Finite Group Databasebeta

← #13138910

Subgroups — #13138910

Order 1728.

531 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 531 conjugacy classes of subgroups — too many to draw. Only its 115 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×C8×#272C211yesmaximal, characteristic
2#11721097C211yesmaximal, characteristic
3C2×#88435C211yesmaximal, characteristic
4C8×(C3⋊Dic3)⋊C3C212yesmaximal, complemented
5#11720312C212yesmaximal, complemented
6#10590692—32nomaximal
7C2×C4×#272C2211yescharacteristic
8C2×C8×H(3,𝔽3)C2211yescharacteristic
9C2×#6180C2211yescharacteristic
10C8×#272C414yescomplemented
11C4×(C3⋊Dic3)⋊C3C2212yes—
12C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
13#88451C412yes—
14#88449C412yes—
15#88435C414yescomplemented
16C2×C3×D3×C8—32no—
17C2×C3×D3×C8—61no—
18#63855—32no—
19C2×#3711—32no—
20C8×C3⋊Dic3D312yescomplemented
21C3×C8×Dic3—32no—
22#63555—32no—
23#63554—32no—
24C22×#272C2×C411yescharacteristic
25C2×C4×H(3,𝔽3)C2311yescharacteristic
26C2×#1084C2×C411yescharacteristic
27C4×#272C2×C412yes—
28C4×#272C2×C412yes—
29C2×(C3⋊Dic3)⋊C3C2×C414yes—
30C8×H(3,𝔽3)C2×C412yes—
31#6180C2×C412yes—
32(C3⋊(C3⋊C8))⋊C3C2×C414yes—
33#5101—91no—
34#4689—92no—
35C2×C3×C4×D3—32no—
36C2×C3×C4×D3—61no—
37C2×C32×C8—21no—
38C2×C32×C8C2×D312yes—
39C4×C3⋊Dic3C2×D312yes—
40C2×C3⋊(C3⋊C8)C2×D312yes—
41C3×C4×Dic3—32no—
42C2×C3×C3⋊C8—32no—
43C2×C3×C3⋊C8—32no—
44C2×C3×C3⋊C8—61no—
45C3×D3×C8—38no—
46C3×D3×C8—64no—
47Dic3⋊Dic3—34no—
48(C2×D3)⋊Dic3—34no—
49#3711—38no—
50C22×H(3,𝔽3)C22×C411yescharacteristic
51C2×#272—22no—
52C2×#272C4212yes—
53C4×H(3,𝔽3)C22×C411yescharacteristic
54C4×H(3,𝔽3)C22×C411yescharacteristic
55#1084C4212yes—
56(C3⋊Dic3)⋊C3—24no—
57C22×C3×C8—91no—
58#926—91no—
59C2×#208—91no—
60C2×ΓL(1,𝔽25)—92no—
61C2×D3×C8—32no—
62C2×D3×C8—61no—
63#872—92no—
64C3×C4×C8—92no—
65#815—92no—
66#815—92no—
67C8×Dic3—32no—
68C8×Dic3—92no—
69#803—92no—
70C22×C3×D3—32no—
71C22×C3×D3—61no—
72C2×C32×C4—21no—
73C2×C32×C4C22×D312yes—
74C2×C3⋊Dic3C4×D314yes—
75C2×C3×Dic3—32no—
76C2×C3×Dic3—34no—
77C2×C3×Dic3—61no—
78C3×C4×D3—34no—
79C3×C4×D3—34no—
80C3×C4×D3—62no—
81C3×C4×D3—62no—
82C32×C8—22no—
83C32×C8C4×D314yes—
84C3⋊(C3⋊C8)C4×D314yes—
85C3×C3⋊C8—34no—
86C3×C3⋊C8—34no—
87C3×C3⋊C8—62no—
88#405—271noSylow
89C2×H(3,𝔽3)C2×C4211yescharacteristic
90C2×H(3,𝔽3)#13112yes—
91#272—22no—
92C22×C3×C4—91no—
93C2×C4×D3—32no—
94C2×C4×D3—61no—
95C2×C3×C8—32no—
96C2×C3×C8—61no—
97C2×C3×C8—92no—
98C2×C3×C8—91no—
99C2×C3×C8—94no—
100C2×C3×C8D3211yescharacteristic
101C3×C42—92no—
102#217—92no—
103(C2×D3)⋊C4—94no—
104C4⋊Dic3—92no—
105Dic3⋊C4—94no—
106C4×Dic3—32no—
107C4×Dic3—92no—
108#208—94no—
109C2×C3⋊C8—32no—
110C2×C3⋊C8—32no—
111C2×C3⋊C8—61no—
112C2×C3⋊C8—92no—
113ΓL(1,𝔽25)—98no—
114D3×C8—38no—
115D3×C8—64no—
116C22×C32—21no—
117C22×C32C2×C4×D312yes—
118C2×C3×D3—34no—
