Finite Group Databasebeta

← #13143845

Subgroups — #13143845

Order 1728.

528 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 117 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#11721826C211yesmaximal, characteristic
2H(3,𝔽3)×C22⋊C8C211yesmaximal, characteristic
3#11720573C211yesmaximal, characteristic
4#11720548C211yesmaximal, characteristic, complemented
5#11720547C211yesmaximal, characteristic, complemented
6#11720547C211yesmaximal, characteristic, complemented
7C8×#1084C211yesmaximal, characteristic, complemented
8C3×#4720—34nomaximal
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12#88552C411yescharacteristic
13#88552C411yescharacteristic
14C4⋊#1084C411yescharacteristic
15C4⋊#1084C411yescharacteristic
16C4×#1084C2211yescharacteristic
17C4×#1084C2211yescharacteristic
18C2×#6180C2211yescharacteristic
19C2×#6180C2211yescharacteristic
20C3×#926—34no—
21C32×C22⋊C8D314yes—
22C3×#830—34no—
23C3×#815—34no—
24C3×Dic3⋊C8—34no—
25C3×Dic3⋊C8—34no—
26C3×C8×Dic3—34no—
27C23×H(3,𝔽3)C2×C411yescharacteristic
28C2×C4×H(3,𝔽3)—21no—
29C2×C4×H(3,𝔽3)—21no—
30C2×C4×H(3,𝔽3)C2×C411yescharacteristic
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×#1084C2×C411yescharacteristic
33C2×#1084C2×C411yescharacteristic
34C2×#1084C2×C411yescharacteristic
35C2×#1084C2×C411yescharacteristic
36C8×H(3,𝔽3)—21no—
37C8×H(3,𝔽3)—21no—
38#6180—21no—
39#6180—21no—
40C3×#433—91no—
41#4720—34no—
42C22×C32×C4C2×D314yes—
43C2×C32×C8C2×D314yes—
44C2×C32×C8C2×D314yes—
45C3×#217—34no—
46C3×#217—34no—
47C3×C4⋊Dic3—34no—
48C3×C4⋊Dic3—34no—
49C3×C4×Dic3—34no—
50C3×C4×Dic3—34no—
51C2×C3×C3⋊C8—34no—
52C2×C3×C3⋊C8—34no—
53C22×H(3,𝔽3)—21no—
54C22×H(3,𝔽3)—21no—
55C22×H(3,𝔽3)—21no—
56C22×H(3,𝔽3)C22×C411yescharacteristic
57C4×H(3,𝔽3)—21no—
58C4×H(3,𝔽3)#4111yescharacteristic
59C4×H(3,𝔽3)#4111yescharacteristic
60#1084—21no—
61#1084—21no—
62#1084—21no—
63#1084—21no—
64C3×#117—91no—
65#926—34no—
66C3×C4⋊C8—91no—
67C3×C4⋊C8—91no—
68C3×C22⋊C8—34no—
69C3×C22⋊C8—91no—
70C3×C22⋊C8C3⋊D311yescharacteristic
71C3×#97—91no—
72C3×C4×C8—91no—
73#830—34no—
74#815—34no—
75Dic3⋊C8—34no—
76Dic3⋊C8—34no—
77C8×Dic3—34no—
78C23×C32C4×D314yes—
79C2×C32×C4—24no—
80C2×C32×C4—24no—
81C2×C32×C4C4×D314yes—
82C2×C32×C4C22×D314yes—
83C2×C3×Dic3—34no—
84C2×C3×Dic3—34no—
85C2×C3×Dic3—34no—
86C2×C3×Dic3—34no—
87C32×C8—24no—
88C32×C8—24no—
89C3×C3⋊C8—64no—
90C3×C3⋊C8—64no—
91#433—91noSylow
92C2×H(3,𝔽3)—41no—
93C2×H(3,𝔽3)#11711yescharacteristic
94C2×H(3,𝔽3)#13111yescharacteristic
95C2×H(3,𝔽3)#13111yescharacteristic
96C22×C3×C4—34no—
97C22×C3×C4C2×C3⋊D311yescharacteristic
98C2×C3×C8—34no—
99C2×C3×C8—34no—
100C2×C3×C8—91no—
101C2×C3×C8—91no—
102C2×C3×C8C2×C3⋊D311yescharacteristic
103C2×C3×C8C2×C3⋊D311yescharacteristic
104C3×C4⋊C4—91no—
105C3×C4⋊C4—91no—
106C3×C22⋊C4—91no—
107C3×C22⋊C4—91no—
108C3×C42—91no—
109C3×C42—91no—
110#217—34no—
111#217—34no—
112C4⋊Dic3—34no—
113C4⋊Dic3—34no—
114C4×Dic3—34no—
115C4×Dic3—34no—
116C2×C3⋊C8—34no—
117C2×C3⋊C8—34no—
118C22×C32—24no—
119C22×C32—24no—
120C22×C32—24no—
121C22×C32C2×C4×D314yes—
122C32×C4—24no—
