Finite Group Databasebeta

← #10596157⋊C3 #13153177

Subgroups — #10596157⋊C3 #13153177

Order 1728.

658 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 202 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)×#125C211yesmaximal, characteristic
2#64099⋊C3C214yesmaximal
3#64098⋊C3C214yesmaximal
4#64098⋊C3C214yesmaximal
5#64079⋊C3C212yesmaximal
6#10596157C311yesmaximal, characteristic, complemented
7C3×C3⋊#564—31nomaximal
8C4⋊C4×H(3,𝔽3)C2212yes—
9C4⋊C4×H(3,𝔽3)C2214yes—
10C42×H(3,𝔽3)C2211yescharacteristic
11C4⋊((C3⋊Dic3)⋊C3)C2212yes—
12C4⋊((C3⋊Dic3)⋊C3)C2212yes—
13C4⋊((C3⋊Dic3)⋊C3)C2214yes—
14#3750⋊C3C2214yes—
15#3750⋊C3C2214yes—
16#3750⋊C3C2214yes—
17#3750⋊C3C2214yes—
18C4×(C3⋊Dic3)⋊C3C2214yes—
19C32×#125—21no—
20C32×#125D311yescharacteristic
21C32×#125C2×C311yescharacteristic
22#64099C2×C314yes—
23#64098C2×C314yes—
24#64098C2×C314yes—
25#64079C2×C312yes—
26C3×#890—34no—
27C3×#889—34no—
28C3×#889—34no—
29C3×#870—32no—
30C2×C4×H(3,𝔽3)C2314yes—
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×C4×H(3,𝔽3)C2312yes—
33C2×(C3⋊Dic3)⋊C3C2314yes—
34C2×(C3⋊Dic3)⋊C3C2314yes—
35C3×#564—91no—
36C3⋊#564—31no—
37C3⋊#564—91no—
38C32×C4⋊C4—22no—
39C32×C4⋊C4—24no—
40C32×C4⋊C4C2×D312yes—
41C32×C4⋊C4C2×D314yes—
42C32×C4⋊C4C22×C312yes—
43C32×C4⋊C4C22×C314yes—
44C32×C42—21no—
45C32×C42C2×D311yescharacteristic
46C32×C42C22×C311yescharacteristic
47C4⋊(C3⋊Dic3)C22×C312yes—
48C4⋊(C3⋊Dic3)C22×C312yes—
49C4⋊(C3⋊Dic3)C22×C314yes—
50#3750C22×C314yes—
51#3750C22×C314yes—
52#3750C22×C314yes—
53#3750C22×C314yes—
54C4×C3⋊Dic3C22×C314yes—
55C3×C4⋊Dic3—32no—
56C3×C4⋊Dic3—34no—
57C3×C4⋊Dic3—32no—
58C3×Dic3⋊C4—34no—
59C3×Dic3⋊C4—34no—
60C3×Dic3⋊C4—34no—
61C3×Dic3⋊C4—34no—
62C3×C4×Dic3—34no—
63C22×H(3,𝔽3)C2411yescharacteristic
64C4×H(3,𝔽3)—21no—
65C4×H(3,𝔽3)—24no—
66C4×H(3,𝔽3)—22no—
67(C3⋊Dic3)⋊C3—24no—
68(C3⋊Dic3)⋊C3—24no—
69C3×#125—31no—
70C3×#125—31no—
71C3×#125—61no—
72C3×#125—92no—
73C3×#125—94no—
74C3×#125—94no—
75C3×#125—94no—
76C3×#125C3×D311yescharacteristic
77#890—34no—
78#890—94no—
79#889—34no—
80#889—34no—
81#889—94no—
82#889—94no—
83#870—32no—
84#870—92no—
85C2×C32×C4—21no—
86C2×C32×C4—22no—
87C2×C32×C4—24no—
88C2×C32×C4C22×D314yes—
89C2×C32×C4C22×D311yescharacteristic
90C2×C32×C4C22×D312yes—
91C2×C32×C4C23×C314yes—
92C2×C32×C4C23×C311yescharacteristic
93C2×C32×C4C23×C312yes—
94C2×C3⋊Dic3C23×C314yes—
95C2×C3⋊Dic3C23×C314yes—
96C2×C3×Dic3—34no—
97C2×C3×Dic3—34no—
98#564—91noSylow
99C2×H(3,𝔽3)2−1+411yescharacteristic
100C2×H(3,𝔽3)2−1+412yes—
101C3×C4⋊C4—32no—
102C3×C4⋊C4—34no—
103C3×C4⋊C4—32no—
104C3×C4⋊C4—34no—
105C3×C4⋊C4—62no—
106C3×C4⋊C4—64no—
107C3×C4⋊C4—94no—
108C3×C4⋊C4—94no—
109C3×C4⋊C4—94no—
110C3×C4⋊C4—94no—
111C3×C4⋊C4—92no—
112C3×C4⋊C4—92no—
113C3×C4⋊C4—94no—
114C3×C4⋊C4C2×C3×D312yes—
115C3×C4⋊C4C2×C3×D314yes—
116C3×C42—31no—
117C3×C42—31no—
118C3×C42—61no—
119C3×C42—94no—
120C3×C42C2×C3×D311yescharacteristic
121C4⋊Dic3—32no—
122C4⋊Dic3—32no—
