Finite Group Databasebeta

← #13137352

Subgroups — #13137352

Order 1728.

380 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 91 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#64081⋊C3C211yesmaximal, characteristic
2C4×#6180C211yesmaximal, characteristic, complemented
3#63630⋊C3C211yesmaximal, characteristic, complemented
4#11720317C212yesmaximal
5#11720311C212yesmaximal, complemented
6#10590298—31nomaximal
7#10590264—31nomaximal
8C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
9C42×H(3,𝔽3)C2211yescharacteristic
10C2×#6180C2212yes—
11#3752⋊C3C412yes—
12#3750⋊C3C412yes—
13C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15#64081D311yescharacteristic
16C3×#872—31no—
17#63630D311yescharacteristic, complemented
18C3×#803—31no—
19C3×C4×C3⋊C8—31no—
20C3×C4×C3⋊C8—31no—
21C3×C4×C3⋊C8—61no—
22#63572—32no—
23Dic3⋊(C3⋊C8)—32no—
24#63557—32no—
25(C2×D3)⋊(C3⋊C8)—32no—
26C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
27C2×C4×H(3,𝔽3)C2×C412yes—
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
30C2×(C3⋊Dic3)⋊C3C2×C412yes—
31C4×(C3⋊D3)⋊C3—22no—
32#6180—22no—
33(C3⋊(C3⋊C8))⋊C3—22no—
34#5000—91no—
35#4709—91no—
36#4690—91no—
37C2×C4×C3⋊D3C2×D311yescharacteristic
38C2×C3×C4×D3—31no—
39C32×C42—21no—
40C32×C42C2×D311yescharacteristic
41C32×C42C2×D311yescharacteristic
42#3752Dic312yes—
43#3750Dic312yes—
44C4×C3⋊Dic3C2×D311yescharacteristic
45C2×C3⋊(C3⋊C8)C2×D312yes—
46C3×(C2×D3)⋊C4—32no—
47C3×Dic3⋊C4—32no—
48C3×C4×Dic3—31no—
49C2×C3×C3⋊C8—32no—
50C2×C3×C3⋊C8—32no—
51C2×C3×C3⋊C8—32no—
52C2×C3×C3⋊C8—62no—
53C22×H(3,𝔽3)C22×C411yescharacteristic
54C2×(C3⋊D3)⋊C3—21no—
55C2×(C3⋊D3)⋊C3—22no—
56C4×H(3,𝔽3)—22no—
57C4×H(3,𝔽3)#4112yes—
58(C3⋊Dic3)⋊C3—21no—
59(C3⋊Dic3)⋊C3—22no—
60C3×#117—91no—
61#872—31no—
62#872—91no—
63C3×#97—91no—
64C3×C4×C8—91no—
65#830—92no—
66(C2×D3)⋊C8—92no—
67(C2×D3)⋊C8—92no—
68Dic3⋊C8—92no—
69Dic3⋊C8—92no—
70C4⋊(C3⋊C8)—92no—
71#803—31no—
72#803—91no—
73C4×C3⋊C8—31no—
74C4×C3⋊C8—31no—
75C4×C3⋊C8—61no—
76C22×C3⋊D3C2×Dic311yescharacteristic
77C22×C3×D3—31no—
78C2×C32×C4—22no—
79C2×C32×C4—21no—
80C2×C32×C4C4×D312yes—
81C2×C32×C4C2×Dic312yes—
82C2×C32×C4C22×D311yescharacteristic
83C2×C32×C4C22×D311yescharacteristic
84C2×C3⋊Dic3C2×Dic312yes—
85C2×C3⋊Dic3C2×Dic311yescharacteristic
86C4×C3⋊D3—22no—
87C2×C3×Dic3—32no—
88C2×C3×Dic3—31no—
89C3×C4×D3—62no—
90C3⋊(C3⋊C8)—22no—
91C3×C3⋊C8—64no—
92C3×C3⋊C8—62no—
93C3×C3⋊C8—62no—
94C3×C3⋊C8—62no—
95#433—271noSylow
96C2×H(3,𝔽3)#11711yescharacteristic
97C2×H(3,𝔽3)#13112yes—
98(C3⋊D3)⋊C3—41no—
99C22×C3×C4—91no—
100C2×C4×D3—31no—
101C2×C4×D3—91no—
102C2×C3×C8—92no—
103C2×C3×C8—92no—
104C3×C4⋊C4—92no—
105C3×C22⋊C4—92no—
106C3×C42—31no—
107C3×C42—31no—
108C3×C42—61no—
109C3×C42—91no—
110C3×C42D3211yescharacteristic
111(C2×D3)⋊C4—32no—
112(C2×D3)⋊C4—92no—
113Dic3⋊C4—32no—
114Dic3⋊C4—92no—
115C4×Dic3—31no—
116C4×Dic3—91no—
117C2×C3⋊C8—32no—
118C2×C3⋊C8—32no—
119C2×C3⋊C8—32no—
120C2×C3⋊C8—62no—
121C2×C3⋊C8—92no—
