Finite Group Databasebeta

← #10588475⋊C3 #13141938

Subgroups — #10588475⋊C3 #13141938

Order 1728.

345 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 345 conjugacy classes of subgroups — too many to draw. Only its 94 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#63506⋊C3C211yesmaximal, characteristic
2C2×#3674⋊C3C211yesmaximal, characteristic
3C2×Dic36⋊C3C211yesmaximal, characteristic
4#63449⋊C3C211yesmaximal, characteristic
53−1+2×#103C211yesmaximal, characteristic
6(3−1+2⋊C4)⋊C8C211yesmaximal, characteristic
7#63368⋊C3C211yesmaximal, characteristic
8C3×#4795—31nomaximal
9#10588475C311yesmaximal, characteristic, complemented
10C2×Q8×3−1+2C2211yescharacteristic
11C2×3−1+2⋊Q8C2211yescharacteristic
12C2×3−1+2⋊Q8C2211yescharacteristic
13C2×C8×3−1+2C2211yescharacteristic
14C4⋊C4×3−1+2C2211yescharacteristic
15#3674⋊C3—22no—
16(3−1+2⋊C4)⋊C4—21no—
17(3−1+2⋊C4)⋊C4—21no—
18(3−1+2⋊C4)⋊C4—21no—
19C4×3−1+2⋊C4—21no—
20C4×3−1+2⋊C4C2211yescharacteristic
21C2×3−1+2⋊C8C2211yescharacteristic
22Dic36⋊C3—22no—
23C3×#944—31no—
24C2×C3×#216—31no—
25C2×C3×Dic12—31no—
26C3×#887—31no—
27C32×#103D311yescharacteristic
28C3×Dic3⋊C8—31no—
29C3×#810—31no—
30#63506C2×C311yescharacteristic
31C2×#3674C2×C311yescharacteristic
32C2×Dic36C2×C311yescharacteristic
33#63449C2×C311yescharacteristic
34C9×#103—21no—
35C9×#103C2×C311yescharacteristic
36Dic9⋊C8C2×C311yescharacteristic
37#63368C2×C311yescharacteristic
38Q8×3−1+2—21no—
39Q8×3−1+2—21no—
40C2×C4×3−1+2—21no—
41C2×C4×3−1+2—21no—
42C2×C4×3−1+2C2311yescharacteristic
43C2×3−1+2⋊C4—21no—
44C2×3−1+2⋊C4—21no—
45C2×3−1+2⋊C4D411yescharacteristic
46C2×3−1+2⋊C4D411yescharacteristic
473−1+2⋊Q8—21no—
483−1+2⋊Q8—21no—
493−1+2⋊Q8—21no—
503−1+2⋊Q8D412yes—
51C8×3−1+2—21no—
523−1+2⋊C8—21no—
53C3×#462—91no—
54#4795—31no—
55C2×C32×Q8C2×D311yescharacteristic
56C2×C3×Dic6—31no—
57C2×C3×Dic6—31no—
58C2×C32×C8C2×D311yescharacteristic
59C32×C4⋊C4C2×D311yescharacteristic
60C3×#216—62no—
61C3×Dic3⋊C4—61no—
62C3×Dic3⋊C4—61no—
63C3×Dic3⋊C4—61no—
64C3×C4×Dic3—31no—
65C3×C4×Dic3—61no—
66C2×C3×C3⋊C8—31no—
67C3×Dic12—62no—
68C2×Q8×C9—21no—
69C2×Q8×C9C22×C311yescharacteristic
70C2×Dic18C22×C311yescharacteristic
71C2×Dic18C22×C311yescharacteristic
72C2×C8×C9—21no—
73C2×C8×C9C22×C311yescharacteristic
74C9×C4⋊C4—21no—
75C9×C4⋊C4C22×C311yescharacteristic
76#3674—22no—
77Dic9⋊C4—21no—
78Dic9⋊C4—21no—
79Dic9⋊C4—21no—
80C4×Dic9—21no—
81C4×Dic9C22×C311yescharacteristic
82C2×C9⋊C8C22×C311yescharacteristic
83Dic36—22no—
84C22×3−1+2C2×D411yescharacteristic
85C4×3−1+2—41no—
86C4×3−1+2—41no—
87C4×3−1+2C2×D411yescharacteristic
88C4×3−1+2#4111yescharacteristic
