Finite Group Databasebeta

← #10588741⋊C3 #13142934

Subgroups — #10588741⋊C3 #13142934

Order 1728.

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

normal subgroups only

This group has 494 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#63710⋊C3C211yesmaximal, characteristic
23−1+2×#124C211yesmaximal, characteristic, complemented
3#63505⋊C3C212yesmaximal, complemented
4#63391⋊C3C211yesmaximal, characteristic, complemented
5#63364⋊C3C212yesmaximal, complemented
6C3×#5061—31nomaximal
7#10588741C311yesmaximal, characteristic, complemented
8C2×Q8×3−1+2C2211yescharacteristic, complemented
9C2×D4×3−1+2C2211yescharacteristic
10C2×#611⋊C3—22no—
11#3699⋊C3—22no—
12D4×AGL(1,ℤ/9)—22no—
13#3697⋊C3C2212yes—
14C22⋊C4×3−1+2—22no—
15C42×3−1+2C2211yescharacteristic, complemented
16#3675⋊C3—22no—
17#3674⋊C3—22no—
18#3667⋊C3C2212yescomplemented
19C3×#1004—31no—
20C32×#124D311yescharacteristic
21C3×#942—32no—
22#63710C2×C311yescharacteristic
23C3×#833—31no—
24C3×#805—32no—
25C9×#124—21no—
26C9×#124C2×C311yescharacteristic, complemented
27#63505C2×C312yescomplemented
28#63391C2×C311yescharacteristic, complemented
29#63364C2×C312yescomplemented
30C23×3−1+2D412yes—
31C22×AGL(1,ℤ/9)—22no—
32Q8×3−1+2—22no—
33D4×3−1+2—22no—
34C2×C4×3−1+2—21no—
35C2×C4×3−1+2—21no—
36C2×C4×3−1+2C2311yescharacteristic
37#611⋊C3—24no—
38#611⋊C3—22no—
39C2×3−1+2⋊C4—22no—
40C4⋊AGL(1,ℤ/9)D412yes—
41C4×AGL(1,ℤ/9)—22no—
423−1+2⋊Q8D412yes—
433−1+2⋊C8—22no—
44C3×#455—91no—
45#5061—31no—
46C2×C32×Q8C2×D311yescharacteristic
47C2×C32×D4C2×D311yescharacteristic
48C2×C3×#67—62no—
49C3×#237—62no—
50C3×D3×D4—62no—
51C3×#235—32no—
52C32×C22⋊C4—22no—
53C32×C42C2×D311yescharacteristic
54C3×#216—62no—
55C3×#215—62no—
56C3×#208—32no—
57C2×Q8×C9—21no—
58C2×Q8×C9C22×C311yescharacteristic, complemented
59C2×D4×C9—21no—
60C2×D4×C9C22×C311yescharacteristic
61C2×#611—22no—
62#3699—22no—
63D4×D9—22no—
64#3697C22×C312yes—
65C9×C22⋊C4—24no—
66C9×C22⋊C4—22no—
67C42×C9—21no—
68C42×C9C22×C311yescharacteristic, complemented
69#3675—22no—
70#3674—22no—
71#3667C22×C312yescomplemented
72C22×3−1+2—22no—
73C22×3−1+2—42no—
74C22×3−1+2C2×D411yescharacteristic
75C2×AGL(1,ℤ/9)—22no—
76C2×AGL(1,ℤ/9)—82no—
77C4×3−1+2—41no—
78C4×3−1+2—41no—
79C4×3−1+2C2×D412yes—
803−1+2⋊C4—22no—
81C3×H(5,𝔽2)—91no—
82#1004—31no—
83C3×#137—92no—
84C3×#124—31no—
85C3×#124C3×D311yescharacteristic
86#942—32no—
87C3×#104—92no—
88C3×#100—91no—
89#833—31no—
90#805—32no—
91C23×C32#6712yes—
92C22×C3×D3—62no—
93C32×Q8—22no—
94C32×D4—22no—
95C2×C32×C4—21no—
96C2×C32×C4—21no—
97C2×C32×C4C22×D311yescharacteristic
98C3×#67—64no—
99C3×#67—62no—
100C2×C3×Dic3—62no—
101C3×D12—32no—
102C3×C4×D3—62no—
103C3×Dic6—32no—
104C23×C9—22no—
105C23×C9C3×D412yes—
106C22×D9—22no—
107C3×C3⋊C8—62no—
108Q8×C9—22no—
109Q8×C9—42no—
110D4×C9—22no—
111D4×C9—42no—
112C2×C4×C9—22no—
113C2×C4×C9—22no—
114C2×C4×C9—21no—
115C2×C4×C9—21no—
116C2×C4×C9—21no—
117C2×C4×C9C23×C311yescharacteristic
118#611—24no—
119#611—22no—
120C2×Dic9—22no—
121D36C3×D412yes—
122C4×D9—22no—
123Dic18C3×D412yes—
124C9⋊C8—22no—
125#455—91noSylow
126C2×3−1+2—21no—
127C2×3−1+2—42no—
128C2×3−1+2#12011yescharacteristic
129AGL(1,ℤ/9)—42no—
130C3×#41—92no—
131C3×#41—182no—
132C2×C3×Q8—31no—
133C2×C3×Q8C2×C3×D311yescharacteristic
134C2×C3×D4—31no—
135C2×C3×D4—182no—
136C2×C3×D4—182no—
137C2×C3×D4C2×C3×D311yescharacteristic
138C2×#67—62no—
139#237—62no—
140D3×D4—62no—
141#235—32no—
142C3×Q16—182no—
143C3×SD16—182no—
144C3×M16—92no—
145C3×C22⋊C4—22no—
146C3×C22⋊C4—62no—
147C3×C42—31no—
148C3×C42C2×C3×D311yescharacteristic
149#216—62no—
150#215—62no—
151#208—32no—
152C22×C32—22no—
153C22×C32—42no—
154C22×C32C2×#6711yescharacteristic
