Finite Group Databasebeta

← #10589741⋊C3 #13152988

Subgroups — #10589741⋊C3 #13152988

Order 1728.

679 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
13−1+2×C22⋊Q8C211yesmaximal, characteristic, complemented
2#63508⋊C3C211yesmaximal, characteristic, complemented
3Q8×3−1+2⋊C4C211yesmaximal, characteristic, complemented
4C4×#611⋊C3C211yesmaximal, characteristic
5#63488⋊C3C211yesmaximal, characteristic, complemented
6#63454⋊C3C212yesmaximal, complemented
7C4⋊C4×AGL(1,ℤ/9)C211yesmaximal, characteristic, complemented
8#63452⋊C3C211yesmaximal, characteristic, complemented
9#63451⋊C3C212yesmaximal, complemented
10#63445⋊C3C212yesmaximal
11#63441⋊C3C212yesmaximal, complemented
12C3×#5639—31nomaximal
13#10589741C311yesmaximal, characteristic, complemented
14C2×Q8×3−1+2C2211yescharacteristic
15C22×C4×3−1+2C2211yescharacteristic
16C2×#611⋊C3C2211yescharacteristic
17C2×C4×AGL(1,ℤ/9)C2212yes—
18C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
19C4⋊C4×3−1+2C2212yes—
20C4⋊C4×3−1+2C2211yescharacteristic
21C22⋊C4×3−1+2C2212yes—
22#3676⋊C3C2212yes—
23#3676⋊C3C2211yescharacteristic
24(C2×AGL(1,ℤ/9))⋊C4C2211yescharacteristic
25(C2×AGL(1,ℤ/9))⋊C4C2212yes—
26C4⋊(3−1+2⋊C4)C2212yes—
27C4⋊(3−1+2⋊C4)C2212yes—
28C4⋊(3−1+2⋊C4)C2211yescharacteristic
29C4⋊(3−1+2⋊C4)C2211yescharacteristic
30(3−1+2⋊C4)⋊C4C2211yescharacteristic
31(3−1+2⋊C4)⋊C4C2212yes—
32(3−1+2⋊C4)⋊C4C2212yes—
33(3−1+2⋊C4)⋊C4C2212yes—
34C4×3−1+2⋊C4C2212yes—
35C4×3−1+2⋊C4C2211yescharacteristic
36C4×3−1+2⋊C4C2212yes—
37C32×C22⋊Q8D311yescharacteristic
38C3×#946—31no—
39C3×Q8×Dic3—31no—
40C3×C4×#67—31no—
41C3×#926—31no—
42C3×#892—32no—
43C3×D3×C4⋊C4—31no—
44C3×#890—31no—
45C3×#889—32no—
46C3×#883—32no—
47C3×#879—32no—
48C9×C22⋊Q8—21no—
49C9×C22⋊Q8C2×C311yescharacteristic, complemented
50#63508C2×C311yescharacteristic, complemented
51Q8×Dic9C2×C311yescharacteristic, complemented
52C4×#611C2×C311yescharacteristic
53#63488C2×C311yescharacteristic, complemented
54#63454C2×C312yescomplemented
55C4⋊C4×D9C2×C311yescharacteristic, complemented
56#63452C2×C311yescharacteristic, complemented
57#63451C2×C312yescomplemented
58#63445C2×C312yes—
59#63441C2×C312yescomplemented
60C23×3−1+2C2311yescharacteristic
61C22×AGL(1,ℤ/9)C2311yescharacteristic
62Q8×3−1+2—22no—
63C2×C4×3−1+2—22no—
64C2×C4×3−1+2C2312yes—
65C2×C4×3−1+2C2311yescharacteristic
66C2×C4×3−1+2C2312yes—
