Finite Group Databasebeta

← #13138914

Subgroups — #13138914

Order 1728.

467 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 467 conjugacy classes of subgroups — too many to draw. Only its 87 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×#88527C211yesmaximal, characteristic
2#11721097C211yesmaximal, characteristic
3C2×#88434C211yesmaximal, characteristic
4#63641⋊C3C212yesmaximal, complemented
5#11720311C212yesmaximal, complemented
6#10590695—32nomaximal
7C2×C4×#272C2211yescharacteristic
8C2×C8×H(3,𝔽3)C2211yescharacteristic
9C2×#6180C2211yescharacteristic
10#88527—22no—
11C4×(C3⋊Dic3)⋊C3C2212yes—
12C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
13#88451C411yescharacteristic
14#88451C411yescharacteristic
15#88449C412yes—
16#88434—22no—
17C2×C3×ΓL(1,𝔽25)—32no—
18C2×C3×ΓL(1,𝔽25)—61no—
19#63855—32no—
20C2×D3×C3⋊C8—32no—
21#63641D312yescomplemented
22C3×Dic3⋊C8—32no—
23#63572—32no—
24Dic3⋊(C3⋊C8)—32no—
25C22×#272C2×C411yescharacteristic
26C2×C4×H(3,𝔽3)C2311yescharacteristic
27C2×#1084C2×C411yescharacteristic
28C4×#272D412yes—
29C4×#272Q812yes—
30C2×(C3⋊Dic3)⋊C3C2×C412yes—
31C2×(C3⋊Dic3)⋊C3C2×C412yes—
32C8×H(3,𝔽3)—21no—
33#6180—21no—
34(C3⋊(C3⋊C8))⋊C3—22no—
35#5119—91no—
36#4839—92no—
37C2×C3×C4×D3—32no—
38C2×C3×C4×D3—61no—
39C2×C32×C8—21no—
40C2×C32×C8C2×D312yes—
41C4×C3⋊Dic3C2×D312yes—
42C2×C3⋊(C3⋊C8)C2×D312yes—
43C3×C4×Dic3—32no—
44C2×C3×C3⋊C8—32no—
45C2×C3×C3⋊C8—32no—
46C2×C3×C3⋊C8—61no—
47C3×ΓL(1,𝔽25)—64no—
48C3×ΓL(1,𝔽25)—64no—
49Dic3⋊Dic3—32no—
50Dic3⋊Dic3—32no—
51(C2×D3)⋊Dic3—32no—
52(C2×D3)⋊Dic3—32no—
53D3×C3⋊C8—64no—
54C22×H(3,𝔽3)C22×C411yescharacteristic
55C2×#272—22no—
56C2×#272C4⋊C412yes—
57C4×H(3,𝔽3)C2×D411yescharacteristic
58C4×H(3,𝔽3)C2×Q811yescharacteristic
59#1084C4⋊C412yes—
60(C3⋊Dic3)⋊C3—22no—
61(C3⋊Dic3)⋊C3—22no—
62C2×C3×M16—91no—
63#926—91no—
64C22×C3⋊C8—91no—
65C2×ΓL(1,𝔽25)—32no—
66C2×ΓL(1,𝔽25)—61no—
67C2×D3×C8—92no—
68#872—92no—
69C3×C4⋊C8—92no—
70Dic3⋊C8—32no—
71Dic3⋊C8—92no—
72Dic3⋊C8—92no—
73Dic3⋊C8—92no—
74C4⋊(C3⋊C8)—92no—
75C22×C3×D3—32no—
76C22×C3×D3—61no—
77C2×C32×C4—21no—
78C2×C32×C4C22×D312yes—
79C2×C3⋊Dic3C4×D312yes—
80C2×C3⋊Dic3C4×D312yes—
81C2×C3×Dic3—32no—
82C2×C3×Dic3—32no—
83C2×C3×Dic3—32no—
84C2×C3×Dic3—61no—
85C3×C4×D3—34no—
86C3×C4×D3—34no—
87C3×C4×D3—62no—
88C3×C4×D3—62no—
89C32×C8—22no—
90C32×C8—22no—
91C3⋊(C3⋊C8)—22no—
92C3×C3⋊C8—62no—
93C3×C3⋊C8—62no—
94C3×C3⋊C8—62no—
95#424—271noSylow
96C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
97C2×H(3,𝔽3)#13112yes—
98#272—22no—
99C22×C3×C4—91no—
100C2×C4×D3—32no—
101C2×C4×D3—61no—
102C3×M16—182no—
103C2×C3×C8—32no—
104C2×C3×C8—61no—
105C2×C3×C8—92no—
106C2×C3×C8—91no—
107C2×C3×C8D3211yescharacteristic
108C3×C42—92no—
109#217—91no—
110#217—91no—
111(C2×D3)⋊C4—92no—
112(C2×D3)⋊C4—92no—
113C4⋊Dic3—92no—
114Dic3⋊C4—92no—
115Dic3⋊C4—92no—
116C4×Dic3—32no—
117C4×Dic3—92no—
118C2×C3⋊C8—32no—
119C2×C3⋊C8—32no—
120C2×C3⋊C8—61no—
121C2×C3⋊C8—92no—
122C2×C3⋊C8—182no—
