Finite Group Databasebeta

← #13137386

Subgroups — #13137386

Order 1728.

541 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 541 conjugacy classes of subgroups — too many to draw. Only its 99 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#11721385C211yesmaximal, characteristic
2#64078⋊C3C211yesmaximal, characteristic, complemented
3C2×#88431C214yesmaximal, complemented
4C4×#6180C211yesmaximal, characteristic, complemented
5#10590373—31nomaximal
6#10590369—31nomaximal
7C2×C4⋊((C3⋊D3)⋊C3)—22no—
8C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
9C2×#634⋊C3C2212yes—
10C42×H(3,𝔽3)C2211yescharacteristic
11C2×#6180C2212yes—
12C4⋊((C3⋊Dic3)⋊C3)—22no—
13#88431—28no—
14#64083D311yescharacteristic
15#64078D311yescharacteristic, complemented
16C3×#874—31no—
17C3×#869—31no—
18C2×#3717—34no—
19C2×#3716—34no—
20C3×C4×C3⋊C8—31no—
21C3×C4×C3⋊C8—31no—
22C3×C4×C3⋊C8—61no—
23C22×(C3⋊D3)⋊C3—22no—
24C2×C4×H(3,𝔽3)D412yes—
25C2×C4×H(3,𝔽3)C2311yescharacteristic
26C2×(C3⋊Dic3)⋊C3—22no—
27C4⋊((C3⋊D3)⋊C3)—24no—
28C4⋊((C3⋊D3)⋊C3)—44no—
29#634⋊C3—24no—
30#6180D414yescomplemented
31#5085—91no—
32#5076—91no—
33#4693—91no—
34C2×C4⋊(C3⋊D3)—22no—
35C2×C4⋊(C3⋊D3)C2×D312yes—
36C2×#634C2×D312yes—
37C2×C3×D12—32no—
38C2×C3×D12—62no—
39C2×C3×Dic6—32no—
40C32×C42—21no—
41C32×C42C2×D311yescharacteristic
42C32×C42C2×D311yescharacteristic
43C4⋊(C3⋊Dic3)—22no—
44C3×C4⋊Dic3—62no—
45C2×C3×C3⋊C8—32no—
46C2×C3×C3⋊C8—32no—
47C2×C3×C3⋊C8—62no—
48#3717—68no—
49#3716—68no—
50C22×H(3,𝔽3)C2×D411yescharacteristic
51C2×(C3⋊D3)⋊C3—42no—
52C2×(C3⋊D3)⋊C3—44no—
53C4×H(3,𝔽3)SD1614yescomplemented
54C4×H(3,𝔽3)C2×D412yes—
55(C3⋊Dic3)⋊C3—42no—
56C3×C4⋊Q8—91no—
57C3×#127—91no—
58C2×#215—94no—
59C2×#214—94no—
60C2×#204—94no—
61#874—31no—
62#874—91no—
63#869—31no—
64#869—91no—
65C3×C4×C8—91no—
66C4×C3⋊C8—31no—
67C4×C3⋊C8—31no—
68C4×C3⋊C8—61no—
69C22×C3⋊D3—22no—
70C22×C3×D3—62no—
71C2×C32×C4—22no—
72C2×C32×C4—21no—
73C2×C32×C4#6712yes—
74C2×C32×C4#6712yes—
75C2×C32×C4C22×D311yescharacteristic
76C2×C32×C4C22×D311yescharacteristic
77C2×C3⋊Dic3—22no—
78C4⋊(C3⋊D3)—24no—
79C4⋊(C3⋊D3)—44no—
80#634—24no—
81C2×C3×Dic3—62no—
82C3×D12—64no—
83C3×D12—124no—
84C3×Dic6—64no—
85C3×C3⋊C8—34no—
86C3×C3⋊C8—34no—
87C3×C3⋊C8—64no—
88#492—271noSylow
89C2×H(3,𝔽3)#12711yescharacteristic
90C2×H(3,𝔽3)C2×SD1612yes—
91(C3⋊D3)⋊C3—82no—
92C2×C3×Q8—92no—
93C2×C3×D4—92no—
94C2×C3×D4—182no—
95C2×D12—32no—
96C2×D12—62no—
97C2×D12—92no—
98C2×D12—182no—
99C2×Dic6—32no—
100C2×Dic6—92no—
101C2×C3×C8—92no—
102C3×C4⋊C4—182no—
103C3×C42—31no—
104C3×C42—31no—
105C3×C42—61no—
106C3×C42D3211yescharacteristic
107#215—188no—
108#214—188no—
109C4⋊Dic3—62no—
110C4⋊Dic3—182no—
111C2×C3⋊C8—32no—
112C2×C3⋊C8—32no—
113C2×C3⋊C8—62no—
114#204—188no—
115C22×C32—21no—
116C22×C32C2×#6711yescharacteristic
117C22×C32C2×#6711yescharacteristic
118C2×C3⋊D3—44no—
119C2×C3⋊D3—42no—
120C2×C3×D3—122no—
