Finite Group Databasebeta

← #13144462

Subgroups — #13144462

Order 1728.

583 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 583 conjugacy classes of subgroups — too many to draw. Only its 109 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
1H(3,𝔽3)×#121C211yesmaximal, characteristic
2#11721826C211yesmaximal, characteristic, complemented
3#11720578C211yesmaximal, characteristic, complemented
4#11720578C211yesmaximal, characteristic, complemented
5#11720573C211yesmaximal, characteristic, complemented
6#11720530C211yesmaximal, characteristic, complemented
7#11720529C211yesmaximal, characteristic, complemented
8C3×#5035—34nomaximal
9C2×D4×H(3,𝔽3)—21no—
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C4⋊C4×H(3,𝔽3)C2211yescharacteristic
13C22⋊C4×H(3,𝔽3)—21no—
14#88552—21no—
15C4⋊#1084C2211yescharacteristic
16C4⋊#1084C2211yescharacteristic
17C4×#1084—21no—
18C2×#6180C2211yescharacteristic
19C2×#6180C2211yescharacteristic
20C32×#121D314yes—
21C3×#926—34no—
22C3×#832—34no—
23C3×#832—34no—
24C3×#830—34no—
25C3×#808—34no—
26C3×#807—34no—
27C23×H(3,𝔽3)—21no—
28C23×H(3,𝔽3)D411yescharacteristic
29D4×H(3,𝔽3)—21no—
30D4×H(3,𝔽3)—21no—
31D4×H(3,𝔽3)—42no—
32C2×C4×H(3,𝔽3)—21no—
33C2×C4×H(3,𝔽3)—21no—
34C2×C4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)D411yescharacteristic
36C2×C4×H(3,𝔽3)C2311yescharacteristic
37C2×#1084—21no—
38C2×#1084—21no—
39#6180—21no—
40#6180—21no—
41C3×#482—91no—
42#5035—34no—
43C2×C32×D4—24no—
44C2×C32×D4C2×D314yes—
45C22×C32×C4C2×D314yes—
46C32×C4⋊C4C2×D314yes—
47C32×C22⋊C4—24no—
48C3×#217—64no—
49C3×C4⋊Dic3—34no—
50C3×C4⋊Dic3—34no—
51C3×C4×Dic3—64no—
52C2×C3×C3⋊C8—34no—
53C2×C3×C3⋊C8—34no—
54C22×H(3,𝔽3)—21no—
55C22×H(3,𝔽3)—21no—
56C22×H(3,𝔽3)—21no—
57C22×H(3,𝔽3)—41no—
58C22×H(3,𝔽3)—41no—
59C22×H(3,𝔽3)—41no—
60C22×H(3,𝔽3)C2×D411yescharacteristic
61C4×H(3,𝔽3)—21no—
62C4×H(3,𝔽3)—41no—
63C4×H(3,𝔽3)#4111yescharacteristic
64C4×H(3,𝔽3)#4111yescharacteristic
65#1084—41no—
66#1084—41no—
67C3×#121—34no—
68C3×#121C3⋊D311yescharacteristic
69C3×#117—91no—
70#926—34no—
71C3×#107—91no—
72C3×#106—91no—
73C3×#102—91no—
74C3×#102—91no—
75C3×C22⋊C8—91no—
76#832—34no—
77#832—34no—
78#830—34no—
79#808—34no—
80#807—34no—
81C23×C32—24no—
82C23×C32#6714yes—
83C32×D4—24no—
84C32×D4—24no—
85C32×D4—48no—
86C2×C32×C4—24no—
87C2×C32×C4—24no—
88C2×C32×C4—24no—
89C2×C32×C4#6714yes—
90C2×C32×C4C22×D314yes—
91C2×C3×Dic3—64no—
92C2×C3×Dic3—64no—
93C3×C3⋊C8—64no—
94C3×C3⋊C8—64no—
95#482—91noSylow
96C2×H(3,𝔽3)—41no—
97C2×H(3,𝔽3)—81no—
98C2×H(3,𝔽3)#12311yescharacteristic
99C2×H(3,𝔽3)#13511yescharacteristic
100C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
101C2×C3×D4—21no—
102C2×C3×D4—34no—
103C2×C3×D4—64no—
104C2×C3×D4C2×C3⋊D311yescharacteristic
105C22×C3×C4—34no—
106C22×C3×C4C2×C3⋊D311yescharacteristic
107C2×C3×C8—91no—
108C2×C3×C8—91no—
109C3×C4⋊C4—34no—
110C3×C4⋊C4—91no—
111C3×C4⋊C4—91no—
112C3×C4⋊C4C2×C3⋊D311yescharacteristic
113C3×C22⋊C4—21no—
114C3×C22⋊C4—64no—
115C3×C22⋊C4—181no—
116C3×C42—181no—
117#217—64no—
118C4⋊Dic3—34no—
119C4⋊Dic3—34no—
120C4×Dic3—64no—
121C2×C3⋊C8—34no—
122C2×C3⋊C8—34no—
123C22×C32—24no—
124C22×C32—24no—
125C22×C32—24no—
126C22×C32—44no—
127C22×C32—44no—
