Finite Group Databasebeta

← #13144077

Subgroups — #13144077

Order 1728.

627 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 627 conjugacy classes of subgroups — too many to draw. Only its 125 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
2C2×C8×#272C211yesmaximal, characteristic
3#11721718C211yesmaximal, characteristic
4H(3,𝔽3)×C4⋊C8C211yesmaximal, characteristic, complemented
5#11720547C211yesmaximal, characteristic, complemented
6#11720547C211yesmaximal, characteristic, complemented
7C4⋊#6180C211yesmaximal, characteristic, complemented
8C3×#4839—34nomaximal
9C2×C4×#272C2211yescharacteristic
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12C42×H(3,𝔽3)C2211yescharacteristic
13#88542C411yescharacteristic
14#88542C411yescharacteristic
15#88540C411yescharacteristic
16#88540C411yescharacteristic
17C4×#1084C2211yescharacteristic
18C2×#6180C2211yescharacteristic
19C2×#6180C2211yescharacteristic
20#88527—22no—
21C8×#272—22no—
22C2×C3×ΓL(1,𝔽25)—34no—
23C2×C3×D3×C8—34no—
24C3×#872—34no—
25C32×C4⋊C8D314yescomplemented
26C3×Dic3⋊C8—34no—
27C3×Dic3⋊C8—34no—
28C3×C4⋊(C3⋊C8)—34no—
29C22×#272C2×C411yescharacteristic
30C2×C4×H(3,𝔽3)C2×C411yescharacteristic
31C2×C4×H(3,𝔽3)C2×C411yescharacteristic
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33C2×#1084C2×C411yescharacteristic
34C2×#1084C2×C411yescharacteristic
35C2×#1084C2×C411yescharacteristic
36C4×#272D412yes—
37C4×#272Q812yes—
38C8×H(3,𝔽3)—21no—
39C8×H(3,𝔽3)—21no—
40#6180—21no—
41#6180—21no—
42C3×#424—91no—
43#4839—34no—
44C2×C3×C4×D3—34no—
45C2×C32×C8C2×D314yes—
46C2×C32×C8C2×D314yes—
47C32×C42C2×D314yes—
48C3×(C2×D3)⋊C4—34no—
49C3×(C2×D3)⋊C4—34no—
50C3×Dic3⋊C4—34no—
51C3×Dic3⋊C4—34no—
52C3×C4×Dic3—34no—
53C2×C3×C3⋊C8—34no—
54C2×C3×C3⋊C8—34no—
55C3×ΓL(1,𝔽25)—68no—
56C3×D3×C8—68no—
57C22×H(3,𝔽3)C22×C411yescharacteristic
58C2×#272—21no—
59C2×#272—21no—
60C2×#272C4⋊C412yes—
61C4×H(3,𝔽3)—21no—
62C4×H(3,𝔽3)—21no—
63C4×H(3,𝔽3)C2×D411yescharacteristic
64C4×H(3,𝔽3)C2×Q811yescharacteristic
65#1084—21no—
66#1084—21no—
67#1084C4⋊C412yes—
68C2×C3×M16—91no—
69C22×C3×C8—91no—
70C3×#117—91no—
71C2×ΓL(1,𝔽25)—34no—
72C2×D3×C8—34no—
73#872—34no—
74C3×C4⋊C8—34no—
75C3×C4⋊C8—91no—
76C3×C4⋊C8—91no—
77C3×C4⋊C8—91no—
78C3×C4⋊C8C3⋊D311yescharacteristic
79Dic3⋊C8—34no—
80Dic3⋊C8—34no—
81C4⋊(C3⋊C8)—34no—
82C22×C3×D3—34no—
83C2×C32×C4C4×D314yes—
84C2×C32×C4C4×D314yes—
85C2×C32×C4C22×D314yes—
86C2×C3×Dic3—34no—
87C2×C3×Dic3—34no—
88C2×C3×Dic3—34no—
89C3×C4×D3—38no—
90C3×C4×D3—38no—
91C32×C8—24no—
92C32×C8—24no—
93C3×C3⋊C8—64no—
94C3×C3⋊C8—64no—
95#424—91noSylow
96C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
97C2×H(3,𝔽3)#13111yescharacteristic
98C2×H(3,𝔽3)#13111yescharacteristic
99#272—22no—
100C22×C3×C4—91no—
101C2×C4×D3—34no—
102C3×M16—182no—
103C2×C3×C8—34no—
104C2×C3×C8—34no—
105C2×C3×C8—91no—
106C2×C3×C8—91no—
107C2×C3×C8—182no—
108C2×C3×C8C2×C3⋊D311yescharacteristic
109C2×C3×C8C2×C3⋊D311yescharacteristic
110C3×C4⋊C4—91no—
111C3×C4⋊C4—91no—
112C3×C22⋊C4—91no—
113C3×C22⋊C4—91no—
114C3×C42—34no—
115C3×C42—91no—
116C3×C42C2×C3⋊D311yescharacteristic
117(C2×D3)⋊C4—34no—
118(C2×D3)⋊C4—34no—
119Dic3⋊C4—34no—