67C2×C4×3−1+2C2311yescharacteristic
68#611⋊C3—22no—
69C2×3−1+2⋊C4C2312yes—
70C2×3−1+2⋊C4C2311yescharacteristic
71C2×3−1+2⋊C4C2311yescharacteristic
72C2×3−1+2⋊C4C2311yescharacteristic
73C2×3−1+2⋊C4C2312yes—
74C4×AGL(1,ℤ/9)—24no—
75C4×AGL(1,ℤ/9)—22no—
76C3×#552—91no—
77#5639—31no—
78C2×C32×Q8C2×D311yescharacteristic
79C22×C32×C4C2×D311yescharacteristic
80C2×C3×#67—31no—
81C2×C3×C4×D3—32no—
82C2×C3×C4×D3—31no—
83C32×C4⋊C4C2×D312yes—
84C32×C4⋊C4C2×D311yescharacteristic
85C32×C22⋊C4C2×D312yes—
86C3×#217—31no—
87C3×#217—32no—
88C3×(C2×D3)⋊C4—32no—
89C3×(C2×D3)⋊C4—31no—
90C3×C4⋊Dic3—32no—
91C3×C4⋊Dic3—32no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—31no—
94C3×Dic3⋊C4—32no—
95C3×Dic3⋊C4—32no—
96C3×Dic3⋊C4—32no—
97C3×Dic3⋊C4—31no—
98C3×C4×Dic3—32no—
99C3×C4×Dic3—32no—
100C3×C4×Dic3—31no—
101C2×Q8×C9—21no—
102C2×Q8×C9C22×C311yescharacteristic
103C22×C4×C9—21no—
104C22×C4×C9C22×C311yescharacteristic
105C2×#611C22×C311yescharacteristic
106C2×C4×D9C22×C311yescharacteristic
107C2×C4×D9C22×C312yes—
108C9×C4⋊C4—21no—
109C9×C4⋊C4—22no—
110C9×C4⋊C4C22×C312yes—
111C9×C4⋊C4C22×C311yescharacteristic
112C9×C22⋊C4—22no—
113C9×C22⋊C4C22×C312yes—
114#3676C22×C312yes—
115#3676C22×C311yescharacteristic
116(C2×D9)⋊C4C22×C311yescharacteristic
117(C2×D9)⋊C4C22×C312yes—
118C4⋊Dic9C22×C311yescharacteristic
119C4⋊Dic9C22×C311yescharacteristic
120C4⋊Dic9C22×C312yes—
121C4⋊Dic9C22×C312yes—
122Dic9⋊C4C22×C311yescharacteristic
123Dic9⋊C4C22×C312yes—
124Dic9⋊C4C22×C312yes—
125Dic9⋊C4C22×C312yes—
126C4×Dic9C22×C312yes—
127C4×Dic9C22×C311yescharacteristic
128C4×Dic9C22×C312yes—
129C22×3−1+2—21no—
130C22×3−1+2—21no—
131C22×3−1+2—21no—
132C22×3−1+2C2411yescharacteristic
133C2×AGL(1,ℤ/9)—21no—
134C2×AGL(1,ℤ/9)—21no—
135C2×AGL(1,ℤ/9)#4112yes—
136C4×3−1+2—22no—
137C4×3−1+2—21no—
138C4×3−1+2—22no—
139C4×3−1+2#4112yes—
1403−1+2⋊C4—21no—
1413−1+2⋊C4—22no—
1423−1+2⋊C4—21no—
1433−1+2⋊C4—21no—
1443−1+2⋊C4—22no—
145C3×#126—92no—
146C3×#125—92no—
147C3×#125—91no—
148C3×#123—92no—
149C3×C22⋊Q8—31no—
150C3×C22⋊Q8—91no—
151C3×C22⋊Q8C3×D311yescharacteristic
152C3×C4×Q8—91no—
153C3×C4×D4—91no—
154C3×#117—92no—
155C3×#117—91no—
156C2×C3×C4⋊C4—91no—
157#946—31no—
158Q8×Dic3—31no—
159C4×#67—31no—
160#926—31no—
161#892—32no—
162D3×C4⋊C4—31no—
163#890—31no—
164#889—32no—
165#883—32no—
166#879—32no—
167C23×C32C22×D311yescharacteristic
