Finite Group Databasebeta

← #13138784

Subgroups — #13138784

Order 1728.

471 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 471 conjugacy classes of subgroups — too many to draw. Only its 91 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×#3743⋊C3C212yesmaximal, complemented
2C4×#6198C211yesmaximal, characteristic
3C8×#1084C211yesmaximal, characteristic, complemented
4#11720317C212yesmaximal, complemented
5#11720304C211yesmaximal, characteristic, complemented
6#10590729—32nomaximal
7C2×C4×(C3⋊D3)⋊C3C2212yes—
8C2×#6198C411yescharacteristic
9C2×C8×H(3,𝔽3)C2211yescharacteristic
10C4×#1084C2211yescharacteristic
11C2×#6180C2211yescharacteristic
12C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
13#3743⋊C3—24no—
14#88450C412yes—
15#88449C411yescharacteristic
16C2×#3743D312yescomplemented
17C2×C3×ΓL(1,𝔽25)—32no—
18C4×#626—32no—
19C3×C8×Dic3—32no—
20C3×C8×Dic3—61no—
21Dic3⋊(C3⋊C8)—32no—
22#63557—32no—
23(C2×D3)⋊(C3⋊C8)—32no—
24C22×(C3⋊D3)⋊C3C2×C412yes—
25C2×C4×H(3,𝔽3)C2311yescharacteristic
26C2×#1084C2×C411yescharacteristic
27C2×#1084C2×C411yescharacteristic
28C2×(C3⋊Dic3)⋊C3C2×C412yes—
29C4×(C3⋊D3)⋊C3—22no—
30C4×(C3⋊D3)⋊C3—22no—
31#6198—22no—
32C8×H(3,𝔽3)D412yescomplemented
33#6180—21no—
34(C3⋊(C3⋊C8))⋊C3—22no—
35#5128—92no—
36#4688—91no—
37C2×C4×C3⋊D3C2×D312yes—
38C2×C3×C4×D3—32no—
39C2×#626—32no—
40C2×C32×C8—21no—
41C2×C32×C8C2×D312yes—
42C2×C3⋊(C3⋊C8)C2×D312yes—
43#3743—24no—
44C3×C4×Dic3—32no—
45C3×C4×Dic3—61no—
46C2×C3×C3⋊C8—32no—
47C2×C3×C3⋊C8—32no—
48C2×C3×C3⋊C8—61no—
49C3×ΓL(1,𝔽25)—64no—
50Dic3⋊Dic3—32no—
51#3722—32no—
52(C2×D3)⋊Dic3—32no—
53C22×H(3,𝔽3)C22×C411yescharacteristic
54C2×(C3⋊D3)⋊C3—22no—
55C2×(C3⋊D3)⋊C3—22no—
56C2×(C3⋊D3)⋊C3—22no—
57C4×H(3,𝔽3)C2×D411yescharacteristic
58C4×H(3,𝔽3)#4111yescharacteristic
59#1084—21no—
60#1084M1612yescomplemented
61(C3⋊Dic3)⋊C3—22no—
62C2×C3×M16—92no—
63C4×#67—92no—
64C2×ΓL(1,𝔽25)—32no—
65C2×ΓL(1,𝔽25)—92no—
66C4×D12—91no—
67C3×C4×C8—91no—
68#830—92no—
69(C2×D3)⋊C8—92no—
70(C2×D3)⋊C8—92no—
71Dic3⋊C8—92no—
72C8×Dic3—32no—
73C8×Dic3—61no—
74C4⋊(C3⋊C8)—91no—
75C22×C3⋊D3C4×D312yes—
76C22×C3×D3—32no—
77C2×C32×C4—21no—
78C2×C32×C4C22×D312yes—
79C2×C3⋊Dic3C4×D312yes—
80C4×C3⋊D3—22no—
81C4×C3⋊D3—22no—
82C2×C3×Dic3—32no—
83C2×C3×Dic3—32no—
84C2×C3×Dic3—32no—
85C2×C3×Dic3—61no—
86C2×C3×Dic3—61no—
87C3×C4×D3—62no—
88C3×C4×D3—62no—
89#626—64no—
90C32×C8—22no—
91C32×C8#6714yescomplemented
92C3⋊(C3⋊C8)—22no—
93C3×C3⋊C8—62no—
94C3×C3⋊C8—62no—
95C3×C3⋊C8—62no—
96#436—271noSylow
97C2×H(3,𝔽3)C4×D411yescharacteristic
98C2×H(3,𝔽3)C2×M1611yescharacteristic
99C2×H(3,𝔽3)#13111yescharacteristic
100(C3⋊D3)⋊C3—42no—
101C22×C3×C4—92no—
102C2×#67—92no—
103C2×D12—91no—
104C2×C4×D3—32no—
105C2×C4×D3—92no—
106C3×M16—184no—
107C2×C3×C8—32no—
108C2×C3×C8—61no—
109C2×C3×C8—92no—
110C2×C3×C8—91no—
111C2×C3×C8D3211yescharacteristic
112C3×C42—91no—
113#217—92no—
114(C2×D3)⋊C4—92no—
115(C2×D3)⋊C4—92no—
116C4⋊Dic3—91no—
117Dic3⋊C4—92no—
118C4×Dic3—32no—
119C4×Dic3—61no—
120C2×C3⋊C8—32no—
121C2×C3⋊C8—32no—
122C2×C3⋊C8—61no—
123C2×C3⋊C8—92no—
124ΓL(1,𝔽25)—64no—
125ΓL(1,𝔽25)—184no—
126C22×C32—21no—
127C22×C32C2×C4×D312yes—
128C2×C3⋊D3—22no—
129C2×C3⋊D3—22no—
130C2×C3⋊D3—22no—
131C2×C3×D3—62no—
