Finite Group Databasebeta

← #13138790

Subgroups — #13138790

Order 1728.

502 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 93 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
2#11721136C212yesmaximal, complemented
3#11720548C211yesmaximal, characteristic, complemented
4#63644⋊C3C211yesmaximal, characteristic, complemented
5#11720318C212yesmaximal, complemented
6#10590771—31nomaximal
7#10590768—31nomaximal
8C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
9C2×((C3⋊D3)⋊C3)⋊C4—22no—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11C4×#1084C2211yescharacteristic
12C2×#6180C2211yescharacteristic
13C4⋊((C3⋊Dic3)⋊C3)C2212yes—
14#3745⋊C3C414yescomplemented
15#88450—22no—
16C2×#3745D311yescharacteristic
17C2×C3×D24—31no—
18#63894—32no—
19Dic3×D12—32no—
20#63644D311yescharacteristic, complemented
21C3×#817—31no—
22C3×#815—31no—
23C3×#815—31no—
24C3×#815—61no—
25#63564—32no—
26#63563—32no—
27C22×(C3⋊D3)⋊C3—22no—
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29C2×#1084D412yes—
30C2×(C3⋊Dic3)⋊C3—22no—
31C4⋊((C3⋊D3)⋊C3)—22no—
32C4⋊((C3⋊D3)⋊C3)C2×C414yescomplemented
33((C3⋊D3)⋊C3)⋊C4—44no—
34C8×H(3,𝔽3)C2×C412yescomplemented
35#6180—21no—
36#5152—91no—
37#4869—91no—
38#4711—91no—
39C2×C4⋊(C3⋊D3)C2×D312yes—
40C2×C3×D12—32no—
41C2×#624—62no—
42C2×D3×Dic3—62no—
43C2×C32×C8—21no—
44C2×C32×C8C2×D311yescharacteristic
45C2×C32×C8C2×D311yescharacteristic
46C4⋊(C3⋊Dic3)C2×D312yes—
47#3745Dic314yescomplemented
48C3×C4⋊Dic3—32no—
49C3×C4×Dic3—31no—
50C3×C4×Dic3—31no—
51C3×C4×Dic3—61no—
52C2×C3×C3⋊C8—31no—
53C2×C3×C3⋊C8—31no—
54C2×C3×C3⋊C8—61no—
55C3×D24—34no—
56#3722—62no—
57(C2×D3)⋊Dic3—62no—
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)C22×C411yescharacteristic
63C4×H(3,𝔽3)#4111yescharacteristic
64#1084—22no—
65(C3⋊Dic3)⋊C3—42no—
66C2×C3×D8—91no—
67D4×Dic3—92no—
68C2×D24—31no—
69C2×D24—91no—
70#893—92no—
71C4×D12—92no—
72C3×#106—91no—
73C3×#97—91no—
74#832—92no—
75#821—92no—
76#817—31no—
77#817—91no—
78#815—31no—
79#815—31no—
80#815—61no—
81#809—92no—
82C22×C3⋊D3—22no—
83C22×C3×D3—62no—
84C2×C32×C4—21no—
85C2×C32×C4C22×D311yescharacteristic
86C2×C32×C4C22×D311yescharacteristic
87C2×C3⋊Dic3—22no—
88C4⋊(C3⋊D3)—22no—
89C4⋊(C3⋊D3)C2×Dic314yescomplemented
90C2×C3×Dic3—32no—
91C2×C3×Dic3—32no—
92C2×C3×Dic3—62no—
93C2×C3×Dic3—62no—
94C3×D12—34no—
95C3×D12—62no—
96#624—124no—
97D3×Dic3—124no—
98C32×C8—22no—
99C32×C8C4×D312yescomplemented
100C32×C8C2×Dic312yescomplemented
101C3×C3⋊C8—62no—
102C3×C3⋊C8—61no—
103C3×C3⋊C8—61no—
104#442—271noSylow
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×Dic3—182no—
110C2×D12—32no—
111C2×D12—92no—
112C2×C4×D3—182no—
113C2×C4×D3—182no—
114C3×D8—94no—
115C2×C3×C8—31no—
116C2×C3×C8—31no—
117C2×C3×C8—61no—
118C2×C3×C8—91no—
119C2×C3×C8D3211yescharacteristic
120C3×C4⋊C4—92no—
121C3×C42—91no—
122#217—182no—
123(C2×D3)⋊C4—182no—
124(C2×D3)⋊C4—182no—
125C4⋊Dic3—32no—
126C4⋊Dic3—92no—
127C4×Dic3—31no—
128C4×Dic3—31no—
129C4×Dic3—61no—
130C2×C3⋊C8—31no—
131C2×C3⋊C8—31no—
132C2×C3⋊C8—61no—
133D24—34no—
134D24—94no—
135C22×C32—21no—
136C22×C32D3×D411yescharacteristic
137C22×C32D3×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—21no—
145C32×C4—21no—
146C32×C4C2×C4×D311yescharacteristic
147C32×C4#23711yescharacteristic
