Finite Group Databasebeta

← #13137411

Subgroups — #13137411

Order 1728.

605 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 605 conjugacy classes of subgroups — too many to draw. Only its 119 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
2C2×#88435C212yesmaximal
3#63630⋊C3C212yesmaximal, complemented
4#11720302C212yesmaximal, complemented
5#10590334—32nomaximal
6C2×C4×#272C412yes—
7C2×C4×#272C2211yescharacteristic
8C42×H(3,𝔽3)C2211yescharacteristic
9C4×#1084C412yes—
10C4×#1084C2211yescharacteristic
11C2×(C3⋊(C3⋊C8))⋊C3C2214yes—
12#88435C418yescomplemented
13C3×C42×D3—32no—
14C3×C42×D3—61no—
15C2×#3711—34no—
16#63630D312yescomplemented
17C3×#803—32no—
18Dic3×C3⋊C8—34no—
19C22×#272C2×C411yescharacteristic
20C2×C4×H(3,𝔽3)C2×C412yes—
21C2×C4×H(3,𝔽3)C2311yescharacteristic
22C2×#1084C2×C411yescharacteristic
23C2×#1084C2×C412yes—
24C4×#272—24no—
25C4×#272C2×C414yes—
26(C3⋊(C3⋊C8))⋊C3C2×C418yescomplemented
27C4×#208—91no—
28#4707—92no—
29C2×C3×C4×D3—32no—
30C2×C3×C4×D3—34no—
31C2×C3×C4×D3—62no—
32C2×C3×C4×D3—61no—
33C32×C42—21no—
34C32×C42C2×D312yes—
35C2×C3⋊(C3⋊C8)C2×D314yes—
36C3×C4×Dic3—34no—
37C3×C4×Dic3—32no—
38C3×C4×Dic3—62no—
39C3×C4×Dic3—61no—
40C2×C3×C3⋊C8—34no—
41#3711—316no—
42C22×H(3,𝔽3)C22×C411yescharacteristic
43C2×#272—22no—
44C2×#272C4212yes—
45C4×H(3,𝔽3)—22no—
46C4×H(3,𝔽3)C22×C412yes—
47#1084C4212yes—
48#1084M1614yescomplemented
49C2×C3×C42—91no—
50C2×#208—92no—
51C2×ΓL(1,𝔽25)—94no—
52C42×D3—32no—
53C42×D3—61no—
54C3×#97—92no—
55C8×Dic3—94no—
56#803—32no—
57#803—92no—
58C4×C3⋊C8—92no—
59C22×C3×D3—32no—
60C22×C3×D3—61no—
61C2×C32×C4—22no—
62C2×C32×C4—21no—
63C2×C32×C4C4×D314yes—
64C2×C32×C4C22×D312yes—
65C2×C3×Dic3—32no—
66C2×C3×Dic3—34no—
67C2×C3×Dic3—61no—
68C2×C3×Dic3—62no—
69C3×C4×D3—38no—
70C3×C4×D3—64no—
71C3×C4×D3—68no—
72C3×C4×D3—68no—
73C3⋊(C3⋊C8)C4×D318yescomplemented
74C3×C3⋊C8—38no—
75C4×M16—271noSylow
76C2×H(3,𝔽3)C2×C4211yescharacteristic
77C2×H(3,𝔽3)C2×M1612yes—
78#272—22no—
79C22×C3×C4—91no—
80C22×C3×C4—92no—
81C2×C4×D3—32no—
82C2×C4×D3—34no—
83C2×C4×D3—61no—
84C2×C4×D3—62no—
85C2×C3×C8—94no—
86C3×C42—32no—
87C3×C42—61no—
88C3×C42—92no—
89C3×C42—91no—
90C3×C42D3211yescharacteristic
91C4×Dic3—32no—
92C4×Dic3—34no—
93C4×Dic3—62no—
94C4×Dic3—61no—
95#208—98no—
96C2×C3⋊C8—34no—
97C2×C3⋊C8—94no—
98ΓL(1,𝔽25)—916no—
99C22×C32—21no—
100C22×C32C2×C4×D312yes—
101C2×C3×D3—34no—
102C2×C3×D3—62no—
103C2×C3×D3—64no—
104C2×C3×D3—64no—
105C32×C4—24no—
106C32×C4—24no—
107C32×C4—22no—
108C32×C4C2×C4×D314yes—