123C4⋊Dic3—34no—
124C4⋊Dic3—92no—
125C4⋊Dic3—92no—
126C4⋊Dic3—94no—
127Dic3⋊C4—34no—
128Dic3⋊C4—34no—
129Dic3⋊C4—34no—
130Dic3⋊C4—34no—
131Dic3⋊C4—94no—
132Dic3⋊C4—94no—
133Dic3⋊C4—94no—
134Dic3⋊C4—94no—
135C4×Dic3—34no—
136C4×Dic3—94no—
137C22×C32—21no—
138C22×C32C23×D311yescharacteristic
139C22×C32C24×C311yescharacteristic
140C32×C4—22no—
141C32×C4—21no—
142C32×C4—24no—
143C32×C4—22no—
144C32×C4—21no—
145C32×C4—24no—
146C32×C4—22no—
147C32×C4—42no—
148C32×C4—44no—
149C3⋊Dic3—24no—
150C3⋊Dic3—24no—
151C3×Dic3—64no—
152C3×Dic3—64no—
153#125—94no—
154#125—94no—
155#125—94no—
156#125—92no—
157#125(C3⋊D3)⋊C311yescharacteristic
158H(3,𝔽3)#56411yescharacteristic, complemented, Sylow
159C2×C3×C4—31no—
160C2×C3×C4—32no—
161C2×C3×C4—34no—
162C2×C3×C4—34no—
163C2×C3×C4—31no—
164C2×C3×C4—32no—
165C2×C3×C4—61no—
166C2×C3×C4—62no—
167C2×C3×C4—64no—
168C2×C3×C4—94no—
169C2×C3×C4—94no—
170C2×C3×C4C22×C3×D314yes—
171C2×C3×C4C22×C3×D311yescharacteristic
172C2×C3×C4C22×C3×D312yes—
173C2×Dic3—34no—
174C2×Dic3—34no—
175C2×Dic3—94no—
176C2×Dic3—94no—
177C2×C32—21no—
178C2×C32—22no—
179C2×C32#100711yescharacteristic
180C2×C32#101012yes—
181C2×C32C3×2−1+411yescharacteristic
182C2×C32C3×2−1+412yes—
183C4⋊C4—94no—
184C4⋊C4—94no—
185C4⋊C4—94no—
186C4⋊C4—94no—
187C4⋊C4—94no—
188C4⋊C4—92no—
189C4⋊C4—92no—
190C4⋊C4C2×(C3⋊D3)⋊C312yes—
191C4⋊C4C2×(C3⋊D3)⋊C314yes—
192C42—94no—
193C42C2×(C3⋊D3)⋊C311yescharacteristic
194C22×C3—31no—
195C22×C3—31no—
196C22×C3—61no—
197C22×C3C23×C3×D311yescharacteristic
198C3×C4—21no—
199C3×C4—24no—
200C3×C4—22no—
201C3×C4—61no—
202C3×C4—61no—
203C3×C4—64no—
204C3×C4—62no—
205C3×C4—62no—
206C3×C4—64no—
207C3×C4—62no—
208C3×C4—122no—
209C3×C4—124no—
210C3×C4—184no—
211C3×C4—184no—
212Dic3—64no—
213Dic3—64no—
214Dic3—184no—
215Dic3—184no—
216C32—21no—
217C32C3⋊#56411yescharacteristic, complemented
218C32C3×#56411yescharacteristic, complemented
219C2×C4—94no—
220C2×C4—94no—
221C2×C4C22×(C3⋊D3)⋊C314yes—
222C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
223C2×C4C22×(C3⋊D3)⋊C312yes—
224C2×C3—31no—
225C2×C3—31no—
226C2×C3—32no—
227C2×C3—32no—
228C2×C3—61no—
229C2×C3—62no—
230C2×C3C3×#100711yescharacteristic
231C2×C3C3×#101012yes—
232C22C23×(C3⋊D3)⋊C311yescharacteristic
233C4—21no—
234C4—24no—
235C4—22no—
236C4—184no—
237C4—184no—
238C3—31no—
239C3—31no—
240C3—61no—
241C3C3×C3⋊#56411yescharacteristic
242C2#1172262711yescharacteristic
243C2#1172263912yes—
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#125C2
prime radicalC22×H(3,𝔽3)C24
hypercenter#125(C3⋊D3)⋊C3
hypocenterC32C3×#564
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #125

Complements

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

NH ≅ G/Ntype
6. #10596157C3semidirect
158. H(3,𝔽3)#564semidirect
217. C32C3⋊#564semidirect
218. C32C3×#564semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1252
#64099⋊C32
#64098⋊C32
#64079⋊C32
#105961573
C3×C3⋊#5643