Finite Group Databasebeta

← C2×C3×#63765 #13173232

Subgroups — C2×C3×#63765 #13173232

Order 1728.

376 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 104 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
1C2×C3×C8×C3⋊D3C211yesmaximal, characteristic
2C2×C3×#3708C212yesmaximal, complemented
3C3×#63765C214yesmaximal, direct product factor
4C2×#63765C311yesmaximal, characteristic, direct product factor
5C2×C3×C4×C3⋊D3C411yescharacteristic
6C2×C33×C8C411yescharacteristic
7C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
8C3×C8×C3⋊D3C412yes—
9C3×C8×C3⋊D3C2212yes—
10C3×#3708C2214yescomplemented
11C2×C8×C3⋊D3C2×C311yescharacteristic
12C2×C3×D3×C8—62no—
13C2×#3708C2×C312yescomplemented
14#63765C2×C314yesdirect product factor
15C22×C3×C3⋊D3C811yescharacteristic
16C2×C33×C4C2×C411yescharacteristic
17C2×C3×C3⋊Dic3C811yescharacteristic
18C3×C4×C3⋊D3C812yes—
19C3×C4×C3⋊D3C2×C412yes—
20C33×C8C2×C412yes—
21C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
22C3×C3⋊(C3⋊C8)C2311yescharacteristic
23C2×C3×M32—91nomaximal
24C2×C4×C3⋊D3C3×C411yescharacteristic
25C2×C3×C4×D3—62no—
26C2×C32×C8—22no—
27C2×C32×C8—42no—
28C2×C32×C8C3×C411yescharacteristic
29C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
30C8×C3⋊D3C3×C412yes—
31C8×C3⋊D3C22×C312yes—
32C2×C3×C3⋊C8—62no—
33C3×D3×C8—64no—
34C3×D3×C8—64no—
35#3708C22×C314yescomplemented
36C22×C33C2×C811yescharacteristic
37C2×C3×C3⋊D3—22no—
38C2×C3×C3⋊D3C2×C812yes—
39C33×C4C2×C811yescharacteristic
40C33×C4C22×C411yescharacteristic
41C3×C3⋊Dic3C2×C812yes—
42C22×C3×C8—91no—
43C2×D3×C8—62no—
44C3×M32—94no—
45C2×C3×C16—92no—
46C22×C3⋊D3C3×C811yescharacteristic
47C22×C3×D3—62no—
48C2×C32×C4—22no—
49C2×C32×C4—42no—
50C2×C32×C4C2×C3×C411yescharacteristic
51C2×C3⋊Dic3C3×C811yescharacteristic
52C4×C3⋊D3C3×C812yes—
53C4×C3⋊D3C2×C3×C412yes—
54C2×C3×Dic3—62no—
55C3×C4×D3—64no—
56C3×C4×D3—64no—
57C32×C8—24no—
58C32×C8—44no—
59C32×C8C2×C3×C412yes—
60C3⋊(C3⋊C8)C2×C3×C411yescharacteristic
61C3⋊(C3⋊C8)C23×C311yescharacteristic
62C3×C3⋊C8—62no—
63C3×C3⋊C8—62no—
64C2×M32—91noSylow
65C2×C33M3212yescomplemented
66C2×C33C22×C811yescharacteristic
67C3×C3⋊D3—22no—
68C22×C3×C4—91no—
69C2×C4×D3—62no—
70C2×C3×C8—22no—
71C2×C3×C8—42no—
72C2×C3×C8—91no—
73C2×C3×C8—92no—
74C2×C3×C8—92no—
75C2×C3×C8#15711yescharacteristic
76C2×C3⋊C8—62no—
77D3×C8—64no—
78D3×C8—64no—
79C3×C16—94no—
80C22×C32—22no—
81C22×C32—42no—
82C22×C32C2×C3×C811yescharacteristic
83C2×C3⋊D3—22no—
84C2×C3⋊D3C2×C3×C812yes—
85C2×C3×D3—64no—
86C2×C3×D3—68no—
87C32×C4—22no—
88C32×C4—22no—
89C32×C4—42no—