122C22×C32—21no—
123C22×C32C2×C4×D311yescharacteristic
124C22×C32C22×Dic311yescharacteristic
125C2×C3⋊D3—21no—
126C2×C3⋊D3—22no—
127C2×C3×D3—61no—
128C2×C3×D3—62no—
129C32×C4—24no—
130C32×C4—22no—
131C32×C4—22no—
132C32×C4—22no—
133C32×C4#23512yes—
134C32×C4#23512yes—
135C3⋊Dic3—21no—
136C3⋊Dic3—22no—
137C3×Dic3—61no—
138C3×Dic3—62no—
139#117—91no—
140C4⋊C8—272no—
141C22⋊C8—272no—
142#97—91no—
143C4×C8—91no—
144H(3,𝔽3)#43311yescharacteristic, complemented, Sylow
145C23×C3—91no—
146C22×D3—31no—
147C22×D3—91no—
148C2×C3×C4—32no—
149C2×C3×C4—32no—
150C2×C3×C4—31no—
151C2×C3×C4—31no—
152C2×C3×C4—62no—
153C2×C3×C4—61no—
154C2×C3×C4—92no—
155C2×C3×C4—91no—
156C2×C3×C4—182no—
157C2×C3×C4D3×Dic312yes—
158C2×C3×C4C2×D3211yescharacteristic
159C2×Dic3—32no—
160C2×Dic3—31no—
161C2×Dic3—92no—
162C2×Dic3—91no—
163C4×D3—62no—
164C4×D3—182no—
165C3×C8—182no—
166C3×C8—182no—
167C3⋊C8—62no—
168C3⋊C8—62no—
169C3⋊C8—64no—
170C3⋊C8—62no—
171C3⋊C8—182no—
172C2×C32—22no—
173C2×C32—21no—
174C2×C32#87211yescharacteristic
175C2×C32#90712yes—
176C2×C32#92611yescharacteristic
177C2×C32#94812yes—
178C3⋊D3—41no—
179C3×D3—121no—
180C22×C4—91no—
181C2×C8—92no—
182C2×C8—92no—
183C4⋊C4—92no—
184C22⋊C4—92no—
185C42—91no—
186C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
187C22×C3—31no—
188C22×C3—31no—
189C22×C3—61no—
190C22×C3—181no—
191C22×C3—182no—
192C22×C3C2×D3×Dic311yescharacteristic
193C2×D3—62no—
194C2×D3—61no—
195C2×D3—181no—
196C2×D3—182no—
197C3×C4—22no—
198C3×C4—32no—
199C3×C4—32no—
200C3×C4—62no—
201C3×C4—62no—
202C3×C4—64no—
203C3×C4—62no—
204C3×C4—181no—
205C3×C4—182no—
206C3×C4#379812yes—
207Dic3—62no—
208Dic3—61no—
209Dic3—181no—
210Dic3—182no—
211C32—21no—
212C32#470911yescharacteristic, complemented
213C32#500011yescharacteristic, complemented
214C23—91no—
215C2×C4—92no—
216C2×C4—91no—
217C2×C4—182no—
218C2×C4((C3⋊D3)⋊C3)⋊C412yes—
219C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
220C8—182no—
221C8—182no—
222C2×C3—32no—
223C2×C3—32no—
224C2×C3—31no—
225C2×C3—31no—
226C2×C3—62no—
227C2×C3—61no—
228C2×C3—361no—
229C2×C3#6381512yes—
230C2×C3#6385511yescharacteristic
231D3—121no—
232D3—361no—
233C22—182no—
234C22—181no—
235C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
236C4—22no—
237C4—182no—
238C4—181no—
239C4#8865012yes—
240C3—31no—
241C3—31no—
242C3—61no—
243C3#1059026511yescharacteristic
244C2—361no—
245C2#1172105512yes—
246C2#1172110511yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×D3×Dic3
prime radicalC22×(C3⋊D3)⋊C3C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#433
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C4×#6180C2semidirect
3. #63630⋊C3C2semidirect
5. #11720311C2semidirect
17. #63630D3semidirect
144. H(3,𝔽3)#433semidirect
212. C32#4709semidirect
213. C32#5000semidirect

Maximal subgroups

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

subgroupindex
#64081⋊C32
C4×#61802
#63630⋊C32
#117203172
#117203112
#105902983
#105902643