Finite Group Databasebeta

← #10592764⋊C3 #13142952

Subgroups — #10592764⋊C3 #13142952

Order 1728.

357 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 100 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#64079⋊C3C211yesmaximal, characteristic
3#63635⋊C3C212yesmaximal
4#63634⋊C3C212yesmaximal
5C4×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic
6#10592764C311yesmaximal, characteristic, complemented
7C3×#5065—31nomaximal
8C4⋊C4×H(3,𝔽3)—22no—
9C4⋊C4×H(3,𝔽3)C2212yes—
10C42×H(3,𝔽3)C2211yescharacteristic
11C4⋊((C3⋊Dic3)⋊C3)C2212yes—
12#3750⋊C3—22no—
13C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
14C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
15C32×#125—21no—
16C32×#125D311yescharacteristic
17C32×#125C2×C311yescharacteristic
18#64079C2×C311yescharacteristic
19C3×#870—31no—
20#63635C2×C312yes—
21#63634C2×C312yes—
22C4×C3⋊(C3⋊C8)C2×C311yescharacteristic
23C3×#808—32no—
24C3×#807—32no—
25C3×C4×C3⋊C8—31no—
26C2×C4×H(3,𝔽3)—22no—
27C2×C4×H(3,𝔽3)D412yes—
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29C2×(C3⋊Dic3)⋊C3—22no—
30(C3⋊(C3⋊C8))⋊C3Q812yes—
31(C3⋊(C3⋊C8))⋊C3Q812yes—
32C3×#499—91no—
33#5065—31no—
34#5065—91no—
35C32×C4⋊C4—24no—
36C32×C4⋊C4—22no—
37C32×C4⋊C4—22no—
38C32×C4⋊C4—22no—
39C32×C4⋊C4C2×D312yes—
40C32×C4⋊C4C22×C312yes—
41C32×C42—21no—
42C32×C42C2×D311yescharacteristic
43C32×C42C22×C311yescharacteristic
44C4⋊(C3⋊Dic3)C22×C312yes—
45#3750—22no—
46C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
47C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
48C3×C4⋊Dic3—32no—
49C3×Dic3⋊C4—62no—
50C2×C3×C3⋊C8—31no—
51C2×C3×C3⋊C8—31no—
52C22×H(3,𝔽3)C2×D411yescharacteristic
53C4×H(3,𝔽3)—22no—
54C4×H(3,𝔽3)—42no—
55C4×H(3,𝔽3)C2×Q811yescharacteristic
56C4×H(3,𝔽3)C2×Q811yescharacteristic
57(C3⋊Dic3)⋊C3—42no—
58C3×#125—31no—
59C3×#125—31no—
60C3×#125—61no—
61C3×#125—91no—
62C3×#125C3×D311yescharacteristic
63#870—31no—
64#870—91no—
65C3×#107—92no—
66C3×#106—92no—
67C3×C4×C8—91no—
68#808—32no—
69#808—92no—
70#807—32no—
71#807—92no—
72C4×C3⋊C8—31no—
73C4×C3⋊C8—91no—
74C2×C32×C4—24no—
75C2×C32×C4—22no—
76C2×C32×C4—22no—
77C2×C32×C4—22no—
78C2×C32×C4—21no—
79C2×C32×C4#6712yes—
80C2×C32×C4C3×D412yes—
81C2×C32×C4C22×D311yescharacteristic
82C2×C32×C4C23×C311yescharacteristic
83C2×C3⋊Dic3—22no—
84C2×C3×Dic3—62no—
85C3⋊(C3⋊C8)C3×Q812yes—
86C3⋊(C3⋊C8)C3×Q812yes—
87C3×C3⋊C8—32no—
88C3×C3⋊C8—32no—
89#499—91noSylow
90C2×H(3,𝔽3)C4⋊Q811yescharacteristic
91C2×H(3,𝔽3)#13512yes—
92C2×C3×C8—91no—
93C2×C3×C8—91no—
94C3×C4⋊C4—22no—
95C3×C4⋊C4—32no—
96C3×C4⋊C4—32no—
97C3×C4⋊C4—62no—
98C3×C4⋊C4—62no—
99C3×C4⋊C4—64no—
100C3×C4⋊C4—62no—
101C3×C4⋊C4—92no—
102C3×C4⋊C4—182no—
103C3×C4⋊C4C2×C3×D312yes—
104C3×C42—31no—
105C3×C42—31no—
106C3×C42—61no—
107C3×C42C2×C3×D311yescharacteristic
108C4⋊Dic3—32no—
109C4⋊Dic3—92no—
110Dic3⋊C4—62no—
111Dic3⋊C4—182no—
