Finite Group Databasebeta

← C2×#63501⋊C3 #13153620

Subgroups — C2×#63501⋊C3 #13153620

Order 1728.

1396 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×3−1+2C211yesmaximal, characteristic, complemented
2C22×#611⋊C3C211yesmaximal, characteristic, complemented
3C22×#611⋊C3C211yesmaximal, characteristic, complemented
4C23×3−1+2⋊C4C211yesmaximal, characteristic, complemented
5C2×#3676⋊C3C211yesmaximal, characteristic, complemented
6#63501⋊C3C218yesmaximal, direct product factor
7C2×(C2×AGL(1,ℤ/9))⋊C4C211yesmaximal, characteristic, complemented
8C2×(3−1+2⋊C4)⋊C4C211yesmaximal, characteristic, complemented
9C2×C3×#939—31nomaximal
10C2×#63501C311yesmaximal, characteristic, complemented
11C24×3−1+2C2211yescharacteristic
12C24×3−1+2C2211yescharacteristic
13C23×AGL(1,ℤ/9)C2211yescharacteristic
14C2×D4×3−1+2—24no—
15C2×D4×3−1+2C2214yescomplemented
16C22×C4×3−1+2C2211yescharacteristic
17C2×#611⋊C3—24no—
18C2×#611⋊C3—24no—
19C2×#611⋊C3C2214yescomplemented
20C2×#611⋊C3C2214yescomplemented
21C22×3−1+2⋊C4—24no—
22C22×3−1+2⋊C4C2211yescharacteristic
23C22×3−1+2⋊C4C2211yescharacteristic
24C22×3−1+2⋊C4C2211yescharacteristic
25C22×3−1+2⋊C4C2214yescomplemented
26#3676⋊C3C2214yescomplemented
27(C2×AGL(1,ℤ/9))⋊C4C2214yescomplemented
28(3−1+2⋊C4)⋊C4C2214yescomplemented
29C22×C32×D4D311yescharacteristic
30C22×C3×#67—31no—
31C22×C3×#67—31no—
32C23×C3×Dic3—31no—
33C2×C3×#217—31no—
34C3×#939—38no—
35C2×C3×(C2×D3)⋊C4—31no—
36C2×C3×Dic3⋊C4—31no—
37C22×D4×C9—21no—
38C22×D4×C9C2×C311yescharacteristic, complemented
39C22×#611C2×C311yescharacteristic, complemented
40C22×#611C2×C311yescharacteristic, complemented
41C23×Dic9C2×C311yescharacteristic, complemented
42C2×#3676C2×C311yescharacteristic, complemented
43#63501C2×C318yescomplemented
44C2×(C2×D9)⋊C4C2×C311yescharacteristic, complemented
45C2×Dic9⋊C4C2×C311yescharacteristic, complemented
46C23×3−1+2—22no—
47C23×3−1+2—22no—
48C23×3−1+2—22no—
49C23×3−1+2—22no—
50C23×3−1+2—22no—
51C23×3−1+2D414yescomplemented
52C23×3−1+2C2312yes—
53C23×3−1+2C2312yes—
54C23×3−1+2C2311yescharacteristic
55C22×AGL(1,ℤ/9)—22no—
56C22×AGL(1,ℤ/9)—22no—
57C22×AGL(1,ℤ/9)—22no—
58C22×AGL(1,ℤ/9)C2312yes—
59D4×3−1+2—28no—
60C2×C4×3−1+2—22no—
61C2×C4×3−1+2C2312yes—
62#611⋊C3—28no—
63#611⋊C3—28no—
64C2×3−1+2⋊C4—22no—
65C2×3−1+2⋊C4—22no—
66C2×3−1+2⋊C4—28no—
67C2×3−1+2⋊C4D414yescomplemented
68C2×3−1+2⋊C4C2312yes—
69C2×3−1+2⋊C4C2312yes—
70C2×3−1+2⋊C4C2312yes—
71C2×C3×#121—91no—
72C2×#939—31no—
73C24×C32C2×D311yescharacteristic
74C24×C32C2×D311yescharacteristic
75C23×C3×D3—31no—
76C2×C32×D4—24no—
77C2×C32×D4C2×D314yes—
78C22×C32×C4C2×D311yescharacteristic
79C2×C3×#67—34no—
80C2×C3×#67—34no—
