Finite Group Databasebeta

← #10589737⋊C3 #13152985

Subgroups — #10589737⋊C3 #13152985

Order 1728.

721 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#63508⋊C3C211yesmaximal, characteristic, complemented
3Q8×3−1+2⋊C4C211yesmaximal, characteristic, complemented
4C4×#611⋊C3C211yesmaximal, characteristic
5C2×C4⋊(3−1+2⋊C4)C211yesmaximal, characteristic, complemented
6#63459⋊C3C212yesmaximal, complemented
7C4⋊C4×AGL(1,ℤ/9)C211yesmaximal, characteristic, complemented
8#63451⋊C3C212yesmaximal, complemented
9#63450⋊C3C211yesmaximal, characteristic, complemented
10(#611⋊C3)⋊C4C212yesmaximal
11#63440⋊C3C212yesmaximal, complemented
12C3×#5635—31nomaximal
13#10589737C311yesmaximal, characteristic, complemented
14C2×Q8×3−1+2C2211yescharacteristic
15C22×C4×3−1+2C2211yescharacteristic
16C2×#611⋊C3C2211yescharacteristic
17C22×3−1+2⋊C4C2212yes—
18C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
19C2×C4×AGL(1,ℤ/9)C2212yes—
20C2×3−1+2⋊Q8C2212yes—
21C4⋊C4×3−1+2C2211yescharacteristic
22C4⋊C4×3−1+2C2212yes—
23C22⋊C4×3−1+2C2212yes—
24#3676⋊C3C2211yescharacteristic
25(C2×AGL(1,ℤ/9))⋊C4C2211yescharacteristic
26(C2×AGL(1,ℤ/9))⋊C4C2212yes—
27C4⋊(3−1+2⋊C4)C2211yescharacteristic
28C4⋊(3−1+2⋊C4)C2212yes—
29C4⋊(3−1+2⋊C4)C2211yescharacteristic
30C4⋊(3−1+2⋊C4)C2212yes—
31(3−1+2⋊C4)⋊C4C2212yes—
32(3−1+2⋊C4)⋊C4C2211yescharacteristic
33(3−1+2⋊C4)⋊C4C2212yes—
34(3−1+2⋊C4)⋊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—
41C2×C3×C4⋊Dic3—31no—
42C3×#897—32no—
43C3×D3×C4⋊C4—31no—
44C3×#889—32no—
45C3×#888—31no—
46C3×#67⋊C4—32no—
47C3×#878—32no—
48C9×C22⋊Q8—21no—
49C9×C22⋊Q8C2×C311yescharacteristic, complemented
50#63508C2×C311yescharacteristic, complemented
51Q8×Dic9C2×C311yescharacteristic, complemented
52C4×#611C2×C311yescharacteristic
53C2×C4⋊Dic9C2×C311yescharacteristic, complemented
54#63459C2×C312yescomplemented
55C4⋊C4×D9C2×C311yescharacteristic, complemented
56#63451C2×C312yescomplemented
57#63450C2×C311yescharacteristic, complemented
58#611⋊C4C2×C312yes—
59#63440C2×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+2C2312yes—
66C2×C4×3−1+2C2311yescharacteristic
67C2×C4×3−1+2C2311yescharacteristic
68#611⋊C3Q814yescomplemented
69C2×3−1+2⋊C4—42no—
70C2×3−1+2⋊C4C2311yescharacteristic
71C2×3−1+2⋊C4C2312yes—
72C2×3−1+2⋊C4C2312yes—
73C2×3−1+2⋊C4C2311yescharacteristic
74C2×3−1+2⋊C4C2311yescharacteristic
