Finite Group Databasebeta

← #13144460

Subgroups — #13144460

Order 1728.

627 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 627 conjugacy classes of subgroups — too many to draw. Only its 123 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
1H(3,𝔽3)×#121C211yesmaximal, characteristic
2C2×C4⋊#1084C211yesmaximal, characteristic, complemented
3#11720578C212yesmaximal, complemented
4#11720573C211yesmaximal, characteristic, complemented
5#11720529C212yesmaximal, complemented
6C3×#5033—34nomaximal
7C2×D4×H(3,𝔽3)—21no—
8C2×D4×H(3,𝔽3)C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084—21no—
11C4⋊C4×H(3,𝔽3)C2211yescharacteristic
12C22⋊C4×H(3,𝔽3)—21no—
13C4⋊#1084—21no—
14C4⋊#1084C2212yes—
15C2×#6180C2212yes—
16C32×#121D314yes—
17C2×C3×C4⋊Dic3—34no—
18C3×#832—38no—
19C3×#830—34no—
20C3×#807—38no—
21C23×H(3,𝔽3)—21no—
22C23×H(3,𝔽3)D411yescharacteristic
23D4×H(3,𝔽3)—22no—
24D4×H(3,𝔽3)—42no—
25C2×C4×H(3,𝔽3)—22no—
26C2×C4×H(3,𝔽3)—21no—
27C2×C4×H(3,𝔽3)D411yescharacteristic
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29C2×#1084—22no—
30C2×#1084—42no—
31#6180—22no—
32C3×#480—91no—
33#5033—34no—
34C2×C32×D4—24no—
35C2×C32×D4C2×D314yes—
36C22×C32×C4C2×D314yes—
37C22×C3×Dic3—64no—
38C32×C4⋊C4C2×D314yes—
39C32×C22⋊C4—24no—
40C3×C4⋊Dic3—38no—
41C3×C4⋊Dic3—64no—
42C2×C3×C3⋊C8—38no—
43C22×H(3,𝔽3)—21no—
44C22×H(3,𝔽3)—21no—
45C22×H(3,𝔽3)—41no—
46C22×H(3,𝔽3)—41no—
47C22×H(3,𝔽3)—41no—
48C22×H(3,𝔽3)D812yes—
49C22×H(3,𝔽3)C2×D411yescharacteristic
50C4×H(3,𝔽3)—21no—
51C4×H(3,𝔽3)—41no—
52C4×H(3,𝔽3)#4112yes—
53#1084—42no—
54C3×#121—34no—
55C3×#121C3⋊D311yescharacteristic
56C2×C3×C4⋊C4—91no—
57C2×C4⋊Dic3—34no—
58C3×#107—92no—
59C3×#102—92no—
60C3×C22⋊C8—91no—
61#832—38no—
62#830—34no—
63#807—38no—
64C23×C32—24no—
65C23×C32#6714yes—
66C32×D4—28no—
67C32×D4—48no—
68C2×C32×C4—28no—
69C2×C32×C4—24no—
70C2×C32×C4#6714yes—
71C2×C32×C4C22×D314yes—
72C2×C3×Dic3—68no—
73C2×C3×Dic3—128no—
74C3×C3⋊C8—68no—
75#480—91noSylow
76C2×H(3,𝔽3)—22no—
77C2×H(3,𝔽3)—81no—
78C2×H(3,𝔽3)#12311yescharacteristic
79C2×H(3,𝔽3)C2×D811yescharacteristic
80C2×H(3,𝔽3)#13711yescharacteristic
81C2×C3×D4—21no—
82C2×C3×D4—34no—
83C2×C3×D4—64no—
84C2×C3×D4C2×C3⋊D311yescharacteristic
85C22×C3×C4—34no—
86C22×C3×C4—181no—
87C22×C3×C4C2×C3⋊D311yescharacteristic
88C22×Dic3—64no—
89C2×C3×C8—92no—
90C3×C4⋊C4—34no—
91C3×C4⋊C4—92no—
92C3×C4⋊C4—181no—
93C3×C4⋊C4C2×C3⋊D311yescharacteristic
94C3×C22⋊C4—21no—
95C3×C22⋊C4—64no—
96C4⋊Dic3—38no—
97C4⋊Dic3—64no—
98C2×C3⋊C8—38no—
99C22×C32—24no—
100C22×C32—24no—
101C22×C32—44no—
102C22×C32—44no—
103C22×C32—44no—
104C22×C32#21318yes—
105C22×C32C2×#6714yes—
106C32×C4—24no—
107C32×C4—44no—
