Finite Group Databasebeta

← #13173067

Subgroups — #13173067

Order 1728.

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

normal subgroups only

This group has 782 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×Q8×A4C211yesmaximal, characteristic, complemented
2C4⋊(C3⋊(C3⋊S4))C211yesmaximal, characteristic
3C3⋊(C3⋊(A4⋊C8))C211yesmaximal, characteristic, complemented
4#10593651—312nomaximal
5#10592926—31nomaximal
6C2×C3⋊(C3⋊S4)—21no—
7C32×C4×A4—21no—
8C32×C4×A4C2211yescharacteristic
9#88862—41nomaximal
10C22×C32×Q8D311yescharacteristic, complemented
11C3×Q8×A4D3112yescomplemented
12C4⋊(C3⋊S4)—312no—
13C2×#3755—31no—
14C2×#3755—31no—
15#64139—31no—
16C3⋊(A4⋊C8)—312no—
17#63662—31no—
18#63662—31no—
19#63657—31no—
20C2×C32×A4D411yescharacteristic
21C3⋊(C3⋊S4)—41no—
22C33×Q8—41no—
23C4⋊(C3⋊(C3⋊D3))—41no—
24C3⋊(C3⋊(C3⋊C8))—41no—
25#5417—99no—
26#5227—94no—
27C2×C3⋊S4—612no—
28C2×C32×Q8—31no—
29C2×C32×Q8—31no—
30C2×C32×Q8—61no—
31C22×C32×C4—21no—
32C22×C32×C4C2×D311yescharacteristic
33C2×#638—61no—
34C2×C4⋊(C3⋊D3)—31no—
35C3×C4×A4—212no—
36C3×C4×A4C2×D3112yes—
37#3755—62no—
38#3755—61no—
39#3755—61no—
40#3755—1212no—
41#3752—61no—
42C4⋊(C3⋊Dic3)—31no—
43C2×C3⋊(C3⋊C8)—31no—
44C2×C3⋊(C3⋊C8)—31no—
45C2×C3⋊(C3⋊D3)—81no—
46C32×A4SD1611yescharacteristic, complemented
47C33×C4—41no—
48C33×C4—81no—
49C22×C3×Q8C3⋊D314yescomplemented
50Q8×A4C3⋊D319yescomplemented
51C4⋊S4—99no—
52C2×#215—94no—
53C2×#215—94no—
54#930—94no—
55A4⋊C8—99no—
56#835—94no—
57#835—94no—
58#830—94no—
59C23×C32#6711yescharacteristic
60C22×C3⋊D3—61no—
61C2×C3×A4#67112yes—
62C3⋊S4—1212no—
63C32×Q8—32no—
64C32×Q8—31no—
65C32×Q8—412no—
66C32×Q8—61no—
67C32×Q8S411yescharacteristic, complemented
68C2×C32×C4—31no—
69C2×C32×C4—31no—
70C2×C32×C4—61no—
71C2×C32×C4—61no—
72#638—121no—
73#638—121no—
74C2×C3⋊Dic3—61no—
75C4⋊(C3⋊D3)—61no—
76C4⋊(C3⋊D3)—61no—
77C4⋊(C3⋊D3)—1212no—
78C3⋊(C3⋊C8)—61no—
79C3⋊(C3⋊C8)—61no—
80C3⋊(C3⋊C8)—1212no—
81#448—271noSylow
82C2×C33—41no—
83C3⋊(C3⋊D3)—161no—
84C2×S4—189no—
85C2×C3×Q8—34no—
86C2×C3×Q8—34no—
87C2×C3×Q8—64no—
88C22×C3×C4—24no—
89C22×C3×C4C2×C3⋊D314yes—
90C2×#67—184no—
91C2×D12—94no—
92C4×A4—29no—
93C4×A4C2×C3⋊D319yes—
94#215—188no—
95#215—184no—
96#215—184no—
97#215—369no—
98(C2×D3)⋊C4—184no—
99C4⋊Dic3—94no—
100C2×C3⋊C8—94no—
101C2×C3⋊C8—94no—
102C22×C32—31no—
103C22×C32—31no—
104C22×C32#21511yescharacteristic, complemented
105C2×C3⋊D3—121no—
106C2×C3⋊D3—121no—
107C2×C3⋊D3—121no—
108C2×C3⋊D3—2412no—
109C3×A4#215112yescomplemented
110C32×C4—21no—
111C32×C4—31no—
112C32×C4—412no—
113C32×C4—61no—
114C32×C4—812no—
115C32×C4C2×S411yescharacteristic
116C3⋊Dic3—121no—
117C22×Q8C3⋊(C3⋊D3)11yescharacteristic, complemented
118C2×SD16—271no—
119C2×SD16—271no—
120#121—271no—
121#103—271no—
