Finite Group Databasebeta

← #10589746⋊C3 #13152991

Subgroups — #10589746⋊C3 #13152991

Order 1728.

787 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
13−1+2×C22⋊Q8C211yesmaximal, characteristic, complemented
2#63509⋊C3C212yesmaximal, complemented
3#63492⋊C3C211yesmaximal, characteristic, complemented
4C2×(C2×AGL(1,ℤ/9))⋊C4C211yesmaximal, characteristic, complemented
5#63460⋊C3C212yesmaximal, complemented
6#63457⋊C3C212yesmaximal, complemented
7#63456⋊C3C212yesmaximal, complemented
8#63444⋊C3C212yesmaximal, complemented
9(#611⋊C3)⋊C4C212yesmaximal, complemented
10C3×#5644—31nomaximal
11#10589746C311yesmaximal, characteristic, complemented
12C23×AGL(1,ℤ/9)C2211yescharacteristic
13C2×Q8×3−1+2C2211yescharacteristic, complemented
14C22×C4×3−1+2C2211yescharacteristic
15C2×#611⋊C3C2212yes—
16C22×3−1+2⋊C4C2211yescharacteristic
17C2×C4⋊AGL(1,ℤ/9)C2212yes—
18C2×C4⋊AGL(1,ℤ/9)C2212yes—
19C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic, complemented
20C2×C4×AGL(1,ℤ/9)C2212yescomplemented
21C4⋊C4×3−1+2C2211yescharacteristic, complemented
22C4⋊C4×3−1+2C2212yescomplemented
23C22⋊C4×3−1+2C2212yes—
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—
28(C2×AGL(1,ℤ/9))⋊C4C2212yescomplemented
29(C2×AGL(1,ℤ/9))⋊C4C2212yescomplemented
30C4⋊(3−1+2⋊C4)C2211yescharacteristic, complemented
31(3−1+2⋊C4)⋊C4C2212yescomplemented
32C4×3−1+2⋊C4C2212yescomplemented
33C32×C22⋊Q8D311yescharacteristic
34C3×#947—32no—
35C3×#930—31no—
36C2×C3×(C2×D3)⋊C4—31no—
37C3×#898—32no—
38C3×#895—32no—
39C3×#894—32no—
40C3×#882—32no—
41C3×#67⋊C4—32no—
42C9×C22⋊Q8—21no—
43C9×C22⋊Q8C2×C311yescharacteristic, complemented
44#63509C2×C312yescomplemented
45#63492C2×C311yescharacteristic, complemented
46C2×(C2×D9)⋊C4C2×C311yescharacteristic, complemented
47#63460C2×C312yescomplemented
48#63457C2×C312yescomplemented
49#63456C2×C312yescomplemented
50#63444C2×C312yescomplemented
51#611⋊C4C2×C312yescomplemented
52C23×3−1+2C2311yescharacteristic
53C22×AGL(1,ℤ/9)—22no—
54C22×AGL(1,ℤ/9)—41no—
55C22×AGL(1,ℤ/9)—41no—
56C22×AGL(1,ℤ/9)C2312yes—
57C22×AGL(1,ℤ/9)C2312yes—
58Q8×3−1+2—41no—
59C2×C4×3−1+2—41no—
60C2×C4×3−1+2C2312yes—
61C2×C4×3−1+2C2312yes—
62C2×C4×3−1+2C2311yescharacteristic
63C2×C4×3−1+2C2311yescharacteristic
64#611⋊C3—24no—
65C2×3−1+2⋊C4—41no—
66C2×3−1+2⋊C4C2312yes—
67C2×3−1+2⋊C4C2312yes—
68C4⋊AGL(1,ℤ/9)—42no—
69C4⋊AGL(1,ℤ/9)—42no—
70C4⋊AGL(1,ℤ/9)—41no—
71C4×AGL(1,ℤ/9)—42no—
72C3×#538—91no—
73#5644—31no—
74C23×C3×D3—31no—
75C2×C32×Q8C2×D311yescharacteristic
