Finite Group Databasebeta

← #10588404⋊C3 #13141629

Subgroups — #10588404⋊C3 #13141629

Order 1728.

499 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 110 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
1C22×3−1+2⋊Q8C211yesmaximal, characteristic, complemented
2#63492⋊C3C211yesmaximal, characteristic
3C2×#3664⋊C3C212yesmaximal, complemented
43−1+2×C22⋊C8C211yesmaximal, characteristic, complemented
5#63376⋊C3C212yesmaximal, complemented
6C3×#4738—31nomaximal
7#10588404C311yesmaximal, characteristic, complemented
8C22×C4×3−1+2C2211yescharacteristic
9C2×#611⋊C3—21no—
10C22×3−1+2⋊C4—21no—
11C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
12C2×3−1+2⋊Q8—21no—
13C2×3−1+2⋊Q8—24no—
14C2×3−1+2⋊Q8C2212yes—
15C2×C8×3−1+2C2212yes—
16(C2×AGL(1,ℤ/9))⋊C4—21no—
17C4⋊(3−1+2⋊C4)C2211yescharacteristic
18#3664⋊C3—24no—
19C22×C3×Dic6—31no—
20C3×#930—31no—
21C2×C3×#204—32no—
22C22×Dic18C2×C311yescharacteristic, complemented
23C32×C22⋊C8D311yescharacteristic
24C3×#816—32no—
25#63492C2×C311yescharacteristic
26C2×#3664C2×C312yescomplemented
27C9×C22⋊C8—21no—
28C9×C22⋊C8C2×C311yescharacteristic, complemented
29#63376C2×C312yescomplemented
30C23×3−1+2D411yescharacteristic
31C22×AGL(1,ℤ/9)—21no—
32C2×C4×3−1+2—22no—
33C2×C4×3−1+2D411yescharacteristic
34C2×C4×3−1+2C2311yescharacteristic
35#611⋊C3—42no—
36C2×3−1+2⋊C4—24no—
37C2×3−1+2⋊C4—22no—
38C2×3−1+2⋊C4—21no—
39C4⋊AGL(1,ℤ/9)—22no—
403−1+2⋊Q8—24no—
413−1+2⋊Q8—22no—
423−1+2⋊Q8D414yescomplemented
43C8×3−1+2—22no—
44C3×#448—91no—
45#4738—31no—
46C22×C32×C4C2×D311yescharacteristic
47C2×C3×#67—61no—
48C22×C3×Dic3—61no—
49C2×C3×D12—31no—
50C2×C3×Dic6—32no—
51C2×C3×Dic6—64no—
52C2×C3×Dic6—61no—
53C2×C32×C8C2×D312yes—
54C3×(C2×D3)⋊C4—61no—
55C3×C4⋊Dic3—31no—
56C3×#204—64no—
57C22×C4×C9—21no—
58C22×C4×C9C22×C311yescharacteristic
59C2×#611—21no—
60C22×Dic9—21no—
61C2×D36C22×C311yescharacteristic
62C2×Dic18—21no—
63C2×Dic18—24no—
64C2×Dic18C22×C312yes—
65C2×C8×C9—22no—
66C2×C8×C9C22×C312yes—
67(C2×D9)⋊C4—21no—
68C4⋊Dic9C22×C311yescharacteristic
69#3664—24no—
70C22×3−1+2—21no—
71C22×3−1+2—21no—
72C22×3−1+2SD1612yescomplemented
73C22×3−1+2C2×D411yescharacteristic
74C2×AGL(1,ℤ/9)—41no—
75C2×AGL(1,ℤ/9)—41no—
76C2×AGL(1,ℤ/9)—41no—
77C4×3−1+2—21no—
78C4×3−1+2C2×D412yes—
793−1+2⋊C4—24no—
803−1+2⋊C4—41no—
81C22×C3×Q8—91no—
82C22×Dic6—31no—
83C2×C3×SD16—92no—
84C3×#121—91no—
85#930—31no—
86C2×#204—32no—
87C3×#103—92no—
88C3×C22⋊C8—31no—
89C3×C22⋊C8C3×D311yescharacteristic
90#816—32no—
91C23×C32D1211yescharacteristic
92C22×C3×D3—61no—
93C2×C32×C4—22no—
94C2×C32×C4D1211yescharacteristic