122#103—271no—
123C22⋊C8—271no—
124C33—41noSylow
125C23×C3#63814yes—
126C22×D3—184no—
127C2×A4#63819yes—
128S4—369no—
129C3×Q8—38no—
130C3×Q8—34no—
131C3×Q8—49no—
132C3×Q8—64no—
133C3×Q8C3⋊S414yescomplemented
134C2×C3×C4—34no—
135C2×C3×C4—34no—
136C2×C3×C4—64no—
137C2×C3×C4—64no—
138#67—364no—
139#67—364no—
140C2×Dic3—184no—
141D12—184no—
142D12—184no—
143D12—369no—
144C3⋊C8—184no—
145C3⋊C8—184no—
146C3⋊C8—369no—
147C2×C32—31no—
148C2×C32—31no—
149C2×C32—412no—
150C2×C32GL(2,ℤ/4)11yescharacteristic
151C3⋊D3—241no—
152C3⋊D3—4812no—
153C2×Q8—31no—
154C2×Q8—31no—
155C2×Q8—61no—
156C2×D4—271no—
157C2×D4—541no—
158C22×C4—21no—
159C22×C4C2×C3⋊(C3⋊D3)11yescharacteristic
160SD16—542no—
161SD16—541no—
162SD16—541no—
163C2×C8—271no—
164C2×C8—271no—
165C4⋊C4—271no—
166C22⋊C4—541no—
167C22×C3—34no—
168C22×C3—34no—
169C22×C3#375514yescomplemented
170C2×D3—364no—
171C2×D3—364no—
172C2×D3—364no—
173C2×D3—729no—
174A4#375519yescomplemented
175C3×C4—24no—
176C3×C4—34no—
177C3×C4—49no—
178C3×C4—64no—
179C3×C4—89no—
180C3×C4C2×C3⋊S414yes—
181Dic3—364no—
182C32—412no—
183C32#541711yescharacteristic, complemented
184C23—541no—
185C23#631211yescharacteristic
186Q8—32no—
187Q8—31no—
188Q8—61no—
189Q8C3⋊(C3⋊S4)11yescharacteristic, complemented
190D4—541no—
191D4—541no—
192D4—1081no—
193D4—1081no—
194C2×C4—31no—
195C2×C4—31no—
196C2×C4—61no—
197C2×C4—61no—
198C2×C4—541no—
199C8—541no—
200C8—541no—
201C2×C3—34no—
202C2×C3—34no—
203C2×C3—49no—
204C2×C3#6426914yes—
205D3—724no—
206D3—1449no—
207C22—31no—
208C22—31no—
209C22—1081no—
210C22—1081no—
211C22—1081no—
212C22#8886211yescharacteristic, complemented
213C4—21no—
214C4—31no—
215C4—61no—
216C4—1081no—
217C4C2×C3⋊(C3⋊S4)11yescharacteristic
218C3—49no—
219C3#1059365114yescomplemented
220C2—31no—
221C2—31no—
222C2—2161no—
223C2#1172447311yescharacteristic
subgroupG/H
centerC2#11724473
commutator, cosocleC32×C4×A4C22
FrattiniC4C2×C3⋊(C3⋊S4)
FittingC22×C32×Q8D3
socleC23×C32#67
prime radicalC4⋊(C3⋊(C3⋊S4))C2
hypercenterQ8C3⋊(C3⋊S4)
hypocenterC32×A4SD16
LCSG ▷ C32×C4×A4 ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. C32×Q8×A4C2semidirect
3. C3⋊(C3⋊(A4⋊C8))C2semidirect
10. C22×C32×Q8D3semidirect
11. C3×Q8×A4D3semidirect
46. C32×A4SD16semidirect
49. C22×C3×Q8C3⋊D3semidirect
50. Q8×A4C3⋊D3semidirect
67. C32×Q8S4semidirect
104. C22×C32#215semidirect
109. C3×A4#215semidirect
117. C22×Q8C3⋊(C3⋊D3)semidirect
133. C3×Q8C3⋊S4semidirect
169. C22×C3#3755semidirect
174. A4#3755semidirect
183. C32#5417semidirect
189. Q8C3⋊(C3⋊S4)semidirect
212. C22#88862semidirect
219. C3#10593651semidirect

Maximal subgroups

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

subgroupindex
C32×Q8×A42
C4⋊(C3⋊(C3⋊S4))2
C3⋊(C3⋊(A4⋊C8))2
#105936513
#105929263
#888624