76C22×C32×C4C2×D311yescharacteristic
77C2×C3×#67—32no—
78C22×C3×Dic3—31no—
79C2×C3×D12—32no—
80C2×C3×D12—32no—
81C2×C3×D12—31no—
82C2×C3×C4×D3—32no—
83C23×D9C22×C311yescharacteristic
84C32×C4⋊C4C2×D311yescharacteristic
85C32×C4⋊C4C2×D312yes—
86C32×C22⋊C4C2×D312yes—
87C3×(C2×D3)⋊C4—32no—
88C3×(C2×D3)⋊C4—32no—
89C3×(C2×D3)⋊C4—32no—
90C3×(C2×D3)⋊C4—32no—
91C3×(C2×D3)⋊C4—32no—
92C3×(C2×D3)⋊C4—32no—
93C3×C4⋊Dic3—31no—
94C3×Dic3⋊C4—32no—
95C3×C4×Dic3—32no—
96C2×Q8×C9—21no—
97C2×Q8×C9C22×C311yescharacteristic, complemented
98C22×C4×C9—21no—
99C22×C4×C9C22×C311yescharacteristic
100C2×#611C22×C312yes—
101C22×Dic9C22×C311yescharacteristic
102C2×D36C22×C312yes—
103C2×D36C22×C312yes—
104C2×D36C22×C311yescharacteristic, complemented
105C2×C4×D9C22×C312yescomplemented
106C9×C4⋊C4—21no—
107C9×C4⋊C4—22no—
108C9×C4⋊C4C22×C311yescharacteristic, complemented
109C9×C4⋊C4C22×C312yescomplemented
110C9×C22⋊C4—22no—
111C9×C22⋊C4C22×C312yes—
112(C2×D9)⋊C4C22×C312yes—
113(C2×D9)⋊C4C22×C312yes—
114(C2×D9)⋊C4C22×C312yes—
115(C2×D9)⋊C4C22×C312yes—
116(C2×D9)⋊C4C22×C312yescomplemented
117(C2×D9)⋊C4C22×C312yescomplemented
118C4⋊Dic9C22×C311yescharacteristic, complemented
119Dic9⋊C4C22×C312yescomplemented
120C4×Dic9C22×C312yescomplemented
121C22×3−1+2—21no—
122C22×3−1+2—21no—
123C22×3−1+2#4112yes—
124C22×3−1+2C2411yescharacteristic
125C2×AGL(1,ℤ/9)—22no—
126C2×AGL(1,ℤ/9)—22no—
127C2×AGL(1,ℤ/9)—22no—
128C2×AGL(1,ℤ/9)—22no—
129C2×AGL(1,ℤ/9)—22no—
130C2×AGL(1,ℤ/9)—22no—
131C2×AGL(1,ℤ/9)—44no—
132C4×3−1+2—21no—
133C4×3−1+2—22no—
134C4×3−1+2—21no—
135C4×3−1+2—22no—
1363−1+2⋊C4—22no—
1373−1+2⋊C4—22no—
138C3×#126—92no—
139C3×#124—92no—
140C3×#123—92no—
141C3×C22⋊Q8—31no—
142C3×C22⋊Q8C3×D311yescharacteristic
143C3×#121—92no—
144C3×#121—91no—
145C3×#120—92no—
146C3×C4×D4—92no—
147C2×C3×C22⋊C4—91no—
148#947—32no—
149#930—31no—
150C2×(C2×D3)⋊C4—31no—
151#898—32no—
152#895—32no—
153#894—32no—
154#882—32no—
155#67⋊C4—32no—
156C23×C32C22×D311yescharacteristic
157C22×C3×D3—32no—
158C22×C3×D3—32no—
159C22×C3×D3—62no—
160C22×C3×D3—121no—
161C22×C3×D3—121no—
162C32×Q8—41no—
163C2×C32×C4—41no—
164C2×C32×C4C22×D312yes—
165C2×C32×C4C22×D312yes—
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C3×#67—64no—
169C2×C3×Dic3—32no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—121no—
172C3×D12—121no—
173C3×D12—122no—
174C3×D12—122no—
175C3×C4×D3—122no—
176C23×C9—21no—
177C23×C9C23×C311yescharacteristic
178C22×D9—22no—
179C22×D9—41no—
180C22×D9—41no—
181C22×D9C23×C312yes—
182C22×D9C23×C312yes—
