Finite Group Databasebeta

← #13158859

Subgroups — #13158859

Order 1728.

880 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 164 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#11723916C211yesmaximal, characteristic
2C3×#64079C211yesmaximal, characteristic, complemented
3C32×C4×C3⋊C8C211yesmaximal, characteristic, complemented
4#11722042C212yesmaximal, complemented
5#11722036C212yesmaximal
6#10592396—31nomaximal
7#10590293—34nomaximal
8C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
9C2×#6308C2211yescharacteristic
10C33×C42C2211yescharacteristic
11#88859—22no—
12C3×C4⋊(C3⋊Dic3)C2212yes—
13C3×#3750—22no—
14C2×C32×C3⋊C8C2212yes—
15#64084—31no—
16#64084—34no—
17#64084—64no—
18#64079D311yescharacteristic, complemented
19C3×#870—34no—
20C32×C4×C8—31no—
21#63648—32no—
22#63643—32no—
23C3×C4×C3⋊C8D314yescomplemented
24#63567—38no—
25#63564—38no—
26C22×C3⋊(C3⋊D3)—21no—
27C2×C33×C4D412yes—
28C2×C33×C4C2311yescharacteristic
29C2×C3⋊(C3⋊Dic3)—21no—
30C4⋊(C3⋊(C3⋊D3))—22no—
31#6308—22no—
32C2×C3×C3⋊Dic3—22no—
33C32×C3⋊C8—22no—
34#5068—91no—
35#4697—94no—
36C2×C4⋊(C3⋊D3)—31no—
37C2×C4⋊(C3⋊D3)—34no—
38C2×C4⋊(C3⋊D3)—64no—
39C2×#634—31no—
40C2×#634—34no—
41C2×#634—64no—
42C2×C32×C8—32no—
43C32×C42—24no—
44C32×C42C2×D314yes—
45C32×C42C2×D311yescharacteristic
46#3752—616no—
47#3752—68no—
48#3752—62no—
49C4⋊(C3⋊Dic3)C2×D312yes—
50#3750—22no—
51C3×C4⋊Dic3—38no—
52C3×Dic3⋊C4—68no—
53C2×C3×C3⋊C8C2×D318yes—
54C22×C33C2×D411yescharacteristic
55C2×C3⋊(C3⋊D3)—41no—
56C2×C3⋊(C3⋊D3)—42no—
57C33×C4—22no—
58C33×C4#4112yes—
59C3⋊(C3⋊Dic3)—41no—
60C3×C3⋊Dic3—42no—
61C3×#125—91no—
62#875—94no—
63#875—91no—
64#875—184no—
65#870—34no—
66C3×C4×C8—34no—
67#821—98no—
68#816—98no—
69#810—92no—
70#809—92no—
71C4×C3⋊C8C3⋊D311yescharacteristic, complemented
72C22×C3⋊D3—68no—
73C22×C3⋊D3—64no—
74C22×C3⋊D3—61no—
75C2×C32×C4—28no—
76C2×C32×C4—24no—
77C2×C32×C4D1218yes—
78C2×C32×C4#6712yes—
79C2×C32×C4C22×D314yes—
80C2×C32×C4C22×D311yescharacteristic
81C2×C3⋊Dic3—22no—
82C2×C3⋊Dic3—68no—
83C2×C3⋊Dic3—64no—
84C2×C3⋊Dic3—61no—
85C4⋊(C3⋊D3)—62no—
86C4⋊(C3⋊D3)—68no—
87C4⋊(C3⋊D3)—128no—
88#634—62no—
89#634—68no—
90#634—128no—
91C2×C3×Dic3—68no—
92C32×C8—62no—
93C3×C3⋊C8—28no—
94#488—271noSylow
95C2×C33#12411yescharacteristic
96C2×C33#13512yes—
97C3⋊(C3⋊D3)—81no—
98C2×D12—94no—
99C2×D12—91no—
100C2×D12—184no—
101C2×Dic6—94no—
102C2×Dic6—91no—
103C2×Dic6—184no—
104C2×C3×C8—38no—
105C3×C4⋊C4—92no—
106C3×C4⋊C4—182no—
107C3×C42—24no—
108C3×C42D3214yes—
109C3×C42C2×C3⋊D311yescharacteristic
110(C2×D3)⋊C4—1816no—
111(C2×D3)⋊C4—188no—
112(C2×D3)⋊C4—182no—
113C4⋊Dic3—38no—
114Dic3⋊C4—68no—
115C2×C3⋊C8C2×C3⋊D312yes—
116C22×C32—24no—
117C22×C32C2×D1214yes—
118C22×C32C2×#6711yescharacteristic
119C2×C3⋊D3—128no—
120C2×C3⋊D3—1216no—
121C2×C3⋊D3—124no—
