Finite Group Databasebeta

← #10588433⋊C3 #13141705

Subgroups — #10588433⋊C3 #13141705

Order 1728.

457 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 457 conjugacy classes of subgroups — too many to draw. Only its 90 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#11722630C211yesmaximal, characteristic, complemented
2#63509⋊C3C211yesmaximal, characteristic, complemented
3#63470⋊C3C212yesmaximal, complemented
43−1+2×#101C211yesmaximal, characteristic, complemented
5#63384⋊C3C212yesmaximal, complemented
6C3×#4753—31nomaximal
7#10588433C311yesmaximal, characteristic, complemented
8C2×Q8×3−1+2C2211yescharacteristic
9#88724—22no—
10Q8×AGL(1,ℤ/9)—22no—
11#3697⋊C3—22no—
12#3697⋊C3C2212yes—
13C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
14M16×3−1+2C2212yes—
15(C2×AGL(1,ℤ/9))⋊C4—22no—
16C4×3−1+2⋊C4C2211yescharacteristic
17D72⋊C3—22no—
18#3664⋊C3—22no—
19C3×#1007—31no—
20C3×#947—31no—
21C3×#908—32no—
22#63713C2×C311yescharacteristic, complemented
23C32×#101D311yescharacteristic
24C3×#825—32no—
25#63509C2×C311yescharacteristic, complemented
26#63470C2×C312yescomplemented
27C9×#101—21no—
28C9×#101C2×C311yescharacteristic, complemented
29#63384C2×C312yescomplemented
30C22×AGL(1,ℤ/9)—21no—
31Q8×3−1+2—22no—
32C2×C4×3−1+2D412yes—
33C2×C4×3−1+2C2311yescharacteristic
34#611⋊C3—22no—
35C2×3−1+2⋊C4—21no—
36C4⋊AGL(1,ℤ/9)—22no—
37C4⋊AGL(1,ℤ/9)—22no—
38C4⋊AGL(1,ℤ/9)D412yes—
39C4×AGL(1,ℤ/9)—24no—
40C4×AGL(1,ℤ/9)—22no—
413−1+2⋊Q8—22no—
423−1+2⋊Q8D412yes—
43C8×3−1+2—22no—
44C3×#454—91no—
45#4753—31no—
46C2×C32×Q8C2×D311yescharacteristic
47C3×#239—62no—
48C3×D3×Q8—62no—
49C3×#235—32no—
50C3×#235—62no—
51C2×C3×D12—31no—
52C32×M16C2×D312yes—
53C3×(C2×D3)⋊C4—62no—
54C3×C4×Dic3—31no—
55C3×D24—62no—
56C3×#204—62no—
57C2×Q8×C9—21no—
58C2×Q8×C9C22×C311yescharacteristic
59#3701—22no—
60Q8×D9—22no—
61#3697—22no—
62#3697C22×C312yes—
63C2×D36C22×C311yescharacteristic
64C9×M16—22no—
65C9×M16C22×C312yes—
66(C2×D9)⋊C4—22no—
67C4×Dic9C22×C311yescharacteristic
68D72—22no—
69#3664—22no—
70C22×3−1+2C2×D411yescharacteristic
71C2×AGL(1,ℤ/9)—22no—
72C2×AGL(1,ℤ/9)—42no—
73C2×AGL(1,ℤ/9)—41no—
74C4×3−1+2—22no—
75C4×3−1+2C2×D412yes—
763−1+2⋊C4—22no—
773−1+2⋊C4—41no—
78C3×2−1+4—91no—
79#1007—31no—
80C3×AGL(1,ℤ/8)—92no—
81C3×#124—91no—
82#947—31no—
83#908—32no—
84C3×#104—92no—
85C3×#101—31no—
86C3×#101C3×D311yescharacteristic
87#825—32no—
88C22×C3×D3—61no—
89C32×Q8—22no—
90C2×C32×C4D1212yes—
91C2×C32×C4C22×D311yescharacteristic
92C3×#67—62no—
93C2×C3×Dic3—61no—
94C3×D12—32no—
95C3×D12—62no—
96C3×D12—62no—
97C3×C4×D3—64no—
98C3×C4×D3—62no—
99C3×Dic6—32no—
100C3×Dic6—62no—
101C22×D9—21no—
102C32×C8—22no—
103Q8×C9—22no—
104Q8×C9—42no—
105C2×C4×C9—22no—
106C2×C4×C9—21no—
107C2×C4×C9C3×D412yes—
108C2×C4×C9C23×C311yescharacteristic
109#611—22no—
110C2×Dic9—21no—
111D36—22no—
112D36—22no—
113D36C3×D412yes—
114C4×D9—24no—
115C4×D9—22no—
116Dic18—22no—
117Dic18C3×D412yes—
118C8×C9—22no—
119C8×C9—42no—
120#454—91noSylow
121C2×3−1+2—21no—
122C2×3−1+2#12011yescharacteristic
123AGL(1,ℤ/9)—42no—
124AGL(1,ℤ/9)—81no—
125C3×#41—92no—
126C3×#41—182no—
127C3×#41—182no—
128C2×C3×Q8—31no—
129C2×C3×Q8—182no—
130C2×C3×Q8C2×C3×D311yescharacteristic
131C2×C3×D4—91no—
132#239—62no—
133D3×Q8—62no—
134#235—32no—
135#235—62no—
136C2×D12—31no—
137C3×SD16—182no—
138C3×D8—182no—
139C3×M16—32no—
140C3×M16C2×C3×D312yes—
141C3×C22⋊C4—182no—
