Finite Group Databasebeta

← #13158827

Subgroups — #13158827

Order 1728.

752 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 164 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
1C3×#64084C211yesmaximal, characteristic
2C32×#870C211yesmaximal, characteristic, complemented
3C4×C3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
4#11722039C212yesmaximal, complemented
5#11722033C212yesmaximal
6#10592767—31nomaximal
7#10590278—34nomaximal
8C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
9C2×C3×#634C2211yescharacteristic
10C33×C42C2211yescharacteristic
11C2×C3⋊(C3⋊(C3⋊C8))C2212yes—
12C3×#3752—22no—
13C32×C4⋊Dic3C2212yes—
14C32×Dic3⋊C4—22no—
15C32×#125—31no—
16#64084D311yescharacteristic
17C3×#875—34no—
18C3×#870D314yescomplemented
19#63637—32no—
20#63636—32no—
21C4×C3⋊(C3⋊C8)—31no—
22C4×C3⋊(C3⋊C8)—34no—
23C4×C3⋊(C3⋊C8)—64no—
24#63565—38no—
25#63562—38no—
26C22×C3×C3⋊D3—21no—
27C2×C33×C4D412yes—
28C2×C33×C4C2311yescharacteristic
29C2×C3×C3⋊Dic3—21no—
30C3×C4⋊(C3⋊D3)—22no—
31C3×#634—22no—
32C2×C32×Dic3—22no—
33C3⋊(C3⋊(C3⋊C8))—22no—
34#5068—94no—
35#5056—91no—
36C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
37C2×#634C2×D311yescharacteristic
38C2×C3×D12—34no—
39C2×C3×Dic6—34no—
40C32×C4⋊C4—32no—
41C32×C4⋊C4—62no—
42C32×C42—24no—
43C32×C42C2×D314yes—
44C32×C42C2×D311yescharacteristic
45#3752—22no—
46C2×C3⋊(C3⋊C8)—32no—
47C2×C3⋊(C3⋊C8)—38no—
48C2×C3⋊(C3⋊C8)—68no—
49C3×(C2×D3)⋊C4—68no—
50C3×C4⋊Dic3C2×D318yes—
51C3×Dic3⋊C4—28no—
52C22×C33C2×D411yescharacteristic
53C2×C3×C3⋊D3—41no—
54C2×C3×C3⋊D3—42no—
55C33×C4—22no—
56C33×C4#4112yes—
57C3×C3⋊Dic3—41no—
58C32×Dic3—42no—
59C3×#125—34no—
60C3×#124—91no—
61#875—34no—
62#870C3⋊D311yescharacteristic, complemented
63#835—92no—
64#832—92no—
65#810—98no—
66#809—98no—
67C4×C3⋊C8—94no—
68C4×C3⋊C8—91no—
69C4×C3⋊C8—184no—
70C22×C3⋊D3—21no—
71C22×C3×D3—64no—
72C2×C32×C4—28no—
73C2×C32×C4—24no—
74C2×C32×C4—62no—
75C2×C32×C4#6718yes—
76C2×C32×C4#6712yes—
77C2×C32×C4C22×D314yes—
78C2×C32×C4C22×D311yescharacteristic
79C2×C3⋊Dic3—21no—
80C4⋊(C3⋊D3)—22no—
81#634—22no—
82C2×C3×Dic3—28no—
83C2×C3×Dic3—64no—
84C3×D12—68no—
85C3×Dic6—68no—
86C3⋊(C3⋊C8)—616no—
87C3⋊(C3⋊C8)—62no—
88C3⋊(C3⋊C8)—68no—
89#488—271noSylow
90C2×C33#12411yescharacteristic
91C2×C33#13512yes—
92C3×C3⋊D3—81no—
93C2×C3×Q8—91no—
94C2×C3×D4—91no—
95C2×D12—34no—
96C2×Dic6—34no—
97C3×C4⋊C4—38no—
98C3×C4⋊C4—68no—
99C3×C22⋊C4—182no—
100C3×C42—24no—
101C3×C42D3214yes—
102C3×C42C2×C3⋊D311yescharacteristic
103(C2×D3)⋊C4—68no—
104C4⋊Dic3C2×C3⋊D312yes—
105Dic3⋊C4—22no—
106C2×C3⋊C8—98no—
107C2×C3⋊C8—92no—
108C2×C3⋊C8—188no—
109C22×C32—24no—
110C22×C32C2×#6714yes—
