Finite Group Databasebeta

← #10588745⋊C3 #13142958

Subgroups — #10588745⋊C3 #13142958

Order 1728.

298 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 96 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
13−1+2×#125C211yesmaximal, characteristic
2C4×3−1+2⋊Q8C211yesmaximal, characteristic
3#63368⋊C3C211yesmaximal, characteristic
4#63368⋊C3C211yesmaximal, characteristic
5#63367⋊C3C211yesmaximal, characteristic
6#63366⋊C3C211yesmaximal, characteristic
7C4⋊(3−1+2⋊C8)C211yesmaximal, characteristic
8C3×#5063—31nomaximal
9#10588745C311yesmaximal, characteristic, complemented
10C2×3−1+2⋊Q8C2211yescharacteristic
11C4⋊C4×3−1+2—21no—
12C4⋊C4×3−1+2—21no—
13C4⋊C4×3−1+2C2211yescharacteristic
14C4⋊C4×3−1+2C2211yescharacteristic
15C42×3−1+2C2211yescharacteristic
16C4⋊(3−1+2⋊C4)C2211yescharacteristic
17(3−1+2⋊C4)⋊C4—21no—
18C4×3−1+2⋊C4—21no—
19C2×3−1+2⋊C8C2211yescharacteristic
20C2×3−1+2⋊C8C2211yescharacteristic
21C32×#125D311yescharacteristic
22C3×C4×Dic6—31no—
23C3×#810—31no—
24C3×#810—31no—
25C3×#808—31no—
26C3×#807—31no—
27C3×C4⋊(C3⋊C8)—31no—
28C9×#125—21no—
29C9×#125C2×C311yescharacteristic
30C4×Dic18C2×C311yescharacteristic
31#63368C2×C311yescharacteristic
32#63368C2×C311yescharacteristic
33#63367C2×C311yescharacteristic
34#63366C2×C311yescharacteristic
35C4⋊(C9⋊C8)C2×C311yescharacteristic
36C2×C4×3−1+2—21no—
37C2×C4×3−1+2—21no—
38C2×C4×3−1+2D411yescharacteristic
39C2×C4×3−1+2D411yescharacteristic
40C2×C4×3−1+2C2311yescharacteristic
41C2×3−1+2⋊C4—21no—
42C2×3−1+2⋊C4—21no—
433−1+2⋊Q8—21no—
443−1+2⋊Q8Q812yes—
453−1+2⋊C8—21no—
463−1+2⋊C8—21no—
47C3×#479—91no—
48#5063—31no—
49C2×C3×Dic6—31no—
50C32×C4⋊C4—21no—
51C32×C4⋊C4—21no—
52C32×C4⋊C4C2×D311yescharacteristic
53C32×C4⋊C4C2×D311yescharacteristic
54C32×C42C2×D311yescharacteristic
55C3×C4⋊Dic3—31no—
56C3×Dic3⋊C4—61no—
57C3×C4×Dic3—61no—
58C2×C3×C3⋊C8—31no—
59C2×C3×C3⋊C8—31no—
60C2×Dic18C22×C311yescharacteristic
61C9×C4⋊C4—22no—
62C9×C4⋊C4—22no—
63C9×C4⋊C4—21no—
64C9×C4⋊C4—21no—
65C9×C4⋊C4—21no—
66C9×C4⋊C4—21no—
67C9×C4⋊C4C22×C311yescharacteristic
68C9×C4⋊C4C22×C311yescharacteristic
69C42×C9—21no—
70C42×C9C22×C311yescharacteristic
71C4⋊Dic9C22×C311yescharacteristic
72Dic9⋊C4—21no—
73C4×Dic9—21no—
74C2×C9⋊C8C22×C311yescharacteristic
75C2×C9⋊C8C22×C311yescharacteristic
76C22×3−1+2C2×D411yescharacteristic
77C4×3−1+2—21no—
78C4×3−1+2—21no—
79C4×3−1+2—41no—
80C4×3−1+2—41no—
81C4×3−1+2C2×Q811yescharacteristic
82C4×3−1+2#4111yescharacteristic
833−1+2⋊C4—22no—
843−1+2⋊C4—41no—
85C3×#125—31no—
86C3×#125C3×D311yescharacteristic
87C3×C4×Q8—91no—
88C4×Dic6—31no—
89C3×#107—91no—
90C3×#106—91no—
91C3×C4⋊C8—91no—
92C3×#103—91no—
93C3×#103—91no—
94#810—31no—
95#810—31no—
96#808—31no—
97#807—31no—
98C4⋊(C3⋊C8)—31no—
99C2×C32×C4—21no—
100C2×C32×C4—21no—
101C2×C32×C4#6711yescharacteristic
102C2×C32×C4#6711yescharacteristic
103C2×C32×C4C22×D311yescharacteristic
104C2×C3×Dic3—61no—
105C2×C3×Dic3—61no—
106C3×Dic6—32no—
107C3×Dic6—61no—
108C3×C3⋊C8—61no—
109C3×C3⋊C8—61no—
110C2×C4×C9—22no—
111C2×C4×C9—22no—
112C2×C4×C9—21no—
113C2×C4×C9—21no—
114C2×C4×C9—21no—
115C2×C4×C9—21no—
116C2×C4×C9—21no—
117C2×C4×C9C3×D411yescharacteristic
118C2×C4×C9C3×D411yescharacteristic
119C2×C4×C9C23×C311yescharacteristic
120C2×Dic9—21no—
121C2×Dic9—21no—
122Dic18—21no—
123Dic18C3×Q812yes—
124C9⋊C8—21no—
125C9⋊C8—21no—
126#479—91noSylow
127C2×3−1+2C22⋊Q811yescharacteristic
128C2×3−1+2#13511yescharacteristic
129C2×3−1+2#13711yescharacteristic
130C2×C3×Q8—91no—
131C2×Dic6—31no—
132C2×C3×C8—91no—
133C2×C3×C8—91no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—31no—
137C3×C4⋊C4—31no—
138C3×C4⋊C4—61no—
139C3×C4⋊C4—61no—
