Finite Group Databasebeta

← #10589613⋊C3 #13152462

Subgroups — #10589613⋊C3 #13152462

Order 1728.

730 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
1C2×C4⋊C4×3−1+2C211yesmaximal, characteristic, complemented
2C4×#611⋊C3C211yesmaximal, characteristic
3C4×#611⋊C3C212yesmaximal
4#63486⋊C3C211yesmaximal, characteristic, complemented
5#63486⋊C3C212yesmaximal, complemented
6#63459⋊C3C211yesmaximal, characteristic, complemented
7#63458⋊C3C212yesmaximal, complemented
8C4⋊C4×AGL(1,ℤ/9)C211yesmaximal, characteristic, complemented
9#63451⋊C3C212yesmaximal, complemented
10#63450⋊C3C211yesmaximal, characteristic, complemented
11#63449⋊C3C211yesmaximal, characteristic, complemented
12C3×#5511—31nomaximal
13#10589613C311yesmaximal, characteristic, complemented
14C22×C4×3−1+2C2211yescharacteristic
15C22×C4×3−1+2C2212yes—
16C2×#611⋊C3C2211yescharacteristic
17C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
18C2×C4×AGL(1,ℤ/9)C2212yes—
19C2×3−1+2⋊Q8C2211yescharacteristic
20C2×3−1+2⋊Q8C2212yes—
21C4⋊C4×3−1+2C2211yescharacteristic
22C4⋊C4×3−1+2C2211yescharacteristic
23C4⋊C4×3−1+2C2212yes—
24#3676⋊C3C2211yescharacteristic
25#3676⋊C3C2212yes—
26(C2×AGL(1,ℤ/9))⋊C4C2211yescharacteristic
27(C2×AGL(1,ℤ/9))⋊C4C2212yes—
28C4⋊(3−1+2⋊C4)C2211yescharacteristic
29C4⋊(3−1+2⋊C4)C2212yes—
30(3−1+2⋊C4)⋊C4C2212yes—
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—
37C2×C32×C4⋊C4D311yescharacteristic
38C3×C4×#67—31no—
39C3×C4×#67—32no—
40C3×#924—32no—
41C3×#924—31no—
42C3×#897—31no—
43C3×#896—32no—
44C3×D3×C4⋊C4—31no—
45C3×#889—32no—
46C3×#888—31no—
47C3×#887—31no—
48C2×C9×C4⋊C4—21no—
49C2×C9×C4⋊C4C2×C311yescharacteristic, complemented
50C4×#611C2×C311yescharacteristic
51C4×#611C2×C312yes—
52#63486C2×C312yescomplemented
53#63486C2×C311yescharacteristic, complemented
54#63459C2×C311yescharacteristic, complemented
55#63458C2×C312yescomplemented
56C4⋊C4×D9C2×C311yescharacteristic, complemented
57#63451C2×C312yescomplemented
58#63450C2×C311yescharacteristic, complemented
59#63449C2×C311yescharacteristic, complemented
60C23×3−1+2C2311yescharacteristic
61C22×AGL(1,ℤ/9)C2311yescharacteristic
62C2×C4×3−1+2—22no—
63C2×C4×3−1+2—42no—
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⋊C4C2311yescharacteristic
70C2×3−1+2⋊C4C2312yes—
71C2×3−1+2⋊C4C2312yes—
72C2×3−1+2⋊C4C2311yescharacteristic
73C2×3−1+2⋊C4C2311yescharacteristic
74C4×AGL(1,ℤ/9)—22no—
75C4×AGL(1,ℤ/9)—42no—
763−1+2⋊Q8—22no—
773−1+2⋊Q8—42no—
78C3×#554—91no—
79#5511—31no—
80C22×C32×C4C2×D311yescharacteristic
81C22×C32×C4C2×D312yes—
