Finite Group Databasebeta

← #10589736⋊C3 #13152983

Subgroups — #10589736⋊C3 #13152983

Order 1728.

763 conjugacy classes of subgroups.

The subgroup lattice has 452 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⋊C3C212yesmaximal, complemented
3C2×(C2×AGL(1,ℤ/9))⋊C4C211yesmaximal, characteristic
4#63486⋊C3C211yesmaximal, characteristic, complemented
5#63459⋊C3C212yesmaximal, complemented
6#63458⋊C3C212yesmaximal, complemented
7#63458⋊C3C212yesmaximal, complemented
8C22⋊C4×AGL(1,ℤ/9)C212yesmaximal
9#63440⋊C3C212yesmaximal, complemented
10C3×#5634—31nomaximal
11#10589736C311yesmaximal, characteristic, complemented
12C23×AGL(1,ℤ/9)C2211yescharacteristic
13C2×Q8×3−1+2C2211yescharacteristic, complemented
14C22×C4×3−1+2C2211yescharacteristic
15C22×3−1+2⋊C4C2211yescharacteristic
16C2×C4×AGL(1,ℤ/9)C2212yes—
17C2×C4×AGL(1,ℤ/9)C2212yes—
18C2×3−1+2⋊Q8C2211yescharacteristic, complemented
19C2×3−1+2⋊Q8C2212yescomplemented
20C4⋊C4×3−1+2C2211yescharacteristic, complemented
21C4⋊C4×3−1+2C2212yescomplemented
22C22⋊C4×3−1+2C2212yes—
23#3676⋊C3C2212yes—
24(C2×AGL(1,ℤ/9))⋊C4C2212yes—
25(C2×AGL(1,ℤ/9))⋊C4C2212yes—
26(C2×AGL(1,ℤ/9))⋊C4C2212yes—
27(C2×AGL(1,ℤ/9))⋊C4C2212yes—
28C4⋊(3−1+2⋊C4)C2211yescharacteristic, complemented
29C4⋊(3−1+2⋊C4)C2212yescomplemented
30(3−1+2⋊C4)⋊C4C2212yescomplemented
31(3−1+2⋊C4)⋊C4C2212yescomplemented
32(3−1+2⋊C4)⋊C4C2212yescomplemented
33C32×C22⋊Q8D311yescharacteristic
34C3×#946—32no—
35C2×C3×(C2×D3)⋊C4—31no—
36C3×#924—31no—
37C3×#897—32no—
38C3×#896—32no—
39C3×#896—32no—
40C3×D3×C22⋊C4—32no—
41C3×#878—32no—
42C9×C22⋊Q8—21no—
43C9×C22⋊Q8C2×C311yescharacteristic, complemented
44#63508C2×C312yescomplemented
45C2×(C2×D9)⋊C4C2×C311yescharacteristic
46#63486C2×C311yescharacteristic, complemented
47#63459C2×C312yescomplemented
48#63458C2×C312yescomplemented
49#63458C2×C312yescomplemented
50C22⋊C4×D9C2×C312yes—
51#63440C2×C312yescomplemented
52C23×3−1+2C2311yescharacteristic
53C22×AGL(1,ℤ/9)—41no—
54C22×AGL(1,ℤ/9)—41no—
55C22×AGL(1,ℤ/9)Q814yescomplemented
56C22×AGL(1,ℤ/9)C2312yes—
57Q8×3−1+2—41no—
58C2×C4×3−1+2—41no—
59C2×C4×3−1+2C2312yes—
60C2×C4×3−1+2C2311yescharacteristic
61C2×C4×3−1+2C2312yes—
62C2×C4×3−1+2C2311yescharacteristic
63C2×3−1+2⋊C4—41no—
64C2×3−1+2⋊C4C2312yes—
65C2×3−1+2⋊C4C2312yes—
66C2×3−1+2⋊C4C2312yes—
67C4×AGL(1,ℤ/9)—42no—
68C4×AGL(1,ℤ/9)—42no—
693−1+2⋊Q8—41no—
703−1+2⋊Q8—42no—
71C3×#543—91no—
72#5634—31no—
73C23×C3×D3—31no—