75C4×AGL(1,ℤ/9)—22no—
76C4×AGL(1,ℤ/9)—42no—
773−1+2⋊Q8—42no—
78C3×#554—91no—
79#5635—31no—
80C2×C32×Q8C2×D311yescharacteristic
81C22×C32×C4C2×D311yescharacteristic
82C2×C3×#67—31no—
83C22×C3×Dic3—32no—
84C2×C3×C4×D3—31no—
85C2×C3×C4×D3—32no—
86C2×C3×Dic6—32no—
87C32×C4⋊C4C2×D311yescharacteristic
88C32×C4⋊C4C2×D312yes—
89C32×C22⋊C4C2×D312yes—
90C3×#217—31no—
91C3×(C2×D3)⋊C4—32no—
92C3×(C2×D3)⋊C4—31no—
93C3×C4⋊Dic3—32no—
94C3×C4⋊Dic3—31no—
95C3×C4⋊Dic3—32no—
96C3×C4⋊Dic3—31no—
97C3×Dic3⋊C4—32no—
98C3×Dic3⋊C4—32no—
99C3×Dic3⋊C4—32no—
100C3×Dic3⋊C4—31no—
101C3×C4×Dic3—32no—
102C3×C4×Dic3—31no—
103C2×Q8×C9—21no—
104C2×Q8×C9C22×C311yescharacteristic
105C22×C4×C9—21no—
106C22×C4×C9C22×C311yescharacteristic
107C2×#611C22×C311yescharacteristic
108C22×Dic9C22×C312yes—
109C2×C4×D9C22×C311yescharacteristic
110C2×C4×D9C22×C312yes—
111C2×Dic18C22×C312yes—
112C9×C4⋊C4—21no—
113C9×C4⋊C4—22no—
114C9×C4⋊C4C22×C311yescharacteristic
115C9×C4⋊C4C22×C312yes—
116C9×C22⋊C4—22no—
117C9×C22⋊C4C22×C312yes—
118#3676C22×C311yescharacteristic
119(C2×D9)⋊C4C22×C311yescharacteristic
120(C2×D9)⋊C4C22×C312yes—
121C4⋊Dic9C22×C312yes—
122C4⋊Dic9C22×C311yescharacteristic
123C4⋊Dic9C22×C311yescharacteristic
124C4⋊Dic9C22×C312yes—
125Dic9⋊C4C22×C311yescharacteristic
126Dic9⋊C4C22×C312yes—
127Dic9⋊C4C22×C312yes—
128Dic9⋊C4C22×C312yes—
129C4×Dic9C22×C311yescharacteristic
130C4×Dic9C22×C312yes—
131C22×3−1+2—21no—
132C22×3−1+2—21no—
133C22×3−1+2C2×Q812yes—
134C22×3−1+2C2411yescharacteristic
135C2×AGL(1,ℤ/9)—21no—
136C2×AGL(1,ℤ/9)—21no—
137C2×AGL(1,ℤ/9)C2×Q812yes—
138C4×3−1+2—21no—
139C4×3−1+2—22no—
140C4×3−1+2—22no—
141C4×3−1+2#4112yes—
1423−1+2⋊C4—21no—
1433−1+2⋊C4—21no—
1443−1+2⋊C4—22no—
1453−1+2⋊C4—22no—
1463−1+2⋊C4C2×Q812yes—
147C3×C4⋊Q8—91no—
148C3×#125—92no—
149C3×C22⋊Q8—31no—
150C3×C22⋊Q8—92no—
151C3×C22⋊Q8—92no—
152C3×C22⋊Q8—91no—
153C3×C22⋊Q8C3×D311yescharacteristic
154C3×C4×Q8—91no—
155C3×C4×D4—92no—
156C3×C4×D4—91no—
157C2×C3×C4⋊C4—91no—
158C2×C3×C4⋊C4—91no—
159#946—31no—
160Q8×Dic3—31no—
161C4×#67—31no—
162C2×C4⋊Dic3—31no—
163#897—32no—
164D3×C4⋊C4—31no—
165#889—32no—
166#888—31no—
167#67⋊C4—32no—
168#878—32no—
169C23×C32C22×D311yescharacteristic
170C22×C3×D3—31no—
171C32×Q8—22no—
172C2×C32×C4—22no—
173C2×C32×C4C22×D312yes—
174C2×C32×C4C22×D312yes—
175C2×C32×C4C22×D311yescharacteristic
