Finite Group Databasebeta

← #13180159

Subgroups — #13180159

Order 1728.

306 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 46 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
1#11724154C211yesmaximal, characteristic, complemented
2#11724124C211yesmaximal, characteristic
3#11724104C211yesmaximal, characteristic, complemented
4#11724101C211yesmaximal, characteristic, complemented
5GO+(4,𝔽2)⋊Dic3C211yesmaximal, characteristic, complemented
6C3×GO+(4,𝔽2)⋊C4C211yesmaximal, characteristic, complemented
7Dic3×AGL(1,𝔽9)C211yesmaximal, characteristic, complemented
8#10596402—31nomaximal
9C2×C3⋊PSU(3,𝔽2)—21no—
10C2×#157⋊D3C2211yescharacteristic
11C2×D32⋊D3—21no—
12C2×C3×GO+(4,𝔽2)C2211yescharacteristic
13C2×C3⋊AGL(1,𝔽9)C2211yescharacteristic
14C2×C3×AGL(1,𝔽9)C2211yescharacteristic
15#88953—21no—
16#88946C2211yescharacteristic
17C3×#3777C2211yescharacteristic
18Dic3⋊#157—21no—
19Dic3×#157C2211yescharacteristic
20C4⋊PSU(3,𝔽2)—31no—
21#64231—31no—
22C4×AGL(1,𝔽9)—31no—
23GO+(4,𝔽2)⋊C4—31no—
24GO+(4,𝔽2)⋊C4—31no—
25GO+(4,𝔽2)⋊C4—31no—
26GO+(4,𝔽2)⋊C4D311yescharacteristic, complemented
27C2×#1113—21no—
28C2×C3×D32—21no—
29C2×C3⋊#157—21no—
30C2×C3⋊#157D411yescharacteristic
31C2×C3×#157—21no—
32C2×C3×#157C2311yescharacteristic
33C3⋊PSU(3,𝔽2)—42no—
34#157⋊D3—21no—
35#157⋊D3—21no—
36D32⋊D3—42no—
37C3×GO+(4,𝔽2)—21no—
38C3×GO+(4,𝔽2)—21no—
39C3⋊AGL(1,𝔽9)—21no—
40C3×AGL(1,𝔽9)—21no—
41#6290—41no—
42Dic3×C3⋊D3D411yescharacteristic
43C3×#626—41no—
44#4760—91nomaximal
45C2×PSU(3,𝔽2)—61no—
46C2×GO+(4,𝔽2)—31no—
47C2×GO+(4,𝔽2)—61no—
48C2×GO+(4,𝔽2)C2×D311yescharacteristic
49C2×AGL(1,𝔽9)—31no—
50C2×AGL(1,𝔽9)C2×D311yescharacteristic
51D3×#67—121no—
52#3802—61no—
53C4⋊#157—61no—
54C4×#157—31no—
55#3777—31no—
56#3777C2×D311yescharacteristic
57C2×C3×C3⋊D3—41no—
58C2×C3×C3⋊D3C2×D411yescharacteristic
59C2×C32×D3—41no—
60#1113—41no—
61#1113—41no—
62C3×D32—41no—
63C3×D32—41no—
64C3⋊#157—41no—
65C3⋊#157D812yes—
66C3×#157—41no—
67C3×#157#4111yescharacteristic
68C3×#157#4111yescharacteristic
69C3⋊(C3⋊Dic3)SD1611yescharacteristic
70C32×Dic3SD1611yescharacteristic
71#940—91no—
72#888—91no—
73C3×#102—91no—
74#832—91no—
75#821—91no—
76C8×Dic3—91no—
77#809—91no—
78C22×C3×D3—121no—
79C2×D32—21no—
80C2×D32—61no—
81C2×D32—121no—
82C2×#157—21no—
83C2×#157—31no—
84C2×#157—61no—
85C2×#157C22×D311yescharacteristic
86PSU(3,𝔽2)—122no—
87GO+(4,𝔽2)—21no—
88GO+(4,𝔽2)—21no—
89GO+(4,𝔽2)—61no—
90GO+(4,𝔽2)—61no—
91GO+(4,𝔽2)—122no—
92AGL(1,𝔽9)—21no—
93AGL(1,𝔽9)—61no—
94#638—121no—
95C4×C3⋊D3—31no—
96C3×#67—121no—
97C3×D12—121no—
98#626—41no—
99#625—121no—
100#625—121no—
101#625—241no—
102D3×Dic3—121no—
103#486—271noSylow
104C2×C33C2×SD1611yescharacteristic
105C3×C3⋊D3—81no—
106C3×C3⋊D3#12411yescharacteristic
107C3×C3⋊D3C2×D811yescharacteristic
108C32×D3—81no—
109C2×C3×D4—91no—
110C2×#67—181no—
111C2×#67—181no—
112D3×D4—361no—
113C2×D12—91no—
114C2×Dic6—181no—
115C2×C3×C8—91no—
116C3×C4⋊C4—91no—
117C4⋊Dic3—91no—
118Dic3⋊C4—181no—
119C4×Dic3—91no—
120C2×C3⋊C8—91no—
121C22×C32—121no—
122C2×C3⋊D3—41no—
123C2×C3⋊D3D3×D411yescharacteristic
124C2×C3×D3—41no—
125C2×C3×D3—41no—
126C2×C3×D3—81no—
127C2×C3×D3—121no—
128C2×C3×D3—121no—
129C2×C3×D3—121no—
130C2×C3×D3—241no—
131C2×C3×D3—241no—
132C2×C3×D3—241no—
133D32—41no—
134D32—41no—
135D32—121no—
136D32—121no—
137D32—241no—