74C2×C32×Q8C2×D311yescharacteristic
75C22×C32×C4C2×D311yescharacteristic
76C22×C3×Dic3—31no—
77C2×C3×C4×D3—32no—
78C2×C3×C4×D3—32no—
79C2×C3×Dic6—32no—
80C2×C3×Dic6—31no—
81C23×D9C22×C311yescharacteristic
82C32×C4⋊C4C2×D311yescharacteristic
83C32×C4⋊C4C2×D312yes—
84C32×C22⋊C4C2×D312yes—
85C3×#217—32no—
86C3×(C2×D3)⋊C4—32no—
87C3×(C2×D3)⋊C4—32no—
88C3×(C2×D3)⋊C4—32no—
89C3×(C2×D3)⋊C4—32no—
90C3×C4⋊Dic3—32no—
91C3×C4⋊Dic3—31no—
92C3×Dic3⋊C4—32no—
93C3×Dic3⋊C4—32no—
94C3×Dic3⋊C4—32no—
95C2×Q8×C9—21no—
96C2×Q8×C9C22×C311yescharacteristic, complemented
97C22×C4×C9—21no—
98C22×C4×C9C22×C311yescharacteristic
99C22×Dic9C22×C311yescharacteristic
100C2×C4×D9C22×C312yes—
101C2×C4×D9C22×C312yes—
102C2×Dic18C22×C312yescomplemented
103C2×Dic18C22×C311yescharacteristic, complemented
104C9×C4⋊C4—21no—
105C9×C4⋊C4—22no—
106C9×C4⋊C4C22×C311yescharacteristic, complemented
107C9×C4⋊C4C22×C312yescomplemented
108C9×C22⋊C4—22no—
109C9×C22⋊C4C22×C312yes—
110#3676C22×C312yes—
111(C2×D9)⋊C4C22×C312yes—
112(C2×D9)⋊C4C22×C312yes—
113(C2×D9)⋊C4C22×C312yes—
114(C2×D9)⋊C4C22×C312yes—
115C4⋊Dic9C22×C312yescomplemented
116C4⋊Dic9C22×C311yescharacteristic, complemented
117Dic9⋊C4C22×C312yescomplemented
118Dic9⋊C4C22×C312yescomplemented
119Dic9⋊C4C22×C312yescomplemented
120C22×3−1+2—21no—
121C22×3−1+2—21no—
122C22×3−1+2C2×Q812yes—
123C22×3−1+2C2411yescharacteristic
124C2×AGL(1,ℤ/9)—22no—
125C2×AGL(1,ℤ/9)—22no—
126C2×AGL(1,ℤ/9)—44no—
127C2×AGL(1,ℤ/9)C2×Q814yes—
128C4×3−1+2—21no—
129C4×3−1+2—22no—
130C4×3−1+2—21no—
131C4×3−1+2—22no—
1323−1+2⋊C4—22no—
1333−1+2⋊C4—22no—
1343−1+2⋊C4—22no—
135C3×C22⋊Q8—31no—
136C3×C22⋊Q8—92no—
137C3×C22⋊Q8—92no—
138C3×C22⋊Q8—92no—
139C3×C22⋊Q8—92no—
140C3×C22⋊Q8—91no—
141C3×C22⋊Q8—92no—
142C3×C22⋊Q8C3×D311yescharacteristic
143C2×C3×C22⋊C4—92no—
144C2×C3×C22⋊C4—91no—
145#946—32no—
146C2×(C2×D3)⋊C4—31no—
147#924—31no—
148#897—32no—
149#896—32no—
150#896—32no—
151D3×C22⋊C4—32no—
152#878—32no—
153C23×C32C22×D311yescharacteristic
154C22×C3×D3—32no—
155C22×C3×D3—34no—
156C22×C3×D3—121no—
157C22×C3×D3—121no—
158C32×Q8—41no—
159C2×C32×C4—41no—
160C2×C32×C4C22×D311yescharacteristic
161C2×C32×C4C22×D312yes—
162C2×C32×C4C22×D312yes—
163C2×C32×C4C22×D311yescharacteristic
164C2×C3×Dic3—32no—
165C2×C3×Dic3—32no—
166C2×C3×Dic3—32no—
167C2×C3×Dic3—121no—
168C3×C4×D3—122no—
169C3×C4×D3—122no—
170C3×Dic6—121no—
171C3×Dic6—122no—
172C23×C9—21no—
173C23×C9C23×C311yescharacteristic
