic
42C3×C3⋊Dic3—41no—
43C3×C3⋊Dic3SD1611yescharacteristic
44C2×C3×Q16—91no—
45D3×Q16—62no—
46#818—91no—
47C2×C3⋊C16—91no—
48C2×#157—121no—
49C32×Q8—21no—
50C32×Q8—24no—
51C32×Q8—44no—
52C4×C3⋊D3—41no—
53C4×C3⋊D3#6711yescharacteristic
54#634—21no—
55#634—41no—
56C3×C4×D3—62no—
57C3×C4×D3—124no—
58C3×Dic6—64no—
59C3×Dic6—124no—
60C32×C8—22no—
61C32×C8—42no—
62C32×C8C2×Dic311yescharacteristic
63C3⋊(C3⋊C8)#6711yescharacteristic
64C3×C3⋊C8—62no—
65#358—271noSylow
66C2×C33#10211yescharacteristic
67C3×C3⋊D3SD3211yescharacteristic
68C3×C3⋊D3Q3211yescharacteristic
69C2×C3×Q8—181no—
70D3×Q8—64no—
71C3×Q16—22no—
72C3×Q16—42no—
73C3×Q16—91no—
74C3×Q16—181no—
75C3×Q16#15711yescharacteristic, complemented
76C2×C3×C8—91no—
77#216—64no—
78C4⋊Dic3—181no—
79Dic12—62no—
80D3×C8—62no—
81C3⋊C16—181no—
82C2×C3⋊D3#21311yescharacteristic
83C2×C3×D3—62no—
84#157—241no—
85C32×C4—22no—
86C32×C4—42no—
87C32×C4—41no—
88C32×C4—44no—
89C32×C4—84no—
90C32×C4#21711yescharacteristic
91C3⋊Dic3—41no—
92C3⋊Dic3#21411yescharacteristic
93C3×Dic3—62no—
94C3×Dic3—124no—
95C2×Q16—91no—
96C2×C16—271no—
97#107—271no—
98C33#35811yescharacteristic, complemented, Sylow
99C3×Q8—24no—
100C3×Q8—21no—
101C3×Q8—44no—
102C3×Q8—181no—
103C3×Q8—361no—
104C2×C3×C4—91no—
105C2×C3×C4—361no—
106C2×Dic3—361no—
107C4×D3—62no—
108C4×D3—124no—
109Dic6—64no—
110Dic6—124no—
111C3×C8—22no—
112C3×C8—42no—
113C3×C8—91no—
114C3×C8C2×#15711yescharacteristic
115C3⋊C8—62no—
116C2×C32—22no—
117C2×C32—42no—
118C2×C32#83211yescharacteristic
119C3⋊D3#82811yescharacteristic
120C3⋊D3#82911yescharacteristic
121C3×D3—62no—
122C3×D3—62no—
123C2×Q8—181no—
124Q16—91no—
125Q16—181no—
126Q16C3⋊#15711yescharacteristic, complemented
127C2×C8—91no—
128C4⋊C4—541no—
129C16—541no—
130C22×C3—91no—
131C2×D3—62no—
132C3×C4—22no—
133C3×C4—44no—
134C3×C4—41no—
135C3×C4—42no—
136C3×C4—84no—
137C3×C4—91no—
138C3×C4—361no—
139C3×C4C22⋊#15711yescharacteristic
140Dic3—62no—
141Dic3—124no—
142Dic3—721no—
143C32—22no—
144C32—42no—
145C32#456411yescharacteristic, complemented
146Q8—21no—
147Q8—181no—
148Q8—361no—
149C2×C4—91no—
150C2×C4—361no—
151C2×C4—1081no—
152C8—91no—
153C8C2×C3⋊#15711yescharacteristic
154C2×C3—22no—
155C2×C3—42no—
156C2×C3—91no—
157C2×C3—91no—
158C2×C3#6378211yescharacteristic
159D3—62no—
160D3—62no—
161C22—91no—
162C4—41no—
163C4—91no—
164C4—361no—
165C4—2161no—
166C4#8899411yescharacteristic
167C3—22no—
168C3—42no—
169C3#1059019711yescharacteristic, complemented
170C2—91no—
171C2—91no—
172C2#1172310711yescharacteristic
subgroupG/H
centerC2#11723107
commutatorC33×C8C2×C4
FrattiniC8C2×C3⋊#157
FittingC33×Q16C4
socleC2×C33#102
prime radicalC2×C3×C3⋊D3D8
cosocleC3×C8×C3⋊D3C22
hypercenterQ16C3⋊#157
hypocenterC33#358
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
7. C33×Q16C4semidirect
9. C3×#3746C4semidirect
30. C32×Q16Dic3semidirect
32. #3746Dic3semidirect
75. C3×Q16#157semidirect
98. C33#358semidirect
126. Q16C3⋊#157semidirect
145. C32#4564semidirect
169. C3#10590197semidirect

Maximal subgroups

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

subgroupindex
C3×Q16×C3⋊D32
#117230942
#117230892
#105901973
#45649