Finite Group Databasebeta

← #10588557⋊C3 #13142237

Subgroups — #10588557⋊C3 #13142237

Order 1728.

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

normal subgroups only

This group has 288 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#63452⋊C3C211yesmaximal, characteristic
2#63449⋊C3C211yesmaximal, characteristic
33−1+2×#107C211yesmaximal, characteristic
4#63376⋊C3C211yesmaximal, characteristic
5(3−1+2⋊C4)⋊C8C211yesmaximal, characteristic
6#63368⋊C3C211yesmaximal, characteristic
7#63367⋊C3C211yesmaximal, characteristic
8C3×#4877—31nomaximal
9#10588557C311yesmaximal, characteristic, complemented
10C2×3−1+2⋊Q8C2211yescharacteristic
11C2×C8×3−1+2C2211yescharacteristic
12C4⋊C4×3−1+2C2211yescharacteristic
13C4⋊C4×3−1+2C2211yescharacteristic
14C4⋊(3−1+2⋊C4)—21no—
15C4⋊(3−1+2⋊C4)C2211yescharacteristic
16(3−1+2⋊C4)⋊C4—21no—
17(3−1+2⋊C4)⋊C4—21no—
18C4×3−1+2⋊C4—21no—
19C4×3−1+2⋊C4C2211yescharacteristic
20C2×3−1+2⋊C8C2211yescharacteristic
21C3×#890—31no—
22C3×#887—31no—
23C32×#107D311yescharacteristic
24C3×#816—31no—
25C3×Dic3⋊C8—31no—
26C3×#810—31no—
27C3×#808—31no—
28#63452C2×C311yescharacteristic
29#63449C2×C311yescharacteristic
30C9×#107—21no—
31C9×#107C2×C311yescharacteristic
32#63376C2×C311yescharacteristic
33Dic9⋊C8C2×C311yescharacteristic
34#63368C2×C311yescharacteristic
35#63367C2×C311yescharacteristic
36C2×C4×3−1+2—21no—
37C2×C4×3−1+2—21no—
38C2×C4×3−1+2C2311yescharacteristic
39C2×3−1+2⋊C4—21no—
40C2×3−1+2⋊C4—21no—
41C2×3−1+2⋊C4D411yescharacteristic
42C2×3−1+2⋊C4D411yescharacteristic
433−1+2⋊Q8—21no—
443−1+2⋊Q8Q812yes—
45C8×3−1+2—21no—
463−1+2⋊C8—21no—
47C3×#479—91no—
48#4877—31no—
49C2×C3×Dic6—31no—
50C2×C32×C8C2×D311yescharacteristic
51C32×C4⋊C4C2×D311yescharacteristic
52C32×C4⋊C4C2×D311yescharacteristic
53C3×C4⋊Dic3—31no—
54C3×C4⋊Dic3—61no—
55C3×Dic3⋊C4—61no—
56C3×Dic3⋊C4—61no—
57C3×C4×Dic3—31no—
58C3×C4×Dic3—61no—
59C2×C3×C3⋊C8—31no—
60C2×Dic18C22×C311yescharacteristic
61C2×C8×C9—21no—
62C2×C8×C9C22×C311yescharacteristic
63C9×C4⋊C4—21no—
64C9×C4⋊C4—21no—
65C9×C4⋊C4C22×C311yescharacteristic
66C9×C4⋊C4C22×C311yescharacteristic
67C4⋊Dic9—21no—
68C4⋊Dic9C22×C311yescharacteristic
69Dic9⋊C4—21no—
70Dic9⋊C4—21no—
71C4×Dic9—21no—
72C4×Dic9C22×C311yescharacteristic
73C2×C9⋊C8C22×C311yescharacteristic
74C22×3−1+2C2×D411yescharacteristic
75C4×3−1+2—41no—
76C4×3−1+2—41no—
77C4×3−1+2C2×Q811yescharacteristic
78C4×3−1+2#4111yescharacteristic
793−1+2⋊C4—22no—
803−1+2⋊C4—21no—
813−1+2⋊C4—21no—
823−1+2⋊C4—41no—
83C3×#125—91no—
84C3×C4×Q8—91no—
85#890—31no—
86#887—31no—
87C3×#107—31no—
88C3×#107C3×D311yescharacteristic
89C3×#106—91no—
90C3×C4⋊C8—91no—
91C3×#103—91no—
92C3×#103—91no—
93#816—31no—
94Dic3⋊C8—31no—
95#810—31no—
96#808—31no—
97C2×C32×C4—21no—
98C2×C32×C4—21no—
99C2×C32×C4C22×D311yescharacteristic
100C2×C3×Dic3—31no—
101C2×C3×Dic3—31no—
102C2×C3×Dic3—61no—
103C2×C3×Dic3—61no—
104C3×Dic6—32no—
105C3×Dic6—61no—
106C32×C8—21no—
107C3×C3⋊C8—61no—
108C2×C4×C9—21no—
109C2×C4×C9—21no—
110C2×C4×C9—21no—
111C2×C4×C9—41no—
112C2×C4×C9—41no—
113C2×C4×C9C23×C311yescharacteristic
114C2×Dic9—21no—
115C2×Dic9—21no—
116C2×Dic9C3×D411yescharacteristic
117C2×Dic9C3×D411yescharacteristic
118Dic18—21no—
119Dic18C3×Q812yes—
120C8×C9—22no—
121C8×C9—21no—
122C9⋊C8—21no—
123#479—91noSylow
124C2×3−1+2C22⋊Q811yescharacteristic
125C2×3−1+2#13511yescharacteristic
126C2×3−1+2#13711yescharacteristic
127C2×C3×Q8—91no—
128C2×Dic6—31no—
129C2×C3×C8—31no—
130C2×C3×C8—91no—
131C2×C3×C8C2×C3×D311yescharacteristic
132C3×C4⋊C4—31no—
133C3×C4⋊C4—31no—
134C3×C4⋊C4—91no—
