Finite Group Databasebeta

← #13144362

Subgroups — #13144362

Order 1728.

825 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 825 conjugacy classes of subgroups — too many to draw. Only its 183 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
3C2×#88540C212yesmaximal
4#11720574C212yesmaximal
5#11720574C211yesmaximal, characteristic
6C3×#4981—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—
14#88540—24no—
15C2×C32×C4⋊C4D314yes—
16C2×C3×C4⋊Dic3—34no—
17C2×C3×Dic3⋊C4—38no—
18C3×C3⋊#95—34no—
19C3×C3⋊#95—38no—
20C23×H(3,𝔽3)C2311yescharacteristic
21C2×C4×H(3,𝔽3)—21no—
22C2×C4×H(3,𝔽3)—21no—
23C2×C4×H(3,𝔽3)—22no—
24C2×C4×H(3,𝔽3)—21no—
25C2×C4×H(3,𝔽3)—22no—
26C2×C4×H(3,𝔽3)D412yes—
27C2×C4×H(3,𝔽3)D412yes—
28C2×#1084—22no—
29C2×#1084—22no—
30C2×#1084—22no—
31C2×#1084—22no—
32C2×#1084—22no—
33C2×#1084Q814yes—
34C3×#398—91no—
35#4981—34no—
36C22×C32×C4C2×D314yes—
37C22×C32×C4C2×D314yes—
38C22×C32×C4C2×D314yes—
39C22×C3×Dic3—38no—
40C22×C3×Dic3—38no—
41C32×C4⋊C4—28no—
42C3×C4⋊Dic3—68no—
43C3×Dic3⋊C4—616no—
44C22×H(3,𝔽3)C2×D411yescharacteristic
45C22×H(3,𝔽3)C2×D411yescharacteristic
46C22×H(3,𝔽3)C2×Q812yes—
47C22×H(3,𝔽3)#4111yescharacteristic
48C22×H(3,𝔽3)#4112yes—
49C4×H(3,𝔽3)—22no—
50C4×H(3,𝔽3)—22no—
51C4×H(3,𝔽3)—41no—
52#1084—24no—
53#1084—42no—
54C2×C3×C4⋊C4—34no—
55C2×C3×C4⋊C4—92no—
56C2×C3×C4⋊C4—91no—
57C2×C3×C4⋊C4C3⋊D311yescharacteristic
58C2×C4⋊Dic3—34no—
59C2×Dic3⋊C4—38no—
60C3×#95—91no—
61C3×#95—92no—
62C3⋊#95—34no—
63C3⋊#95—38no—
64C23×C32C22×D314yes—
65C2×C32×C4—24no—
66C2×C32×C4—28no—
67C2×C32×C4—24no—
68C2×C32×C4—28no—
69C2×C32×C4—24no—
70C2×C32×C4D1218yes—
71C2×C32×C4#6718yes—
72C2×C3×Dic3—316no—
73C2×C3×Dic3—68no—
74C2×C3×Dic3—68no—
75C2×C3×Dic3—68no—
76C2×C3×Dic3—68no—
77C2×C3×Dic3—68no—
78#398—91noSylow
79C2×H(3,𝔽3)#12111yescharacteristic
80C2×H(3,𝔽3)C22⋊Q812yes—
81C2×H(3,𝔽3)#12311yescharacteristic
82C2×H(3,𝔽3)#12512yes—
83C2×H(3,𝔽3)C4⋊Q811yescharacteristic
84C22×C3×C4—34no—
85C22×C3×C4—34no—
86C22×C3×C4—34no—
87C22×C3×C4—92no—
88C22×C3×C4—92no—
89C22×C3×C4C2×C3⋊D311yescharacteristic
90C22×C3×C4C2×C3⋊D311yescharacteristic
91C22×C3×C4C2×C3⋊D311yescharacteristic
92C22×Dic3—38no—
93C22×Dic3—38no—
94C3×C4⋊C4—22no—
95C3×C4⋊C4—68no—
96C3×C4⋊C4—184no—
97C3×C4⋊C4—182no—
98C4⋊Dic3—68no—
99Dic3⋊C4—616no—
100C22×C32C2×D1214yes—
101C22×C32#23514yes—
102C22×C32#23718yes—
103C22×C32D3×Q818yes—
104C22×C32C2×#6714yes—
105C32×C4—28no—
106C32×C4—28no—
107C32×C4—44no—
108C3×Dic3—616no—
109C3×Dic3—128no—
110C2×C4⋊C4—92no—