111C22×C32C2×#6711yescharacteristic
112C2×C3⋊D3—41no—
113C2×C3⋊D3—42no—
114C2×C3×D3—124no—
115C2×C3×D3—128no—
116C32×C4—216no—
117C32×C4—28no—
118C32×C4—22no—
119C32×C4—28no—
120C32×C4—122no—
121C32×C4#23712yes—
122C32×C4#23918yes—
123C3⋊Dic3—41no—
124C3×Dic3—48no—
125C3×Dic3—124no—
126#125—31no—
127#124—91no—
128#103—272no—
129#102—272no—
130C4×C8—271no—
131C33#48811yescharacteristic, complemented, Sylow
132C23×C3—181no—
133C22×D3—64no—
134C3×Q8—182no—
135C3×D4—182no—
136C2×C3×C4—28no—
137C2×C3×C4—24no—
138C2×C3×C4—68no—
139C2×C3×C4—181no—
140C2×C3×C4#62518yes—
141C2×C3×C4#63812yes—
142C2×C3×C4C2×D3214yes—
143C2×C3×C4C22×C3⋊D311yescharacteristic
144C2×Dic3—22no—
145C2×Dic3—64no—
146D12—68no—
147Dic6—68no—
148C3⋊C8—1816no—
149C3⋊C8—188no—
150C3⋊C8—182no—
151C2×C32—28no—
152C2×C32—24no—
153C2×C32#93611yescharacteristic
154C2×C32#94714yes—
155C2×C32#95018yes—
156C2×C32#95012yes—
157C3⋊D3—81no—
158C3×D3—244no—
159C2×Q8—91no—
160C2×D4—91no—
161C2×C8—272no—
162C4⋊C4—32no—
163C4⋊C4—62no—
164C22⋊C4—182no—
165C42D3×C3⋊D311yescharacteristic
166C22×C3—24no—
167C22×C3—361no—
168C22×C3—362no—
169C22×C3C2×#62514yes—
170C22×C3C2×#63811yescharacteristic
171C2×D3—128no—
172C2×D3—124no—
173C3×C4—216no—
174C3×C4—28no—
175C3×C4—22no—
176C3×C4—28no—
177C3×C4—128no—
178C3×C4—361no—
179C3×C4#379618yes—
180C3×C4#383212yes—
181Dic3—42no—
182Dic3—124no—
183C32—24no—
184C32#505611yescharacteristic, complemented
185C32#506814yescomplemented
186C23—181no—
187Q8—182no—
188D4—182no—
189C2×C4—62no—
190C2×C4—181no—
191C2×C4#629012yes—
192C2×C4C2×D3×C3⋊D311yescharacteristic
193C8—542no—
194C2×C3—28no—
195C2×C3—24no—
196C2×C3—721no—
197C2×C3#6382218yes—
198C2×C3#6386314yes—
199C2×C3#6415611yescharacteristic
200C2×C3#6415912yes—
201D3—244no—
202C22—362no—
203C22—361no—
204C22C2×#629011yescharacteristic
205C4—22no—
206C4—122no—
207C4—361no—
208C4#8901712yes—
209C3—24no—
210C3#1059027814yescomplemented
211C3#1059276711yescharacteristic, complemented
212C2—721no—
213C2#1172336312yes—
214C2#1172343611yescharacteristic
subgroupG/H
centerC22C2×#6290
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC42D3×C3⋊D3
hypocenterC33#488
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C32×#870C2semidirect
3. C4×C3⋊(C3⋊(C3⋊C8))C2semidirect
4. #11722039C2semidirect
18. C3×#870D3semidirect
62. #870C3⋊D3semidirect
131. C33#488semidirect
184. C32#5056semidirect
185. C32#5068semidirect
210. C3#10590278semidirect
211. C3#10592767semidirect

Maximal subgroups

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

subgroupindex
C3×#640842
C32×#8702
C4×C3⋊(C3⋊(C3⋊C8))2
#117220392
#117220332
#105927673
#105902783