Finite Group Databasebeta

← #13147055

Subgroups — #13147055

Order 1728.

436 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×C3×Q8⋊C9C211yesmaximal, characteristic
2C3⋊#63421C212yesmaximal, complemented
3#10592924—31nomaximal
4#10589522—33nomaximal
5C2×C3×Q8⋊C9C412yescomplemented
6C2×C3×Q8⋊C9C2211yescharacteristic
7#88551—41nomaximal
8C22×C32×Q8D311yescharacteristic
9#64135—31no—
10C2×#3748—31no—
11C22×Q8⋊C9D313yes—
12#63662—32no—
13#63662—32no—
14#63421—36no—
15C23×C3×C9—41no—
16C2×C3⋊Dic9—42no—
17C3×Q8⋊C9C2×C411yescharacteristic
18#5225—93no—
19#5225—91no—
20C2×C32×Q8—32no—
21C2×C32×Q8—31no—
22C2×C32×Q8Dic312yes—
23C2×C32×Q8C2×D311yescharacteristic
24C22×C32×C4—31no—
25#3757—61no—
26C4⋊(C3⋊Dic3)—32no—
27C4×C3⋊Dic3—61no—
28#3748—62no—
29C2×C3⋊(C3⋊C8)—32no—
30C2×Q8⋊C9Dic316yescomplemented
31C2×Q8⋊C9C2×D313yes—
32#3676—123no—
33C22×C3×C9—41no—
34C22×C3×C9—42no—
35C22×C3×C9—82no—
36C3⋊Dic9—82no—
37C22×C3×Q8D913yes—
38C22×C3×Q8C3⋊D311yescharacteristic
39#926—93no—
40#926—91no—
41C2×#208—93no—
42C2×#208—91no—
43#835—96no—
44#835—92no—
45#835—96no—
46#835—92no—
47C23×C32S411yescharacteristic
48C32×Q8—32no—
49C32×Q8—31no—
50C32×Q8—61no—
51C32×Q8C2×Dic311yescharacteristic
52C2×C32×C4—32no—
53C2×C32×C4—32no—
54C2×C32×C4—31no—
55C2×C32×C4—31no—
56C2×C3⋊Dic3—62no—
57C23×C9—43no—
58C3⋊(C3⋊C8)—62no—
59C2×Dic9—126no—
60Q8⋊C9C2×Dic313yes—
61#419—271noSylow
62C2×C3×C9—42no—
63C2×C3×C9—41no—
64C2×C3×C9—82no—
65C2×C3×Q8—33no—
66C2×C3×Q8—31no—
67C2×C3×Q8—36no—
68C2×C3×Q8—32no—
69C2×C3×Q8Dic916yescomplemented
70C2×C3×Q8C2×D913yes—
71C2×C3×Q8C3⋊Dic312yes—
72C2×C3×Q8C2×C3⋊D311yescharacteristic
73C22×C3×C4—33no—
74C22×C3×C4—31no—
75#217—183no—
76#217—181no—
77C4⋊Dic3—92no—
78C4⋊Dic3—96no—
79C4×Dic3—181no—
80C4×Dic3—183no—
81#208—186no—
82#208—182no—
83C2×C3⋊C8—92no—
84C2×C3⋊C8—96no—
85C22×C32—22no—
86C22×C32SL(2,ℤ/4)12yes—
87C22×C32C2×S411yescharacteristic
88C32×C4—31no—
89C32×C4—31no—
90C32×C4—32no—
91C3⋊Dic3—122no—
92C22×C9—43no—
93C22×C9—46no—
94C22×C9—86no—
95Dic9—246no—
96C22×Q8C3⋊D911yescharacteristic
97C2×M16—271no—
98#117—271no—
99#103—272no—
100#103—272no—
101C3×C9—41noSylow
102C23×C3(C22⋊C9)⋊C213yes—
103C23×C3C3⋊S411yescharacteristic
104C3×Q8—36no—
