Finite Group Databasebeta

← #10588151⋊C3 #13135347

Subgroups — #10588151⋊C3 #13135347

Order 1728.

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

normal subgroups only

This group has 284 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
1#63502⋊C3C211yesmaximal, characteristic
23−1+2×#99C211yesmaximal, characteristic, complemented
3#63393⋊C3C211yesmaximal, characteristic, complemented
4C3×#4474—31nomaximal
5#10588151C311yesmaximal, characteristic, complemented
6C2×D4×3−1+2C2211yescharacteristic
7C2×#611⋊C3—21no—
8C2×#611⋊C3—21no—
9C2×C4⋊AGL(1,ℤ/9)C411yescharacteristic, complemented
10C22⋊C4×3−1+2—21no—
11#3676⋊C3—21no—
12C4×3−1+2⋊C4C411yescharacteristic, complemented
13C3×#940—31no—
14C32×#99D311yescharacteristic
15C3×#834—31no—
16#63502C2×C311yescharacteristic
17C9×#99—21no—
18C9×#99C2×C311yescharacteristic, complemented
19#63393C2×C311yescharacteristic, complemented
20C23×3−1+2D411yescharacteristic
21C23×3−1+2D411yescharacteristic
22C22×AGL(1,ℤ/9)—21no—
23D4×3−1+2—41no—
24C2×C4×3−1+2—41no—
25C2×C4×3−1+2C2×C411yescharacteristic
26#611⋊C3—42no—
27#611⋊C3—42no—
28C2×3−1+2⋊C4—21no—
29C2×3−1+2⋊C4—41no—
30C4⋊AGL(1,ℤ/9)—41no—
31C3×#353—91no—
32#4474—31no—
33C2×C32×D4C2×D311yescharacteristic
34C2×C3×#67—61no—
35C2×C3×#67—61no—
36C2×C3×D12—31no—
37C32×C22⋊C4—21no—
38C3×#217—61no—
39C3×C4×Dic3—31no—
40C2×D4×C9—21no—
41C2×D4×C9C22×C311yescharacteristic
42C2×#611—21no—
43C2×#611—21no—
44C2×D36C3×C411yescharacteristic, complemented
45C9×C22⋊C4—22no—
46C9×C22⋊C4—21no—
47#3676—21no—
48C4×Dic9C3×C411yescharacteristic, complemented
49C22×3−1+2—21no—
50C22×3−1+2—21no—
51C22×3−1+2—41no—
52C22×3−1+2—41no—
53C22×3−1+2C22⋊C411yescharacteristic
54C2×AGL(1,ℤ/9)—41no—
55C2×AGL(1,ℤ/9)—41no—
56C2×AGL(1,ℤ/9)—41no—
57C4×3−1+2—21no—
58C4×3−1+2—81no—
593−1+2⋊C4—22no—
603−1+2⋊C4—81no—
61C3×#127—91no—
62#940—31no—
63C3×#99—31no—
64C3×#99—91no—
65C3×#99C3×D311yescharacteristic
66#834—31no—
67C23×C32D1211yescharacteristic
68C23×C32#6711yescharacteristic
69C22×C3×D3—61no—
70C32×D4—41no—
71C2×C32×C4—41no—
72C2×C32×C4C4×D311yescharacteristic
73C3×#67—122no—
74C3×#67—122no—
75C2×C3×Dic3—61no—
76C2×C3×Dic3—121no—
77C3×D12—121no—
78C23×C9—21no—
79C23×C9—21no—
80C23×C9C3×D411yescharacteristic
81C23×C9C3×D411yescharacteristic
82C22×D9—21no—
83D4×C9—41no—
84D4×C9—42no—
85C2×C4×C9—21no—
86C2×C4×C9—42no—
87C2×C4×C9—41no—
88C2×C4×C9C2×C3×C411yescharacteristic
89#611—42no—
90#611—42no—
91C2×Dic9—21no—
92C2×Dic9—41no—
93D36—41no—
94#353—91noSylow
95C2×3−1+2—21no—
96C2×3−1+2—41no—
97C2×3−1+2—41no—
98C2×3−1+2#9911yescharacteristic
99AGL(1,ℤ/9)—81no—
100C2×C3×D4—31no—
101C2×C3×D4—91no—
102C2×C3×D4—181no—
103C2×C3×D4—181no—
104C2×C3×D4C2×C3×D311yescharacteristic
105C2×#67—61no—
106C2×#67—61no—
107C2×D12—31no—
108C3×C22⋊C4—21no—
109C3×C22⋊C4—61no—
110C3×C22⋊C4—181no—
111C3×C42—91no—
112#217—61no—
113C4×Dic3—31no—
114C22×C32—21no—
115C22×C32—21no—
116C22×C32—41no—
117C22×C32—41no—
118C22×C32(C2×D3)⋊C411yescharacteristic
119C2×C3×D3—121no—
120C2×C3×D3—121no—
121C2×C3×D3—121no—
122C32×C4—21no—
123C32×C4—81no—
124C3×Dic3—62no—
125C3×Dic3—241no—
126C22×C9—22no—
127C22×C9—21no—
128C22×C9—21no—
129C22×C9—21no—
130C22×C9—41no—
131C22×C9—41no—
132C22×C9—41no—
133C22×C9—81no—
