Finite Group Databasebeta

← #13138900

Subgroups — #13138900

Order 1728.

509 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 509 conjugacy classes of subgroups — too many to draw. Only its 95 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
2C4×#6199C211yesmaximal, characteristic
3C2×#88435C211yesmaximal, characteristic
4#63641⋊C3C212yesmaximal, complemented
5#11720311C212yesmaximal, complemented
6#10590663—32nomaximal
7C2×C4×#272C2211yescharacteristic
8C2×#6199C412yes—
9C2×C8×H(3,𝔽3)C2211yescharacteristic
10C2×#6180C2211yescharacteristic
11#88527—22no—
12C4×(C3⋊Dic3)⋊C3C2212yes—
13C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
14#88447C411yescharacteristic
15#88447C411yescharacteristic
16#88435—22no—
17C2×C3×ΓL(1,𝔽25)—32no—
18C2×C3×ΓL(1,𝔽25)—61no—
19C4×D3×Dic3—32no—
20C2×#3711—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—
32#6199—24no—
33C8×H(3,𝔽3)—21no—
34#6180—21no—
35(C3⋊(C3⋊C8))⋊C3—22no—
36#5118—91no—
37#4837—92no—
38C2×C3×C4×D3—32no—
39C2×C3×C4×D3—61no—
40C2×D3×Dic3—32no—
41C2×D3×Dic3—32no—
42C2×C32×C8—21no—
43C2×C32×C8C2×D312yes—
44C4×C3⋊Dic3C2×D312yes—
45C2×C3⋊(C3⋊C8)C2×D312yes—
46C3×C4×Dic3—32no—
47C2×C3×C3⋊C8—32no—
48C2×C3×C3⋊C8—32no—
49C2×C3×C3⋊C8—61no—
50C3×ΓL(1,𝔽25)—64no—
51C3×ΓL(1,𝔽25)—64no—
52Dic32—32no—
53Dic32—32no—
54#3711—64no—
55C22×H(3,𝔽3)C22×C411yescharacteristic
56C2×#272—22no—
57C2×#272C4⋊C412yes—
58C4×H(3,𝔽3)C2×D411yescharacteristic
59C4×H(3,𝔽3)C2×Q811yescharacteristic
60#1084C4⋊C412yes—
61(C3⋊Dic3)⋊C3—22no—
62(C3⋊Dic3)⋊C3M1614yescomplemented
63C2×C3×M16—91no—
64C2×C4×Dic3—91no—
65C2×#208—91no—
66C2×ΓL(1,𝔽25)—32no—
67C2×ΓL(1,𝔽25)—61no—
68C2×ΓL(1,𝔽25)—92no—
69C42×D3—92no—
70C3×C4⋊C8—92no—
71Dic3⋊C8—32no—
72Dic3⋊C8—92no—
73Dic3⋊C8—92no—
74Dic3⋊C8—92no—
75C4⋊(C3⋊C8)—92no—
76C22×C3×D3—32no—
77C22×C3×D3—61no—
78C2×C32×C4—21no—
79C2×C32×C4C22×D312yes—
80C2×C3⋊Dic3C4×D312yes—
81C2×C3⋊Dic3C4×D312yes—
82C2×C3×Dic3—32no—
83C2×C3×Dic3—32no—
84C2×C3×Dic3—32no—
85C2×C3×Dic3—61no—
86C3×C4×D3—34no—
87C3×C4×D3—34no—
88C3×C4×D3—62no—
89C3×C4×D3—62no—
90D3×Dic3—64no—
91D3×Dic3—64no—
92C32×C8—22no—
93C32×C8—22no—
94C3⋊(C3⋊C8)—22no—
95C3×C3⋊C8—62no—
96C3×C3⋊C8—62no—
97C3×C3⋊C8—62no—
98#423—271noSylow
99C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
100C2×H(3,𝔽3)C2×M1612yes—
101#272—22no—
102C22×C3×C4—91no—
103C22×Dic3—92no—
104C2×C4×D3—32no—
105C2×C4×D3—61no—
106C2×C4×D3—92no—
107C2×C4×D3—92no—
108C3×M16—182no—
109C2×C3×C8—32no—
110C2×C3×C8—61no—
111C2×C3×C8—92no—
112C2×C3×C8—91no—
113C2×C3×C8D3211yescharacteristic
114C3×C42—92no—
115C4×Dic3—32no—
116C4×Dic3—92no—
117C4×Dic3—91no—
118C4×Dic3—91no—
119C4×Dic3—92no—
120C4×Dic3—92no—
121#208—182no—
122C2×C3⋊C8—32no—
123C2×C3⋊C8—32no—
124C2×C3⋊C8—61no—
125C2×C3⋊C8—92no—
126ΓL(1,𝔽25)—64no—
127ΓL(1,𝔽25)—64no—
128ΓL(1,𝔽25)—184no—
129C22×C32—21no—
130C22×C32C2×C4×D312yes—