81C2×C3×#67—64no—
82C2×C3×#67—64no—
83C22×C3×Dic3—31no—
84C22×C3×Dic3—31no—
85C22×C3×Dic3—31no—
86C22×C3×Dic3—34no—
87C22×C3×Dic3—64no—
88C24×C9—21no—
89C24×C9—21no—
90C24×C9C22×C311yescharacteristic
91C24×C9C22×C311yescharacteristic
92C23×D9C22×C311yescharacteristic
93C3×#217—34no—
94C3×(C2×D3)⋊C4—34no—
95C3×Dic3⋊C4—34no—
96C2×D4×C9—28no—
97C2×D4×C9—24no—
98C2×D4×C9—24no—
99C2×D4×C9C22×C314yescomplemented
100C22×C4×C9—21no—
101C22×C4×C9C22×C311yescharacteristic
102C2×#611—24no—
103C2×#611—24no—
104C2×#611C22×C314yescomplemented
105C2×#611C22×C314yescomplemented
106C22×Dic9—24no—
107C22×Dic9C22×C311yescharacteristic
108C22×Dic9C22×C311yescharacteristic
109C22×Dic9C22×C311yescharacteristic
110C22×Dic9C22×C314yescomplemented
111#3676C22×C314yescomplemented
112(C2×D9)⋊C4C22×C314yescomplemented
113Dic9⋊C4C22×C314yescomplemented
114C22×3−1+2—28no—
115C22×3−1+2—22no—
116C22×3−1+2—22no—
117C22×3−1+2—22no—
118C22×3−1+2—22no—
119C22×3−1+2—22no—
120C22×3−1+2—44no—
121C22×3−1+2C2×D412yes—
122C22×3−1+2C2×D412yes—
123C22×3−1+2C2×D414yescomplemented
124C22×3−1+2#4112yes—
125C22×3−1+2C2411yescharacteristic
126C2×AGL(1,ℤ/9)—22no—
127C2×AGL(1,ℤ/9)—22no—
128C2×AGL(1,ℤ/9)—22no—
129C2×AGL(1,ℤ/9)—44no—
130C4×3−1+2—22no—
1313−1+2⋊C4—22no—
1323−1+2⋊C4—22no—
1333−1+2⋊C4C2×D414yescomplemented
134C22×C3×D4—31no—
135C22×C3×D4—91no—
136C22×C3×D4—91no—
137C22×C3×D4C3×D311yescharacteristic
138C23×C3×C4—91no—
139C22×#67—31no—
140C22×#67—31no—
141C23×Dic3—31no—
142C3×#121—98no—
143C2×C3×C4⋊C4—91no—
144C2×C3×C22⋊C4—91no—
145C2×C3×C22⋊C4—91no—
146C2×#217—31no—
147#939—38no—
148C2×(C2×D3)⋊C4—31no—
149C2×Dic3⋊C4—31no—
150C23×C32—22no—
151C23×C32—22no—
152C23×C32—22no—
153C23×C32—22no—
154C23×C32—22no—
155C23×C32#6714yes—
156C23×C32C22×D312yes—
157C23×C32C22×D312yes—
158C23×C32C22×D311yescharacteristic
159C22×C3×D3—32no—
160C22×C3×D3—62no—
161C22×C3×D3—62no—
162C22×C3×D3—62no—
163C32×D4—28no—
164C2×C32×C4—22no—
165C2×C32×C4C22×D312yes—
166C3×#67—68no—
167C3×#67—68no—
168C2×C3×Dic3—34no—
169C2×C3×Dic3—32no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—68no—
173C2×C3×Dic3—62no—
174C2×C3×Dic3—62no—
175C23×C9—22no—
176C23×C9—22no—
177C23×C9—24no—
178C23×C9—22no—
179C23×C9—22no—
180C23×C9—22no—
181C23×C9—24no—
182C23×C9—21no—
183C23×C9—22no—
184C23×C9—22no—
185C23×C9—42no—
186C23×C9—42no—
187C23×C9—42no—
188C23×C9—42no—
189C23×C9C3×D414yescomplemented
190C23×C9C23×C312yes—
191C23×C9C23×C312yes—
192C23×C9C23×C311yescharacteristic
193C22×D9—22no—
194C22×D9—22no—
195C22×D9—22no—
196C22×D9C23×C312yes—
197D4×C9—216no—
198D4×C9—28no—
199C2×C4×C9—24no—
200C2×C4×C9—22no—
201C2×C4×C9—22no—
202C2×C4×C9C23×C312yes—
