Finite Group Databasebeta

← #13147025

Subgroups — #13147025

Order 1728.

346 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 59 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
1#11722390C211yesmaximal, characteristic
2C3×#131⋊C9C211yesmaximal, characteristic, complemented
3C3⋊#63422C211yesmaximal, characteristic, complemented
4#10592843—31nomaximal
5#10589506—33nomaximal
6C3×#41⋊C9C2211yescharacteristic
7C3⋊#3689C411yescharacteristic, complemented
8C3⋊#3688C411yescharacteristic, complemented
9C8×C3⋊D9—41nomaximal
10C32×#131D311yescharacteristic
11#64159—31no—
12#64110—31no—
13#63691—33no—
14#63664—31no—
15#63652—31no—
16#63649—31no—
17C8×C3⋊Dic3—31no—
18#131⋊C9D313yescomplemented
19#63422—33no—
20C4×C3⋊D9—41no—
21C3×Q8⋊C9C2×C411yescharacteristic
22C3×C8×C9—41no—
23C3⋊(C9⋊C8)—41no—
24#5144—93no—
25#5144—91no—
26C32×#41C2×D311yescharacteristic
27#3828—31no—
28C32×M16—31no—
29C2×C32×C8—31no—
30#3756—31no—
31#3755—31no—
32#3754—31no—
33#3753—31no—
34C4×C3⋊Dic3—31no—
35#3748—31no—
36C2×C3⋊(C3⋊C8)—31no—
37#3743—61no—
38C8×C3⋊D3—61no—
39#41⋊C9C2×D313yes—
40#3689—33no—
41#3688—33no—
42C8×D9—123no—
43C2×C3⋊D9—41no—
44C3×C4×C9—41no—
45C3⋊Dic9—41no—
46C3×#131D913yescomplemented
47C3×#131C3⋊D311yescharacteristic
48#950—93no—
49#950—91no—
50#901—93no—
51#901—91no—
52#837—93no—
53#837—91no—
54#825—93no—
55#825—91no—
56#822—93no—
57#822—91no—
58C8×Dic3—93no—
59C8×Dic3—91no—
60C32×Q8C4×D311yescharacteristic
61C32×D4—31no—
62C2×C32×C4—31no—
63#638—61no—
64C2×C3⋊Dic3—61no—
65C4⋊(C3⋊D3)—31no—
66C4×C3⋊D3—61no—
67#634—31no—
68C32×C8—31no—
69C32×C8S411yescharacteristic
70C3⋊(C3⋊C8)—31no—
71C3⋊(C3⋊C8)—31no—
72C3⋊(C3⋊C8)—61no—
73C4×D9—123no—
74Q8⋊C9C4×D313yes—
75C8×C9—43no—
76C9⋊C8—123no—
77#443—271noSylow
78C2×C3×C9—41no—
79C3⋊D9—42no—
80C3×#41C2×D913yes—
81C3×#41C2×C3⋊D311yescharacteristic
82#235—91no—
83#235—93no—
84C3×M16—33no—
85C3×M16—31no—
86C2×C3×C8—33no—
87C2×C3×C8—31no—
88#216—93no—
89#216—91no—
90#215—93no—
91#215—91no—
92#214—93no—
93#214—91no—
94#213—93no—
95#213—91no—
96C4×Dic3—91no—
97C4×Dic3—93no—
98#208—91no—
99#208—93no—
100C2×C3⋊C8—91no—
101C2×C3⋊C8—93no—
102ΓL(1,𝔽25)—181no—
103ΓL(1,𝔽25)—183no—
104D3×C8—183no—
105D3×C8—181no—
106C22×C32—31no—
107C2×C3⋊D3—61no—
108C32×C4—31no—
109C32×C4C2×S411yescharacteristic
110C3⋊Dic3—62no—
111C3⋊Dic3—61no—
112C2×D9—123no—
113C4×C9—43no—
114Dic9—123no—
115#135—271no—
116#131—271no—
