Finite Group Databasebeta

← #13143728

Subgroups — #13143728

Order 1728.

737 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 737 conjugacy classes of subgroups — too many to draw. Only its 167 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
2C2×#88540C211yesmaximal, characteristic
3H(3,𝔽3)×#95C211yesmaximal, characteristic
4#11720574C212yesmaximal
5#11720574C212yesmaximal
6C3×#4659—34nomaximal
7C22×C4×H(3,𝔽3)C2212yes—
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084C2211yescharacteristic
10C22×#1084C2212yes—
11C22×#1084C2211yescharacteristic
12C4⋊#1084—22no—
13#88540—22no—
14C2×C3×C4⋊Dic3—34no—
15C2×C3×Dic3⋊C4—34no—
16C32×#95D314yes—
17C3×C3⋊#95—38no—
18C3×C3⋊#95—38no—
19C23×H(3,𝔽3)C2311yescharacteristic
20C2×C4×H(3,𝔽3)—22no—
21C2×C4×H(3,𝔽3)—22no—
22C2×C4×H(3,𝔽3)—22no—
23C2×C4×H(3,𝔽3)—22no—
24C2×C4×H(3,𝔽3)D412yes—
25C2×#1084—21no—
26C2×#1084—21no—
27C2×#1084—21no—
28C2×#1084—22no—
29C2×#1084—22no—
30C2×#1084—24no—
31C2×#1084Q812yes—
32C3×#400—91no—
33#4659—34no—
34C22×C32×C4C2×D318yes—
35C22×C32×C4C2×D314yes—
36C22×C3×Dic3—34no—
37C22×C3×Dic3—34no—
38C22×C3×Dic3—38no—
39C3×C4⋊Dic3—68no—
40C3×Dic3⋊C4—68no—
41C22×H(3,𝔽3)C2×D411yescharacteristic
42C22×H(3,𝔽3)C2×Q811yescharacteristic
43C22×H(3,𝔽3)#4111yescharacteristic
44C22×H(3,𝔽3)#4112yes—
45C22×H(3,𝔽3)#4112yes—
46C4×H(3,𝔽3)—22no—
47C4×H(3,𝔽3)—42no—
48#1084—22no—
49#1084—41no—
50#1084—42no—
51C2×C3×C4⋊C4—91no—
52C2×C3×C4⋊C4—91no—
53C2×C4⋊Dic3—34no—
54C2×Dic3⋊C4—34no—
55C3×#95—34no—
56C3×#95—92no—
57C3×#95—92no—
58C3×#95C3⋊D311yescharacteristic
59C3⋊#95—38no—
60C3⋊#95—38no—
61C23×C32C22×D314yes—
62C2×C32×C4—28no—
63C2×C32×C4—28no—
64C2×C32×C4—28no—
65C2×C32×C4—28no—
66C2×C32×C4D1218yes—
67C2×C3×Dic3—38no—
68C2×C3×Dic3—64no—
69C2×C3×Dic3—64no—
70C2×C3×Dic3—64no—
71C2×C3×Dic3—68no—
72C2×C3×Dic3—616no—
73C2×C3×Dic3—68no—
74#400—91noSylow
75C2×H(3,𝔽3)C22⋊Q811yescharacteristic
76C2×H(3,𝔽3)#12311yescharacteristic
77C2×H(3,𝔽3)#12411yescharacteristic
78C2×H(3,𝔽3)#12512yes—
79C2×H(3,𝔽3)#12612yes—
80C22×C3×C4—34no—
81C22×C3×C4—38no—
82C22×C3×C4—91no—
83C22×C3×C4—92no—
84C22×C3×C4—91no—
85C22×C3×C4C2×C3⋊D312yes—
86C22×C3×C4C2×C3⋊D311yescharacteristic
87C22×Dic3—34no—
88C22×Dic3—34no—
89C22×Dic3—38no—
90C3×C4⋊C4—182no—
91C3×C4⋊C4—182no—
92C4⋊Dic3—68no—
93Dic3⋊C4—68no—
94C22×C32C2×D1214yes—
95C22×C32#23518yes—
96C22×C32#23714yes—
97C22×C32#23718yes—
98C22×C32D3×Q814yes—
99C32×C4—28no—
100C32×C4—48no—
101C3×Dic3—68no—
102C3×Dic3—128no—
103C3×Dic3—124no—
104C2×C4⋊C4—91no—
105C2×C4⋊C4—91no—
