Finite Group Databasebeta

← #10588720⋊C3 #13142855

Subgroups — #10588720⋊C3 #13142855

Order 1728.

533 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 533 conjugacy classes of subgroups — too many to draw. Only its 100 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
1C2×#3697⋊C3C211yesmaximal, characteristic, complemented
23−1+2×#121C211yesmaximal, characteristic, complemented
3C2×#3673⋊C3C211yesmaximal, characteristic, complemented
4C2×#3672⋊C3C211yesmaximal, characteristic, complemented
5#63389⋊C3C211yesmaximal, characteristic, complemented
6#63369⋊C3C211yesmaximal, characteristic, complemented
7#63368⋊C3C211yesmaximal, characteristic, complemented
8C3×#5037—31nomaximal
9#10588720C311yesmaximal, characteristic, complemented
10C2×D4×3−1+2—21no—
11C2×D4×3−1+2C2211yescharacteristic
12C22×C4×3−1+2C2211yescharacteristic
13C2×#611⋊C3—21no—
14#3697⋊C3—22no—
15#3697⋊C3—22no—
16C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
17C2×C4×AGL(1,ℤ/9)—21no—
18C2×3−1+2⋊Q8C2211yescharacteristic
19C4⋊C4×3−1+2C2211yescharacteristic
20C22⋊C4×3−1+2—21no—
21#3673⋊C3—22no—
22#3672⋊C3—22no—
23C2×3−1+2⋊C8C2211yescharacteristic
24C2×3−1+2⋊C8C2211yescharacteristic
25C2×C3×#235—31no—
26C32×#121D311yescharacteristic
27C2×C3×#214—31no—
28C2×C3×#213—31no—
29C2×#3697C2×C311yescharacteristic, complemented
30C3×#830—31no—
31C3×#810—31no—
32C3×#809—31no—
33C9×#121—21no—
34C9×#121C2×C311yescharacteristic, complemented
35C2×#3673C2×C311yescharacteristic, complemented
36C2×#3672C2×C311yescharacteristic, complemented
37#63389C2×C311yescharacteristic, complemented
38#63369C2×C311yescharacteristic, complemented
39#63368C2×C311yescharacteristic, complemented
40C23×3−1+2—21no—
41C23×3−1+2D411yescharacteristic
42C22×AGL(1,ℤ/9)—21no—
43D4×3−1+2—21no—
44D4×3−1+2—21no—
45D4×3−1+2—42no—
46C2×C4×3−1+2—21no—
47C2×C4×3−1+2—21no—
48C2×C4×3−1+2—21no—
49C2×C4×3−1+2D411yescharacteristic
50C2×C4×3−1+2C2311yescharacteristic
51#611⋊C3—24no—
52C2×3−1+2⋊C4—21no—
53C4⋊AGL(1,ℤ/9)—21no—
54C4⋊AGL(1,ℤ/9)D412yescomplemented
55C4×AGL(1,ℤ/9)—24no—
563−1+2⋊Q8—21no—
573−1+2⋊Q8D412yescomplemented
583−1+2⋊C8—21no—
593−1+2⋊C8—21no—
60C3×#449—91no—
61#5037—31no—
62C2×C32×D4—21no—
63C2×C32×D4C2×D311yescharacteristic
64C22×C32×C4C2×D311yescharacteristic
65C2×C3×#67—61no—
66C3×#235—62no—
67C3×#235—62no—
68C2×C3×D12—31no—
69C2×C3×C4×D3—61no—
70C2×C3×Dic6—31no—
71C32×C4⋊C4C2×D311yescharacteristic
72C32×C22⋊C4—21no—
73C3×#214—62no—
74C3×#213—62no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—31no—
77C2×D4×C9—22no—
78C2×D4×C9—21no—
79C2×D4×C9—21no—
80C2×D4×C9C22×C311yescharacteristic
