Finite Group Databasebeta

← #10592410⋊C3 #13141534

Subgroups — #10592410⋊C3 #13141534

Order 1728.

502 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 118 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×#3745⋊C3C211yesmaximal, characteristic
2C4×C4⋊((C3⋊D3)⋊C3)C212yesmaximal, complemented
3H(3,𝔽3)×#97C211yesmaximal, characteristic, complemented
4#63648⋊C3C212yesmaximal, complemented
5#63644⋊C3C211yesmaximal, characteristic, complemented
6#10592410C311yesmaximal, characteristic, complemented
7C3×#4711—31nomaximal
8C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
9C2×C4×(C3⋊D3)⋊C3—22no—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12C42×H(3,𝔽3)C2211yescharacteristic
13#3752⋊C3—22no—
14C4⋊((C3⋊Dic3)⋊C3)C2212yes—
15#3745⋊C3C414yescomplemented
16C2×#3745C2×C311yescharacteristic
17C4×C4⋊(C3⋊D3)C2×C312yescomplemented
18C2×C3×D24—31no—
19C3×C4×D12—32no—
20C32×#97—21no—
21C32×#97D311yescharacteristic, complemented
22C32×#97C2×C311yescharacteristic, complemented
23#63648C2×C312yescomplemented
24#63644C2×C311yescharacteristic, complemented
25C3×#821—32no—
26C3×#817—31no—
27C22×(C3⋊D3)⋊C3—22no—
28C2×C4×H(3,𝔽3)D412yes—
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30C2×(C3⋊Dic3)⋊C3—22no—
31C4⋊((C3⋊D3)⋊C3)—22no—
32C4⋊((C3⋊D3)⋊C3)C2×C414yescomplemented
33C4×(C3⋊D3)⋊C3—44no—
34C8×H(3,𝔽3)—21no—
35C8×H(3,𝔽3)C2×C412yescomplemented
36C3×#442—91no—
37#4711—31no—
38#4711—91no—
39C2×C4⋊(C3⋊D3)C22×C312yes—
40C2×C4×C3⋊D3—22no—
41C2×C3×D12—32no—
42C2×C3×C4×D3—62no—
43C2×C32×C8—21no—
44C2×C32×C8—21no—
45C2×C32×C8C2×D311yescharacteristic
46C2×C32×C8C2×D311yescharacteristic
47C2×C32×C8C22×C311yescharacteristic
48C2×C32×C8C22×C311yescharacteristic
49C32×C42—21no—
50C32×C42C2×D311yescharacteristic
51C32×C42C22×C311yescharacteristic
52#3752—22no—
53C4⋊(C3⋊Dic3)C22×C312yes—
54#3745C3×C414yescomplemented
55C3×(C2×D3)⋊C4—62no—
56C3×C4⋊Dic3—32no—
57C3×D24—34no—
58C22×H(3,𝔽3)C2×D411yescharacteristic
59C2×(C3⋊D3)⋊C3—24no—
60C2×(C3⋊D3)⋊C3—42no—
61C2×(C3⋊D3)⋊C3—42no—
62C4×H(3,𝔽3)—22no—
63C4×H(3,𝔽3)C22×C411yescharacteristic
64C4×H(3,𝔽3)#4111yescharacteristic
65(C3⋊Dic3)⋊C3—42no—
66C2×C3×D8—91no—
67C3×C4×D4—92no—
68C2×D24—31no—
69C2×D24—91no—
70C4×D12—32no—
71C4×D12—92no—
72C3×#106—91no—
73C3×#102—92no—
74C3×#97—31no—
75C3×#97—31no—
76C3×#97—61no—
77C3×#97C3×D311yescharacteristic
78#821—32no—
79#821—92no—
80#817—31no—
81#817—91no—
82C22×C3⋊D3—22no—
83C22×C3×D3—62no—
84C2×C32×C4—22no—
85C2×C32×C4—21no—
86C2×C32×C4D1212yes—
87C2×C32×C4C3×D412yes—
88C2×C32×C4C22×D311yescharacteristic