893−1+2⋊C4—22no—
903−1+2⋊C4—21no—
913−1+2⋊C4—41no—
923−1+2⋊C4Q1612yescomplemented
93C2×C3×Q16—91no—
94C2×C3×Q16—91no—
95C3×C4⋊Q8—91no—
96C3×C4×Q8—91no—
97#944—31no—
98C2×#216—31no—
99C2×Dic12—31no—
100#887—31no—
101C3×C4⋊C8—91no—
102C3×#103—31no—
103C3×#103—91no—
104C3×#103C3×D311yescharacteristic
105Dic3⋊C8—31no—
106#810—31no—
107C32×Q8—21no—
108C32×Q8—21no—
109C2×C32×C4—21no—
110C2×C32×C4—21no—
111C2×C32×C4C22×D311yescharacteristic
112C2×C3×Dic3—31no—
113C2×C3×Dic3—31no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C3×Dic6—32no—
117C3×Dic6—61no—
118C3×Dic6—61no—
119C3×Dic6—61no—
120C32×C8—21no—
121C3×C3⋊C8—61no—
122Q8×C9—22no—
123Q8×C9—21no—
124Q8×C9—21no—
125Q8×C9—41no—
126C2×C4×C9—21no—
127C2×C4×C9—21no—
128C2×C4×C9—21no—
129C2×C4×C9—41no—
130C2×C4×C9—41no—
131C2×C4×C9C23×C311yescharacteristic
132C2×Dic9—21no—
133C2×Dic9—21no—
134C2×Dic9C3×D411yescharacteristic
135C2×Dic9C3×D411yescharacteristic
136Dic18—21no—
137Dic18—21no—
138Dic18—21no—
139Dic18C3×D412yes—
140C8×C9—21no—
141C8×C9—41no—
142C9⋊C8—21no—
143#462—91noSylow
144C2×3−1+2#12111yescharacteristic
145C2×3−1+2C2×Q1611yescharacteristic
146C2×3−1+2#13711yescharacteristic
147C2×C3×Q8—31no—
148C2×C3×Q8—91no—
149C2×C3×Q8—91no—
150C2×C3×Q8C2×C3×D311yescharacteristic
151C2×Dic6—31no—
152C2×Dic6—31no—
153C3×Q16—182no—
154C3×Q16—182no—
155C2×C3×C8—31no—
156C2×C3×C8—91no—
157C2×C3×C8C2×C3×D311yescharacteristic
158C3×C4⋊C4—31no—
159C3×C4⋊C4—181no—
160C3×C4⋊C4—181no—
161C3×C4⋊C4—181no—
162C3×C4⋊C4C2×C3×D311yescharacteristic
163C3×C42—91no—
164C3×C42—181no—
165#216—62no—
166Dic3⋊C4—61no—
167Dic3⋊C4—61no—
168Dic3⋊C4—61no—
169C4×Dic3—31no—
170C4×Dic3—61no—
171C2×C3⋊C8—31no—
172Dic12—62no—
173C22×C32D3×D411yescharacteristic
174C32×C4—41no—
175C32×C4—41no—
176C32×C4#23511yescharacteristic
177C32×C4D3×D411yescharacteristic
178C3×Dic3—32no—
179C3×Dic3—62no—
180C3×Dic3—61no—
181C3×Dic3—121no—
182C22×C9—21no—
183C22×C9C2×C3×D411yescharacteristic
184C4×C9—21no—
185C4×C9—21no—
186C4×C9—42no—
187C4×C9—41no—
188C4×C9—41no—
189C4×C9—81no—
190C4×C9C2×C3×D411yescharacteristic
191C4×C9C3×#4111yescharacteristic
192Dic9—22no—
193Dic9—21no—
194Dic9—41no—
195Dic9C3×Q1612yescomplemented
196C2×Q16—91no—
197C2×Q16—91no—
198C4⋊Q8—91no—
199C4×Q8—91no—
200C4⋊C8—91no—
201#103—91no—
202#103AGL(1,ℤ/9)11yescharacteristic
2033−1+2#46211yescharacteristic, complemented, Sylow
204C3×Q8—21no—
205C3×Q8—21no—
206C3×Q8—61no—
207C3×Q8—61no—
208C3×Q8—92no—
209C3×Q8—181no—
210C3×Q8—181no—
211C3×Q8—181no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—31no—
215C2×C3×C4—61no—
216C2×C3×C4—61no—
217C2×C3×C4—91no—
218C2×C3×C4—91no—
