Finite Group Databasebeta

← #13144361

Subgroups — #13144361

Order 1728.

737 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 737 conjugacy classes of subgroups — too many to draw. Only its 179 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⋊C4×H(3,𝔽3)C211yesmaximal, characteristic
2C2×C4⋊#1084C211yesmaximal, characteristic
3#11720574C212yesmaximal
4#11720574C212yesmaximal
5#11720574C211yesmaximal, characteristic
6C3×#4986—34nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084C2212yes—
11C22×#1084C2212yes—
12C4⋊C4×H(3,𝔽3)—22no—
13C4⋊#1084—22no—
14C2×C32×C4⋊C4D314yes—
15C2×C3×C4⋊Dic3—34no—
16C3×C3⋊#95—38no—
17C3×C3⋊#95—38no—
18C3×C3⋊#95—34no—
19C23×H(3,𝔽3)C2311yescharacteristic
20C2×C4×H(3,𝔽3)—21no—
21C2×C4×H(3,𝔽3)—21no—
22C2×C4×H(3,𝔽3)—22no—
23C2×C4×H(3,𝔽3)—21no—
24C2×C4×H(3,𝔽3)—22no—
25C2×C4×H(3,𝔽3)D412yes—
26C2×C4×H(3,𝔽3)Q812yes—
27C2×#1084—22no—
28C2×#1084—24no—
29C2×#1084—22no—
30C2×#1084—22no—
31C2×#1084—22no—
32C3×#400—91no—
33#4986—34no—
34C22×C32×C4C2×D314yes—
35C22×C32×C4C2×D314yes—
36C22×C32×C4C2×D314yes—
37C22×C3×Dic3—38no—
38C22×C3×Dic3—38no—
39C32×C4⋊C4—28no—
40C3×C4⋊Dic3—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)—22no—
48C4×H(3,𝔽3)—41no—
49#1084—42no—
50#1084—42no—
51C2×C3×C4⋊C4—34no—
52C2×C3×C4⋊C4—91no—
53C2×C3×C4⋊C4C3⋊D311yescharacteristic
54C2×C4⋊Dic3—34no—
55C3×#95—91no—
56C3×#95—92no—
57C3×#95—92no—
58C3⋊#95—34no—
59C3⋊#95—38no—
60C3⋊#95—38no—
61C23×C32C22×D314yes—
62C2×C32×C4—24no—
63C2×C32×C4—28no—
64C2×C32×C4—24no—
65C2×C32×C4—28no—
66C2×C32×C4—24no—
67C2×C32×C4Dic618yes—
68C2×C32×C4#6718yes—
69C2×C3×Dic3—68no—
70C2×C3×Dic3—68no—
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—34no—
82C22×C3×C4—34no—
83C22×C3×C4—92no—
84C22×C3×C4—92no—
85C22×C3×C4C2×C3⋊D311yescharacteristic
86C22×C3×C4C2×C3⋊D311yescharacteristic
87C22×C3×C4C2×C3⋊D311yescharacteristic
88C22×Dic3—38no—
89C22×Dic3—38no—
90C3×C4⋊C4—22no—
91C3×C4⋊C4—68no—
92C3×C4⋊C4—182no—
93C4⋊Dic3—68no—
94C22×C32C2×Dic614yes—
95C22×C32#23514yes—
96C22×C32#23718yes—
97C22×C32#23918yes—
98C22×C32C2×#6714yes—
99C32×C4—28no—
100C32×C4—28no—
101C32×C4—44no—
102C3×Dic3—128no—
103C3×Dic3—128no—
104C2×C4⋊C4—91no—
105C2×C4⋊C4#27211yescharacteristic
106#95—91no—
107#95—92no—
108#95—92no—
109H(3,𝔽3)#40011yescharacteristic, complemented, Sylow
110C23×C3—34no—
111C23×C3C22×C3⋊D311yescharacteristic
112C2×C3×C4—21no—
113C2×C3×C4—22no—
114C2×C3×C4—21no—
115C2×C3×C4—22no—
116C2×C3×C4—21no—
117C2×C3×C4—38no—
118C2×C3×C4—38no—
119C2×C3×C4—64no—
120C2×C3×C4—64no—
121C2×C3×C4—68no—
122C2×C3×C4—64no—
123C2×C3×C4—68no—
124C2×C3×C4—182no—
125C2×C3×C4—182no—
126C2×C3×C4—182no—
127C2×C3×C4—184no—
128C2×C3×C4—182no—
129C2×C3×C4#63412yes—
130C2×C3×C4#63812yes—
131C2×Dic3—68no—
132C2×Dic3—68no—
133C2×Dic3—68no—
134C2×Dic3—68no—
135C2×Dic3—616no—
136C2×C32#89018yes—
137C2×C32#89818yes—
138C2×C32#92414yes—
139C2×C32#93514yes—
140C2×C32#94714yes—
141C22×C4—92no—
142C22×C4—92no—
143C22×C4C2×#27211yescharacteristic
144C22×C4C2×#27211yescharacteristic
145C22×C4C2×#27211yescharacteristic
146C4⋊C4—22no—
147C4⋊C4—182no—
148C22×C3—34no—
149C22×C3—34no—
150C22×C3—38no—
151C22×C3—38no—
152C22×C3—34no—
153C22×C3C2×#63411yescharacteristic
154C22×C3#382811yescharacteristic
155C22×C3#383012yes—
156C22×C3#383212yes—
157C22×C3C2×#63811yescharacteristic
158C3×C4—22no—
159C3×C4—22no—
160C3×C4—41no—
161C3×C4—68no—
162C3×C4—68no—
163C3×C4—124no—
164C3×C4—362no—
165C3×C4—362no—
166Dic3—128no—
167Dic3—128no—
168C32#498614yescomplemented
169C23C22×#27211yescharacteristic
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—182no—
176C2×C4—182no—
177C2×C4—182no—
178C2×C4—184no—
179C2×C4—182no—
180C2×C4#622912yes—
181C2×C4#623512yes—
182C2×C3—38no—
183C2×C3—38no—
184C2×C3—34no—
185C2×C3—34no—
186C2×C3—34no—
187C2×C3#6409912yes—
188C2×C3#6410712yes—
189C2×C3#6413311yescharacteristic
190C2×C3#6414411yescharacteristic
191C2×C3#6415611yescharacteristic
192C22C2×#622911yescharacteristic
193C22#8874011yescharacteristic
194C22#8874412yes—
195C22#8874812yes—
196C22C2×#623511yescharacteristic
197C4—22no—
198C4—22no—
199C4—41no—
200C4—362no—
201C4—362no—
202C3—34no—
203C3#1059268511yescharacteristic
204C2#1172175812yes—
205C2#1172176612yes—
206C2#1172182411yescharacteristic
207C2#1172184511yescharacteristic
208C2#1172186411yescharacteristic
subgroupG/H
center, Frattini, socleC23×C3C22×C3⋊D3
commutator, prime radical, cosocleC23×H(3,𝔽3)C23
FittingC2×C4⋊C4×H(3,𝔽3)C2
hypercenterC2×C3×C4⋊C4C3⋊D3
hypocenterH(3,𝔽3)#400
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
109. H(3,𝔽3)#400semidirect
168. C32#4986semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×H(3,𝔽3)2
C2×C4⋊#10842
#117205742
C3×#49863