82C2×C3×#67—31no—
83C2×C3×C4×D3—31no—
84C2×C3×C4×D3—32no—
85C2×C3×Dic6—31no—
86C2×C3×Dic6—32no—
87C32×C4⋊C4C2×D311yescharacteristic
88C32×C4⋊C4C2×D311yescharacteristic
89C32×C4⋊C4C2×D312yes—
90C3×#217—32no—
91C3×#217—31no—
92C3×(C2×D3)⋊C4—31no—
93C3×(C2×D3)⋊C4—32no—
94C3×C4⋊Dic3—31no—
95C3×C4⋊Dic3—32no—
96C3×Dic3⋊C4—32no—
97C3×Dic3⋊C4—32no—
98C3×Dic3⋊C4—31no—
99C3×Dic3⋊C4—32no—
100C3×Dic3⋊C4—32no—
101C3×C4×Dic3—31no—
102C3×C4×Dic3—32no—
103C22×C4×C9—21no—
104C22×C4×C9—22no—
105C22×C4×C9C22×C311yescharacteristic
106C22×C4×C9C22×C312yes—
107C2×#611C22×C311yescharacteristic
108C2×C4×D9C22×C311yescharacteristic
109C2×C4×D9C22×C312yes—
110C2×Dic18C22×C311yescharacteristic
111C2×Dic18C22×C312yes—
112C9×C4⋊C4—21no—
113C9×C4⋊C4—21no—
114C9×C4⋊C4—22no—
115C9×C4⋊C4C22×C311yescharacteristic
116C9×C4⋊C4C22×C311yescharacteristic
117C9×C4⋊C4C22×C312yes—
118#3676C22×C312yes—
119#3676C22×C311yescharacteristic
120(C2×D9)⋊C4C22×C311yescharacteristic
121(C2×D9)⋊C4C22×C312yes—
122C4⋊Dic9C22×C311yescharacteristic
123C4⋊Dic9C22×C312yes—
124Dic9⋊C4C22×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—22no—
1433−1+2⋊C4—21no—
1443−1+2⋊C4—21no—
1453−1+2⋊C4—22no—
1463−1+2⋊C4C2×Q812yes—
147C3×C4⋊Q8—91no—
148C3×#125—92no—
149C3×C22⋊Q8—92no—
150C3×C22⋊Q8—91no—
151C3×C22⋊Q8—92no—
152C3×C22⋊Q8—91no—
153C3×C4×Q8—91no—
154C3×C4×D4—92no—
155C3×C4×D4—91no—
156C2×C3×C4⋊C4—31no—
157C2×C3×C4⋊C4—91no—
158C2×C3×C4⋊C4C3×D311yescharacteristic
159C4×#67—31no—
160C4×#67—32no—
161#924—32no—
162#924—31no—
163#897—31no—
164#896—32no—
165D3×C4⋊C4—31no—
166#889—32no—
167#888—31no—
168#887—31no—
169C23×C32C22×D311yescharacteristic
170C22×C3×D3—31no—
171C2×C32×C4—22no—
172C2×C32×C4—42no—
173C2×C32×C4C22×D312yes—
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C22×D312yes—
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—
183C3×C4×D3—62no—
184C3×C4×D3—122no—
185C3×Dic6—62no—
186C3×Dic6—122no—
187C23×C9—21no—
188C23×C9C23×C311yescharacteristic
189C22×D9C23×C311yescharacteristic
190C2×C4×C9—24no—
191C2×C4×C9—22no—
192C2×C4×C9—21no—
193C2×C4×C9—21no—
194C2×C4×C9—22no—
195C2×C4×C9—22no—
196C2×C4×C9—42no—
197C2×C4×C9—44no—
198C2×C4×C9C23×C312yes—
199C2×C4×C9C23×C311yescharacteristic
200C2×C4×C9C23×C312yes—
201C2×C4×C9C23×C311yescharacteristic
202#611C3×Q814yescomplemented
203C2×Dic9C23×C312yes—
204C2×Dic9C23×C311yescharacteristic
205C2×Dic9C23×C311yescharacteristic
206C2×Dic9C23×C312yes—
207C2×Dic9C23×C311yescharacteristic
208C4×D9—22no—
209C4×D9—42no—
210Dic18—22no—
