Finite Group Databasebeta

← #13173065

Subgroups — #13173065

Order 1728.

874 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 874 conjugacy classes of subgroups — too many to draw. Only its 139 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
1C32×D4×A4C211yesmaximal, characteristic, complemented
2C4⋊(C3⋊(C3⋊S4))C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊(A4⋊C8))C211yesmaximal, characteristic, complemented
4#10593649—312nomaximal
5#10592916—31nomaximal
6C22×C32×A4—21no—
7C2×C3⋊(C3⋊S4)—21no—
8C32×C4×A4C2211yescharacteristic
9#88860—41nomaximal
10C22×C32×D4D311yescharacteristic, complemented
11C3×D4×A4D3112yescomplemented
12C4⋊(C3⋊S4)—312no—
13C2×#3753—31no—
14C2×#3753—31no—
15#64139—31no—
16C3⋊(A4⋊C8)—312no—
17#63659—31no—
18#63659—31no—
19#63657—31no—
20C2×C32×A4—41no—
21C2×C32×A4D411yescharacteristic
22C3⋊(C3⋊S4)—41no—
23C33×D4—41no—
24C4⋊(C3⋊(C3⋊D3))—41no—
25C3⋊(C3⋊(C3⋊C8))—41no—
26#5415—99no—
27#5217—94no—
28C24×C32—21no—
29C22×C3×A4—212no—
30C2×C3⋊S4—612no—
31C2×C32×D4—31no—
32C2×C32×D4—31no—
33C2×C32×D4—61no—
34C22×C32×C4C2×D311yescharacteristic
35C2×#638—61no—
36C2×C4⋊(C3⋊D3)—31no—
37C3×C4×A4C2×D3112yes—
38#3753—62no—
39#3753—61no—
40#3753—61no—
41#3753—1212no—
42#3752—61no—
43C4⋊(C3⋊Dic3)—31no—
44C2×C3⋊(C3⋊C8)—31no—
45C2×C3⋊(C3⋊C8)—31no—
46C22×C33—81no—
47C2×C3⋊(C3⋊D3)—81no—
48C32×A4D811yescharacteristic, complemented
49C33×C4—41no—
50C22×C3×D4C3⋊D314yescomplemented
51D4×A4C3⋊D319yescomplemented
52C4⋊S4—99no—
53C2×#213—94no—
54C2×#213—94no—
55#930—94no—
56A4⋊C8—99no—
57#832—94no—
58#832—94no—
59#830—94no—
60C23×C32—41no—
61C23×C32—61no—
62C23×C32—61no—
63C23×C32—121no—
64C23×C32#6711yescharacteristic
65C22×C3⋊D3—61no—
66C2×C3×A4—412no—
67C2×C3×A4#67112yes—
68C3⋊S4—1212no—
69C32×D4—32no—
70C32×D4—31no—
71C32×D4—412no—
72C32×D4—61no—
73C32×D4S411yescharacteristic, complemented
74C2×C32×C4—31no—
75C2×C32×C4—31no—
76#638—121no—
77#638—121no—
78C2×C3⋊Dic3—61no—
79C4⋊(C3⋊D3)—61no—
80C4⋊(C3⋊D3)—61no—
81C4⋊(C3⋊D3)—1212no—
82C3⋊(C3⋊C8)—61no—
83C3⋊(C3⋊C8)—61no—
84C3⋊(C3⋊C8)—1212no—
85#447—271noSylow
86C2×C33—41no—
87C2×C33—161no—
88C3⋊(C3⋊D3)—161no—
89C24×C3—24no—
90C22×A4—29no—
91C2×S4—189no—
92C2×C3×D4—34no—
93C2×C3×D4—34no—
94C2×C3×D4—64no—
95C22×C3×C4C2×C3⋊D314yes—
96C2×#67—184no—
97C2×D12—94no—
98C4×A4C2×C3⋊D319yes—
99#213—188no—
100#213—184no—
101#213—184no—
102#213—369no—
103(C2×D3)⋊C4—184no—
104C4⋊Dic3—94no—
105C2×C3⋊C8—94no—
106C2×C3⋊C8—94no—
107C22×C32—21no—
108C22×C32—31no—
109C22×C32—31no—
110C22×C32—61no—
111C22×C32—812no—
112C22×C32—121no—
113C22×C32—121no—
114C22×C32—121no—
115C22×C32—121no—
116C22×C32#21311yescharacteristic, complemented
117C2×C3⋊D3—121no—
118C2×C3⋊D3—121no—
119C2×C3⋊D3—121no—
120C2×C3⋊D3—2412no—
121C3×A4#213112yescomplemented
122C32×C4—31no—
123C32×C4—412no—
124C32×C4C2×S411yescharacteristic
125C3⋊Dic3—121no—
126C22×D4C3⋊(C3⋊D3)11yescharacteristic, complemented
127C2×D8—271no—
128C2×D8—271no—
129#121—271no—
130#102—271no—
131#102—271no—
132C22⋊C8—271no—
133C33—41noSylow