174C22×D9—41no—
175C22×D9—41no—
176C22×D9C3×Q814yescomplemented
177C22×D9C23×C312yes—
178Q8×C9—41no—
179Q8×C9—42no—
180C2×C4×C9—21no—
181C2×C4×C9—22no—
182C2×C4×C9—22no—
183C2×C4×C9—21no—
184C2×C4×C9—41no—
185C2×C4×C9—42no—
186C2×C4×C9C23×C311yescharacteristic
187C2×C4×C9C23×C312yes—
188C2×C4×C9C23×C312yes—
189C2×C4×C9C23×C311yescharacteristic
190C2×Dic9—41no—
191C2×Dic9C23×C312yes—
192C2×Dic9C23×C312yes—
193C2×Dic9C23×C312yes—
194C4×D9—42no—
195C4×D9—42no—
196Dic18—41no—
197Dic18—42no—
198#543—91noSylow
199C2×3−1+2—22no—
200C2×3−1+2C22×Q811yescharacteristic
201C2×3−1+2H(5,𝔽2)11yescharacteristic
202C2×3−1+2H(5,𝔽2)11yescharacteristic
203AGL(1,ℤ/9)—24no—
204C24×C3—91no—
205C23×D3—31no—
206C2×C3×Q8—31no—
207C2×C3×Q8—92no—
208C2×C3×Q8—91no—
209C2×C3×Q8C2×C3×D311yescharacteristic
210C22×C3×C4—31no—
211C22×C3×C4—92no—
212C22×C3×C4—92no—
213C22×C3×C4—91no—
214C22×C3×C4C2×C3×D311yescharacteristic
215C22×Dic3—31no—
216C2×C4×D3—32no—
217C2×C4×D3—32no—
218C2×Dic6—32no—
219C2×Dic6—31no—
220C3×C4⋊C4—31no—
221C3×C4⋊C4—32no—
222C3×C4⋊C4—92no—
223C3×C4⋊C4—92no—
224C3×C4⋊C4—92no—
225C3×C4⋊C4—92no—
226C3×C4⋊C4—91no—
227C3×C4⋊C4C2×C3×D311yescharacteristic
228C3×C4⋊C4C2×C3×D312yes—
229C3×C22⋊C4—32no—
230C3×C22⋊C4—92no—
231C3×C22⋊C4—92no—
232C3×C22⋊C4—92no—
233C3×C22⋊C4—92no—
234C3×C22⋊C4—92no—
235C3×C22⋊C4C2×C3×D312yes—
236#217—32no—
237(C2×D3)⋊C4—32no—
238(C2×D3)⋊C4—32no—
239(C2×D3)⋊C4—32no—
240(C2×D3)⋊C4—32no—
241C4⋊Dic3—32no—
242C4⋊Dic3—31no—
243Dic3⋊C4—32no—
244Dic3⋊C4—32no—
245Dic3⋊C4—32no—
246C22×C32—21no—
247C22×C32—21no—
248C22×C32D3×Q812yes—
249C22×C32C23×D311yescharacteristic
250C2×C3×D3—34no—
251C2×C3×D3—62no—
252C2×C3×D3—62no—
253C2×C3×D3—124no—
254C32×C4—21no—
255C32×C4—22no—
256C32×C4—21no—
257C32×C4—22no—
258C3×Dic3—62no—
259C3×Dic3—62no—
260C3×Dic3—62no—
261C22×C9—21no—
262C22×C9—21no—
263C22×C9—21no—
264C22×C9—22no—
265C22×C9—41no—
266C22×C9—41no—
267C22×C9C2×C3×Q812yes—
268C22×C9C24×C311yescharacteristic
269C2×D9—22no—
270C2×D9—22no—
271C2×D9—44no—
272C2×D9C2×C3×Q814yes—
273C4×C9—22no—
274C4×C9—22no—
275C4×C9—21no—
276C4×C9—22no—
277C4×C9—21no—
278C4×C9—42no—
279C4×C9—41no—
280C4×C9—42no—
281Dic9—22no—
282Dic9—22no—
283Dic9—22no—
284C22⋊Q8—92no—
285C22⋊Q8—91no—
286C22⋊Q8—92no—
287C22⋊Q8—92no—
288C22⋊Q8—92no—
289C22⋊Q8—92no—
290C22⋊Q8AGL(1,ℤ/9)11yescharacteristic, complemented
291C2×C22⋊C4—91no—
292C2×C22⋊C4—92no—
2933−1+2#54311yescharacteristic, complemented, Sylow
