Finite Group Databasebeta

← #10588704⋊C3 #13142804

Subgroups — #10588704⋊C3 #13142804

Order 1728.

305 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C4×Q8×3−1+2C211yesmaximal, characteristic
2#63432⋊C3C211yesmaximal, characteristic
3#63395⋊C3C211yesmaximal, characteristic
4#63395⋊C3C211yesmaximal, characteristic
5#63367⋊C3C211yesmaximal, characteristic
6#63366⋊C3C211yesmaximal, characteristic
7C4⋊(3−1+2⋊C8)C211yesmaximal, characteristic
8C3×#5022—31nomaximal
9#10588704C311yesmaximal, characteristic, complemented
10C2×Q8×3−1+2C2211yescharacteristic
11C4⋊C4×3−1+2—21no—
12C4⋊C4×3−1+2C2211yescharacteristic
13C42×3−1+2—21no—
14C42×3−1+2C2211yescharacteristic
15C4⋊(3−1+2⋊C4)C2211yescharacteristic
16C4⋊(3−1+2⋊C4)C2211yescharacteristic
17(3−1+2⋊C4)⋊C4—21no—
18(3−1+2⋊C4)⋊C4—21no—
19C2×3−1+2⋊C8C2211yescharacteristic
20C2×3−1+2⋊C8C2211yescharacteristic
21C32×C4×Q8D311yescharacteristic
22C3×#870—31no—
23C3×#835—31no—
24C3×#835—31no—
25C3×#808—31no—
26C3×#807—31no—
27C3×C4⋊(C3⋊C8)—31no—
28C4×Q8×C9—21no—
29C4×Q8×C9C2×C311yescharacteristic
30#63432C2×C311yescharacteristic
31#63395C2×C311yescharacteristic
32#63395C2×C311yescharacteristic
33#63367C2×C311yescharacteristic
34#63366C2×C311yescharacteristic
35C4⋊(C9⋊C8)C2×C311yescharacteristic
36Q8×3−1+2—21no—
37Q8×3−1+2Q812yes—
38C2×C4×3−1+2—21no—
39C2×C4×3−1+2—21no—
40C2×C4×3−1+2D411yescharacteristic
41C2×C4×3−1+2D411yescharacteristic
42C2×C4×3−1+2C2311yescharacteristic
43C2×3−1+2⋊C4—21no—
44C2×3−1+2⋊C4—21no—
453−1+2⋊C8—21no—
463−1+2⋊C8—21no—
47C3×#479—91no—
48#5022—31no—
49C2×C32×Q8C2×D311yescharacteristic
50C32×C4⋊C4—21no—
51C32×C4⋊C4C2×D311yescharacteristic
52C32×C42—21no—
53C32×C42C2×D311yescharacteristic
54C3×C4⋊Dic3—31no—
55C3×C4⋊Dic3—31no—
56C3×Dic3⋊C4—61no—
57C3×Dic3⋊C4—61no—
58C2×C3×C3⋊C8—31no—
59C2×C3×C3⋊C8—31no—
60C2×Q8×C9—21no—
61C2×Q8×C9C22×C311yescharacteristic
62C9×C4⋊C4—22no—
63C9×C4⋊C4—21no—
64C9×C4⋊C4—21no—
65C9×C4⋊C4C22×C311yescharacteristic
66C42×C9—22no—
67C42×C9—21no—
68C42×C9—21no—
69C42×C9C22×C311yescharacteristic
70C4⋊Dic9C22×C311yescharacteristic
71C4⋊Dic9C22×C311yescharacteristic
72Dic9⋊C4—21no—
73Dic9⋊C4—21no—
74C2×C9⋊C8C22×C311yescharacteristic
75C2×C9⋊C8C22×C311yescharacteristic
76C22×3−1+2C2×D411yescharacteristic
77C4×3−1+2—22no—
78C4×3−1+2—21no—
79C4×3−1+2—21no—
80C4×3−1+2—41no—
81C4×3−1+2C2×Q811yescharacteristic
82C4×3−1+2#4111yescharacteristic
833−1+2⋊C4—41no—
843−1+2⋊C4—41no—
85C3×#125—91no—
86C3×C4×Q8—31no—
87C3×C4×Q8C3×D311yescharacteristic
88#870—31no—
89C3×#107—91no—
90C3×#106—91no—
91C3×C4⋊C8—91no—
92C3×#103—91no—
93C3×#103—91no—
94#835—31no—
95#835—31no—
96#808—31no—
97#807—31no—
98C4⋊(C3⋊C8)—31no—
99C32×Q8—21no—
100C32×Q8Dic612yes—
101C2×C32×C4—21no—
102C2×C32×C4—21no—
103C2×C32×C4#6711yescharacteristic
104C2×C32×C4#6711yescharacteristic
105C2×C32×C4C22×D311yescharacteristic
106C2×C3×Dic3—61no—
107C2×C3×Dic3—61no—
108C3×C3⋊C8—61no—
109C3×C3⋊C8—61no—
110Q8×C9—22no—
111Q8×C9—21no—
112Q8×C9—22no—
113Q8×C9C3×Q812yes—
114C2×C4×C9—22no—
115C2×C4×C9—21no—
116C2×C4×C9—22no—
117C2×C4×C9—21no—
118C2×C4×C9—21no—
119C2×C4×C9—21no—
120C2×C4×C9—21no—
121C2×C4×C9C3×D411yescharacteristic
122C2×C4×C9C3×D411yescharacteristic
123C2×C4×C9C23×C311yescharacteristic
124C2×Dic9—21no—
125C2×Dic9—21no—
126C9⋊C8—21no—
127C9⋊C8—21no—
128#479—91noSylow
129C2×3−1+2C22⋊Q811yescharacteristic
130C2×3−1+2#13511yescharacteristic
131C2×3−1+2#13711yescharacteristic
132C2×C3×Q8—31no—
133C2×C3×Q8C2×C3×D311yescharacteristic
134C2×C3×C8—91no—
135C2×C3×C8—91no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—31no—
138C3×C4⋊C4—61no—
139C3×C4⋊C4—91no—
