Finite Group Databasebeta

← C3×#10589596 #13152342

Subgroups — C3×#10589596 #13152342

Order 1728.

882 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C9×C22⋊C4C211yesmaximal, characteristic, complemented
2C3×#63515C212yesmaximal, complemented
3C3×#63492C211yesmaximal, characteristic, complemented
4C3×C4×#611C212yesmaximal, complemented
5C3×#63486C211yesmaximal, characteristic, complemented
6C3×#63447C212yesmaximal, complemented
7C3×#63446C212yesmaximal, complemented
8C3×#63445C212yesmaximal, complemented
9C3×#63441C212yesmaximal, complemented
10C3×#5494—31nomaximal
11#10589596C311yesmaximal, characteristic, direct product factor
12C24×C3×C9C2211yescharacteristic
13C22×C3×C4×C9C2211yescharacteristic
14C22×C3×C4×C9C2211yescharacteristic
15C2×C3×#611C2212yes—
16C2×C3×#611C2212yes—
17C2×C3×#611C2212yes—
18C2×C3×D36C2211yescharacteristic, complemented
19C2×C3×C4×D9C2212yescomplemented
20C2×C3×Dic18C2211yescharacteristic, complemented
21C3×C9×C22⋊C4C2212yes—
22C3×C9×C22⋊C4C2212yes—
23C3×#3676C2212yes—
24C3×#3676C2212yes—
25C3×#3676C2212yes—
26C3×(C2×D9)⋊C4C2212yescomplemented
27C3×(C2×D9)⋊C4C2212yescomplemented
28C3×C4⋊Dic9C2211yescharacteristic, complemented
29C3×C4⋊Dic9C2211yescharacteristic, complemented
30C3×Dic9⋊C4C2212yescomplemented
31C3×Dic9⋊C4C2212yescomplemented
32C3×C4×Dic9C2212yescomplemented
33C2×C32×C22⋊C4D311yescharacteristic
34C3×#953—32no—
35C3×#930—31no—
36C3×C4×#67—32no—
37C3×#924—31no—
38C3×#885—32no—
39C3×#884—32no—
40C3×#883—32no—
41C3×#879—32no—
42C2×C9×C22⋊C4—21no—
43C2×C9×C22⋊C4C2×C311yescharacteristic, complemented
44#63515C2×C312yescomplemented
45#63492C2×C311yescharacteristic, complemented
46C4×#611C2×C312yescomplemented
47#63486C2×C311yescharacteristic, complemented
48#63447C2×C312yescomplemented
49#63446C2×C312yescomplemented
50#63445C2×C312yescomplemented
51#63441C2×C312yescomplemented
52C23×C3×C9—22no—
53C23×C3×C9—41no—
54C23×C3×C9—41no—
55C23×C3×C9C2312yes—
56C23×C3×C9C2311yescharacteristic
57C22×C3×D9C2312yes—
58C2×C3×C4×C9—41no—
59C2×C3×C4×C9—41no—
60C2×C3×C4×C9C2311yescharacteristic
61C2×C3×C4×C9C2312yes—
62C2×C3×C4×C9C2311yescharacteristic
63C3×#611—24no—
64C3×#611—42no—
65C3×#611—42no—
66C2×C3×Dic9C2312yes—
67C2×C3×Dic9C2312yes—
68C2×C3×Dic9C2312yes—
69C3×D36—41no—
70C3×C4×D9—42no—
71C3×Dic18—41no—
72C3×#538—91no—
73#5494—31no—
74C24×C32C2×D311yescharacteristic
75C22×C32×C4C2×D311yescharacteristic
76C22×C32×C4C2×D311yescharacteristic
77C2×C3×#67—32no—
78C2×C3×#67—32no—
79C2×C3×#67—32no—
80C2×C3×D12—31no—