131C2×C3×D3—34no—
132C2×C3×D3—62no—
133C2×C3×D3—62no—
134C2×C3×D3—62no—
135C2×C3×D3—122no—
136C32×C4—21no—
137C32×C4—21no—
138C32×C4D3×D412yes—
139C32×C4D3×Q812yes—
140C3⋊Dic3—22no—
141C3⋊Dic3ΓL(1,𝔽25)14yescomplemented
142C3×Dic3—34no—
143C3×Dic3—34no—
144C3×Dic3—62no—
145C3×Dic3—62no—
146C2×M16—91no—
147C2×M16—271no—
148C2×C42—271no—
149C4⋊C8—92no—
150C4⋊C8—272no—
151H(3,𝔽3)#42311yescharacteristic, complemented, Sylow
152C23×C3—91no—
153C22×D3—32no—
154C22×D3—61no—
155C2×C3×C4—32no—
156C2×C3×C4—61no—
157C2×C3×C4—92no—
158C2×C3×C4—91no—
159C2×C3×C4—92no—
160C2×C3×C4—92no—
161C2×C3×C4—92no—
162C2×C3×C4C2×D3211yescharacteristic
163C2×Dic3—32no—
164C2×Dic3—32no—
165C2×Dic3—32no—
166C2×Dic3—61no—
167C2×Dic3—92no—
168C2×Dic3—92no—
169C2×Dic3—184no—
170C4×D3—34no—
171C4×D3—34no—
172C4×D3—62no—
173C4×D3—62no—
174C4×D3—184no—
175C4×D3—184no—
176C3×C8—21no—
177C3×C8—62no—
178C3×C8—62no—
179C3×C8—182no—
180C3×C8—181no—
181C3⋊C8—62no—
182C3⋊C8—62no—
183C3⋊C8—62no—
184C3⋊C8—182no—
185C2×C32—22no—
186C2×C32—21no—
187C2×C32D3×C4⋊C412yes—
188C2×C32C2×ΓL(1,𝔽25)12yes—
189C2×C32D3×M1612yes—
190C3×D3—64no—
191C3×D3—122no—
192C22×C4—91no—
193C22×C4—272no—
194M16—182no—
195M16—542no—
196C2×C8—92no—
197C2×C8—91no—
198C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
199C42—92no—
200C42—271no—
201C42—271no—
202C22×C3—32no—
203C22×C3—61no—
204C22×C3—92no—
205C22×C3—182no—
206C22×C3C4×D3211yescharacteristic
207C2×D3—34no—
208C2×D3—62no—
209C2×D3—62no—
210C2×D3—62no—
211C2×D3—122no—
212C3×C4—32no—
213C3×C4—32no—
214C3×C4—61no—
215C3×C4—61no—
216C3×C4—94no—
217C3×C4—92no—
218C3×C4—182no—
219C3×C4#379711yescharacteristic
220C3×C4#381111yescharacteristic
221Dic3—34no—
222Dic3—34no—
223Dic3—62no—
224Dic3—62no—
225Dic3—94no—
226Dic3—182no—
227C32—21no—
228C32#483712yescomplemented
229C23—91no—
230C2×C4—92no—
231C2×C4—91no—
232C2×C4—92no—
233C2×C4—92no—
234C2×C4—92no—
235C2×C4—544no—
236C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
237C8—21no—
238C8—181no—
239C8—182no—
240C2×C3—32no—
241C2×C3—32no—
242C2×C3—32no—
243C2×C3—62no—
244C2×C3—61no—
245C2×C3—182no—
246C2×C3D3×ΓL(1,𝔽25)12yes—
247C2×C3#6388311yescharacteristic
248D3—64no—
249D3—122no—
250C22—92no—
251C22—182no—
252C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
253C4—94no—
254C4—92no—
255C4—182no—
256C4#8865111yescharacteristic
257C4#8867711yescharacteristic
258C3—32no—
259C3—61no—
260C3#1059066611yescharacteristic
261C2—182no—
262C2#1172083512yes—
263C2#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)#423
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
62. (C3⋊Dic3)⋊C3M16semidirect
141. C3⋊Dic3ΓL(1,𝔽25)semidirect
151. H(3,𝔽3)#423semidirect
228. C32#4837semidirect

Maximal subgroups

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

subgroupindex
C2×#885272
C4×#61992
C2×#884352
#63641⋊C32
#117203112
#105906633