81C22×C4×C9—21no—
82C22×C4×C9C22×C311yescharacteristic
83C2×#611—21no—
84#3697—22no—
85#3697—22no—
86C2×D36C22×C311yescharacteristic
87C2×C4×D9—21no—
88C2×Dic18C22×C311yescharacteristic
89C9×C4⋊C4—21no—
90C9×C4⋊C4C22×C311yescharacteristic
91C9×C22⋊C4—22no—
92C9×C22⋊C4—21no—
93#3673—22no—
94#3672—22no—
95C2×C9⋊C8C22×C311yescharacteristic
96C2×C9⋊C8C22×C311yescharacteristic
97C22×3−1+2—21no—
98C22×3−1+2—21no—
99C22×3−1+2—21no—
100C22×3−1+2—41no—
101C22×3−1+2—41no—
102C22×3−1+2—41no—
103C22×3−1+2C2×D411yescharacteristic
104C2×AGL(1,ℤ/9)—22no—
105C2×AGL(1,ℤ/9)—41no—
106C2×AGL(1,ℤ/9)—41no—
107C4×3−1+2—21no—
108C4×3−1+2—41no—
109C4×3−1+2C2×D411yescharacteristic
110C4×3−1+2C2×D411yescharacteristic
1113−1+2⋊C4—22no—
112C2×C3×#41—91no—
113C2×#235—31no—
114C2×C3×SD16—91no—
115C2×C3×D8—91no—
116C3×#121—31no—
117C3×#121C3×D311yescharacteristic
118C2×#214—31no—
119C2×#213—31no—
120C3×#103—91no—
121C3×#102—91no—
122C3×C22⋊C8—91no—
123#830—31no—
124#810—31no—
125#809—31no—
126C23×C32—21no—
127C23×C32#6711yescharacteristic
128C22×C3×D3—61no—
129C32×D4—21no—
130C32×D4—21no—
131C32×D4—42no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4#6711yescharacteristic
136C2×C32×C4C22×D311yescharacteristic
137C3×#67—64no—
138C2×C3×Dic3—61no—
139C3×D12—32no—
140C3×D12—61no—
141C3×C4×D3—64no—
142C3×Dic6—32no—
143C3×Dic6—61no—
144C23×C9—22no—
145C23×C9—21no—
146C23×C9—21no—
147C23×C9C3×D411yescharacteristic
148C22×D9—21no—
149C3×C3⋊C8—61no—
150C3×C3⋊C8—61no—
151D4×C9—21no—
152D4×C9—21no—
153D4×C9—41no—
154D4×C9—41no—
155D4×C9—44no—
156D4×C9—42no—
157C2×C4×C9—21no—
158C2×C4×C9—21no—
159C2×C4×C9—22no—
160C2×C4×C9—21no—
161C2×C4×C9—21no—
162C2×C4×C9—21no—
163C2×C4×C9—41no—
164C2×C4×C9—41no—
165C2×C4×C9C3×D411yescharacteristic
166C2×C4×C9C23×C311yescharacteristic
167#611—24no—
168C2×Dic9—21no—
169D36—21no—
170D36C3×D412yescomplemented
171C4×D9—24no—
172Dic18—21no—
173Dic18C3×D412yescomplemented
174C9⋊C8—21no—
175C9⋊C8—21no—
176#449—91noSylow
177C2×3−1+2—41no—
178C2×3−1+2—81no—
179C2×3−1+2#12011yescharacteristic
180C2×3−1+2#13511yescharacteristic
181C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
182AGL(1,ℤ/9)—42no—
183C3×#41—182no—
184C3×#41—182no—
185C2×C3×Q8—91no—
186C2×C3×D4—21no—
187C2×C3×D4—31no—
188C2×C3×D4—61no—
189C2×C3×D4—91no—
190C2×C3×D4—181no—
191C2×C3×D4C2×C3×D311yescharacteristic
192C22×C3×C4—31no—
193C22×C3×C4—181no—
194C22×C3×C4C2×C3×D311yescharacteristic
195C2×#67—61no—
196#235—62no—
197#235—62no—
198C2×D12—31no—
199C2×C4×D3—61no—
200C2×Dic6—31no—
201C3×SD16—182no—
202C3×D8—182no—
203C2×C3×C8—91no—
204C2×C3×C8—91no—
