Finite Group Databasebeta

← #13143950

Subgroups — #13143950

Order 1728.

583 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 583 conjugacy classes of subgroups — too many to draw. Only its 97 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#11721848C211yesmaximal, characteristic
2#11721760C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#102C211yesmaximal, characteristic, complemented
4#11720578C211yesmaximal, characteristic, complemented
5#11720553C211yesmaximal, characteristic, complemented
6#11720550C211yesmaximal, characteristic, complemented
7#11720529C211yesmaximal, characteristic, complemented
8C3×#4779—34nomaximal
9C2×D4×H(3,𝔽3)C2211yescharacteristic
10C2×#6235—21no—
11C2×C4×#272C2211yescharacteristic
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14#88552—21no—
15#88542—21no—
16C4⋊#1084C2211yescharacteristic
17C4⋊#1084C2211yescharacteristic
18C4×#1084—21no—
19C2×#6180C2211yescharacteristic
20C3×#938—34no—
21C3×#892—34no—
22C32×#102D314yescomplemented
23C3×#832—34no—
24C3×(C2×D3)⋊C8—34no—
25C3×#817—34no—
26C3×#807—34no—
27C23×H(3,𝔽3)—21no—
28C22×#272D411yescharacteristic
29D4×H(3,𝔽3)—21no—
30D4×H(3,𝔽3)—21no—
31C2×C4×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33#6235—42no—
34C2×#1084—21no—
35C2×#1084—21no—
36C2×#1084D411yescharacteristic
37C4×#272—21no—
38C4×#272—21no—
39C8×H(3,𝔽3)—21no—
40#6180—21no—
41C3×#482—91no—
42#4779—34no—
43C2×C32×D4C2×D314yes—
44C2×C3×#67—64no—
45C2×C3×C4×D3—34no—
46C2×C32×C8C2×D314yes—
47C32×C4⋊C4C2×D314yes—
48C3×#217—64no—
49C3×(C2×D3)⋊C4—64no—
50C3×C4⋊Dic3—34no—
51C3×C4⋊Dic3—34no—
52C3×C4×Dic3—64no—
53C2×C3×C3⋊C8—34no—
54C22×H(3,𝔽3)—41no—
55C22×H(3,𝔽3)—41no—
56C22×H(3,𝔽3)—41no—
57C22×H(3,𝔽3)C2×D411yescharacteristic
58C2×#272—21no—
59C2×#272—21no—
60C2×#272—21no—
61C4×H(3,𝔽3)—41no—
62C4×H(3,𝔽3)#4111yescharacteristic
63C4×H(3,𝔽3)#4111yescharacteristic
64#1084—21no—
65#1084—41no—
66#1084—41no—
67C3×#121—91no—
68C3×#117—91no—
69#938—34no—
70#892—34no—
71C3×#107—91no—
72C3×#106—91no—
73C3×#102—34no—
74C3×#102—91no—
75C3×#102C3⋊D311yescharacteristic
76C3×C22⋊C8—91no—
77#832—34no—
78(C2×D3)⋊C8—34no—
79#817—34no—
80#807—34no—
81C23×C32—24no—
82C22×C3×D3—34no—
83C32×D4—24no—
84C32×D4—24no—
85C2×C32×C4—24no—
86C2×C32×C4C22×D314yes—
87C3×#67—128no—
88C2×C3×Dic3—34no—
89C2×C3×Dic3—64no—
90C2×C3×Dic3—64no—
91C3×C4×D3—64no—
92C3×C4×D3—64no—
93C32×C8—24no—
94C3×C3⋊C8—64no—
95#482—91noSylow
96C2×H(3,𝔽3)—81no—
97C2×H(3,𝔽3)#12311yescharacteristic
98C2×H(3,𝔽3)#13511yescharacteristic
99C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
100#272—41no—
101C2×C3×D4—34no—
102C2×C3×D4—181no—
103C2×C3×D4C2×C3⋊D311yescharacteristic
104C22×C3×C4—91no—
105C2×#67—64no—
106C2×C4×D3—34no—
107C2×C3×C8—34no—
108C2×C3×C8—91no—
109C2×C3×C8C2×C3⋊D311yescharacteristic
110C3×C4⋊C4—34no—
111C3×C4⋊C4—91no—
112C3×C4⋊C4—91no—
113C3×C4⋊C4C2×C3⋊D311yescharacteristic
114C3×C22⋊C4—181no—
115C3×C22⋊C4—181no—
116C3×C42—181no—
117#217—64no—
118(C2×D3)⋊C4—64no—
119C4⋊Dic3—34no—
120C4⋊Dic3—34no—
121C4×Dic3—64no—
122C2×C3⋊C8—34no—
123C22×C32—44no—
124C22×C32—44no—
125C22×C32—44no—
126C22×C32D3×D414yes—
127C2×C3×D3—64no—
128C2×C3×D3—64no—