183Q8×C9—41no—
184Q8×C9—42no—
185C2×C4×C9—21no—
186C2×C4×C9—22no—
187C2×C4×C9—22no—
188C2×C4×C9—21no—
189C2×C4×C9—41no—
190C2×C4×C9—42no—
191C2×C4×C9C23×C312yes—
192C2×C4×C9C23×C312yes—
193C2×C4×C9C23×C311yescharacteristic
194C2×C4×C9C23×C311yescharacteristic
195#611—24no—
196C2×Dic9—41no—
197C2×Dic9C23×C312yes—
198C2×Dic9C23×C312yes—
199D36—42no—
200D36—42no—
201D36—41no—
202C4×D9—42no—
203#538—91noSylow
204C2×3−1+2—22no—
205C2×3−1+2C2×#4111yescharacteristic
206C2×3−1+2H(5,𝔽2)11yescharacteristic
207C2×3−1+2H(5,𝔽2)11yescharacteristic
208AGL(1,ℤ/9)—42no—
209AGL(1,ℤ/9)—42no—
210C24×C3—91no—
211C23×D3—31no—
212C2×C3×Q8—31no—
213C2×C3×Q8C2×C3×D311yescharacteristic
214C2×C3×D4—92no—
215C2×C3×D4—92no—
216C2×C3×D4—91no—
217C2×C3×D4—92no—
218C22×C3×C4—31no—
219C22×C3×C4—92no—
220C22×C3×C4—91no—
221C22×C3×C4C2×C3×D311yescharacteristic
222C2×#67—32no—
223C22×Dic3—31no—
224C2×D12—32no—
225C2×D12—32no—
226C2×D12—31no—
227C2×C4×D3—32no—
228C3×C4⋊C4—32no—
229C3×C4⋊C4—31no—
230C3×C4⋊C4—92no—
231C3×C4⋊C4—91no—
232C3×C4⋊C4C2×C3×D311yescharacteristic
233C3×C4⋊C4C2×C3×D312yes—
234C3×C22⋊C4—32no—
235C3×C22⋊C4—92no—
236C3×C22⋊C4—92no—
237C3×C22⋊C4—92no—
238C3×C22⋊C4—92no—
239C3×C22⋊C4—92no—
240C3×C22⋊C4—92no—
241C3×C22⋊C4C2×C3×D312yes—
242C3×C42—92no—
243(C2×D3)⋊C4—32no—
244(C2×D3)⋊C4—32no—
245(C2×D3)⋊C4—32no—
246(C2×D3)⋊C4—32no—
247(C2×D3)⋊C4—32no—
248(C2×D3)⋊C4—32no—
249C4⋊Dic3—31no—
250Dic3⋊C4—32no—
251C4×Dic3—32no—
252C22×C32—21no—
253C22×C32—21no—
254C22×C32#23912yes—
255C22×C32C23×D311yescharacteristic
256C2×C3×D3—62no—
257C2×C3×D3—62no—
258C2×C3×D3—62no—
259C2×C3×D3—62no—
260C2×C3×D3—62no—
261C2×C3×D3—62no—
262C2×C3×D3—124no—
263C32×C4—21no—
264C32×C4—22no—
265C32×C4—21no—
266C32×C4—22no—
267C3×Dic3—62no—
268C3×Dic3—62no—
269C22×C9—21no—
270C22×C9—21no—
271C22×C9—21no—
272C22×C9—22no—
273C22×C9—41no—
274C22×C9—41no—
275C22×C9C3×#4112yes—
276C22×C9C24×C311yescharacteristic
277C2×D9—22no—
278C2×D9—22no—
279C2×D9—22no—
280C2×D9—22no—
281C2×D9—22no—
282C2×D9—22no—
283C2×D9—44no—
284C4×C9—22no—
285C4×C9—22no—
286C4×C9—21no—
287C4×C9—22no—
288C4×C9—21no—
289C4×C9—42no—
290C4×C9—41no—
291C4×C9—42no—
292Dic9—22no—
293Dic9—22no—
294#126—92no—
295#124—92no—
296#123—92no—
297C22⋊Q8AGL(1,ℤ/9)11yescharacteristic, complemented
298#121—91no—
299#121—92no—
300#120—92no—
301C4×D4—92no—
302C2×C22⋊C4—91no—
3033−1+2#53811yescharacteristic, complemented, Sylow
304C23×C3—31no—
305C23×C3—92no—
306C23×C3—92no—
307C23×C3—182no—
308C23×C3—361no—
309C23×C3—361no—