176C2×C32×C4C22×D311yescharacteristic
177C3×#67—34no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3—32no—
180C2×C3×Dic3—31no—
181C2×C3×Dic3—31no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—122no—
184C3×C4×D3—62no—
185C3×C4×D3—122no—
186C3×Dic6—122no—
187C23×C9—21no—
188C23×C9C23×C311yescharacteristic
189C22×D9C23×C311yescharacteristic
190Q8×C9—22no—
191Q8×C9—42no—
192C2×C4×C9—22no—
193C2×C4×C9—21no—
194C2×C4×C9—22no—
195C2×C4×C9—22no—
196C2×C4×C9—21no—
197C2×C4×C9—42no—
198C2×C4×C9C23×C312yes—
199C2×C4×C9C23×C312yes—
200C2×C4×C9C23×C311yescharacteristic
201C2×C4×C9C23×C311yescharacteristic
202#611C3×Q814yescomplemented
203C2×Dic9—42no—
204C2×Dic9C23×C312yes—
205C2×Dic9C23×C311yescharacteristic
206C2×Dic9C23×C311yescharacteristic
207C2×Dic9C23×C312yes—
208C2×Dic9C23×C311yescharacteristic
209C4×D9—22no—
210C4×D9—42no—
211Dic18—42no—
212#554—91noSylow
213C2×3−1+2—22no—
214C2×3−1+2C22×Q811yescharacteristic
215C2×3−1+2C2×#4111yescharacteristic
216C2×3−1+2H(5,𝔽2)11yescharacteristic
217AGL(1,ℤ/9)—22no—
218C2×C3×Q8—31no—
219C2×C3×Q8—92no—
220C2×C3×Q8C2×C3×D311yescharacteristic
221C2×C3×D4—91no—
222C22×C3×C4—31no—
223C22×C3×C4—91no—
224C22×C3×C4—92no—
225C22×C3×C4—92no—
226C22×C3×C4C2×C3×D311yescharacteristic
227C2×#67—31no—
228C22×Dic3—32no—
229C2×C4×D3—31no—
230C2×C4×D3—32no—
231C2×Dic6—32no—
232C3×C4⋊C4—31no—
233C3×C4⋊C4—32no—
234C3×C4⋊C4—92no—
235C3×C4⋊C4—91no—
236C3×C4⋊C4—92no—
237C3×C4⋊C4—92no—
238C3×C4⋊C4—91no—
239C3×C4⋊C4—92no—
240C3×C4⋊C4—91no—
241C3×C4⋊C4—92no—
242C3×C4⋊C4C2×C3×D311yescharacteristic
243C3×C4⋊C4C2×C3×D312yes—
244C3×C22⋊C4—32no—
245C3×C22⋊C4—92no—
246C3×C22⋊C4—91no—
247C3×C22⋊C4—91no—
248C3×C22⋊C4C2×C3×D312yes—
249C3×C42—92no—
250C3×C42—91no—
251#217—31no—
252(C2×D3)⋊C4—32no—
253(C2×D3)⋊C4—31no—
254C4⋊Dic3—32no—
255C4⋊Dic3—31no—
256C4⋊Dic3—32no—
257C4⋊Dic3—31no—
258Dic3⋊C4—32no—
259Dic3⋊C4—32no—
260Dic3⋊C4—32no—
261Dic3⋊C4—31no—
262C4×Dic3—32no—
263C4×Dic3—31no—
264C22×C32—21no—
265C22×C32—21no—
266C22×C32D3×Q812yes—
267C22×C32C23×D311yescharacteristic
268C2×C3×D3—32no—
269C2×C3×D3—61no—
270C2×C3×D3—61no—
271C32×C4—21no—
272C32×C4—22no—
273C32×C4—22no—
274C32×C4#23712yes—
275C3×Dic3—32no—
276C3×Dic3—61no—
277C3×Dic3—61no—
278C3×Dic3—62no—
279C3×Dic3—62no—
280C22×C9—21no—
281C22×C9—21no—
282C22×C9—21no—
283C22×C9—22no—
284C22×C9—41no—
285C22×C9—41no—