203#611—28no—
204#611—28no—
205C2×Dic9—22no—
206C2×Dic9—22no—
207C2×Dic9—28no—
208C2×Dic9C3×D414yescomplemented
209C2×Dic9C23×C312yes—
210C2×Dic9C23×C312yes—
211C2×Dic9C23×C312yes—
212C2×#121—91noSylow
213C2×3−1+2—24no—
214C2×3−1+2—42no—
215C2×3−1+2#12114yescomplemented
216C2×3−1+2C22×D411yescharacteristic
217C2×3−1+2C22×D411yescharacteristic
218C2×3−1+2C2×#4111yescharacteristic
219AGL(1,ℤ/9)—42no—
220C24×C3—31no—
221C24×C3—31no—
222C24×C3—91no—
223C24×C3C2×C3×D311yescharacteristic
224C24×C3C2×C3×D311yescharacteristic
225C23×D3—31no—
226C2×C3×D4—24no—
227C2×C3×D4—34no—
228C2×C3×D4—64no—
229C2×C3×D4—94no—
230C2×C3×D4—94no—
231C2×C3×D4—184no—
232C2×C3×D4—184no—
233C2×C3×D4C2×C3×D314yes—
234C22×C3×C4—31no—
235C22×C3×C4—91no—
236C22×C3×C4—91no—
237C22×C3×C4—91no—
238C22×C3×C4—94no—
239C22×C3×C4—184no—
240C22×C3×C4C2×C3×D311yescharacteristic
241C2×#67—34no—
242C2×#67—34no—
243C2×#67—64no—
244C2×#67—64no—
245C22×Dic3—31no—
246C22×Dic3—31no—
247C22×Dic3—31no—
248C22×Dic3—34no—
249C22×Dic3—64no—
250C3×C4⋊C4—94no—
251C3×C22⋊C4—94no—
252C3×C22⋊C4—94no—
253#217—34no—
254(C2×D3)⋊C4—34no—
255Dic3⋊C4—34no—
256C22×C32—28no—
257C22×C32—22no—
258C22×C32—22no—
259C22×C32—22no—
260C22×C32—22no—
261C22×C32—22no—
262C22×C32—44no—
263C22×C32D3×D412yes—
264C22×C32#23712yes—
265C22×C32C2×#6714yes—
266C22×C32C2×#6712yes—
267C22×C32C23×D311yescharacteristic
268C2×C3×D3—62no—
269C2×C3×D3—62no—
270C2×C3×D3—62no—
271C2×C3×D3—124no—
272C32×C4—22no—
273C3×Dic3—34no—
274C3×Dic3—62no—
275C3×Dic3—62no—
276C22×C9—28no—
277C22×C9—22no—
278C22×C9—22no—
279C22×C9—24no—
280C22×C9—22no—
281C22×C9—22no—
282C22×C9—22no—
283C22×C9—21no—
284C22×C9—22no—
285C22×C9—22no—
286C22×C9—22no—
287C22×C9—24no—
288C22×C9—48no—
289C22×C9—42no—
290C22×C9—42no—
291C22×C9—42no—
292C22×C9—42no—
293C22×C9—48no—
294C22×C9—44no—
295C22×C9C2×C3×D412yes—
296C22×C9C2×C3×D412yes—
297C22×C9C2×C3×D414yescomplemented
298C22×C9C3×#4112yes—
299C22×C9C24×C311yescharacteristic
300C2×D9—22no—
301C2×D9—22no—
302C2×D9—22no—
303C2×D9—44no—
304C4×C9—24no—
305C4×C9—22no—
306Dic9—22no—
307Dic9—22no—
308Dic9C2×C3×D414yescomplemented
309C22×D4—91no—
310C22×D4—91no—
311C22×D4AGL(1,ℤ/9)11yescharacteristic, complemented
312C23×C4—91no—
313#121—98no—
314C2×C4⋊C4—91no—
315C2×C22⋊C4—91no—
316C2×C22⋊C4—91no—
3173−1+2C2×#12111yescharacteristic, complemented, Sylow
318C23×C3—22no—
319C23×C3—22no—
320C23×C3—22no—
321C23×C3—22no—
322C23×C3—22no—
323C23×C3—34no—
324C23×C3—32no—
325C23×C3—32no—
326C23×C3—31no—
327C23×C3—62no—
328C23×C3—62no—
329C23×C3—62no—
330C23×C3—62no—
331C23×C3—62no—
332C23×C3—92no—
333C23×C3—182no—
334C23×C3—182no—
335C23×C3—182no—
336C23×C3C3×#6714yes—
