Finite Group Databasebeta

← #13173184

Subgroups — #13173184

Order 1728.

626 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 83 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×C32×#41⋊C3C211yesmaximal, characteristic, complemented
2C2×C3⋊(C3⋊(#41⋊C3))C211yesmaximal, characteristic, complemented
3SL(2,𝔽3)×C3⋊Dic3C211yesmaximal, characteristic, complemented
4#10596305C311yesmaximal, characteristic, complemented
5#5834⋊C3—34nomaximal
6C32×#41⋊C3—21no—
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3⋊(C3⋊(#41⋊C3))—21no—
9C3×#3750—41nomaximal
10C2×C32×#41C2×C311yescharacteristic
11C2×#3832C2×C311yescharacteristic
12C2×C3×#41⋊C3—24no—
13C2×C3×#41⋊C3D314yescomplemented
14C2×C3⋊(#41⋊C3)—34no—
15Q8×C3⋊Dic3C2×C311yescharacteristic
16#64149—31no—
17#64147—31no—
18#64139—31no—
19C4×#638—31no—
20Dic3×SL(2,𝔽3)—34no—
21C2×C33×C4—41no—
22C2×C3×C3⋊Dic3—41no—
23C2×C3×C3⋊Dic3—41no—
24C32×SL(2,𝔽3)D411yescharacteristic, complemented
25#5834—34no—
26C4⋊(#41⋊C3)—91no—
27C32×#41—21no—
28C32×#41—61no—
29C2×C32×Q8C22×C311yescharacteristic
30C2×C32×D4—31no—
31C22×C32×C4—31no—
32C2×#638—32no—
33C2×#638—31no—
34#3832—21no—
35#3832—61no—
36C2×C4⋊(C3⋊D3)—31no—
37C2×C4×C3⋊D3—31no—
38C3×#41⋊C3—28no—
39C3×#41⋊C3—24no—
40C2×C3×SL(2,𝔽3)—24no—
41C2×C3×SL(2,𝔽3)C2×D314yes—
42C3⋊(#41⋊C3)—64no—
43#3757—31no—
44#3752—31no—
45C4⋊(C3⋊Dic3)—31no—
46#3750A411yescharacteristic
47C4×C3⋊Dic3—31no—
48C3×Dic3⋊C4—124no—
49C22×C33—41no—
50C33×C4—81no—
51C3×C3⋊Dic3—42no—
52C3×C3⋊Dic3—81no—
53C2×C3×#41C3×D314yes—
54C2×#239—34no—
55C2×#41⋊C3—24no—
56C2×#41⋊C3—91no—
57C2×#41⋊C3C3⋊D311yescharacteristic, complemented
58Q8×Dic3—34no—
59#940—94no—
60#938—94no—
61#930—94no—
62C4×#67—94no—
63C4×SL(2,𝔽3)—91no—
64C23×C32—31no—
65C22×C3⋊D3—31no—
66C32×Q8—31no—
67C32×Q8C3×D411yescharacteristic
68C32×D4—61no—
69C32×D4—61no—
70C2×C32×C4—31no—
71C2×C32×C4—44no—
72C2×C32×C4—61no—
73C2×C32×C4—61no—
74C2×C32×C4—84no—
75C2×C32×C4C2×A411yescharacteristic
76#638—62no—
77#638—122no—
78C2×C3⋊Dic3—31no—
79C2×C3⋊Dic3C2×A411yescharacteristic
80C2×C3⋊Dic3C2×A411yescharacteristic
81C4⋊(C3⋊D3)—61no—
82C4⋊(C3⋊D3)—61no—
83C4×C3⋊D3—61no—
84C4×C3⋊D3—61no—
85C2×C3×Dic3—124no—
86C2×C3×Dic3—124no—
87C3×SL(2,𝔽3)—24no—
88C3×SL(2,𝔽3)#6714yescomplemented
89#550—91noSylow
90C2×C33—41no—
91C2×C33—41no—
92C2×C33—41no—
93C3×#41—24no—
94C3×#41—64no—
95C2×C3×Q8C2×C3×D314yes—
96C2×C3×D4—34no—
97C22×C3×C4—34no—
98C2×#67—94no—
99C2×#67—98no—
100#239—64no—
101#239—184no—
102C2×D12—94no—
103C2×C4×D3—94no—
104#41⋊C3—28no—
105#41⋊C3—21no—
106#41⋊C3—181no—
107C2×SL(2,𝔽3)—24no—
108C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic
109C3×C4⋊C4—361no—
110#217—94no—
111(C2×D3)⋊C4—94no—
112C4⋊Dic3—94no—
113Dic3⋊C4—34no—
114C4×Dic3—94no—
115C22×C32—44no—
116C22×C32—61no—
117C22×C32—61no—
118C22×C32—61no—
119C22×C32—84no—
120C22×C32C22×A411yescharacteristic
121C2×C3⋊D3—61no—
122C2×C3⋊D3—61no—
123C2×C3⋊D3—61no—
124C32×C4—21no—
125C32×C4—31no—
126C32×C4—31no—
127C32×C4—84no—
128C32×C4—88no—
129C3⋊Dic3—21no—
130C3⋊Dic3—61no—
131C3⋊Dic3#41⋊C312yescomplemented
132C3×Dic3—128no—
133C3×Dic3—244no—
