Finite Group Databasebeta

← #13143809

Subgroups — #13143809

Order 1728.

605 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 605 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C42×#272C211yesmaximal, characteristic
2H(3,𝔽3)×#97C211yesmaximal, characteristic, complemented
3#11720553C212yesmaximal
4#11720547C212yesmaximal, complemented
5#11720520C211yesmaximal, characteristic, complemented
6C3×#4705—34nomaximal
7C2×C4×#272C411yescharacteristic
8C2×C4×#272C411yescharacteristic
9C2×C4×#272C2211yescharacteristic
10C2×C8×H(3,𝔽3)C2212yes—
11C42×H(3,𝔽3)C2211yescharacteristic
12C4×#1084C411yescharacteristic
13C4×#1084C411yescharacteristic
14C4×#1084C2211yescharacteristic
15C2×#6180C2212yes—
16C3×C42×D3—34no—
17C32×#97D314yescomplemented
18C3×(C2×D3)⋊C8—38no—
19C3×Dic3⋊C8—38no—
20C3×#803—34no—
21C22×#272C2×C411yescharacteristic
22C2×C4×H(3,𝔽3)C2×C411yescharacteristic
23C2×C4×H(3,𝔽3)C2×C411yescharacteristic
24C2×C4×H(3,𝔽3)C2311yescharacteristic
25C2×#1084C2×C411yescharacteristic
26C2×#1084C2×C411yescharacteristic
27C2×#1084C2×C411yescharacteristic
28C4×#272—22no—
29C4×#272—22no—
30C4×#272—22no—
31C8×H(3,𝔽3)—22no—
32#6180—22no—
33C3×#432—91no—
34#4705—34no—
35C2×C3×C4×D3—34no—
36C2×C3×C4×D3—34no—
37C2×C3×C4×D3—34no—
38C2×C32×C8C2×D318yes—
39C32×C42C2×D314yes—
40C3×C4×Dic3—34no—
41C3×C4×Dic3—34no—
42C3×C4×Dic3—34no—
43C2×C3×C3⋊C8—38no—
44C22×H(3,𝔽3)C22×C411yescharacteristic
45C2×#272—21no—
46C2×#272—21no—
47C2×#272M1612yes—
48C4×H(3,𝔽3)—21no—
49C4×H(3,𝔽3)—21no—
50C4×H(3,𝔽3)#4112yes—
51#1084—21no—
52#1084—21no—
53#1084M1612yes—
54C2×C3×C42—91no—
55C42×D3—34no—
56C3×C4⋊C8—92no—
57C3×C22⋊C8—92no—
58C3×#97—34no—
59C3×#97—91no—
60C3×#97C3⋊D311yescharacteristic
61(C2×D3)⋊C8—38no—
62Dic3⋊C8—38no—
63#803—34no—
64C22×C3×D3—34no—
65C2×C32×C4C4×D314yes—
66C2×C32×C4C4×D314yes—
67C2×C32×C4C22×D314yes—
68C2×C3×Dic3—34no—
69C2×C3×Dic3—34no—
70C2×C3×Dic3—34no—
71C3×C4×D3—68no—
72C3×C4×D3—68no—
73C3×C4×D3—68no—
74C32×C8—28no—
75C3×C3⋊C8—68no—
76#432—91noSylow
77C2×H(3,𝔽3)#11711yescharacteristic
78C2×H(3,𝔽3)C2×M1611yescharacteristic
79C2×H(3,𝔽3)C2×M1611yescharacteristic
80#272—22no—
81C22×C3×C4—91no—
82C22×C3×C4—91no—
83C22×C3×C4—91no—
84C2×C4×D3—34no—
85C2×C4×D3—34no—
86C2×C4×D3—34no—
87C2×C3×C8—38no—
88C2×C3×C8—92no—
89C2×C3×C8C2×C3⋊D312yes—
90C3×C42—34no—
91C3×C42—91no—
92C3×C42—91no—
93C3×C42—91no—
94C3×C42C2×C3⋊D311yescharacteristic
95C4×Dic3—34no—
96C4×Dic3—34no—
97C4×Dic3—34no—
98C2×C3⋊C8—38no—
99C22×C32C2×C4×D314yes—
100C2×C3×D3—38no—
101C2×C3×D3—64no—
102C2×C3×D3—64no—
103C32×C4—24no—
104C32×C4—24no—
105C32×C4#23518yes—
106C3×Dic3—38no—
107C3×Dic3—64no—
108C3×Dic3—64no—
109C2×C42—91no—
110C4⋊C8—92no—
