Finite Group Databasebeta

← #13173163

Subgroups — #13173163

Order 1728.

494 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 75 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⋊D3×#41⋊C3C211yesmaximal, characteristic
2C32×#131⋊C3C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊(#131⋊C3))C211yesmaximal, characteristic, complemented
4#10596221C311yesmaximal, characteristic, complemented
5#5750⋊C3—34nomaximal
6C32×#41⋊C3C2211yescharacteristic
7C3⋊D3×SL(2,𝔽3)C411yescharacteristic
8C3⋊(C3⋊(#41⋊C3))C411yescharacteristic
9C3×#3743—41nomaximal
10#41×C3⋊D3C2×C311yescharacteristic
11D3×#41⋊C3—34no—
12C32×#131C2×C311yescharacteristic, complemented
13#64157C2×C311yescharacteristic, complemented
14#64116—31no—
15M16×C3⋊D3—31no—
16#64110—31no—
17C2×#3743—31no—
18C3×#131⋊C3—24no—
19C3×#131⋊C3D314yescomplemented
20C3⋊(#131⋊C3)—34no—
21C3×C4×C3⋊D3—41no—
22C32×SL(2,𝔽3)C2×C411yescharacteristic
23C33×C8—41no—
24C3×C3⋊(C3⋊C8)—41no—
25#5750—34no—
26#568⋊C3—91no—
27C32×#41C22×C311yescharacteristic
28#3832C3×C411yescharacteristic
29Q8×C3⋊D3C3×C411yescharacteristic
30#3830—31no—
31D4×C3⋊D3—31no—
32#3828—31no—
33C2×C4×C3⋊D3—31no—
34C3×#41⋊C3—24no—
35C3×#41⋊C3C2×D314yes—
36D3×SL(2,𝔽3)—34no—
37C3⋊(#41⋊C3)—34no—
38C32×M16—31no—
39C2×C32×C8—31no—
40#3748—31no—
41C2×C3⋊(C3⋊C8)—31no—
42#3743—31no—
43#3743—31no—
44#3743—31no—
45#3743A411yescharacteristic
46C8×C3⋊D3—32no—
47C3×ΓL(1,𝔽25)—124no—
48C2×C3×C3⋊D3—41no—
49C33×C4—41no—
50C3×C3⋊Dic3—41no—
51D3×#41—34no—
52C2×#41⋊C3—91no—
53C3×#131C3×D314yescomplemented
54#948—34no—
55#907—94no—
56D3×M16—94no—
57#901—94no—
58C2×ΓL(1,𝔽25)—94no—
59#131⋊C3—24no—
60#131⋊C3—91no—
61#131⋊C3C3⋊D311yescharacteristic, complemented
62C22×C3⋊D3—31no—
63C32×Q8C2×C3×C411yescharacteristic
64C32×D4—31no—
65C2×C32×C4—31no—
66#638—32no—
67C2×C3⋊Dic3—31no—
68C4⋊(C3⋊D3)—31no—
69C4×C3⋊D3—31no—
70C4×C3⋊D3—31no—
71C4×C3⋊D3—31no—
72C4×C3⋊D3C2×A411yescharacteristic
73#634—31no—
74C3×C4×D3—124no—
75C3×SL(2,𝔽3)—24no—
76C3×SL(2,𝔽3)C4×D314yes—
77C32×C8—31no—
78C32×C8—44no—
79C32×C8—84no—
80C32×C8C2×A411yescharacteristic
81C3⋊(C3⋊C8)—31no—
82C3⋊(C3⋊C8)C2×A411yescharacteristic
83C3×C3⋊C8—124no—
84#568—91noSylow
85C2×C33—41no—
86C3×C3⋊D3—81no—
87C3×#41C2×C3×D314yes—
88#239—34no—
89D3×Q8—34no—
90#237—94no—
91D3×D4—94no—
92#235—94no—
93C2×C4×D3—94no—
94#41⋊C3—24no—
95#41⋊C3—91no—
96#41⋊C3C2×C3⋊D311yescharacteristic
97C2×SL(2,𝔽3)—91no—
98C3×M16—34no—
99C3×M16—361no—
100C2×C3×C8—34no—
101#208—94no—
102C2×C3⋊C8—94no—
103ΓL(1,𝔽25)—34no—
104ΓL(1,𝔽25)—94no—
105ΓL(1,𝔽25)—94no—
106ΓL(1,𝔽25)—94no—
107D3×C8—98no—
108C22×C32—31no—
109C2×C3⋊D3—31no—
110C2×C3⋊D3—121no—
111C2×C3⋊D3C4×A411yescharacteristic
112C2×C3×D3—124no—