148C32×C4#23911yescharacteristic
149C32×C4C22×Dic311yescharacteristic
150C3⋊Dic3—42no—
151C3×Dic3—62no—
152C3×Dic3—62no—
153C3×Dic3—122no—
154C3×Dic3—122no—
155C2×D8—91no—
156C4×D4—272no—
157#106—91no—
158#102—272no—
159#97—91no—
160H(3,𝔽3)#44211yescharacteristic, complemented, Sylow
161C23×C3—182no—
162C22×D3—62no—
163C22×D3—182no—
164C3×D4—94no—
165C3×D4—182no—
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4—61no—
169C2×C3×C4—92no—
170C2×C3×C4—182no—
171C2×C3×C4C2×D3211yescharacteristic
172C2×Dic3—32no—
173C2×Dic3—32no—
174C2×Dic3—62no—
175C2×Dic3—62no—
176C2×Dic3—182no—
177C2×Dic3—364no—
178D12—34no—
179D12—62no—
180D12—94no—
181D12—182no—
182C4×D3—364no—
183C4×D3—364no—
184C3×C8—32no—
185C3×C8—32no—
186C3×C8—62no—
187C3×C8—181no—
188C3×C8D3×Dic312yes—
189C3⋊C8—61no—
190C3⋊C8—61no—
191C3⋊C8—62no—
192C2×C32—22no—
193C2×C32—21no—
194C2×C32#89311yescharacteristic
195C2×C32#91112yes—
196C2×C32#91412yes—
197C2×C32D4×Dic311yescharacteristic
198C3⋊D3—44no—
199C3×D3—124no—
200C2×D4—92no—
201C22×C4—542no—
202D8—94no—
203C2×C8—91no—
204C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
205C4⋊C4—92no—
206C22⋊C4—542no—
207C42—91no—
208C22×C3—31no—
209C22×C3—31no—
210C22×C3—61no—
211C22×C3—184no—
212C22×C3—362no—
213C22×C3—362no—
214C22×C3#380211yescharacteristic
215C2×D3—64no—
216C2×D3—122no—
217C2×D3—122no—
218C2×D3—184no—
219C2×D3—362no—
220C2×D3—362no—
221C3×C4—31no—
222C3×C4—31no—
223C3×C4—31no—
224C3×C4—31no—
225C3×C4—61no—
226C3×C4—61no—
227C3×C4—182no—
228C3×C4—362no—
229C3×C4#379611yescharacteristic
230C3×C4C2×D3×Dic311yescharacteristic
231Dic3—62no—
232Dic3—62no—
233Dic3—122no—
234Dic3—122no—
235Dic3—362no—
236C32—21no—
237C32#486911yescharacteristic, complemented
238C32#515211yescharacteristic, complemented
239C23—182no—
240D4—94no—
241D4—182no—
242C2×C4—92no—
243C2×C4—182no—
244C2×C4—1084no—
245C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
246C8—181no—
247C8((C3⋊D3)⋊C3)⋊C412yescomplemented
248C2×C3—32no—
249C2×C3—32no—
250C2×C3—31no—
251C2×C3—31no—
252C2×C3—62no—
253C2×C3—61no—
254C2×C3—364no—
255C2×C3#6379812yes—
256C2×C3#6389811yescharacteristic
257D3—124no—
258D3—364no—
259C22—184no—
260C22—362no—
261C22—362no—
262C22#8865511yescharacteristic
263C4—182no—
264C4—362no—
265C4#8864911yescharacteristic
266C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
267C3—31no—
268C3—31no—
269C3—61no—
270C3#1059078111yescharacteristic
271C2—364no—
272C2#1172083912yes—
273C2#1172113611yescharacteristic
subgroupG/H
centerC22#88655
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC2×#3745⋊C3C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#442
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11721136C2semidirect
3. #11720548C2semidirect
4. #63644⋊C3C2semidirect
5. #11720318C2semidirect
14. #3745⋊C3C4semidirect
20. #63644D3semidirect
32. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
34. C8×H(3,𝔽3)C2×C4semidirect
47. #3745Dic3semidirect
89. C4⋊(C3⋊D3)C2×Dic3semidirect
99. C32×C8C4×D3semidirect
100. C32×C8C2×Dic3semidirect
160. H(3,𝔽3)#442semidirect
237. C32#4869semidirect
238. C32#5152semidirect
247. C8((C3⋊D3)⋊C3)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3745⋊C32
#117211362
#117205482
#63644⋊C32
#117203182
#105907713
#105907683