286C22×C9C2×C3×Q812yes—
287C22×C9C24×C311yescharacteristic
288C2×D9—21no—
289C2×D9—21no—
290C2×D9C2×C3×Q812yes—
291C4×C9—21no—
292C4×C9—22no—
293C4×C9—22no—
294C4×C9—22no—
295C4×C9—42no—
296C4×C9—41no—
297C4×C9—42no—
298C4×C9C3×#4112yes—
299Dic9—21no—
300Dic9—21no—
301Dic9—22no—
302Dic9—22no—
303Dic9C2×C3×Q812yes—
304C4⋊Q8—91no—
305#125—92no—
306C22⋊Q8—91no—
307C22⋊Q8—92no—
308C22⋊Q8—92no—
309C22⋊Q8AGL(1,ℤ/9)11yescharacteristic, complemented
310C4×Q8—91no—
311C4×D4—91no—
312C4×D4—92no—
313C2×C4⋊C4—91no—
314C2×C4⋊C4—91no—
3153−1+2#55411yescharacteristic, complemented, Sylow
316C23×C3—31no—
317C23×C3—91no—
318C23×C3C22×C3×D311yescharacteristic
319C22×D3—31no—
320C3×Q8—22no—
321C3×Q8—62no—
322C3×Q8—362no—
323C3×D4—94no—
324C2×C3×C4—22no—
325C2×C3×C4—31no—
326C2×C3×C4—32no—
327C2×C3×C4—32no—
328C2×C3×C4—31no—
329C2×C3×C4—62no—
330C2×C3×C4—91no—
331C2×C3×C4—91no—
332C2×C3×C4—92no—
333C2×C3×C4—91no—
334C2×C3×C4—92no—
335C2×C3×C4—182no—
336C2×C3×C4—362no—
337C2×C3×C4—362no—
338C2×C3×C4C22×C3×D312yes—
339C2×C3×C4C22×C3×D312yes—
340C2×C3×C4C22×C3×D311yescharacteristic
341C2×C3×C4C22×C3×D311yescharacteristic
342#67—34no—
343C2×Dic3—31no—
344C2×Dic3—32no—
345C2×Dic3—31no—
346C2×Dic3—31no—
347C2×Dic3—32no—
348C2×Dic3—122no—
349C4×D3—62no—
350C4×D3—122no—
351Dic6—122no—
352C2×C32—22no—
353C2×C32C2×#23711yescharacteristic
354C2×C32C2×D3×Q811yescharacteristic
355C2×C32#100911yescharacteristic
356C3×D3—62no—
357C2×C9—22no—
358C2×C9—21no—
359C2×C9—21no—
360C2×C9—21no—
361C2×C9—42no—
362C2×C9C22×C3×Q811yescharacteristic
363C2×C9C2×C3×#4111yescharacteristic
364C2×C9C3×H(5,𝔽2)11yescharacteristic
365D9—22no—
366C2×Q8—92no—
367C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
368C2×D4—91no—
369C22×C4—92no—
370C22×C4—91no—
371C22×C4—92no—
372C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
373C4⋊C4—91no—
374C4⋊C4—92no—
375C4⋊C4—92no—
376C4⋊C4—91no—
377C4⋊C4—91no—
378C4⋊C4—92no—
379C4⋊C4—92no—
380C4⋊C4—92no—
381C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
382C4⋊C4C2×AGL(1,ℤ/9)12yes—
383C22⋊C4—91no—
384C22⋊C4—91no—
385C22⋊C4—92no—
386C22⋊C4C2×AGL(1,ℤ/9)12yes—
387C42—92no—
388C42—91no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—31no—
392C22×C3—32no—
393C22×C3—61no—
394C22×C3—61no—
395C22×C3—92no—
396C22×C3—181no—
397C22×C3—181no—
398C22×C3C3×D3×Q812yes—
399C22×C3C23×C3×D311yescharacteristic