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—91no—
219C2×C3×Q8—92no—
220C2×C3×D4—91no—
221C22×C3×C4—31no—
222C22×C3×C4—32no—
223C22×C3×C4—91no—
224C22×C3×C4—92no—
225C22×C3×C4C2×C3×D311yescharacteristic
226C22×C3×C4C2×C3×D312yes—
227C2×#67—31no—
228C2×C4×D3—31no—
229C2×C4×D3—32no—
230C2×Dic6—31no—
231C2×Dic6—32no—
232C3×C4⋊C4—31no—
233C3×C4⋊C4—32no—
234C3×C4⋊C4—31no—
235C3×C4⋊C4—92no—
236C3×C4⋊C4—91no—
237C3×C4⋊C4—92no—
238C3×C4⋊C4—92no—
239C3×C4⋊C4—92no—
240C3×C4⋊C4—91no—
241C3×C4⋊C4—92no—
242C3×C4⋊C4C2×C3×D311yescharacteristic
243C3×C4⋊C4C2×C3×D311yescharacteristic
244C3×C4⋊C4C2×C3×D312yes—
245C3×C22⋊C4—91no—
246C3×C22⋊C4—92no—
247C3×C22⋊C4—92no—
248C3×C22⋊C4—91no—
249C3×C42—92no—
250C3×C42—91no—
251#217—32no—
252#217—31no—
253(C2×D3)⋊C4—31no—
254(C2×D3)⋊C4—32no—
255C4⋊Dic3—31no—
256C4⋊Dic3—32no—
257Dic3⋊C4—32no—
258Dic3⋊C4—32no—
259Dic3⋊C4—31no—
260Dic3⋊C4—32no—
261Dic3⋊C4—32no—
262C4×Dic3—31no—
263C4×Dic3—32no—
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#23512yes—
275C3×Dic3—32no—
276C3×Dic3—62no—
277C3×Dic3—61no—
278C3×Dic3—61no—
279C3×Dic3—62no—
280C22×C9—22no—
281C22×C9—22no—
282C22×C9—21no—
283C22×C9—21no—
284C22×C9—21no—
285C22×C9—22no—
286C22×C9C2×C3×Q812yes—
287C22×C9C24×C311yescharacteristic
288C2×D9—21no—
289C2×D9—21no—
290C2×D9C2×C3×Q812yes—
291C4×C9—22no—
292C4×C9—21no—
293C4×C9—22no—
294C4×C9—22no—
295C4×C9—22no—
296C4×C9—42no—
297C4×C9—42no—
298C4×C9C3×#4112yes—
299Dic9—22no—
300Dic9—21no—
301Dic9—21no—
302Dic9—22no—
303Dic9C2×C3×Q812yes—
304C4⋊Q8—91no—
305#125—92no—
306C22⋊Q8—92no—
307C22⋊Q8—91no—
308C22⋊Q8—92no—
309C22⋊Q8—91no—
310C4×Q8—91no—
311C4×D4—92no—
312C4×D4—91no—
313C2×C4⋊C4—91no—
314C2×C4⋊C4AGL(1,ℤ/9)11yescharacteristic, complemented
3153−1+2#55411yescharacteristic, complemented, Sylow
316C23×C3—31no—
317C23×C3—91no—
318C23×C3C22×C3×D311yescharacteristic
319C22×D3—31no—
320C3×Q8—182no—
321C3×Q8—362no—
322C3×D4—94no—
323C2×C3×C4—22no—
324C2×C3×C4—31no—
325C2×C3×C4—32no—
326C2×C3×C4—31no—
327C2×C3×C4—32no—
328C2×C3×C4—42no—
329C2×C3×C4—62no—
330C2×C3×C4—91no—
331C2×C3×C4—92no—
332C2×C3×C4—91no—
333C2×C3×C4—91no—
334C2×C3×C4—92no—
335C2×C3×C4—122no—
336C2×C3×C4—182no—
337C2×C3×C4—362no—
338C2×C3×C4C22×C3×D312yes—
339C2×C3×C4C22×C3×D311yescharacteristic
340C2×C3×C4C22×C3×D312yes—
341C2×C3×C4C22×C3×D311yescharacteristic
342#67—34no—
343C2×Dic3—31no—
344C2×Dic3—32no—
345C2×Dic3—31no—
346C2×Dic3—31no—
347C2×Dic3—32no—
348C4×D3—62no—
349C4×D3—122no—
350Dic6—62no—
351Dic6—122no—
352C2×C32—22no—
353C2×C32C2×#23511yescharacteristic
354C2×C32#100411yescharacteristic
