Finite Group Databasebeta

← C3×#10588218 #13135609

Subgroups — C3×#10588218 #13135609

Order 1728.

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

normal subgroups only

This group has 286 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
1C3×C9×#124C211yesmaximal, characteristic
2C3×#63393C212yesmaximal
3C3×#4541—31nomaximal
4#10588218C311yesmaximal, characteristic, direct product factor
5C2×C3×Q8×C9C411yescharacteristic, complemented
6C2×C3×D4×C9C2211yescharacteristic
7C3×C9×C22⋊C4—22no—
8C3×C42×C9C411yescharacteristic, complemented
9C3×#3676—22no—
10C32×#124D311yescharacteristic
11C3×#834—32no—
12C9×#124—21no—
13C9×#124C2×C311yescharacteristic
14#63393C2×C312yes—
15C23×C3×C9D412yes—
16C3×Q8×C9—41no—
17C3×D4×C9—41no—
18C2×C3×C4×C9—21no—
19C2×C3×C4×C9—21no—
20C2×C3×C4×C9C2×C411yescharacteristic
21C2×C3×Dic9—42no—
22C3×#354—91no—
23#4541—31no—
24C2×C32×Q8Dic311yescharacteristic
25C2×C32×D4C2×D311yescharacteristic
26C32×C22⋊C4—22no—
27C32×C42Dic311yescharacteristic
28C3×#217—62no—
29C2×Q8×C9—21no—
30C2×Q8×C9C3×C411yescharacteristic, complemented
31C2×D4×C9—21no—
32C2×D4×C9C22×C311yescharacteristic
33C9×C22⋊C4—24no—
34C9×C22⋊C4—22no—
35C42×C9—21no—
36C42×C9C3×C411yescharacteristic, complemented
37#3676—22no—
38C22×C3×C9—22no—
39C22×C3×C9—42no—
40C22×C3×C9C22⋊C411yescharacteristic
41C3×C4×C9—21no—
42C3×C4×C9—41no—
43C3×C4×C9—41no—
44C3×Dic9—82no—
45C3×#124—21no—
46C3×#124D911yescharacteristic
47C3×#124C3×D311yescharacteristic
48C3×#99—92no—
49#834—32no—
50C23×C32#6712yes—
51C32×Q8—41no—
52C32×D4—41no—
53C2×C32×C4—21no—
54C2×C32×C4—21no—
55C2×C32×C4C2×Dic311yescharacteristic
56C2×C3×Dic3—122no—
57C23×C9—22no—
58C23×C9C3×D412yes—
59Q8×C9—41no—
60Q8×C9—42no—
61D4×C9—41no—
62D4×C9—42no—
63C2×C4×C9—22no—
64C2×C4×C9—22no—
65C2×C4×C9—21no—
66C2×C4×C9—21no—
67C2×C4×C9—21no—
68C2×C4×C9C2×C3×C411yescharacteristic
69C2×Dic9—42no—
70#354—91noSylow
71C2×C3×C9—21no—
72C2×C3×C9—42no—
73C2×C3×C9#9911yescharacteristic
74C2×C3×Q8—21no—
75C2×C3×Q8Dic911yescharacteristic, complemented
76C2×C3×Q8C3×Dic311yescharacteristic
77C2×C3×D4—21no—
78C2×C3×D4C2×D911yescharacteristic
79C2×C3×D4C2×C3×D311yescharacteristic
80C3×C22⋊C4—24no—
81C3×C22⋊C4—22no—
82C3×C22⋊C4—22no—
83C3×C22⋊C4—182no—
84C3×C42—21no—
85C3×C42Dic911yescharacteristic, complemented
86C3×C42C3×Dic311yescharacteristic
87#217—62no—
88C22×C32—22no—
89C22×C32—42no—
90C22×C32#21711yescharacteristic
91C32×C4—21no—
92C32×C4—41no—
93C32×C4—41no—
94C3×Dic3—242no—
95C22×C9—22no—
96C22×C9—21no—
97C22×C9—42no—
98C22×C9—42no—
99C22×C9—44no—
100C22×C9C3×C22⋊C411yescharacteristic
101C4×C9—22no—
102C4×C9—21no—
103C4×C9—42no—
104C4×C9—41no—
105C4×C9—42no—
106C4×C9—41no—
