Finite Group Databasebeta

← #13143715

Subgroups — #13143715

Order 1728.

759 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 759 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C2×C4×#1084C211yesmaximal, characteristic
2H(3,𝔽3)×#95C211yesmaximal, characteristic
3#11720574C211yesmaximal, characteristic
4#11720574C212yesmaximal
5#11720574C212yesmaximal
6C3×#4651—34nomaximal
7C22×C4×H(3,𝔽3)C2212yes—
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084C2212yes—
10C22×#1084C2211yescharacteristic
11C22×#1084C2211yescharacteristic
12C4×#1084C414yescomplemented
13C2×C3×C4×Dic3—34no—
14C32×#95D314yes—
15C3×C3⋊#95—38no—
16C3×C3⋊#95—38no—
17C3×C3⋊#95—34no—
18C23×H(3,𝔽3)C2311yescharacteristic
19C2×C4×H(3,𝔽3)—22no—
20C2×C4×H(3,𝔽3)—22no—
21C2×C4×H(3,𝔽3)—22no—
22C2×C4×H(3,𝔽3)—22no—
23C2×C4×H(3,𝔽3)C2×C412yes—
24C2×#1084—22no—
25C2×#1084—21no—
26C2×#1084—21no—
27C2×#1084—22no—
28C2×#1084—22no—
29C2×#1084—22no—
30C2×#1084C2×C412yes—
31C2×#1084C2×C412yes—
32C3×#383—91no—
33#4651—34no—
34C22×C32×C4C2×D318yes—
35C22×C32×C4C2×D314yes—
36C22×C3×Dic3—34no—
37C22×C3×Dic3—34no—
38C22×C3×Dic3—38no—
39C3×C4×Dic3—316no—
40C22×H(3,𝔽3)C22×C411yescharacteristic
41C22×H(3,𝔽3)#4112yes—
42C22×H(3,𝔽3)#4112yes—
43C22×H(3,𝔽3)#4111yescharacteristic
44C22×H(3,𝔽3)#4111yescharacteristic
45C4×H(3,𝔽3)—22no—
46C4×H(3,𝔽3)—42no—
47#1084—22no—
48#1084—22no—
49#1084—42no—
50C2×C3×C42—91no—
51C2×C4×Dic3—34no—
52C3×#95—34no—
53C3×#95—92no—
54C3×#95—92no—
55C3×#95—91no—
56C3×#95C3⋊D311yescharacteristic
57C3⋊#95—38no—
58C3⋊#95—38no—
59C3⋊#95—34no—
60C23×C32C22×D314yes—
61C2×C32×C4—28no—
62C2×C32×C4—28no—
63C2×C32×C4—28no—
64C2×C32×C4—28no—
65C2×C32×C4C4×D318yes—
66C2×C3×Dic3—38no—
67C2×C3×Dic3—38no—
68C2×C3×Dic3—68no—
69C2×C3×Dic3—64no—
70C2×C3×Dic3—64no—
71C2×C3×Dic3—68no—
72C2×C3×Dic3—68no—
73C2×C3×Dic3—68no—
74#383—91noSylow
75C2×H(3,𝔽3)#11711yescharacteristic
76C2×H(3,𝔽3)#11711yescharacteristic
77C2×H(3,𝔽3)#11711yescharacteristic
78C2×H(3,𝔽3)#12612yes—
79C2×H(3,𝔽3)#12612yes—
80C22×C3×C4—34no—
81C22×C3×C4—38no—
82C22×C3×C4—91no—
83C22×C3×C4—91no—
84C22×C3×C4—92no—
85C22×C3×C4C2×C3⋊D312yes—
86C22×C3×C4C2×C3⋊D311yescharacteristic
87C22×Dic3—34no—
88C22×Dic3—34no—
89C22×Dic3—38no—
90C3×C42—94no—
91C4×Dic3—316no—
92C22×C32C2×C4×D314yes—
93C22×C32#23518yes—
94C22×C32#23718yes—
95C22×C32#23714yes—
96C22×C32#23914yes—
97C32×C4—28no—
98C32×C4—48no—
99C3×Dic3—68no—
100C3×Dic3—68no—
101C3×Dic3—128no—
102C2×C42—91no—
103#95—92no—
104#95—92no—
105#95—91no—
106#95#27211yescharacteristic
107H(3,𝔽3)#38311yescharacteristic, complemented, Sylow