123C32×C4#23714yes—
124C32×C4#23714yes—
125C3×Dic3—64no—
126C3×Dic3—64no—
127C3×Dic3—64no—
128C3×Dic3—64no—
129#117—91no—
130C4⋊C8—91no—
131C4⋊C8—91no—
132C22⋊C8—91no—
133C22⋊C8#27211yescharacteristic
134#97—91no—
135C4×C8—91no—
136H(3,𝔽3)#43311yescharacteristic, complemented, Sylow
137C23×C3—34no—
138C23×C3C4×C3⋊D311yescharacteristic
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—34no—
142C2×C3×C4—34no—
143C2×C3×C4—64no—
144C2×C3×C4—64no—
145C2×C3×C4—91no—
146C2×C3×C4—91no—
147C2×C3×C4—91no—
148C2×C3×C4—91no—
149C2×C3×C4C4×C3⋊D311yescharacteristic
150C2×C3×C4C22×C3⋊D311yescharacteristic
151C2×Dic3—34no—
152C2×Dic3—34no—
153C2×Dic3—34no—
154C2×Dic3—34no—
155C3×C8—21no—
156C3×C8—21no—
157C3×C8—64no—
158C3×C8—64no—
159C3×C8—181no—
160C3×C8—181no—
161C3⋊C8—64no—
162C3⋊C8—64no—
163C2×C32—44no—
164C2×C32#87714yes—
165C2×C32#90114yes—
166C2×C32#90714yes—
167C22×C4C2×#27211yescharacteristic
168C2×C8—91no—
169C2×C8—91no—
170C2×C8C2×#27211yescharacteristic
171C2×C8C2×#27211yescharacteristic
172C4⋊C4—91no—
173C4⋊C4—91no—
174C22⋊C4—91no—
175C22⋊C4—91no—
176C42—91no—
177C42—91no—
178C22×C3—21no—
179C22×C3—21no—
180C22×C3—21no—
181C22×C3—34no—
182C22×C3—64no—
183C22×C3—64no—
184C22×C3—64no—
185C22×C3C2×C4×C3⋊D311yescharacteristic
186C3×C4—21no—
187C3×C4—34no—
188C3×C4—34no—
189C3×C4—64no—
190C3×C4—181no—
191C3×C4—181no—
192C3×C4—181no—
193C3×C4—181no—
194C3×C4#383011yescharacteristic
195C3×C4#383011yescharacteristic
196Dic3—64no—
197Dic3—64no—
198Dic3—64no—
199Dic3—64no—
200C32#472014yescomplemented
201C23C4×#27211yescharacteristic
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—91no—
205C2×C4—91no—
206C2×C4—91no—
207C2×C4—91no—
208C2×C4C4×#27211yescharacteristic
209C2×C4C22×#27211yescharacteristic
210C8—21no—
211C8—21no—
212C8—181no—
213C8—181no—
214C2×C3—34no—
215C2×C3—34no—
216C2×C3—34no—
217C2×C3—41no—
218C2×C3—124no—
219C2×C3#6408611yescharacteristic
220C2×C3#6411011yescharacteristic
221C2×C3#6411611yescharacteristic
222C22—21no—
223C22—21no—
224C22—21no—
225C22C2×C4×#27211yescharacteristic
226C4—21no—
227C4—181no—
228C4—181no—
229C4—181no—
230C4—181no—
231C4#8874411yescharacteristic
232C4#8874411yescharacteristic
233C3—34no—
234C3#1059241911yescharacteristic
235C2—41no—
236C2#1172173311yescharacteristic
237C2#1172177611yescharacteristic
238C2#1172178611yescharacteristic
subgroupG/H
center, FrattiniC2×C3×C4C22×C3⋊D3
commutatorC22×H(3,𝔽3)C22×C4
FittingH(3,𝔽3)×C22⋊C8C2
socleC22×C3C2×C4×C3⋊D3
prime radicalC23×H(3,𝔽3)C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC3×C22⋊C8C3⋊D3
hypocenterH(3,𝔽3)#433
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3×C4 ◁ C3×C22⋊C8

Complements

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

NH ≅ G/Ntype
4. #11720548C2semidirect
5. #11720547C2semidirect
6. #11720547C2semidirect
7. C8×#1084C2semidirect
136. H(3,𝔽3)#433semidirect
200. C32#4720semidirect

Maximal subgroups

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

subgroupindex
#117218262
H(3,𝔽3)×C22⋊C82
#117205732
#117205482
#117205472
C8×#10842
C3×#47203