Finite Group Databasebeta

← C3×#10590184 #13157109

Subgroups — C3×#10590184 #13157109

Order 1728.

368 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 68 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×#64286C212yesmaximal
2C2×C3×#3743C211yesmaximal, characteristic
3#10590184C311yesmaximal, characteristic, direct product factor
4C2×C3×C4×C3⋊D3C2211yescharacteristic
5C3×C22⋊#157—22no—
6C3×C4⋊#157—22no—
7C3×C4×#157C414yescomplemented
8C2×C33×C8C411yescharacteristic, complemented
9C2×C3×C3⋊(C3⋊C8)C411yescharacteristic, complemented
10C3×#3743—22no—
11#64286C2×C312yes—
12C2×#3743C2×C311yescharacteristic
13C2×C3×ΓL(1,𝔽25)—62no—
14C22×C3×C3⋊D3D411yescharacteristic
15C2×C3×#157—24no—
16C2×C33×C4C2×C411yescharacteristic
17C2×C3×C3⋊Dic3Q811yescharacteristic
18C3×C4×C3⋊D3C2×C412yes—
19C3×C4×C3⋊D3D412yes—
20C33×C8—21no—
21C3×C3⋊(C3⋊C8)—21no—
22C3×H(3,ℤ/4)—91nomaximal
23C2×C4×C3⋊D3C22×C311yescharacteristic
24C2×C3×C4×D3—62no—
25C22⋊#157—22no—
26C4⋊#157—22no—
27C4×#157C3×C414yescomplemented
28C2×C32×C8—22no—
29C2×C32×C8—42no—
30C2×C32×C8C3×C411yescharacteristic, complemented
31C2×C3⋊(C3⋊C8)C3×C411yescharacteristic, complemented
32#3743—22no—
33C2×C3×C3⋊C8—62no—
34C3×ΓL(1,𝔽25)—68no—
35C22×C33C4⋊C411yescharacteristic
36C2×C3×C3⋊D3—41no—
37C2×C3×C3⋊D3C22⋊C412yes—
38C3×#157—44no—
39C33×C4C4211yescharacteristic
40C33×C4C22⋊C411yescharacteristic
41C3×C3⋊Dic3C4⋊C412yes—
42C2×C3×M16—91no—
43C3×#117—92no—
44C2×ΓL(1,𝔽25)—62no—
45C22×C3⋊D3C3×D411yescharacteristic
46C22×C3×D3—62no—
47C2×#157—24no—
48C2×C32×C4—22no—
49C2×C32×C4—42no—
50C2×C32×C4C2×C3×C411yescharacteristic
51C2×C3⋊Dic3C3×Q811yescharacteristic
52C4×C3⋊D3C2×C3×C412yes—
53C4×C3⋊D3C3×D412yes—
54C2×C3×Dic3—62no—
55C3×C4×D3—64no—
56C3×C4×D3—64no—
57C32×C8—21no—
58C32×C8—24no—
59C32×C8—44no—
60C3⋊(C3⋊C8)—21no—
61C3×C3⋊C8—64no—
62H(3,ℤ/4)—91noSylow
63C2×C33—21no—
64C2×C33#9511yescharacteristic
65C3×C3⋊D3—22no—
66C22×C3×C4—91no—
67C2×C4×D3—62no—
68C3×M16—182no—
69C2×C3×C8—22no—
70C2×C3×C8—42no—
71C2×C3×C8—91no—
72C2×C3×C8#15711yescharacteristic, complemented
73C3×C4⋊C4—182no—
74C3×C22⋊C4—182no—
75C3×C42—94no—
76C2×C3⋊C8—62no—
77ΓL(1,𝔽25)—68no—
78C22×C32—22no—
79C22×C32—42no—
80C22×C32C3×C4⋊C411yescharacteristic
81C2×C3⋊D3—41no—
82C2×C3⋊D3C3×C22⋊C412yes—
83C2×C3×D3—64no—
84C2×C3×D3—124no—
85#157—44no—
86C32×C4—22no—
87C32×C4—22no—
88C32×C4—42no—
89C32×C4—42no—
90C32×C4C3×C4211yescharacteristic
91C32×C4C3×C22⋊C411yescharacteristic