108C32×C4#23718yes—
109C3×Dic3—128no—
110#121#27211yescharacteristic
111C2×C4⋊C4—91no—
112#107—92no—
113#102—92no—
114C22⋊C8—91no—
115H(3,𝔽3)#48011yescharacteristic, complemented, Sylow
116C23×C3—21no—
117C23×C3—34no—
118C23×C3—64no—
119C23×C3#63811yescharacteristic
120C3×D4—22no—
121C3×D4—42no—
122C3×D4—68no—
123C3×D4—128no—
124C2×C3×C4—22no—
125C2×C3×C4—21no—
126C2×C3×C4—34no—
127C2×C3×C4—34no—
128C2×C3×C4—68no—
129C2×C3×C4—64no—
130C2×C3×C4—182no—
131C2×C3×C4—362no—
132C2×C3×C4#63811yescharacteristic
133C2×C3×C4C22×C3⋊D311yescharacteristic
134C2×Dic3—68no—
135C2×Dic3—128no—
136C3×C8—182no—
137C3⋊C8—68no—
138C2×C32—28no—
139C2×C32—84no—
140C2×C32C2×#21314yes—
141C2×C32#93514yes—
142C2×C32#95114yes—
143C2×D4—21no—
144C2×D4C2×#27211yescharacteristic
145C22×C4—181no—
146C22×C4C2×#27211yescharacteristic
147C2×C8—92no—
148C4⋊C4—92no—
149C4⋊C4—181no—
150C4⋊C4C2×#27211yescharacteristic
151C22⋊C4—21no—
152C22×C3—21no—
153C22×C3—21no—
154C22×C3—34no—
155C22×C3—38no—
156C22×C3—41no—
157C22×C3—41no—
158C22×C3—41no—
159C22×C3—64no—
160C22×C3—64no—
161C22×C3—124no—
162C22×C3—124no—
163C22×C3—124no—
164C22×C3#375312yes—
165C22×C3C2×#63811yescharacteristic
166C3×C4—21no—
167C3×C4—38no—
168C3×C4—41no—
169C3×C4—64no—
170C3×C4—124no—
171C3×C4—362no—
172C3×C4#383012yes—
173Dic3—128no—
174C32#503314yescomplemented
175C23—21no—
176C23#623511yescharacteristic
177D4—22no—
178D4—42no—
179C2×C4—22no—
180C2×C4—21no—
181C2×C4—182no—
182C2×C4—362no—
183C2×C4#623511yescharacteristic
184C2×C4C22×#27211yescharacteristic
185C8—182no—
186C2×C3—22no—
187C2×C3—34no—
188C2×C3—34no—
189C2×C3—34no—
190C2×C3—68no—
191C2×C3—81no—
192C2×C3—244no—
193C2×C3C2×#375311yescharacteristic
194C2×C3#6414411yescharacteristic
195C2×C3#6416011yescharacteristic
196C22—21no—
197C22—21no—
198C22—41no—
199C22—41no—
200C22—41no—
201C22#8854512yes—
202C22C2×#623511yescharacteristic
203C4—21no—
204C4—41no—
205C4—362no—
206C4#8874412yes—
207C3—34no—
208C3#1059273211yescharacteristic
209C2—22no—
210C2—81no—
211C2C2×#8854511yescharacteristic
212C2#1172184511yescharacteristic
213C2#1172187211yescharacteristic
subgroupG/H
center, socleC22×C3C2×#638
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
Fitting, prime radicalH(3,𝔽3)×#121C2
hypercenterC3×#121C3⋊D3
hypocenterH(3,𝔽3)#480
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#121

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊#1084C2semidirect
3. #11720578C2semidirect
4. #11720573C2semidirect
5. #11720529C2semidirect
115. H(3,𝔽3)#480semidirect
174. C32#5033semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1212
C2×C4⋊#10842
#117205782
#117205732
#117205292
C3×#50333