Finite Group Databasebeta

← #13144366

Subgroups — #13144366

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
1C2×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#11721830C211yesmaximal, characteristic
3#11720573C211yesmaximal, characteristic, complemented
4#11720531C212yesmaximal, complemented
5#11720530C212yesmaximal, complemented
6C3×#4968—34nomaximal
7C22×C4×H(3,𝔽3)—21no—
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C2×#6235—21no—
10C2×C4⋊#272C2211yescharacteristic
11C4⋊C4×H(3,𝔽3)—21no—
12C4⋊C4×H(3,𝔽3)C2212yes—
13#88542—21no—
14C4⋊#1084C2211yescharacteristic
15C2×#6180C2212yes—
16C2×C32×C4⋊C4D314yescomplemented
17C3×#930—34no—
18C3×#830—34no—
19C3×#809—38no—
20C3×#808—38no—
21C23×H(3,𝔽3)D411yescharacteristic
22C22×#272—21no—
23C2×C4×H(3,𝔽3)—22no—
24C2×C4×H(3,𝔽3)—22no—
25C2×C4×H(3,𝔽3)—42no—
26C2×C4×H(3,𝔽3)D411yescharacteristic
27C2×C4×H(3,𝔽3)C2311yescharacteristic
28#6235—42no—
29C2×#1084—21no—
30C4⋊#272—22no—
31#6180—22no—
32C3×#481—91no—
33#4968—34no—
34C22×C32×C4—24no—
35C22×C32×C4C2×D314yes—
36C2×C3×#67—64no—
37C2×C3×D12—34no—
38C32×C4⋊C4—24no—
39C32×C4⋊C4C2×D318yes—
40C3×(C2×D3)⋊C4—64no—
41C3×C4⋊Dic3—34no—
42C2×C3×C3⋊C8—38no—
43C22×H(3,𝔽3)—21no—
44C22×H(3,𝔽3)—21no—
45C22×H(3,𝔽3)SD1612yes—
46C22×H(3,𝔽3)C2×D411yescharacteristic
47C2×#272—41no—
48C2×#272—41no—
49C2×#272—41no—
50C4×H(3,𝔽3)—21no—
51C4×H(3,𝔽3)—42no—
52C4×H(3,𝔽3)#4112yes—
53#1084—41no—
54C3×#121—91no—
55C2×C3×C4⋊C4—34no—
56C2×C3×C4⋊C4C3⋊D311yescharacteristic
57#930—34no—
58C3×#106—92no—
59C3×#102—92no—
60C3×C22⋊C8—91no—
61#830—34no—
62#809—38no—
63#808—38no—
64C23×C32#6714yes—
65C22×C3×D3—64no—
66C2×C32×C4—28no—
67C2×C32×C4—28no—
68C2×C32×C4—48no—
69C2×C32×C4#6714yes—
70C2×C32×C4C22×D314yes—
71C3×#67—128no—
72C2×C3×Dic3—64no—
73C3×D12—68no—
74C3×C3⋊C8—68no—
75#481—91noSylow
76C2×H(3,𝔽3)—22no—
77C2×H(3,𝔽3)#12311yescharacteristic
78C2×H(3,𝔽3)C2×SD1611yescharacteristic
79C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
80#272—81no—
81C2×C3×D4—91no—
82C2×C3×D4—181no—
83C22×C3×C4—21no—
84C22×C3×C4—34no—
85C22×C3×C4—64no—
86C22×C3×C4C2×C3⋊D311yescharacteristic
87C2×#67—64no—
88C2×D12—34no—
89C2×C3×C8—92no—
90C3×C4⋊C4—21no—
91C3×C4⋊C4—38no—
92C3×C4⋊C4—64no—
93C3×C4⋊C4—91no—
94C3×C4⋊C4C2×C3⋊D312yes—
95C3×C22⋊C4—181no—
96(C2×D3)⋊C4—64no—
97C4⋊Dic3—34no—
98C2×C3⋊C8—38no—
99C22×C32—24no—
100C22×C32—24no—
101C22×C32#21518yes—
102C22×C32C2×#6714yes—
103C2×C3×D3—124no—
104C2×C3×D3—124no—
105C2×C3×D3—124no—
106C32×C4—24no—
107C32×C4—48no—
108C32×C4#23518yes—
109C3×Dic3—124no—