337C23×C3C22×C3×D312yes—
338C23×C3C22×C3×D312yes—
339C23×C3C22×C3×D311yescharacteristic
340C22×D3—32no—
341C22×D3—62no—
342C22×D3—62no—
343C22×D3—62no—
344C3×D4—28no—
345C3×D4—68no—
346C3×D4—188no—
347C3×D4—188no—
348C2×C3×C4—22no—
349C2×C3×C4—32no—
350C2×C3×C4—62no—
351C2×C3×C4—92no—
352C2×C3×C4—94no—
353C2×C3×C4—92no—
354C2×C3×C4—92no—
355C2×C3×C4—188no—
356C2×C3×C4—182no—
357C2×C3×C4—182no—
358C2×C3×C4C22×C3×D312yes—
359#67—68no—
360#67—68no—
361C2×Dic3—32no—
362C2×Dic3—34no—
363C2×Dic3—32no—
364C2×Dic3—32no—
365C2×Dic3—68no—
366C2×Dic3—62no—
367C2×Dic3—62no—
368C2×C32—24no—
369C2×C32—42no—
370C2×C32#93914yes—
371C2×C32C2×D3×D411yescharacteristic
372C2×C32C2×#23711yescharacteristic
373C2×C32C22×#6711yescharacteristic
374C3×D3—122no—
375C2×C9—24no—
376C2×C9—24no—
377C2×C9—21no—
378C2×C9—21no—
379C2×C9—21no—
380C2×C9—44no—
381C2×C9—44no—
382C2×C9—42no—
383C2×C9C3×#12114yescomplemented
384C2×C9C22×C3×D411yescharacteristic
385C2×C9C22×C3×D411yescharacteristic
386C2×C9C2×C3×#4111yescharacteristic
387D9—42no—
388C24—91no—
389C24C2×AGL(1,ℤ/9)11yescharacteristic
390C24C2×AGL(1,ℤ/9)11yescharacteristic
391C2×D4—24no—
392C2×D4—94no—
393C2×D4—94no—
394C2×D4—184no—
395C2×D4—184no—
396C2×D4C2×AGL(1,ℤ/9)14yescomplemented
397C22×C4—91no—
398C22×C4—94no—
399C22×C4—91no—
400C22×C4—91no—
401C22×C4—184no—
402C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
403C4⋊C4—94no—
404C22⋊C4—94no—
405C22⋊C4—94no—
406C22×C3—28no—
407C22×C3—22no—
408C22×C3—22no—
409C22×C3—22no—
410C22×C3—22no—
411C22×C3—22no—
412C22×C3—31no—
413C22×C3—32no—
414C22×C3—32no—
415C22×C3—32no—
416C22×C3—34no—
417C22×C3—44no—
418C22×C3—62no—
419C22×C3—62no—
420C22×C3—68no—
421C22×C3—62no—
422C22×C3—62no—
423C22×C3—62no—
424C22×C3—124no—
425C22×C3—182no—
426C22×C3—182no—
427C22×C3—182no—
428C22×C3—364no—
429C22×C3C3×D3×D412yes—
430C22×C3C3×#23712yes—
431C22×C3C2×C3×#6712yes—
432C22×C3C2×C3×#6714yes—
433C22×C3C23×C3×D311yescharacteristic
434C2×D3—62no—
435C2×D3—62no—
436C2×D3—62no—
437C2×D3—124no—
438C3×C4—22no—
439C3×C4—62no—
440C3×C4—94no—
441C3×C4—182no—
442C3×C4—182no—
443Dic3—34no—
444Dic3—62no—
445Dic3—62no—
446C32C2×#93911yescharacteristic
447C9—21no—
448C9C2×C3×#12111yescharacteristic, complemented
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—92no—
455C23—182no—
456C23—182no—
457C23—182no—
458C23#611⋊C314yescomplemented
459C23C22×AGL(1,ℤ/9)12yes—
460C23C22×AGL(1,ℤ/9)12yes—
461C23C22×AGL(1,ℤ/9)11yescharacteristic
462D4—28no—
463D4—188no—
464D4—188no—
465C2×C4—22no—
466C2×C4—92no—
467C2×C4—94no—
468C2×C4—92no—
469C2×C4—92no—
470C2×C4—188no—
471C2×C4—182no—
472C2×C4—182no—
473C2×C4C22×AGL(1,ℤ/9)12yes—
474C2×C3—24no—