355C2×C32C2×D3×Q811yescharacteristic
356C3×D3—62no—
357C2×C9—24no—
358C2×C9—22no—
359C2×C9—21no—
360C2×C9—21no—
361C2×C9—21no—
362C2×C9C22×C3×Q811yescharacteristic
363C2×C9C2×C3×#4111yescharacteristic
364C2×C9C3×H(5,𝔽2)11yescharacteristic
365D9—22no—
366C2×Q8—91no—
367C2×Q8—92no—
368C2×D4—91no—
369C22×C4—91no—
370C22×C4—92no—
371C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
372C22×C4C2×AGL(1,ℤ/9)12yes—
373C4⋊C4—92no—
374C4⋊C4—91no—
375C4⋊C4—91no—
376C4⋊C4—92no—
377C4⋊C4—92no—
378C4⋊C4—92no—
379C4⋊C4—92no—
380C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
381C4⋊C4C2×AGL(1,ℤ/9)12yes—
382C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
383C22⋊C4—92no—
384C22⋊C4—91no—
385C22⋊C4—92no—
386C22⋊C4—91no—
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×#23512yes—
416Dic3—32no—
417Dic3—62no—
418Dic3—61no—
419Dic3—61no—
420Dic3—62no—
421C32#551111yescharacteristic
422C9—21no—
423C9C3×#55411yescharacteristic, complemented
424C23—91no—
425C23C22×AGL(1,ℤ/9)11yescharacteristic
426Q8—182no—
427Q8—362no—
428D4—94no—
429C2×C4—22no—
430C2×C4—42no—
431C2×C4—91no—
432C2×C4—92no—
433C2×C4—91no—
434C2×C4—91no—
435C2×C4—92no—
436C2×C4—182no—
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×#23511yescharacteristic
449C2×C3C3×#100411yescharacteristic
450C2×C3C2×C3×D3×Q811yescharacteristic
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#3697⋊C312yes—
468C3—31no—
469C3C3×#551111yescharacteristic
470C2—22no—
471C2—182no—
472C2C2×#3697⋊C311yescharacteristic
473C2#63710⋊C311yescharacteristic
474C2C2×Q8×AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingC2×C4⋊C4×3−1+2C2
prime radicalC2×#611⋊C3C22
hypercenterC2×C4⋊C4AGL(1,ℤ/9)
hypocenterC9C3×#554
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×3−1+2C2semidirect
4. #63486⋊C3C2semidirect
5. #63486⋊C3C2semidirect
6. #63459⋊C3C2semidirect
7. #63458⋊C3C2semidirect
8. C4⋊C4×AGL(1,ℤ/9)C2semidirect
9. #63451⋊C3C2semidirect
10. #63450⋊C3C2semidirect
11. #63449⋊C3C2semidirect
13. #10589613C3semidirect
49. C2×C9×C4⋊C4C2×C3semidirect
52. #63486C2×C3semidirect
53. #63486C2×C3semidirect
54. #63459C2×C3semidirect
55. #63458C2×C3semidirect
56. C4⋊C4×D9C2×C3semidirect
57. #63451C2×C3semidirect
58. #63450C2×C3semidirect
59. #63449C2×C3semidirect
68. #611⋊C3Q8semidirect
202. #611C3×Q8semidirect
314. C2×C4⋊C4AGL(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
C2×C4⋊C4×3−1+22
C4×#611⋊C32
#63486⋊C32
#63459⋊C32
#63458⋊C32
C4⋊C4×AGL(1,ℤ/9)2
#63451⋊C32
#63450⋊C32
#63449⋊C32
C3×#55113
#105896133