294C23×C3—31no—
295C23×C3—92no—
296C23×C3—94no—
297C23×C3—361no—
298C23×C3—361no—
299C23×C3C22×C3×D311yescharacteristic
300C22×D3—32no—
301C22×D3—34no—
302C22×D3—121no—
303C22×D3—121no—
304C3×Q8—41no—
305C3×Q8—121no—
306C3×Q8—361no—
307C3×Q8—362no—
308C2×C3×C4—32no—
309C2×C3×C4—32no—
310C2×C3×C4—31no—
311C2×C3×C4—31no—
312C2×C3×C4—41no—
313C2×C3×C4—92no—
314C2×C3×C4—92no—
315C2×C3×C4—92no—
316C2×C3×C4—121no—
317C2×C3×C4—362no—
318C2×C3×C4—362no—
319C2×C3×C4—361no—
320C2×C3×C4C22×C3×D311yescharacteristic
321C2×C3×C4C22×C3×D312yes—
322C2×C3×C4C22×C3×D312yes—
323C2×C3×C4C22×C3×D311yescharacteristic
324C2×Dic3—32no—
325C2×Dic3—32no—
326C2×Dic3—32no—
327C2×Dic3—121no—
328C4×D3—122no—
329C4×D3—122no—
330Dic6—121no—
331Dic6—122no—
332C2×C32—22no—
333C2×C32#100411yescharacteristic
334C2×C32C2×D3×Q811yescharacteristic
335C2×C32#100911yescharacteristic
336C3×D3—64no—
337C2×C9—22no—
338C2×C9—21no—
339C2×C9—21no—
340C2×C9—21no—
341C2×C9—42no—
342C2×C9C22×C3×Q811yescharacteristic
343C2×C9C3×H(5,𝔽2)11yescharacteristic
344C2×C9C3×H(5,𝔽2)11yescharacteristic
345D9—24no—
346C24—91no—
347C2×Q8—91no—
348C2×Q8—92no—
349C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic, complemented
350C22×C4—91no—
351C22×C4—92no—
352C22×C4—92no—
353C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
354C4⋊C4—92no—
355C4⋊C4—92no—
356C4⋊C4—92no—
357C4⋊C4—91no—
358C4⋊C4—92no—
359C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic, complemented
360C4⋊C4C2×AGL(1,ℤ/9)12yescomplemented
361C22⋊C4—92no—
362C22⋊C4—92no—
363C22⋊C4—92no—
364C22⋊C4—92no—
365C22⋊C4—92no—
366C22⋊C4C2×AGL(1,ℤ/9)12yes—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—31no—
370C22×C3—32no—
371C22×C3—61no—
372C22×C3—61no—
373C22×C3—94no—
374C22×C3—182no—
375C22×C3—182no—
376C22×C3—364no—
377C22×C3C3×D3×Q812yes—
378C22×C3C23×C3×D311yescharacteristic
379C2×D3—34no—
380C2×D3—62no—
381C2×D3—62no—
382C2×D3—124no—
383C3×C4—21no—
384C3×C4—22no—
385C3×C4—21no—
386C3×C4—22no—
387C3×C4—61no—
388C3×C4—61no—
389C3×C4—62no—
390C3×C4—62no—
391C3×C4—182no—
392C3×C4—182no—
393C3×C4—182no—
394Dic3—62no—
395Dic3—62no—
396Dic3—62no—
397C32#563411yescharacteristic
398C9—21no—
399C9C3×#54311yescharacteristic, complemented
400C23—92no—
401C23—94no—
402C23—361no—
403C23—361no—
404C23C22×AGL(1,ℤ/9)11yescharacteristic
405Q8—41no—
406Q8—361no—
407Q8—362no—
408C2×C4—41no—
409C2×C4—92no—
410C2×C4—92no—
411C2×C4—92no—
412C2×C4—362no—
413C2×C4—362no—