90C32×C4—42no—
91C32×C4C2×C3×C811yescharacteristic
92C32×C4C22×C3×C411yescharacteristic
93C3⋊Dic3C2×C3×C812yes—
94C3×Dic3—64no—
95C22×C8—91no—
96M32—94no—
97C2×C16—92no—
98C33C2×M3211yescharacteristic, complemented, Sylow
99C23×C3—91no—
100C22×D3—62no—
101C2×C3×C4—22no—
102C2×C3×C4—42no—
103C2×C3×C4—91no—
104C2×C3×C4—92no—
105C2×C3×C4—92no—
106C2×C3×C4C2×#15711yescharacteristic
107C2×Dic3—62no—
108C4×D3—64no—
109C4×D3—64no—
110C3×C8—24no—
111C3×C8—44no—
112C3×C8—91no—
113C3×C8—91no—
114C3×C8C2×#15712yes—
115C3⋊C8—62no—
116C3⋊C8—62no—
117C2×C32—24no—
118C2×C32—22no—
119C2×C32—44no—
120C2×C32—42no—
121C2×C32C3×M3212yescomplemented
122C2×C32C22×C3×C811yescharacteristic
123C3⋊D3—22no—
124C3×D3—68no—
125C22×C4—91no—
126C2×C8—91no—
127C2×C8—92no—
128C2×C8—92no—
129C2×C8C3×#15711yescharacteristic
130C16—94no—
131C22×C3—22no—
132C22×C3—42no—
133C22×C3—92no—
134C22×C3—182no—
135C22×C3#378711yescharacteristic
136C2×D3—64no—
137C2×D3—68no—
138C3×C4—22no—
139C3×C4—22no—
140C3×C4—42no—
141C3×C4—42no—
142C3×C4—92no—
143C3×C4#378711yescharacteristic
144C3×C4C22×#15711yescharacteristic
145Dic3—64no—
146C32—22no—
147C32—42no—
148C32C2×C3×M3211yescharacteristic, complemented
149C23—91no—
150C2×C4—91no—
151C2×C4—92no—
152C2×C4—92no—
153C2×C4C2×C3×#15711yescharacteristic
154C8—91no—
155C8—91no—
156C8C2×C3×#15712yes—
157C2×C3—24no—
158C2×C3—22no—
159C2×C3—44no—
160C2×C3—42no—
161C2×C3—182no—
162C2×C3#6376512yesdirect product factor
163C2×C3C2×#378711yescharacteristic
164D3—68no—
165C22—92no—
166C22—182no—
167C22C3×#378711yescharacteristic
168C4—92no—
169C4C3×#378711yescharacteristic
170C4C22×C3×#15711yescharacteristic
171C3—22no—
172C3—42no—
173C3C2×#6376511yescharacteristic, direct product factor
174C2—182no—
175C2C3×#6376512yesdirect product factor
176C2C2×C3×#378711yescharacteristic
subgroupG/H
centerC2×C3×C4C2×#157
commutatorC2×C32C22×C3×C8
FrattiniC4C22×C3×#157
FittingC2×C33×C8C4
socleC22×C33C2×C8
prime radicalC22×C3×C3⋊D3C8
cosocleC3⋊(C3⋊C8)C23×C3
hypercenterC2×C3×C8#157
hypocenterC32C2×C3×M32
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3708C2semidirect
3. C3×#63765C2direct (N × H)
4. C2×#63765C3direct (N × H)
10. C3×#3708C22semidirect
13. C2×#3708C2×C3semidirect
14. #63765C2×C3direct (N × H)
35. #3708C22×C3semidirect
65. C2×C33M32semidirect
98. C33C2×M32semidirect
121. C2×C32C3×M32semidirect
148. C32C2×C3×M32semidirect
162. C2×C3#63765direct (N × H)
173. C3C2×#63765direct (N × H)
175. C2C3×#63765direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C8×C3⋊D32
C2×C3×#37082
C3×#637652
C2×#637653
C2×C3×M329