Finite Group Databasebeta

← #13147069

Subgroups — #13147069

Order 1728.

365 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 56 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×#41⋊H(3,𝔽3)C211yesmaximal, characteristic
2#11720740C212yesmaximal, complemented
3C3×#5423—33nomaximal
4C3×#5236—31nomaximal
5#41⋊H(3,𝔽3)C411yescharacteristic, complemented
6#41⋊H(3,𝔽3)C2211yescharacteristic
7C2×Q8⋊H(3,𝔽3)C411yescharacteristic, complemented
8#88538—41nomaximal
9C2×C32×#41D311yescharacteristic
10C2×C3×#41⋊C3D313yes—
11C3×#926—31no—
12C2×C3×#208—31no—
13C3×GU(2,𝔽3)—36no—
14C3×#837—32no—
15C3×#837—32no—
16C2×C4×H(3,𝔽3)—41no—
17Q8⋊H(3,𝔽3)C2×C411yescharacteristic
18#6180—42no—
19#5423—33no—
20C3×#421—91no—
21#5236—31no—
22C32×#41—31no—
23C32×#41—31no—
24C32×#41Dic311yescharacteristic
25C32×#41C2×D311yescharacteristic
26C2×C32×Q8Dic311yescharacteristic
27C2×C32×D4—31no—
28C22×C32×C4—31no—
29C3×#41⋊C3Dic313yescomplemented
30C3×#41⋊C3C2×D313yes—
31C2×C3×SL(2,𝔽3)Dic313yescomplemented
32C3×#217—61no—
33C3×C4⋊Dic3—61no—
34C3×C4×Dic3—32no—
35C3×#208—32no—
36C3×#208—61no—
37C3×#208—123no—
38C2×C3×C3⋊C8—61no—
39C22×H(3,𝔽3)—41no—
40C4×H(3,𝔽3)—41no—
41C4×H(3,𝔽3)—41no—
42C2×C3×#41—31no—
43C2×C3×#41C3⋊D311yescharacteristic
44C2×#41⋊C3—33no—
45C2×C3×M16—91no—
46C3×#117—91no—
47#926—31no—
48C2×#208—31no—
49GU(2,𝔽3)—36no—
50C3×#104—92no—
51C3×#104—92no—
52#837—32no—
53#837—32no—
54C23×C32—31no—
55C32×Q8—31no—
56C32×Q8C2×Dic311yescharacteristic
57C32×D4—31no—
58C32×D4—31no—
59C32×D4—61no—
60C2×C32×C4—31no—
61C2×C32×C4—31no—
62C2×C32×C4—31no—
63C2×C32×C4—31no—
64C2×C32×C4—31no—
65C2×C32×C4—43no—
66C2×C32×C4S411yescharacteristic
67C2×C3×Dic3—62no—
68C3×SL(2,𝔽3)C2×Dic313yes—
69C3×C3⋊C8—62no—
70C3×C3⋊C8—126no—
71#421—91noSylow
72C2×H(3,𝔽3)—41no—
73C2×H(3,𝔽3)—81no—
74C3×#41—31no—
75C3×#41—31no—
76C3×#41—31no—
77C3×#41—31no—
78C3×#41—31no—
79C3×#41—31no—
80C3×#41—61no—
81C3×#41—61no—
82C3×#41C3⋊Dic311yescharacteristic
83C3×#41C2×C3⋊D311yescharacteristic
84C2×C3×Q8—31no—
85C2×C3×Q8C3⋊Dic311yescharacteristic
86C2×C3×D4—31no—
87C2×C3×D4—31no—
88C2×C3×D4—61no—
89C22×C3×C4—31no—
90C22×C3×C4—31no—
91C22×C3×C4—61no—
92#41⋊C3—33no—
93#41⋊C3—33no—
94C2×SL(2,𝔽3)—33no—
95C3×M16—92no—
96C3×M16—181no—
97C2×C3×C8—181no—
98C3×C4⋊C4—181no—
99C3×C22⋊C4—181no—
100C3×C42—92no—
101#217—61no—
102C4⋊Dic3—61no—
103C4×Dic3—32no—
104#208—32no—
105#208—61no—
106#208—123no—
107C2×C3⋊C8—61no—
108C22×C32—31no—
109C22×C32—31no—
110C22×C32—43no—
111C22×C32—121no—
112C22×C32SL(2,ℤ/4)11yescharacteristic
113C32×C4—31no—
114C32×C4—31no—
115C32×C4—43no—
116C32×C4—43no—
117C32×C4SL(2,ℤ/4)11yescharacteristic
118C32×C4C2×S411yescharacteristic
119C3×Dic3—122no—
120C2×#41#27211yescharacteristic
121C2×M16—91no—
122#117—91no—
123#104—92no—
124#104—92no—
125H(3,𝔽3)—41noSylow
126C23×C3—31no—
127C23×C3—31no—
128C23×C3—61no—
129C3×Q8—31no—
130C3×Q8—31no—
131C3×Q8—31no—
132C3×Q8—61no—