168C22×C3×D3—31no—
169C32×Q8—22no—
170C2×C32×C4—22no—
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D312yes—
173C2×C32×C4C22×D312yes—
174C2×C32×C4C22×D311yescharacteristic
175C3×#67—62no—
176C2×C3×Dic3—31no—
177C2×C3×Dic3—32no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3—32no—
180C2×C3×Dic3—31no—
181C3×C4×D3—64no—
182C3×C4×D3—62no—
183C23×C9—21no—
184C23×C9C23×C311yescharacteristic
185C22×D9C23×C311yescharacteristic
186Q8×C9—22no—
187Q8×C9—42no—
188C2×C4×C9—22no—
189C2×C4×C9—21no—
190C2×C4×C9—22no—
191C2×C4×C9—22no—
192C2×C4×C9—21no—
193C2×C4×C9—42no—
194C2×C4×C9C23×C311yescharacteristic
195C2×C4×C9C23×C312yes—
196C2×C4×C9C23×C312yes—
197C2×C4×C9C23×C311yescharacteristic
198#611—22no—
199C2×Dic9C23×C311yescharacteristic
200C2×Dic9C23×C312yes—
201C2×Dic9C23×C311yescharacteristic
202C2×Dic9C23×C312yes—
203C2×Dic9C23×C311yescharacteristic
204C4×D9—24no—
205C4×D9—22no—
206#552—91noSylow
207C2×3−1+2—41no—
208C2×3−1+2C2×#4111yescharacteristic
209C2×3−1+2C2×#4111yescharacteristic
210C2×3−1+22−1+411yescharacteristic
211AGL(1,ℤ/9)—22no—
212C2×C3×Q8—31no—
213C2×C3×Q8C2×C3×D311yescharacteristic
214C2×C3×D4—91no—
215C22×C3×C4—31no—
216C22×C3×C4—92no—
217C22×C3×C4—91no—
218C22×C3×C4C2×C3×D311yescharacteristic
219C2×#67—31no—
220C2×C4×D3—32no—
221C2×C4×D3—31no—
222C3×C4⋊C4—31no—
223C3×C4⋊C4—32no—
224C3×C4⋊C4—92no—
225C3×C4⋊C4—91no—
226C3×C4⋊C4—92no—
227C3×C4⋊C4—92no—
228C3×C4⋊C4—91no—
229C3×C4⋊C4—91no—
230C3×C4⋊C4—92no—
231C3×C4⋊C4—92no—
232C3×C4⋊C4C2×C3×D312yes—
233C3×C4⋊C4C2×C3×D311yescharacteristic
234C3×C22⋊C4—32no—
235C3×C22⋊C4—91no—
236C3×C22⋊C4—92no—
237C3×C22⋊C4—92no—
238C3×C22⋊C4—91no—
239C3×C22⋊C4C2×C3×D312yes—
240C3×C42—92no—
241C3×C42—92no—
242C3×C42—91no—
243#217—31no—
244#217—32no—
245(C2×D3)⋊C4—32no—
246(C2×D3)⋊C4—31no—
247C4⋊Dic3—32no—
248C4⋊Dic3—32no—
249C4⋊Dic3—31no—
250C4⋊Dic3—31no—
251Dic3⋊C4—32no—
252Dic3⋊C4—32no—
253Dic3⋊C4—32no—
254Dic3⋊C4—31no—
255C4×Dic3—32no—
256C4×Dic3—32no—
257C4×Dic3—31no—
258C22×C32—21no—
259C22×C32—21no—
260C22×C32—21no—
261C22×C32C23×D311yescharacteristic
262C2×C3×D3—32no—
263C2×C3×D3—61no—
264C2×C3×D3—61no—
265C32×C4—21no—
266C32×C4—22no—
267C32×C4—22no—
268C32×C4#23712yes—
269C3×Dic3—61no—
270C3×Dic3—61no—
271C3×Dic3—62no—
272C3×Dic3—62no—
273C3×Dic3—61no—
274C22×C9—22no—
275C22×C9—21no—
276C22×C9—21no—
277C22×C9—21no—
278C22×C9—21no—
279C22×C9—41no—
