Finite Group Databasebeta

← #13147845

Subgroups — #13147845

Order 1728.

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

normal subgroups only

This group has 894 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#11721715C211yesmaximal, characteristic, complemented
2C4×C4⋊((C3⋊D3)⋊C3)C212yesmaximal, complemented
3#11721143C214yesmaximal, complemented
4#11721141C214yesmaximal, complemented
5#11721081C214yesmaximal, complemented
6#10593987—32nomaximal
7C2×#6229C2212yes—
8C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
9C2×C4×(C3⋊D3)⋊C3C2214yes—
10C2×#6198C2214yes—
11C42×H(3,𝔽3)C2211yescharacteristic
12C4⋊#1084C2214yes—
13#3752⋊C3C2214yes—
14C4⋊((C3⋊Dic3)⋊C3)C2212yes—
15#88450C2218yes—
16#88447C2214yes—
17C4×C4⋊(C3⋊D3)D312yescomplemented
18C3×C4×D12—32no—
19C3×#869—32no—
20C3×#869—61no—
21#63916—34no—
22#63909—34no—
23#63907—38no—
24#63849—34no—
25#63841—34no—
26C22×(C3⋊D3)⋊C3C2314yes—
27C2×C4×H(3,𝔽3)C2311yescharacteristic
28C2×C4×H(3,𝔽3)C2312yes—
29C2×#1084C2314yes—
30#6229—42no—
31C2×(C3⋊Dic3)⋊C3C2314yes—
32C4⋊((C3⋊D3)⋊C3)—24no—
33C4×(C3⋊D3)⋊C3—28no—
34#6198—44no—
35#5734—91no—
36#5563—92no—
37C2×C4⋊(C3⋊D3)C2×D312yes—
38C2×C4×C3⋊D3C2×D314yes—
39C2×C3×D12—32no—
40C2×C3×C4×D3—34no—
41C2×C3×Dic6—34no—
42C2×C3×Dic6—62no—
43C2×#626—38no—
44C32×C42—21no—
45C32×C42C2×D312yes—
46#3752C2×D314yes—
47C4⋊(C3⋊Dic3)C2×D312yes—
48C3×(C2×D3)⋊C4—34no—
49C3×C4⋊Dic3—34no—
50C3×C4⋊Dic3—32no—
51C3×C4⋊Dic3—34no—
52C3×C4⋊Dic3—64no—
53#3722—38no—
54(C2×D3)⋊Dic3—38no—
55Dic32—38no—
56C22×H(3,𝔽3)C2411yescharacteristic
57C2×(C3⋊D3)⋊C3—24no—
58C2×(C3⋊D3)⋊C3—24no—
59C2×(C3⋊D3)⋊C3—24no—
60C4×H(3,𝔽3)—21no—
61C4×H(3,𝔽3)#4114yes—
62#1084—24no—
63(C3⋊Dic3)⋊C3—24no—
64C3×C4⋊Q8—91no—
65C3×C4×D4—92no—
66#947—94no—
67#938—94no—
68#930—94no—
69#926—94no—
70#895—94no—
71#892—94no—
72#892—94no—
73#885—98no—
74C4×D12—32no—
75C4×D12—92no—
76#869—32no—
77#869—61no—
78C22×C3⋊D3C22×D314yes—
79C22×C3×D3—34no—
80C2×C32×C4—22no—
81C2×C32×C4—21no—
82C2×C32×C4C22×D312yes—
83C2×C32×C4C22×D312yes—
84C2×C32×C4C22×D312yes—
85C2×C3⋊Dic3C22×D314yes—
86C4⋊(C3⋊D3)—24no—
87C4×C3⋊D3—28no—
88C2×C3×Dic3—34no—
89C2×C3×Dic3—38no—
90C2×C3×Dic3—64no—
91C3×D12—64no—
92C3×C4×D3—68no—
93C3×Dic6—124no—
94C3×Dic6—124no—
95#626—128no—
96#555—271noSylow
97C2×H(3,𝔽3)C2×#4112yes—
98C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
99(C3⋊D3)⋊C3—44no—
100C2×C3×Q8—92no—
101C2×C3×D4—92no—
102C22×C3×C4—94no—
103C2×#67—98no—
104C2×D12—32no—
105C2×D12—92no—
106C2×D12—94no—
107C2×C4×D3—34no—
108C2×C4×D3—94no—
109C2×Dic6—34no—
110C2×Dic6—62no—
111C3×C4⋊C4—92no—
112C3×C4⋊C4—94no—
113C3×C22⋊C4—94no—
114C3×C42—32no—
115C3×C42—61no—
116C3×C42D3211yescharacteristic
117#217—98no—
118(C2×D3)⋊C4—34no—
119(C2×D3)⋊C4—94no—
120(C2×D3)⋊C4—98no—
121(C2×D3)⋊C4—98no—
122C4⋊Dic3—34no—
123C4⋊Dic3—34no—
124C4⋊Dic3—32no—
125C4⋊Dic3—64no—
126C4⋊Dic3—92no—
127C4×Dic3—94no—
128C4×Dic3—98no—