89C2×C32×C4C23×C311yescharacteristic
90C2×C3⋊Dic3—22no—
91C4⋊(C3⋊D3)—22no—
92C4⋊(C3⋊D3)C2×C3×C414yescomplemented
93C4×C3⋊D3—44no—
94C2×C3×Dic3—62no—
95C3×D12—34no—
96C3×D12—62no—
97C3×C4×D3—124no—
98C32×C8—22no—
99C32×C8—21no—
100C32×C8—21no—
101C32×C8—22no—
102C32×C8C4×D312yescomplemented
103C32×C8C2×C3×C412yescomplemented
104#442—91noSylow
105C2×H(3,𝔽3)C4×D411yescharacteristic
106C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
107(C3⋊D3)⋊C3—44no—
108C2×C3×D4—92no—
109C22×C3×C4—182no—
110C2×D12—32no—
111C2×D12—92no—
112C2×C4×D3—62no—
113C2×C4×D3—182no—
114C3×D8—94no—
115C2×C3×C8—31no—
116C2×C3×C8—31no—
117C2×C3×C8—31no—
118C2×C3×C8—31no—
119C2×C3×C8—61no—
120C2×C3×C8—61no—
121C2×C3×C8C2×C3×D311yescharacteristic
122C2×C3×C8C2×C3×D311yescharacteristic
123C3×C4⋊C4—92no—
124C3×C22⋊C4—182no—
125C3×C42—31no—
126C3×C42—31no—
127C3×C42—61no—
128C3×C42C2×C3×D311yescharacteristic
129(C2×D3)⋊C4—62no—
130(C2×D3)⋊C4—182no—
131C4⋊Dic3—32no—
132C4⋊Dic3—92no—
133D24—34no—
134D24—94no—
135C22×C32—21no—
136C22×C32C2×D1211yescharacteristic
137C22×C32C2×C3×D411yescharacteristic
138C2×C3⋊D3—24no—
139C2×C3⋊D3—42no—
140C2×C3⋊D3—42no—
141C2×C3×D3—64no—
142C2×C3×D3—122no—
143C2×C3×D3—122no—
144C32×C4—22no—
145C32×C4—22no—
146C32×C4—21no—
147C32×C4—21no—
148C32×C4—42no—
149C32×C4C2×C4×D311yescharacteristic
150C32×C4#23511yescharacteristic
151C32×C4C22×C3×C411yescharacteristic
152C32×C4C3×#4111yescharacteristic
153C3⋊Dic3—42no—
154C3×Dic3—122no—
155C2×D8—91no—
156C4×D4—92no—
157#106—91no—
158#102—92no—
159#97(C3⋊D3)⋊C311yescharacteristic, complemented
160H(3,𝔽3)#44211yescharacteristic, complemented, Sylow
161C23×C3—182no—
162C22×D3—62no—
163C22×D3—182no—
164C3×D4—94no—
165C3×D4—182no—
166C2×C3×C4—32no—
167C2×C3×C4—32no—
168C2×C3×C4—31no—
169C2×C3×C4—31no—
170C2×C3×C4—62no—
171C2×C3×C4—61no—
172C2×C3×C4—182no—
173C2×C3×C4—364no—
174C2×C3×C4C3×D1212yes—
175C2×C3×C4C22×C3×D311yescharacteristic
176C2×Dic3—62no—
177C2×Dic3—182no—
178D12—34no—
179D12—62no—
180D12—94no—
181D12—182no—
182C4×D3—124no—
183C4×D3—364no—
184C3×C8—21no—
185C3×C8—32no—
186C3×C8—32no—
187C3×C8—61no—
188C3×C8—61no—
189C3×C8—62no—
190C3×C8—62no—
191C3×C8C3×C4×D312yes—
192C2×C32—22no—
193C2×C32—21no—
194C2×C32C4×D1211yescharacteristic
195C2×C32#90812yes—
196C2×C32C3×C4×D411yescharacteristic
197C2×C32C3×AGL(1,ℤ/8)12yes—
198C3⋊D3—44no—
199C3×D3—124no—
200C2×D4—92no—
201C22×C4—182no—
202D8—94no—
203C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
204C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
205C4⋊C4—92no—
206C22⋊C4—182no—
207C42C2×(C3⋊D3)⋊C311yescharacteristic
