Finite Group Databasebeta

← #13180145

Subgroups — #13180145

Order 1728.

310 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 48 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×#89091C211yesmaximal, characteristic
2C2×C3×AΓL(1,𝔽9)C211yesmaximal, characteristic
3#11724153C211yesmaximal, characteristic, complemented
4Dic3×GO+(4,𝔽2)C211yesmaximal, characteristic
5#11724105C211yesmaximal, characteristic, complemented
6#11724102C211yesmaximal, characteristic, complemented
7Dic3⋊AGL(1,𝔽9)C211yesmaximal, characteristic, complemented
8#10596409—31nomaximal
9C2×C3⋊PSU(3,𝔽2)C2211yescharacteristic
10C2×C3×PSU(3,𝔽2)C2211yescharacteristic
11C2×C3×GO+(4,𝔽2)C2211yescharacteristic
12C2×C3⋊AGL(1,𝔽9)C2211yescharacteristic
13C2×C3×AGL(1,𝔽9)C2211yescharacteristic
14#89091—22no—
15C3×AΓL(1,𝔽9)—22no—
16D32×Dic3—21no—
17#88945—21no—
18#88940C2211yescharacteristic
19D32⋊Dic3—21no—
20Dic3⋊#157—21no—
21Dic3×#157C2211yescharacteristic
22C2×AΓL(1,𝔽9)—31no—
23C2×AΓL(1,𝔽9)D311yescharacteristic
24C4⋊PSU(3,𝔽2)—31no—
25C4×GO+(4,𝔽2)—31no—
26C4⋊AGL(1,𝔽9)—31no—
27PSU(3,𝔽2)⋊C4—31no—
28PSU(3,𝔽2)⋊C4—31no—
29C2×C3×D32—21no—
30C2×C3⋊#157—21no—
31C2×C3⋊#157D411yescharacteristic
32C2×C3×#157—21no—
33C2×C3×#157C2311yescharacteristic
34C3⋊PSU(3,𝔽2)—21no—
35C3⋊PSU(3,𝔽2)—21no—
36C3×PSU(3,𝔽2)—21no—
37C3×PSU(3,𝔽2)—21no—
38C3×GO+(4,𝔽2)—21no—
39C3×GO+(4,𝔽2)D412yes—
40C3⋊AGL(1,𝔽9)—21no—
41C3×AGL(1,𝔽9)—21no—
42#6294—21no—
43Dic3×C3⋊D3D411yescharacteristic
44D3×C3⋊Dic3—41no—
45C3×D3×Dic3—41no—
46#5162—91nomaximal
47C2×PSU(3,𝔽2)—31no—
48C2×PSU(3,𝔽2)C2×D311yescharacteristic
49C2×GO+(4,𝔽2)C2×D311yescharacteristic
50C2×AGL(1,𝔽9)—31no—
51C2×AGL(1,𝔽9)C2×D311yescharacteristic
52AΓL(1,𝔽9)—22no—
53AΓL(1,𝔽9)—62no—
54C2×D3×Dic3—121no—
55C4×D32—61no—
56C4⋊#157—61no—
57C4×#157—31no—
58#3777—61no—
59#3773—31no—
60D32⋊C4—61no—
61C2×C3×C3⋊D3C2×D411yescharacteristic
62C2×C32×D3—41no—
63C3×D32—22no—
64C3×D32—41no—
65C3⋊#157—21no—
66C3⋊#157—41no—
67C3×#157—41no—
68C3×#157C2×D411yescharacteristic
69C3×#157#4111yescharacteristic
70C3⋊(C3⋊Dic3)SD1611yescharacteristic, complemented
71C3×C3⋊Dic3—41no—
72C32×Dic3SD1611yescharacteristic, complemented
73C2×C3×SD16—91no—
74#944—91no—
75D4×Dic3—91no—
76C2×#214—91no—
77#835—91no—
78#816—91no—
79Dic3⋊C8—91no—
80C22×C3×D3—121no—
81C2×D32—21no—
82C2×#157—21no—
83C2×#157—31no—
84C2×#157—61no—
85C2×#157C22×D311yescharacteristic
86PSU(3,𝔽2)—21no—
87PSU(3,𝔽2)—21no—
88PSU(3,𝔽2)—61no—
89PSU(3,𝔽2)—61no—
90GO+(4,𝔽2)—21no—
91GO+(4,𝔽2)#6712yes—
92AGL(1,𝔽9)—21no—
93AGL(1,𝔽9)—61no—
94C2×C3⋊Dic3—121no—
95C4×C3⋊D3—31no—
96C2×C3×Dic3—121no—
97C3×C4×D3—121no—
98#624—61no—
99D3×Dic3—41no—
100D3×Dic3—121no—
101D3×Dic3—121no—
102D3×Dic3—121no—
103D3×Dic3—121no—
104D3×Dic3—241no—
105#461—271noSylow
106C2×C33C2×SD1611yescharacteristic
107C3×C3⋊D3#12111yescharacteristic
108C3×C3⋊D3#13711yescharacteristic
109C32×D3—42no—
110C2×C3×Q8—91no—
111C2×C3×D4—91no—
112C22×Dic3—181no—
113C2×C4×D3—361no—
114C2×Dic6—91no—
115C3×SD16—182no—
116C2×C3×C8—91no—
117#217—181no—
118#214—182no—
119C4⋊Dic3—91no—
120Dic3⋊C4—181no—
121Dic3⋊C4—181no—
122C4×Dic3—91no—
123C2×C3⋊C8—91no—
124C22×C32—121no—
125C2×C3⋊D3D3×D411yescharacteristic
126C2×C3×D3—41no—
127C2×C3×D3—41no—
128C2×C3×D3—81no—
129C2×C3×D3—121no—
130C2×C3×D3—122no—
