Finite Group Databasebeta

← #10588716⋊C3 #13142853

Subgroups — #10588716⋊C3 #13142853

Order 1728.

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

normal subgroups only

This group has 521 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, complemented
23−1+2×#121C211yesmaximal, characteristic
3C2×#3672⋊C3C212yesmaximal, complemented
4#63389⋊C3C211yesmaximal, characteristic, complemented
5#63368⋊C3C212yesmaximal, complemented
6C3×#5040—31nomaximal
7#10588716C311yesmaximal, characteristic, complemented
8C2×D4×3−1+2—21no—
9C2×D4×3−1+2C2211yescharacteristic
10C22×C4×3−1+2C2211yescharacteristic
11C22×3−1+2⋊C4—21no—
12C2×3−1+2⋊Q8—21no—
13C2×3−1+2⋊Q8—24no—
14C2×3−1+2⋊Q8C2212yes—
15C4⋊C4×3−1+2C2211yescharacteristic
16C22⋊C4×3−1+2—21no—
17#3672⋊C3—24no—
18C2×3−1+2⋊C8C2212yes—
19C22×C3×Dic6—31no—
20C32×#121D311yescharacteristic
21C2×C3×#214—32no—
22C22×Dic18C2×C311yescharacteristic, complemented
23C3×#830—31no—
24C3×#810—32no—
25C9×#121—21no—
26C9×#121C2×C311yescharacteristic
27C2×#3672C2×C312yescomplemented
28#63389C2×C311yescharacteristic, complemented
29#63368C2×C312yescomplemented
30C23×3−1+2—21no—
31C23×3−1+2D411yescharacteristic
32D4×3−1+2—22no—
33D4×3−1+2—42no—
34C2×C4×3−1+2—22no—
35C2×C4×3−1+2—21no—
36C2×C4×3−1+2D411yescharacteristic
37C2×C4×3−1+2C2311yescharacteristic
38C2×3−1+2⋊C4—24no—
39C2×3−1+2⋊C4—22no—
403−1+2⋊Q8—24no—
413−1+2⋊Q8—22no—
423−1+2⋊Q8D414yescomplemented
433−1+2⋊C8—22no—
44C3×#448—91no—
45#5040—31no—
46C2×C32×D4—21no—
47C2×C32×D4C2×D311yescharacteristic
48C22×C32×C4C2×D311yescharacteristic
49C22×C3×Dic3—61no—
50C2×C3×Dic6—32no—
51C2×C3×Dic6—64no—
52C2×C3×Dic6—61no—
53C32×C4⋊C4C2×D311yescharacteristic
54C32×C22⋊C4—21no—
55C3×#214—64no—
56C2×C3×C3⋊C8—32no—
57C2×D4×C9—22no—
58C2×D4×C9—21no—
59C2×D4×C9—21no—
60C2×D4×C9C22×C311yescharacteristic
61C22×C4×C9—21no—
62C22×C4×C9C22×C311yescharacteristic
63C22×Dic9—21no—
64C2×Dic18—21no—
65C2×Dic18—24no—
66C2×Dic18C22×C312yes—
67C9×C4⋊C4—21no—
68C9×C4⋊C4C22×C311yescharacteristic
69C9×C22⋊C4—22no—
70C9×C22⋊C4—21no—
71#3672—24no—
72C2×C9⋊C8C22×C312yes—
73C22×3−1+2—21no—
74C22×3−1+2—21no—
75C22×3−1+2—41no—
76C22×3−1+2—41no—
77C22×3−1+2—41no—
78C22×3−1+2SD1612yescomplemented
79C22×3−1+2C2×D411yescharacteristic
80C4×3−1+2—21no—
81C4×3−1+2—41no—
82C4×3−1+2C2×D412yes—
833−1+2⋊C4—24no—
84C22×C3×Q8—91no—
85C22×Dic6—31no—
86C2×C3×SD16—92no—
87C3×#121—31no—
88C3×#121C3×D311yescharacteristic
89C2×#214—32no—
90C3×#103—92no—
91C3×C22⋊C8—91no—
92#830—31no—
93#810—32no—
94C23×C32—21no—
95C23×C32#6711yescharacteristic
96C32×D4—22no—
97C32×D4—42no—
98C2×C32×C4—22no—
99C2×C32×C4—21no—
100C2×C32×C4#6711yescharacteristic