111C22⋊C8—92no—
112#97—91no—
113#97#27211yescharacteristic, complemented
114H(3,𝔽3)#43211yescharacteristic, complemented, Sylow
115C23×C3—91no—
116C22×D3—34no—
117C2×C3×C4—34no—
118C2×C3×C4—34no—
119C2×C3×C4—34no—
120C2×C3×C4—91no—
121C2×C3×C4—91no—
122C2×C3×C4—91no—
123C2×C3×C4—182no—
124C2×C3×C4—182no—
125C2×C3×C4—182no—
126C2×C3×C4C4×C3⋊D311yescharacteristic
127C2×C3×C4C4×C3⋊D311yescharacteristic
128C2×C3×C4C22×C3⋊D311yescharacteristic
129C2×Dic3—34no—
130C2×Dic3—34no—
131C2×Dic3—34no—
132C4×D3—68no—
133C4×D3—68no—
134C4×D3—68no—
135C3×C8—22no—
136C3×C8—68no—
137C3×C8—182no—
138C3⋊C8—68no—
139C2×C32#87214yes—
140C2×C32D3×M1614yes—
141C2×C32D3×M1614yes—
142C3×D3—68no—
143C22×C4—91no—
144C22×C4—91no—
145C22×C4—91no—
146C2×C8—92no—
147C2×C8C2×#27212yes—
148C42—91no—
149C42—91no—
150C42—91no—
151C42C2×#27211yescharacteristic
152C22×C3—34no—
153C22×C3—92no—
154C22×C3—181no—
155C22×C3—181no—
156C22×C3C2×C4×C3⋊D311yescharacteristic
157C2×D3—38no—
158C2×D3—64no—
159C2×D3—64no—
160C3×C4—21no—
161C3×C4—21no—
162C3×C4—38no—
163C3×C4—64no—
164C3×C4—64no—
165C3×C4—92no—
166C3×C4—181no—
167C3×C4—181no—
168C3×C4#382812yes—
169Dic3—38no—
170Dic3—64no—
171Dic3—64no—
172C32#470514yescomplemented
173C23—91no—
174C2×C4—91no—
175C2×C4—91no—
176C2×C4—91no—
177C2×C4—182no—
178C2×C4—182no—
179C2×C4—182no—
180C2×C4C4×#27211yescharacteristic
181C2×C4C4×#27211yescharacteristic
182C2×C4C22×#27211yescharacteristic
183C8—22no—
184C8—182no—
185C2×C3—34no—
186C2×C3—34no—
187C2×C3—34no—
188C2×C3—182no—
189C2×C3#6408111yescharacteristic
190C2×C3M16×C3⋊D311yescharacteristic
191C2×C3M16×C3⋊D311yescharacteristic
192D3—68no—
193C22—92no—
194C22—181no—
195C22—181no—
196C22C2×C4×#27211yescharacteristic
197C4—21no—
198C4—21no—
199C4—92no—
200C4—181no—
201C4—181no—
202C4#8874012yes—
203C3—34no—
204C3#1059240411yescharacteristic
205C2—182no—
206C2#1172171811yescharacteristic
207C2M16×#27211yescharacteristic
208C2M16×#27211yescharacteristic
subgroupG/H
center, FrattiniC2×C3×C4C22×C3⋊D3
commutatorC22×H(3,𝔽3)C22×C4
FittingH(3,𝔽3)×#97C2
socleC22×C3C2×C4×C3⋊D3
prime radicalC22×#272C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC3×#97C3⋊D3
hypocenterH(3,𝔽3)#432
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3×C4 ◁ C3×#97

Complements

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

NH ≅ G/Ntype
2. H(3,𝔽3)×#97C2semidirect
4. #11720547C2semidirect
5. #11720520C2semidirect
17. C32×#97D3semidirect
113. #97#272semidirect
114. H(3,𝔽3)#432semidirect
172. C32#4705semidirect

Maximal subgroups

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

subgroupindex
C42×#2722
H(3,𝔽3)×#972
#117205532
#117205472
#117205202
C3×#47053