475C2×C3—31no—
476C2×C3—31no—
477C2×C3—31no—
478C2×C3—34no—
479C2×C3—42no—
480C2×C3—64no—
481C2×C3—122no—
482C2×C3—362no—
483C2×C3C3×#93914yes—
484C2×C3C2×C3×D3×D411yescharacteristic
485C2×C3C2×C3×#23711yescharacteristic
486C2×C3C22×C3×#6711yescharacteristic
487D3—122no—
488C22—22no—
489C22—22no—
490C22—28no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—44no—
495C22—182no—
496C22—182no—
497C22—182no—
498C22—364no—
499C22D4×AGL(1,ℤ/9)12yes—
500C22#3699⋊C312yes—
501C22C2×#611⋊C312yes—
502C22C2×#611⋊C314yescomplemented
503C22C23×AGL(1,ℤ/9)11yescharacteristic
504C4—22no—
505C4—94no—
506C4—182no—
507C4—182no—
508C3—31no—
509C3C2×C3×#93911yescharacteristic
510C2—24no—
511C2—42no—
512C2—362no—
513C2#63501⋊C314yesdirect product factor
514C2C2×D4×AGL(1,ℤ/9)11yescharacteristic
515C2C2×#3699⋊C311yescharacteristic
516C2C22×#611⋊C311yescharacteristic
subgroupG/H
centerC23C22×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC22×D4×3−1+2C2
socleC23×C3C22×C3×D3
hypercenterC22×D4AGL(1,ℤ/9)
hypocenterC9C2×C3×#121
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C22×D4×3−1+2C2semidirect
2. C22×#611⋊C3C2semidirect
3. C22×#611⋊C3C2semidirect
4. C23×3−1+2⋊C4C2semidirect
5. C2×#3676⋊C3C2semidirect
6. #63501⋊C3C2direct (N × H)
7. C2×(C2×AGL(1,ℤ/9))⋊C4C2semidirect
8. C2×(3−1+2⋊C4)⋊C4C2semidirect
10. C2×#63501C3semidirect
15. C2×D4×3−1+2C22semidirect
19. C2×#611⋊C3C22semidirect
20. C2×#611⋊C3C22semidirect
25. C22×3−1+2⋊C4C22semidirect
26. #3676⋊C3C22semidirect
27. (C2×AGL(1,ℤ/9))⋊C4C22semidirect
28. (3−1+2⋊C4)⋊C4C22semidirect
38. C22×D4×C9C2×C3semidirect
39. C22×#611C2×C3semidirect
40. C22×#611C2×C3semidirect
41. C23×Dic9C2×C3semidirect
42. C2×#3676C2×C3semidirect
43. #63501C2×C3semidirect
44. C2×(C2×D9)⋊C4C2×C3semidirect
45. C2×Dic9⋊C4C2×C3semidirect
51. C23×3−1+2D4semidirect
67. C2×3−1+2⋊C4D4semidirect
99. C2×D4×C9C22×C3semidirect
104. C2×#611C22×C3semidirect
105. C2×#611C22×C3semidirect
110. C22×Dic9C22×C3semidirect
111. #3676C22×C3semidirect
112. (C2×D9)⋊C4C22×C3semidirect
113. Dic9⋊C4C22×C3semidirect
123. C22×3−1+2C2×D4semidirect
133. 3−1+2⋊C4C2×D4semidirect
189. C23×C9C3×D4semidirect
208. C2×Dic9C3×D4semidirect
215. C2×3−1+2#121semidirect
297. C22×C9C2×C3×D4semidirect
308. Dic9C2×C3×D4semidirect
311. C22×D4AGL(1,ℤ/9)semidirect
317. 3−1+2C2×#121semidirect
383. C2×C9C3×#121semidirect
396. C2×D4C2×AGL(1,ℤ/9)semidirect
448. C9C2×C3×#121semidirect
458. C23#611⋊C3semidirect
502. C22C2×#611⋊C3semidirect
513. C2#63501⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×3−1+22
C22×#611⋊C32
C23×3−1+2⋊C42
C2×#3676⋊C32
#63501⋊C32
C2×(C2×AGL(1,ℤ/9))⋊C42
C2×(3−1+2⋊C4)⋊C42
C2×C3×#9393
C2×#635013