205C3×C4⋊C4—31no—
206C3×C4⋊C4C2×C3×D311yescharacteristic
207C3×C22⋊C4—21no—
208C3×C22⋊C4—61no—
209#214—62no—
210#213—62no—
211C2×C3⋊C8—31no—
212C2×C3⋊C8—31no—
213C22×C32—21no—
214C22×C32—21no—
215C22×C32—21no—
216C22×C32—41no—
217C22×C32—41no—
218C22×C32—41no—
219C22×C32C2×#6711yescharacteristic
220C2×C3×D3—62no—
221C2×C3×D3—121no—
222C2×C3×D3—121no—
223C32×C4—21no—
224C32×C4—41no—
225C32×C4D3×D411yescharacteristic
226C32×C4D3×D411yescharacteristic
227C3×Dic3—62no—
228C22×C9—22no—
229C22×C9—21no—
230C22×C9—21no—
231C22×C9—21no—
232C22×C9—21no—
233C22×C9—41no—
234C22×C9—41no—
235C22×C9—42no—
236C22×C9—42no—
237C22×C9—42no—
238C22×C9—41no—
239C22×C9—41no—
240C22×C9—41no—
241C22×C9C2×C3×D411yescharacteristic
242C2×D9—22no—
243C2×D9—41no—
244C2×D9—41no—
245C4×C9—21no—
246C4×C9—21no—
247C4×C9—21no—
248C4×C9—41no—
249C4×C9—42no—
250C4×C9—41no—
251C4×C9C2×C3×D411yescharacteristic
252C4×C9C2×C3×D411yescharacteristic
253Dic9—22no—
254C2×#41—91no—
255C2×SD16—91no—
256C2×D8—91no—
257#121AGL(1,ℤ/9)11yescharacteristic, complemented
258#103—91no—
259#102—91no—
260C22⋊C8—91no—
2613−1+2#44911yescharacteristic, complemented, Sylow
262C23×C3—21no—
263C23×C3—31no—
264C23×C3—61no—
265C23×C3—181no—
266C23×C3C3×#6711yescharacteristic
267C22×D3—61no—
268C3×Q8—92no—
269C3×Q8—181no—
270C3×D4—21no—
271C3×D4—21no—
272C3×D4—42no—
273C3×D4—61no—
274C3×D4—61no—
275C3×D4—92no—
276C3×D4—122no—
277C3×D4—181no—
278C3×D4—184no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—31no—
283C2×C3×C4—31no—
284C2×C3×C4—61no—
285C2×C3×C4—61no—
286C2×C3×C4—61no—
287C2×C3×C4—184no—
288C2×C3×C4—181no—
289C2×C3×C4C3×#6711yescharacteristic
290C2×C3×C4C22×C3×D311yescharacteristic
291#67—64no—
292C2×Dic3—61no—
293D12—32no—
294D12—61no—
295C4×D3—64no—
296Dic6—32no—
297Dic6—61no—
298C3×C8—181no—
299C3×C8—181no—
300C3⋊C8—61no—
301C3⋊C8—61no—
302C2×C32—41no—
303C2×C32—81no—
304C2×C32#93211yescharacteristic
305C2×C32#93711yescharacteristic
306C2×C32#95011yescharacteristic
307C3×D3—122no—
308C2×C9—21no—
309C2×C9—21no—
310C2×C9—21no—
311C2×C9—42no—
312C2×C9—41no—
313C2×C9—82no—
314C2×C9—81no—
315C2×C9C3×#12011yescharacteristic
316C2×C9C3×#13511yescharacteristic
317C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
318D9—42no—
319#41—182no—
320#41—182no—
321C2×Q8—91no—
322C2×D4—21no—
323C2×D4—91no—
324C2×D4—181no—
325C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
326C22×C4—181no—
327C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
328SD16—182no—
329D8—182no—
330C2×C8—91no—
331C2×C8—91no—
332C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
