Finite Group Databasebeta

← #13181403

Subgroups — #13181403

Order 1728.

1618 conjugacy classes of subgroups; only the 215 normal subgroups are shown.

normal subgroups only

This group has 1618 conjugacy classes of subgroups — too many to draw. Only its 215 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))C213yesmaximal, complemented
3C4×C3⋊(C3⋊S4)C213yesmaximal, complemented
4#10596582—312nomaximal
5#10596288—31nomaximal
6C2×C3⋊(C3⋊S4)C2213yes—
7#89080—41nomaximal
8C32×C4×A4C2213yes—
9C3⋊(C3⋊SL(2,ℤ/4))C2211yescharacteristic
10C22×C32×Q8D311yescharacteristic, complemented
11C2×#3832—31no—
12C3×Q8×A4D3112yescomplemented
13C4⋊(C3⋊S4)—336no—
14C4×C3⋊S4—336no—
15#64156—33no—
16#64155—33no—
17Q8×C3⋊Dic3—31no—
18#64139—33no—
19C4×#638—33no—
20C2×C32×A4C2311yescharacteristic
21C3⋊(C3⋊S4)—23no—
22C33×Q8—41no—
23C4⋊(C3⋊(C3⋊D3))—43no—
24C4×C3⋊(C3⋊D3)—43no—
25#5919—99no—
26#5817—94no—
27C2×C3⋊S4—336no—
28C2×C32×Q8—31no—
29C2×C32×Q8—33no—
30C22×C32×C4C2×D313yes—
31C2×#638—33no—
32#3832—63no—
33#3832—61no—
34#3832—1212no—
35C2×C4⋊(C3⋊D3)—33no—
36C2×C4×C3⋊D3—33no—
37C3×C4×A4C2×D3136yes—
38C3⋊SL(2,ℤ/4)—312no—
39#3757—31no—
40#3752—33no—
41#3752—36no—
42C4⋊(C3⋊Dic3)—33no—
43#3750—33no—
44C4×C3⋊Dic3—33no—
45C2×C3⋊(C3⋊D3)—43no—
46C32×A4#4111yescharacteristic, complemented
47C33×C4—43no—
48C3⋊(C3⋊Dic3)—41no—
49C22×C3×Q8C3⋊D314yescomplemented
50C2×#239—94no—
51Q8×A4C3⋊D319yescomplemented
52C4⋊S4—927no—
53C4×S4—927no—
54#947—912no—
55#946—912no—
56Q8×Dic3—94no—
57#930—912no—
58C4×#67—912no—
59C23×C32C22×D311yescharacteristic
60C22×C3⋊D3—33no—
61C2×C3×A4C22×D3112yes—
62C3⋊S4—636no—
63C32×Q8—33no—
64C32×Q8—412no—
65C32×Q8—61no—
66C32×Q8S411yescharacteristic, complemented
67C2×C32×C4—33no—
68C2×C32×C4—33no—
69#638—63no—
70#638—63no—
71C2×C3⋊Dic3—31no—
72C2×C3⋊Dic3—33no—
73C2×C3⋊Dic3—31no—
74C4⋊(C3⋊D3)—63no—
75C4⋊(C3⋊D3)—63no—
76C4⋊(C3⋊D3)—1236no—
77C4×C3⋊D3—63no—
78C4×C3⋊D3—63no—
79C4×C3⋊D3—1236no—
80#548—271noSylow
81C2×C33—41no—
82C3⋊(C3⋊D3)—83no—
83C2×S4—927no—
84C2×C3×Q8—34no—
85C2×C3×Q8—312no—
86C22×C3×C4C2×C3⋊D3112yes—
87C2×#67—912no—
88#239—1812no—
89#239—184no—
90#239—369no—
91C2×D12—912no—
92C2×C4×D3—912no—
93C4×A4C2×C3⋊D3127yes—
94SL(2,ℤ/4)—99no—
95#217—94no—
96(C2×D3)⋊C4—912no—
97(C2×D3)⋊C4—924no—
98C4⋊Dic3—912no—
99Dic3⋊C4—912no—
100C4×Dic3—912no—
101C22×C32—31no—
102C22×C32—31no—
103C22×C32#23911yescharacteristic, complemented
104C2×C3⋊D3—63no—
105C2×C3⋊D3—63no—
106C2×C3⋊D3—63no—
107C2×C3⋊D3—1236no—
108C3×A4#239112yescomplemented
109C32×C4—33no—
110C32×C4—436no—
111C32×C4C2×S413yes—
112C3⋊Dic3—61no—
113C3⋊Dic3—63no—
114C3⋊Dic3—61no—
115C3⋊Dic3—1212no—
116C2×#41—271no—
117C22×Q8C3⋊(C3⋊D3)11yescharacteristic, complemented
118#124—273no—
119C22⋊Q8—273no—
