Finite Group Databasebeta

← #10588717⋊C3 #13142848

Subgroups — #10588717⋊C3 #13142848

Order 1728.

419 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 419 conjugacy classes of subgroups — too many to draw. Only its 106 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×#121C211yesmaximal, characteristic
2C2×#3672⋊C3C211yesmaximal, characteristic, complemented
3#63486⋊C3C211yesmaximal, characteristic, complemented
4C22×3−1+2⋊C8C211yesmaximal, characteristic, complemented
5#63392⋊C3C211yesmaximal, characteristic, complemented
6#63368⋊C3C211yesmaximal, characteristic, complemented
7#63367⋊C3C211yesmaximal, characteristic, complemented
8C3×#5041—31nomaximal
9#10588717C311yesmaximal, characteristic, complemented
10C2×D4×3−1+2—21no—
11C2×D4×3−1+2C2211yescharacteristic
12C22×C4×3−1+2C2211yescharacteristic
13C2×3−1+2⋊Q8C2211yescharacteristic
14C4⋊C4×3−1+2C2211yescharacteristic
15C22⋊C4×3−1+2—21no—
16#3676⋊C3—21no—
17#3672⋊C3—22no—
18C4⋊(3−1+2⋊C4)C2211yescharacteristic
19(3−1+2⋊C4)⋊C4—21no—
20C2×3−1+2⋊C8—22no—
21C2×3−1+2⋊C8C2211yescharacteristic
22C2×3−1+2⋊C8C2211yescharacteristic
23C32×#121D311yescharacteristic
24C2×C3×#214—31no—
25C3×#924—31no—
26C22×C3×C3⋊C8—31no—
27C3×#832—31no—
28C3×#810—31no—
29C3×#808—31no—
30C9×#121—21no—
31C9×#121C2×C311yescharacteristic
32C2×#3672C2×C311yescharacteristic, complemented
33#63486C2×C311yescharacteristic, complemented
34C22×C9⋊C8C2×C311yescharacteristic, complemented
35#63392C2×C311yescharacteristic, complemented
36#63368C2×C311yescharacteristic, complemented
37#63367C2×C311yescharacteristic, complemented
38C23×3−1+2—21no—
39C23×3−1+2D411yescharacteristic
40D4×3−1+2—21no—
41D4×3−1+2—21no—
42D4×3−1+2—42no—
43C2×C4×3−1+2—21no—
44C2×C4×3−1+2—21no—
45C2×C4×3−1+2—21no—
46C2×C4×3−1+2D411yescharacteristic
47C2×C4×3−1+2C2311yescharacteristic
48C2×3−1+2⋊C4—21no—
49C2×3−1+2⋊C4—21no—
503−1+2⋊Q8—21no—
513−1+2⋊Q8—21no—
523−1+2⋊C8—21no—
533−1+2⋊C8D412yescomplemented
54C3×#465—91no—
55#5041—31no—
56C2×C32×D4—21no—
57C2×C32×D4C2×D311yescharacteristic
58C22×C32×C4C2×D311yescharacteristic
59C2×C3×Dic6—31no—
60C32×C4⋊C4C2×D311yescharacteristic
61C32×C22⋊C4—21no—
62C3×#217—61no—
63C3×#214—62no—
64C3×C4⋊Dic3—31no—
65C3×Dic3⋊C4—61no—
66C2×C3×C3⋊C8—31no—
67C2×C3×C3⋊C8—31no—
68C2×C3×C3⋊C8—62no—
69C2×D4×C9—22no—
70C2×D4×C9—21no—
71C2×D4×C9—21no—
72C2×D4×C9C22×C311yescharacteristic
73C22×C4×C9—21no—
74C22×C4×C9C22×C311yescharacteristic
75C2×Dic18C22×C311yescharacteristic
76C9×C4⋊C4—21no—
77C9×C4⋊C4C22×C311yescharacteristic
78C9×C22⋊C4—22no—
79C9×C22⋊C4—21no—
80#3676—21no—
81#3672—22no—
82C4⋊Dic9C22×C311yescharacteristic
83Dic9⋊C4—21no—
84C2×C9⋊C8—22no—
85C2×C9⋊C8C22×C311yescharacteristic
86C2×C9⋊C8C22×C311yescharacteristic
87C22×3−1+2—21no—
88C22×3−1+2—21no—
89C22×3−1+2—41no—
90C22×3−1+2—41no—
91C22×3−1+2—41no—
92C22×3−1+2SD1612yescomplemented
93C22×3−1+2C2×D411yescharacteristic
94C4×3−1+2—21no—
95C4×3−1+2—41no—
96C4×3−1+2C2×D411yescharacteristic
97C4×3−1+2#4111yescharacteristic
983−1+2⋊C4—41no—
993−1+2⋊C4—41no—
100C2×C3×SD16—91no—
101C22×C3×C8—91no—
102C3×C22⋊Q8—91no—
103C3×#121—31no—
104C3×#121C3×D311yescharacteristic