105C3×Q8—33no—
106C3×Q8—32no—
107C3×Q8—31no—
108C3×Q8—61no—
109C3×Q8—63no—
110C3×Q8C2×Dic913yes—
111C3×Q8C2×C3⋊Dic311yescharacteristic
112C2×C3×C4—31no—
113C2×C3×C4—31no—
114C2×C3×C4—36no—
115C2×C3×C4—36no—
116C2×C3×C4—33no—
117C2×C3×C4—33no—
118C2×C3×C4—32no—
119C2×C3×C4—32no—
120C2×Dic3—182no—
121C2×Dic3—186no—
122C3⋊C8—182no—
123C3⋊C8—186no—
124C2×C32—22no—
125C2×C32#98312yes—
126C2×C32C2×SL(2,ℤ/4)11yescharacteristic
127C2×C9—46no—
128C2×C9—43no—
129C2×C9—86no—
130C2×Q8—31no—
131C2×Q8—32no—
132C2×Q8C3⋊Dic912yescomplemented
133C2×Q8C2×C3⋊D911yescharacteristic
134C22×C4—31no—
135M16—542no—
136C2×C8—272no—
137C4⋊C4—272no—
138C22⋊C4—541no—
139C42—541no—
140C22×C3—22no—
141C22×C3—26no—
142C22×C3(C22⋊C9)⋊C416yes—
143C22×C3C2×(C22⋊C9)⋊C213yes—
144C22×C3C3⋊SL(2,ℤ/4)12yes—
145C22×C3C2×C3⋊S411yescharacteristic
146C3×C4—31no—
147C3×C4—31no—
148C3×C4—32no—
149C3×C4—33no—
150C3×C4—33no—
151C3×C4—36no—
152Dic3—362no—
153Dic3—366no—
154C32#542011yescharacteristic
155C9—43no—
156C23C3⋊((C22⋊C9)⋊C2)11yescharacteristic
157Q8—32no—
158Q8—31no—
159Q8—61no—
160Q8C2×C3⋊Dic911yescharacteristic
161C2×C4—32no—
162C2×C4—32no—
163C2×C4—31no—
164C2×C4—31no—
165C2×C4—542no—
166C8—542no—
167C2×C3—26no—
168C2×C3—22no—
169C2×C3#6368916yes—
170C2×C3C2×(C22⋊C9)⋊C413yes—
171C2×C3#6426412yes—
172C2×C3C2×C3⋊SL(2,ℤ/4)11yescharacteristic
173C22—22no—
174C22C3⋊((C22⋊C9)⋊C4)12yes—
175C22C2×C3⋊((C22⋊C9)⋊C2)11yescharacteristic
176C4—31no—
177C4—31no—
178C4—32no—
179C4—1082no—
180C3#1058952213yescomplemented
181C3#1059365411yescharacteristic
182C2—22no—
183C2#1172238512yes—
184C2C2×C3⋊((C22⋊C9)⋊C4)11yescharacteristic
subgroupG/H
centerC22C2×C3⋊((C22⋊C9)⋊C2)
commutator, hypocenterC3×Q8⋊C9C2×C4
FrattiniC22×C3C2×C3⋊S4
FittingC22×C32×Q8D3
socleC22×C32C2×S4
prime radicalC23×C32S4
cosocleC2×C3×Q8⋊C9C22
hypercenterC23C3⋊((C22⋊C9)⋊C2)
LCSG ▷ C3×Q8⋊C9
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C3⋊#63421C2semidirect
5. C2×C3×Q8⋊C9C4semidirect
30. C2×Q8⋊C9Dic3semidirect
69. C2×C3×Q8Dic9semidirect
132. C2×Q8C3⋊Dic9semidirect
180. C3#10589522semidirect

Maximal subgroups

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

subgroupindex
C22×C3×Q8⋊C92
C3⋊#634212
#105929243
#105895223
#885514