112C2×C3⋊C8—31no—
113C2×C3⋊C8—31no—
114C2×C3⋊C8—91no—
115C2×C3⋊C8—91no—
116C22×C32—21no—
117C22×C32C2×#6711yescharacteristic
118C22×C32C2×C3×D411yescharacteristic
119C32×C4—22no—
120C32×C4—22no—
121C32×C4—21no—
122C32×C4—21no—
123C32×C4—42no—
124C32×C4—44no—
125C32×C4—42no—
126C32×C4—42no—
127C32×C4D3×Q811yescharacteristic
128C32×C4D3×Q811yescharacteristic
129C32×C4C2×C3×Q811yescharacteristic
130C32×C4C2×C3×Q811yescharacteristic
131C3⋊Dic3—42no—
132C3×Dic3—122no—
133#125—91no—
134#125(C3⋊D3)⋊C311yescharacteristic
135#107—92no—
136#106—92no—
137C4×C8—91no—
138H(3,𝔽3)#49911yescharacteristic, complemented, Sylow
139C2×C3×C4—22no—
140C2×C3×C4—32no—
141C2×C3×C4—32no—
142C2×C3×C4—31no—
143C2×C3×C4—31no—
144C2×C3×C4—62no—
145C2×C3×C4—62no—
146C2×C3×C4—64no—
147C2×C3×C4—62no—
148C2×C3×C4—61no—
149C2×C3×C4—182no—
150C2×C3×C4C3×#6712yes—
151C2×C3×C4C22×C3×D311yescharacteristic
152C2×Dic3—62no—
153C2×Dic3—182no—
154C3×C8—92no—
155C3×C8—92no—
156C3⋊C8—32no—
157C3⋊C8—32no—
158C3⋊C8—92no—
159C3⋊C8—92no—
160C2×C32—21no—
161C2×C32—22no—
162C2×C32#94411yescharacteristic
163C2×C32#95012yes—
164C2×C32C3×C4⋊Q811yescharacteristic
165C2×C32C3×#13512yes—
166C2×C8—91no—
167C2×C8—91no—
168C4⋊C4—22no—
169C4⋊C4—92no—
170C4⋊C4—182no—
171C4⋊C4C2×(C3⋊D3)⋊C312yes—
172C42C2×(C3⋊D3)⋊C311yescharacteristic
173C22×C3—31no—
174C22×C3—31no—
175C22×C3—61no—
176C22×C3C2×C3×#6711yescharacteristic
177C3×C4—22no—
178C3×C4—31no—
179C3×C4—31no—
180C3×C4—31no—
181C3×C4—31no—
182C3×C4—42no—
183C3×C4—62no—
184C3×C4—62no—
185C3×C4—61no—
186C3×C4—61no—
187C3×C4—122no—
188C3×C4—122no—
189C3×C4—122no—
190C3×C4—124no—
191C3×C4—362no—
192C3×C4C3×D3×Q811yescharacteristic
193C3×C4C3×D3×Q811yescharacteristic
194Dic3—122no—
195Dic3—362no—
196C32—21no—
197C32#506511yescharacteristic, complemented
198C32C3×#49911yescharacteristic, complemented
199C2×C4—22no—
200C2×C4—182no—
201C2×C4#638⋊C312yes—
202C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
203C8—92no—
204C8—92no—
205C2×C3—31no—
206C2×C3—31no—
207C2×C3—32no—
208C2×C3—32no—
209C2×C3—61no—
210C2×C3—62no—
211C2×C3C3×#94411yescharacteristic
212C2×C3C3×#95012yes—
213C22C2×#638⋊C311yescharacteristic
214C4—22no—
215C4—42no—
216C4—362no—
217C4Q8×(C3⋊D3)⋊C311yescharacteristic
218C4Q8×(C3⋊D3)⋊C311yescharacteristic
219C3—31no—
220C3—31no—
221C3—61no—
222C3C3×#506511yescharacteristic
223C2#64153⋊C311yescharacteristic
224C2#64159⋊C312yes—
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#125C2
socleC22×C3C2×C3×#67
prime radicalC22×H(3,𝔽3)C2×D4
hypercenter#125(C3⋊D3)⋊C3
hypocenterC32C3×#499
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
6. #10592764C3semidirect
138. H(3,𝔽3)#499semidirect
197. C32#5065semidirect
198. C32C3×#499semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1252
#64079⋊C32
#63635⋊C32
#63634⋊C32
C4×(C3⋊(C3⋊C8))⋊C32
#105927643
C3×#50653