Finite Group Databasebeta

← C2×#63446⋊C3 #13152381

Subgroups — C2×#63446⋊C3 #13152381

Order 1728.

1344 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#611⋊C3C211yesmaximal, characteristic, complemented
2C22×#611⋊C3C211yesmaximal, characteristic, complemented
3C22×C4⋊AGL(1,ℤ/9)C211yesmaximal, characteristic, complemented
4C22×C4×AGL(1,ℤ/9)C211yesmaximal, characteristic, complemented
5C2×C22⋊C4×3−1+2C211yesmaximal, characteristic, complemented
6C2×(C2×AGL(1,ℤ/9))⋊C4C211yesmaximal, characteristic, complemented
7C2×(3−1+2⋊C4)⋊C4C211yesmaximal, characteristic, complemented
8#63446⋊C3C218yesmaximal, direct product factor
9C2×C3×#884—31nomaximal
10C2×#63446C311yesmaximal, characteristic, complemented
11C24×3−1+2C2211yescharacteristic
12C23×AGL(1,ℤ/9)C2211yescharacteristic
13C23×AGL(1,ℤ/9)C2211yescharacteristic
14C22×C4×3−1+2C2211yescharacteristic
15C22×C4×3−1+2C2211yescharacteristic
16C2×#611⋊C3—24no—
17C2×#611⋊C3—24no—
18C2×#611⋊C3C2214yescomplemented
19C2×#611⋊C3C2214yescomplemented
20C22×3−1+2⋊C4C2211yescharacteristic
21C22×3−1+2⋊C4C2211yescharacteristic
22C2×C4⋊AGL(1,ℤ/9)—24no—
23C2×C4⋊AGL(1,ℤ/9)C2214yescomplemented
24C2×C4×AGL(1,ℤ/9)—24no—
25C2×C4×AGL(1,ℤ/9)C2214yescomplemented
26C22⋊C4×3−1+2C2214yescomplemented
27(C2×AGL(1,ℤ/9))⋊C4C2214yescomplemented
28(3−1+2⋊C4)⋊C4C2214yescomplemented
29C22×C3×#67—31no—
30C22×C3×#67—31no—
31C22×C3×D12—31no—
32C22×C3×C4×D3—31no—
33C2×C32×C22⋊C4D311yescharacteristic
34C2×C3×(C2×D3)⋊C4—31no—
35C2×C3×Dic3⋊C4—31no—
36C3×#884—38no—
37C22×#611C2×C311yescharacteristic, complemented
38C22×#611C2×C311yescharacteristic, complemented
39C22×D36C2×C311yescharacteristic, complemented
40C22×C4×D9C2×C311yescharacteristic, complemented
41C2×C9×C22⋊C4—21no—
42C2×C9×C22⋊C4C2×C311yescharacteristic, complemented
43C2×(C2×D9)⋊C4C2×C311yescharacteristic, complemented
44C2×Dic9⋊C4C2×C311yescharacteristic, complemented
45#63446C2×C318yescomplemented
46C23×3−1+2—22no—
47C23×3−1+2—22no—
48C23×3−1+2—22no—
49C23×3−1+2C2312yes—
50C23×3−1+2C2311yescharacteristic
51C22×AGL(1,ℤ/9)—22no—
52C22×AGL(1,ℤ/9)—22no—
53C22×AGL(1,ℤ/9)—22no—
54C22×AGL(1,ℤ/9)—22no—
55C22×AGL(1,ℤ/9)—22no—
56C22×AGL(1,ℤ/9)D414yescomplemented
57C22×AGL(1,ℤ/9)C2312yes—
58C22×AGL(1,ℤ/9)C2312yes—
59C2×C4×3−1+2—22no—
60C2×C4×3−1+2—22no—
61C2×C4×3−1+2C2312yes—
62C2×C4×3−1+2C2312yes—
63#611⋊C3—28no—
64#611⋊C3—28no—
65C2×3−1+2⋊C4—22no—
66C2×3−1+2⋊C4D414yescomplemented
67C2×3−1+2⋊C4C2312yes—
68C2×3−1+2⋊C4C2312yes—
69C4⋊AGL(1,ℤ/9)—28no—
70C4×AGL(1,ℤ/9)—28no—
71C2×C3×#121—91no—
72C2×#884—31no—
73C24×C32C2×D311yescharacteristic
74C23×C3×D3—31no—
75C23×C3×D3—31no—
76C22×C32×C4C2×D311yescharacteristic