134C23×C3—44no—
135C23×C3—64no—
136C23×C3—64no—
137C23×C3—124no—
138C23×C3#63814yes—
139C22×D3—184no—
140C2×A4—49no—
141C2×A4#63819yes—
142S4—369no—
143C3×D4—38no—
144C3×D4—34no—
145C3×D4—49no—
146C3×D4—64no—
147C3×D4C3⋊S414yescomplemented
148C2×C3×C4—34no—
149C2×C3×C4—34no—
150#67—364no—
151#67—364no—
152C2×Dic3—184no—
153D12—184no—
154D12—184no—
155D12—369no—
156C3⋊C8—184no—
157C3⋊C8—184no—
158C3⋊C8—369no—
159C2×C32—31no—
160C2×C32—31no—
161C2×C32—41no—
162C2×C32—412no—
163C2×C32—121no—
164C2×C32—1612no—
165C2×C32GL(2,ℤ/4)11yescharacteristic
166C3⋊D3—241no—
167C3⋊D3—4812no—
168C24—21no—
169C2×D4—31no—
170C2×D4—31no—
171C2×D4—61no—
172C2×D4—271no—
173C2×D4—541no—
174C22×C4C2×C3⋊(C3⋊D3)11yescharacteristic
175D8—542no—
176D8—541no—
177D8—541no—
178C2×C8—271no—
179C2×C8—271no—
180C4⋊C4—271no—
181C22⋊C4—541no—
182C22×C3—24no—
183C22×C3—34no—
184C22×C3—34no—
185C22×C3—64no—
186C22×C3—89no—
187C22×C3—124no—
188C22×C3—124no—
189C22×C3—124no—
190C22×C3—124no—
191C22×C3#375314yescomplemented
192C2×D3—364no—
193C2×D3—364no—
194C2×D3—364no—
195C2×D3—729no—
196A4#375319yescomplemented
197C3×C4—34no—
198C3×C4—49no—
199C3×C4C2×C3⋊S414yes—
200Dic3—364no—
201C32—412no—
202C32#541511yescharacteristic, complemented
203C23—41no—
204C23—61no—
205C23—61no—
206C23—121no—
207C23—541no—
208C23#631211yescharacteristic
209D4—32no—
210D4—31no—
211D4—61no—
212D4—541no—
213D4—541no—
214D4—1081no—
215D4—1081no—
216D4C3⋊(C3⋊S4)11yescharacteristic, complemented
217C2×C4—31no—
218C2×C4—31no—
219C2×C4—541no—
220C8—541no—
221C8—541no—
222C2×C3—34no—
223C2×C3—34no—
224C2×C3—44no—
225C2×C3—49no—
226C2×C3—124no—
227C2×C3—169no—
228C2×C3#6426914yes—
229D3—724no—
230D3—1449no—
231C22—21no—
232C22—31no—
233C22—31no—
234C22—61no—
235C22—121no—
236C22—121no—
237C22—121no—
238C22—121no—
239C22—1081no—
240C22—1081no—
241C22—1081no—
242C22#8886011yescharacteristic, complemented
243C4—31no—
244C4—1081no—
245C4C2×C3⋊(C3⋊S4)11yescharacteristic
246C3—49no—
247C3#1059364914yescomplemented
248C2—31no—
249C2—31no—
250C2—41no—
251C2—121no—
252C2—2161no—
253C2#1172447311yescharacteristic
subgroupG/H
centerC2#11724473
commutator, cosocleC32×C4×A4C22
FrattiniC4C2×C3⋊(C3⋊S4)
FittingC22×C32×D4D3
socleC23×C32#67
hypercenterD4C3⋊(C3⋊S4)
hypocenterC32×A4D8
LCSG ▷ C32×C4×A4 ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
1. C32×D4×A4C2semidirect
2. C4⋊(C3⋊(C3⋊S4))C2semidirect
3. C3⋊(C3⋊(A4⋊C8))C2semidirect
10. C22×C32×D4D3semidirect
11. C3×D4×A4D3semidirect
48. C32×A4D8semidirect
50. C22×C3×D4C3⋊D3semidirect
51. D4×A4C3⋊D3semidirect
73. C32×D4S4semidirect
116. C22×C32#213semidirect
121. C3×A4#213semidirect
126. C22×D4C3⋊(C3⋊D3)semidirect
147. C3×D4C3⋊S4semidirect
191. C22×C3#3753semidirect
196. A4#3753semidirect
202. C32#5415semidirect
216. D4C3⋊(C3⋊S4)semidirect
242. C22#88860semidirect
247. C3#10593649semidirect

Maximal subgroups

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

subgroupindex
C32×D4×A42
C4⋊(C3⋊(C3⋊S4))2
C3⋊(C3⋊(A4⋊C8))2
#105936493
#105929163
#888604