106#95—92no—
107#95—92no—
108#95#27211yescharacteristic
109H(3,𝔽3)#40011yescharacteristic, complemented, Sylow
110C23×C3—34no—
111C23×C3C22×C3⋊D311yescharacteristic
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4—22no—
115C2×C3×C4—22no—
116C2×C3×C4—38no—
117C2×C3×C4—68no—
118C2×C3×C4—68no—
119C2×C3×C4—68no—
120C2×C3×C4—68no—
121C2×C3×C4—92no—
122C2×C3×C4—181no—
123C2×C3×C4—181no—
124C2×C3×C4—182no—
125C2×C3×C4—181no—
126C2×C3×C4—184no—
127C2×C3×C4—182no—
128C2×C3×C4C4⋊(C3⋊D3)12yes—
129C2×Dic3—38no—
130C2×Dic3—64no—
131C2×Dic3—64no—
132C2×Dic3—64no—
133C2×Dic3—68no—
134C2×Dic3—616no—
135C2×Dic3—68no—
136C2×C32#87514yes—
137C2×C32#87918yes—
138C2×C32#88614yes—
139C2×C32#88918yes—
140C2×C32#89714yes—
141C22×C4—91no—
142C22×C4—92no—
143C22×C4—91no—
144C22×C4C2×#27212yes—
145C22×C4C2×#27211yescharacteristic
146C4⋊C4—182no—
147C4⋊C4—182no—
148C22×C3—34no—
149C22×C3—38no—
150C22×C3—34no—
151C22×C3—38no—
152C22×C3—34no—
153C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
154C22×C3#382812yes—
155C22×C3#383011yescharacteristic
156C22×C3#383012yes—
157C22×C3Q8×C3⋊D311yescharacteristic
158C3×C4—22no—
159C3×C4—42no—
160C3×C4—68no—
161C3×C4—128no—
162C3×C4—182no—
163C3×C4—362no—
164C3×C4—361no—
165Dic3—68no—
166Dic3—124no—
167Dic3—128no—
168C32#465914yescomplemented
169C23C22×#27211yescharacteristic
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—92no—
175C2×C4—181no—
176C2×C4—181no—
177C2×C4—181no—
178C2×C4—182no—
179C2×C4—184no—
180C2×C4—182no—
181C2×C4C4⋊#27212yes—
182C2×C3—38no—
183C2×C3—34no—
184C2×C3—38no—
185C2×C3—34no—
186C2×C3—34no—
187C2×C3#6408411yescharacteristic
188C2×C3#6408812yes—
189C2×C3#6409511yescharacteristic
190C2×C3#6409812yes—
191C2×C3#6410611yescharacteristic
192C22C2×C4⋊#27211yescharacteristic
193C22#8874012yes—
194C22#8874411yescharacteristic
195C22#8874412yes—
196C22Q8×#27211yescharacteristic
197C4—22no—
198C4—42no—
199C4—182no—
200C4—361no—
201C4—362no—
202C3—34no—
203C3#1059235811yescharacteristic
204C2#1172172111yescharacteristic
205C2#1172173512yes—
206C2#1172174211yescharacteristic
207C2#1172175712yes—
208C2#1172176511yescharacteristic
subgroupG/H
center, Frattini, socleC23×C3C22×C3⋊D3
commutator, prime radical, cosocleC23×H(3,𝔽3)C23
FittingH(3,𝔽3)×#95C2
hypercenterC3×#95C3⋊D3
hypocenterH(3,𝔽3)#400
LCSG ▷ C23×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
109. H(3,𝔽3)#400semidirect
168. C32#4659semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#10842
C2×#885402
H(3,𝔽3)×#952
#117205742
C3×#46593