77C22×C32×C4C2×D311yescharacteristic
78C2×C3×#67—34no—
79C2×C3×#67—34no—
80C2×C3×#67—64no—
81C2×C3×#67—64no—
82C22×C3×Dic3—31no—
83C22×C3×Dic3—31no—
84C2×C3×D12—34no—
85C2×C3×D12—64no—
86C2×C3×C4×D3—34no—
87C2×C3×C4×D3—64no—
88C24×C9—21no—
89C24×C9C22×C311yescharacteristic
90C23×D9C22×C311yescharacteristic
91C23×D9C22×C311yescharacteristic
92C32×C22⋊C4C2×D314yes—
93C3×(C2×D3)⋊C4—34no—
94C3×Dic3⋊C4—34no—
95C22×C4×C9—21no—
96C22×C4×C9—21no—
97C22×C4×C9C22×C311yescharacteristic
98C22×C4×C9C22×C311yescharacteristic
99C2×#611—24no—
100C2×#611—24no—
101C2×#611C22×C314yescomplemented
102C2×#611C22×C314yescomplemented
103C22×Dic9C22×C311yescharacteristic
104C22×Dic9C22×C311yescharacteristic
105C2×D36—24no—
106C2×D36C22×C314yescomplemented
107C2×C4×D9—24no—
108C2×C4×D9C22×C314yescomplemented
109C9×C22⋊C4—24no—
110C9×C22⋊C4C22×C314yescomplemented
111(C2×D9)⋊C4C22×C314yescomplemented
112Dic9⋊C4C22×C314yescomplemented
113C22×3−1+2—22no—
114C22×3−1+2—22no—
115C22×3−1+2—22no—
116C22×3−1+2—44no—
117C22×3−1+2C2×D412yes—
118C22×3−1+2C2×D412yes—
119C22×3−1+2#4112yes—
120C22×3−1+2C2411yescharacteristic
121C2×AGL(1,ℤ/9)—28no—
122C2×AGL(1,ℤ/9)—22no—
123C2×AGL(1,ℤ/9)—22no—
124C2×AGL(1,ℤ/9)—22no—
125C2×AGL(1,ℤ/9)—22no—
126C2×AGL(1,ℤ/9)—22no—
127C2×AGL(1,ℤ/9)—44no—
128C2×AGL(1,ℤ/9)C2×D414yescomplemented
129C4×3−1+2—22no—
130C4×3−1+2—22no—
1313−1+2⋊C4—22no—
1323−1+2⋊C4C2×D414yescomplemented
133C22×C3×D4—91no—
134C22×C3×D4—91no—
135C22×C3×D4—91no—
136C23×C3×C4—91no—
137C22×#67—31no—
138C22×#67—31no—
139C22×D12—31no—
140C22×C4×D3—31no—
141C3×#121—98no—
142C2×C3×C4⋊C4—91no—
143C2×C3×C22⋊C4—31no—
144C2×C3×C22⋊C4—91no—
145C2×C3×C22⋊C4C3×D311yescharacteristic
146C2×(C2×D3)⋊C4—31no—
147C2×Dic3⋊C4—31no—
148#884—38no—
149C23×C32—22no—
150C23×C32—22no—
151C23×C32—22no—
152C23×C32C22×D312yes—
153C23×C32C22×D311yescharacteristic
154C22×C3×D3—34no—
155C22×C3×D3—32no—
156C22×C3×D3—32no—
157C22×C3×D3—62no—
158C22×C3×D3—62no—
159C22×C3×D3—62no—
160C22×C3×D3—62no—
161C22×C3×D3—62no—
162C2×C32×C4—22no—
163C2×C32×C4—22no—
164C2×C32×C4C22×D312yes—
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—62no—
172C3×D12—68no—
173C3×C4×D3—68no—
174C23×C9—24no—
175C23×C9—22no—
176C23×C9—22no—
177C23×C9—22no—
178C23×C9—21no—
179C23×C9—22no—
180C23×C9—42no—
181C23×C9—42no—
182C23×C9C23×C312yes—
183C23×C9C23×C311yescharacteristic
184C22×D9—22no—
185C22×D9—22no—
186C22×D9—22no—
187C22×D9—22no—
188C22×D9—22no—
189C22×D9C3×D414yescomplemented
190C22×D9C23×C312yes—
191C22×D9C23×C312yes—
192C2×C4×C9—22no—
193C2×C4×C9—22no—
194C2×C4×C9—22no—
195C2×C4×C9—22no—
196C2×C4×C9—42no—
197C2×C4×C9—42no—
198C2×C4×C9C23×C312yes—