131C2×C3×D3—122no—
132D32—22no—
133D32—41no—
134#157—41no—
135#157—61no—
136#157—121no—
137#157#23711yescharacteristic
138#157C2×#6711yescharacteristic
139C32×C4—31no—
140C3⋊Dic3—31no—
141C3⋊Dic3—41no—
142C3⋊Dic3—121no—
143C3⋊Dic3—241no—
144C3×Dic3—41no—
145C3×Dic3—121no—
146C3×Dic3—121no—
147C3×Dic3—241no—
148C2×SD16—91no—
149C2×SD16—271no—
150C4⋊Q8—271no—
151C4×D4—271no—
152C4⋊C8—271no—
153#103—271no—
154#103—271no—
155C33#46111yescharacteristic, complemented, Sylow
156C23×C3—181no—
157C22×D3—121no—
158C3×Q8—181no—
159C3×Q8—181no—
160C3×D4—92no—
161C3×D4—181no—
162C2×C3×C4—91no—
163C2×C3×C4—181no—
164C2×C3×C4—361no—
165C2×Dic3—91no—
166C2×Dic3—91no—
167C2×Dic3—121no—
168C2×Dic3—181no—
169C2×Dic3—181no—
170C2×Dic3—361no—
171C2×Dic3—361no—
172C2×Dic3—721no—
173C4×D3—121no—
174C4×D3—361no—
175C4×D3—361no—
176C4×D3—361no—
177Dic6—181no—
178Dic6—181no—
179C3×C8—181no—
180C3⋊C8—181no—
181C2×C32—41no—
182C2×C32—81no—
183C2×C32—122no—
184C2×C32D3×SD1611yescharacteristic
185C3⋊D3#91511yescharacteristic
186C3⋊D3#93911yescharacteristic
187C3×D3—42no—
188C3×D3—42no—
189C3×D3—82no—
190C3×D3—121no—
191C3×D3—121no—
192C2×Q8—91no—
193C2×Q8—271no—
194C2×D4—91no—
195C22×C4—541no—
196SD16—182no—
197SD16—542no—
198C2×C8—91no—
199C2×C8—271no—
200C4⋊C4—271no—
201C4⋊C4—541no—
202C4⋊C4—541no—
203C22⋊C4—541no—
204C42—271no—
205C22×C3—91no—
206C22×C3—121no—
207C22×C3—121no—
208C22×C3—182no—
209C22×C3—241no—
210C22×C3—361no—
211C2×D3—41no—
212C2×D3—121no—
213C2×D3—122no—
214C2×D3—122no—
215C3×C4—91no—
216C3×C4—91no—
217C3×C4—121no—
218C3×C4—361no—
219C3×C4—361no—
220Dic3—91no—
221Dic3—121no—
222Dic3—121no—
223Dic3—181no—
224Dic3—241no—
225Dic3—361no—
226Dic3—361no—
227Dic3—721no—
228Dic3AΓL(1,𝔽9)11yescharacteristic, complemented
229C32—41no—
230C32—81no—
231C32#516211yescharacteristic, complemented
232C23—181no—
233Q8—181no—
234Q8—181no—
235Q8—541no—
236Q8—541no—
237D4—92no—
238D4—181no—
239C2×C4—91no—
240C2×C4—181no—
241C2×C4—271no—
242C2×C4—271no—
243C2×C4—361no—
244C2×C4—541no—
245C2×C4—541no—
246C2×C4—1081no—
247C8—181no—
248C8—541no—
249C2×C3—41no—
250C2×C3—81no—
251C2×C3—91no—
252C2×C3—91no—
253C2×C3—122no—
254C2×C3—122no—
255C2×C3—242no—
256C2×C3C2×AΓL(1,𝔽9)11yescharacteristic
257D3—42no—
258D3—121no—
259D3—121no—
260C22—91no—
261C22—121no—
262C22—182no—
263C22—361no—
264C4—31no—
265C4—91no—
266C4—91no—
267C4—271no—
268C4—361no—
269C4—361no—
270C4—541no—
271C4—1081no—
272C3—41no—
273C3—81no—
274C3#1059640911yescharacteristic, complemented
275C2—91no—
276C2—91no—
277C2—122no—
278C2D3×AΓL(1,𝔽9)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D3×AΓL(1,𝔽9)
commutator, cosocleC2×C3×#157C23
Fitting, socleC2×C33C2×SD16
prime radicalC2×C3×GO+(4,𝔽2)C22
hypocenterC33#461
LCSG ▷ C2×C3×#157 ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
3. #11724153C2semidirect
5. #11724105C2semidirect
6. #11724102C2semidirect
7. Dic3⋊AGL(1,𝔽9)C2semidirect
70. C3⋊(C3⋊Dic3)SD16semidirect
72. C32×Dic3SD16semidirect
155. C33#461semidirect
228. Dic3AΓL(1,𝔽9)semidirect
231. C32#5162semidirect
274. C3#10596409semidirect

Maximal subgroups

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

subgroupindex
C2×#890912
C2×C3×AΓL(1,𝔽9)2
#117241532
Dic3×GO+(4,𝔽2)2
#117241052
#117241022
Dic3⋊AGL(1,𝔽9)2
#105964093
#51629