108C23×C3—34no—
109C23×C3C22×C3⋊D311yescharacteristic
110C2×C3×C4—22no—
111C2×C3×C4—22no—
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4—38no—
115C2×C3×C4—68no—
116C2×C3×C4—68no—
117C2×C3×C4—68no—
118C2×C3×C4—68no—
119C2×C3×C4—92no—
120C2×C3×C4—92no—
121C2×C3×C4—182no—
122C2×C3×C4—181no—
123C2×C3×C4—181no—
124C2×C3×C4—182no—
125C2×C3×C4—182no—
126C2×C3×C4—182no—
127C2×C3×C4C4×C3⋊D312yes—
128C2×Dic3—38no—
129C2×Dic3—38no—
130C2×Dic3—68no—
131C2×Dic3—64no—
132C2×Dic3—64no—
133C2×Dic3—68no—
134C2×Dic3—68no—
135C2×Dic3—68no—
136C2×C32#87214yes—
137C2×C32#87714yes—
138C2×C32#87918yes—
139C2×C32#89214yes—
140C2×C32#89818yes—
141C22×C4—91no—
142C22×C4—91no—
143C22×C4—92no—
144C22×C4C2×#27212yes—
145C22×C4C2×#27211yescharacteristic
146C42—94no—
147C22×C3—34no—
148C22×C3—38no—
149C22×C3—38no—
150C22×C3—34no—
151C22×C3—34no—
152C22×C3C2×C4×C3⋊D311yescharacteristic
153C22×C3#382812yes—
154C22×C3#383012yes—
155C22×C3#383011yescharacteristic
156C22×C3#383211yescharacteristic
157C3×C4—22no—
158C3×C4—42no—
159C3×C4—68no—
160C3×C4—128no—
161C3×C4—182no—
162C3×C4—182no—
163C3×C4—362no—
164Dic3—68no—
165Dic3—68no—
166Dic3—128no—
167C32#465114yescomplemented
168C23C22×#27211yescharacteristic
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—92no—
174C2×C4—92no—
175C2×C4—182no—
176C2×C4—181no—
177C2×C4—181no—
178C2×C4—182no—
179C2×C4—182no—
180C2×C4—182no—
181C2×C4C4×#27212yes—
182C2×C3—34no—
183C2×C3—38no—
184C2×C3—34no—
185C2×C3—38no—
186C2×C3—34no—
187C2×C3#6408111yescharacteristic
188C2×C3#6408611yescharacteristic
189C2×C3#6408812yes—
190C2×C3#6410111yescharacteristic
191C2×C3#6410712yes—
192C22C2×C4×#27211yescharacteristic
193C22#8874012yes—
194C22#8874412yes—
195C22#8874411yescharacteristic
196C22#8874811yescharacteristic
197C4—22no—
198C4—42no—
199C4—182no—
200C4—182no—
201C4—362no—
202C3—34no—
203C3#1059235011yescharacteristic
204C2#1172171811yescharacteristic
205C2#1172173311yescharacteristic
206C2#1172173512yes—
207C2#1172176011yescharacteristic
208C2#1172176612yes—
subgroupG/H
center, Frattini, socleC23×C3C22×C3⋊D3
commutatorC22×H(3,𝔽3)C22×C4
FittingH(3,𝔽3)×#95C2
prime radical, cosocleC23×H(3,𝔽3)C23
hypercenterC3×#95C3⋊D3
hypocenterH(3,𝔽3)#383
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C3×#95

Complements

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

NH ≅ G/Ntype
12. C4×#1084C4semidirect
107. H(3,𝔽3)#383semidirect
167. C32#4651semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#10842
H(3,𝔽3)×#952
#117205742
C3×#46513