110#121—91no—
111C2×C4⋊C4#27211yescharacteristic, complemented
112#106—92no—
113#102—92no—
114C22⋊C8—91no—
115H(3,𝔽3)#48111yescharacteristic, complemented, Sylow
116C23×C3—34no—
117C23×C3—181no—
118C23×C3#63811yescharacteristic
119C22×D3—64no—
120C3×D4—182no—
121C3×D4—362no—
122C2×C3×C4—22no—
123C2×C3×C4—22no—
124C2×C3×C4—34no—
125C2×C3×C4—34no—
126C2×C3×C4—42no—
127C2×C3×C4—68no—
128C2×C3×C4—68no—
129C2×C3×C4—128no—
130C2×C3×C4—181no—
131C2×C3×C4#63811yescharacteristic
132C2×C3×C4C22×C3⋊D311yescharacteristic
133#67—128no—
134C2×Dic3—64no—
135D12—68no—
136C3×C8—182no—
137C3⋊C8—68no—
138C2×C32—28no—
139C2×C32#92914yes—
140C2×C32#93214yes—
141C2×C32C2×#21514yes—
142C3×D3—244no—
143C2×D4—91no—
144C2×D4—181no—
145C22×C4—21no—
146C22×C4C2×#27211yescharacteristic
147C2×C8—92no—
148C4⋊C4—21no—
149C4⋊C4—91no—
150C4⋊C4C2×#27212yes—
151C22⋊C4—181no—
152C22×C3—21no—
153C22×C3—21no—
154C22×C3—34no—
155C22×C3—38no—
156C22×C3—64no—
157C22×C3—64no—
158C22×C3—361no—
159C22×C3—361no—
160C22×C3—361no—
161C22×C3#375512yes—
162C22×C3C2×#63811yescharacteristic
163C2×D3—124no—
164C2×D3—124no—
165C2×D3—124no—
166C3×C4—21no—
167C3×C4—38no—
168C3×C4—42no—
169C3×C4—64no—
170C3×C4—128no—
171C3×C4—361no—
172C3×C4#382812yes—
173Dic3—124no—
174C32#496814yescomplemented
175C23—181no—
176C23#623511yescharacteristic
177D4—182no—
178D4—362no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—42no—
182C2×C4—181no—
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—721no—
192C2×C3#6413811yescharacteristic
193C2×C3#6414111yescharacteristic
194C2×C3C2×#375511yescharacteristic
195D3—244no—
196C22—21no—
197C22—21no—
198C22—361no—
199C22—361no—
200C22—361no—
201C22#8854912yes—
202C22C2×#623511yescharacteristic
203C4—21no—
204C4—42no—
205C4—361no—
206C4#8874012yes—
207C3—34no—
208C3#1059266711yescharacteristic
209C2—22no—
210C2—721no—
211C2#1172182911yescharacteristic
212C2#1172184211yescharacteristic
213C2C2×#8854911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#638
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC2×C4⋊C4×H(3,𝔽3)C2
prime radical#11721830C2
hypercenterC2×C3×C4⋊C4C3⋊D3
hypocenterH(3,𝔽3)#481
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×H(3,𝔽3)C2semidirect
3. #11720573C2semidirect
4. #11720531C2semidirect
5. #11720530C2semidirect
16. C2×C32×C4⋊C4D3semidirect
111. C2×C4⋊C4#272semidirect
115. H(3,𝔽3)#481semidirect
174. C32#4968semidirect

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
#117218302
#117205732
#117205312
#117205302
C3×#49683