129C22×C32—21no—
130C22×C32C23×D312yes—
131C2×C3⋊D3—24no—
132C2×C3⋊D3—24no—
133C2×C3⋊D3—24no—
134C2×C3×D3—64no—
135C2×C3×D3—64no—
136C2×C3×D3—64no—
137C32×C4—22no—
138C32×C4—22no—
139C32×C4—24no—
140C32×C4#23514yes—
141C32×C4#23714yes—
142C3⋊Dic3—24no—
143C3×Dic3—64no—
144C3×Dic3—68no—
145C3×Dic3—124no—
146C4⋊Q8—91no—
147#124—274no—
148#121—274no—
149C4×D4—92no—
150#117—274no—
151H(3,𝔽3)#55511yescharacteristic, complemented, Sylow
152C23×C3—94no—
153C22×D3—34no—
154C22×D3—94no—
155C3×Q8—362no—
156C3×D4—184no—
157C2×C3×C4—32no—
158C2×C3×C4—32no—
159C2×C3×C4—32no—
160C2×C3×C4—62no—
161C2×C3×C4—61no—
162C2×C3×C4—94no—
163C2×C3×C4—94no—
164C2×C3×C4—188no—
165C2×C3×C4C2×D3211yescharacteristic
166C2×C3×C4C2×D3212yes—
167#67—368no—
168C2×Dic3—38no—
169C2×Dic3—34no—
170C2×Dic3—64no—
171C2×Dic3—94no—
172D12—64no—
173D12—184no—
174D12—364no—
175C4×D3—68no—
176C4×D3—188no—
177Dic6—124no—
178Dic6—124no—
179C2×C32—22no—
180C2×C32—21no—
181C2×C32C2×#23512yes—
182C2×C32C2×#23712yes—
183C2×C32#100912yes—
184C3⋊D3—44no—
185C3×D3—124no—
186C2×Q8—92no—
187C2×D4—92no—
188C2×D4—274no—
189C22×C4—94no—
190C4⋊C4—92no—
191C4⋊C4—94no—
192C22⋊C4—94no—
193C22⋊C4—278no—
194C42—274no—
195C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
196C22×C3—32no—
197C22×C3—61no—
198C22×C3—184no—
199C22×C3—184no—
200C22×C3—184no—
201C22×C3C22×D3211yescharacteristic
202C2×D3—64no—
203C2×D3—64no—
204C2×D3—64no—
205C2×D3—184no—
206C2×D3—184no—
207C2×D3—184no—
208C3×C4—21no—
209C3×C4—34no—
210C3×C4—34no—
211C3×C4—62no—
212C3×C4—62no—
213C3×C4—64no—
214C3×C4—184no—
215C3×C4—184no—
216C3×C4#379514yes—
217Dic3—68no—
218Dic3—64no—
219Dic3—124no—
220Dic3—184no—
221C32—21no—
222C32#556312yescomplemented
223C23—94no—
224Q8—362no—
225D4—184no—
226D4—1084no—
227C2×C4—94no—
228C2×C4—94no—
229C2×C4—188no—
230C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
231C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
232C2×C3—32no—
233C2×C3—32no—
234C2×C3—32no—
235C2×C3—62no—
236C2×C3—61no—
237C2×C3—364no—
238C2×C3C2×#379512yes—
239C2×C3#6430811yescharacteristic
240D3—124no—
241D3—364no—
242C22—184no—
243C22—184no—
244C22—184no—
245C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
246C4—21no—
247C4—184no—
248C4—184no—
249C4#8864814yes—
250C3—32no—
251C3—61no—
252C3#1059398811yescharacteristic
253C2—364no—
254C2C2×#8864812yes—
255C2#1172249311yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#555
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
1. #11721715C2semidirect
2. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
3. #11721143C2semidirect
4. #11721141C2semidirect
5. #11721081C2semidirect
17. C4×C4⋊(C3⋊D3)D3semidirect
151. H(3,𝔽3)#555semidirect
222. C32#5563semidirect

Maximal subgroups

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

subgroupindex
#117217152
C4×C4⋊((C3⋊D3)⋊C3)2
#117211432
#117211412
#117210812
#105939873