400C2×D3—32no—
401C2×D3—61no—
402C2×D3—61no—
403C3×C4—21no—
404C3×C4—22no—
405C3×C4—22no—
406C3×C4—32no—
407C3×C4—61no—
408C3×C4—62no—
409C3×C4—62no—
410C3×C4—92no—
411C3×C4—181no—
412C3×C4—182no—
413C3×C4—182no—
414C3×C4—181no—
415C3×C4C3×#23712yes—
416Dic3—32no—
417Dic3—61no—
418Dic3—61no—
419Dic3—62no—
420Dic3—62no—
421C32#563511yescharacteristic
422C9—21no—
423C9C3×#55411yescharacteristic, complemented
424C23—91no—
425C23C22×AGL(1,ℤ/9)11yescharacteristic
426Q8—22no—
427Q8—362no—
428D4—94no—
429C2×C4—22no—
430C2×C4—91no—
431C2×C4—91no—
432C2×C4—92no—
433C2×C4—91no—
434C2×C4—92no—
435C2×C4—182no—
436C2×C4—362no—
437C2×C4—362no—
438C2×C4C22×AGL(1,ℤ/9)12yes—
439C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
440C2×C4C22×AGL(1,ℤ/9)12yes—
441C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
442C2×C3—22no—
443C2×C3—31no—
444C2×C3—31no—
445C2×C3—31no—
446C2×C3—62no—
447C2×C3—182no—
448C2×C3C2×C3×#23711yescharacteristic
449C2×C3C2×C3×D3×Q811yescharacteristic
450C2×C3C3×#100911yescharacteristic
451D3—62no—
452C22—21no—
453C22—21no—
454C22—92no—
455C22—181no—
456C22—181no—
457C22Q8×AGL(1,ℤ/9)12yes—
458C22C23×AGL(1,ℤ/9)11yescharacteristic
459C4—21no—
460C4—22no—
461C4—22no—
462C4—92no—
463C4—181no—
464C4—182no—
465C4—182no—
466C4—181no—
467C4#3699⋊C312yes—
468C3—31no—
469C3C3×#563511yescharacteristic
470C2—22no—
471C2—182no—
472C2C2×#3699⋊C311yescharacteristic
473C2C2×Q8×AGL(1,ℤ/9)11yescharacteristic
474C2#1172264111yescharacteristic
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×#554
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. C2×C4⋊(3−1+2⋊C4)C2semidirect
6. #63459⋊C3C2semidirect
7. C4⋊C4×AGL(1,ℤ/9)C2semidirect
8. #63451⋊C3C2semidirect
9. #63450⋊C3C2semidirect
11. #63440⋊C3C2semidirect
13. #10589737C3semidirect
49. C9×C22⋊Q8C2×C3semidirect
50. #63508C2×C3semidirect
51. Q8×Dic9C2×C3semidirect
53. C2×C4⋊Dic9C2×C3semidirect
54. #63459C2×C3semidirect
55. C4⋊C4×D9C2×C3semidirect
56. #63451C2×C3semidirect
57. #63450C2×C3semidirect
59. #63440C2×C3semidirect
68. #611⋊C3Q8semidirect
202. #611C3×Q8semidirect
309. C22⋊Q8AGL(1,ℤ/9)semidirect
315. 3−1+2#554semidirect
423. C9C3×#554semidirect

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
C2×C4⋊(3−1+2⋊C4)2
#63459⋊C32
C4⋊C4×AGL(1,ℤ/9)2
#63451⋊C32
#63450⋊C32
(#611⋊C3)⋊C42
#63440⋊C32
C3×#56353
#105897373