155C2×C3×D3—62no—
156C2×C3×D3—242no—
157C32×C4—41no—
158C32×C4—41no—
159C32×C4D3×D412yes—
160C3×Dic3—62no—
161C22×C9—22no—
162C22×C9—21no—
163C22×C9—42no—
164C22×C9—42no—
165C22×C9—44no—
166C22×C9C2×C3×D411yescharacteristic
167C2×D9—22no—
168C2×D9—82no—
169C4×C9—22no—
170C4×C9—42no—
171C4×C9—41no—
172C4×C9—42no—
173C4×C9—41no—
174C4×C9C2×C3×D412yes—
175Dic9—22no—
176H(5,𝔽2)—91no—
177#137—92no—
178#124AGL(1,ℤ/9)11yescharacteristic, complemented
179#104—92no—
180#100—91no—
1813−1+2#45511yescharacteristic, complemented, Sylow
182C23×C3—32no—
183C23×C3—182no—
184C23×C3C3×#6712yes—
185C22×D3—62no—
186C3×Q8—22no—
187C3×Q8—62no—
188C3×Q8—92no—
189C3×D4—22no—
190C3×D4—62no—
191C3×D4—92no—
192C3×D4—182no—
193C3×D4—184no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—31no—
197C2×C3×C4—61no—
198C2×C3×C4—61no—
199C2×C3×C4—182no—
200C2×C3×C4—182no—
201C2×C3×C4C22×C3×D311yescharacteristic
202#67—64no—
203#67—62no—
204C2×Dic3—62no—
205D12—32no—
206C4×D3—62no—
207Dic6—32no—
208C3×C8—182no—
209C3⋊C8—62no—
210C2×C32—21no—
211C2×C32—42no—
212C2×C32#93711yescharacteristic
213C3×D3—122no—
214C2×C9—22no—
215C2×C9—21no—
216C2×C9—21no—
217C2×C9—42no—
218C2×C9—82no—
219C2×C9C3×#12011yescharacteristic
220D9—42no—
221#41—92no—
222#41—182no—
223C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic, complemented
224C2×D4—182no—
225C2×D4—182no—
226C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
227Q16—182no—
228SD16—182no—
229M16—92no—
230C22⋊C4—22no—
231C42C2×AGL(1,ℤ/9)11yescharacteristic, complemented
232C22×C3—22no—
233C22×C3—31no—
234C22×C3—42no—
235C22×C3—62no—
236C22×C3—122no—
237C22×C3—182no—
238C22×C3—722no—
239C22×C3C2×C3×#6711yescharacteristic
240C2×D3—62no—
241C2×D3—242no—
242C3×C4—32no—
243C3×C4—41no—
244C3×C4—41no—
245C3×C4—121no—
246C3×C4—121no—
247C3×C4—182no—
248C3×C4C3×D3×D412yes—
249Dic3—62no—
250C32#506111yescharacteristic
251C9—21no—
252C9C3×#45511yescharacteristic, complemented
253C23—182no—
254C23#611⋊C312yes—
255Q8—22no—
256Q8—92no—
257D4—22no—
258D4—92no—
259D4—184no—
260D4—182no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—182no—
264C2×C4—182no—
265C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
266C8—182no—
267C2×C3—21no—
268C2×C3—31no—
269C2×C3—42no—
270C2×C3—61no—
271C2×C3—122no—
272C2×C3—362no—
273C2×C3C3×#93711yescharacteristic
274D3—122no—
275C22—22no—
276C22—42no—
277C22—182no—
278C22—722no—
279C22C2×#611⋊C311yescharacteristic
280C4—41no—
281C4—41no—
282C4—182no—
283C4D4×AGL(1,ℤ/9)12yes—
284C3—31no—
285C3C3×#506111yescharacteristic
286C2—21no—
287C2—42no—
288C2—362no—
289C2#63499⋊C311yescharacteristic
subgroupG/H
centerC2#63499⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#124C2
socleC2×C3C3×#937
prime radical#63710⋊C3C2
hypercenter#124AGL(1,ℤ/9)
hypocenterC9C3×#455
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ #124

Complements

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

NH ≅ G/Ntype
2. 3−1+2×#124C2semidirect
3. #63505⋊C3C2semidirect
4. #63391⋊C3C2semidirect
5. #63364⋊C3C2semidirect
7. #10588741C3semidirect
8. C2×Q8×3−1+2C22semidirect
15. C42×3−1+2C22semidirect
18. #3667⋊C3C22semidirect
26. C9×#124C2×C3semidirect
27. #63505C2×C3semidirect
28. #63391C2×C3semidirect
29. #63364C2×C3semidirect
58. C2×Q8×C9C22×C3semidirect
68. C42×C9C22×C3semidirect
71. #3667C22×C3semidirect
178. #124AGL(1,ℤ/9)semidirect
181. 3−1+2#455semidirect
223. C2×Q8C2×AGL(1,ℤ/9)semidirect
231. C42C2×AGL(1,ℤ/9)semidirect
252. C9C3×#455semidirect

Maximal subgroups

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

subgroupindex
#63710⋊C32
3−1+2×#1242
#63505⋊C32
#63391⋊C32
#63364⋊C32
C3×#50613
#105887413