Finite Group Databasebeta

← #13139000

Subgroups — #13139000

Order 1728.

487 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 487 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C2×#88530C211yesmaximal, characteristic
2C2×#3746⋊C3C211yesmaximal, characteristic
3#11721090C212yesmaximal
4C2×#88439C212yesmaximal
5C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
6#10590905—31nomaximal
7#10590896—31nomaximal
8C2×#6229C2212yes—
9C2×#634⋊C3C2212yes—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11#88530—22no—
12C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
13C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
14#3746⋊C3—22no—
15#88448—24no—
16#88439—24no—
17C2×#3746D311yescharacteristic
18C2×C3×Dic12—31no—
19C2×C3×Dic12—31no—
20C2×C3×Dic12—31no—
21C2×C3×Dic12—61no—
22#63873—32no—
23#63866—32no—
24C2×#3719—32no—
25C2×#3718—32no—
26C8×C3⋊Dic3D311yescharacteristic
27C3×C8×Dic3—31no—
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29C2×#1084—22no—
30#6229—22no—
31#6229—22no—
32C2×(C3⋊Dic3)⋊C3—22no—
33C2×(C3⋊Dic3)⋊C3D412yes—
34#634⋊C3—22no—
35#634⋊C3—22no—
36C8×H(3,𝔽3)D412yes—
37(C3⋊(C3⋊C8))⋊C3D412yes—
38#5183—91no—
39#5183—91no—
40#4699—91no—
41C2×#634C2×D312yes—
42C2×C3×Dic6—32no—
43C2×C3×Dic6—32no—
44C2×C3×Dic6—32no—
45C2×C3×Dic6—62no—
46C2×C32×C8—21no—
47C2×C32×C8C2×D311yescharacteristic
48C2×C32×C8C2×D311yescharacteristic
49C4×C3⋊Dic3C2×D311yescharacteristic
50C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
51#3746—22no—
52C3×C4×Dic3—31no—
53C2×C3×C3⋊C8—31no—
54C3×Dic12—64no—
55C3×Dic12—62no—
56C3×Dic12—62no—
57C3×Dic12—62no—
58#3724—64no—
59Dic3⋊Dic3—64no—
60#3719—64no—
61#3718—64no—
62C22×H(3,𝔽3)C2×D411yescharacteristic
63C4×H(3,𝔽3)C2×D411yescharacteristic
64C4×H(3,𝔽3)C2×D411yescharacteristic
65#1084—42no—
66(C3⋊Dic3)⋊C3—42no—
67(C3⋊Dic3)⋊C3Q1614yescomplemented
68C2×C3×Q16—91no—
69C2×C3×Q16—91no—
70#944—92no—
71#944—92no—
72C2×#216—92no—
73C2×#216—92no—
74C2×Dic12—31no—
75C2×Dic12—31no—
76C2×Dic12—31no—
77C2×Dic12—61no—
78C2×Dic12—91no—
79C2×Dic12—92no—
80#869—92no—
81C3×C4×C8—91no—
82C8×Dic3—31no—
83C8×Dic3—91no—
84C2×C32×C4—21no—
85C2×C32×C4C22×D311yescharacteristic
86C2×C32×C4C22×D311yescharacteristic
87C2×C3⋊Dic3—22no—
88C2×C3⋊Dic3D1212yes—
89#634—22no—
90#634—22no—
91C2×C3×Dic3—32no—
92C2×C3×Dic3—62no—
93C2×C3×Dic3—62no—
94C2×C3×Dic3—62no—
95C2×C3×Dic3—122no—
96C3×Dic6—64no—
97C3×Dic6—62no—
98C3×Dic6—62no—
99C3×Dic6—62no—
100C3×Dic6—62no—
101C3×Dic6—62no—
102C3×Dic6—62no—
103C3×Dic6—122no—
104C32×C8—22no—
105C32×C8D1212yes—
106C32×C8#6712yes—
107C3⋊(C3⋊C8)D1212yes—
108C3×C3⋊C8—32no—
109#494—271noSylow
110C2×H(3,𝔽3)#12711yescharacteristic
111C2×H(3,𝔽3)C2×Q1612yes—
112C2×C3×Q8—92no—
113C2×C3×Q8—92no—
114C2×Dic6—32no—
115C2×Dic6—32no—
116C2×Dic6—32no—
117C2×Dic6—62no—
118C2×Dic6—92no—
119C3×Q16—182no—
120C3×Q16—182no—
121C2×C3×C8—31no—
122C2×C3×C8—31no—
123C2×C3×C8—61no—
124C2×C3×C8—91no—
125C2×C3×C8D3211yescharacteristic
126C3×C42—91no—
127#216—184no—
128#216—184no—
129C4⋊Dic3—184no—
130Dic3⋊C4—184no—
131Dic3⋊C4—184no—
132C4×Dic3—31no—
133C4×Dic3—91no—
134C2×C3⋊C8—31no—