199C2×C4×C9C23×C312yes—
200#611—28no—
201#611—28no—
202C2×Dic9—22no—
203C2×Dic9C3×D414yescomplemented
204C2×Dic9C23×C312yes—
205C2×Dic9C23×C312yes—
206D36—28no—
207C4×D9—28no—
208C2×#121—91noSylow
209C2×3−1+2—42no—
210C2×3−1+2#12114yescomplemented
211C2×3−1+2C22×D411yescharacteristic
212C2×3−1+2C22×D411yescharacteristic
213C2×3−1+2C2×#4111yescharacteristic
214AGL(1,ℤ/9)—24no—
215AGL(1,ℤ/9)—42no—
216C24×C3—31no—
217C24×C3—91no—
218C24×C3—91no—
219C24×C3C2×C3×D311yescharacteristic
220C23×D3—31no—
221C23×D3—31no—
222C2×C3×D4—94no—
223C2×C3×D4—94no—
224C2×C3×D4—94no—
225C2×C3×D4—184no—
226C2×C3×D4—184no—
227C2×C3×D4—184no—
228C22×C3×C4—31no—
229C22×C3×C4—31no—
230C22×C3×C4—94no—
231C22×C3×C4—91no—
232C22×C3×C4—91no—
233C22×C3×C4—184no—
234C22×C3×C4C2×C3×D311yescharacteristic
235C22×C3×C4C2×C3×D311yescharacteristic
236C2×#67—34no—
237C2×#67—34no—
238C2×#67—64no—
239C2×#67—64no—
240C22×Dic3—31no—
241C22×Dic3—31no—
242C2×D12—34no—
243C2×D12—64no—
244C2×C4×D3—34no—
245C2×C4×D3—64no—
246C3×C4⋊C4—94no—
247C3×C22⋊C4—34no—
248C3×C22⋊C4—94no—
249C3×C22⋊C4C2×C3×D314yes—
250(C2×D3)⋊C4—34no—
251Dic3⋊C4—34no—
252C22×C32—22no—
253C22×C32—22no—
254C22×C32—22no—
255C22×C32—44no—
256C22×C32#23512yes—
257C22×C32D3×D412yes—
258C22×C32D3×D412yes—
259C22×C32C23×D311yescharacteristic
260C2×C3×D3—34no—
261C2×C3×D3—68no—
262C2×C3×D3—62no—
263C2×C3×D3—62no—
264C2×C3×D3—62no—
265C2×C3×D3—62no—
266C2×C3×D3—62no—
267C2×C3×D3—124no—
268C32×C4—22no—
269C32×C4—22no—
270C3×Dic3—34no—
271C3×Dic3—62no—
272C22×C9—24no—
273C22×C9—22no—
274C22×C9—22no—
275C22×C9—22no—
276C22×C9—21no—
277C22×C9—22no—
278C22×C9—22no—
279C22×C9—22no—
280C22×C9—42no—
281C22×C9—42no—
282C22×C9—48no—
283C22×C9—44no—
284C22×C9C2×C3×D412yes—
285C22×C9C2×C3×D412yes—
286C22×C9C3×#4112yes—
287C22×C9C24×C311yescharacteristic
288C2×D9—28no—
289C2×D9—22no—
290C2×D9—22no—
291C2×D9—22no—
292C2×D9—22no—
293C2×D9—22no—
294C2×D9—44no—
295C2×D9C2×C3×D414yescomplemented
296C4×C9—22no—
297C4×C9—22no—
298C4×C9—42no—
299C4×C9—42no—
300Dic9—22no—
301Dic9C2×C3×D414yescomplemented
302C22×D4—91no—
303C22×D4—91no—
304C22×D4—91no—
305C23×C4—91no—
306#121—98no—
307C2×C4⋊C4—91no—
308C2×C22⋊C4—91no—
309C2×C22⋊C4AGL(1,ℤ/9)11yescharacteristic, complemented
3103−1+2C2×#12111yescharacteristic, complemented, Sylow
311C23×C3—22no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—32no—
315C23×C3—31no—
316C23×C3—62no—
317C23×C3—62no—
318C23×C3—62no—
319C23×C3—92no—
320C23×C3—94no—
321C23×C3—92no—
322C23×C3—182no—
323C23×C3—182no—
324C23×C3—182no—
325C23×C3—182no—
326C23×C3—182no—
327C23×C3C22×C3×D312yes—
328C23×C3C22×C3×D311yescharacteristic
329C22×D3—32no—
