Finite Group Databasebeta

← #10588726⋊C3 #13142893

Subgroups — #10588726⋊C3 #13142893

Order 1728.

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

normal subgroups only

This group has 481 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
23−1+2×C22⋊Q8C211yesmaximal, characteristic
3C2×#3674⋊C3C212yesmaximal, complemented
4#63389⋊C3C211yesmaximal, characteristic
5#63368⋊C3C212yesmaximal, complemented
6C3×#5050—31nomaximal
7#10588726C311yesmaximal, characteristic, complemented
8C2×Q8×3−1+2C2211yescharacteristic
9C22×C4×3−1+2C2211yescharacteristic
10C22×3−1+2⋊C4—21no—
11C2×3−1+2⋊Q8—21no—
12C2×3−1+2⋊Q8—24no—
13C2×3−1+2⋊Q8C2212yes—
14C4⋊C4×3−1+2—21no—
15C4⋊C4×3−1+2C2211yescharacteristic
16C22⋊C4×3−1+2—21no—
17#3674⋊C3—24no—
18C2×3−1+2⋊C8C2212yes—
19C22×C3×Dic6—31no—
20C32×C22⋊Q8D311yescharacteristic
21C2×C3×#216—32no—
22C22×Dic18C2×C311yescharacteristic
23C3×#830—31no—
24C3×#810—32no—
25C9×C22⋊Q8—21no—
26C9×C22⋊Q8C2×C311yescharacteristic
27C2×#3674C2×C312yescomplemented
28#63389C2×C311yescharacteristic
29#63368C2×C312yescomplemented
30C23×3−1+2D411yescharacteristic
31Q8×3−1+2—22no—
32C2×C4×3−1+2—22no—
33C2×C4×3−1+2—21no—
34C2×C4×3−1+2—21no—
35C2×C4×3−1+2D411yescharacteristic
36C2×C4×3−1+2C2311yescharacteristic
37C2×3−1+2⋊C4—24no—
38C2×3−1+2⋊C4—22no—
393−1+2⋊Q8—24no—
403−1+2⋊Q8—22no—
413−1+2⋊Q8D414yes—
423−1+2⋊C8—22no—
43C3×#451—91no—
44#5050—31no—
45C2×C32×Q8C2×D311yescharacteristic
46C22×C32×C4C2×D311yescharacteristic
47C22×C3×Dic3—61no—
48C2×C3×Dic6—32no—
49C2×C3×Dic6—64no—
50C2×C3×Dic6—61no—
51C32×C4⋊C4—21no—
52C32×C4⋊C4C2×D311yescharacteristic
53C32×C22⋊C4—21no—
54C3×#216—64no—
55C2×C3×C3⋊C8—32no—
56C2×Q8×C9—21no—
57C2×Q8×C9C22×C311yescharacteristic
58C22×C4×C9—21no—
59C22×C4×C9C22×C311yescharacteristic
60C22×Dic9—21no—
61C2×Dic18—21no—
62C2×Dic18—24no—
63C2×Dic18C22×C312yes—
64C9×C4⋊C4—22no—
65C9×C4⋊C4—21no—
66C9×C4⋊C4—21no—
67C9×C4⋊C4C22×C311yescharacteristic
68C9×C22⋊C4—22no—
69C9×C22⋊C4—21no—
70#3674—24no—
71C2×C9⋊C8C22×C312yes—
72C22×3−1+2—21no—
73C22×3−1+2—21no—
74C22×3−1+2Q1612yescomplemented
75C22×3−1+2C2×D411yescharacteristic
76C4×3−1+2—21no—
77C4×3−1+2—41no—
78C4×3−1+2—41no—
79C4×3−1+2C2×D412yes—
803−1+2⋊C4—24no—
81C22×C3×Q8—91no—
82C22×Dic6—31no—
83C2×C3×Q16—92no—
84C3×C22⋊Q8—31no—
85C3×C22⋊Q8C3×D311yescharacteristic
86C2×#216—32no—
87C3×#103—92no—
88C3×C22⋊C8—91no—
89#830—31no—
90#810—32no—
91C23×C32#6711yescharacteristic
92C32×Q8—22no—
93C2×C32×C4—22no—
94C2×C32×C4—21no—
95C2×C32×C4—21no—
96C2×C32×C4#6711yescharacteristic
97C2×C32×C4C22×D311yescharacteristic
98C2×C3×Dic3—64no—
99C2×C3×Dic3—62no—
100C3×Dic6—34no—
101C3×Dic6—64no—
102C3×Dic6—62no—
103C23×C9—21no—
104C23×C9C3×D411yescharacteristic
105C3×C3⋊C8—62no—
106Q8×C9—22no—
107Q8×C9—42no—
108C2×C4×C9—22no—
109C2×C4×C9—22no—
110C2×C4×C9—22no—
111C2×C4×C9—21no—
112C2×C4×C9—21no—
113C2×C4×C9—21no—
114C2×C4×C9—21no—
115C2×C4×C9—42no—
116C2×C4×C9C3×D411yescharacteristic
117C2×C4×C9C23×C311yescharacteristic
118C2×Dic9—24no—
119C2×Dic9—22no—
120Dic18—24no—
121Dic18—22no—
122Dic18C3×D414yes—
123C9⋊C8—22no—
124#451—91noSylow
125C2×3−1+2—22no—
126C2×3−1+2#12011yescharacteristic
127C2×3−1+2C2×Q1611yescharacteristic
128C2×3−1+2#13711yescharacteristic
129C2×C3×Q8—31no—
130C2×C3×Q8—92no—
131C2×C3×Q8—181no—
132C2×C3×Q8—184no—
133C2×C3×Q8C2×C3×D311yescharacteristic
134C22×C3×C4—31no—
135C22×C3×C4—181no—
136C22×C3×C4C2×C3×D311yescharacteristic
137C22×Dic3—61no—
138C2×Dic6—32no—
139C2×Dic6—64no—
140C2×Dic6—61no—
141C3×Q16—184no—
142C2×C3×C8—92no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—31no—
145C3×C4⋊C4—61no—
146C3×C4⋊C4C2×C3×D311yescharacteristic
