Finite Group Databasebeta

← #10592408⋊C3 #13141532

Subgroups — #10592408⋊C3 #13141532

Order 1728.

376 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 108 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
2H(3,𝔽3)×#97C211yesmaximal, characteristic, complemented
3#63647⋊C3C212yesmaximal
4#63641⋊C3C212yesmaximal, complemented
5C4×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
6#10592408C311yesmaximal, characteristic, complemented
7C3×#4709—31nomaximal
8C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
9C2×C8×H(3,𝔽3)C2212yes—
10C42×H(3,𝔽3)C2211yescharacteristic
11#3752⋊C3C412yes—
12#3750⋊C3C412yes—
13C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15#64081C2×C311yescharacteristic
16C3×#872—31no—
17C32×#97—21no—
18C32×#97D311yescharacteristic, complemented
19C32×#97C2×C311yescharacteristic, complemented
20#63647C2×C312yes—
21#63641C2×C312yescomplemented
22C4×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
23C3×(C2×D3)⋊C8—32no—
24C3×Dic3⋊C8—32no—
25C3×C4×C3⋊C8—31no—
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—
32C8×H(3,𝔽3)—22no—
33(C3⋊(C3⋊C8))⋊C3—22no—
34C3×#433—91no—
35#4709—31no—
36#4709—91no—
37C2×C4×C3⋊D3C22×C311yescharacteristic
38C2×C3×C4×D3—31no—
39C2×C32×C8—22no—
40C2×C32×C8C2×D312yes—
41C2×C32×C8C22×C312yes—
42C32×C42—21no—
43C32×C42C2×D311yescharacteristic
44C32×C42C22×C311yescharacteristic
45#3752C3×C412yes—
46#3750C3×C412yes—
47C4×C3⋊Dic3C22×C311yescharacteristic
48C2×C3⋊(C3⋊C8)C22×C312yes—
49C3×(C2×D3)⋊C4—32no—
50C3×Dic3⋊C4—32no—
51C3×C4×Dic3—31no—
52C2×C3×C3⋊C8—32no—
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×C4⋊C8—92no—
64C3×C22⋊C8—92no—
65C3×#97—31no—
66C3×#97—31no—
67C3×#97—61no—
68C3×#97C3×D311yescharacteristic
69C3×C4×C8—91no—
70(C2×D3)⋊C8—32no—
71(C2×D3)⋊C8—92no—
72Dic3⋊C8—32no—
73Dic3⋊C8—92no—
74C4×C3⋊C8—31no—
75C4×C3⋊C8—91no—
76C22×C3⋊D3C2×C3×C411yescharacteristic
77C22×C3×D3—31no—
78C2×C32×C4—22no—
79C2×C32×C4—21no—
80C2×C32×C4C4×D312yes—
81C2×C32×C4C2×C3×C412yes—
82C2×C32×C4C22×D311yescharacteristic
83C2×C32×C4C23×C311yescharacteristic
84C2×C3⋊Dic3C2×C3×C412yes—
85C2×C3⋊Dic3C2×C3×C411yescharacteristic
86C4×C3⋊D3—22no—
87C2×C3×Dic3—32no—
88C2×C3×Dic3—31no—
89C3×C4×D3—62no—
90C32×C8—24no—
91C32×C8—22no—
92C32×C8—22no—
93C3⋊(C3⋊C8)—22no—
94C3×C3⋊C8—62no—
95#433—91noSylow
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—32no—
103C2×C3×C8—32no—
104C2×C3×C8—62no—
105C2×C3×C8—92no—
106C2×C3×C8C2×C3×D312yes—
107C3×C4⋊C4—92no—
108C3×C22⋊C4—92no—
109C3×C42—31no—
110C3×C42—31no—
111C3×C42—61no—
112C3×C42—91no—
113C3×C42C2×C3×D311yescharacteristic
114(C2×D3)⋊C4—32no—
115(C2×D3)⋊C4—92no—
116Dic3⋊C4—32no—
117Dic3⋊C4—92no—
118C4×Dic3—31no—
119C4×Dic3—91no—
120C2×C3⋊C8—32no—
121C2×C3⋊C8—92no—
122C22×C32—21no—
123C22×C32C2×C4×D311yescharacteristic
124C22×C32C22×C3×C411yescharacteristic
125C2×C3⋊D3—21no—
126C2×C3⋊D3—22no—
127C2×C3×D3—61no—