105C2×#214—31no—
106#924—31no—
107C22×C3⋊C8—31no—
108C3×#106—91no—
109C3×#103—91no—
110C3×#102—91no—
111#832—31no—
112#810—31no—
113#808—31no—
114C23×C32—21no—
115C23×C32#6711yescharacteristic
116C32×D4—21no—
117C32×D4—21no—
118C32×D4—42no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4#6711yescharacteristic
123C2×C32×C4C22×D311yescharacteristic
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C3×Dic6—61no—
127C3×Dic6—61no—
128C23×C9—22no—
129C23×C9—21no—
130C23×C9—21no—
131C23×C9C3×D411yescharacteristic
132C3×C3⋊C8—32no—
133C3×C3⋊C8—61no—
134D4×C9—21no—
135D4×C9—21no—
136D4×C9—41no—
137D4×C9—41no—
138D4×C9—44no—
139D4×C9—42no—
140C2×C4×C9—21no—
141C2×C4×C9—21no—
142C2×C4×C9—22no—
143C2×C4×C9—21no—
144C2×C4×C9—21no—
145C2×C4×C9—21no—
146C2×C4×C9—41no—
147C2×C4×C9—41no—
148C2×C4×C9C3×D411yescharacteristic
149C2×C4×C9C23×C311yescharacteristic
150C2×Dic9—21no—
151C2×Dic9—21no—
152Dic18—21no—
153Dic18—21no—
154C9⋊C8—21no—
155C9⋊C8C3×D412yescomplemented
156#465—91noSylow
157C2×3−1+2—22no—
158C2×3−1+2—81no—
159C2×3−1+2#12111yescharacteristic
160C2×3−1+2C2×SD1611yescharacteristic
161C2×3−1+2#13511yescharacteristic
162C2×C3×Q8—91no—
163C2×C3×D4—21no—
164C2×C3×D4—31no—
165C2×C3×D4—61no—
166C2×C3×D4C2×C3×D311yescharacteristic
167C22×C3×C4—31no—
168C22×C3×C4C2×C3×D311yescharacteristic
169C2×Dic6—31no—
170C3×SD16—182no—
171C2×C3×C8—91no—
172C2×C3×C8—91no—
173C2×C3×C8—182no—
174C3×C4⋊C4—31no—
175C3×C4⋊C4—91no—
176C3×C4⋊C4—181no—
177C3×C4⋊C4C2×C3×D311yescharacteristic
178C3×C22⋊C4—21no—
179C3×C22⋊C4—61no—
180C3×C22⋊C4—181no—
181#217—61no—
182#214—62no—
183C4⋊Dic3—31no—
184Dic3⋊C4—61no—
185C2×C3⋊C8—31no—
186C2×C3⋊C8—31no—
187C2×C3⋊C8—62no—
188C22×C32—21no—
189C22×C32—21no—
190C22×C32—41no—
191C22×C32—41no—
192C22×C32—41no—
193C22×C32#21412yes—
194C22×C32C2×#6711yescharacteristic
195C32×C4—21no—
196C32×C4—41no—
197C32×C4D3×D411yescharacteristic
198C32×C4#23711yescharacteristic
199C3×Dic3—121no—
200C3×Dic3—121no—
201C22×C9—21no—
202C22×C9—21no—
203C22×C9—22no—
204C22×C9—21no—
205C22×C9—41no—
206C22×C9—41no—
207C22×C9—42no—
208C22×C9—42no—
209C22×C9—42no—
210C22×C9—41no—
211C22×C9—41no—
212C22×C9—41no—
213C22×C9C3×SD1612yescomplemented
214C22×C9C2×C3×D411yescharacteristic
215C4×C9—21no—
216C4×C9—21no—
217C4×C9—21no—
218C4×C9—41no—
219C4×C9—42no—
220C4×C9—41no—
221C4×C9C2×C3×D411yescharacteristic
222C4×C9C3×#4111yescharacteristic
223Dic9—41no—
224Dic9—41no—
225C2×SD16—91no—
226C22×C8—91no—
227C22⋊Q8—91no—
228#121AGL(1,ℤ/9)11yescharacteristic
229#106—91no—
230#103—91no—
231#102—91no—
2323−1+2#46511yescharacteristic, complemented, Sylow
233C23×C3—21no—
234C23×C3—31no—
235C23×C3—61no—
236C23×C3C3×#6711yescharacteristic
237C3×Q8—181no—
238C3×Q8—181no—
239C3×D4—21no—
240C3×D4—21no—
241C3×D4—42no—
242C3×D4—61no—
243C3×D4—61no—
244C3×D4—122no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—31no—
249C2×C3×C4—31no—
250C2×C3×C4—61no—
251C2×C3×C4—61no—
252C2×C3×C4—61no—
253C2×C3×C4—181no—
254C2×C3×C4—181no—
255C2×C3×C4C3×#6711yescharacteristic
256C2×C3×C4C22×C3×D311yescharacteristic
257C2×Dic3—61no—
258C2×Dic3—61no—
259Dic6—61no—