310C23×C3C22×C3×D311yescharacteristic
311C22×D3—32no—
312C22×D3—32no—
313C22×D3—62no—
314C22×D3—121no—
315C22×D3—121no—
316C3×Q8—41no—
317C3×Q8—121no—
318C3×D4—184no—
319C3×D4—361no—
320C3×D4—362no—
321C3×D4—362no—
322C2×C3×C4—32no—
323C2×C3×C4—32no—
324C2×C3×C4—31no—
325C2×C3×C4—31no—
326C2×C3×C4—41no—
327C2×C3×C4—92no—
328C2×C3×C4—92no—
329C2×C3×C4—121no—
330C2×C3×C4—362no—
331C2×C3×C4—361no—
332C2×C3×C4C22×C3×D312yes—
333C2×C3×C4C22×C3×D312yes—
334C2×C3×C4C22×C3×D311yescharacteristic
335C2×C3×C4C22×C3×D311yescharacteristic
336#67—64no—
337C2×Dic3—32no—
338C2×Dic3—32no—
339C2×Dic3—121no—
340D12—121no—
341D12—122no—
342D12—122no—
343C4×D3—122no—
344C2×C32—22no—
345C2×C32#100411yescharacteristic
346C2×C32C2×#23911yescharacteristic
347C2×C32#100911yescharacteristic
348C3×D3—122no—
349C3×D3—122no—
350C2×C9—22no—
351C2×C9—21no—
352C2×C9—21no—
353C2×C9—21no—
354C2×C9—42no—
355C2×C9C2×C3×#4111yescharacteristic
356C2×C9C3×H(5,𝔽2)11yescharacteristic
357C2×C9C3×H(5,𝔽2)11yescharacteristic
358D9—42no—
359D9—42no—
360C24—91no—
361C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic, complemented
362C2×D4—92no—
363C2×D4—92no—
364C2×D4—92no—
365C2×D4—91no—
366C22×C4—91no—
367C22×C4—92no—
368C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
369C4⋊C4—92no—
370C4⋊C4—91no—
371C4⋊C4C2×AGL(1,ℤ/9)12yescomplemented
372C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic, complemented
373C22⋊C4—92no—
374C22⋊C4—92no—
375C22⋊C4—92no—
376C22⋊C4—92no—
377C22⋊C4—92no—
378C22⋊C4—92no—
379C22⋊C4C2×AGL(1,ℤ/9)12yes—
380C42—92no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—31no—
384C22×C3—32no—
385C22×C3—61no—
386C22×C3—61no—
387C22×C3—182no—
388C22×C3—182no—
389C22×C3—182no—
390C22×C3—182no—
391C22×C3—182no—
392C22×C3—182no—
393C22×C3—364no—
394C22×C3C3×#23912yes—
395C22×C3C23×C3×D311yescharacteristic
396C2×D3—62no—
397C2×D3—62no—
398C2×D3—62no—
399C2×D3—62no—
400C2×D3—62no—
401C2×D3—62no—
402C2×D3—124no—
403C3×C4—21no—
404C3×C4—22no—
405C3×C4—21no—
406C3×C4—22no—
407C3×C4—61no—
408C3×C4—61no—
409C3×C4—62no—
410C3×C4—62no—
411C3×C4—182no—
412C3×C4—182no—
413Dic3—62no—
414Dic3—62no—
415C32#564411yescharacteristic
416C9—21no—
417C9C3×#53811yescharacteristic, complemented
418C23—92no—
419C23—92no—
420C23—182no—
421C23—361no—
422C23—361no—
423C23C22×AGL(1,ℤ/9)11yescharacteristic
424Q8—41no—
425D4—184no—
426D4—361no—
427D4—362no—
428D4—362no—
429C2×C4—41no—
430C2×C4—92no—
431C2×C4—92no—
432C2×C4—362no—
433C2×C4—361no—
434C2×C4C22×AGL(1,ℤ/9)12yes—