95C2×C32×C4C22×D311yescharacteristic
96C3×#67—122no—
97C2×C3×Dic3—64no—
98C2×C3×Dic3—62no—
99C2×C3×Dic3—61no—
100C3×D12—62no—
101C3×Dic6—34no—
102C3×Dic6—64no—
103C3×Dic6—62no—
104C23×C9—21no—
105C23×C9C3×D411yescharacteristic
106C22×D9—21no—
107C32×C8—22no—
108C2×C4×C9—22no—
109C2×C4×C9—21no—
110C2×C4×C9—21no—
111C2×C4×C9—42no—
112C2×C4×C9C3×D411yescharacteristic
113C2×C4×C9C23×C311yescharacteristic
114#611—42no—
115C2×Dic9—24no—
116C2×Dic9—22no—
117C2×Dic9—21no—
118D36—22no—
119Dic18—24no—
120Dic18—22no—
121Dic18C3×D414yescomplemented
122C8×C9—22no—
123C8×C9—42no—
124#448—91noSylow
125C2×3−1+2—22no—
126C2×3−1+2#12011yescharacteristic
127C2×3−1+2C2×SD1611yescharacteristic
128C2×3−1+2#13711yescharacteristic
129AGL(1,ℤ/9)—81no—
130C2×C3×Q8—92no—
131C2×C3×Q8—181no—
132C2×C3×Q8—184no—
133C2×C3×D4—91no—
134C2×C3×D4—181no—
135C22×C3×C4—31no—
136C22×C3×C4—181no—
137C22×C3×C4C2×C3×D311yescharacteristic
138C2×#67—61no—
139C22×Dic3—61no—
140C2×D12—31no—
141C2×Dic6—32no—
142C2×Dic6—64no—
143C2×Dic6—61no—
144C3×SD16—184no—
145C2×C3×C8—32no—
146C2×C3×C8C2×C3×D312yes—
147C3×C4⋊C4—91no—
148C3×C22⋊C4—181no—
149(C2×D3)⋊C4—61no—
150C4⋊Dic3—31no—
151#204—64no—
152C22×C32—21no—
153C22×C32—21no—
154C22×C32#20412yes—
155C22×C32C2×D1211yescharacteristic
156C2×C3×D3—121no—
157C2×C3×D3—121no—
158C2×C3×D3—121no—
159C32×C4—21no—
160C32×C4D3×D412yes—
161C3×Dic3—64no—
162C3×Dic3—121no—
163C22×C9—21no—
164C22×C9—21no—
165C22×C9—22no—
166C22×C9—21no—
167C22×C9—41no—
168C22×C9—41no—
169C22×C9C3×SD1612yescomplemented
170C22×C9C2×C3×D411yescharacteristic
171C2×D9—41no—
172C2×D9—41no—
173C2×D9—41no—
174C4×C9—21no—
175C4×C9—22no—
176C4×C9—41no—
177C4×C9C2×C3×D412yes—
178Dic9—24no—
179Dic9—41no—
180C22×Q8—91no—
181C2×SD16—92no—
182#121—91no—
183#103—92no—
184C22⋊C8AGL(1,ℤ/9)11yescharacteristic, complemented
1853−1+2#44811yescharacteristic, complemented, Sylow
186C23×C3—31no—
187C23×C3—181no—
188C23×C3C3×D1211yescharacteristic
189C22×D3—61no—
190C3×Q8—94no—
191C3×Q8—182no—
192C3×Q8—184no—
193C3×D4—182no—
194C3×D4—362no—
195C2×C3×C4—22no—
196C2×C3×C4—31no—
197C2×C3×C4—31no—
198C2×C3×C4—62no—
199C2×C3×C4—182no—
200C2×C3×C4—184no—
201C2×C3×C4—181no—
202C2×C3×C4C3×D1211yescharacteristic
203C2×C3×C4C22×C3×D311yescharacteristic
204#67—122no—
205C2×Dic3—64no—
206C2×Dic3—62no—
207C2×Dic3—61no—
208D12—62no—
209Dic6—34no—
210Dic6—64no—
211Dic6—62no—
212C3×C8—22no—
213C3×C8—62no—
214C2×C32—22no—
215C2×C32#88211yescharacteristic
216C2×C32C2×#20411yescharacteristic
217C2×C32#90911yescharacteristic
218C3×D3—241no—
219C2×C9—22no—
220C2×C9—21no—
