Finite Group Databasebeta

← #106×C3⋊D9 #13144096

Subgroups — #106×C3⋊D9 #13144096

Order 1728.

650 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 175 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×C8×C3⋊D9C211yesmaximal, characteristic
2C4⋊C4×C3⋊D9C212yesmaximal
3C3×C9×#106C211yesmaximal, characteristic, complemented
4#11720540C211yesmaximal, characteristic, complemented
5#11720526C212yesmaximal, complemented
6C3⋊D3×#106—31nomaximal
7D9×#106—33nomaximal
8C2×C4×C3⋊D9—22no—
9C2×C4×C3⋊D9C2211yescharacteristic
10C2×C3×C8×C9C2211yescharacteristic
11C3×C9×C4⋊C4C2212yes—
12C4⋊(C3⋊Dic9)C2212yes—
13#88534—22no—
14C2×C3⋊(C9⋊C8)C2211yescharacteristic
15C8×C3⋊D9C414yescomplemented
16C2×C8×C3⋊D3—31no—
17C4⋊C4×C3⋊D3—32no—
18C32×#106D311yescharacteristic
19#63644—31no—
20#63635—32no—
21C2×C8×D9—33no—
22C4⋊C4×D9—36no—
23C9×#106D313yescomplemented
24#63377—33no—
25#63367—36no—
26C22×C3⋊D9D411yescharacteristic
27C2×C3×C4×C9—22no—
28C2×C3×C4×C9C2311yescharacteristic
29C2×C3⋊Dic9—22no—
30C2×C3⋊Dic9D411yescharacteristic
31C4×C3⋊D9—44no—
32C4×C3⋊D9C2×C412yes—
33C4×C3⋊D9Q812yes—
34C3×C8×C9C2×C412yescomplemented
35C3⋊(C9⋊C8)C2×C412yescomplemented
36D3×#106—93no—
37D3×#106—91no—
38C2×C4×C3⋊D3—31no—
39C2×C4×C3⋊D3—62no—
40C2×C32×C8C2×D311yescharacteristic
41C32×C4⋊C4C2×D312yes—
42C4⋊(C3⋊Dic3)—32no—
43#3750—62no—
44C2×C3⋊(C3⋊C8)—31no—
45C8×C3⋊D3—34no—
46C2×C4×D9—33no—
47C2×C4×D9—66no—
48C2×C8×C9C2×D313yes—
49C9×C4⋊C4C2×D316yes—
50C4⋊Dic9—36no—
51Dic9⋊C4—66no—
52C2×C9⋊C8—33no—
53C8×D9—312no—
54C22×C3×C9C2×D411yescharacteristic
55C2×C3⋊D9C4⋊C412yes—
56C2×C3⋊D9SD1614yes—
57C3×C4×C9—42no—
58C3×C4×C9C22×C411yescharacteristic
59C3×C4×C9C2×Q811yescharacteristic
60C3⋊Dic9—42no—
61C3⋊Dic9C4⋊C412yes—
62C2×D3×C8—93no—
63C2×D3×C8—91no—
64D3×C4⋊C4—96no—
65D3×C4⋊C4—92no—
66C3×#106D913yescomplemented
67C3×#106C3⋊D311yescharacteristic
68#817—93no—
69#817—91no—
70#808—96no—
71#808—92no—
72C22×C3⋊D3—31no—
73C2×C32×C4—22no—
74C2×C32×C4C22×D311yescharacteristic
75C2×C3⋊Dic3—31no—
76C2×C3⋊Dic3—62no—
77C4×C3⋊D3—32no—
78C4×C3⋊D3—32no—
79C4×C3⋊D3—124no—
80C22×D9—33no—
81C32×C8C4×D312yes—
82C3⋊(C3⋊C8)—32no—
83C2×C4×C9—26no—
84C2×C4×C9C22×D313yes—
85C2×Dic9—33no—
86C2×Dic9—66no—
87C4×D9—36no—
88C4×D9—36no—
89C4×D9—1212no—
90C8×C9C4×D316yescomplemented
91C9⋊C8—36no—
92C2×#106—271noSylow
93C2×C3×C9C2×C4⋊C411yescharacteristic
94C2×C3×C9C2×SD1612yes—
95C3⋊D9#10614yesdirect product factor
96C2×C4×D3—93no—
97C2×C4×D3—91no—
98C2×C4×D3—182no—
99C2×C4×D3—186no—
100C2×C3×C8C2×D913yes—
101C2×C3×C8C2×C3⋊D311yescharacteristic
102C3×C4⋊C4C2×D916yes—
103C3×C4⋊C4C2×C3⋊D312yes—
104C4⋊Dic3—96no—
105C4⋊Dic3—92no—
106Dic3⋊C4—182no—
107Dic3⋊C4—186no—
108C2×C3⋊C8—93no—
109C2×C3⋊C8—91no—
110D3×C8—912no—
111D3×C8—94no—
112C22×C32D3×D411yescharacteristic
113C2×C3⋊D3—32no—
114C2×C3⋊D3—34no—
115C32×C4—42no—
116C32×C4C2×C4×D311yescharacteristic
117C32×C4D3×Q811yescharacteristic
118C3⋊Dic3—32no—
119C3⋊Dic3—122no—
120C22×C9D3×D413yes—
