Finite Group Databasebeta

← #13147082

Subgroups — #13147082

Order 1728.

365 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 36 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
12−1+4⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#11722394C211yesmaximal, characteristic, complemented
3#11720740C211yesmaximal, characteristic, complemented
4C3×#5431—33nomaximal
5C3×#5245—31nomaximal
6#41⋊H(3,𝔽3)C2211yescharacteristic
7C2×Q8⋊H(3,𝔽3)—21no—
8#88611—21no—
9#88545—41nomaximal
10C32×2−1+4D311yescharacteristic, complemented
11C3×2−1+4⋊C3D313yescomplemented
12C3×#986—33no—
13C3×#949—31no—
14C3×#949—31no—
15C3×#947—31no—
16C3×GU(2,𝔽3)—33no—
17C3×#837—31no—
18C3×#837—31no—
19C3×#836—31no—
20D4×H(3,𝔽3)—41no—
21C4⋊#272—41no—
22Q8⋊H(3,𝔽3)D411yescharacteristic
23#6180—41no—
24#5431—33no—
25C3×#454—91no—
26#5245—31no—
27C32×#41—31no—
28C32×#41—61no—
29C32×#41C2×D311yescharacteristic
30C2×C32×Q8—21no—
31C2×C32×Q8—31no—
32C2×C3×D12—31no—
33C3×#41⋊C3C2×D313yes—
34C2×C3×SL(2,𝔽3)—23no—
35C3×GL(2,𝔽3)—63no—
36C3×#215—61no—
37C3×#215—61no—
38C3×#213—61no—
39C3×#213—61no—
40C3×#213—123no—
41C3×(C2×D3)⋊C4—62no—
42C3×C4×Dic3—31no—
43C3×#208—31no—
44C3×#208—31no—
45C22×H(3,𝔽3)—81no—
46C2×#272—81no—
47C4×H(3,𝔽3)—41no—
48C3×2−1+4—31no—
49C3×2−1+4C3⋊D311yescharacteristic
502−1+4⋊C3—33no—
51#986—33no—
52C3×AGL(1,ℤ/8)—91no—
53C3×AGL(1,ℤ/8)—91no—
54C3×#124—91no—
55#949—31no—
56#949—31no—
57#947—31no—
58GU(2,𝔽3)—33no—
59C3×#104—91no—
60C3×#104—91no—
61C3×#101—91no—
62#837—31no—
63#837—31no—
64#836—31no—
65C22×C3×D3—61no—
66C32×Q8—31no—
67C32×Q8—61no—
68C32×Q8#6711yescharacteristic
69C32×D4—31no—
70C32×D4—43no—
71C32×D4—61no—
72C32×D4S411yescharacteristic
73C2×C32×C4—31no—
74C2×C32×C4—32no—
75C2×C32×C4—61no—
76C2×C3×Dic3—61no—
77C3×D12—61no—
78C3×D12—61no—
79C3×D12—123no—
80C3×SL(2,𝔽3)#6713yes—
81C3×C3⋊C8—61no—
82C3×C3⋊C8—61no—
83C3×C3⋊C8—123no—
84#454—91noSylow
85C2×H(3,𝔽3)—41no—
86C2×H(3,𝔽3)—161no—
87#272—161no—
88C3×#41—31no—
89C3×#41—31no—
90C3×#41—31no—
91C3×#41—61no—
92C3×#41—61no—
93C3×#41—62no—
94C3×#41—61no—
95C3×#41C2×C3⋊D311yescharacteristic
96C2×C3×Q8—21no—
97C2×C3×Q8—31no—
98C2×C3×Q8—31no—
99C2×C3×Q8—61no—
100C2×C3×Q8—61no—
101C2×C3×D4—91no—
102C2×D12—31no—
103#41⋊C3—33no—
104C2×SL(2,𝔽3)—63no—
105GL(2,𝔽3)—63no—
106C3×SD16—181no—
107C3×SD16—181no—
108C3×D8—181no—
109C3×D8—181no—
110C3×M16—91no—
111C3×M16—91no—
112C3×C22⋊C4—182no—
113C3×C42—91no—
114#215—61no—
115#215—61no—
116#213—61no—
117#213—61no—
118#213—123no—
119(C2×D3)⋊C4—62no—
120C4×Dic3—31no—
121#208—31no—
122#208—31no—
123C22×C32—21no—
124C22×C32—31no—
125C22×C32—83no—
126C2×C3×D3—122no—
127C2×C3×D3—121no—
128C2×C3×D3—243no—
129C32×C4—31no—
130C32×C4—43no—
131C32×C4—61no—
132C32×C4C2×S411yescharacteristic
133C3×Dic3—121no—
1342−1+4#27211yescharacteristic, complemented
135AGL(1,ℤ/8)—91no—
136AGL(1,ℤ/8)—91no—
137#124—91no—
138#104—91no—
139#104—91no—
