Finite Group Databasebeta

← #10588427⋊C3 #13141681

Subgroups — #10588427⋊C3 #13141681

Order 1728.

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

normal subgroups only

This group has 527 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, complemented
2#63502⋊C3C211yesmaximal, characteristic, complemented
3#63470⋊C3C212yesmaximal, complemented
43−1+2×#100C211yesmaximal, characteristic, complemented
5#63384⋊C3C212yesmaximal, complemented
6C3×#4747—31nomaximal
7#10588427C311yesmaximal, characteristic, complemented
8C2×D4×3−1+2C2211yescharacteristic
9C2×#611⋊C3—22no—
10C2×#611⋊C3—22no—
11#3699⋊C3—22no—
12D4×AGL(1,ℤ/9)—22no—
13#3697⋊C3C2212yes—
14C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic, complemented
15M16×3−1+2C2212yescomplemented
16C4×3−1+2⋊C4C2211yescharacteristic, complemented
17D72⋊C3—22no—
18#3664⋊C3—22no—
19C3×#1004—31no—
20C3×#940—31no—
21C3×#908—32no—
22#63710C2×C311yescharacteristic, complemented
23C32×#100D311yescharacteristic
24C3×#825—32no—
25#63502C2×C311yescharacteristic, complemented
26#63470C2×C312yescomplemented
27C9×#100—21no—
28C9×#100C2×C311yescharacteristic, complemented
29#63384C2×C312yescomplemented
30C23×3−1+2D412yes—
31C22×AGL(1,ℤ/9)—22no—
32C22×AGL(1,ℤ/9)—21no—
33D4×3−1+2—22no—
34C2×C4×3−1+2C2311yescharacteristic
35#611⋊C3—24no—
36#611⋊C3—22no—
37#611⋊C3—44no—
38C2×3−1+2⋊C4—22no—
39C2×3−1+2⋊C4—21no—
40C4⋊AGL(1,ℤ/9)—22no—
41C4⋊AGL(1,ℤ/9)D412yescomplemented
42C4×AGL(1,ℤ/9)—22no—
433−1+2⋊Q8D412yescomplemented
44C8×3−1+2—22no—
45C3×#453—91no—
46#4747—31no—
47C2×C32×D4C2×D311yescharacteristic
48C2×C3×#67—62no—
49C2×C3×#67—62no—
50C3×#237—62no—
51C3×D3×D4—62no—
52C3×#235—32no—
53C2×C3×D12—31no—
54C32×M16C2×D312yes—
55C3×C4×Dic3—31no—
56C3×D24—62no—
57C3×#204—62no—
58C2×D4×C9—21no—
59C2×D4×C9C22×C311yescharacteristic
60C2×#611—22no—
61C2×#611—22no—
62#3699—22no—
63D4×D9—22no—
64#3697C22×C312yes—
65C2×D36C22×C311yescharacteristic, complemented
66C9×M16—22no—
67C9×M16C22×C312yescomplemented
68C4×Dic9C22×C311yescharacteristic, complemented
69D72—22no—
70#3664—22no—
71C22×3−1+2—22no—
72C22×3−1+2—42no—
73C22×3−1+2C2×D411yescharacteristic
74C2×AGL(1,ℤ/9)—22no—
75C2×AGL(1,ℤ/9)—42no—
76C2×AGL(1,ℤ/9)—41no—
77C2×AGL(1,ℤ/9)—82no—
78C4×3−1+2C2×D412yes—
793−1+2⋊C4—22no—
803−1+2⋊C4—22no—
81C3×H(5,𝔽2)—91no—
82#1004—31no—
83C3×AGL(1,ℤ/8)—92no—
84C3×#127—91no—
85#940—31no—
86#908—32no—
87C3×#104—92no—
88C3×#100—31no—
89C3×#100C3×D311yescharacteristic
90#825—32no—
91C23×C32D1212yes—
92C22×C3×D3—62no—
93C22×C3×D3—61no—
94C32×D4—22no—
95C2×C32×C4C22×D311yescharacteristic
96C3×#67—64no—
97C3×#67—62no—
98C3×#67—124no—
99C2×C3×Dic3—62no—
100C2×C3×Dic3—61no—
101C3×D12—32no—
102C3×D12—62no—
103C3×C4×D3—62no—
104C3×Dic6—32no—
105C23×C9—22no—
106C23×C9C3×D412yes—
107C22×D9—22no—
108C22×D9—21no—
109C32×C8—22no—
110D4×C9—22no—
111D4×C9—42no—
112C2×C4×C9—21no—
113C2×C4×C9C23×C311yescharacteristic
114#611—24no—
115#611—22no—
116#611—44no—
117C2×Dic9—22no—
118C2×Dic9—21no—
119D36—22no—
120D36C3×D412yescomplemented
121C4×D9—22no—
122Dic18C3×D412yescomplemented
123C8×C9—22no—
124C8×C9—42no—
125#453—91noSylow
126C2×3−1+2—21no—
127C2×3−1+2—42no—
128C2×3−1+2#12011yescharacteristic
129AGL(1,ℤ/9)—42no—
130AGL(1,ℤ/9)—81no—
131C3×#41—92no—
132C3×#41—182no—
133C2×C3×D4—31no—
134C2×C3×D4—91no—
135C2×C3×D4—182no—
