Finite Group Databasebeta

← #10592402⋊C3 #13141526

Subgroups — #10592402⋊C3 #13141526

Order 1728.

355 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 355 conjugacy classes of subgroups — too many to draw. Only its 92 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#64079⋊C3C211yesmaximal, characteristic
2#64078⋊C3C211yesmaximal, characteristic
3H(3,𝔽3)×#97C211yesmaximal, characteristic
4#63643⋊C3C214yesmaximal
5#10592402C311yesmaximal, characteristic, complemented
6C3×#4703—31nomaximal
7C2×#634⋊C3C2212yes—
8C2×C8×H(3,𝔽3)C2212yes—
9C42×H(3,𝔽3)C2211yescharacteristic
10C4⋊((C3⋊Dic3)⋊C3)—22no—
11C4⋊((C3⋊Dic3)⋊C3)C2212yes—
12#3750⋊C3—22no—
13#64079C2×C311yescharacteristic
14#64078C2×C311yescharacteristic
15C3×#870—31no—
16C3×#869—31no—
17C32×#97—21no—
18C32×#97D311yescharacteristic
19C32×#97C2×C311yescharacteristic
20#63643C2×C314yes—
21C3×#816—34no—
22C2×C4×H(3,𝔽3)D412yes—
23C2×C4×H(3,𝔽3)C2311yescharacteristic
24C2×(C3⋊Dic3)⋊C3—22no—
25C2×(C3⋊Dic3)⋊C3—22no—
26#634⋊C3—24no—
27C8×H(3,𝔽3)—22no—
28C3×#491—91no—
29#4703—31no—
30#4703—91no—
31C2×#634C22×C312yes—
32C2×C3×Dic6—32no—
33C2×C32×C8—22no—
34C2×C32×C8C2×D312yes—
35C2×C32×C8C22×C312yes—
36C32×C42—21no—
37C32×C42C2×D311yescharacteristic
38C32×C42C22×C311yescharacteristic
39C4⋊(C3⋊Dic3)—22no—
40C4⋊(C3⋊Dic3)C22×C312yes—
41#3750—22no—
42C3×C4⋊Dic3—32no—
43C3×C4⋊Dic3—62no—
44C3×Dic3⋊C4—62no—
45C22×H(3,𝔽3)C2×D411yescharacteristic
46C4×H(3,𝔽3)—22no—
47C4×H(3,𝔽3)#4112yes—
48(C3⋊Dic3)⋊C3—42no—
49(C3⋊Dic3)⋊C3—42no—
50C3×C4⋊Q8—91no—
51C3×#125—91no—
52#870—31no—
53#870—91no—
54#869—31no—
55#869—91no—
56C3×#103—94no—
57C3×#97—31no—
58C3×#97—31no—
59C3×#97—61no—
60C3×#97C3×D311yescharacteristic
61#816—34no—
62#816—94no—
63C2×C32×C4—22no—
64C2×C32×C4—21no—
65C2×C32×C4D1212yes—
66C2×C32×C4C3×D412yes—
67C2×C32×C4C22×D311yescharacteristic
68C2×C32×C4C23×C311yescharacteristic
69C2×C3⋊Dic3—22no—
70C2×C3⋊Dic3—22no—
71#634—24no—
72C2×C3×Dic3—62no—
73C2×C3×Dic3—62no—
74C3×Dic6—64no—
75C32×C8—24no—
76C32×C8—22no—
77C32×C8—22no—
78#491—91noSylow
79C2×H(3,𝔽3)#12411yescharacteristic
80C2×H(3,𝔽3)#13712yes—
81C2×C3×Q8—92no—
82C2×Dic6—32no—
83C2×Dic6—92no—
84C2×C3×C8—32no—
85C2×C3×C8—32no—
86C2×C3×C8—62no—
87C2×C3×C8C2×C3×D312yes—
88C3×C4⋊C4—92no—
89C3×C4⋊C4—182no—
90C3×C4⋊C4—182no—
91C3×C42—31no—
92C3×C42—31no—
93C3×C42—61no—
94C3×C42C2×C3×D311yescharacteristic
95C4⋊Dic3—32no—
96C4⋊Dic3—62no—
97C4⋊Dic3—92no—
98C4⋊Dic3—182no—
99Dic3⋊C4—62no—
100Dic3⋊C4—182no—
101C22×C32—21no—
102C22×C32C2×D1211yescharacteristic
103C22×C32C2×C3×D411yescharacteristic
