Finite Group Databasebeta

← #13147872

Subgroups — #13147872

Order 1728.

934 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 934 conjugacy classes of subgroups — too many to draw. Only its 149 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#11721720C211yesmaximal, characteristic
2C4×#634⋊C3C212yesmaximal, complemented
3#11721151C214yesmaximal, complemented
4#11721149C214yesmaximal, complemented
5#11721114C214yesmaximal, complemented
6#10593951—32nomaximal
7C2×C4⋊#272C2214yes—
8C2×C4⋊#272C2212yes—
9C2×#634⋊C3C2212yes—
10C2×#6199C2214yes—
11C42×H(3,𝔽3)C2211yescharacteristic
12C4⋊((C3⋊Dic3)⋊C3)C2212yes—
13#3750⋊C3C2214yes—
14C4×(C3⋊Dic3)⋊C3C2214yes—
15#88451C2214yes—
16#88451C2218yes—
17C4×#634D312yescomplemented
18C3×#874—32no—
19C3×#874—61no—
20C3×C4×Dic6—32no—
21#63898—34no—
22Dic3×D12—34no—
23#63884—38no—
24#63863—34no—
25#63860—34no—
26C22×#272C2314yes—
27C2×C4×H(3,𝔽3)C2311yescharacteristic
28C2×C4×H(3,𝔽3)C2312yes—
29C4⋊#272—28no—
30C4⋊#272—42no—
31C2×(C3⋊Dic3)⋊C3C2314yes—
32C2×(C3⋊Dic3)⋊C3C2314yes—
33#634⋊C3—24no—
34#6199—44no—
35#5713—91no—
36#5585—92no—
37C2×#634C2×D312yes—
38C2×C3×D12—34no—
39C2×C3×D12—34no—
40C2×C3×D12—34no—
41C2×C3×D12—64no—
42C2×C3×D12—62no—
43C2×C3×Dic6—32no—
44C2×D3×Dic3—38no—
45C32×C42—21no—
46C32×C42C2×D312yes—
47C4⋊(C3⋊Dic3)C2×D312yes—
48#3750C2×D314yes—
49C4×C3⋊Dic3C2×D314yes—
50C3×C4⋊Dic3—32no—
51C3×Dic3⋊C4—34no—
52C3×C4×Dic3—34no—
53(C2×D3)⋊Dic3—38no—
54(C2×D3)⋊Dic3—38no—
55(C2×D3)⋊Dic3—38no—
56C22×H(3,𝔽3)C2411yescharacteristic
57C2×#272—28no—
58C2×#272—24no—
59C4×H(3,𝔽3)—21no—
60C4×H(3,𝔽3)#4114yes—
61(C3⋊Dic3)⋊C3—24no—
62(C3⋊Dic3)⋊C3—24no—
63C3×#127—91no—
64C3×C4×Q8—92no—
65#947—94no—
66#936—94no—
67#935—94no—
68D4×Dic3—94no—
69#894—98no—
70#893—94no—
71#875—94no—
72#874—32no—
73#874—61no—
74C4×D12—94no—
75C4×Dic6—32no—
76C4×Dic6—92no—
77C22×C3×D3—38no—
78C22×C3×D3—64no—
79C2×C32×C4—21no—
80C2×C32×C4—22no—
81C2×C32×C4C22×D312yes—
82C2×C32×C4C22×D312yes—
83C2×C32×C4C22×D312yes—
84C2×C3⋊Dic3C22×D314yes—
85C2×C3⋊Dic3C22×D314yes—
86#634—24no—
87C2×C3×Dic3—34no—
88C2×C3×Dic3—34no—
89C3×D12—68no—
90C3×D12—68no—
91C3×D12—128no—
92C3×D12—124no—
93C3×D12—124no—
94C3×Dic6—64no—
95D3×Dic3—128no—
96#559—271noSylow
97C2×H(3,𝔽3)C2×#4112yes—
98C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
99#272—44no—
100C2×C3×Q8—92no—
101C2×C3×D4—94no—
102C2×C3×D4—92no—
103C22×Dic3—94no—
104C2×D12—34no—
105C2×D12—34no—
106C2×D12—34no—
107C2×D12—64no—
108C2×D12—62no—
109C2×C4×D3—98no—
110C2×Dic6—32no—
111C2×Dic6—92no—
112C3×C4⋊C4—92no—
113C3×C4⋊C4—94no—
114C3×C42—32no—
115C3×C42—61no—
116C3×C42—94no—
117C3×C42D3211yescharacteristic
118#217—98no—
119#217—94no—
120(C2×D3)⋊C4—98no—
121(C2×D3)⋊C4—98no—
122(C2×D3)⋊C4—98no—
123C4⋊Dic3—32no—
124C4⋊Dic3—92no—
125Dic3⋊C4—34no—
126Dic3⋊C4—94no—
127C4×Dic3—34no—
128C4×Dic3—94no—
