Finite Group Databasebeta

← #13137398

Subgroups — #13137398

Order 1728.

404 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 71 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#11721060C211yesmaximal, characteristic
2#11721055C212yesmaximal, complemented
3#11720520C211yesmaximal, characteristic, complemented
4#63632⋊C3C212yesmaximal, complemented
5#11720316C211yesmaximal, characteristic, complemented
6#10590342—32nomaximal
7#3828⋊C3C2212yes—
8C42×H(3,𝔽3)C2211yescharacteristic
9C2×#6180C2211yescharacteristic
10C2×#6180C2211yescharacteristic
11#3748⋊C3C2212yes—
12#88433C411yescharacteristic, complemented
13#88432C411yescharacteristic, complemented
14#88431C412yescomplemented
15#88430—22no—
16((C3⋊D3)⋊C3)⋊C8—22no—
17#63829—32no—
18#63817—32no—
19#63814—32no—
20#63632D312yescomplemented
21C3×#805—32no—
22C3×#803—32no—
23C3×#803—61no—
24#63577—32no—
25C2×C4×H(3,𝔽3)—21no—
26C2×C4×H(3,𝔽3)C2311yescharacteristic
27#638⋊C3—22no—
28C4⋊((C3⋊D3)⋊C3)C2×C412yes—
29C4×(C3⋊D3)⋊C3—22no—
30#634⋊C3C2×C412yes—
31#6180C2×C411yescharacteristic
32#6180C2×C411yescharacteristic
33#6180D412yes—
34(C3⋊(C3⋊C8))⋊C3—22no—
35#4823—92no—
36#4717—91no—
37#3828C2×D312yes—
38C3×#235—32no—
39C32×C42—21no—
40C32×C42C2×D312yes—
41#3748C2×D312yes—
42C3×#208—32no—
43C2×C3×C3⋊C8—32no—
44C2×C3×C3⋊C8—32no—
45C2×C3×C3⋊C8—61no—
46C2×C3×C3⋊C8—61no—
47#3719—32no—
48#3717—32no—
49#3716—32no—
50#3714—32no—
51#3712—62no—
52#3711—62no—
53#3710—62no—
54D3×C3⋊C8—62no—
55C22×H(3,𝔽3)#4111yescharacteristic
56C2×(C3⋊D3)⋊C3—22no—
57C4×H(3,𝔽3)—41no—
58C4×H(3,𝔽3)C22×C411yescharacteristic
59C4×H(3,𝔽3)C2×D411yescharacteristic
60(C3⋊Dic3)⋊C3—22no—
61#950—92no—
62#948—92no—
63#907—92no—
64#904—91no—
65#901—92no—
66C3×#104—92no—
67C3×#97—91no—
68#822—92no—
69#819—91no—
70#805—32no—
71#805—92no—
72#803—32no—
73#803—61no—
74C2×C32×C4—22no—
75C2×C32×C4—22no—
76C2×C32×C4—21no—
77C2×C32×C4C22×D312yes—
78#638—22no—
79C4⋊(C3⋊D3)C4×D312yes—
80C4×C3⋊D3—22no—
81#634C4×D312yes—
82C3×#67—62no—
83C3×D12—32no—
84C3×C4×D3—62no—
85C3×Dic6—32no—
86C3⋊(C3⋊C8)—22no—
87C3×C3⋊C8—32no—
88C3×C3⋊C8—32no—
89C3×C3⋊C8—32no—
90C3×C3⋊C8—32no—
91C3×C3⋊C8—61no—
92C3×C3⋊C8—61no—
93C3×C3⋊C8—62no—
94C3×C3⋊C8—62no—
95#444—271noSylow
96C2×H(3,𝔽3)—21no—
97C2×H(3,𝔽3)C4×D411yescharacteristic
98(C3⋊D3)⋊C3—42no—
99C3×#41—92no—
100#235—32no—
101#235—92no—
102C3×M16—92no—
103C2×C3×C8—91no—
104C2×C3×C8—91no—
105C3×C42—32no—
106C3×C42—61no—
107C3×C42D3211yescharacteristic
108#216—92no—
109#215—92no—
110#214—92no—
111#213—92no—
112#208—32no—
113#208—92no—
114#208—182no—
115C2×C3⋊C8—32no—
116C2×C3⋊C8—32no—
117C2×C3⋊C8—61no—
118C2×C3⋊C8—61no—
119C2×C3⋊C8—182no—
120Dic12—91no—
121D24—91no—
122#204—92no—
123ΓL(1,𝔽25)—182no—
124ΓL(1,𝔽25)—182no—
125D3×C8—182no—
126D3×C8—182no—