128C22×C32—44no—
129C22×C32C2×#6714yes—
130C32×C4—24no—
131C32×C4—44no—
132C32×C4#23714yes—
133C32×C4#23714yes—
134C3×Dic3—124no—
135C3×Dic3—124no—
136#121#27211yescharacteristic
137#117—91no—
138#107—91no—
139#106—91no—
140#102—91no—
141#102—91no—
142C22⋊C8—91no—
143H(3,𝔽3)#48211yescharacteristic, complemented, Sylow
144C23×C3—21no—
145C23×C3—34no—
146C23×C3—64no—
147C23×C3#63811yescharacteristic
148C3×D4—21no—
149C3×D4—21no—
150C3×D4—42no—
151C3×D4—64no—
152C3×D4—64no—
153C3×D4—128no—
154C2×C3×C4—21no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—34no—
158C2×C3×C4—34no—
159C2×C3×C4—64no—
160C2×C3×C4—64no—
161C2×C3×C4—64no—
162C2×C3×C4—181no—
163C2×C3×C4—181no—
164C2×C3×C4#63811yescharacteristic
165C2×C3×C4C22×C3⋊D311yescharacteristic
166C2×Dic3—64no—
167C2×Dic3—64no—
168C3×C8—181no—
169C3×C8—181no—
170C3⋊C8—64no—
171C3⋊C8—64no—
172C2×C32—44no—
173C2×C32—84no—
174C2×C32#93214yes—
175C2×C32#93514yes—
176C2×C32#95014yes—
177C2×D4—21no—
178C2×D4C2×#27211yescharacteristic
179C22×C4C2×#27211yescharacteristic
180C2×C8—91no—
181C2×C8—91no—
182C4⋊C4—91no—
183C4⋊C4—91no—
184C4⋊C4C2×#27211yescharacteristic
185C22⋊C4—21no—
186C22⋊C4—181no—
187C42—181no—
188C22×C3—21no—
189C22×C3—21no—
190C22×C3—21no—
191C22×C3—34no—
192C22×C3—41no—
193C22×C3—41no—
194C22×C3—41no—
195C22×C3—64no—
196C22×C3—64no—
197C22×C3—64no—
198C22×C3—124no—
199C22×C3—124no—
200C22×C3—124no—
201C22×C3C2×#63811yescharacteristic
202C3×C4—21no—
203C3×C4—34no—
204C3×C4—34no—
205C3×C4—41no—
206C3×C4—64no—
207C3×C4—124no—
208C3×C4—361no—
209C3×C4—361no—
210C3×C4#383011yescharacteristic
211C3×C4#383011yescharacteristic
212Dic3—124no—
213Dic3—124no—
214C32#503514yescomplemented
215C23—21no—
216C23#623511yescharacteristic
217D4—21no—
218D4—21no—
219D4—42no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—181no—
224C2×C4—181no—
225C2×C4#623511yescharacteristic
226C2×C4C22×#27211yescharacteristic
227C8—181no—
228C8—181no—
229C2×C3—34no—
230C2×C3—34no—
231C2×C3—34no—
232C2×C3—41no—
233C2×C3—81no—
234C2×C3—124no—
235C2×C3—244no—
236C2×C3#6414111yescharacteristic
237C2×C3#6414411yescharacteristic
238C2×C3#6415911yescharacteristic
239C22—21no—
240C22—21no—
241C22—21no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22C2×#623511yescharacteristic
246C4—21no—
247C4—41no—
248C4—361no—
249C4—361no—
250C4#8874411yescharacteristic
251C4#8874411yescharacteristic
252C3—34no—
253C3#1059273411yescharacteristic
254C2—41no—
255C2—81no—
256C2#1172184211yescharacteristic
257C2#1172184511yescharacteristic
258C2#1172187111yescharacteristic
subgroupG/H
center, socleC22×C3C2×#638
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
Fitting, prime radicalH(3,𝔽3)×#121C2
hypercenterC3×#121C3⋊D3
hypocenterH(3,𝔽3)#482
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#121

Complements

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

NH ≅ G/Ntype
2. #11721826C2semidirect
3. #11720578C2semidirect
4. #11720578C2semidirect
5. #11720573C2semidirect
6. #11720530C2semidirect
7. #11720529C2semidirect
143. H(3,𝔽3)#482semidirect
214. C32#5035semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1212
#117218262
#117205782
#117205732
#117205302
#117205292
C3×#50353