111C2×C4⋊C4—91no—
112C2×C4⋊C4#27211yescharacteristic
113#95—91no—
114#95—92no—
115H(3,𝔽3)#39811yescharacteristic, complemented, Sylow
116C23×C3—34no—
117C23×C3C22×C3⋊D311yescharacteristic
118C2×C3×C4—21no—
119C2×C3×C4—22no—
120C2×C3×C4—21no—
121C2×C3×C4—22no—
122C2×C3×C4—21no—
123C2×C3×C4—38no—
124C2×C3×C4—38no—
125C2×C3×C4—64no—
126C2×C3×C4—64no—
127C2×C3×C4—68no—
128C2×C3×C4—64no—
129C2×C3×C4—68no—
130C2×C3×C4—94no—
131C2×C3×C4—182no—
132C2×C3×C4—182no—
133C2×C3×C4—182no—
134C2×C3×C4—182no—
135C2×C3×C4—182no—
136C2×C3×C4C4⋊(C3⋊D3)12yes—
137C2×C3×C4#63812yes—
138C2×Dic3—316no—
139C2×Dic3—68no—
140C2×Dic3—68no—
141C2×Dic3—68no—
142C2×Dic3—68no—
143C2×Dic3—68no—
144C2×C32#88918yes—
145C2×C32#89718yes—
146C2×C32#93014yes—
147C2×C32#93514yes—
148C2×C32#94414yes—
149C22×C4—92no—
150C22×C4—92no—
151C22×C4C2×#27211yescharacteristic
152C22×C4C2×#27211yescharacteristic
153C22×C4C2×#27211yescharacteristic
154C4⋊C4—22no—
155C4⋊C4—184no—
156C4⋊C4—182no—
157C22×C3—34no—
158C22×C3—34no—
159C22×C3—38no—
160C22×C3—38no—
161C22×C3—34no—
162C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
163C22×C3#382811yescharacteristic
164C22×C3#383012yes—
165C22×C3Q8×C3⋊D312yes—
166C22×C3C2×#63811yescharacteristic
167C3×C4—22no—
168C3×C4—22no—
169C3×C4—41no—
170C3×C4—68no—
171C3×C4—68no—
172C3×C4—124no—
173C3×C4—184no—
174C3×C4—362no—
175Dic3—616no—
176Dic3—128no—
177C32#498114yescomplemented
178C23C22×#27211yescharacteristic
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—94no—
185C2×C4—182no—
186C2×C4—182no—
187C2×C4—182no—
188C2×C4—182no—
189C2×C4—182no—
190C2×C4C4⋊#27212yes—
191C2×C4#623512yes—
192C2×C3—38no—
193C2×C3—38no—
194C2×C3—34no—
195C2×C3—34no—
196C2×C3—34no—
197C2×C3#6409812yes—
198C2×C3#6410612yes—
199C2×C3#6413911yescharacteristic
200C2×C3#6414411yescharacteristic
201C2×C3#6415311yescharacteristic
202C22C2×C4⋊#27211yescharacteristic
203C22#8874011yescharacteristic
204C22#8874412yes—
205C22Q8×#27212yes—
206C22C2×#623511yescharacteristic
207C4—22no—
208C4—22no—
209C4—41no—
210C4—184no—
211C4—362no—
212C3—34no—
213C3#1059268011yescharacteristic
214C2#1172175712yes—
215C2#1172176512yes—
216C2#1172183011yescharacteristic
217C2#1172184511yescharacteristic
218C2#1172186111yescharacteristic
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)#398
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
115. H(3,𝔽3)#398semidirect
177. C32#4981semidirect

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
C2×#885402
#117205742
C3×#49813