Finite Group Databasebeta

← #13157209

Subgroups — #13157209

Order 1728.

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

normal subgroups only

This group has 284 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×#64121C211yesmaximal, characteristic
2#11723093C211yesmaximal, characteristic, complemented
3#11723090C211yesmaximal, characteristic, complemented
4#10590192—31nomaximal
5C3×#3830—21no—
6C4⋊(C3⋊#157)—21no—
7C33×D8C411yescharacteristic, complemented
8C3×#3754—21no—
9C3×#3746C411yescharacteristic, complemented
10C3×C8×C3⋊D3C2211yescharacteristic
11C3⋊#3708—21no—
12#64121D311yescharacteristic
13C3×#912—62no—
14#63768—31no—
15#63765—31no—
16C2×C3⋊#157—41no—
17C33×D4—21no—
18C3×#638—41no—
19C2×C3×C3⋊Dic3—41no—
20C3×C4×C3⋊D3D411yescharacteristic
21C3×#634—21no—
22C33×C8C2×C411yescharacteristic
23C3×C3⋊(C3⋊C8)D411yescharacteristic
24#4566—91nomaximal
25#3830—21no—
26C3×#237—64no—
27C4⋊#157—61no—
28C32×D8—22no—
29C32×D8—42no—
30C32×D8Dic311yescharacteristic, complemented
31#3754—21no—
32#3746Dic311yescharacteristic, complemented
33C8×C3⋊D3C2×D311yescharacteristic
34C3×#214—64no—
35C3×Dic12—62no—
36C3×D3×C8—62no—
37#3708—61no—
38C22×C33—41no—
39C2×C3×C3⋊D3D811yescharacteristic
40C3⋊#157—81no—
41C33×C4C22⋊C411yescharacteristic
42C3×C3⋊Dic3—41no—
43C3×C3⋊Dic3SD1611yescharacteristic
44C3×#135—91no—
45#912—62no—
46#817—91no—
47#812—91no—
48C2×#157—121no—
49C32×D4—21no—
50C32×D4—24no—
51C32×D4—44no—
52#638—41no—
53C2×C3⋊Dic3—41no—
54C4×C3⋊D3#6711yescharacteristic
55#634—21no—
56C3×#67—124no—
57C2×C3×Dic3—124no—
58C3×C4×D3—62no—
59C3×Dic6—64no—
60C32×C8—22no—
61C32×C8—42no—
62C32×C8C2×Dic311yescharacteristic
63C3⋊(C3⋊C8)#6711yescharacteristic
64C3×C3⋊C8—62no—
65#360—271noSylow
66C2×C33—81no—
67C2×C33#10211yescharacteristic
68C3×C3⋊D3—21no—
69C3×#41—181no—
70#237—64no—
71C3×Q16—91no—
72C3×SD16—181no—
73C3×D8—22no—
74C3×D8—42no—
75C3×D8#15711yescharacteristic, complemented
76C2×C3×C8—91no—
77#214—64no—
78C4⋊Dic3—181no—
79Dic12—62no—
80D3×C8—62no—
81C3⋊C16—181no—
82C22×C32—41no—
83C22×C32—44no—
84C22×C32—84no—
85C2×C3⋊D3#21311yescharacteristic
86C2×C3×D3—62no—
87#157—241no—
88C32×C4—22no—
89C32×C4—42no—
90C32×C4#21711yescharacteristic
91C3⋊Dic3—41no—
92C3⋊Dic3#21411yescharacteristic
93C3×Dic3—62no—
94C3×Dic3—124no—
95#135—91no—
96M32—271no—
97#106—271no—
98C33#36011yescharacteristic, complemented, Sylow
99C3×Q8—181no—
100C3×D4—24no—
101C3×D4—21no—
102C3×D4—44no—
103C3×D4—361no—
104C2×C3×C4—91no—
105C2×C3×C4—361no—
106#67—124no—
107C2×Dic3—124no—
108C2×Dic3—361no—
109C4×D3—62no—
110Dic6—64no—
111C3×C8—22no—
112C3×C8—42no—
113C3×C8—91no—
114C3×C8C2×#15711yescharacteristic
115C3⋊C8—62no—
116C2×C32—22no—
117C2×C32—42no—
118C2×C32—81no—
119C2×C32—84no—
120C2×C32—164no—
121C2×C32#83211yescharacteristic
122C3⋊D3—21no—
123C3×D3—122no—
124#41—181no—
125Q16—91no—
126SD16—181no—
127D8C3⋊#15711yescharacteristic, complemented
128C2×C8—91no—
129C4⋊C4—541no—
130C16—541no—
131C22×C3—44no—
132C22×C3—41no—
133C22×C3—84no—
134C22×C3—91no—
135C2×D3—62no—
136C3×C4—22no—
137C3×C4—42no—
138C3×C4—91no—
139C3×C4—361no—
140C3×C4C22⋊#15711yescharacteristic
141Dic3—62no—
142Dic3—124no—
143Dic3—721no—
144C32—22no—
145C32—42no—
146C32#456611yescharacteristic, complemented
147Q8—181no—
148D4—21no—
149D4—361no—
150C2×C4—91no—
151C2×C4—361no—
152C2×C4—1081no—
153C8—91no—
154C8C2×C3⋊#15711yescharacteristic
155C2×C3—22no—
156C2×C3—42no—
157C2×C3—84no—
158C2×C3—81no—
159C2×C3—164no—
160C2×C3—181no—
161C2×C3#6378211yescharacteristic
162D3—122no—
163C22—41no—
164C22—91no—
165C4—91no—
166C4—361no—
167C4—2161no—
168C4#8899411yescharacteristic
169C3—22no—
170C3—42no—
171C3#1059019211yescharacteristic, complemented
172C2—81no—
173C2—181no—
174C2#1172310711yescharacteristic
subgroupG/H
centerC2#11723107
commutatorC33×C8C2×C4
FrattiniC8C2×C3⋊#157
FittingC33×D8C4
socleC2×C33#102
prime radicalC3×#64121C2
cosocleC3×C8×C3⋊D3C22
hypercenterD8C3⋊#157
hypocenterC33#360
LCSG ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ D8

Complements

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

NH ≅ G/Ntype
2. #11723093C2semidirect
3. #11723090C2semidirect
7. C33×D8C4semidirect
9. C3×#3746C4semidirect
30. C32×D8Dic3semidirect
32. #3746Dic3semidirect
75. C3×D8#157semidirect
98. C33#360semidirect
127. D8C3⋊#157semidirect
146. C32#4566semidirect
171. C3#10590192semidirect

Maximal subgroups

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

subgroupindex
C3×#641212
#117230932
#117230902
#105901923
#45669