113C32×C4—31no—
114C32×C4—44no—
115C32×C4—84no—
116C32×C4C22×A411yescharacteristic
117C3⋊Dic3—31no—
118C3⋊Dic3C4×A411yescharacteristic
119C3×Dic3—124no—
120C2×#41—91no—
121#131—91no—
122#131—271no—
123#131—271no—
124#131C3×C3⋊D311yescharacteristic, complemented
125C2×M16—271no—
126C2×M16—271no—
127C33—41noSylow
128C22×D3—94no—
129C3×Q8C3×C4×D314yes—
130C3×D4—34no—
131C2×C3×C4—34no—
132C2×C3×C4—361no—
133#67—98no—
134C2×Dic3—94no—
135D12—94no—
136C4×D3—34no—
137C4×D3—94no—
138C4×D3—94no—
139C4×D3—94no—
140Dic6—94no—
141SL(2,𝔽3)—24no—
142SL(2,𝔽3)C4×C3⋊D311yescharacteristic
143C3×C8—34no—
144C3×C8—41no—
145C3×C8—84no—
146C3×C8—361no—
147C3×C8D3×A414yes—
148C3⋊C8—34no—
149C3⋊C8—94no—
150C2×C32—44no—
151C2×C32—61no—
152C2×C32—84no—
153C2×C32C2×C4×A411yescharacteristic
154C3⋊D3—21no—
155C3⋊D3—61no—
156C3×D3—244no—
157#41—91no—
158#41—271no—
159#41—271no—
160#41C2×C3×C3⋊D311yescharacteristic
161C2×Q8—91no—
162C2×D4—271no—
163C22×C4—271no—
164M16—31no—
165M16—91no—
166M16—271no—
167M16—271no—
168M16—271no—
169M16—271no—
170C2×C8—31no—
171C2×C8—272no—
172C2×C8—271no—
173C22×C3—34no—
174C22×C3—361no—
175C2×D3—34no—
176C2×D3—94no—
177C2×D3—364no—
178C3×C4—34no—
179C3×C4—41no—
180C3×C4—84no—
181C3×C4—361no—
182C3×C4C2×D3×A414yes—
183Dic3—34no—
184Dic3—94no—
185C32—44no—
186C32—84no—
187C32#568⋊C311yescharacteristic, complemented
188C23—271no—
189Q8—271no—
190Q8C3×C4×C3⋊D311yescharacteristic
191D4—31no—
192D4—271no—
193D4—272no—
194C2×C4—31no—
195C2×C4—91no—
196C2×C4—271no—
197C2×C4—271no—
198C2×C4—271no—
199C2×C4—271no—
200C8—31no—
201C8—91no—
202C8—271no—
203C8A4×C3⋊D311yescharacteristic
204C2×C3—41no—
205C2×C3—64no—
206C2×C3—84no—
207C2×C3—721no—
208C2×C3C4×D3×A414yes—
209D3—64no—
210D3—184no—
211C22—31no—
212C22—91no—
213C22—271no—
214C22—1081no—
215C4—31no—
216C4—91no—
217C4—271no—
218C4C2×A4×C3⋊D311yescharacteristic
219C3—41no—
220C3—84no—
221C3#5750⋊C314yescomplemented
222C2—61no—
223C2—181no—
224C2—541no—
225C2C4×A4×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC4C2×A4×C3⋊D3
commutator, hypocenterC32×Q8C2×C3×C4
FittingC32×#131C2×C3
socleC2×C32C2×C4×A4
prime radicalC3⋊D3×#41⋊C3C2
cosocleC32×#41C22×C3
hypercenterC8A4×C3⋊D3
LCSG ▷ C32×Q8
UCSC1 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
2. C32×#131⋊C3C2semidirect
3. C3⋊(C3⋊(#131⋊C3))C2semidirect
4. #10596221C3semidirect
12. C32×#131C2×C3semidirect
13. #64157C2×C3semidirect
19. C3×#131⋊C3D3semidirect
53. C3×#131C3×D3semidirect
61. #131⋊C3C3⋊D3semidirect
124. #131C3×C3⋊D3semidirect
187. C32#568⋊C3semidirect
221. C3#5750⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3⋊D3×#41⋊C32
C32×#131⋊C32
C3⋊(C3⋊(#131⋊C3))2
#105962213
#5750⋊C33
C3×#37434