221C2×C9—21no—
222C2×C9—21no—
223C2×C9—42no—
224C2×C9C3×#12011yescharacteristic
225C2×C9C2×C3×SD1611yescharacteristic
226C2×C9C3×#13711yescharacteristic
227D9—81no—
228C2×Q8—92no—
229C2×Q8—184no—
230C2×Q8—181no—
231C2×D4—91no—
232C2×D4—181no—
233C22×C4—181no—
234C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
235SD16—184no—
236C2×C8C2×AGL(1,ℤ/9)12yes—
237C4⋊C4—91no—
238C22⋊C4—181no—
239C22×C3—21no—
240C22×C3—21no—
241C22×C3—31no—
242C22×C3—32no—
243C22×C3—61no—
244C22×C3—61no—
245C22×C3—361no—
246C22×C3—361no—
247C22×C3—361no—
248C22×C3C3×#20412yes—
249C22×C3C2×C3×D1211yescharacteristic
250C2×D3—121no—
251C2×D3—121no—
252C2×D3—121no—
253C3×C4—21no—
254C3×C4—32no—
255C3×C4—61no—
256C3×C4—184no—
257C3×C4—361no—
258C3×C4C3×D3×D412yes—
259Dic3—64no—
260Dic3—121no—
261C32#473811yescharacteristic
262C9—21no—
263C9C3×#44811yescharacteristic, complemented
264C23—181no—
265C23C4⋊AGL(1,ℤ/9)11yescharacteristic
266Q8—94no—
267Q8—184no—
268Q8—182no—
269D4—182no—
270D4—362no—
271C2×C4—22no—
272C2×C4—184no—
273C2×C4—182no—
274C2×C4—181no—
275C2×C4C4⋊AGL(1,ℤ/9)11yescharacteristic
276C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
277C8—22no—
278C2×C3—22no—
279C2×C3—31no—
280C2×C3—31no—
281C2×C3—31no—
282C2×C3—62no—
283C2×C3—721no—
284C2×C3C3×#88211yescharacteristic
285C2×C3C2×C3×#20411yescharacteristic
286C2×C3C3×#90911yescharacteristic
287D3—241no—
288C22—21no—
289C22—21no—
290C22—361no—
291C22—361no—
292C22—361no—
293C22#3664⋊C312yescomplemented
294C22C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
295C4—21no—
296C4—184no—
297C4—361no—
298C4D4×AGL(1,ℤ/9)12yes—
299C3—31no—
300C3C3×#473811yescharacteristic
301C2—22no—
302C2—721no—
303C2#63444⋊C311yescharacteristic
304C2C2×#3664⋊C311yescharacteristic
305C2#63471⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×C22⋊C8C2
socleC22×C3C2×C3×D12
prime radical#63492⋊C3C2
hypercenterC22⋊C8AGL(1,ℤ/9)
hypocenterC9C3×#448
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
1. C22×3−1+2⋊Q8C2semidirect
3. C2×#3664⋊C3C2semidirect
4. 3−1+2×C22⋊C8C2semidirect
5. #63376⋊C3C2semidirect
7. #10588404C3semidirect
22. C22×Dic18C2×C3semidirect
26. C2×#3664C2×C3semidirect
28. C9×C22⋊C8C2×C3semidirect
29. #63376C2×C3semidirect
42. 3−1+2⋊Q8D4semidirect
72. C22×3−1+2SD16semidirect
121. Dic18C3×D4semidirect
169. C22×C9C3×SD16semidirect
184. C22⋊C8AGL(1,ℤ/9)semidirect
185. 3−1+2#448semidirect
263. C9C3×#448semidirect
293. C22#3664⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×3−1+2⋊Q82
#63492⋊C32
C2×#3664⋊C32
3−1+2×C22⋊C82
#63376⋊C32
C3×#47383
#105884043