104C32×C4—22no—
105C32×C4—22no—
106C32×C4—22no—
107C32×C4—42no—
108C32×C4#23512yes—
109C32×C4C3×#4112yes—
110C3⋊Dic3—42no—
111C3⋊Dic3—42no—
112C3×Dic3—122no—
113C3×Dic3—122no—
114C4⋊Q8—91no—
115#125—91no—
116#103—94no—
117#97(C3⋊D3)⋊C311yescharacteristic
118H(3,𝔽3)#49111yescharacteristic, complemented, Sylow
119C3×Q8—184no—
120C2×C3×C4—32no—
121C2×C3×C4—32no—
122C2×C3×C4—31no—
123C2×C3×C4—31no—
124C2×C3×C4—62no—
125C2×C3×C4—61no—
126C2×C3×C4—182no—
127C2×C3×C4—182no—
128C2×C3×C4C3×D1212yes—
129C2×C3×C4C22×C3×D311yescharacteristic
130C2×Dic3—62no—
131C2×Dic3—62no—
132C2×Dic3—182no—
133C2×Dic3—182no—
134Dic6—64no—
135Dic6—184no—
136C3×C8—22no—
137C3×C8—62no—
138C3×C8—62no—
139C3×C8—64no—
140C2×C32—22no—
141C2×C32—21no—
142C2×C32#87511yescharacteristic
143C2×C32#90912yes—
144C2×C32C3×#12411yescharacteristic
145C2×C32C3×#13712yes—
146C2×Q8—92no—
147C2×C8C2×(C3⋊D3)⋊C312yes—
148C4⋊C4—92no—
149C4⋊C4—182no—
150C4⋊C4—182no—
151C42C2×(C3⋊D3)⋊C311yescharacteristic
152C22×C3—31no—
153C22×C3—31no—
154C22×C3—61no—
155C22×C3C2×C3×D1211yescharacteristic
156C3×C4—22no—
157C3×C4—32no—
158C3×C4—32no—
159C3×C4—62no—
160C3×C4—62no—
161C3×C4—62no—
162C3×C4—122no—
163C3×C4—362no—
164C3×C4—362no—
165C3×C4C3×#23512yes—
166Dic3—122no—
167Dic3—122no—
168Dic3—362no—
169Dic3—362no—
170C32—21no—
171C32#470311yescharacteristic, complemented
172C32C3×#49111yescharacteristic, complemented
173Q8—184no—
174C2×C4—182no—
175C2×C4—182no—
176C2×C4C4⋊((C3⋊D3)⋊C3)12yes—
177C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
178C8—22no—
179C2×C3—32no—
180C2×C3—32no—
181C2×C3—31no—
182C2×C3—31no—
183C2×C3—62no—
184C2×C3—61no—
185C2×C3C3×#87511yescharacteristic
186C2×C3C3×#90912yes—
187C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
188C4—22no—
189C4—362no—
190C4—362no—
191C4#3828⋊C312yes—
192C3—31no—
193C3—31no—
194C3—61no—
195C3C3×#470311yescharacteristic
196C2#64084⋊C311yescharacteristic
197C2#64118⋊C312yes—
subgroupG/H
centerC22C2×C4⋊((C3⋊D3)⋊C3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#97C2
socleC22×C3C2×C3×D12
prime radicalC22×H(3,𝔽3)C2×D4
hypercenter#97(C3⋊D3)⋊C3
hypocenterC32C3×#491
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #97

Complements

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

NH ≅ G/Ntype
5. #10592402C3semidirect
118. H(3,𝔽3)#491semidirect
171. C32#4703semidirect
172. C32C3×#491semidirect

Maximal subgroups

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

subgroupindex
#64079⋊C32
#64078⋊C32
H(3,𝔽3)×#972
#63643⋊C32
#105924023
C3×#47033