120#121—273no—
121C4×Q8—271no—
122C4×D4—273no—
123C33—41noSylow
124C23×C3C22×C3⋊D314yes—
125C22×D3—912no—
126C2×A4C22×C3⋊D319yes—
127S4—1827no—
128C3×Q8—312no—
129C3×Q8—49no—
130C3×Q8—64no—
131C3×Q8C3⋊S414yescomplemented
132C2×C3×C4—312no—
133C2×C3×C4—312no—
134#67—1812no—
135#67—1812no—
136C2×Dic3—94no—
137C2×Dic3—912no—
138C2×Dic3—94no—
139D12—1812no—
140D12—1812no—
141D12—3627no—
142C4×D3—1812no—
143C4×D3—1812no—
144C4×D3—3627no—
145C2×C32—31no—
146C2×C32—31no—
147C2×C32—412no—
148C2×C32C22×S411yescharacteristic
149C3⋊D3—123no—
150C3⋊D3—2436no—
151#41—543no—
152#41—541no—
153C2×Q8—31no—
154C2×Q8—33no—
155C2×D4—273no—
156C2×D4—273no—
157C22×C4—273no—
158C22×C4C2×C3⋊(C3⋊D3)13yes—
159C4⋊C4—273no—
160C4⋊C4—273no—
161C22⋊C4—273no—
162C22⋊C4—276no—
163C22⋊C4—271no—
164C42—273no—
165C22×C3—34no—
166C22×C3—34no—
167C22×C3#383214yescomplemented
168C2×D3—1812no—
169C2×D3—1812no—
170C2×D3—1812no—
171C2×D3—3627no—
172A4#383219yescomplemented
173C3×C4—312no—
174C3×C4—427no—
175C3×C4C2×C3⋊S4112yes—
176Dic3—1812no—
177Dic3—184no—
178Dic3—184no—
179Dic3—369no—
180C32—412no—
181C32#591911yescharacteristic, complemented
182C23—273no—
183C23C22×C3⋊(C3⋊D3)11yescharacteristic
184Q8—33no—
185Q8—61no—
186Q8C3⋊(C3⋊S4)11yescharacteristic, complemented
187D4—543no—
188D4—543no—
189D4—543no—
190D4—543no—
191C2×C4—33no—
192C2×C4—33no—
193C2×C4—271no—
194C2×C4—273no—
195C2×C4—271no—
196C2×C4—543no—
197C2×C4—543no—
198C2×C3—34no—
199C2×C3—34no—
200C2×C3—49no—
201C2×C3C22×C3⋊S414yes—
202D3—3612no—
203D3—7227no—
204C22—31no—
205C22—31no—
206C22—543no—
207C22—543no—
208C22—543no—
209C22#8908011yescharacteristic, complemented
210C4—33no—
211C4—543no—
212C4—541no—
213C4—541no—
214C4C2×C3⋊(C3⋊S4)13yes—
215C3—49no—
216C3#1059658214yescomplemented
217C2—31no—
218C2—31no—
219C2—1083no—
220C2C22×C3⋊(C3⋊S4)11yescharacteristic
subgroupG/H
center, FrattiniC2C22×C3⋊(C3⋊S4)
commutator, cosocleC2×C32×A4C23
FittingC22×C32×Q8D3
socleC23×C32C22×D3
hypercenterQ8C3⋊(C3⋊S4)
hypocenterC32×A4#41
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C2 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. C32×Q8×A4C2semidirect
2. C4⋊(C3⋊(C3⋊S4))C2semidirect
3. C4×C3⋊(C3⋊S4)C2semidirect
10. C22×C32×Q8D3semidirect
12. C3×Q8×A4D3semidirect
46. C32×A4#41semidirect
49. C22×C3×Q8C3⋊D3semidirect
51. Q8×A4C3⋊D3semidirect
66. C32×Q8S4semidirect
103. C22×C32#239semidirect
108. C3×A4#239semidirect
117. C22×Q8C3⋊(C3⋊D3)semidirect
131. C3×Q8C3⋊S4semidirect
167. C22×C3#3832semidirect
172. A4#3832semidirect
181. C32#5919semidirect
186. Q8C3⋊(C3⋊S4)semidirect
209. C22#89080semidirect
216. C3#10596582semidirect

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
C4×C3⋊(C3⋊S4)2
#105965823
#105962883
#890804