140C3×C4⋊C4—91no—
141C3×C4⋊C4—181no—
142C3×C4⋊C4C2×C3×D311yescharacteristic
143C3×C4⋊C4C2×C3×D311yescharacteristic
144C3×C42—31no—
145C3×C42—181no—
146C3×C42C2×C3×D311yescharacteristic
147C4⋊Dic3—31no—
148Dic3⋊C4—61no—
149C4×Dic3—61no—
150C2×C3⋊C8—31no—
151C2×C3⋊C8—31no—
152C22×C32C2×#6711yescharacteristic
153C32×C4—21no—
154C32×C4—21no—
155C32×C4—41no—
156C32×C4—41no—
157C32×C4D3×Q811yescharacteristic
158C32×C4#23911yescharacteristic
159C3×Dic3—62no—
160C3×Dic3—121no—
161C22×C9—21no—
162C22×C9C2×C3×D411yescharacteristic
163C4×C9—21no—
164C4×C9—21no—
165C4×C9—21no—
166C4×C9—21no—
167C4×C9—41no—
168C4×C9—41no—
169C4×C9—42no—
170C4×C9—42no—
171C4×C9—41no—
172C4×C9—41no—
173C4×C9C2×C3×Q811yescharacteristic
174C4×C9C3×#4111yescharacteristic
175Dic9—22no—
176Dic9—41no—
177#125AGL(1,ℤ/9)11yescharacteristic
178C4×Q8—91no—
179#107—91no—
180#106—91no—
181C4⋊C8—91no—
182#103—91no—
183#103—91no—
1843−1+2#47911yescharacteristic, complemented, Sylow
185C3×Q8—92no—
186C3×Q8—181no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—31no—
190C2×C3×C4—31no—
191C2×C3×C4—31no—
192C2×C3×C4—61no—
193C2×C3×C4—61no—
194C2×C3×C4—181no—
195C2×C3×C4—181no—
196C2×C3×C4C3×#6711yescharacteristic
197C2×C3×C4C3×#6711yescharacteristic
198C2×C3×C4C22×C3×D311yescharacteristic
199C2×Dic3—61no—
200C2×Dic3—61no—
201Dic6—32no—
202Dic6—61no—
203C3×C8—181no—
204C3×C8—181no—
205C3⋊C8—61no—
206C3⋊C8—61no—
207C2×C32#94611yescharacteristic
208C2×C32#95011yescharacteristic
209C2×C32#95111yescharacteristic
210C2×C9—21no—
211C2×C9—21no—
212C2×C9—21no—
213C2×C9C3×C22⋊Q811yescharacteristic
214C2×C9C3×#13511yescharacteristic
215C2×C9C3×#13711yescharacteristic
216C2×Q8—91no—
217C2×C8—91no—
218C2×C8—91no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—91no—
222C4⋊C4—181no—
223C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
224C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
225C42—181no—
226C42C2×AGL(1,ℤ/9)11yescharacteristic
227C22×C3—31no—
228C22×C3C2×C3×#6711yescharacteristic
229C3×C4—21no—
230C3×C4—21no—
231C3×C4—31no—
232C3×C4—31no—
233C3×C4—41no—
234C3×C4—41no—
235C3×C4—61no—
236C3×C4—61no—
237C3×C4—121no—
238C3×C4—121no—
239C3×C4—182no—
240C3×C4—361no—
241C3×C4C3×D3×Q811yescharacteristic
242C3×C4C3×#23911yescharacteristic
243Dic3—62no—
244Dic3—121no—
245C32#506311yescharacteristic
246C9—21no—
247C9C3×#47911yescharacteristic, complemented
248Q8—92no—
249Q8—181no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—181no—
253C2×C4—181no—
254C2×C4#611⋊C311yescharacteristic
255C2×C4#611⋊C311yescharacteristic
256C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
257C8—181no—
258C8—181no—
259C2×C3—31no—
260C2×C3—31no—
261C2×C3—31no—
262C2×C3C3×#94611yescharacteristic
263C2×C3C3×#95011yescharacteristic
264C2×C3C3×#95111yescharacteristic
265C22C2×#611⋊C311yescharacteristic
266C4—21no—
267C4—21no—
268C4—41no—
269C4—41no—
270C4—182no—
271C4—361no—
272C4Q8×AGL(1,ℤ/9)11yescharacteristic
273C4#8872411yescharacteristic
274C3—31no—
275C3C3×#506311yescharacteristic
276C2#63508⋊C311yescharacteristic
277C2#63511⋊C311yescharacteristic
278C2#63513⋊C311yescharacteristic
subgroupG/H
centerC22C2×#611⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#125C2
socleC22×C3C2×C3×#67
prime radicalC22×C32C2×#67
hypercenter#125AGL(1,ℤ/9)
hypocenterC9C3×#479
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
9. #10588745C3semidirect
184. 3−1+2#479semidirect
247. C9C3×#479semidirect

Maximal subgroups

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

subgroupindex
3−1+2×#1252
C4×3−1+2⋊Q82
#63368⋊C32
#63367⋊C32
#63366⋊C32
C4⋊(3−1+2⋊C8)2
C3×#50633
#105887453