280C22×C9—41no—
281C22×C9C24×C311yescharacteristic
282C2×D9—21no—
283C2×D9—21no—
284C2×D9C3×#4112yes—
285C4×C9—22no—
286C4×C9—21no—
287C4×C9—22no—
288C4×C9—22no—
289C4×C9—42no—
290C4×C9—41no—
291C4×C9—42no—
292C4×C9C3×#4112yes—
293Dic9—21no—
294Dic9—22no—
295Dic9—21no—
296Dic9—21no—
297Dic9—22no—
298#126—92no—
299#125—92no—
300#125—91no—
301#123—92no—
302C22⋊Q8—91no—
303C22⋊Q8AGL(1,ℤ/9)11yescharacteristic, complemented
304C4×Q8—91no—
305C4×D4—91no—
306#117—91no—
307#117—92no—
308C2×C4⋊C4—91no—
3093−1+2#55211yescharacteristic, complemented, Sylow
310C23×C3—31no—
311C23×C3—91no—
312C23×C3C22×C3×D311yescharacteristic
313C22×D3—31no—
314C3×Q8—22no—
315C3×Q8—62no—
316C3×D4—182no—
317C2×C3×C4—22no—
318C2×C3×C4—31no—
319C2×C3×C4—31no—
320C2×C3×C4—32no—
321C2×C3×C4—32no—
322C2×C3×C4—62no—
323C2×C3×C4—91no—
324C2×C3×C4—92no—
325C2×C3×C4—92no—
326C2×C3×C4—91no—
327C2×C3×C4—91no—
328C2×C3×C4—182no—
329C2×C3×C4—184no—
330C2×C3×C4C22×C3×D311yescharacteristic
331C2×C3×C4C22×C3×D312yes—
332C2×C3×C4C22×C3×D312yes—
333C2×C3×C4C22×C3×D311yescharacteristic
334#67—62no—
335C2×Dic3—31no—
336C2×Dic3—32no—
337C2×Dic3—31no—
338C2×Dic3—32no—
339C2×Dic3—31no—
340C4×D3—64no—
341C4×D3—62no—
342C2×C32—41no—
343C2×C32C2×#23711yescharacteristic
344C2×C32#100711yescharacteristic
345C2×C32D3×#4111yescharacteristic
346C3×D3—62no—
347C2×C9—21no—
348C2×C9—21no—
349C2×C9—21no—
350C2×C9—42no—
351C2×C9—41no—
352C2×C9C2×C3×#4111yescharacteristic
353C2×C9C2×C3×#4111yescharacteristic
354C2×C9C3×2−1+411yescharacteristic
355D9—22no—
356C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
357C2×D4—91no—
358C22×C4—92no—
359C22×C4—91no—
360C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
361C4⋊C4—91no—
362C4⋊C4—92no—
363C4⋊C4—91no—
364C4⋊C4—92no—
365C4⋊C4—91no—
366C4⋊C4—92no—
367C4⋊C4—92no—
368C4⋊C4—92no—
369C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
370C4⋊C4C2×AGL(1,ℤ/9)12yes—
371C22⋊C4—92no—
372C22⋊C4—91no—
373C22⋊C4—91no—
374C22⋊C4—92no—
375C22⋊C4C2×AGL(1,ℤ/9)12yes—
376C42—92no—
377C42—92no—
378C42—91no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—31no—
383C22×C3—61no—
384C22×C3—61no—
385C22×C3—61no—
386C22×C3—92no—
387C22×C3—181no—
388C22×C3—181no—
389C22×C3C23×C3×D311yescharacteristic
390C2×D3—32no—
391C2×D3—61no—
392C2×D3—61no—
393C3×C4—21no—
394C3×C4—22no—
395C3×C4—22no—
396C3×C4—32no—
397C3×C4—61no—
