Finite Group Databasebeta

← #72259951

Subgroups — #72259951

Order 2592.

353 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 50 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×C3×#157⋊D3C211yesmaximal, characteristic
2C2×C3⋊(C3⋊AGL(1,𝔽9))C211yesmaximal, characteristic, complemented
3C3×#88946C211yesmaximal, characteristic, complemented
4#11724101—31nomaximal
5GO+(4,𝔽2)⋊Dic3—31nomaximal
6C2×C3×#1113—21no—
7C2×C3×C3⋊#157—21no—
8C2×C32×#157C2211yescharacteristic
9C3×#157⋊D3—21no—
10C3×#157⋊D3C412yescomplemented
11C3⋊(C3⋊AGL(1,𝔽9))—21no—
12C2×#157⋊D3D311yescharacteristic
13C2×C3×GO+(4,𝔽2)—31no—
14C2×C3⋊AGL(1,𝔽9)—31no—
15C2×C3⋊AGL(1,𝔽9)—31no—
16C2×C3⋊AGL(1,𝔽9)—61no—
17#88946D311yescharacteristic, complemented
18C3×#3777—31no—
19C2×C32×C3⋊D3—41no—
20C2×C32×C3⋊D3D411yescharacteristic
21C3×#1113—22no—
22C3×#1113—41no—
23C3×C3⋊#157—41no—
24C32×#157C2×C411yescharacteristic
25C32×#157D411yescharacteristic
26GO+(4,𝔽2)⋊C4—91no—
27#63562—91nomaximal
28C2×#1113—21no—
29C2×C3×D32—61no—
30C2×C3×D32—121no—
31C2×C3⋊#157—21no—
32C2×C3×#157—21no—
33C2×C3×#157—61no—
34C2×C3×#157C2×D311yescharacteristic
35C2×C3×#157C2×D311yescharacteristic
36#157⋊D3—21no—
37#157⋊D3Dic312yescomplemented
38C3×GO+(4,𝔽2)—32no—
39C3×GO+(4,𝔽2)—61no—
40C3⋊AGL(1,𝔽9)—62no—
41C3⋊AGL(1,𝔽9)—61no—
42C3⋊AGL(1,𝔽9)—61no—
43C2×C34SD1611yescharacteristic
44C32×C3⋊D3—42no—
45C32×C3⋊D3C22⋊C411yescharacteristic
46C32×C3⋊D3D811yescharacteristic
47C2×GO+(4,𝔽2)—31no—
48C2×AGL(1,𝔽9)—91no—
49C2×C3×D12—91no—
50#3777—31no—
51C2×C3⋊(C3⋊C8)—91no—
52C3×C4⋊Dic3—91no—
53C2×C3×C3⋊D3—21no—
54C2×C3×C3⋊D3—41no—
55C2×C3×C3⋊D3—41no—
56C2×C3×C3⋊D3—81no—
57C2×C3×C3⋊D3#6711yescharacteristic
58C2×C3×C3⋊D3#6711yescharacteristic
59C2×C32×D3—121no—
60C2×C32×D3—121no—
61C2×C32×D3—121no—
62C2×C32×D3—241no—
63#1113—22no—
64#1113—41no—
65C3×D32—62no—
66C3×D32—122no—
67C3×D32—122no—
68C3×D32—121no—
69C3⋊#157—41no—
70C3×#157—21no—
71C3×#157—21no—
72C3×#157—121no—
73C3×#157C4×D311yescharacteristic
74C3×#157D1211yescharacteristic
75C3×#157C2×Dic311yescharacteristic
76C3×#157#6711yescharacteristic
77#832—271no—
78#809—271no—
79C34#10211yescharacteristic, complemented, Sylow
80C22×C3×D3—181no—
81C22×C3×D3—361no—
82C2×D32—61no—
83C2×D32—121no—
84C2×#157—61no—
85C2×#157D3211yescharacteristic
86GO+(4,𝔽2)—32no—
87GO+(4,𝔽2)—61no—
88AGL(1,𝔽9)—181no—
89C2×C32×C4—91no—
90C2×C3×Dic3—181no—
91C3×D12—92no—
92C3×D12—181no—
93C3⋊(C3⋊C8)—181no—
94C2×C33—21no—
95C2×C33—41no—
96C2×C33—81no—
97C2×C33—81no—
98C2×C33—161no—
99C2×C33#21411yescharacteristic
100C2×C33#21511yescharacteristic
101C3×C3⋊D3—21no—
102C3×C3⋊D3—21no—
103C3×C3⋊D3—42no—
104C3×C3⋊D3—42no—
105C3×C3⋊D3—82no—
106C3×C3⋊D3(C2×D3)⋊C411yescharacteristic
107C3×C3⋊D3#21311yescharacteristic
108C3×C3⋊D3#21311yescharacteristic
109C3×C3⋊D3#21711yescharacteristic
110C32×D3—121no—
111C32×D3—121no—
112C32×D3—122no—
113C32×D3—122no—
114C32×D3—242no—
115C2×C3×D4—271no—
116C2×D12—91no—
117C3×C4⋊C4—271no—
118C4⋊Dic3—91no—
119C2×C3⋊C8—271no—
120C2×C3⋊C8—271no—
121C2×C3⋊C8—541no—
122C22×C32—91no—
123C22×C32—361no—
124C2×C3⋊D3—41no—
125C2×C3⋊D3#62511yescharacteristic
126C2×C3×D3—121no—
127C2×C3×D3—121no—
128C2×C3×D3—121no—
129C2×C3×D3—121no—
130C2×C3×D3—121no—
131C2×C3×D3—121no—
132C2×C3×D3—182no—
133C2×C3×D3—241no—
134C2×C3×D3—241no—
135C2×C3×D3—241no—
136C2×C3×D3—241no—
137C2×C3×D3—241no—
138C2×C3×D3—242no—
139C2×C3×D3—362no—
140C2×C3×D3—361no—
141C2×C3×D3—362no—
