Finite Group Databasebeta

← #13138837

Subgroups — #13138837

Order 1728.

435 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 435 conjugacy classes of subgroups — too many to draw. Only its 107 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×C8×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2#11721081C212yesmaximal
3#11720550C211yesmaximal, characteristic, complemented
4#63644⋊C3C211yesmaximal, characteristic, complemented
5#11720306C212yesmaximal, complemented
6#10590712—31nomaximal
7#10590706—31nomaximal
8C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
9C2×C8×H(3,𝔽3)C2211yescharacteristic
10C4⋊#1084C2212yes—
11C4⋊((C3⋊Dic3)⋊C3)C2212yes—
12C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
13C8×(C3⋊D3)⋊C3C414yescomplemented
14#88450—22no—
15#88447—22no—
16C2×C8×C3⋊D3D311yescharacteristic
17C2×C3×D3×C8—31no—
18#63849—32no—
19#63841—32no—
20#63644D311yescharacteristic, complemented
21C3×#817—31no—
22C3×#817—31no—
23C3×#817—31no—
24C3×#817—61no—
25#63576—32no—
26#63575—32no—
27C22×(C3⋊D3)⋊C3D411yescharacteristic
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29C2×#1084—22no—
30C2×(C3⋊Dic3)⋊C3—22no—
31C2×(C3⋊Dic3)⋊C3D411yescharacteristic
32C4×(C3⋊D3)⋊C3C2×C412yes—
33C4×(C3⋊D3)⋊C3Q812yes—
34C8×H(3,𝔽3)C2×C412yes—
35(C3⋊(C3⋊C8))⋊C3C2×C412yes—
36#5106—91no—
37#4860—91no—
38#4860—91no—
39C2×C4×C3⋊D3C2×D311yescharacteristic
40C2×C3×C4×D3—31no—
41C2×C32×C8—21no—
42C2×C32×C8C2×D311yescharacteristic
43C2×C32×C8C2×D311yescharacteristic
44C4⋊(C3⋊Dic3)C2×D312yes—
45C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
46C8×C3⋊D3Dic314yescomplemented
47C3×C4⋊Dic3—32no—
48C3×C4⋊Dic3—32no—
49C3×C4⋊Dic3—32no—
50C3×C4⋊Dic3—62no—
51C2×C3×C3⋊C8—31no—
52C3×D3×C8—34no—
53#3722—62no—
54(C2×D3)⋊Dic3—62no—
55Dic32—62no—
56Dic32—62no—
57C22×H(3,𝔽3)C2×D411yescharacteristic
58C2×(C3⋊D3)⋊C3—22no—
59C2×(C3⋊D3)⋊C3C4⋊C412yes—
60C4×H(3,𝔽3)C22×C411yescharacteristic
61C4×H(3,𝔽3)C2×Q811yescharacteristic
62#1084—42no—
63(C3⋊Dic3)⋊C3—42no—
64(C3⋊Dic3)⋊C3C4⋊C412yes—
65C22×C3×C8—91no—
66#926—92no—
67C2×D3×C8—31no—
68C2×D3×C8—91no—
69#892—92no—
70#892—92no—
71C3×#106—91no—
72C3×#106—91no—
73#818—92no—
74#817—31no—
75#817—31no—
76#817—31no—
77#817—61no—
78#817—91no—
79#807—92no—
80#807—92no—
81C22×C3⋊D3D1211yescharacteristic
82C22×C3×D3—31no—
83C2×C32×C4—21no—
84C2×C32×C4C22×D311yescharacteristic
85C2×C32×C4C22×D311yescharacteristic
86C2×C3⋊Dic3—22no—
87C2×C3⋊Dic3D1211yescharacteristic
88C4×C3⋊D3Dic612yes—
89C4×C3⋊D3C2×Dic312yes—
90C2×C3×Dic3—31no—
91C2×C3×Dic3—62no—
92C2×C3×Dic3—62no—
93C2×C3×Dic3—62no—
94C2×C3×Dic3—122no—
95C3×C4×D3—32no—
96C3×C4×D3—32no—
97C32×C8—22no—
98C32×C8C4×D312yes—
99C32×C8C2×Dic312yes—
100C3⋊(C3⋊C8)C2×Dic312yes—
101C3×C3⋊C8—32no—
102#427—271noSylow
103C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
104C2×H(3,𝔽3)#13512yes—
105(C3⋊D3)⋊C3—22no—
106C22×C3×C4—91no—
107C2×C4×D3—31no—
108C2×C4×D3—91no—
109C2×C3×C8—31no—
110C2×C3×C8—31no—
111C2×C3×C8—61no—
112C2×C3×C8—94no—
113C2×C3×C8—91no—
114C2×C3×C8D3211yescharacteristic
115C3×C4⋊C4—92no—
116C3×C4⋊C4—92no—
117#217—182no—
118(C2×D3)⋊C4—182no—
119(C2×D3)⋊C4—182no—
120C4⋊Dic3—32no—
121C4⋊Dic3—32no—
122C4⋊Dic3—32no—
123C4⋊Dic3—62no—
124C4⋊Dic3—92no—
125C4×Dic3—182no—
126C4×Dic3—182no—
127C4×Dic3—182no—
128C2×C3⋊C8—31no—
129C2×C3⋊C8—91no—
130D3×C8—34no—
131D3×C8—94no—
132C22×C32—21no—
133C22×C32C2×D1211yescharacteristic