134C22×C9—81no—
135C22×C9C3×C22⋊C411yescharacteristic
136C2×D9—41no—
137C2×D9—41no—
138C2×D9—41no—
139C4×C9—21no—
140C4×C9—41no—
141C4×C9—82no—
142C4×C9—81no—
143Dic9—22no—
144Dic9—81no—
145#127—91no—
146#99—91no—
147#99AGL(1,ℤ/9)11yescharacteristic, complemented
1483−1+2#35311yescharacteristic, complemented, Sylow
149C23×C3—31no—
150C23×C3—31no—
151C23×C3—181no—
152C23×C3C3×D1211yescharacteristic
153C23×C3C3×#6711yescharacteristic
154C22×D3—61no—
155C3×D4—41no—
156C3×D4—121no—
157C3×D4—361no—
158C3×D4—362no—
159C3×D4—362no—
160C2×C3×C4—31no—
161C2×C3×C4—41no—
162C2×C3×C4—121no—
163C2×C3×C4—181no—
164C2×C3×C4—361no—
165C2×C3×C4C3×C4×D311yescharacteristic
166#67—122no—
167#67—122no—
168C2×Dic3—61no—
169C2×Dic3—121no—
170D12—121no—
171C2×C32—21no—
172C2×C32—41no—
173C2×C32—41no—
174C2×C32#80611yescharacteristic
175C3×D3—241no—
176C2×C9—21no—
177C2×C9—21no—
178C2×C9—41no—
179C2×C9—42no—
180C2×C9—41no—
181C2×C9—41no—
182C2×C9—81no—
183C2×C9C3×#9911yescharacteristic
184D9—81no—
185C2×D4—91no—
186C2×D4—181no—
187C2×D4—181no—
188C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
189C22⋊C4—21no—
190C22⋊C4—181no—
191C42—91no—
192C22×C3—21no—
193C22×C3—21no—
194C22×C3—31no—
195C22×C3—41no—
196C22×C3—41no—
197C22×C3—61no—
198C22×C3—61no—
199C22×C3—121no—
200C22×C3—121no—
201C22×C3—361no—
202C22×C3—361no—
203C22×C3—361no—
204C22×C3C3×(C2×D3)⋊C411yescharacteristic
205C2×D3—121no—
206C2×D3—121no—
207C2×D3—121no—
208C3×C4—21no—
209C3×C4—61no—
210C3×C4—81no—
211C3×C4—182no—
212C3×C4—241no—
213C3×C4—721no—
214Dic3—62no—
215Dic3—241no—
216C32#447411yescharacteristic
217C9—21no—
218C9C3×#35311yescharacteristic, complemented
219C23—181no—
220C23C4⋊AGL(1,ℤ/9)11yescharacteristic
221C23#611⋊C311yescharacteristic
222D4—41no—
223D4—361no—
224D4—362no—
225D4—362no—
226C2×C4—41no—
227C2×C4—181no—
228C2×C4—361no—
229C2×C4C4×AGL(1,ℤ/9)11yescharacteristic
230C2×C3—21no—
231C2×C3—31no—
232C2×C3—41no—
233C2×C3—41no—
234C2×C3—61no—
235C2×C3—121no—
236C2×C3—121no—
237C2×C3—721no—
238C2×C3C3×#80611yescharacteristic
239D3—241no—
240C22—21no—
241C22—21no—
242C22—41no—
243C22—41no—
244C22—361no—
245C22—361no—
246C22—361no—
247C22(C2×AGL(1,ℤ/9))⋊C411yescharacteristic
248C4—21no—
249C4—81no—
250C4—182no—
251C4—721no—
252C3—31no—
253C3C3×#447411yescharacteristic
254C2—21no—
255C2—41no—
256C2—41no—
257C2—721no—
258C2#63365⋊C311yescharacteristic
subgroupG/H
centerC2#63365⋊C3
commutatorC2×C4×C9C2×C3×C4
FrattiniC2×C3×D4C2×C3×D3
Fitting3−1+2×#99C2
socleC2×C3C3×#806
prime radical#63502⋊C3C2
cosocleC2×D4×C9C22×C3
hypercenter#99AGL(1,ℤ/9)
hypocenterC9C3×#353
LCSG ▷ C2×C4×C9 ▷ C22×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #99

Complements

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

NH ≅ G/Ntype
2. 3−1+2×#99C2semidirect
3. #63393⋊C3C2semidirect
5. #10588151C3semidirect
9. C2×C4⋊AGL(1,ℤ/9)C4semidirect
12. C4×3−1+2⋊C4C4semidirect
18. C9×#99C2×C3semidirect
19. #63393C2×C3semidirect
44. C2×D36C3×C4semidirect
48. C4×Dic9C3×C4semidirect
147. #99AGL(1,ℤ/9)semidirect
148. 3−1+2#353semidirect
218. C9C3×#353semidirect

Maximal subgroups

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

subgroupindex
#63502⋊C32
3−1+2×#992
#63393⋊C32
C3×#44743
#105881513