142D32—62no—
143D32—122no—
144D32—122no—
145D32—121no—
146#157—121no—
147#157D3×Dic311yescharacteristic
148#157#62611yescharacteristic
149C32×C4—91no—
150C32×C4—91no—
151C3×Dic3—361no—
152#102—811noSylow
153C33—21no—
154C33—41no—
155C33—81no—
156C33—81no—
157C33—161no—
158C33#80911yescharacteristic, complemented
159C33#83211yescharacteristic, complemented
160C23×C3—541no—
161C22×D3—181no—
162C22×D3—361no—
163C3×D4—272no—
164C3×D4—541no—
165C2×C3×C4—91no—
166C2×C3×C4—91no—
167C2×C3×C4—181no—
168C2×C3×C4—541no—
169C2×Dic3—181no—
170D12—92no—
171D12—181no—
172C3⋊C8—541no—
173C3⋊C8—541no—
174C3⋊C8—542no—
175C2×C32—41no—
176C2×C32—41no—
177C2×C32—81no—
178C2×C32—81no—
179C2×C32—81no—
180C2×C32—82no—
181C2×C32—81no—
182C2×C32—81no—
183C2×C32—91no—
184C2×C32—91no—
185C2×C32—161no—
186C2×C32—161no—
187C2×C32—162no—
188C2×C32—362no—
189C2×C32#371511yescharacteristic
190C2×C32AΓL(1,𝔽9)11yescharacteristic
191C3⋊D3—42no—
192C3⋊D3#371311yescharacteristic
193C3⋊D3(C2×D3)⋊Dic311yescharacteristic
194C3×D3—121no—
195C3×D3—121no—
196C3×D3—121no—
197C3×D3—121no—
198C3×D3—122no—
199C3×D3—122no—
200C3×D3—122no—
201C3×D3—122no—
202C3×D3—241no—
203C3×D3—241no—
204C3×D3—242no—
205C3×D3—242no—
206C3×D3—242no—
207C3×D3—242no—
208C3×D3—244no—
209C2×D4—271no—
210C2×C8—811no—
211C4⋊C4—271no—
212C22×C3—91no—
213C22×C3—91no—
214C22×C3—181no—
215C22×C3—361no—
216C22×C3—361no—
217C22×C3—542no—
218C22×C3—721no—
219C22×C3—1081no—
220C2×D3—121no—
221C2×D3—121no—
222C2×D3—121no—
223C2×D3—182no—
224C2×D3—241no—
225C2×D3—362no—
226C2×D3—361no—
227C2×D3—362no—
228C3×C4—91no—
229C3×C4—91no—
230C3×C4—91no—
231C3×C4—91no—
232C3×C4—181no—
233C3×C4—181no—
234C3×C4—1081no—
235Dic3—361no—
236C32—41no—
237C32—41no—
238C32—81no—
239C32—81no—
240C32—81no—
241C32—82no—
242C32—81no—
243C32—81no—
244C32—161no—
245C32—161no—
246C32—162no—
247C32#6356211yescharacteristic, complemented
248C32GO+(4,𝔽2)⋊C411yescharacteristic, complemented
249C23—541no—
250D4—272no—
251D4—541no—
252C2×C4—91no—
253C2×C4—541no—
254C8—1621no—
255C2×C3—21no—
256C2×C3—41no—
257C2×C3—81no—
258C2×C3—81no—
259C2×C3—91no—
260C2×C3—91no—
261C2×C3—91no—
262C2×C3—91no—
263C2×C3—161no—
264C2×C3—181no—
265C2×C3—181no—
266C2×C3—362no—
267C2×C3—362no—
268C2×C3—722no—
269C2×C3#8909111yescharacteristic
270C2×C3#8909211yescharacteristic
271D3—121no—
272D3—121no—
273D3—122no—
274D3—122no—
275D3—242no—
276C22—91no—
277C22—361no—
278C22—542no—
279C22—1081no—
280C4—91no—
281C4—91no—
282C4—1081no—
283C3—21no—
284C3—41no—
285C3—81no—
286C3—81no—
287C3—161no—
288C3GO+(4,𝔽2)⋊Dic311yescharacteristic, complemented
289C3#1172410111yescharacteristic, complemented
290C2—91no—
291C2—91no—
292C2—362no—
293C2#1304979711yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2#13049797
commutatorC32×#157C2×C4
Fitting, socleC2×C34SD16
prime radicalC2×C3×#157⋊D3C2
cosocleC2×C32×#157C22
hypocenterC34#102
LCSG ▷ C32×#157 ▷ C32×C3⋊D3 ▷ C34
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊(C3⋊AGL(1,𝔽9))C2semidirect
3. C3×#88946C2semidirect
10. C3×#157⋊D3C4semidirect
17. #88946D3semidirect
37. #157⋊D3Dic3semidirect
79. C34#102semidirect
158. C33#809semidirect
159. C33#832semidirect
247. C32#63562semidirect
248. C32GO+(4,𝔽2)⋊C4semidirect
288. C3GO+(4,𝔽2)⋊Dic3semidirect
289. C3#11724101semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#157⋊D32
C2×C3⋊(C3⋊AGL(1,𝔽9))2
C3×#889462
#117241013
GO+(4,𝔽2)⋊Dic33
#635629