135C2×C3⋊C8—91no—
136Dic12—62no—
137Dic12—62no—
138Dic12—64no—
139Dic12—62no—
140Dic12—182no—
141Dic12—184no—
142C22×C32—21no—
143C22×C32C2×D1211yescharacteristic
144C22×C32D3×D411yescharacteristic
145C32×C4—21no—
146C32×C4—21no—
147C32×C4C2×D1211yescharacteristic
148C32×C4C2×D1211yescharacteristic
149C32×C4D3×D411yescharacteristic
150C32×C4C2×#6711yescharacteristic
151C3⋊Dic3—42no—
152C3⋊Dic3Dic1214yescomplemented
153C3×Dic3—34no—
154C3×Dic3—124no—
155C3×Dic3—122no—
156C3×Dic3—122no—
157C3×Dic3—122no—
158C2×Q16—91no—
159C2×Q16—91no—
160C2×Q16—272no—
161C4⋊Q8—272no—
162C4×C8—91no—
163H(3,𝔽3)#49411yescharacteristic, complemented, Sylow
164C3×Q8—182no—
165C3×Q8—182no—
166C3×Q8—182no—
167C3×Q8—182no—
168C2×C3×C4—31no—
169C2×C3×C4—31no—
170C2×C3×C4—61no—
171C2×C3×C4—92no—
172C2×C3×C4—182no—
173C2×C3×C4—182no—
174C2×C3×C4C2×D3211yescharacteristic
175C2×Dic3—32no—
176C2×Dic3—62no—
177C2×Dic3—62no—
178C2×Dic3—62no—
179C2×Dic3—92no—
180C2×Dic3—122no—
181C2×Dic3—182no—
182Dic6—62no—
183Dic6—62no—
184Dic6—62no—
185Dic6—62no—
186Dic6—64no—
187Dic6—62no—
188Dic6—62no—
189Dic6—122no—
190Dic6—182no—
191Dic6—182no—
192C3×C8—32no—
193C3×C8—32no—
194C3×C8—62no—
195C3×C8—92no—
196C3×C8#62612yes—
197C3⋊C8—32no—
198C3⋊C8—92no—
199C2×C32—22no—
200C2×C32—21no—
201C2×C32#87411yescharacteristic
202C2×C32C2×Dic1212yes—
203C2×C32D3×Q1612yes—
204C2×C32#94011yescharacteristic
205C2×Q8—92no—
206C2×Q8—92no—
207Q16—182no—
208Q16—182no—
209Q16—544no—
210C2×C8—91no—
211C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
212C4⋊C4—544no—
213C42—91no—
214C22×C3—31no—
215C22×C3—31no—
216C22×C3—61no—
217C22×C3D3×D1211yescharacteristic
218C3×C4—31no—
219C3×C4—31no—
220C3×C4—31no—
221C3×C4—31no—
222C3×C4—61no—
223C3×C4—61no—
224C3×C4—94no—
225C3×C4—362no—
226C3×C4—362no—
227C3×C4D3×D1211yescharacteristic
228C3×C4C2×#62611yescharacteristic
229Dic3—34no—
230Dic3—94no—
231Dic3—122no—
232Dic3—122no—
233Dic3—124no—
234Dic3—122no—
235Dic3—362no—
236C32—21no—
237C32#469911yescharacteristic, complemented
238C32#518311yescharacteristic, complemented
239Q8—182no—
240Q8—182no—
241Q8—182no—
242Q8—182no—
243C2×C4—92no—
244C2×C4—182no—
245C2×C4—182no—
246C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
247C8—92no—
248C8#620012yes—
249C2×C3—32no—
250C2×C3—32no—
251C2×C3—31no—
252C2×C3—31no—
253C2×C3—62no—
254C2×C3—61no—
255C2×C3D3×Dic1212yes—
256C2×C3#6391111yescharacteristic
257C22#8865411yescharacteristic
258C4—94no—
259C4—362no—
260C4—362no—
261C4#8865411yescharacteristic
262C4C2×#620011yescharacteristic
263C3—31no—
264C3—31no—
265C3—61no—
266C3#1059089611yescharacteristic
267C2#1172085112yes—
268C2#1172115811yescharacteristic
subgroupG/H
centerC22#88654
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#494
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
67. (C3⋊Dic3)⋊C3Q16semidirect
152. C3⋊Dic3Dic12semidirect
163. H(3,𝔽3)#494semidirect
237. C32#4699semidirect
238. C32#5183semidirect

Maximal subgroups

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

subgroupindex
C2×#885302
C2×#3746⋊C32
#117210902
C2×#884392
C8×(C3⋊Dic3)⋊C32
#105909053
#105908963