Finite Group Databasebeta

← #13144072

Subgroups — #13144072

Order 1728.

649 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 649 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C42×#272C211yesmaximal, characteristic
2H(3,𝔽3)×C4⋊C8C211yesmaximal, characteristic, complemented
3#11720553C212yesmaximal
4C8×#1084C212yesmaximal, complemented
5C4⋊#6180C211yesmaximal, characteristic, complemented
6C3×#4833—34nomaximal
7C2×C4×#272C411yescharacteristic
8C2×C4×#272C411yescharacteristic
9C2×C4×#272C2211yescharacteristic
10C2×C8×H(3,𝔽3)C2212yes—
11C42×H(3,𝔽3)C2211yescharacteristic
12C4×#1084C411yescharacteristic
13C4×#1084C411yescharacteristic
14C4×#1084C2211yescharacteristic
15C2×#6180C2212yes—
16C3×C42×D3—34no—
17C32×C4⋊C8D314yescomplemented
18C3×(C2×D3)⋊C8—38no—
19C3×C8×Dic3—38no—
20C3×C4⋊(C3⋊C8)—34no—
21C22×#272C2×C411yescharacteristic
22C2×C4×H(3,𝔽3)C2×C411yescharacteristic
23C2×C4×H(3,𝔽3)C2×C411yescharacteristic
24C2×C4×H(3,𝔽3)C2311yescharacteristic
25C2×#1084C2×C411yescharacteristic
26C2×#1084C2×C411yescharacteristic
27C2×#1084C2×C411yescharacteristic
28C4×#272—21no—
29C4×#272—21no—
30C4×#272—22no—
31C4×#272C814yescomplemented
32C8×H(3,𝔽3)—22no—
33#6180—22no—
34C3×#431—91no—
35#4833—34no—
36C2×C3×C4×D3—34no—
37C2×C3×C4×D3—34no—
38C2×C3×C4×D3—34no—
39C2×C32×C8C2×D318yes—
40C32×C42C2×D314yes—
41C3×C4×Dic3—34no—
42C3×C4×Dic3—34no—
43C3×C4×Dic3—34no—
44C2×C3×C3⋊C8—38no—
45C22×H(3,𝔽3)C22×C411yescharacteristic
46C2×#272—21no—
47C2×#272—21no—
48C2×#272C2×C812yes—
49C4×H(3,𝔽3)—21no—
50C4×H(3,𝔽3)C2×C812yes—
51C4×H(3,𝔽3)#4111yescharacteristic
52C4×H(3,𝔽3)#4111yescharacteristic
53#1084—21no—
54#1084C2×C812yes—
55#1084M1612yes—
56C2×C3×C42—91no—
57C42×D3—34no—
58C3×C4⋊C8—34no—
59C3×C4⋊C8—91no—
60C3×C4⋊C8C3⋊D311yescharacteristic
61C3×C22⋊C8—92no—
62C3×C4×C8—92no—
63(C2×D3)⋊C8—38no—
64C8×Dic3—38no—
65C4⋊(C3⋊C8)—34no—
66C22×C3×D3—34no—
67C2×C32×C4C4×D314yes—
68C2×C32×C4C4×D314yes—
69C2×C32×C4C22×D314yes—
70C2×C3×Dic3—34no—
71C2×C3×Dic3—34no—
72C2×C3×Dic3—34no—
73C3×C4×D3—316no—
74C3×C4×D3—68no—
75C3×C4×D3—64no—
76C3×C4×D3—64no—
77C32×C8—28no—
78C3×C3⋊C8—68no—
79#431—91noSylow
80C2×H(3,𝔽3)#11711yescharacteristic
81C2×H(3,𝔽3)C22×C811yescharacteristic
82C2×H(3,𝔽3)C2×M1611yescharacteristic
83#272—22no—
84C22×C3×C4—91no—
85C22×C3×C4—91no—
86C22×C3×C4—91no—
87C2×C4×D3—34no—
88C2×C4×D3—34no—
89C2×C4×D3—34no—
90C2×C3×C8—38no—
91C2×C3×C8—92no—
92C2×C3×C8C2×C3⋊D312yes—
93C3×C42—34no—
94C3×C42—91no—
95C3×C42—91no—
96C3×C42—91no—
97C3×C42C2×C3⋊D311yescharacteristic
98C4×Dic3—34no—
99C4×Dic3—34no—
100C4×Dic3—34no—
101C2×C3⋊C8—38no—
102C22×C32C2×C4×D314yes—
103C2×C3×D3—38no—
104C2×C3×D3—64no—
105C2×C3×D3—64no—
106C32×C4—24no—
107C32×C4D3×C818yes—
108C32×C4#23714yes—
109C32×C4#23914yes—
110C3×Dic3—38no—
111C3×Dic3—38no—
112C3×Dic3—64no—
113C2×C42—91no—
114C4⋊C8—91no—
115C4⋊C8#27211yescharacteristic, complemented