136C2×C3×D4—182no—
137C2×C3×D4—182no—
138C2×C3×D4C2×C3×D311yescharacteristic
139C2×#67—62no—
140C2×#67—62no—
141#237—62no—
142D3×D4—62no—
143#235—32no—
144C2×D12—31no—
145C3×SD16—182no—
146C3×D8—182no—
147C3×M16—32no—
148C3×M16C2×C3×D312yes—
149C3×C42—91no—
150C4×Dic3—31no—
151D24—62no—
152#204—62no—
153C22×C32—22no—
154C22×C32—42no—
155C22×C32C2×D1211yescharacteristic
156C2×C3×D3—62no—
157C2×C3×D3—122no—
158C2×C3×D3—121no—
159C2×C3×D3—242no—
160C32×C4D3×D412yes—
161C3×Dic3—62no—
162C3×Dic3—62no—
163C22×C9—22no—
164C22×C9—21no—
165C22×C9—42no—
166C22×C9—42no—
167C22×C9—82no—
168C22×C9C2×C3×D411yescharacteristic
169C2×D9—22no—
170C2×D9—42no—
171C2×D9—41no—
172C2×D9—82no—
173C4×C9—22no—
174C4×C9C2×C3×D412yes—
175Dic9—22no—
176Dic9—22no—
177H(5,𝔽2)—91no—
178AGL(1,ℤ/8)—92no—
179#127—91no—
180#104—92no—
181#100AGL(1,ℤ/9)11yescharacteristic, complemented
1823−1+2#45311yescharacteristic, complemented, Sylow
183C23×C3—32no—
184C23×C3—181no—
185C23×C3—182no—
186C23×C3C3×D1212yes—
187C22×D3—62no—
188C22×D3—61no—
189C3×Q8—92no—
190C3×D4—22no—
191C3×D4—62no—
192C3×D4—92no—
193C3×D4—182no—
194C3×D4—182no—
195C3×D4—184no—
196C3×D4—364no—
197C2×C3×C4—31no—
198C2×C3×C4—182no—
199C2×C3×C4—181no—
200C2×C3×C4—182no—
201C2×C3×C4C22×C3×D311yescharacteristic
202#67—64no—
203#67—62no—
204#67—124no—
205C2×Dic3—62no—
206C2×Dic3—61no—
207D12—32no—
208D12—62no—
209C4×D3—62no—
210Dic6—32no—
211C3×C8—22no—
212C3×C8—62no—
213C2×C32—21no—
214C2×C32—42no—
215C2×C32#88211yescharacteristic
216C3×D3—122no—
217C3×D3—241no—
218C2×C9—21no—
219C2×C9—21no—
220C2×C9—41no—
221C2×C9—42no—
222C2×C9—82no—
223C2×C9C3×#12011yescharacteristic
224D9—42no—
225D9—81no—
226#41—92no—
227#41—182no—
228C2×D4—91no—
229C2×D4—182no—
230C2×D4—182no—
231C2×D4—182no—
232C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
233SD16—182no—
234D8—182no—
235M16C2×AGL(1,ℤ/9)12yescomplemented
236C42—91no—
237C22×C3—22no—
238C22×C3—31no—
239C22×C3—42no—
240C22×C3—62no—
241C22×C3—122no—
242C22×C3—182no—
243C22×C3—361no—
244C22×C3—362no—
245C22×C3—722no—
246C22×C3C2×C3×D1211yescharacteristic
247C2×D3—62no—
248C2×D3—122no—
249C2×D3—121no—
250C2×D3—242no—
251C3×C4—32no—
252C3×C4—182no—
253C3×C4—182no—
254C3×C4C3×D3×D412yes—
255Dic3—62no—
256Dic3—62no—
257C32#474711yescharacteristic
258C9—21no—
259C9C3×#45311yescharacteristic, complemented
260C23—182no—
261C23—181no—
262C23C4⋊AGL(1,ℤ/9)12yes—
263Q8—92no—
264D4—22no—
265D4—92no—
266D4—184no—
267D4—182no—
268D4—182no—
269D4—364no—
270C2×C4—182no—
271C2×C4—181no—
272C2×C4—182no—
273C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
274C8—22no—
275C2×C3—21no—
276C2×C3—31no—
277C2×C3—42no—
278C2×C3—61no—
279C2×C3—122no—
280C2×C3—362no—
281C2×C3—721no—
282C2×C3C3×#88211yescharacteristic
283D3—122no—
284D3—241no—
285C22—22no—
286C22—42no—
287C22—182no—
288C22—362no—
289C22—361no—
290C22—722no—
291C22C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
292C4—182no—
293C4—182no—
294C4D4×AGL(1,ℤ/9)12yes—
295C3—31no—
296C3C3×#474711yescharacteristic
297C2—21no—
298C2—42no—
299C2—362no—
300C2—721no—
301C2#63444⋊C311yescharacteristic
subgroupG/H
centerC2#63444⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#100C2
socleC2×C3C3×#882
hypercenter#100AGL(1,ℤ/9)
hypocenterC9C3×#453
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ #100