147C3×C22⋊C4—21no—
148C3×C22⋊C4—61no—
149#216—64no—
150C2×C3⋊C8—32no—
151C22×C32—21no—
152C22×C32—21no—
153C22×C32#21612yes—
154C22×C32C2×#6711yescharacteristic
155C32×C4—21no—
156C32×C4—41no—
157C32×C4—41no—
158C32×C4D3×D412yes—
159C3×Dic3—64no—
160C22×C9—21no—
161C22×C9—21no—
162C22×C9—22no—
163C22×C9—21no—
164C22×C9—41no—
165C22×C9—41no—
166C22×C9C3×Q1612yescomplemented
167C22×C9C2×C3×D411yescharacteristic
168C4×C9—21no—
169C4×C9—22no—
170C4×C9—41no—
171C4×C9—42no—
172C4×C9—42no—
173C4×C9—41no—
174C4×C9—41no—
175C4×C9C2×C3×D412yes—
176Dic9—24no—
177C22×Q8—91no—
178C2×Q16—92no—
179C22⋊Q8AGL(1,ℤ/9)11yescharacteristic
180#103—92no—
181C22⋊C8—91no—
1823−1+2#45111yescharacteristic, complemented, Sylow
183C23×C3—31no—
184C23×C3C3×#6711yescharacteristic
185C3×Q8—22no—
186C3×Q8—62no—
187C3×Q8—94no—
188C3×Q8—182no—
189C3×Q8—184no—
190C2×C3×C4—22no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—31no—
194C2×C3×C4—31no—
195C2×C3×C4—62no—
196C2×C3×C4—61no—
197C2×C3×C4—61no—
198C2×C3×C4—182no—
199C2×C3×C4—184no—
200C2×C3×C4C3×#6711yescharacteristic
201C2×C3×C4C22×C3×D311yescharacteristic
202C2×Dic3—64no—
203C2×Dic3—62no—
204Dic6—34no—
205Dic6—64no—
206Dic6—62no—
207C3×C8—182no—
208C3⋊C8—62no—
209C2×C32—22no—
210C2×C32#93711yescharacteristic
211C2×C32C2×#21611yescharacteristic
212C2×C32#95111yescharacteristic
213C2×C9—22no—
214C2×C9—21no—
215C2×C9—21no—
216C2×C9—21no—
217C2×C9—42no—
218C2×C9C3×#12011yescharacteristic
219C2×C9C2×C3×Q1611yescharacteristic
220C2×C9C3×#13711yescharacteristic
221C2×Q8—92no—
222C2×Q8—184no—
223C2×Q8—181no—
224C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
225C22×C4—181no—
226C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
227Q16—184no—
228C2×C8—92no—
229C4⋊C4—21no—
230C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
231C22⋊C4—21no—
232C22×C3—21no—
233C22×C3—21no—
234C22×C3—31no—
235C22×C3—32no—
236C22×C3—61no—
237C22×C3—61no—
238C22×C3C3×#21612yes—
239C22×C3C2×C3×#6711yescharacteristic
240C3×C4—21no—
241C3×C4—32no—
242C3×C4—41no—
243C3×C4—41no—
244C3×C4—61no—
245C3×C4—121no—
246C3×C4—121no—
247C3×C4—184no—
248C3×C4C3×D3×D412yes—
249Dic3—64no—
250C32#505011yescharacteristic
251C9—21no—
252C9C3×#45111yescharacteristic, complemented
253C23#611⋊C311yescharacteristic
254Q8—22no—
255Q8—94no—
256Q8—184no—
257Q8—182no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—184no—
262C2×C4—182no—
263C2×C4#611⋊C311yescharacteristic
264C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
265C8—182no—
266C2×C3—22no—
267C2×C3—31no—
268C2×C3—31no—
269C2×C3—31no—
270C2×C3—62no—
271C2×C3C3×#93711yescharacteristic
272C2×C3C2×C3×#21611yescharacteristic
273C2×C3C3×#95111yescharacteristic
274C22—21no—
275C22—21no—
276C22#3674⋊C312yescomplemented
277C22C2×#611⋊C311yescharacteristic
278C4—21no—
279C4—41no—
280C4—41no—
281C4—184no—
282C4D4×AGL(1,ℤ/9)12yes—
283C3—31no—
284C3C3×#505011yescharacteristic
285C2—22no—
286C2#63499⋊C311yescharacteristic
287C2C2×#3674⋊C311yescharacteristic
288C2#63511⋊C311yescharacteristic
subgroupG/H
centerC22C2×#611⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×C22⋊Q8C2
socleC22×C3C2×C3×#67
prime radicalC23×C32#67
hypercenterC22⋊Q8AGL(1,ℤ/9)
hypocenterC9C3×#451
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#3674⋊C3C2semidirect
5. #63368⋊C3C2semidirect
7. #10588726C3semidirect
27. C2×#3674C2×C3semidirect
29. #63368C2×C3semidirect
74. C22×3−1+2Q16semidirect
166. C22×C9C3×Q16semidirect
182. 3−1+2#451semidirect
252. C9C3×#451semidirect
276. C22#3674⋊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
3−1+2×C22⋊Q82
C2×#3674⋊C32
#63389⋊C32
#63368⋊C32
C3×#50503
#105887263