142C3×C42—91no—
143(C2×D3)⋊C4—62no—
144C4×Dic3—31no—
145D24—62no—
146#204—62no—
147C22×C32C2×D1211yescharacteristic
148C2×C3×D3—62no—
149C2×C3×D3—122no—
150C2×C3×D3—121no—
151C32×C4—22no—
152C32×C4D3×D412yes—
153C3×Dic3—62no—
154C3×Dic3—121no—
155C22×C9—21no—
156C22×C9C2×C3×D411yescharacteristic
157C2×D9—22no—
158C2×D9—42no—
159C2×D9—41no—
160C4×C9—22no—
161C4×C9—22no—
162C4×C9—42no—
163C4×C9C2×C3×D412yes—
164Dic9—22no—
165Dic9—41no—
1662−1+4—91no—
167AGL(1,ℤ/8)—92no—
168#124—91no—
169#104—92no—
170#101AGL(1,ℤ/9)11yescharacteristic, complemented
1713−1+2#45411yescharacteristic, complemented, Sylow
172C23×C3—181no—
173C22×D3—61no—
174C3×Q8—22no—
175C3×Q8—62no—
176C3×Q8—92no—
177C3×Q8—182no—
178C3×D4—92no—
179C3×D4—182no—
180C3×D4—182no—
181C3×D4—182no—
182C2×C3×C4—32no—
183C2×C3×C4—31no—
184C2×C3×C4—182no—
185C2×C3×C4—184no—
186C2×C3×C4—181no—
187C2×C3×C4C3×D1212yes—
188C2×C3×C4C22×C3×D311yescharacteristic
189#67—62no—
190C2×Dic3—61no—
191D12—32no—
192D12—62no—
193D12—62no—
194C4×D3—64no—
195C4×D3—62no—
196Dic6—32no—
197Dic6—62no—
198C3×C8—22no—
199C3×C8—62no—
200C2×C32—21no—
201C2×C32#88211yescharacteristic
202C3×D3—122no—
203C3×D3—241no—
204C2×C9—21no—
205C2×C9—21no—
206C2×C9—41no—
207C2×C9C3×#12011yescharacteristic
208D9—42no—
209D9—81no—
210#41—92no—
211#41—182no—
212#41—182no—
213C2×Q8—182no—
214C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
215C2×D4—91no—
216SD16—182no—
217D8—182no—
218M16C2×AGL(1,ℤ/9)12yes—
219C22⋊C4—182no—
220C42—91no—
221C22×C3—31no—
222C22×C3—182no—
223C22×C3—361no—
224C22×C3—362no—
225C22×C3C2×C3×D1211yescharacteristic
226C2×D3—62no—
227C2×D3—122no—
228C2×D3—121no—
229C3×C4—22no—
230C3×C4—32no—
231C3×C4—62no—
232C3×C4—182no—
233C3×C4—361no—
234C3×C4C3×D3×D412yes—
235Dic3—62no—
236Dic3—121no—
237C32#475311yescharacteristic
238C9—21no—
239C9C3×#45411yescharacteristic, complemented
240C23—181no—
241Q8—22no—
242Q8—92no—
243Q8—182no—
244D4—92no—
245D4—182no—
246D4—182no—
247D4—182no—
248C2×C4—184no—
249C2×C4—182no—
250C2×C4—181no—
251C2×C4C4⋊AGL(1,ℤ/9)12yes—
252C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
253C8—22no—
254C2×C3—21no—
255C2×C3—31no—
256C2×C3—61no—
257C2×C3—362no—
258C2×C3—721no—
259C2×C3C3×#88211yescharacteristic
260D3—122no—
261D3—241no—
262C22—182no—
263C22—362no—
264C22—361no—
265C22C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
266C4—22no—
267C4—182no—
268C4—361no—
269C4D4×AGL(1,ℤ/9)12yes—
270C3—31no—
271C3C3×#475311yescharacteristic
272C2—21no—
273C2—362no—
274C2—721no—
275C2#63444⋊C311yescharacteristic
subgroupG/H
centerC2#63444⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#101C2
socleC2×C3C3×#882
hypercenter#101AGL(1,ℤ/9)
hypocenterC9C3×#454
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ #101

Complements

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

NH ≅ G/Ntype
1. #11722630C2semidirect
2. #63509⋊C3C2semidirect
3. #63470⋊C3C2semidirect
4. 3−1+2×#101C2semidirect
5. #63384⋊C3C2semidirect
7. #10588433C3semidirect
22. #63713C2×C3semidirect
25. #63509C2×C3semidirect
26. #63470C2×C3semidirect
28. C9×#101C2×C3semidirect
29. #63384C2×C3semidirect
170. #101AGL(1,ℤ/9)semidirect
171. 3−1+2#454semidirect
239. C9C3×#454semidirect

Maximal subgroups

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

subgroupindex
#117226302
#63509⋊C32
#63470⋊C32
3−1+2×#1012
#63384⋊C32
C3×#47533
#105884333