129C22×C32—21no—
130C22×C32C23×D312yes—
131C2×C3×D3—68no—
132C2×C3×D3—68no—
133C2×C3×D3—68no—
134C2×C3×D3—128no—
135C2×C3×D3—124no—
136C32×C4—22no—
137C32×C4—22no—
138C32×C4—24no—
139C32×C4#23514yes—
140C32×C4#23914yes—
141C3⋊Dic3—24no—
142C3⋊Dic3—24no—
143C3×Dic3—64no—
144C3×Dic3—64no—
145#127—91no—
146#124—274no—
147#123—274no—
148C4×Q8—92no—
149C4×D4—274no—
150H(3,𝔽3)#55911yescharacteristic, complemented, Sylow
151C23×C3—94no—
152C22×D3—38no—
153C22×D3—64no—
154C3×Q8—184no—
155C3×D4—188no—
156C3×D4—362no—
157C2×C3×C4—32no—
158C2×C3×C4—32no—
159C2×C3×C4—32no—
160C2×C3×C4—61no—
161C2×C3×C4—62no—
162C2×C3×C4—94no—
163C2×C3×C4—94no—
164C2×C3×C4C2×D3211yescharacteristic
165C2×C3×C4C2×D3212yes—
166C2×Dic3—34no—
167C2×Dic3—34no—
168C2×Dic3—94no—
169C2×Dic3—94no—
170C2×Dic3—364no—
171D12—68no—
172D12—68no—
173D12—128no—
174D12—124no—
175D12—124no—
176C4×D3—368no—
177Dic6—64no—
178Dic6—184no—
179C2×C32—22no—
180C2×C32—21no—
181C2×C32C2×#23512yes—
182C2×C32C2×#23912yes—
183C2×C32#100912yes—
184C3×D3—128no—
185C3×D3—244no—
186C2×Q8—92no—
187C2×D4—94no—
188C2×D4—92no—
189C22×C4—274no—
190C4⋊C4—92no—
191C4⋊C4—94no—
192C22⋊C4—278no—
193C22⋊C4—274no—
194C42—94no—
195C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
196C22×C3—32no—
197C22×C3—61no—
198C22×C3—188no—
199C22×C3—184no—
200C22×C3C22×D3211yescharacteristic
201C2×D3—68no—
202C2×D3—68no—
203C2×D3—68no—
204C2×D3—128no—
205C2×D3—124no—
206C3×C4—21no—
207C3×C4—34no—
208C3×C4—34no—
209C3×C4—62no—
210C3×C4—62no—
211C3×C4—64no—
212C3×C4—184no—
213C3×C4—184no—
214C3×C4#379914yes—
215Dic3—64no—
216Dic3—64no—
217Dic3—184no—
218Dic3—184no—
219C32—21no—
220C32#558512yescomplemented
221C23—94no—
222Q8—184no—
223D4—188no—
224D4—362no—
225C2×C4—94no—
226C2×C4—94no—
227C2×C4—1084no—
228C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
229C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
230C2×C3—32no—
231C2×C3—32no—
232C2×C3—32no—
233C2×C3—62no—
234C2×C3—61no—
235C2×C3—364no—
236C2×C3C2×#379912yes—
237C2×C3#6430811yescharacteristic
238D3—128no—
239D3—244no—
240C22—188no—
241C22—184no—
242C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
243C4—21no—
244C4—184no—
245C4—184no—
246C4#8865214yes—
247C3—32no—
248C3—61no—
249C3#1059396311yescharacteristic
250C2—364no—
251C2C2×#8865212yes—
252C2#1172249311yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radical#11721720C2
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#559
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
2. C4×#634⋊C3C2semidirect
3. #11721151C2semidirect
4. #11721149C2semidirect
5. #11721114C2semidirect
17. C4×#634D3semidirect
150. H(3,𝔽3)#559semidirect
220. C32#5585semidirect

Maximal subgroups

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

subgroupindex
#117217202
C4×#634⋊C32
#117211512
#117211492
#117211142
#105939513