123ΓL(1,𝔽25)—64no—
124ΓL(1,𝔽25)—64no—
125D3×C8—184no—
126C22×C32—21no—
127C22×C32C2×C4×D312yes—
128C2×C3×D3—34no—
129C2×C3×D3—62no—
130C2×C3×D3—62no—
131C2×C3×D3—62no—
132C2×C3×D3—122no—
133C32×C4—21no—
134C32×C4—21no—
135C32×C4D3×D412yes—
136C32×C4D3×Q812yes—
137C3⋊Dic3—22no—
138C3⋊Dic3—22no—
139C3×Dic3—34no—
140C3×Dic3—62no—
141C3×Dic3—62no—
142C3×Dic3—62no—
143C2×M16—91no—
144C22×C8—271no—
145#117—271no—
146C4⋊C8—92no—
147C4⋊C8—272no—
148H(3,𝔽3)#42411yescharacteristic, complemented, Sylow
149C23×C3—91no—
150C22×D3—32no—
151C22×D3—61no—
152C2×C3×C4—32no—
153C2×C3×C4—61no—
154C2×C3×C4—91no—
155C2×C3×C4—92no—
156C2×C3×C4—92no—
157C2×C3×C4—92no—
158C2×C3×C4—92no—
159C2×C3×C4C2×D3211yescharacteristic
160C2×Dic3—32no—
161C2×Dic3—32no—
162C2×Dic3—32no—
163C2×Dic3—61no—
164C2×Dic3—92no—
165C2×Dic3—92no—
166C4×D3—34no—
167C4×D3—34no—
168C4×D3—62no—
169C4×D3—62no—
170C3×C8—21no—
171C3×C8—62no—
172C3×C8—62no—
173C3×C8—182no—
174C3×C8—181no—
175C3⋊C8—62no—
176C3⋊C8—62no—
177C3⋊C8—62no—
178C3⋊C8—182no—
179C2×C32—22no—
180C2×C32—21no—
181C2×C32D3×C4⋊C412yes—
182C2×C32#90112yes—
183C2×C32#90712yes—
184C3×D3—64no—
185C3×D3—122no—
186C22×C4—91no—
187M16—182no—
188C2×C8—92no—
189C2×C8—91no—
190C2×C8—542no—
191C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
192C4⋊C4—272no—
193C22⋊C4—271no—
194C22⋊C4—271no—
195C42—92no—
196C22×C3—32no—
197C22×C3—61no—
198C22×C3—92no—
199C22×C3—182no—
200C22×C3C4×D3211yescharacteristic
201C2×D3—34no—
202C2×D3—62no—
203C2×D3—62no—
204C2×D3—62no—
205C2×D3—122no—
206C3×C4—32no—
207C3×C4—32no—
208C3×C4—61no—
209C3×C4—61no—
210C3×C4—92no—
211C3×C4—182no—
212C3×C4—182no—
213C3×C4#379711yescharacteristic
214C3×C4#381111yescharacteristic
215Dic3—34no—
216Dic3—62no—
217Dic3—62no—
218Dic3—62no—
219Dic3—182no—
220Dic3—182no—
221C32—21no—
222C32#483912yescomplemented
223C23—91no—
224C2×C4—91no—
225C2×C4—92no—
226C2×C4—92no—
227C2×C4—92no—
228C2×C4—92no—
229C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
230C8—21no—
231C8—182no—
232C8—181no—
233C2×C3—32no—
234C2×C3—32no—
235C2×C3—32no—
236C2×C3—62no—
237C2×C3—61no—
238C2×C3—182no—
239C2×C3#6380412yes—
240C2×C3#6388311yescharacteristic
241D3—64no—
242D3—122no—
243C22—92no—
244C22—182no—
245C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
246C4—92no—
247C4—182no—
248C4—182no—
249C4#8865111yescharacteristic
250C4#8867711yescharacteristic
251C3—32no—
252C3—61no—
253C3#1059070211yescharacteristic
254C2—182no—
255C2#1172084112yes—
256C2#1172114711yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3C4×D32
prime radicalC22×#272C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#424
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
4. #63641⋊C3C2semidirect
5. #11720311C2semidirect
21. #63641D3semidirect
148. H(3,𝔽3)#424semidirect
222. C32#4839semidirect

Maximal subgroups

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

subgroupindex
C2×#885272
#117210972
C2×#884342
#63641⋊C32
#117203112
#105906953