128C2×C3×D3—62no—
129C32×C4—22no—
130C32×C4—22no—
131C32×C4—22no—
132C32×C4—42no—
133C32×C4#23512yes—
134C32×C4C3×#4112yes—
135C3⋊Dic3—21no—
136C3⋊Dic3—22no—
137C3×Dic3—61no—
138C3×Dic3—62no—
139#117—91no—
140C4⋊C8—92no—
141C22⋊C8—92no—
142#97(C3⋊D3)⋊C311yescharacteristic, complemented
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×C4C3×C4×D312yes—
158C2×C3×C4C22×C3×D311yescharacteristic
159C2×Dic3—32no—
160C2×Dic3—31no—
161C2×Dic3—92no—
162C2×Dic3—91no—
163C4×D3—62no—
164C4×D3—182no—
165C3×C8—22no—
166C3×C8—62no—
167C3×C8—62no—
168C3×C8—64no—
169C3×C8—182no—
170C3⋊C8—62no—
171C3⋊C8—182no—
172C2×C32—22no—
173C2×C32—21no—
174C2×C32#87211yescharacteristic
175C2×C32#90712yes—
176C2×C32C3×#11711yescharacteristic
177C2×C32C3×#13112yes—
178C3⋊D3—41no—
179C3×D3—121no—
180C22×C4—91no—
181C2×C8—92no—
182C2×C8C2×(C3⋊D3)⋊C312yes—
183C4⋊C4—92no—
184C22⋊C4—92no—
185C42—91no—
186C42C2×(C3⋊D3)⋊C311yescharacteristic
187C22×C3—31no—
188C22×C3—31no—
189C22×C3—61no—
190C22×C3—181no—
191C22×C3—182no—
192C22×C3C2×C3×C4×D311yescharacteristic
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—62no—
203C3×C4—122no—
204C3×C4—181no—
205C3×C4—182no—
206C3×C4C3×#23512yes—
207Dic3—62no—
208Dic3—61no—
209Dic3—181no—
210Dic3—182no—
211C32—21no—
212C32#470911yescharacteristic, complemented
213C32C3×#43311yescharacteristic, complemented
214C23—91no—
215C2×C4—92no—
216C2×C4—91no—
217C2×C4—182no—
218C2×C4C4×(C3⋊D3)⋊C312yes—
219C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
220C8—22no—
221C8—182no—
222C2×C3—32no—
223C2×C3—32no—
224C2×C3—31no—
225C2×C3—31no—
226C2×C3—62no—
227C2×C3—61no—
228C2×C3—361no—
229C2×C3C3×#87211yescharacteristic
230C2×C3C3×#90712yes—
231D3—121no—
232D3—361no—
233C22—182no—
234C22—181no—
235C22C2×C4×(C3⋊D3)⋊C311yescharacteristic
236C4—22no—
237C4—182no—
238C4—181no—
239C4#3828⋊C312yes—
240C3—31no—
241C3—31no—
242C3—61no—
243C3C3×#470911yescharacteristic
244C2—361no—
245C2#64081⋊C311yescharacteristic
246C2#64116⋊C312yes—
subgroupG/H
centerC2×C4C22×(C3⋊D3)⋊C3
commutatorC22×C32C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#97C2
socleC22×C3C2×C3×C4×D3
prime radicalC22×(C3⋊D3)⋊C3C2×C4
cosocleC2×C32×C4C23×C3
hypercenter#97(C3⋊D3)⋊C3
hypocenterC32C3×#433
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ #97

Complements

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

NH ≅ G/Ntype
2. H(3,𝔽3)×#97C2semidirect
4. #63641⋊C3C2semidirect
5. C4×(C3⋊(C3⋊C8))⋊C3C2semidirect
6. #10592408C3semidirect
18. C32×#97D3semidirect
19. C32×#97C2×C3semidirect
21. #63641C2×C3semidirect
22. C4×C3⋊(C3⋊C8)C2×C3semidirect
142. #97(C3⋊D3)⋊C3semidirect
144. H(3,𝔽3)#433semidirect
212. C32#4709semidirect
213. C32C3×#433semidirect

Maximal subgroups

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

subgroupindex
#64081⋊C32
H(3,𝔽3)×#972
#63647⋊C32
#63641⋊C32
C4×(C3⋊(C3⋊C8))⋊C32
#105924083
C3×#47093