140C3×C4⋊C4—91no—
141C3×C4⋊C4—181no—
142C3×C4⋊C4—181no—
143C3×C4⋊C4C2×C3×D311yescharacteristic
144C3×C42—21no—
145C3×C42—31no—
146C3×C42—61no—
147C3×C42C2×C3×D311yescharacteristic
148C4⋊Dic3—31no—
149C4⋊Dic3—31no—
150Dic3⋊C4—61no—
151Dic3⋊C4—61no—
152C2×C3⋊C8—31no—
153C2×C3⋊C8—31no—
154C22×C32C2×#6711yescharacteristic
155C32×C4—22no—
156C32×C4—21no—
157C32×C4—21no—
158C32×C4—41no—
159C32×C4C2×Dic611yescharacteristic
160C32×C4#23511yescharacteristic
161C3×Dic3—121no—
162C3×Dic3—121no—
163C22×C9—21no—
164C22×C9C2×C3×D411yescharacteristic
165C4×C9—22no—
166C4×C9—24no—
167C4×C9—22no—
168C4×C9—21no—
169C4×C9—21no—
170C4×C9—21no—
171C4×C9—21no—
172C4×C9—41no—
173C4×C9—42no—
174C4×C9—41no—
175C4×C9C2×C3×Q811yescharacteristic
176C4×C9C3×#4111yescharacteristic
177Dic9—41no—
178Dic9—41no—
179#125—91no—
180C4×Q8AGL(1,ℤ/9)11yescharacteristic
181#107—91no—
182#106—91no—
183C4⋊C8—91no—
184#103—91no—
185#103—91no—
1863−1+2#47911yescharacteristic, complemented, Sylow
187C3×Q8—21no—
188C3×Q8—32no—
189C3×Q8—61no—
190C3×Q8C3×Dic612yes—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—31no—
194C2×C3×C4—31no—
195C2×C3×C4—31no—
196C2×C3×C4—61no—
197C2×C3×C4—61no—
198C2×C3×C4—181no—
199C2×C3×C4—181no—
200C2×C3×C4C3×#6711yescharacteristic
201C2×C3×C4C3×#6711yescharacteristic
202C2×C3×C4C22×C3×D311yescharacteristic
203C2×Dic3—61no—
204C2×Dic3—61no—
205C3×C8—181no—
206C3×C8—181no—
207C3⋊C8—61no—
208C3⋊C8—61no—
209C2×C32#92411yescharacteristic
210C2×C32#94211yescharacteristic
211C2×C32#95011yescharacteristic
212C2×C9—21no—
213C2×C9—21no—
214C2×C9—21no—
215C2×C9C3×C22⋊Q811yescharacteristic
216C2×C9C3×#13511yescharacteristic
217C2×C9C3×#13711yescharacteristic
218C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
219C2×C8—91no—
220C2×C8—91no—
221C4⋊C4—21no—
222C4⋊C4—91no—
223C4⋊C4—91no—
224C4⋊C4—181no—
225C4⋊C4—181no—
226C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
227C42—21no—
228C42C2×AGL(1,ℤ/9)11yescharacteristic
229C22×C3—31no—
230C22×C3C2×C3×#6711yescharacteristic
231C3×C4—22no—
232C3×C4—21no—
233C3×C4—21no—
234C3×C4—31no—
235C3×C4—31no—
236C3×C4—41no—
237C3×C4—61no—
238C3×C4—61no—
239C3×C4—62no—
240C3×C4—121no—
241C3×C4—361no—
242C3×C4—361no—
243C3×C4C2×C3×Dic611yescharacteristic
244C3×C4C3×#23511yescharacteristic
245Dic3—121no—
246Dic3—121no—
247C32#502211yescharacteristic
248C9—21no—
249C9C3×#47911yescharacteristic, complemented
250Q8—21no—
251Q83−1+2⋊Q812yes—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—181no—
255C2×C4—181no—
256C2×C4#611⋊C311yescharacteristic
257C2×C4#611⋊C311yescharacteristic
258C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
259C8—181no—
260C8—181no—
261C2×C3—31no—
262C2×C3—31no—
263C2×C3—31no—
264C2×C3C3×#92411yescharacteristic
265C2×C3C3×#94211yescharacteristic
266C2×C3C3×#95011yescharacteristic
267C22C2×#611⋊C311yescharacteristic
268C4—22no—
269C4—21no—
270C4—21no—
271C4—41no—
272C4—361no—
273C4—361no—
274C4C2×3−1+2⋊Q811yescharacteristic
275C4#3697⋊C311yescharacteristic
276C3—31no—
277C3C3×#502211yescharacteristic
278C2#63486⋊C311yescharacteristic
279C2#63505⋊C311yescharacteristic
280C2#63513⋊C311yescharacteristic
subgroupG/H
centerC22C2×#611⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC4×Q8×3−1+2C2
socleC22×C3C2×C3×#67
prime radicalC22×C32C2×#67
hypercenterC4×Q8AGL(1,ℤ/9)
hypocenterC9C3×#479
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
9. #10588704C3semidirect
186. 3−1+2#479semidirect
249. C9C3×#479semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×3−1+22
#63432⋊C32
#63395⋊C32
#63367⋊C32
#63366⋊C32
C4⋊(3−1+2⋊C8)2
C3×#50223
#105887043