219C2×C3×C4—181no—
220C2×C3×C4—181no—
221C2×C3×C4C22×C3×D311yescharacteristic
222C2×Dic3—31no—
223C2×Dic3—31no—
224C2×Dic3—61no—
225C2×Dic3—61no—
226Dic6—32no—
227Dic6—61no—
228Dic6—61no—
229Dic6—61no—
230C3×C8—21no—
231C3×C8—61no—
232C3×C8—181no—
233C3⋊C8—61no—
234C2×C32#88411yescharacteristic
235C2×C32#91511yescharacteristic
236C2×C32D3×Q1611yescharacteristic
237C2×C9—21no—
238C2×C9—21no—
239C2×C9—21no—
240C2×C9C3×#12111yescharacteristic
241C2×C9C2×C3×Q1611yescharacteristic
242C2×C9C3×#13711yescharacteristic
243C2×Q8—91no—
244C2×Q8—91no—
245C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
246Q16—182no—
247Q16—182no—
248C2×C8—91no—
249C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
250C4⋊C4—181no—
251C4⋊C4—181no—
252C4⋊C4—181no—
253C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
254C42—91no—
255C42—181no—
256C22×C3—31no—
257C22×C3C3×D3×D411yescharacteristic
258C3×C4—31no—
259C3×C4—31no—
260C3×C4—41no—
261C3×C4—41no—
262C3×C4—92no—
263C3×C4—121no—
264C3×C4—121no—
265C3×C4—181no—
266C3×C4—182no—
267C3×C4—361no—
268C3×C4C3×#23511yescharacteristic
269C3×C4C3×D3×D411yescharacteristic
270Dic3—32no—
271Dic3—62no—
272Dic3—61no—
273Dic3—121no—
274C32#479511yescharacteristic
275C9—21no—
276C9C3×#46211yescharacteristic, complemented
277Q8—21no—
278Q8—21no—
279Q8—92no—
280Q8—181no—
281Q8—181no—
282Q8—181no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—91no—
286C2×C4—91no—
287C2×C4—181no—
288C2×C4—181no—
289C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
290C8—21no—
291C8—181no—
292C2×C3—31no—
293C2×C3—31no—
294C2×C3—31no—
295C2×C3C3×#88411yescharacteristic
296C2×C3C3×#91511yescharacteristic
297C2×C3C3×D3×Q1611yescharacteristic
298C22D4×AGL(1,ℤ/9)11yescharacteristic
299C4—41no—
300C4—41no—
301C4—92no—
302C4—181no—
303C4—182no—
304C4—361no—
305C4#3697⋊C311yescharacteristic
306C4D4×AGL(1,ℤ/9)11yescharacteristic
307C3—31no—
308C3C3×#479511yescharacteristic
309C2#63446⋊C311yescharacteristic
310C2#63477⋊C311yescharacteristic
311C2Q16×AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
centerC22D4×AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#103C2
socleC22×C3C3×D3×D4
prime radicalC22×C32D3×D4
hypercenter#103AGL(1,ℤ/9)
hypocenterC9C3×#462
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
9. #10588475C3semidirect
92. 3−1+2⋊C4Q16semidirect
195. Dic9C3×Q16semidirect
203. 3−1+2#462semidirect
276. C9C3×#462semidirect

Maximal subgroups

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

subgroupindex
#63506⋊C32
C2×#3674⋊C32
C2×Dic36⋊C32
#63449⋊C32
3−1+2×#1032
(3−1+2⋊C4)⋊C82
#63368⋊C32
C3×#47953
#105884753