107Dic9—82no—
108#124C3×D911yescharacteristic
109#99—92no—
110C3×C9#35411yescharacteristic, complemented, Sylow
111C23×C3—22no—
112C23×C3#61112yes—
113C23×C3C3×#6712yes—
114C3×Q8—41no—
115C3×Q8—41no—
116C3×Q8—42no—
117C3×D4—41no—
118C3×D4—41no—
119C3×D4—42no—
120C2×C3×C4—22no—
121C2×C3×C4—22no—
122C2×C3×C4—21no—
123C2×C3×C4—21no—
124C2×C3×C4—21no—
125C2×C3×C4—21no—
126C2×C3×C4—21no—
127C2×C3×C4—362no—
128C2×C3×C4C2×Dic911yescharacteristic
129C2×C3×C4C2×C3×Dic311yescharacteristic
130C2×Dic3—122no—
131C2×C32—21no—
132C2×C32—42no—
133C2×C32#83411yescharacteristic
134C2×C9—22no—
135C2×C9—21no—
136C2×C9—21no—
137C2×C9—42no—
138C2×C9—82no—
139C2×C9C3×#9911yescharacteristic
140C2×Q8C3×Dic911yescharacteristic, complemented
141C2×D4C2×C3×D911yescharacteristic
142C22⋊C4—22no—
143C22⋊C4—182no—
144C42C3×Dic911yescharacteristic, complemented
145C22×C3—22no—
146C22×C3—22no—
147C22×C3—21no—
148C22×C3—42no—
149C22×C3—42no—
150C22×C3—42no—
151C22×C3—44no—
152C22×C3#367611yescharacteristic
153C22×C3C3×#21711yescharacteristic
154C3×C4—22no—
155C3×C4—21no—
156C3×C4—21no—
157C3×C4—42no—
158C3×C4—41no—
159C3×C4—42no—
160C3×C4—41no—
161C3×C4—41no—
162C3×C4—41no—
163C3×C4—722no—
164Dic3—242no—
165C32#454111yescharacteristic
166C9—21no—
167C9C3×#35411yescharacteristic, complemented
168C23C3×#61112yes—
169Q8—41no—
170D4—41no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—362no—
174C2×C4C2×C3×Dic911yescharacteristic
175C2×C3—22no—
176C2×C3—21no—
177C2×C3—21no—
178C2×C3—21no—
179C2×C3—42no—
180C2×C3—42no—
181C2×C3—82no—
182C2×C3#6339311yescharacteristic
183C2×C3C3×#83411yescharacteristic
184C22—22no—
185C22—42no—
186C22C3×#367611yescharacteristic
187C4—21no—
188C4—41no—
189C4—41no—
190C4—722no—
191C3—21no—
192C3#1058821811yescharacteristic, direct product factor
193C3C3×#454111yescharacteristic
194C2—21no—
195C2—42no—
196C2C3×#6339311yescharacteristic
subgroupG/H
centerC2×C3#63393
commutatorC2×C4×C9C2×C3×C4
FrattiniC2×C3×D4C2×C3×D3
FittingC3×C9×#124C2
socleC2×C32#834
prime radicalC2×C32×D4C2×D3
cosocleC2×D4×C9C22×C3
hypercenterC3×#124D9
hypocenterC9C3×#354
LCSG ▷ C2×C4×C9 ▷ C22×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×D4 ◁ C3×#124

Complements

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

NH ≅ G/Ntype
4. #10588218C3direct (N × H)
5. C2×C3×Q8×C9C4semidirect
8. C3×C42×C9C4semidirect
30. C2×Q8×C9C3×C4semidirect
36. C42×C9C3×C4semidirect
75. C2×C3×Q8Dic9semidirect
85. C3×C42Dic9semidirect
110. C3×C9#354semidirect
140. C2×Q8C3×Dic9semidirect
144. C42C3×Dic9semidirect
167. C9C3×#354semidirect
192. C3#10588218direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C9×#1242
C3×#633932
C3×#45413
#105882183