109C3×Dic3—34no—
110C3×Dic3—38no—
111C3×Dic3—62no—
112C3×Dic3—64no—
113C2×M16—272no—
114C2×C42—91no—
115#97—92no—
116C4×C8—272no—
117H(3,𝔽3)C4×M1611yescharacteristic, complemented, Sylow
118C23×C3—91no—
119C22×D3—32no—
120C22×D3—61no—
121C2×C3×C4—34no—
122C2×C3×C4—32no—
123C2×C3×C4—62no—
124C2×C3×C4—61no—
125C2×C3×C4—92no—
126C2×C3×C4—91no—
127C2×C3×C4—94no—
128C2×C3×C4—184no—
129C2×C3×C4#62412yes—
130C2×C3×C4C2×D3211yescharacteristic
131C2×Dic3—32no—
132C2×Dic3—34no—
133C2×Dic3—62no—
134C2×Dic3—61no—
135C4×D3—38no—
136C4×D3—64no—
137C4×D3—68no—
138C4×D3—68no—
139C3×C8—98no—
140C3⋊C8—38no—
141C3⋊C8—98no—
142C2×C32—22no—
143C2×C32—21no—
144C2×C32C42×D312yes—
145C2×C32D3×M1614yes—
146C3×D3—64no—
147C3×D3—64no—
148C22×C4—91no—
149C22×C4—92no—
150M16—278no—
151C2×C8—94no—
152C42—92no—
153C42—91no—
154C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
155C22×C3—32no—
156C22×C3—61no—
157C22×C3—92no—
158C22×C3—182no—
159C22×C3C2×#62411yescharacteristic
160C2×D3—34no—
161C2×D3—62no—
162C2×D3—64no—
163C2×D3—64no—
164C3×C4—22no—
165C3×C4—34no—
166C3×C4—64no—
167C3×C4—64no—
168C3×C4—62no—
169C3×C4—94no—
170C3×C4—92no—
171C3×C4C4×D3212yes—
172Dic3—34no—
173Dic3—38no—
174Dic3—64no—
175Dic3—62no—
176C32—21no—
177C32#470712yescomplemented
178C23—91no—
179C2×C4—92no—
180C2×C4—91no—
181C2×C4—94no—
182C2×C4—184no—
183C2×C4#619912yes—
184C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
185C8—98no—
186C2×C3—34no—
187C2×C3—32no—
188C2×C3—62no—
189C2×C3—61no—
190C2×C3—182no—
191C2×C3#6381812yes—
192C2×C3C4×#62411yescharacteristic
193D3—64no—
194D3—64no—
195C22—92no—
196C22—182no—
197C22C2×#619911yescharacteristic
198C4—22no—
199C4—94no—
200C4—92no—
201C4C4×((C3⋊D3)⋊C3)⋊C212yes—
202C3—32no—
203C3—61no—
204C3#1059024711yescharacteristic
205C2—182no—
206C2#1172106512yes—
207C2C4×#619911yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C2×C42
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#624
prime radicalC22×#272C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C4×M16
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
3. #63630⋊C3C2semidirect
4. #11720302C2semidirect
12. #88435C4semidirect
16. #63630D3semidirect
26. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
48. #1084M16semidirect
73. C3⋊(C3⋊C8)C4×D3semidirect
117. H(3,𝔽3)C4×M16semidirect
177. C32#4707semidirect

Maximal subgroups

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

subgroupindex
C42×#2722
C2×#884352
#63630⋊C32
#117203022
#105903343