414C2×C4—361no—
415C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
416C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
417C2×C4C22×AGL(1,ℤ/9)12yes—
418C2×C4C22×AGL(1,ℤ/9)12yes—
419C2×C3—22no—
420C2×C3—31no—
421C2×C3—31no—
422C2×C3—31no—
423C2×C3—62no—
424C2×C3—184no—
425C2×C3C3×#100411yescharacteristic
426C2×C3C2×C3×D3×Q811yescharacteristic
427C2×C3C3×#100911yescharacteristic
428D3—64no—
429C22—21no—
430C22—21no—
431C22—94no—
432C22—182no—
433C22—182no—
434C22—364no—
435C22Q8×AGL(1,ℤ/9)12yes—
436C22C23×AGL(1,ℤ/9)11yescharacteristic
437C4—21no—
438C4—21no—
439C4—22no—
440C4—22no—
441C4—182no—
442C4—182no—
443C4—182no—
444C3—31no—
445C3C3×#563411yescharacteristic
446C2—22no—
447C2—184no—
448C2#63710⋊C311yescharacteristic
449C2C2×Q8×AGL(1,ℤ/9)11yescharacteristic
450C2#1172264111yescharacteristic
subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
Frattini, socleC22×C3C23×C3×D3
Fitting3−1+2×C22⋊Q8C2
prime radicalC23×AGL(1,ℤ/9)C22
hypercenterC22⋊Q8AGL(1,ℤ/9)
hypocenterC9C3×#543
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
4. #63486⋊C3C2semidirect
5. #63459⋊C3C2semidirect
6. #63458⋊C3C2semidirect
7. #63458⋊C3C2semidirect
9. #63440⋊C3C2semidirect
11. #10589736C3semidirect
13. C2×Q8×3−1+2C22semidirect
18. C2×3−1+2⋊Q8C22semidirect
19. C2×3−1+2⋊Q8C22semidirect
20. C4⋊C4×3−1+2C22semidirect
21. C4⋊C4×3−1+2C22semidirect
28. C4⋊(3−1+2⋊C4)C22semidirect
29. C4⋊(3−1+2⋊C4)C22semidirect
30. (3−1+2⋊C4)⋊C4C22semidirect
31. (3−1+2⋊C4)⋊C4C22semidirect
32. (3−1+2⋊C4)⋊C4C22semidirect
43. C9×C22⋊Q8C2×C3semidirect
44. #63508C2×C3semidirect
46. #63486C2×C3semidirect
47. #63459C2×C3semidirect
48. #63458C2×C3semidirect
49. #63458C2×C3semidirect
51. #63440C2×C3semidirect
55. C22×AGL(1,ℤ/9)Q8semidirect
96. C2×Q8×C9C22×C3semidirect
102. C2×Dic18C22×C3semidirect
103. C2×Dic18C22×C3semidirect
106. C9×C4⋊C4C22×C3semidirect
107. C9×C4⋊C4C22×C3semidirect
115. C4⋊Dic9C22×C3semidirect
116. C4⋊Dic9C22×C3semidirect
117. Dic9⋊C4C22×C3semidirect
118. Dic9⋊C4C22×C3semidirect
119. Dic9⋊C4C22×C3semidirect
176. C22×D9C3×Q8semidirect
290. C22⋊Q8AGL(1,ℤ/9)semidirect
293. 3−1+2#543semidirect
349. C2×Q8C2×AGL(1,ℤ/9)semidirect
359. C4⋊C4C2×AGL(1,ℤ/9)semidirect
360. C4⋊C4C2×AGL(1,ℤ/9)semidirect
399. C9C3×#543semidirect

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
C2×(C2×AGL(1,ℤ/9))⋊C42
#63486⋊C32
#63459⋊C32
#63458⋊C32
C22⋊C4×AGL(1,ℤ/9)2
#63440⋊C32
C3×#56343
#105897363