119C2×C3×D3—62no—
120C2×C3×D3—64no—
121C2×C3×D3—64no—
122C32×C4—21no—
123C32×C4—21no—
124C32×C4C2×C4×D312yes—
125C32×C4C2×C4×D312yes—
126C3⋊Dic3—24no—
127C3×Dic3—34no—
128C3×Dic3—62no—
129C3×Dic3—64no—
130C2×M16—271no—
131C22×C8—91no—
132#117—271no—
133#97—272no—
134C4×C8—92no—
135H(3,𝔽3)#40511yescharacteristic, complemented, Sylow
136C23×C3—91no—
137C22×D3—32no—
138C22×D3—61no—
139C2×C3×C4—32no—
140C2×C3×C4—61no—
141C2×C3×C4—91no—
142C2×C3×C4—92no—
143C2×C3×C4—92no—
144C2×C3×C4—94no—
145C2×C3×C4C2×D3211yescharacteristic
146C2×Dic3—32no—
147C2×Dic3—34no—
148C2×Dic3—61no—
149C2×Dic3—94no—
150C4×D3—34no—
151C4×D3—34no—
152C4×D3—62no—
153C4×D3—62no—
154C3×C8—34no—
155C3×C8—62no—
156C3×C8—94no—
157C3×C8—92no—
158C3×C8#62412yes—
159C3⋊C8—34no—
160C3⋊C8—34no—
161C3⋊C8—62no—
162C3⋊C8—94no—
163C2×C32—22no—
164C2×C32—21no—
165C2×C32C42×D312yes—
166C2×C32#90114yes—
167C3×D3—64no—
168C3×D3—64no—
169C22×C4—91no—
170M16—274no—
171C2×C8—92no—
172C2×C8—91no—
173C2×C8—94no—
174C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
175C4⋊C4—272no—
176C22⋊C4—272no—
177C42—92no—
178C22×C3—32no—
179C22×C3—61no—
180C22×C3—92no—
181C22×C3—182no—
182C22×C3C4×D3211yescharacteristic
183C2×D3—34no—
184C2×D3—62no—
185C2×D3—64no—
186C2×D3—64no—
187C3×C4—32no—
188C3×C4—32no—
189C3×C4—61no—
190C3×C4—61no—
191C3×C4—92no—
192C3×C4—184no—
193C3×C4C4×D3211yescharacteristic
194C3×C4C2×#62411yescharacteristic
195Dic3—34no—
196Dic3—62no—
197Dic3—64no—
198Dic3—184no—
199C32—21no—
200C32#468912yescomplemented
201C23—91no—
202C2×C4—91no—
203C2×C4—92no—
204C2×C4—92no—
205C2×C4—94no—
206C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
207C8—94no—
208C8—92no—
209C8#619912yes—
210C2×C3—34no—
211C2×C3—32no—
212C2×C3—62no—
213C2×C3—61no—
214C2×C3—182no—
215C2×C3#6380312yes—
216C2×C3C4×#62411yescharacteristic
217D3—64no—
218D3—64no—
219C22—92no—
220C22—182no—
221C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
222C4—92no—
223C4—184no—
224C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
225C4C2×#619911yescharacteristic
226C3—32no—
227C3—61no—
228C3#1059070011yescharacteristic
229C2—182no—
230C2#1172084012yes—
231C2C4×#619911yescharacteristic
subgroupG/H
center, hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C2×C42
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3C4×D32
prime radicalC22×#272C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypocenterH(3,𝔽3)#405
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
4. C8×(C3⋊Dic3)⋊C3C2semidirect
5. #11720312C2semidirect
10. C8×#272C4semidirect
15. #88435C4semidirect
20. C8×C3⋊Dic3D3semidirect
135. H(3,𝔽3)#405semidirect
200. C32#4689semidirect

Maximal subgroups

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

subgroupindex
C2×C8×#2722
#117210972
C2×#884352
C8×(C3⋊Dic3)⋊C32
#117203122
#105906923