Complements

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

NH ≅ G/Ntype
1. #63710⋊C3C2semidirect
2. #63502⋊C3C2semidirect
3. #63470⋊C3C2semidirect
4. 3−1+2×#100C2semidirect
5. #63384⋊C3C2semidirect
7. #10588427C3semidirect
14. C2×C4⋊AGL(1,ℤ/9)C22semidirect
15. M16×3−1+2C22semidirect
16. C4×3−1+2⋊C4C22semidirect
22. #63710C2×C3semidirect
25. #63502C2×C3semidirect
26. #63470C2×C3semidirect
28. C9×#100C2×C3semidirect
29. #63384C2×C3semidirect
41. C4⋊AGL(1,ℤ/9)D4semidirect
43. 3−1+2⋊Q8D4semidirect
65. C2×D36C22×C3semidirect
67. C9×M16C22×C3semidirect
68. C4×Dic9C22×C3semidirect
120. D36C3×D4semidirect
122. Dic18C3×D4semidirect
181. #100AGL(1,ℤ/9)semidirect
182. 3−1+2#453semidirect
235. M16C2×AGL(1,ℤ/9)semidirect
259. C9C3×#453semidirect

Maximal subgroups

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

subgroupindex
#63710⋊C32
#63502⋊C32
#63470⋊C32
3−1+2×#1002
#63384⋊C32
C3×#47473
#105884273