92C3⋊Dic3C3×C4⋊C412yes—
93C3×Dic3—64no—
94C2×M16—91no—
95#117—92no—
96C33H(3,ℤ/4)11yescharacteristic, complemented, Sylow
97C23×C3—91no—
98C22×D3—62no—
99C2×C3×C4—22no—
100C2×C3×C4—42no—
101C2×C3×C4—91no—
102C2×C3×C4—92no—
103C2×C3×C4—92no—
104C2×C3×C4—184no—
105C2×C3×C4C2×#15711yescharacteristic
106C2×Dic3—62no—
107C4×D3—64no—
108C4×D3—64no—
109C3×C8—24no—
110C3×C8—21no—
111C3×C8—44no—
112C3×C8—181no—
113C3⋊C8—64no—
114C2×C32—21no—
115C2×C32—24no—
116C2×C32—22no—
117C2×C32—44no—
118C2×C32—42no—
119C2×C32C3×#9511yescharacteristic
120C3⋊D3—22no—
121C3×D3—124no—
122C22×C4—91no—
123M16—182no—
124C2×C8—91no—
125C2×C8C3×#15711yescharacteristic, complemented
126C4⋊C4—182no—
127C22⋊C4—182no—
128C42—94no—
129C22×C3—22no—
130C22×C3—42no—
131C22×C3—92no—
132C22×C3—361no—
133C22×C3C4⋊#15711yescharacteristic
134C2×D3—64no—
135C2×D3—124no—
136C3×C4—22no—
137C3×C4—22no—
138C3×C4—42no—
139C3×C4—42no—
140C3×C4—92no—
141C3×C4—364no—
142C3×C4C4×#15711yescharacteristic
143C3×C4C22⋊#15711yescharacteristic
144Dic3—64no—
145C32—22no—
146C32—42no—
147C32C3×H(3,ℤ/4)11yescharacteristic, complemented
148C23—91no—
149C2×C4—91no—
150C2×C4—92no—
151C2×C4—92no—
152C2×C4—184no—
153C2×C4C2×C3×#15711yescharacteristic
154C8—21no—
155C8—181no—
156C2×C3—24no—
157C2×C3—21no—
158C2×C3—22no—
159C2×C3—44no—
160C2×C3—42no—
161C2×C3—182no—
162C2×C3#6378111yescharacteristic
163D3—124no—
164C22—92no—
165C22—361no—
166C22C3×C4⋊#15711yescharacteristic
167C4—92no—
168C4—364no—
169C4C3×C4×#15711yescharacteristic
170C4C3×C22⋊#15711yescharacteristic
171C3—22no—
172C3—42no—
173C3#1059018411yescharacteristic, direct product factor
174C2—21no—
175C2—182no—
176C2C3×#6378111yescharacteristic
subgroupG/H
centerC3×C4C4×#157
commutatorC32×C4C3×C42
FrattiniC2×C4C2×C3×#157
FittingC2×C33×C8C4
socleC2×C33#95
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C4×C3⋊D3C22×C3
hypercenterC2×C3×C8#157
hypocenterC32C3×H(3,ℤ/4)
LCSG ▷ C32×C4 ▷ C32
UCSC1 ◁ C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
3. #10590184C3direct (N × H)
7. C3×C4×#157C4semidirect
8. C2×C33×C8C4semidirect
9. C2×C3×C3⋊(C3⋊C8)C4semidirect
27. C4×#157C3×C4semidirect
30. C2×C32×C8C3×C4semidirect
31. C2×C3⋊(C3⋊C8)C3×C4semidirect
72. C2×C3×C8#157semidirect
96. C33H(3,ℤ/4)semidirect
125. C2×C8C3×#157semidirect
147. C32C3×H(3,ℤ/4)semidirect
173. C3#10590184direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#642862
C2×C3×#37432
#105901843
C3×H(3,ℤ/4)9