132C2×C3×D3—62no—
133C2×C3×D3—62no—
134C32×C4—21no—
135C32×C4—21no—
136C32×C4#23512yes—
137C32×C4C2×#6712yes—
138C3⋊Dic3—22no—
139C3×Dic3—34no—
140C3×Dic3—62no—
141C3×Dic3—62no—
142C3×Dic3—62no—
143C3×Dic3—62no—
144C2×M16—92no—
145C4×D4—271no—
146C4⋊C8—271no—
147C22⋊C8—272no—
148C4×C8—91no—
149H(3,𝔽3)#43611yescharacteristic, complemented, Sylow
150C23×C3—92no—
151C22×D3—32no—
152C22×D3—92no—
153C2×C3×C4—32no—
154C2×C3×C4—61no—
155C2×C3×C4—91no—
156C2×C3×C4—92no—
157C2×C3×C4—91no—
158C2×C3×C4—182no—
159C2×C3×C4—182no—
160C2×C3×C4C2×D3211yescharacteristic
161#67—184no—
162C2×Dic3—32no—
163C2×Dic3—32no—
164C2×Dic3—32no—
165C2×Dic3—61no—
166C2×Dic3—61no—
167C2×Dic3—92no—
168D12—182no—
169C4×D3—62no—
170C4×D3—62no—
171C4×D3—182no—
172C4×D3—182no—
173C3×C8—34no—
174C3×C8—62no—
175C3×C8—182no—
176C3×C8—181no—
177C3×C8#62512yes—
178C3⋊C8—62no—
179C3⋊C8—62no—
180C3⋊C8—62no—
181C3⋊C8—182no—
182C2×C32—21no—
183C2×C32—21no—
184C2×C32—21no—
185C2×C32D3×M1612yes—
186C2×C32#90712yes—
187C2×C32C4×#6712yes—
188C3⋊D3—42no—
189C3×D3—122no—
190C2×D4—271no—
191C22×C4—92no—
192M16—184no—
193C2×C8—91no—
194C2×C8—92no—
195C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
196C4⋊C4—271no—
197C22⋊C4—272no—
198C42—91no—
199C22×C3—32no—
200C22×C3—61no—
201C22×C3—182no—
202C22×C3—182no—
203C22×C3—182no—
204C22×C3C4×D3211yescharacteristic
205C2×D3—62no—
206C2×D3—62no—
207C2×D3—62no—
208C2×D3—182no—
209C2×D3—182no—
210C2×D3—182no—
211C3×C4—32no—
212C3×C4—32no—
213C3×C4—61no—
214C3×C4—61no—
215C3×C4—92no—
216C3×C4—182no—
217C3×C4—181no—
218C3×C4#379811yescharacteristic
219C3×C4C2×#62511yescharacteristic
220Dic3—34no—
221Dic3—62no—
222Dic3—62no—
223Dic3—62no—
224Dic3—62no—
225Dic3—182no—
226C32—21no—
227C32#512812yescomplemented
228C23—92no—
229D4—542no—
230C2×C4—91no—
231C2×C4—91no—
232C2×C4—92no—
233C2×C4—182no—
234C2×C4—182no—
235C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
236C8—181no—
237C8—182no—
238C8#619812yescomplemented
239C2×C3—32no—
240C2×C3—32no—
241C2×C3—32no—
242C2×C3—61no—
243C2×C3—61no—
244C2×C3—61no—
245C2×C3—362no—
246C2×C3#6379111yescharacteristic
247C2×C3#6380511yescharacteristic
248C2×C3C4×#62511yescharacteristic
249D3—122no—
250D3—362no—
251C22—182no—
252C22—182no—
253C22—182no—
254C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
255C4—92no—
256C4—181no—
257C4—182no—
258C4#8865011yescharacteristic
259C4C2×#619811yescharacteristic
260C3—32no—
261C3—61no—
262C3#1059073311yescharacteristic
263C2—362no—
264C2#1172083611yescharacteristic
265C2#1172084211yescharacteristic
266C2C4×#619811yescharacteristic
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 radicalC2×#6198C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#436
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
1. C2×#3743⋊C3C2semidirect
3. C8×#1084C2semidirect
4. #11720317C2semidirect
5. #11720304C2semidirect
16. C2×#3743D3semidirect
32. C8×H(3,𝔽3)D4semidirect
60. #1084M16semidirect
91. C32×C8#67semidirect
149. H(3,𝔽3)#436semidirect
227. C32#5128semidirect
238. C8#6198semidirect

Maximal subgroups

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

subgroupindex
C2×#3743⋊C32
C4×#61982
C8×#10842
#117203172
#117203042
#105907293