134C22×C32D3×D411yescharacteristic
135C2×C3⋊D3—22no—
136C2×C3⋊D3C4⋊Dic312yes—
137C2×C3×D3—32no—
138C2×C3×D3—62no—
139C32×C4—21no—
140C32×C4—21no—
141C32×C4C2×Dic611yescharacteristic
142C32×C4C2×C4×D311yescharacteristic
143C32×C4D3×Q811yescharacteristic
144C32×C4C22×Dic311yescharacteristic
145C3⋊Dic3—42no—
146C3⋊Dic3C4⋊Dic312yes—
147C3×Dic3—32no—
148C3×Dic3—122no—
149C3×Dic3—122no—
150C3×Dic3—122no—
151C3×Dic3—242no—
152C22×C8—91no—
153#117—272no—
154#107—272no—
155#106—91no—
156#106—91no—
157H(3,𝔽3)#42711yescharacteristic, complemented, Sylow
158C23×C3—91no—
159C22×D3—31no—
160C22×D3—91no—
161C2×C3×C4—31no—
162C2×C3×C4—31no—
163C2×C3×C4—61no—
164C2×C3×C4—92no—
165C2×C3×C4—92no—
166C2×C3×C4—91no—
167C2×C3×C4—182no—
168C2×C3×C4—182no—
169C2×C3×C4C2×D3211yescharacteristic
170C2×Dic3—31no—
171C2×Dic3—62no—
172C2×Dic3—62no—
173C2×Dic3—62no—
174C2×Dic3—91no—
175C2×Dic3—122no—
176C2×Dic3—182no—
177C4×D3—32no—
178C4×D3—32no—
179C4×D3—92no—
180C4×D3—92no—
181C3×C8—32no—
182C3×C8—32no—
183C3×C8—62no—
184C3×C8—92no—
185C3×C8D3×Dic312yes—
186C3⋊C8—32no—
187C3⋊C8—92no—
188C2×C32—22no—
189C2×C32—21no—
190C2×C32D3×C4⋊C411yescharacteristic
191C2×C32#90412yes—
192C2×C32#91612yes—
193C2×C32C2×C4⋊Dic311yescharacteristic
194C3⋊D3—22no—
195C3×D3—62no—
196C22×C4—91no—
197C2×C8—91no—
198C2×C8—94no—
199C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
200C4⋊C4—92no—
201C4⋊C4—92no—
202C22⋊C4—542no—
203C42—542no—
204C22×C3—31no—
205C22×C3—31no—
206C22×C3—61no—
207C22×C3—92no—
208C22×C3—182no—
209C22×C3D3×D1211yescharacteristic
210C2×D3—32no—
211C2×D3—62no—
212C2×D3—92no—
213C2×D3—182no—
214C3×C4—31no—
215C3×C4—31no—
216C3×C4—31no—
217C3×C4—31no—
218C3×C4—61no—
219C3×C4—61no—
220C3×C4—92no—
221C3×C4—362no—
222C3×C4—362no—
223C3×C4D3×Dic611yescharacteristic
224C3×C4C2×D3×Dic311yescharacteristic
225Dic3—32no—
226Dic3—92no—
227Dic3—122no—
228Dic3—122no—
229Dic3—122no—
230Dic3—242no—
231Dic3—362no—
232C32—21no—
233C32#486011yescharacteristic, complemented
234C32#510611yescharacteristic, complemented
235C23—91no—
236C2×C4—91no—
237C2×C4—92no—
238C2×C4—92no—
239C2×C4—182no—
240C2×C4—182no—
241C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
242C8—92no—
243C8((C3⋊D3)⋊C3)⋊C412yes—
244C2×C3—32no—
245C2×C3—32no—
246C2×C3—31no—
247C2×C3—31no—
248C2×C3—62no—
249C2×C3—61no—
250C2×C3—182no—
251C2×C3#6380612yes—
252C2×C3D3×C4⋊Dic311yescharacteristic
253D3—62no—
254D3—182no—
255C22—92no—
256C22—182no—
257C22#8865411yescharacteristic
258C4—92no—
259C4—362no—
260C4—362no—
261C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
262C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
263C3—31no—
264C3—31no—
265C3—61no—
266C3#1059070611yescharacteristic
267C2—182no—
268C2#1172085412yes—
269C2#1172112911yescharacteristic
subgroupG/H
centerC22#88654
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC22×(C3⋊D3)⋊C3D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#427
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
3. #11720550C2semidirect
4. #63644⋊C3C2semidirect
5. #11720306C2semidirect
13. C8×(C3⋊D3)⋊C3C4semidirect
20. #63644D3semidirect
46. C8×C3⋊D3Dic3semidirect
157. H(3,𝔽3)#427semidirect
233. C32#4860semidirect
234. C32#5106semidirect

Maximal subgroups

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

subgroupindex
C2×C8×(C3⋊D3)⋊C32
#117210812
#117205502
#63644⋊C32
#117203062
#105907123
#105907063