Finite Group Databasebeta

← C3×#10590073 #13154923

Subgroups — C3×#10590073 #13154923

Order 1728.

408 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 66 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×#3792C211yesmaximal, characteristic
2C3×#63957C212yesmaximal
3#10590073C311yesmaximal, characteristic, direct product factor
4C22×C3×C3⋊Dic3C2211yescharacteristic
5C3×#3792—22no—
6C2×C3×#622C412yescomplemented
7C32×#217—22no—
8C3×#3724—22no—
9C3×Dic32C414yescomplemented
10C2×#3792C2×C311yescharacteristic
11C3×#877—62no—
12#63957C2×C312yes—
13C23×C33D411yescharacteristic
14C2×C3×C3⋊Dic3C2×C411yescharacteristic
15C2×C3×C3⋊Dic3C2×C412yes—
16C2×C3×C3⋊Dic3D412yes—
17C2×C3×C3⋊Dic3Q811yescharacteristic
18C2×C32×Dic3—24no—
19C3×#622—22no—
20C3×H(3,ℤ/4)—91nomaximal
21C22×C3⋊Dic3C22×C311yescharacteristic
22C22×C3×Dic3—62no—
23#3792—22no—
24C2×#622C3×C412yescomplemented
25C32×C22⋊C4—62no—
26C3×#217—22no—
27C3×#217—22no—
28C3×#217—42no—
29C3×Dic3⋊C4—64no—
30C3×C4×Dic3—64no—
31#3724—22no—
32Dic32C3×C414yescomplemented
33C22×C33—41no—
34C22×C33C22⋊C412yes—
35C22×C33C4⋊C411yescharacteristic
36C3×C3⋊Dic3C4211yescharacteristic
37C3×C3⋊Dic3C22⋊C411yescharacteristic
38C3×C3⋊Dic3C4⋊C412yes—
39C32×Dic3—44no—
40C2×C3×M16—91no—
41C3×#117—92no—
42#877—62no—
43C23×C32—22no—
44C23×C32—42no—
45C23×C32C3×D411yescharacteristic
46C2×C32×C4—64no—
47C2×C3⋊Dic3C2×C3×C411yescharacteristic
48C2×C3⋊Dic3C2×C3×C412yes—
49C2×C3⋊Dic3C3×D412yes—
50C2×C3⋊Dic3C3×Q811yescharacteristic
51C2×C3×Dic3—24no—
52C2×C3×Dic3—24no—
53C2×C3×Dic3—44no—
54C2×C3×Dic3—62no—
55C2×C3×Dic3—62no—
56C2×C3×Dic3—62no—
57C2×C3×Dic3—62no—
58C2×C3×Dic3—62no—
59C2×C3×Dic3—62no—
60#622—22no—
61H(3,ℤ/4)—91noSylow
62C2×C33—22no—
63C2×C33—21no—
64C2×C33#9511yescharacteristic
65C22×C3×C4—91no—
66C22×Dic3—62no—
67C3×M16—182no—
68C2×C3×C8—92no—
69C3×C4⋊C4—182no—
70C3×C22⋊C4—62no—
71C3×C22⋊C4—62no—
72C3×C22⋊C4—122no—
73C3×C42—94no—
74#217—22no—
75Dic3⋊C4—64no—
76C4×Dic3—64no—
77C22×C32—22no—
78C22×C32—22no—
79C22×C32—22no—
80C22×C32—42no—
81C22×C32—42no—
82C22×C32—42no—
83C22×C32—41no—
84C22×C32—44no—
85C22×C32—84no—
86C22×C32C3×C22⋊C412yes—
87C22×C32C3×C4⋊C411yescharacteristic
88C32×C4—124no—
89C3⋊Dic3C3×C4211yescharacteristic
90C3⋊Dic3C3×C22⋊C411yescharacteristic
91C3⋊Dic3C3×C4⋊C412yes—
92C3×Dic3—44no—
93C3×Dic3—44no—
94C3×Dic3—62no—
95C3×Dic3—62no—
96C3×Dic3—62no—