330C22×D3—34no—
331C22×D3—32no—
332C22×D3—62no—
333C22×D3—62no—
334C22×D3—62no—
335C22×D3—62no—
336C22×D3—62no—
337C3×D4—188no—
338C3×D4—188no—
339C3×D4—188no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—32no—
343C2×C3×C4—32no—
344C2×C3×C4—62no—
345C2×C3×C4—62no—
346C2×C3×C4—92no—
347C2×C3×C4—94no—
348C2×C3×C4—92no—
349C2×C3×C4—188no—
350C2×C3×C4—182no—
351C2×C3×C4C22×C3×D312yes—
352C2×C3×C4C22×C3×D312yes—
353#67—68no—
354#67—68no—
355C2×Dic3—32no—
356C2×Dic3—34no—
357C2×Dic3—32no—
358C2×Dic3—62no—
359D12—68no—
360C4×D3—68no—
361C2×C32—42no—
362C2×C32#88414yes—
363C2×C32C2×#23511yescharacteristic
364C2×C32C2×D3×D411yescharacteristic
365C2×C32C2×D3×D411yescharacteristic
366C3×D3—64no—
367C3×D3—122no—
368C2×C9—24no—
369C2×C9—21no—
370C2×C9—21no—
371C2×C9—21no—
372C2×C9—44no—
373C2×C9—42no—
374C2×C9C3×#12114yescomplemented
375C2×C9C22×C3×D411yescharacteristic
376C2×C9C22×C3×D411yescharacteristic
377C2×C9C2×C3×#4111yescharacteristic
378D9—24no—
379D9—42no—
380C24—91no—
381C24—91no—
382C24C2×AGL(1,ℤ/9)11yescharacteristic
383C2×D4—94no—
384C2×D4—94no—
385C2×D4—94no—
386C2×D4—184no—
387C2×D4—184no—
388C2×D4—184no—
389C22×C4—91no—
390C22×C4—91no—
391C22×C4—94no—
392C22×C4—184no—
393C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
394C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
395C4⋊C4—94no—
396C22⋊C4—94no—
397C22⋊C4C2×AGL(1,ℤ/9)14yescomplemented
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—31no—
402C22×C3—32no—
403C22×C3—32no—
404C22×C3—32no—
405C22×C3—44no—
406C22×C3—62no—
407C22×C3—62no—
408C22×C3—62no—
409C22×C3—94no—
410C22×C3—124no—
411C22×C3—182no—
412C22×C3—182no—
413C22×C3—188no—
414C22×C3—182no—
415C22×C3—182no—
416C22×C3—182no—
417C22×C3—364no—
418C22×C3C3×#23512yes—
419C22×C3C3×D3×D412yes—
420C22×C3C3×D3×D412yes—
421C22×C3C23×C3×D311yescharacteristic
422C2×D3—34no—
423C2×D3—68no—
424C2×D3—62no—
425C2×D3—62no—
426C2×D3—62no—
427C2×D3—62no—
428C2×D3—62no—
429C2×D3—124no—
430C3×C4—22no—
431C3×C4—22no—
432C3×C4—62no—
433C3×C4—62no—
434C3×C4—94no—
435C3×C4—182no—
436Dic3—34no—
437Dic3—62no—
438C32C2×#88411yescharacteristic
439C9—21no—
440C9C2×C3×#12111yescharacteristic, complemented
441C23—22no—
442C23—22no—
443C23—22no—
444C23—92no—
445C23—94no—
446C23—92no—
447C23—182no—
448C23—182no—
449C23—182no—
450C23—182no—
451C23—182no—
452C23C22×AGL(1,ℤ/9)12yes—
453C23C22×AGL(1,ℤ/9)11yescharacteristic
454D4—188no—
455D4—188no—
456D4—188no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—92no—
460C2×C4—94no—
461C2×C4—92no—
462C2×C4—188no—
463C2×C4—182no—
464C2×C4C22×AGL(1,ℤ/9)12yes—
465C2×C4C22×AGL(1,ℤ/9)12yes—
466C2×C3—31no—
467C2×C3—31no—