121C2×C3×D3—124no—
122C32×C4—24no—
123C32×C4—22no—
124C32×C4#21414yescomplemented
125C32×C4#21514yescomplemented
126C32×C4D3×D412yes—
127C32×C4D3×D412yes—
128C3⋊Dic3—42no—
129C3×Dic3—122no—
130C2×SD16—274no—
131C4⋊Q8—91no—
132#127—91no—
133C4×C8—91no—
134H(3,𝔽3)#49211yescharacteristic, complemented, Sylow
135C23×C3—182no—
136C22×D3—62no—
137C22×D3—182no—
138C3×Q8—184no—
139C3×D4—184no—
140C3×D4—364no—
141C2×C3×C4—32no—
142C2×C3×C4—32no—
143C2×C3×C4—31no—
144C2×C3×C4—31no—
145C2×C3×C4—62no—
146C2×C3×C4—61no—
147C2×C3×C4—182no—
148C2×C3×C4#62512yes—
149C2×C3×C4C2×D3211yescharacteristic
150C2×Dic3—62no—
151C2×Dic3—182no—
152D12—64no—
153D12—124no—
154D12—184no—
155D12—364no—
156Dic6—64no—
157Dic6—184no—
158C3×C8—94no—
159C3⋊C8—34no—
160C3⋊C8—34no—
161C3⋊C8—64no—
162C2×C32—22no—
163C2×C32—21no—
164C2×C32C2×#21412yes—
165C2×C32#94011yescharacteristic
166C2×C32#94011yescharacteristic
167C2×C32C2×#21512yes—
168C3⋊D3—82no—
169C3×D3—242no—
170C2×Q8—92no—
171C2×D4—92no—
172C2×D4—182no—
173SD16—548no—
174C2×C8—92no—
175C4⋊C4—182no—
176C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
177C22×C3—31no—
178C22×C3—31no—
179C22×C3—61no—
180C22×C3—362no—
181C22×C3—364no—
182C22×C3C2×#62511yescharacteristic
183C2×D3—124no—
184C2×D3—122no—
185C2×D3—362no—
186C2×D3—364no—
187C3×C4—32no—
188C3×C4—32no—
189C3×C4—34no—
190C3×C4—34no—
191C3×C4—62no—
192C3×C4—64no—
193C3×C4—362no—
194C3×C4#371514yes—
195C3×C4#380212yes—
196Dic3—122no—
197Dic3—362no—
198C32—21no—
199C32#507611yescharacteristic, complemented
200C32#508511yescharacteristic, complemented
201C23—182no—
202Q8—184no—
203D4—184no—
204D4—364no—
205C2×C4—182no—
206C2×C4#619812yes—
207C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
208C8—94no—
209C2×C3—32no—
210C2×C3—32no—
211C2×C3—31no—
212C2×C3—31no—
213C2×C3—62no—
214C2×C3—61no—
215C2×C3—722no—
216C2×C3C2×#371512yes—
217C2×C3#6391411yescharacteristic
218D3—242no—
219D3—722no—
220C22—364no—
221C22—362no—
222C22C2×#619811yescharacteristic
223C4—362no—
224C4#8843114yescomplemented
225C4#8865512yes—
226C3—31no—
227C3—31no—
228C3—61no—
229C3#1059037511yescharacteristic
230C2—722no—
231C2C2×#8843112yes—
232C2#1172114411yescharacteristic
subgroupG/H
centerC22C2×#6198
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#625
prime radical#11721385C2
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#492
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #64078⋊C3C2semidirect
3. C2×#88431C2semidirect
4. C4×#6180C2semidirect
15. #64078D3semidirect
30. #6180D4semidirect
53. C4×H(3,𝔽3)SD16semidirect
124. C32×C4#214semidirect
125. C32×C4#215semidirect
134. H(3,𝔽3)#492semidirect
199. C32#5076semidirect
200. C32#5085semidirect
224. C4#88431semidirect

Maximal subgroups

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

subgroupindex
#117213852
#64078⋊C32
C2×#884312
C4×#61802
#105903733
#105903693