435C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
436C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
437C2×C4C22×AGL(1,ℤ/9)12yes—
438C2×C3—22no—
439C2×C3—31no—
440C2×C3—31no—
441C2×C3—31no—
442C2×C3—62no—
443C2×C3—362no—
444C2×C3—362no—
445C2×C3C3×#100411yescharacteristic
446C2×C3C2×C3×#23911yescharacteristic
447C2×C3C3×#100911yescharacteristic
448D3—122no—
449D3—122no—
450C22—21no—
451C22—21no—
452C22—182no—
453C22—182no—
454C22—182no—
455C22—182no—
456C22—182no—
457C22—182no—
458C22—364no—
459C22#8872412yes—
460C22C23×AGL(1,ℤ/9)11yescharacteristic
461C4—21no—
462C4—21no—
463C4—22no—
464C4—22no—
465C4—182no—
466C4—182no—
467C3—31no—
468C3C3×#564411yescharacteristic
469C2—22no—
470C2—362no—
471C2—362no—
472C2#63710⋊C311yescharacteristic
473C2C2×#8872411yescharacteristic
474C2#1172264111yescharacteristic
subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
Frattini, socleC22×C3C23×C3×D3
Fitting3−1+2×C22⋊Q8C2
hypercenterC22⋊Q8AGL(1,ℤ/9)
hypocenterC9C3×#538
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. #63509⋊C3C2semidirect
3. #63492⋊C3C2semidirect
4. C2×(C2×AGL(1,ℤ/9))⋊C4C2semidirect
5. #63460⋊C3C2semidirect
6. #63457⋊C3C2semidirect
7. #63456⋊C3C2semidirect
8. #63444⋊C3C2semidirect
9. (#611⋊C3)⋊C4C2semidirect
11. #10589746C3semidirect
13. C2×Q8×3−1+2C22semidirect
19. C2×C4⋊AGL(1,ℤ/9)C22semidirect
20. C2×C4×AGL(1,ℤ/9)C22semidirect
21. C4⋊C4×3−1+2C22semidirect
22. C4⋊C4×3−1+2C22semidirect
28. (C2×AGL(1,ℤ/9))⋊C4C22semidirect
29. (C2×AGL(1,ℤ/9))⋊C4C22semidirect
30. C4⋊(3−1+2⋊C4)C22semidirect
31. (3−1+2⋊C4)⋊C4C22semidirect
32. C4×3−1+2⋊C4C22semidirect
43. C9×C22⋊Q8C2×C3semidirect
44. #63509C2×C3semidirect
45. #63492C2×C3semidirect
46. C2×(C2×D9)⋊C4C2×C3semidirect
47. #63460C2×C3semidirect
48. #63457C2×C3semidirect
49. #63456C2×C3semidirect
50. #63444C2×C3semidirect
51. #611⋊C4C2×C3semidirect
97. C2×Q8×C9C22×C3semidirect
104. C2×D36C22×C3semidirect
105. C2×C4×D9C22×C3semidirect
108. C9×C4⋊C4C22×C3semidirect
109. C9×C4⋊C4C22×C3semidirect
116. (C2×D9)⋊C4C22×C3semidirect
117. (C2×D9)⋊C4C22×C3semidirect
118. C4⋊Dic9C22×C3semidirect
119. Dic9⋊C4C22×C3semidirect
120. C4×Dic9C22×C3semidirect
297. C22⋊Q8AGL(1,ℤ/9)semidirect
303. 3−1+2#538semidirect
361. C2×Q8C2×AGL(1,ℤ/9)semidirect
371. C4⋊C4C2×AGL(1,ℤ/9)semidirect
372. C4⋊C4C2×AGL(1,ℤ/9)semidirect
417. C9C3×#538semidirect

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
#63509⋊C32
#63492⋊C32
C2×(C2×AGL(1,ℤ/9))⋊C42
#63460⋊C32
#63457⋊C32
#63456⋊C32
#63444⋊C32
(#611⋊C3)⋊C42
C3×#56443
#105897463