101C2×C32×C4C22×D311yescharacteristic
102C2×C3×Dic3—64no—
103C2×C3×Dic3—62no—
104C3×Dic6—34no—
105C3×Dic6—64no—
106C3×Dic6—62no—
107C23×C9—22no—
108C23×C9—21no—
109C23×C9—21no—
110C23×C9C3×D411yescharacteristic
111C3×C3⋊C8—62no—
112D4×C9—22no—
113D4×C9—42no—
114D4×C9—44no—
115D4×C9—42no—
116C2×C4×C9—22no—
117C2×C4×C9—22no—
118C2×C4×C9—21no—
119C2×C4×C9—21no—
120C2×C4×C9—21no—
121C2×C4×C9—42no—
122C2×C4×C9C3×D411yescharacteristic
123C2×C4×C9C23×C311yescharacteristic
124C2×Dic9—24no—
125C2×Dic9—22no—
126Dic18—24no—
127Dic18—22no—
128Dic18C3×D414yescomplemented
129C9⋊C8—22no—
130#448—91noSylow
131C2×3−1+2—22no—
132C2×3−1+2—81no—
133C2×3−1+2#12011yescharacteristic
134C2×3−1+2C2×SD1611yescharacteristic
135C2×3−1+2#13711yescharacteristic
136C2×C3×Q8—92no—
137C2×C3×Q8—181no—
138C2×C3×Q8—184no—
139C2×C3×D4—21no—
140C2×C3×D4—31no—
141C2×C3×D4—61no—
142C2×C3×D4C2×C3×D311yescharacteristic
143C22×C3×C4—31no—
144C22×C3×C4—181no—
145C22×C3×C4C2×C3×D311yescharacteristic
146C22×Dic3—61no—
147C2×Dic6—32no—
148C2×Dic6—64no—
149C2×Dic6—61no—
150C3×SD16—184no—
151C2×C3×C8—92no—
152C3×C4⋊C4—31no—
153C3×C4⋊C4C2×C3×D311yescharacteristic
154C3×C22⋊C4—21no—
155C3×C22⋊C4—61no—
156#214—64no—
157C2×C3⋊C8—32no—
158C22×C32—21no—
159C22×C32—21no—
160C22×C32—41no—
161C22×C32—41no—
162C22×C32—41no—
163C22×C32#21412yes—
164C22×C32C2×#6711yescharacteristic
165C32×C4—21no—
166C32×C4—41no—
167C32×C4D3×D412yes—
168C3×Dic3—64no—
169C22×C9—21no—
170C22×C9—21no—
171C22×C9—22no—
172C22×C9—21no—
173C22×C9—41no—
174C22×C9—41no—
175C22×C9—42no—
176C22×C9—42no—
177C22×C9—42no—
178C22×C9—41no—
179C22×C9—41no—
180C22×C9—41no—
181C22×C9C3×SD1612yescomplemented
182C22×C9C2×C3×D411yescharacteristic
183C4×C9—21no—
184C4×C9—22no—
185C4×C9—41no—
186C4×C9—42no—
187C4×C9—41no—
188C4×C9C2×C3×D412yes—
189Dic9—24no—
190C22×Q8—91no—
191C2×SD16—92no—
192#121AGL(1,ℤ/9)11yescharacteristic
193#103—92no—
194C22⋊C8—91no—
1953−1+2#44811yescharacteristic, complemented, Sylow
196C23×C3—21no—
197C23×C3—31no—
198C23×C3—61no—
199C23×C3C3×#6711yescharacteristic
200C3×Q8—94no—
201C3×Q8—182no—
202C3×Q8—184no—
203C3×D4—22no—
204C3×D4—42no—
205C3×D4—62no—
206C3×D4—122no—
207C2×C3×C4—22no—
208C2×C3×C4—21no—
209C2×C3×C4—31no—
210C2×C3×C4—31no—
211C2×C3×C4—62no—
212C2×C3×C4—61no—
213C2×C3×C4—182no—
214C2×C3×C4—184no—
215C2×C3×C4C3×#6711yescharacteristic
216C2×C3×C4C22×C3×D311yescharacteristic
217C2×Dic3—64no—
218C2×Dic3—62no—
219Dic6—34no—
220Dic6—64no—
221Dic6—62no—
222C3×C8—182no—
223C3⋊C8—62no—
224C2×C32—22no—
225C2×C32—81no—
226C2×C32C2×#21411yescharacteristic
227C2×C32#93711yescharacteristic
228C2×C32#95111yescharacteristic
229C2×C9—22no—