122C2×C3⋊D3—128no—
123C2×C3⋊D3—121no—
124C2×C3⋊D3—122no—
125C32×C4—22no—
126C32×C4—28no—
127C32×C4—28no—
128C32×C4—48no—
129C32×C4#23518yes—
130C32×C4#23912yes—
131C3⋊Dic3—42no—
132C3⋊Dic3—128no—
133C3⋊Dic3—124no—
134C3⋊Dic3—121no—
135C3×Dic3—128no—
136#125—91no—
137#124—271no—
138#103—272no—
139#102—272no—
140C4×C8—31no—
141C33#48811yescharacteristic, complemented, Sylow
142C22×D3—188no—
143C22×D3—184no—
144C22×D3—181no—
145C2×C3×C4—28no—
146C2×C3×C4—24no—
147C2×C3×C4—182no—
148C2×C3×C4#62618yes—
149C2×C3×C4C4⋊(C3⋊D3)12yes—
150C2×C3×C4C2×D3214yes—
151C2×C3×C4C22×C3⋊D311yescharacteristic
152C2×Dic3—68no—
153C2×Dic3—188no—
154C2×Dic3—184no—
155C2×Dic3—181no—
156D12—188no—
157D12—182no—
158D12—368no—
159Dic6—188no—
160Dic6—182no—
161Dic6—368no—
162C3×C8—68no—
163C3⋊C8—22no—
164C2×C32—28no—
165C2×C32—24no—
166C2×C32#87514yes—
167C2×C32#90418yes—
168C2×C32#94711yescharacteristic
169C2×C32#95012yes—
170C3⋊D3—248no—
171C3⋊D3—244no—
172C3⋊D3—241no—
173C2×Q8—271no—
174C2×D4—271no—
175C2×C8—32no—
176C4⋊C4—92no—
177C4⋊C4—182no—
178C22⋊C4—542no—
179C42D3×C3⋊D311yescharacteristic
180C22×C3—24no—
181C22×C3C2×#62614yes—
182C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
183C2×D3—368no—
184C2×D3—3616no—
185C2×D3—361no—
186C2×D3—368no—
187C2×D3—362no—
188C2×D3—364no—
189C3×C4—28no—
190C3×C4—22no—
191C3×C4—28no—
192C3×C4—48no—
193C3×C4—362no—
194C3×C4#379918yes—
195C3×C4#382812yes—
196Dic3—128no—
197Dic3—368no—
198Dic3—361no—
199Dic3—364no—
200C32—24no—
201C32#469714yescomplemented
202C32#506811yescharacteristic, complemented
203C23—541no—
204Q8—542no—
205D4—542no—
206C2×C4—182no—
207C2×C4—541no—
208C2×C4#629212yes—
209C2×C4C2×D3×C3⋊D311yescharacteristic
210C8—62no—
211C2×C3—28no—
212C2×C3—24no—
213C2×C3#6382918yes—
214C2×C3#6386414yes—
215C2×C3#6408411yescharacteristic
216C2×C3#6411312yes—
217D3—728no—
218D3—724no—
219D3—721no—
220C22—1082no—
221C22—1081no—
222C22C2×#629211yescharacteristic
223C4—22no—
224C4—362no—
225C4—1081no—
226C4#8902212yes—
227C3—24no—
228C3#1059029314yescomplemented
229C3#1059239611yescharacteristic, complemented
230C2—2161no—
231C2#1172336912yes—
232C2#1172344111yescharacteristic
subgroupG/H
centerC22C2×#6292
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypercenterC42D3×C3⋊D3
hypocenterC33#488
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C3×#64079C2semidirect
3. C32×C4×C3⋊C8C2semidirect
4. #11722042C2semidirect
18. #64079D3semidirect
23. C3×C4×C3⋊C8D3semidirect
71. C4×C3⋊C8C3⋊D3semidirect
141. C33#488semidirect
201. C32#4697semidirect
202. C32#5068semidirect
228. C3#10590293semidirect
229. C3#10592396semidirect

Maximal subgroups

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

subgroupindex
#117239162
C3×#640792
C32×C4×C3⋊C82
#117220422
#117220362
#105923963
#105902933