127C22×C32—21no—
128C22×C32#23712yes—
129C2×C3⋊D3—22no—
130C2×C3×D3—62no—
131C32×C4—21no—
132C32×C4—21no—
133C32×C4—42no—
134C32×C4—42no—
135C32×C4C2×C4×D312yes—
136C32×C4D3×D412yes—
137C3⋊Dic3—22no—
138C3×Dic3—62no—
139#135—271no—
140#131—272no—
141#108—271no—
142#104—92no—
143#97—91no—
144H(3,𝔽3)#44411yescharacteristic, complemented, Sylow
145C3×Q8—92no—
146C3×D4—92no—
147C3×D4—182no—
148C2×C3×C4—21no—
149C2×C3×C4—32no—
150C2×C3×C4—62no—
151C2×C3×C4—62no—
152C2×C3×C4—61no—
153C2×C3×C4—182no—
154C2×C3×C4C2×D3211yescharacteristic
155#67—62no—
156#67—182no—
157D12—32no—
158D12—92no—
159C4×D3—62no—
160C4×D3—182no—
161Dic6—32no—
162Dic6—92no—
163C3×C8—91no—
164C3×C8—91no—
165C3×C8—92no—
166C3×C8—182no—
167C3⋊C8—32no—
168C3⋊C8—32no—
169C3⋊C8—32no—
170C3⋊C8—32no—
171C3⋊C8—61no—
172C3⋊C8—61no—
173C3⋊C8—62no—
174C3⋊C8—62no—
175C3⋊C8—182no—
176C2×C32—22no—
177C2×C32—22no—
178C2×C32—21no—
179C2×C32#67⋊C412yes—
180C3⋊D3—42no—
181C3×D3—122no—
182#41—92no—
183Q16—271no—
184SD16—272no—
185D8—271no—
186M16—92no—
187M16—542no—
188C2×C8—91no—
189C2×C8—91no—
190C2×C8—542no—
191C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
192C22×C3—32no—
193C22×C3—61no—
194C22×C3—182no—
195C22×C3#380511yescharacteristic
196C2×D3—62no—
197C2×D3—182no—
198C3×C4—32no—
199C3×C4—32no—
200C3×C4—41no—
201C3×C4—61no—
202C3×C4—61no—
203C3×C4—122no—
204C3×C4—122no—
205C3×C4—182no—
206C3×C4C4×D3211yescharacteristic
207C3×C4#380211yescharacteristic
208Dic3—62no—
209Dic3—182no—
210C32—21no—
211C32#482312yescomplemented
212Q8—92no—
213D4—92no—
214D4—182no—
215C2×C4—21no—
216C2×C4—182no—
217C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
218C8—91no—
219C8—91no—
220C8—92no—
221C8—182no—
222C2×C3—21no—
223C2×C3—32no—
224C2×C3—62no—
225C2×C3—62no—
226C2×C3—61no—
227C2×C3—362no—
228C2×C3#6390211yescharacteristic
229D3—122no—
230D3—362no—
231C22—182no—
232C22#8867211yescharacteristic
233C4—41no—
234C4—182no—
235C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
236C4#8865511yescharacteristic
237C3—32no—
238C3—61no—
239C3#1059039011yescharacteristic
240C2—21no—
241C2—362no—
242C2#1172113811yescharacteristic
subgroupG/H
centerC4#88655
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC2×C3#63902
prime radical#11721060C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#444
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #11721055C2semidirect
3. #11720520C2semidirect
4. #63632⋊C3C2semidirect
5. #11720316C2semidirect
12. #88433C4semidirect
13. #88432C4semidirect
14. #88431C4semidirect
20. #63632D3semidirect
144. H(3,𝔽3)#444semidirect
211. C32#4823semidirect

Maximal subgroups

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

subgroupindex
#117210602
#117210552
#117205202
#63632⋊C32
#117203162
#105903423