398C3×C4—62no—
399C3×C4—62no—
400C3×C4—181no—
401C3×C4—182no—
402C3×C4—181no—
403C3×C4—181no—
404C3×C4—182no—
405C3×C4C3×#23712yes—
406Dic3—61no—
407Dic3—61no—
408Dic3—62no—
409Dic3—62no—
410Dic3—61no—
411C32#563911yescharacteristic
412C9—21no—
413C9C3×#55211yescharacteristic, complemented
414C23—91no—
415C23C22×AGL(1,ℤ/9)11yescharacteristic
416Q8—22no—
417D4—182no—
418C2×C4—22no—
419C2×C4—91no—
420C2×C4—92no—
421C2×C4—92no—
422C2×C4—91no—
423C2×C4—91no—
424C2×C4—184no—
425C2×C4—182no—
426C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
427C2×C4C22×AGL(1,ℤ/9)12yes—
428C2×C4C22×AGL(1,ℤ/9)12yes—
429C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
430C2×C3—31no—
431C2×C3—31no—
432C2×C3—31no—
433C2×C3—41no—
434C2×C3—121no—
435C2×C3—182no—
436C2×C3C2×C3×#23711yescharacteristic
437C2×C3C3×#100711yescharacteristic
438C2×C3C3×D3×#4111yescharacteristic
439D3—62no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—92no—
444C22—181no—
445C22—181no—
446C22C23×AGL(1,ℤ/9)11yescharacteristic
447C4—21no—
448C4—22no—
449C4—22no—
450C4—181no—
451C4—181no—
452C4—182no—
453C4—182no—
454C4—181no—
455C4#3699⋊C312yes—
456C3—31no—
457C3C3×#563911yescharacteristic
458C2—41no—
459C2—182no—
460C2C2×#3699⋊C311yescharacteristic
461C2#1172263011yescharacteristic
462C2#41×AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
Frattini, socleC22×C3C23×C3×D3
Fitting3−1+2×C22⋊Q8C2
prime radicalC2×#611⋊C3C22
hypercenterC22⋊Q8AGL(1,ℤ/9)
hypocenterC9C3×#552
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. 3−1+2×C22⋊Q8C2semidirect
2. #63508⋊C3C2semidirect
3. Q8×3−1+2⋊C4C2semidirect
5. #63488⋊C3C2semidirect
6. #63454⋊C3C2semidirect
7. C4⋊C4×AGL(1,ℤ/9)C2semidirect
8. #63452⋊C3C2semidirect
9. #63451⋊C3C2semidirect
11. #63441⋊C3C2semidirect
13. #10589741C3semidirect
49. C9×C22⋊Q8C2×C3semidirect
50. #63508C2×C3semidirect
51. Q8×Dic9C2×C3semidirect
53. #63488C2×C3semidirect
54. #63454C2×C3semidirect
55. C4⋊C4×D9C2×C3semidirect
56. #63452C2×C3semidirect
57. #63451C2×C3semidirect
59. #63441C2×C3semidirect
303. C22⋊Q8AGL(1,ℤ/9)semidirect
309. 3−1+2#552semidirect
413. C9C3×#552semidirect

Maximal subgroups

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

subgroupindex
3−1+2×C22⋊Q82
#63508⋊C32
Q8×3−1+2⋊C42
C4×#611⋊C32
#63488⋊C32
#63454⋊C32
C4⋊C4×AGL(1,ℤ/9)2
#63452⋊C32
#63451⋊C32
#63445⋊C32
#63441⋊C32
C3×#56393
#105897413