230C2×C9—21no—
231C2×C9—21no—
232C2×C9—21no—
233C2×C9—42no—
234C2×C9—82no—
235C2×C9—81no—
236C2×C9C3×#12011yescharacteristic
237C2×C9C2×C3×SD1611yescharacteristic
238C2×C9C3×#13711yescharacteristic
239C2×Q8—92no—
240C2×Q8—184no—
241C2×Q8—181no—
242C2×D4—21no—
243C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
244C22×C4—181no—
245C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
246SD16—184no—
247C2×C8—92no—
248C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
249C22⋊C4—21no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—31no—
253C22×C3—32no—
254C22×C3—41no—
255C22×C3—41no—
256C22×C3—41no—
257C22×C3—61no—
258C22×C3—61no—
259C22×C3—121no—
260C22×C3—121no—
261C22×C3—121no—
262C22×C3C3×#21412yes—
263C22×C3C2×C3×#6711yescharacteristic
264C3×C4—21no—
265C3×C4—32no—
266C3×C4—41no—
267C3×C4—61no—
268C3×C4—121no—
269C3×C4—184no—
270C3×C4C3×D3×D412yes—
271Dic3—64no—
272C32#504011yescharacteristic
273C9—21no—
274C9C3×#44811yescharacteristic, complemented
275C23—21no—
276C23#611⋊C311yescharacteristic
277Q8—94no—
278Q8—184no—
279Q8—182no—
280D4—22no—
281D4—42no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—184no—
285C2×C4—182no—
286C2×C4#611⋊C311yescharacteristic
287C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
288C8—182no—
289C2×C3—22no—
290C2×C3—31no—
291C2×C3—31no—
292C2×C3—31no—
293C2×C3—62no—
294C2×C3—81no—
295C2×C3—241no—
296C2×C3C2×C3×#21411yescharacteristic
297C2×C3C3×#93711yescharacteristic
298C2×C3C3×#95111yescharacteristic
299C22—21no—
300C22—21no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22#3672⋊C312yescomplemented
305C22C2×#611⋊C311yescharacteristic
306C4—21no—
307C4—41no—
308C4—184no—
309C4D4×AGL(1,ℤ/9)12yes—
310C3—31no—
311C3C3×#504011yescharacteristic
312C2—22no—
313C2—81no—
314C2C2×#3672⋊C311yescharacteristic
315C2#63499⋊C311yescharacteristic
316C2#63511⋊C311yescharacteristic
subgroupG/H
centerC22C2×#611⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#121C2
socleC22×C3C2×C3×#67
prime radicalC32×#121D3
hypercenter#121AGL(1,ℤ/9)
hypocenterC9C3×#448
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C22×3−1+2⋊Q8C2semidirect
3. C2×#3672⋊C3C2semidirect
4. #63389⋊C3C2semidirect
5. #63368⋊C3C2semidirect
7. #10588716C3semidirect
22. C22×Dic18C2×C3semidirect
27. C2×#3672C2×C3semidirect
28. #63389C2×C3semidirect
29. #63368C2×C3semidirect
42. 3−1+2⋊Q8D4semidirect
78. C22×3−1+2SD16semidirect
128. Dic18C3×D4semidirect
181. C22×C9C3×SD16semidirect
195. 3−1+2#448semidirect
274. C9C3×#448semidirect
304. C22#3672⋊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×#1212
C2×#3672⋊C32
#63389⋊C32
#63368⋊C32
C3×#50403
#105887163