140#101—91no—
141H(3,𝔽3)—41noSylow
142C23×C3—181no—
143C22×D3—61no—
144C3×Q8—31no—
145C3×Q8—31no—
146C3×Q8—31no—
147C3×Q8—61no—
148C3×Q8—61no—
149C3×Q8—61no—
150C3×Q8—62no—
151C3×Q8#63811yescharacteristic
152C3×D4—31no—
153C3×D4—31no—
154C3×D4—31no—
155C3×D4—61no—
156C3×D4—61no—
157C3×D4—61no—
158C3×D4—62no—
159C3×D4—123no—
160C3×D4—181no—
161C3×D4—181no—
162C3×D4C3⋊S411yescharacteristic
163C2×C3×C4—31no—
164C2×C3×C4—31no—
165C2×C3×C4—32no—
166C2×C3×C4—32no—
167C2×C3×C4—61no—
168C2×C3×C4—62no—
169C2×C3×C4—61no—
170C2×C3×C4—61no—
171C2×C3×C4—62no—
172C2×C3×C4—181no—
173C2×Dic3—61no—
174D12—61no—
175D12—61no—
176D12—123no—
177SL(2,𝔽3)—33no—
178C3×C8—181no—
179C3×C8—181no—
180C3⋊C8—61no—
181C3⋊C8—61no—
182C3⋊C8—123no—
183C2×C32—41no—
184C2×C32—43no—
185C2×C32—61no—
186C2×C32—163no—
187C2×C32GL(2,ℤ/4)11yescharacteristic
188C3×D3—241no—
189C3×D3—483no—
190#41—31no—
191#41—61no—
192#41C2×#27211yescharacteristic
193C2×Q8—21no—
194C2×Q8—31no—
195C2×D4—91no—
196SD16—181no—
197SD16—181no—
198D8—181no—
199D8—181no—
200M16—91no—
201M16—91no—
202C22⋊C4—182no—
203C42—91no—
204C22×C3—21no—
205C22×C3—31no—
206C22×C3—31no—
207C22×C3—61no—
208C22×C3—61no—
209C22×C3—243no—
210C22×C3—362no—
211C22×C3—361no—
212C2×D3—121no—
213C2×D3—122no—
214C2×D3—243no—
215C3×C4—31no—
216C3×C4—31no—
217C3×C4—31no—
218C3×C4—61no—
219C3×C4—62no—
220C3×C4—61no—
221C3×C4—61no—
222C3×C4—123no—
223C3×C4—361no—
224C3×C4C2×C3⋊S411yescharacteristic
225Dic3—121no—
226C32—43no—
227C32#543111yescharacteristic, complemented
228C23—181no—
229Q8—31no—
230Q8—61no—
231Q8#623511yescharacteristic
232D4—31no—
233D4—61no—
234D4—181no—
235D4—181no—
236D4#625811yescharacteristic
237C2×C4—31no—
238C2×C4—32no—
239C2×C4—61no—
240C2×C4—181no—
241C8—181no—
242C8—181no—
243C2×C3—31no—
244C2×C3—41no—
245C2×C3—61no—
246C2×C3—61no—
247C2×C3—121no—
248C2×C3—121no—
249C2×C3—123no—
250C2×C3—483no—
251C2×C3—721no—
252C2×C3#6426911yescharacteristic
253D3—241no—
254D3—483no—
255C22—21no—
256C22—31no—
257C22—362no—
258C22—361no—
259C4—31no—
260C4—61no—
261C4—361no—
262C4C2×#625811yescharacteristic
263C3—31no—
264C3—123no—
265C3#1059366511yescharacteristic
266C2—41no—
267C2—61no—
268C2—721no—
269C2#1172239811yescharacteristic
subgroupG/H
center, socleC2×C3#64269
commutator, cosocle#41⋊H(3,𝔽3)C22
FrattiniC3×C4C2×C3⋊S4
FittingC32×2−1+4D3
hypercenterC3×D4C3⋊S4
hypocenterQ8⋊H(3,𝔽3)D4
LCSG ▷ #41⋊H(3,𝔽3) ▷ Q8⋊H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×D4

Complements

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

NH ≅ G/Ntype
1. 2−1+4⋊H(3,𝔽3)C2semidirect
2. #11722394C2semidirect
3. #11720740C2semidirect
10. C32×2−1+4D3semidirect
11. C3×2−1+4⋊C3D3semidirect
134. 2−1+4#272semidirect
227. C32#5431semidirect

Maximal subgroups

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

subgroupindex
2−1+4⋊H(3,𝔽3)2
#117223942
#117207402
C3×#54313
C3×#52453
#885454