129C2×C3×D3—64no—
130C32×C4—44no—
131C32×C4#23514yes—
132C32×C4#23714yes—
133C3×Dic3—64no—
134C3×Dic3—124no—
135C3×Dic3—124no—
136#121—91no—
137#117—91no—
138#107—91no—
139#106—91no—
140#102—91no—
141#102#27211yescharacteristic, complemented
142C22⋊C8—91no—
143H(3,𝔽3)#48211yescharacteristic, complemented, Sylow
144C23×C3—21no—
145C23×C3—64no—
146C23×C3—91no—
147C22×D3—34no—
148C3×D4—21no—
149C3×D4—21no—
150C3×D4—64no—
151C3×D4—64no—
152C3×D4—362no—
153C2×C3×C4—21no—
154C2×C3×C4—34no—
155C2×C3×C4—64no—
156C2×C3×C4—91no—
157C2×C3×C4—181no—
158C2×C3×C4—181no—
159C2×C3×C4—181no—
160C2×C3×C4—181no—
161C2×C3×C4C22×C3⋊D311yescharacteristic
162#67—128no—
163C2×Dic3—34no—
164C2×Dic3—64no—
165C2×Dic3—64no—
166C4×D3—64no—
167C4×D3—64no—
168C3×C8—21no—
169C3×C8—64no—
170C3×C8—181no—
171C3⋊C8—64no—
172C2×C32—84no—
173C2×C32#88314yes—
174C2×C32#91114yes—
175C2×C32#91614yes—
176C3×D3—124no—
177C2×D4—181no—
178C2×D4C2×#27211yescharacteristic
179C22×C4—91no—
180C2×C8—91no—
181C2×C8C2×#27211yescharacteristic
182C4⋊C4—91no—
183C4⋊C4—91no—
184C4⋊C4C2×#27211yescharacteristic
185C22⋊C4—181no—
186C22⋊C4—181no—
187C42—181no—
188C22×C3—34no—
189C22×C3—41no—
190C22×C3—41no—
191C22×C3—41no—
192C22×C3—124no—
193C22×C3—124no—
194C22×C3—124no—
195C22×C3—181no—
196C22×C3—181no—
197C22×C3—181no—
198C22×C3D4×C3⋊D311yescharacteristic
199C2×D3—64no—
200C2×D3—64no—
201C2×D3—64no—
202C3×C4—34no—
203C3×C4—34no—
204C3×C4—41no—
205C3×C4—124no—
206C3×C4—181no—
207C3×C4—361no—
208C3×C4—361no—
209C3×C4#382811yescharacteristic
210C3×C4#383011yescharacteristic
211Dic3—64no—
212Dic3—124no—
213Dic3—124no—
214C32#477914yescomplemented
215C23—21no—
216C23—91no—
217D4—21no—
218D4—21no—
219D4—362no—
220C2×C4—21no—
221C2×C4—91no—
222C2×C4—181no—
223C2×C4—181no—
224C2×C4—181no—
225C2×C4—181no—
226C2×C4C22×#27211yescharacteristic
227C8—21no—
228C8—181no—
229C2×C3—34no—
230C2×C3—34no—
231C2×C3—34no—
232C2×C3—81no—
233C2×C3—244no—
234C2×C3—361no—
235C2×C3#6409211yescharacteristic
236C2×C3#6412011yescharacteristic
237C2×C3#6412511yescharacteristic
238D3—124no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—181no—
243C22—181no—
244C22—181no—
245C22D4×#27211yescharacteristic
246C4—41no—
247C4—181no—
248C4—361no—
249C4—361no—
250C4#8874011yescharacteristic
251C4#8874411yescharacteristic
252C3—34no—
253C3#1059247811yescharacteristic
254C2—81no—
255C2—361no—
256C2#1172173911yescharacteristic
257C2#1172179311yescharacteristic
258C2#1172180211yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#102C2
prime radical#11721848C2
hypercenterC3×#102C3⋊D3
hypocenterH(3,𝔽3)#482
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#102

Complements

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

NH ≅ G/Ntype
2. #11721760C2semidirect
3. H(3,𝔽3)×#102C2semidirect
4. #11720578C2semidirect
5. #11720553C2semidirect
6. #11720550C2semidirect
7. #11720529C2semidirect
22. C32×#102D3semidirect
141. #102#272semidirect
143. H(3,𝔽3)#482semidirect
214. C32#4779semidirect

Maximal subgroups

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

subgroupindex
#117218482
#117217602
H(3,𝔽3)×#1022
#117205782
#117205532
#117205502
#117205292
C3×#47793