97C3×Dic3—62no—
98C3×Dic3—84no—
99C2×M16—91no—
100#117—92no—
101C33H(3,ℤ/4)11yescharacteristic, complemented, Sylow
102C23×C3—22no—
103C23×C3—42no—
104C23×C3GO+(4,𝔽2)11yescharacteristic
105C2×C3×C4—64no—
106C2×C3×C4—64no—
107C2×C3×C4—92no—
108C2×C3×C4—92no—
109C2×C3×C4—91no—
110C2×C3×C4—91no—
111C2×C3×C4—124no—
112C2×Dic3—24no—
113C2×Dic3—62no—
114C2×Dic3—62no—
115C2×Dic3—62no—
116C2×Dic3—62no—
117C2×Dic3—62no—
118C2×Dic3—62no—
119C3×C8—182no—
120C2×C32—24no—
121C2×C32—22no—
122C2×C32—22no—
123C2×C32—21no—
124C2×C32—42no—
125C2×C32—42no—
126C2×C32—42no—
127C2×C32—44no—
128C2×C32—82no—
129C2×C32—82no—
130C2×C32C3×#9511yescharacteristic
131C22×C4—91no—
132M16—182no—
133C2×C8—92no—
134C4⋊C4—182no—
135C22⋊C4—62no—
136C42—94no—
137C22×C3—22no—
138C22×C3—22no—
139C22×C3—22no—
140C22×C3—41no—
141C22×C3—44no—
142C22×C3—42no—
143C22×C3—42no—
144C22×C3—42no—
145C22×C3—84no—
146C22×C3D32⋊C412yes—
147C22×C3#377311yescharacteristic
148C3×C4—91no—
149C3×C4—91no—
150C3×C4—92no—
151C3×C4—124no—
152C3×C4—124no—
153C3×C4—244no—
154Dic3—44no—
155Dic3—62no—
156Dic3—62no—
157Dic3—62no—
158Dic3—62no—
159C32—22no—
160C32—42no—
161C32C3×H(3,ℤ/4)11yescharacteristic, complemented
162C23C3×GO+(4,𝔽2)11yescharacteristic
163C2×C4—64no—
164C2×C4—92no—
165C2×C4—92no—
166C2×C4—91no—
167C2×C4—91no—
168C8—182no—
169C2×C3—24no—
170C2×C3—22no—
171C2×C3—21no—
172C2×C3—22no—
173C2×C3—42no—
174C2×C3—42no—
175C2×C3—44no—
176C2×C3—42no—
177C2×C3—82no—
178C2×C3—82no—
179C2×C3#6373711yescharacteristic
180C22—41no—
181C22C3×D32⋊C412yes—
182C22C3×#377311yescharacteristic
183C4—91no—
184C4—91no—
185C4—92no—
186C4—124no—
187C3—22no—
188C3—42no—
189C3#1059007311yescharacteristic, direct product factor
190C2—22no—
191C2—21no—
192C2C3×#6373711yescharacteristic
subgroupG/H
centerC2×C3#63737
commutatorC3⋊Dic3C3×C42
FrattiniC23C3×GO+(4,𝔽2)
Fitting, prime radicalC23×C33D4
socleC2×C33#95
cosocleC22×C3⋊Dic3C22×C3
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C3×H(3,ℤ/4)
LCSG ▷ C3⋊Dic3 ▷ C32
UCSC1 ◁ C2×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
3. #10590073C3direct (N × H)
6. C2×C3×#622C4semidirect
9. C3×Dic32C4semidirect
24. C2×#622C3×C4semidirect
32. Dic32C3×C4semidirect
101. C33H(3,ℤ/4)semidirect
161. C32C3×H(3,ℤ/4)semidirect
189. C3#10590073direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37922
C3×#639572
#105900733
C3×H(3,ℤ/4)9