121C2×D9—36no—
122C2×D9—312no—
123C4×C9—46no—
124C4×C9C2×C4×D313yes—
125C4×C9D3×Q813yes—
126Dic9—36no—
127Dic9—126no—
128C22×C8—271no—
129C2×C4⋊C4—272no—
130#106—271no—
131#106—272no—
132#106C3⋊D911yescharacteristic, direct product factor
133C3×C9C2×#10611yescharacteristic, complemented, Sylow
134C22×D3—93no—
135C22×D3—91no—
136C2×C3×C4—26no—
137C2×C3×C4—22no—
138C2×C3×C4C22×D913yes—
139C2×C3×C4C22×C3⋊D311yescharacteristic
140C2×Dic3—93no—
141C2×Dic3—91no—
142C2×Dic3—186no—
143C2×Dic3—182no—
144C4×D3—96no—
145C4×D3—96no—
146C4×D3—92no—
147C4×D3—92no—
148C4×D3—364no—
149C4×D3—3612no—
150C3×C8C4×D916yescomplemented
151C3×C8C4×C3⋊D312yes—
152C3⋊C8—96no—
153C3⋊C8—92no—
154C2×C32D3×C4⋊C411yescharacteristic
155C2×C32D3×SD1612yes—
156C3⋊D3—34no—
157C2×C9D3×C4⋊C413yes—
158C2×C9D3×SD1616yes—
159D9—312no—
160C22×C4—271no—
161C22×C4—542no—
162C2×C8—271no—
163C2×C8—274no—
164C2×C8C2×C3⋊D911yescharacteristic
165C4⋊C4—272no—
166C4⋊C4—542no—
167C4⋊C4C2×C3⋊D912yes—
168C22×C3D4×D913yes—
169C22×C3D4×C3⋊D311yescharacteristic
170C2×D3—92no—
171C2×D3—94no—
172C2×D3—96no—
173C2×D3—912no—
174C3×C4—42no—
175C3×C4—46no—
176C3×C4C2×C4×D913yes—
177C3×C4Q8×D913yes—
178C3×C4C2×C4×C3⋊D311yescharacteristic
179C3×C4Q8×C3⋊D311yescharacteristic
180Dic3—92no—
181Dic3—96no—
182Dic3—362no—
183Dic3—366no—
184C32D3×#10611yescharacteristic
185C9D3×#10613yescomplemented
186C23—271no—
187C2×C4—22no—
188C2×C4—271no—
189C2×C4—272no—
190C2×C4—272no—
191C2×C4—542no—
192C2×C4—1084no—
193C2×C4C22×C3⋊D911yescharacteristic
194C8—272no—
195C8C4×C3⋊D912yescomplemented
196C2×C3C4⋊C4×D913yes—
197C2×C3SD16×D916yes—
198C2×C3C4⋊C4×C3⋊D311yescharacteristic
199C2×C3SD16×C3⋊D312yes—
200D3—94no—
201D3—912no—
202C22—272no—
203C22—274no—
204C22D4×C3⋊D911yescharacteristic
205C4—42no—
206C4—272no—
207C4—1082no—
208C4C2×C4×C3⋊D911yescharacteristic
209C4Q8×C3⋊D911yescharacteristic
210C3D9×#10613yescomplemented
211C3C3⋊D3×#10611yescharacteristic
212C2—274no—
213C2C4⋊C4×C3⋊D911yescharacteristic
214C2SD16×C3⋊D912yes—
subgroupG/H
centerC22D4×C3⋊D9
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#106C2
socleC22×C32D3×D4
prime radicalC22×C3⋊D9D4
cosocleC2×C3×C4×C9C23
hypercenter#106C3⋊D9
hypocenterC3×C9C2×#106
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #106

Complements

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

NH ≅ G/Ntype
3. C3×C9×#106C2semidirect
4. #11720540C2semidirect
5. #11720526C2semidirect
15. C8×C3⋊D9C4semidirect
23. C9×#106D3semidirect
34. C3×C8×C9C2×C4semidirect
35. C3⋊(C9⋊C8)C2×C4semidirect
66. C3×#106D9semidirect
90. C8×C9C4×D3semidirect
95. C3⋊D9#106direct (N × H)
132. #106C3⋊D9direct (N × H)
133. C3×C9C2×#106semidirect
150. C3×C8C4×D9semidirect
185. C9D3×#106semidirect
195. C8C4×C3⋊D9semidirect
210. C3D9×#106semidirect

Maximal subgroups

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

subgroupindex
C2×C8×C3⋊D92
C4⋊C4×C3⋊D92
C3×C9×#1062
#117205402
#117205262
C3⋊D3×#1063
D9×#1063