120Dic3⋊C4—34no—
121C4×Dic3—34no—
122C2×C3⋊C8—34no—
123C2×C3⋊C8—34no—
124ΓL(1,𝔽25)—68no—
125D3×C8—68no—
126C22×C32C2×C4×D314yes—
127C2×C3×D3—38no—
128C2×C3×D3—64no—
129C2×C3×D3—64no—
130C32×C4—24no—
131C32×C4—24no—
132C32×C4D3×D414yes—
133C32×C4D3×Q814yes—
134C3×Dic3—38no—
135C3×Dic3—64no—
136C3×Dic3—64no—
137C2×M16—91no—
138C22×C8—91no—
139#117—91no—
140C4⋊C8—91no—
141C4⋊C8—91no—
142C4⋊C8—91no—
143C4⋊C8#27211yescharacteristic, complemented
144H(3,𝔽3)#42411yescharacteristic, complemented, Sylow
145C23×C3—91no—
146C22×D3—34no—
147C2×C3×C4—34no—
148C2×C3×C4—34no—
149C2×C3×C4—34no—
150C2×C3×C4—91no—
151C2×C3×C4—92no—
152C2×C3×C4—92no—
153C2×C3×C4—91no—
154C2×C3×C4—91no—
155C2×C3×C4C4×C3⋊D311yescharacteristic
156C2×C3×C4C4×C3⋊D311yescharacteristic
157C2×C3×C4C22×C3⋊D311yescharacteristic
158C2×Dic3—34no—
159C2×Dic3—34no—
160C2×Dic3—34no—
161C4×D3—38no—
162C4×D3—38no—
163C3×C8—21no—
164C3×C8—21no—
165C3×C8—64no—
166C3×C8—64no—
167C3×C8—181no—
168C3×C8—181no—
169C3⋊C8—64no—
170C3⋊C8—64no—
171C2×C32D3×C4⋊C414yes—
172C2×C32#90114yes—
173C2×C32#90714yes—
174C3×D3—68no—
175C22×C4—91no—
176M16—182no—
177C2×C8—91no—
178C2×C8—91no—
179C2×C8—182no—
180C2×C8C2×#27211yescharacteristic
181C2×C8C2×#27211yescharacteristic
182C4⋊C4—91no—
183C4⋊C4—91no—
184C22⋊C4—91no—
185C22⋊C4—91no—
186C42—91no—
187C42C2×#27211yescharacteristic
188C22×C3—34no—
189C22×C3—92no—
190C22×C3—181no—
191C22×C3—181no—
192C22×C3C2×C4×C3⋊D311yescharacteristic
193C2×D3—38no—
194C2×D3—64no—
195C2×D3—64no—
196C3×C4—21no—
197C3×C4—21no—
198C3×C4—34no—
199C3×C4—34no—
200C3×C4—64no—
201C3×C4—64no—
202C3×C4—92no—
203C3×C4—181no—
204C3×C4—181no—
205C3×C4D4×C3⋊D311yescharacteristic
206C3×C4Q8×C3⋊D311yescharacteristic
207Dic3—38no—
208Dic3—64no—
209Dic3—64no—
210C32#483914yescomplemented
211C23—91no—
212C2×C4—91no—
213C2×C4—92no—
214C2×C4—92no—
215C2×C4—91no—
216C2×C4—91no—
217C2×C4C4×#27211yescharacteristic
218C2×C4C4×#27211yescharacteristic
219C2×C4C22×#27211yescharacteristic
220C8—21no—
221C8—21no—
222C8—181no—
223C8—181no—
224C2×C3—34no—
225C2×C3—34no—
226C2×C3—34no—
227C2×C3—182no—
228C2×C3C4⋊C4×C3⋊D311yescharacteristic
229C2×C3#6411011yescharacteristic
230C2×C3#6411611yescharacteristic
231D3—68no—
232C22—92no—
233C22—181no—
234C22—181no—
235C22C2×C4×#27211yescharacteristic
236C4—21no—
237C4—21no—
238C4—92no—
239C4—181no—
240C4—181no—
241C4D4×#27211yescharacteristic
242C4Q8×#27211yescharacteristic
243C3—34no—
244C3#1059253811yescharacteristic
245C2—182no—
246C2C4⋊C4×#27211yescharacteristic
247C2#1172177611yescharacteristic
248C2#1172178611yescharacteristic
subgroupG/H
center, FrattiniC2×C3×C4C22×C3⋊D3
commutatorC22×H(3,𝔽3)C22×C4
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)#424
LCSG ▷ C22×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
4. H(3,𝔽3)×C4⋊C8C2semidirect
5. #11720547C2semidirect
6. #11720547C2semidirect
7. C4⋊#6180C2semidirect
25. C32×C4⋊C8D3semidirect
143. C4⋊C8#272semidirect
144. H(3,𝔽3)#424semidirect
210. C32#4839semidirect

Maximal subgroups

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

subgroupindex
C2×#885272
C2×C8×#2722
#117217182
H(3,𝔽3)×C4⋊C82
#117205472
C4⋊#61802
C3×#48393