Finite Group Databasebeta

← #13147756

Subgroups — #13147756

Order 1728.

658 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 129 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#11721716C211yesmaximal, characteristic
2#64079⋊C3C212yesmaximal
3#11721087C214yesmaximal
4#11721086C214yesmaximal
5#11721085C214yesmaximal
6#10593868—32nomaximal
7C42×H(3,𝔽3)C2211yescharacteristic
8C4⋊#1084C2212yes—
9#88540C2214yes—
10C4⋊((C3⋊Dic3)⋊C3)C2212yes—
11C4⋊((C3⋊Dic3)⋊C3)C2212yes—
12#3750⋊C3C2214yes—
13#3750⋊C3C2214yes—
14#88449C2214yes—
15#88448C2214yes—
16#88448C2214yes—
17#88447C2214yes—
18#64079D312yes—
19C3×#870—32no—
20C3×#870—32no—
21C3×#870—61no—
22#63847—34no—
23#63847—34no—
24#63846—34no—
25#63846—34no—
26#63845—34no—
27#63845—34no—
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29C2×C4×H(3,𝔽3)C2312yes—
30C2×#1084C2314yes—
31C2×(C3⋊Dic3)⋊C3C2314yes—
32C2×(C3⋊Dic3)⋊C3C2314yes—
33C3⋊#564—91no—
34C3⋊#564—92no—
35C32×C42—21no—
36C32×C42C2×D312yes—
37C4⋊(C3⋊Dic3)C2×D312yes—
38C4⋊(C3⋊Dic3)C2×D312yes—
39#3750C2×D314yes—
40#3750C2×D314yes—
41C3×C4⋊Dic3—32no—
42C3×C4⋊Dic3—32no—
43C3×C4⋊Dic3—32no—
44C3×C4⋊Dic3—32no—
45C3×C4⋊Dic3—62no—
46C3×Dic3⋊C4—34no—
47C3×Dic3⋊C4—34no—
48C3×Dic3⋊C4—34no—
49C3×Dic3⋊C4—34no—
50C3×Dic3⋊C4—64no—
51#3724—34no—
52#3724—34no—
53Dic3⋊Dic3—34no—
54Dic3⋊Dic3—34no—
55Dic3⋊Dic3—34no—
56Dic3⋊Dic3—34no—
57Dic32—34no—
58Dic32—34no—
59C22×H(3,𝔽3)C2411yescharacteristic
60C4×H(3,𝔽3)—21no—
61C4×H(3,𝔽3)—22no—
62#1084—24no—
63(C3⋊Dic3)⋊C3—24no—
64(C3⋊Dic3)⋊C3—24no—
65C3×#125—92no—
66C3×#125—91no—
67#890—94no—
68#890—94no—
69#890—94no—
70#889—94no—
71#889—94no—
72#889—94no—
73#889—94no—
74#889—94no—
75#889—94no—
76#870—32no—
77#870—32no—
78#870—61no—
79#870—92no—
80C2×C32×C4—21no—
81C2×C32×C4—22no—
82C2×C32×C4C22×D312yes—
83C2×C32×C4C22×D312yes—
84C2×C32×C4C22×D312yes—
85C2×C3⋊Dic3C22×D314yes—
86C2×C3⋊Dic3C22×D314yes—
87C2×C3×Dic3—34no—
88C2×C3×Dic3—34no—
89C2×C3×Dic3—34no—
90C2×C3×Dic3—34no—
91C2×C3×Dic3—64no—
92#564—271noSylow
93C2×H(3,𝔽3)2−1+412yes—
94C2×H(3,𝔽3)2−1+411yescharacteristic
95C3×C4⋊C4—92no—
96C3×C4⋊C4—92no—
97C3×C4⋊C4—94no—
98C3×C4⋊C4—94no—
99C3×C4⋊C4—92no—
100C3×C4⋊C4—94no—
101C3×C42—32no—
102C3×C42—61no—
103C3×C42D3211yescharacteristic
104C4⋊Dic3—32no—
105C4⋊Dic3—32no—
106C4⋊Dic3—32no—
107C4⋊Dic3—32no—
108C4⋊Dic3—62no—
109C4⋊Dic3—92no—
110C4⋊Dic3—92no—
111C4⋊Dic3—94no—
112C4⋊Dic3—94no—
113C4⋊Dic3—94no—
114Dic3⋊C4—34no—
115Dic3⋊C4—34no—
116Dic3⋊C4—34no—
117Dic3⋊C4—34no—