116C22⋊C8—92no—
117C4×C8—92no—
118H(3,𝔽3)#43111yescharacteristic, complemented, Sylow
119C23×C3—91no—
120C22×D3—34no—
121C2×C3×C4—34no—
122C2×C3×C4—34no—
123C2×C3×C4—34no—
124C2×C3×C4—91no—
125C2×C3×C4—91no—
126C2×C3×C4—94no—
127C2×C3×C4—91no—
128C2×C3×C4—182no—
129C2×C3×C4—181no—
130C2×C3×C4—181no—
131C2×C3×C4C4×C3⋊D311yescharacteristic
132C2×C3×C4C4×C3⋊D311yescharacteristic
133C2×C3×C4C22×C3⋊D311yescharacteristic
134C2×Dic3—34no—
135C2×Dic3—34no—
136C2×Dic3—34no—
137C4×D3—316no—
138C4×D3—68no—
139C4×D3—64no—
140C4×D3—64no—
141C3×C8—22no—
142C3×C8—68no—
143C3×C8—182no—
144C3⋊C8—68no—
145C2×C32#89214yes—
146C2×C32C2×D3×C814yes—
147C2×C32D3×M1614yes—
148C3×D3—68no—
149C22×C4—91no—
150C22×C4—91no—
151C22×C4—91no—
152C2×C8—92no—
153C2×C8C2×#27212yes—
154C42—91no—
155C42—91no—
156C42—91no—
157C42C2×#27211yescharacteristic
158C22×C3—34no—
159C22×C3—92no—
160C22×C3—181no—
161C22×C3—181no—
162C22×C3C2×C4×C3⋊D311yescharacteristic
163C2×D3—38no—
164C2×D3—64no—
165C2×D3—64no—
166C3×C4—21no—
167C3×C4—34no—
168C3×C4—34no—
169C3×C4—38no—
170C3×C4—64no—
171C3×C4—92no—
172C3×C4—92no—
173C3×C4—181no—
174C3×C4C8×C3⋊D312yes—
175C3×C4#383011yescharacteristic
176C3×C4#383211yescharacteristic
177Dic3—38no—
178Dic3—38no—
179Dic3—64no—
180C32#483314yescomplemented
181C23—91no—
182C2×C4—91no—
183C2×C4—91no—
184C2×C4—94no—
185C2×C4—91no—
186C2×C4—182no—
187C2×C4—181no—
188C2×C4—181no—
189C2×C4C4×#27211yescharacteristic
190C2×C4C4×#27211yescharacteristic
191C2×C4C22×#27211yescharacteristic
192C8—22no—
193C8—182no—
194C2×C3—34no—
195C2×C3—34no—
196C2×C3—34no—
197C2×C3—182no—
198C2×C3#6410111yescharacteristic
199C2×C3C2×C8×C3⋊D311yescharacteristic
200C2×C3M16×C3⋊D311yescharacteristic
201D3—68no—
202C22—92no—
203C22—181no—
204C22—181no—
205C22C2×C4×#27211yescharacteristic
206C4—21no—
207C4—92no—
208C4—92no—
209C4—181no—
210C4C8×#27212yes—
211C4#8874411yescharacteristic
212C4#8874811yescharacteristic
213C3—34no—
214C3#1059253211yescharacteristic
215C2—182no—
216C2#1172176011yescharacteristic
217C2C2×C8×#27211yescharacteristic
218C2M16×#27211yescharacteristic
subgroupG/H
center, FrattiniC2×C3×C4C22×C3⋊D3
commutatorC2×H(3,𝔽3)C22×C8
FittingH(3,𝔽3)×C4⋊C8C2
socleC22×C3C2×C4×C3⋊D3
prime radicalC22×#272C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC3×C4⋊C8C3⋊D3
hypocenterH(3,𝔽3)#431
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3×C4 ◁ C3×C4⋊C8

Complements

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

NH ≅ G/Ntype
2. H(3,𝔽3)×C4⋊C8C2semidirect
4. C8×#1084C2semidirect
5. C4⋊#6180C2semidirect
17. C32×C4⋊C8D3semidirect
31. C4×#272C8semidirect
115. C4⋊C8#272semidirect
118. H(3,𝔽3)#431semidirect
180. C32#4833semidirect

Maximal subgroups

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

subgroupindex
C42×#2722
H(3,𝔽3)×C4⋊C82
#117205532
C8×#10842
C4⋊#61802
C3×#48333