208C22×C3—31no—
209C22×C3—31no—
210C22×C3—61no—
211C22×C3—184no—
212C22×C3—362no—
213C22×C3—362no—
214C22×C3C2×C3×D1211yescharacteristic
215C2×D3—64no—
216C2×D3—122no—
217C2×D3—122no—
218C2×D3—184no—
219C2×D3—362no—
220C2×D3—362no—
221C3×C4—22no—
222C3×C4—31no—
223C3×C4—31no—
224C3×C4—31no—
225C3×C4—31no—
226C3×C4—62no—
227C3×C4—62no—
228C3×C4—61no—
229C3×C4—61no—
230C3×C4—122no—
231C3×C4—362no—
232C3×C4C2×C3×C4×D311yescharacteristic
233C3×C4C3×#23511yescharacteristic
234Dic3—122no—
235Dic3—362no—
236C32—21no—
237C32#471111yescharacteristic, complemented
238C32C3×#44211yescharacteristic, complemented
239C23—182no—
240D4—94no—
241D4—182no—
242C2×C4—182no—
243C2×C4—364no—
244C2×C4C4⋊((C3⋊D3)⋊C3)12yes—
245C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
246C8—21no—
247C8C4×(C3⋊D3)⋊C312yescomplemented
248C2×C3—32no—
249C2×C3—32no—
250C2×C3—31no—
251C2×C3—31no—
252C2×C3—62no—
253C2×C3—61no—
254C2×C3—364no—
255C2×C3C3×C4×D1211yescharacteristic
256C2×C3C3×#90812yes—
257D3—124no—
258D3—364no—
259C22—184no—
260C22—362no—
261C22—362no—
262C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
263C4—22no—
264C4—362no—
265C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
266C4#3828⋊C311yescharacteristic
267C3—31no—
268C3—31no—
269C3—61no—
270C3C3×#471111yescharacteristic
271C2—364no—
272C2C4×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
273C2#64117⋊C312yes—
subgroupG/H
centerC22C2×C4⋊((C3⋊D3)⋊C3)
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#97C2
socleC22×C3C2×C3×D12
prime radicalC2×#3745⋊C3C2
cosocleC2×C32×C4C23×C3
hypercenter#97(C3⋊D3)⋊C3
hypocenterC32C3×#442
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #97

Complements

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

NH ≅ G/Ntype
2. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
3. H(3,𝔽3)×#97C2semidirect
4. #63648⋊C3C2semidirect
5. #63644⋊C3C2semidirect
6. #10592410C3semidirect
15. #3745⋊C3C4semidirect
17. C4×C4⋊(C3⋊D3)C2×C3semidirect
21. C32×#97D3semidirect
22. C32×#97C2×C3semidirect
23. #63648C2×C3semidirect
24. #63644C2×C3semidirect
32. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
35. C8×H(3,𝔽3)C2×C4semidirect
54. #3745C3×C4semidirect
92. C4⋊(C3⋊D3)C2×C3×C4semidirect
102. C32×C8C4×D3semidirect
103. C32×C8C2×C3×C4semidirect
159. #97(C3⋊D3)⋊C3semidirect
160. H(3,𝔽3)#442semidirect
237. C32#4711semidirect
238. C32C3×#442semidirect
247. C8C4×(C3⋊D3)⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#3745⋊C32
C4×C4⋊((C3⋊D3)⋊C3)2
H(3,𝔽3)×#972
#63648⋊C32
#63644⋊C32
#105924103
C3×#47113