117#131C3⋊D911yescharacteristic, complemented
118#108—271no—
119#104—271no—
120#104—271no—
121C4×C8—271no—
122C3×C9—41noSylow
123C3×Q8C4×D913yes—
124C3×Q8C4×C3⋊D311yescharacteristic
125C3×D4—33no—
126C3×D4—31no—
127C2×C3×C4—31no—
128C2×C3×C4—33no—
129#67—181no—
130#67—183no—
131C2×Dic3—181no—
132C2×Dic3—183no—
133D12—91no—
134D12—93no—
135C4×D3—181no—
136C4×D3—183no—
137Dic6—91no—
138Dic6—93no—
139C3×C8—31no—
140C3×C8—33no—
141C3×C8(C22⋊C9)⋊C213yes—
142C3×C8C3⋊S411yescharacteristic
143C3⋊C8—91no—
144C3⋊C8—93no—
145C3⋊C8—91no—
146C3⋊C8—93no—
147C3⋊C8—181no—
148C3⋊C8—183no—
149C2×C32—61no—
150C2×C32C4×S411yescharacteristic
151C3⋊D3—121no—
152C2×C9—43no—
153D9—126no—
154#41—271no—
155#41C2×C3⋊D911yescharacteristic
156Q16—271no—
157SD16—271no—
158SD16—271no—
159D8—271no—
160M16—31no—
161M16—271no—
162M16—541no—
163C2×C8—31no—
164C2×C8—271no—
165C2×C8—541no—
166C42—271no—
167C22×C3—31no—
168C22×C3—33no—
169C2×D3—181no—
170C2×D3—183no—
171C3×C4—31no—
172C3×C4—33no—
173C3×C4C2×(C22⋊C9)⋊C213yes—
174C3×C4C2×C3⋊S411yescharacteristic
175Dic3—186no—
176Dic3—182no—
177Dic3—181no—
178Dic3—183no—
179C32#540411yescharacteristic
180C9—43no—
181Q8—271no—
182Q8C4×C3⋊D911yescharacteristic
183D4—31no—
184D4—271no—
185D4—541no—
186C2×C4—31no—
187C2×C4—541no—
188C2×C4—541no—
189C8—31no—
190C8—271no—
191C8—271no—
192C8—541no—
193C8C3⋊((C22⋊C9)⋊C2)11yescharacteristic
194C2×C3—63no—
195C2×C3—61no—
196C2×C3C4×(C22⋊C9)⋊C213yes—
197C2×C3C4×C3⋊S411yescharacteristic
198D3—363no—
199D3—361no—
200C22—31no—
201C22—541no—
202C4—31no—
203C4—542no—
204C4—541no—
205C4C2×C3⋊((C22⋊C9)⋊C2)11yescharacteristic
206C3#1058950613yescomplemented
207C3#1059363811yescharacteristic
208C2—61no—
209C2—1081no—
210C2C4×C3⋊((C22⋊C9)⋊C2)11yescharacteristic
subgroupG/H
center, hypercenterC8C3⋊((C22⋊C9)⋊C2)
commutator, hypocenterC3×Q8⋊C9C2×C4
FrattiniC3×C4C2×C3⋊S4
FittingC32×#131D3
socleC2×C32C4×S4
prime radical#11722390C2
cosocleC3×#41⋊C9C22
LCSG ▷ C3×Q8⋊C9
UCSC1 ◁ C8

Complements

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

NH ≅ G/Ntype
2. C3×#131⋊C9C2semidirect
3. C3⋊#63422C2semidirect
7. C3⋊#3689C4semidirect
8. C3⋊#3688C4semidirect
18. #131⋊C9D3semidirect
46. C3×#131D9semidirect
117. #131C3⋊D9semidirect
206. C3#10589506semidirect

Maximal subgroups

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

subgroupindex
#117223902
C3×#131⋊C92
C3⋊#634222
#105928433
#105895063
C8×C3⋊D94