333C22⋊C4—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—31no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—41no—
341C22×C3—61no—
342C22×C3—61no—
343C22×C3—61no—
344C22×C3—121no—
345C22×C3—121no—
346C22×C3—121no—
347C22×C3—182no—
348C22×C3—361no—
349C22×C3—361no—
350C22×C3C2×C3×#6711yescharacteristic
351C2×D3—62no—
352C2×D3—121no—
353C2×D3—121no—
354C3×C4—21no—
355C3×C4—31no—
356C3×C4—31no—
357C3×C4—41no—
358C3×C4—61no—
359C3×C4—121no—
360C3×C4—182no—
361C3×C4C3×D3×D411yescharacteristic
362C3×C4C3×D3×D411yescharacteristic
363Dic3—62no—
364C32#503711yescharacteristic
365C9—21no—
366C9C3×#44911yescharacteristic, complemented
367C23—21no—
368C23—181no—
369C23#611⋊C311yescharacteristic
370Q8—92no—
371Q8—181no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—92no—
376D4—184no—
377D4—181no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—184no—
382C2×C4—181no—
383C2×C4#611⋊C311yescharacteristic
384C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
385C8—181no—
386C8—181no—
387C2×C3—31no—
388C2×C3—31no—
389C2×C3—31no—
390C2×C3—41no—
391C2×C3—81no—
392C2×C3—121no—
393C2×C3—241no—
394C2×C3—362no—
395C2×C3C3×#93211yescharacteristic
396C2×C3C3×#93711yescharacteristic
397C2×C3C3×#95011yescharacteristic
398D3—122no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—182no—
406C22—361no—
407C22—361no—
408C22C2×#611⋊C311yescharacteristic
409C4—21no—
410C4—41no—
411C4—182no—
412C4D4×AGL(1,ℤ/9)11yescharacteristic
413C4D4×AGL(1,ℤ/9)11yescharacteristic
414C3—31no—
415C3C3×#503711yescharacteristic
416C2—41no—
417C2—81no—
418C2—362no—
419C2#63495⋊C311yescharacteristic
420C2#63499⋊C311yescharacteristic
421C2#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
hypercenter#121AGL(1,ℤ/9)
hypocenterC9C3×#449
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. C2×#3697⋊C3C2semidirect
2. 3−1+2×#121C2semidirect
3. C2×#3673⋊C3C2semidirect
4. C2×#3672⋊C3C2semidirect
5. #63389⋊C3C2semidirect
6. #63369⋊C3C2semidirect
7. #63368⋊C3C2semidirect
9. #10588720C3semidirect
29. C2×#3697C2×C3semidirect
34. C9×#121C2×C3semidirect
35. C2×#3673C2×C3semidirect
36. C2×#3672C2×C3semidirect
37. #63389C2×C3semidirect
38. #63369C2×C3semidirect
39. #63368C2×C3semidirect
54. C4⋊AGL(1,ℤ/9)D4semidirect
57. 3−1+2⋊Q8D4semidirect
170. D36C3×D4semidirect
173. Dic18C3×D4semidirect
257. #121AGL(1,ℤ/9)semidirect
261. 3−1+2#449semidirect
366. C9C3×#449semidirect

Maximal subgroups

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

subgroupindex
C2×#3697⋊C32
3−1+2×#1212
C2×#3673⋊C32
C2×#3672⋊C32
#63389⋊C32
#63369⋊C32
#63368⋊C32
C3×#50373
#105887203