135C3×C4⋊C4—181no—
136C3×C4⋊C4—181no—
137C3×C4⋊C4—181no—
138C3×C4⋊C4C2×C3×D311yescharacteristic
139C3×C4⋊C4C2×C3×D311yescharacteristic
140C3×C42—91no—
141C3×C42—181no—
142C4⋊Dic3—31no—
143C4⋊Dic3—61no—
144Dic3⋊C4—61no—
145Dic3⋊C4—61no—
146C4×Dic3—31no—
147C4×Dic3—61no—
148C2×C3⋊C8—31no—
149C22×C32D3×D411yescharacteristic
150C32×C4—41no—
151C32×C4—41no—
152C32×C4#23511yescharacteristic
153C32×C4D3×Q811yescharacteristic
154C3×Dic3—62no—
155C3×Dic3—61no—
156C3×Dic3—61no—
157C3×Dic3—121no—
158C22×C9—21no—
159C22×C9C2×C3×D411yescharacteristic
160C4×C9—21no—
161C4×C9—21no—
162C4×C9—41no—
163C4×C9—41no—
164C4×C9—81no—
165C4×C9—81no—
166C4×C9C2×C3×Q811yescharacteristic
167C4×C9C3×#4111yescharacteristic
168Dic9—22no—
169Dic9—21no—
170Dic9—21no—
171Dic9—41no—
172#125—91no—
173C4×Q8—91no—
174#107AGL(1,ℤ/9)11yescharacteristic
175#106—91no—
176C4⋊C8—91no—
177#103—91no—
178#103—91no—
1793−1+2#47911yescharacteristic, complemented, Sylow
180C3×Q8—92no—
181C3×Q8—181no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—31no—
185C2×C3×C4—61no—
186C2×C3×C4—61no—
187C2×C3×C4—91no—
188C2×C3×C4—91no—
189C2×C3×C4—181no—
190C2×C3×C4—181no—
191C2×C3×C4C22×C3×D311yescharacteristic
192C2×Dic3—31no—
193C2×Dic3—31no—
194C2×Dic3—61no—
195C2×Dic3—61no—
196Dic6—32no—
197Dic6—61no—
198C3×C8—21no—
199C3×C8—61no—
200C3×C8—181no—
201C3⋊C8—61no—
202C2×C32#89611yescharacteristic
203C2×C32#91211yescharacteristic
204C2×C32#91811yescharacteristic
205C2×C9—21no—
206C2×C9—21no—
207C2×C9—21no—
208C2×C9C3×C22⋊Q811yescharacteristic
209C2×C9C3×#13511yescharacteristic
210C2×C9C3×#13711yescharacteristic
211C2×Q8—91no—
212C2×C8—91no—
213C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
214C4⋊C4—91no—
215C4⋊C4—181no—
216C4⋊C4—181no—
217C4⋊C4—181no—
218C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
219C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
220C42—91no—
221C42—181no—
222C22×C3—31no—
223C22×C3C3×D3×D411yescharacteristic
224C3×C4—31no—
225C3×C4—31no—
226C3×C4—41no—
227C3×C4—41no—
228C3×C4—121no—
229C3×C4—121no—
230C3×C4—181no—
231C3×C4—181no—
232C3×C4—182no—
233C3×C4—361no—
234C3×C4C3×#23511yescharacteristic
235C3×C4C3×D3×Q811yescharacteristic
236Dic3—62no—
237Dic3—61no—
238Dic3—61no—
239Dic3—121no—
240C32#487711yescharacteristic
241C9—21no—
242C9C3×#47911yescharacteristic, complemented
243Q8—92no—
244Q8—181no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—91no—
248C2×C4—91no—
249C2×C4—181no—
250C2×C4—181no—
251C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
252C8—21no—
253C8—181no—
254C2×C3—31no—
255C2×C3—31no—
256C2×C3—31no—
257C2×C3C3×#89611yescharacteristic
258C2×C3C3×#91211yescharacteristic
259C2×C3C3×#91811yescharacteristic
260C22D4×AGL(1,ℤ/9)11yescharacteristic
261C4—41no—
262C4—41no—
263C4—182no—
264C4—181no—
265C4—181no—
266C4—361no—
267C4#3697⋊C311yescharacteristic
268C4Q8×AGL(1,ℤ/9)11yescharacteristic
269C3—31no—
270C3C3×#487711yescharacteristic
271C2#63458⋊C311yescharacteristic
272C2#63474⋊C311yescharacteristic
273C2#63480⋊C311yescharacteristic
subgroupG/H
centerC22D4×AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#107C2
socleC22×C3C3×D3×D4
prime radicalC22×C32D3×D4
hypercenter#107AGL(1,ℤ/9)
hypocenterC9C3×#479
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
9. #10588557C3semidirect
179. 3−1+2#479semidirect
242. C9C3×#479semidirect

Maximal subgroups

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

subgroupindex
#63452⋊C32
#63449⋊C32
3−1+2×#1072
#63376⋊C32
(3−1+2⋊C4)⋊C82
#63368⋊C32
#63367⋊C32
C3×#48773
#105885573