260Dic6—61no—
261C3×C8—92no—
262C3×C8—181no—
263C3⋊C8—32no—
264C3⋊C8—61no—
265C2×C32—22no—
266C2×C32—81no—
267C2×C32C2×#21411yescharacteristic
268C2×C32#93911yescharacteristic
269C2×C32#95011yescharacteristic
270C2×C9—22no—
271C2×C9—21no—
272C2×C9—21no—
273C2×C9—21no—
274C2×C9—42no—
275C2×C9—82no—
276C2×C9—81no—
277C2×C9C3×#12111yescharacteristic
278C2×C9C2×C3×SD1611yescharacteristic
279C2×C9C3×#13511yescharacteristic
280C2×Q8—91no—
281C2×D4—21no—
282C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
283C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
284SD16—182no—
285C2×C8—91no—
286C2×C8—91no—
287C2×C8—182no—
288C4⋊C4—91no—
289C4⋊C4—181no—
290C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
291C22⋊C4—21no—
292C22⋊C4—181no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—31no—
296C22×C3—32no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—61no—
301C22×C3—61no—
302C22×C3—121no—
303C22×C3—121no—
304C22×C3—121no—
305C22×C3C3×#21412yes—
306C22×C3C2×C3×#6711yescharacteristic
307C3×C4—21no—
308C3×C4—31no—
309C3×C4—31no—
310C3×C4—41no—
311C3×C4—61no—
312C3×C4—121no—
313C3×C4—361no—
314C3×C4—361no—
315C3×C4C3×D3×D411yescharacteristic
316C3×C4C3×#23711yescharacteristic
317Dic3—121no—
318Dic3—121no—
319C32#504111yescharacteristic
320C9—21no—
321C9C3×#46511yescharacteristic, complemented
322C23—21no—
323C23#611⋊C311yescharacteristic
324Q8—181no—
325Q8—181no—
326D4—21no—
327D4—21no—
328D4—42no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—181no—
333C2×C4—181no—
334C2×C4#611⋊C311yescharacteristic
335C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
336C8—92no—
337C8—181no—
338C2×C3—22no—
339C2×C3—31no—
340C2×C3—31no—
341C2×C3—31no—
342C2×C3—62no—
343C2×C3—81no—
344C2×C3—241no—
345C2×C3C2×C3×#21411yescharacteristic
346C2×C3C3×#93911yescharacteristic
347C2×C3C3×#95011yescharacteristic
348C22—21no—
349C22—21no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22#3672⋊C312yescomplemented
354C22C2×#611⋊C311yescharacteristic
355C4—21no—
356C4—41no—
357C4—361no—
358C4—361no—
359C4D4×AGL(1,ℤ/9)11yescharacteristic
360C4#3699⋊C311yescharacteristic
361C3—31no—
362C3C3×#504111yescharacteristic
363C2—22no—
364C2—81no—
365C2C2×#3672⋊C311yescharacteristic
366C2#63501⋊C311yescharacteristic
367C2#63513⋊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×#465
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
2. C2×#3672⋊C3C2semidirect
3. #63486⋊C3C2semidirect
4. C22×3−1+2⋊C8C2semidirect
5. #63392⋊C3C2semidirect
6. #63368⋊C3C2semidirect
7. #63367⋊C3C2semidirect
9. #10588717C3semidirect
32. C2×#3672C2×C3semidirect
33. #63486C2×C3semidirect
34. C22×C9⋊C8C2×C3semidirect
35. #63392C2×C3semidirect
36. #63368C2×C3semidirect
37. #63367C2×C3semidirect
53. 3−1+2⋊C8D4semidirect
92. C22×3−1+2SD16semidirect
155. C9⋊C8C3×D4semidirect
213. C22×C9C3×SD16semidirect
232. 3−1+2#465semidirect
321. C9C3×#465semidirect
353. C22#3672⋊C3semidirect

Maximal subgroups

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

subgroupindex
3−1+2×#1212
C2×#3672⋊C32
#63486⋊C32
C22×3−1+2⋊C82
#63392⋊C32
#63368⋊C32
#63367⋊C32
C3×#50413
#105887173