468C2×C3—31no—
469C2×C3—34no—
470C2×C3—42no—
471C2×C3—122no—
472C2×C3—184no—
473C2×C3—362no—
474C2×C3C3×#88414yes—
475C2×C3C2×C3×#23511yescharacteristic
476C2×C3C2×C3×D3×D411yescharacteristic
477C2×C3C2×C3×D3×D411yescharacteristic
478D3—64no—
479D3—122no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—44no—
484C22—94no—
485C22—182no—
486C22—182no—
487C22—188no—
488C22—182no—
489C22—182no—
490C22—182no—
491C22—364no—
492C22#3697⋊C312yes—
493C22D4×AGL(1,ℤ/9)12yes—
494C22D4×AGL(1,ℤ/9)12yes—
495C22C23×AGL(1,ℤ/9)11yescharacteristic
496C4—22no—
497C4—22no—
498C4—94no—
499C4—182no—
500C3—31no—
501C3C2×C3×#88411yescharacteristic
502C2—42no—
503C2—184no—
504C2—362no—
505C2#63446⋊C314yesdirect product factor
506C2C2×#3697⋊C311yescharacteristic
507C2C2×D4×AGL(1,ℤ/9)11yescharacteristic
508C2C2×D4×AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
centerC23C22×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC2×C22⋊C4×3−1+2C2
socleC23×C3C22×C3×D3
hypercenterC2×C22⋊C4AGL(1,ℤ/9)
hypocenterC9C2×C3×#121
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#611⋊C3C2semidirect
2. C22×#611⋊C3C2semidirect
3. C22×C4⋊AGL(1,ℤ/9)C2semidirect
4. C22×C4×AGL(1,ℤ/9)C2semidirect
5. C2×C22⋊C4×3−1+2C2semidirect
6. C2×(C2×AGL(1,ℤ/9))⋊C4C2semidirect
7. C2×(3−1+2⋊C4)⋊C4C2semidirect
8. #63446⋊C3C2direct (N × H)
10. C2×#63446C3semidirect
18. C2×#611⋊C3C22semidirect
19. C2×#611⋊C3C22semidirect
23. C2×C4⋊AGL(1,ℤ/9)C22semidirect
25. C2×C4×AGL(1,ℤ/9)C22semidirect
26. C22⋊C4×3−1+2C22semidirect
27. (C2×AGL(1,ℤ/9))⋊C4C22semidirect
28. (3−1+2⋊C4)⋊C4C22semidirect
37. C22×#611C2×C3semidirect
38. C22×#611C2×C3semidirect
39. C22×D36C2×C3semidirect
40. C22×C4×D9C2×C3semidirect
42. C2×C9×C22⋊C4C2×C3semidirect
43. C2×(C2×D9)⋊C4C2×C3semidirect
44. C2×Dic9⋊C4C2×C3semidirect
45. #63446C2×C3semidirect
56. C22×AGL(1,ℤ/9)D4semidirect
66. C2×3−1+2⋊C4D4semidirect
101. C2×#611C22×C3semidirect
102. C2×#611C22×C3semidirect
106. C2×D36C22×C3semidirect
108. C2×C4×D9C22×C3semidirect
110. C9×C22⋊C4C22×C3semidirect
111. (C2×D9)⋊C4C22×C3semidirect
112. Dic9⋊C4C22×C3semidirect
128. C2×AGL(1,ℤ/9)C2×D4semidirect
132. 3−1+2⋊C4C2×D4semidirect
189. C22×D9C3×D4semidirect
203. C2×Dic9C3×D4semidirect
210. C2×3−1+2#121semidirect
295. C2×D9C2×C3×D4semidirect
301. Dic9C2×C3×D4semidirect
309. C2×C22⋊C4AGL(1,ℤ/9)semidirect
310. 3−1+2C2×#121semidirect
374. C2×C9C3×#121semidirect
397. C22⋊C4C2×AGL(1,ℤ/9)semidirect
440. C9C2×C3×#121semidirect
505. C2#63446⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#611⋊C32
C22×C4⋊AGL(1,ℤ/9)2
C22×C4×AGL(1,ℤ/9)2
C2×C22⋊C4×3−1+22
C2×(C2×AGL(1,ℤ/9))⋊C42
C2×(3−1+2⋊C4)⋊C42
#63446⋊C32
C2×C3×#8843
C2×#634463