Finite Group Databasebeta

← #13138879

Subgroups — #13138879

Order 1728.

404 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 79 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#64110⋊C3C212yesmaximal, complemented
2#11721060C211yesmaximal, characteristic
3#11720548C211yesmaximal, characteristic, complemented
4#11720316C211yesmaximal, characteristic, complemented
5#11720305C212yesmaximal, complemented
6#10590924—32nomaximal
7#3828⋊C3C2212yes—
8C2×C8×H(3,𝔽3)C2211yescharacteristic
9C4×#1084C2211yescharacteristic
10C2×#6180C2211yescharacteristic
11#3748⋊C3C2212yes—
12#3743⋊C3—22no—
13C8×(C3⋊D3)⋊C3—22no—
14#88433C411yescharacteristic, complemented
15#88432C411yescharacteristic, complemented
16#88431C412yescomplemented
17#64110D312yescomplemented
18C3×#901—32no—
19#63829—32no—
20C3×#815—32no—
21C3×#815—61no—
22#63577—32no—
23#63570—32no—
24#63569—32no—
25C2×C4×H(3,𝔽3)C2311yescharacteristic
26C2×#1084—21no—
27#638⋊C3—22no—
28C4⋊((C3⋊D3)⋊C3)C2×C412yes—
29C4×(C3⋊D3)⋊C3—22no—
30#634⋊C3C2×C412yes—
31C8×H(3,𝔽3)D412yes—
32#6180C2×C411yescharacteristic
33#6180C2×C411yescharacteristic
34(C3⋊(C3⋊C8))⋊C3—22no—
35#5145—92no—
36#4717—91no—
37#3828C2×D312yes—
38C3×#235—32no—
39C2×C32×C8—21no—
40C2×C32×C8C2×D312yes—
41#3748C2×D312yes—
42#3743—22no—
43C8×C3⋊D3—22no—
44C3×C4×Dic3—32no—
45C3×C4×Dic3—61no—
46C3×#208—32no—
47C2×C3×C3⋊C8—32no—
48C2×C3×C3⋊C8—61no—
49C3×ΓL(1,𝔽25)—62no—
50C3×D3×C8—62no—
51#3719—32no—
52#3717—32no—
53#3716—32no—
54#3714—32no—
55C22×H(3,𝔽3)#4111yescharacteristic
56C2×(C3⋊D3)⋊C3—22no—
57C4×H(3,𝔽3)C22×C411yescharacteristic
58C4×H(3,𝔽3)C2×D411yescharacteristic
59#1084—41no—
60(C3⋊Dic3)⋊C3—22no—
61C3×#131—92no—
62#950—92no—
63#904—91no—
64#901—32no—
65#901—92no—
66C3×#97—91no—
67#837—92no—
68#825—92no—
69#822—92no—
70#819—91no—
71#815—32no—
72#815—61no—
73#805—92no—
74C2×C32×C4—21no—
75C2×C32×C4C22×D312yes—
76#638—22no—
77C4⋊(C3⋊D3)C4×D312yes—
78C4×C3⋊D3—22no—
79#634C4×D312yes—
80C3×#67—62no—
81C2×C3×Dic3—62no—
82C2×C3×Dic3—62no—
83C3×D12—32no—
84C3×C4×D3—62no—
85C3×Dic6—32no—
86C32×C8—22no—
87C32×C8#6712yes—
88C32×C8#6712yes—
89C3⋊(C3⋊C8)—22no—
90C3×C3⋊C8—32no—
91C3×C3⋊C8—32no—
92C3×C3⋊C8—61no—
93C3×C3⋊C8—61no—
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—
103C3×M16—182no—
104C2×C3×C8—32no—
105C2×C3×C8—61no—
106C2×C3×C8—91no—
107C2×C3×C8—182no—
108C2×C3×C8D3211yescharacteristic
109C3×C42—91no—
110#216—92no—
111#215—92no—
112#214—92no—
113#213—92no—
114C4×Dic3—32no—
115C4×Dic3—61no—
116#208—32no—
117#208—92no—
118C2×C3⋊C8—32no—
119C2×C3⋊C8—61no—
120Dic12—91no—
121D24—91no—
122#204—92no—
123ΓL(1,𝔽25)—62no—
124ΓL(1,𝔽25)—182no—
125D3×C8—62no—
126D3×C8—182no—
127C22×C32—21no—
128C22×C32#23512yes—
129C2×C3⋊D3—22no—