118Dic3⋊C4—64no—
119Dic3⋊C4—94no—
120Dic3⋊C4—94no—
121Dic3⋊C4—94no—
122Dic3⋊C4—94no—
123Dic3⋊C4—94no—
124Dic3⋊C4—94no—
125Dic3⋊C4—94no—
126Dic3⋊C4—94no—
127C4×Dic3—94no—
128C4×Dic3—94no—
129C4×Dic3—94no—
130C22×C32—21no—
131C22×C32C23×D312yes—
132C32×C4—22no—
133C32×C4—22no—
134C32×C4—22no—
135C32×C4—22no—
136C32×C4—42no—
137C3⋊Dic3—24no—
138C3⋊Dic3—24no—
139C3×Dic3—64no—
140C3×Dic3—64no—
141C3×Dic3—64no—
142C3×Dic3—64no—
143C3×Dic3—124no—
144#125—91no—
145#125—92no—
146#125—274no—
147#125—274no—
148#125—274no—
149H(3,𝔽3)#56411yescharacteristic, complemented, Sylow
150C2×C3×C4—32no—
151C2×C3×C4—32no—
152C2×C3×C4—32no—
153C2×C3×C4—61no—
154C2×C3×C4—62no—
155C2×C3×C4—94no—
156C2×C3×C4—94no—
157C2×C3×C4—94no—
158C2×C3×C4C2×D3211yescharacteristic
159C2×C3×C4C2×D3212yes—
160C2×Dic3—34no—
161C2×Dic3—34no—
162C2×Dic3—34no—
163C2×Dic3—34no—
164C2×Dic3—64no—
165C2×Dic3—94no—
166C2×Dic3—94no—
167C2×C32—22no—
168C2×C32—21no—
169C2×C32#100712yes—
170C2×C32#101012yes—
171C2×C32#101012yes—
172C4⋊C4—94no—
173C4⋊C4—94no—
174C4⋊C4—94no—
175C4⋊C4—92no—
176C4⋊C4—92no—
177C4⋊C4—92no—
178C4⋊C4—274no—
179C4⋊C4—274no—
180C4⋊C4—274no—
181C42—274no—
182C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
183C22×C3—32no—
184C22×C3—61no—
185C22×C3C22×D3211yescharacteristic
186C3×C4—22no—
187C3×C4—21no—
188C3×C4—62no—
189C3×C4—62no—
190C3×C4—62no—
191C3×C4—62no—
192C3×C4—122no—
193C3×C4—184no—
194C3×C4—184no—
195C3×C4—184no—
196Dic3—64no—
197Dic3—64no—
198Dic3—64no—
199Dic3—64no—
200Dic3—124no—
201Dic3—184no—
202Dic3—184no—
203C32—21no—
204C32C3⋊#56412yescomplemented
205C2×C4—94no—
206C2×C4—94no—
207C2×C4—94no—
208C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
209C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
210C2×C3—32no—
211C2×C3—32no—
212C2×C3—32no—
213C2×C3—62no—
214C2×C3—61no—
215C2×C3#6429712yes—
216C2×C3#6429811yescharacteristic
217C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
218C4—22no—
219C4—21no—
220C4—184no—
221C4—184no—
222C4—184no—
223C3—32no—
224C3—61no—
225C3#1059386811yescharacteristic
226C2#1172248212yes—
227C2#1172248311yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, prime radical, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#564
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
149. H(3,𝔽3)#564semidirect
204. C32C3⋊#564semidirect

Maximal subgroups

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

subgroupindex
#117217162
#64079⋊C32
#117210872
#117210862
#117210852
#105938683