134C2×#41—91no—
135C2×#41C3×C3⋊D311yescharacteristic
136#127—271no—
137#121—271no—
138#121—271no—
139C4×Q8—91no—
140C4×D4—271no—
141C33—41noSylow
142C23×C3—34no—
143C22×D3—94no—
144C3×Q8—34no—
145C3×Q8C3×#6714yes—
146C3×D4—64no—
147C3×D4—64no—
148C2×C3×C4—34no—
149C2×C3×C4—41no—
150C2×C3×C4—64no—
151C2×C3×C4—64no—
152C2×C3×C4—84no—
153C2×C3×C4—361no—
154C2×C3×C4—361no—
155C2×C3×C4D3×A414yes—
156#67—188no—
157#67—368no—
158C2×Dic3—34no—
159C2×Dic3—34no—
160C2×Dic3—94no—
161D12—184no—
162D12—184no—
163C4×D3—184no—
164C4×D3—184no—
165SL(2,𝔽3)—24no—
166SL(2,𝔽3)#63811yescharacteristic, complemented
167C2×C32—44no—
168C2×C32—44no—
169C2×C32—44no—
170C2×C32—84no—
171C2×C32—84no—
172C2×C32—84no—
173C2×C32—121no—
174C2×C32D4×A411yescharacteristic
175C2×C32C2×#41⋊C311yescharacteristic
176C2×C32#99411yescharacteristic
177C3⋊D3—121no—
178#41—21no—
179#41—61no—
180#41—181no—
181#41—541no—
182C2×Q8C2×C3×C3⋊D311yescharacteristic
183C2×D4—31no—
184C2×D4—271no—
185C2×D4—271no—
186C2×D4—272no—
187C22×C4—31no—
188C22×C4—271no—
189C4⋊C4—91no—
190C4⋊C4—271no—
191C22⋊C4—271no—
192C22⋊C4—271no—
193C42—271no—
194C22×C3—41no—
195C22×C3—64no—
196C22×C3—64no—
197C22×C3—64no—
198C22×C3—84no—
199C22×C3C2×D3×A414yes—
200C2×D3—184no—
201C2×D3—184no—
202C2×D3—184no—
203C3×C4—24no—
204C3×C4—34no—
205C3×C4—34no—
206C3×C4—81no—
207C3×C4—88no—
208C3×C4—362no—
209C3×C4—721no—
210Dic3—38no—
211Dic3—64no—
212Dic3—184no—
213C32—44no—
214C32—84no—
215C32C4⋊(#41⋊C3)11yescharacteristic, complemented
216C23—31no—
217C23—271no—
218Q8—31no—
219Q8C3×#63811yescharacteristic
220D4—61no—
221D4—61no—
222D4—541no—
223D4—541no—
224D4—542no—
225D4—1082no—
226C2×C4—31no—
227C2×C4—61no—
228C2×C4—61no—
229C2×C4—91no—
230C2×C4—91no—
231C2×C4—271no—
232C2×C4—541no—
233C2×C4—541no—
234C2×C4A4×C3⋊D311yescharacteristic
235C2×C3—41no—
236C2×C3—41no—
237C2×C3—41no—
238C2×C3—84no—
239C2×C3—84no—
240C2×C3—84no—
241C2×C3—124no—
242C2×C3C3⋊#99414yes—
243C2×C3D3×#41⋊C314yes—
244C2×C3A4×#6714yes—
245D3—364no—
246C22—61no—
247C22—61no—
248C22—61no—
249C22—541no—
250C22—541no—
251C22—541no—
252C22C2×A4×C3⋊D311yescharacteristic
253C4—21no—
254C4—31no—
255C4—31no—
256C4—92no—
257C4—181no—
258C4—541no—
259C3—41no—
260C3—84no—
261C3#5834⋊C314yescomplemented
262C2—121no—
263C2—1081no—
264C2#1172449111yescharacteristic
265C2C3⋊D3×#41⋊C311yescharacteristic
266C2A4×#63811yescharacteristic
subgroupG/H
center, FrattiniC22C2×A4×C3⋊D3
commutator, cosocleC2×C32×Q8C22×C3
FittingC2×C32×#41C2×C3
socleC22×C32C22×A4
hypercenterC2×C4A4×C3⋊D3
hypocenterC32×Q8C3×D4
LCSG ▷ C2×C32×Q8 ▷ C32×Q8
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C32×#41⋊C3C2semidirect
2. C2×C3⋊(C3⋊(#41⋊C3))C2semidirect
3. SL(2,𝔽3)×C3⋊Dic3C2semidirect
4. #10596305C3semidirect
13. C2×C3×#41⋊C3D3semidirect
24. C32×SL(2,𝔽3)D4semidirect
57. C2×#41⋊C3C3⋊D3semidirect
88. C3×SL(2,𝔽3)#67semidirect
131. C3⋊Dic3#41⋊C3semidirect
166. SL(2,𝔽3)#638semidirect
215. C32C4⋊(#41⋊C3)semidirect
261. C3#5834⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#41⋊C32
C2×C3⋊(C3⋊(#41⋊C3))2
SL(2,𝔽3)×C3⋊Dic32
#105963053
#5834⋊C33
C3×#37504