133C3×Q8C2×C3⋊Dic311yescharacteristic
134C3×D4—31no—
135C3×D4—31no—
136C3×D4—31no—
137C3×D4—31no—
138C3×D4—61no—
139C3×D4—61no—
140C3×D4—61no—
141C3×D4—61no—
142C3×D4—61no—
143C3×D4—61no—
144C2×C3×C4—31no—
145C2×C3×C4—31no—
146C2×C3×C4—31no—
147C2×C3×C4—31no—
148C2×C3×C4—31no—
149C2×C3×C4—31no—
150C2×C3×C4—31no—
151C2×C3×C4—31no—
152C2×C3×C4—31no—
153C2×C3×C4—31no—
154C2×C3×C4—31no—
155C2×C3×C4—61no—
156C2×C3×C4—61no—
157C2×C3×C4—61no—
158C2×C3×C4—61no—
159C2×C3×C4—61no—
160C2×C3×C4—123no—
161C2×C3×C4—182no—
162C2×C3×C4C3⋊S411yescharacteristic
163C2×Dic3—62no—
164SL(2,𝔽3)—33no—
165C3×C8—182no—
166C3⋊C8—62no—
167C3⋊C8—126no—
168C2×C32—21no—
169C2×C32—43no—
170C2×C32—61no—
171C2×C32—61no—
172C2×C32—83no—
173C2×C32C2×SL(2,ℤ/4)11yescharacteristic
174#41—31no—
175#41—31no—
176#41#108411yescharacteristic
177#41C2×#27211yescharacteristic
178C2×Q8#108411yescharacteristic
179C2×D4—31no—
180C22×C4—31no—
181M16—92no—
182M16—181no—
183C2×C8—181no—
184C4⋊C4—181no—
185C22⋊C4—181no—
186C42—92no—
187C22×C3—31no—
188C22×C3—31no—
189C22×C3—31no—
190C22×C3—31no—
191C22×C3—31no—
192C22×C3—61no—
193C22×C3—61no—
194C22×C3—121no—
195C22×C3—121no—
196C22×C3—122no—
197C22×C3—123no—
198C22×C3C3⋊SL(2,ℤ/4)11yescharacteristic
199C3×C4—31no—
200C3×C4—31no—
201C3×C4—31no—
202C3×C4—31no—
203C3×C4—31no—
204C3×C4—31no—
205C3×C4—61no—
206C3×C4—61no—
207C3×C4—123no—
208C3×C4—123no—
209C3×C4—362no—
210C3×C4C3⋊SL(2,ℤ/4)11yescharacteristic
211C3×C4C2×C3⋊S411yescharacteristic
212Dic3—122no—
213C32—43no—
214C32#542311yescharacteristic, complemented
215C23—31no—
216Q8—31no—
217Q8C2×#108411yescharacteristic
218D4—31no—
219D4—31no—
220D4—61no—
221C2×C4—31no—
222C2×C4—31no—
223C2×C4—31no—
224C2×C4—31no—
225C2×C4—31no—
226C2×C4—182no—
227C2×C4#625811yescharacteristic
228C8—182no—
229C2×C3—21no—
230C2×C3—31no—
231C2×C3—61no—
232C2×C3—61no—
233C2×C3—61no—
234C2×C3—61no—
235C2×C3—61no—
236C2×C3—121no—
237C2×C3—121no—
238C2×C3—123no—
239C2×C3—243no—
240C2×C3C2×C3⋊SL(2,ℤ/4)11yescharacteristic
241C22—31no—
242C22—31no—
243C22—121no—
244C22#8861311yescharacteristic
245C4—31no—
246C4—31no—
247C4—362no—
248C4#8861311yescharacteristic
249C4C2×#625811yescharacteristic
250C3—31no—
251C3—123no—
252C3#1059365711yescharacteristic
253C2—21no—
254C2—61no—
255C2—61no—
256C2C2×#8861311yescharacteristic
subgroupG/H
center, FrattiniC3×C4C2×C3⋊S4
commutator, hypocenterQ8⋊H(3,𝔽3)C2×C4
FittingC2×C32×#41D3
socleC2×C3C2×C3⋊SL(2,ℤ/4)
prime radicalC2×#41⋊H(3,𝔽3)C2
cosocle#41⋊H(3,𝔽3)C22
hypercenterC2×C3×C4C3⋊S4
LCSG ▷ Q8⋊H(3,𝔽3)
UCSC1 ◁ C3×C4 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. #11720740C2semidirect
5. #41⋊H(3,𝔽3)C4semidirect
7. C2×Q8⋊H(3,𝔽3)C4semidirect
29. C3×#41⋊C3Dic3semidirect
31. C2×C3×SL(2,𝔽3)Dic3semidirect
214. C32#5423semidirect

Maximal subgroups

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

subgroupindex
C2×#41⋊H(3,𝔽3)2
#117207402
C3×#54233
C3×#52363
#885384