130C2×C3×D3—62no—
131C32×C4—21no—
132C32×C4—21no—
133C32×C4C2×C4×D312yes—
134C32×C4C2×#6712yes—
135C3⋊Dic3—22no—
136C3×Dic3—62no—
137C3×Dic3—122no—
138C3×Dic3—122no—
139#135—271no—
140#131—92no—
141#108—271no—
142#104—272no—
143#97—91no—
144H(3,𝔽3)#44411yescharacteristic, complemented, Sylow
145C3×Q8—92no—
146C3×D4—92no—
147C3×D4—182no—
148C2×C3×C4—32no—
149C2×C3×C4—61no—
150C2×C3×C4—182no—
151C2×C3×C4—181no—
152C2×C3×C4C2×D3211yescharacteristic
153#67—62no—
154#67—182no—
155C2×Dic3—62no—
156C2×Dic3—62no—
157D12—32no—
158D12—92no—
159C4×D3—62no—
160C4×D3—182no—
161Dic6—32no—
162Dic6—92no—
163C3×C8—32no—
164C3×C8—32no—
165C3×C8—62no—
166C3×C8—91no—
167C3×C8—91no—
168C3×C8—182no—
169C3×C8#62512yes—
170C3⋊C8—32no—
171C3⋊C8—32no—
172C3⋊C8—61no—
173C3⋊C8—61no—
174C3⋊C8—62no—
175C3⋊C8—182no—
176C2×C32—22no—
177C2×C32—22no—
178C2×C32—21no—
179C2×C32C4×#6712yes—
180C3⋊D3—42no—
181C3×D3—122no—
182#41—92no—
183Q16—271no—
184SD16—272no—
185D8—271no—
186M16—92no—
187M16—182no—
188C2×C8—91no—
189C2×C8—182no—
190C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
191C42—91no—
192C22×C3—32no—
193C22×C3—61no—
194C22×C3—182no—
195C22×C3#379811yescharacteristic
196C2×D3—62no—
197C2×D3—182no—
198C3×C4—32no—
199C3×C4—32no—
200C3×C4—61no—
201C3×C4—61no—
202C3×C4—182no—
203C3×C4—361no—
204C3×C4C4×D3211yescharacteristic
205C3×C4C2×#62511yescharacteristic
206Dic3—62no—
207Dic3—122no—
208Dic3—122no—
209Dic3—182no—
210C32—21no—
211C32#514512yescomplemented
212Q8—92no—
213D4—92no—
214D4—182no—
215C2×C4—182no—
216C2×C4—181no—
217C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
218C8—91no—
219C8—91no—
220C8—182no—
221C8#619812yes—
222C2×C3—21no—
223C2×C3—32no—
224C2×C3—62no—
225C2×C3—62no—
226C2×C3—61no—
227C2×C3—362no—
228C2×C3C4×#62511yescharacteristic
229D3—122no—
230D3—362no—
231C22—182no—
232C22#8865011yescharacteristic
233C4—182no—
234C4—361no—
235C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
236C4C2×#619811yescharacteristic
237C3—32no—
238C3—61no—
239C3#1059092211yescharacteristic
240C2—21no—
241C2—362no—
242C2C4×#619811yescharacteristic
subgroupG/H
centerC4C2×#6198
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC2×C3C4×#625
prime radical#11721060C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#444
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. #64110⋊C3C2semidirect
3. #11720548C2semidirect
4. #11720316C2semidirect
5. #11720305C2semidirect
14. #88433C4semidirect
15. #88432C4semidirect
16. #88431C4semidirect
17. #64110D3semidirect
144. H(3,𝔽3)#444semidirect
211. C32#5145semidirect

Maximal subgroups

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

subgroupindex
#64110⋊C32
#117210602
#117205482
#117203162
#117203052
#105909243