138D32—241no—
139#157—32no—
140#157—41no—
141#157—121no—
142#157#23511yescharacteristic
143#157#23711yescharacteristic
144C32×C4—31no—
145C3⋊Dic3—31no—
146C3⋊Dic3—121no—
147C3⋊Dic3—241no—
148C3×Dic3—41no—
149C4⋊Q8—271no—
150#127—271no—
151#102—91no—
152#102—271no—
153#102—271no—
154#102—271no—
155C4×C8—271no—
156C33#48611yescharacteristic, complemented, Sylow
157C23×C3—181no—
158C22×D3—121no—
159C22×D3—181no—
160C22×D3—361no—
161C3×D4—181no—
162C3×D4—181no—
163C3×D4—361no—
164C2×C3×C4—91no—
165C2×C3×C4—181no—
166#67—121no—
167#67—361no—
168#67—362no—
169#67—361no—
170#67—361no—
171#67—721no—
172C2×Dic3—91no—
173C2×Dic3—91no—
174C2×Dic3—181no—
175D12—121no—
176D12—181no—
177D12—181no—
178C4×D3—361no—
179Dic6—362no—
180C3×C8—181no—
181C3⋊C8—181no—
182C2×C32—41no—
183C2×C32—81no—
184C2×C32—241no—
185C2×C32D3×SD1611yescharacteristic
186C3⋊D3—81no—
187C3⋊D3#88511yescharacteristic
188C3⋊D3D3×D811yescharacteristic
189C3×D3—81no—
190C3×D3—81no—
191C3×D3—121no—
192C3×D3—121no—
193C3×D3—161no—
194C3×D3—241no—
195C3×D3—241no—
196C3×D3—481no—
197C2×Q8—541no—
198C2×D4—91no—
199C2×D4—271no—
200C2×D4—541no—
201C2×D4—541no—
202C2×C8—91no—
203C2×C8—271no—
204C4⋊C4—91no—
205C4⋊C4—271no—
206C4⋊C4—541no—
207C42—271no—
208C22×C3—91no—
209C22×C3—121no—
210C22×C3—121no—
211C22×C3—241no—
212C22×C3—361no—
213C22×C3—361no—
214C22×C3—361no—
215C2×D3—41no—
216C2×D3—121no—
217C2×D3—121no—
218C2×D3—121no—
219C2×D3—241no—
220C2×D3—241no—
221C2×D3—241no—
222C2×D3—361no—
223C2×D3—361no—
224C2×D3—721no—
225C2×D3—721no—
226C3×C4—91no—
227C3×C4—91no—
228C3×C4—121no—
229C3×C4—361no—
230Dic3—92no—
231Dic3—91no—
232Dic3—361no—
233Dic3—361no—
234Dic3—721no—
235Dic3AΓL(1,𝔽9)11yescharacteristic
236C32—41no—
237C32—81no—
238C32#476011yescharacteristic, complemented
239C23—181no—
240C23—541no—
241Q8—1082no—
242D4—181no—
243D4—181no—
244D4—361no—
245D4—541no—
246D4—541no—
247D4—1081no—
248D4—1082no—
249C2×C4—91no—
250C2×C4—181no—
251C2×C4—271no—
252C2×C4—271no—
253C2×C4—541no—
254C8—181no—
255C8—541no—
256C2×C3—41no—
257C2×C3—81no—
258C2×C3—91no—
259C2×C3—91no—
260C2×C3—241no—
261C2×C3—241no—
262C2×C3—481no—
263C2×C3—721no—
264C2×C3C2×AΓL(1,𝔽9)11yescharacteristic
265D3—81no—
266D3—121no—
267D3—121no—
268D3—241no—
269D3—241no—
270D3—481no—
271C22—91no—
272C22—121no—
273C22—361no—
274C22—361no—
275C22—361no—
276C22—1081no—
277C22—1081no—
278C4—31no—
279C4—91no—
280C4—91no—
281C4—272no—
282C4—271no—
283C4—361no—
284C4—1081no—
285C3—41no—
286C3—81no—
287C3#1059640211yescharacteristic, complemented
288C2—91no—
289C2—91no—
290C2—241no—
291C2—721no—
292C2D3×AΓL(1,𝔽9)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D3×AΓL(1,𝔽9)
commutator, cosocleC2×C3×#157C23
Fitting, socleC2×C33C2×SD16
prime radical#11724124C2
hypocenterC33#486
LCSG ▷ C2×C3×#157 ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. #11724154C2semidirect
3. #11724104C2semidirect
4. #11724101C2semidirect
5. GO+(4,𝔽2)⋊Dic3C2semidirect
6. C3×GO+(4,𝔽2)⋊C4C2semidirect
7. Dic3×AGL(1,𝔽9)C2semidirect
26. GO+(4,𝔽2)⋊C4D3semidirect
156. C33#486semidirect
238. C32#4760semidirect
287. C3#10596402semidirect

Maximal subgroups

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

subgroupindex
#117241542
#117241242
#117241042
#117241012
GO+(4,𝔽2)⋊Dic32
C3×GO+(4,𝔽2)⋊C42
Dic3×AGL(1,𝔽9)2
#105964023
#47609