81C2×C3×C4×D3—32no—
82C2×C3×Dic6—31no—
83C24×C9—21no—
84C24×C9C22×C311yescharacteristic
85C32×C22⋊C4C2×D312yes—
86C32×C22⋊C4C2×D312yes—
87C3×#217—32no—
88C3×#217—32no—
89C3×#217—32no—
90C3×(C2×D3)⋊C4—32no—
91C3×(C2×D3)⋊C4—32no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—31no—
94C3×Dic3⋊C4—32no—
95C3×Dic3⋊C4—32no—
96C3×C4×Dic3—32no—
97C22×C4×C9—21no—
98C22×C4×C9—21no—
99C22×C4×C9C22×C311yescharacteristic
100C22×C4×C9C22×C311yescharacteristic
101C2×#611C22×C312yes—
102C2×#611C22×C312yes—
103C2×#611C22×C312yes—
104C2×D36C22×C311yescharacteristic, complemented
105C2×C4×D9C22×C312yescomplemented
106C2×Dic18C22×C311yescharacteristic, complemented
107C9×C22⋊C4—22no—
108C9×C22⋊C4—22no—
109C9×C22⋊C4C22×C312yes—
110C9×C22⋊C4C22×C312yes—
111#3676C22×C312yes—
112#3676C22×C312yes—
113#3676C22×C312yes—
114(C2×D9)⋊C4C22×C312yescomplemented
115(C2×D9)⋊C4C22×C312yescomplemented
116C4⋊Dic9C22×C311yescharacteristic, complemented
117C4⋊Dic9C22×C311yescharacteristic, complemented
118Dic9⋊C4C22×C312yescomplemented
119Dic9⋊C4C22×C312yescomplemented
120C4×Dic9C22×C312yescomplemented
121C22×C3×C9—21no—
122C22×C3×C9—21no—
123C22×C3×C9—22no—
124C22×C3×C9—22no—
125C22×C3×C9—22no—
126C22×C3×C9—44no—
127C22×C3×C9#4112yes—
128C22×C3×C9C2411yescharacteristic
129C2×C3×D9—22no—
130C2×C3×D9—22no—
131C2×C3×D9—22no—
132C3×C4×C9—21no—
133C3×C4×C9—22no—
134C3×C4×C9—21no—
135C3×Dic9—22no—
136C3×Dic9—22no—
137C3×Dic9—22no—
138C3×#126—92no—
139C3×#124—92no—
140C3×#123—92no—
141C3×C22⋊Q8—91no—
142C3×#121—91no—
143C3×#121—92no—
144C3×#120—92no—
145C3×C4×D4—92no—
146C2×C3×C22⋊C4—21no—
147C2×C3×C22⋊C4D911yescharacteristic, complemented
148C2×C3×C22⋊C4C3×D311yescharacteristic
149#953—32no—
150#930—31no—
151C4×#67—32no—
152#924—31no—
153#885—32no—
154#884—32no—
155#883—32no—
156#879—32no—
157C23×C32—22no—
158C23×C32—41no—
159C23×C32—41no—
160C23×C32C22×D312yes—
161C23×C32C22×D311yescharacteristic
162C22×C3×D3—32no—
163C2×C32×C4—41no—
164C2×C32×C4—41no—
165C2×C32×C4C22×D312yes—
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C3×#67—64no—
169C3×#67—122no—
170C3×#67—122no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3—32no—
174C3×D12—121no—
175C3×C4×D3—122no—
176C3×Dic6—121no—
177C23×C9—24no—
178C23×C9—22no—
179C23×C9—21no—
180C23×C9—22no—
181C23×C9—41no—
182C23×C9—41no—
183C23×C9—42no—
184C23×C9—42no—
185C23×C9C23×C312yes—
186C23×C9C23×C311yescharacteristic
187C22×D9C23×C312yes—
188C2×C4×C9—22no—
189C2×C4×C9—21no—
190C2×C4×C9—21no—
191C2×C4×C9—41no—
192C2×C4×C9—41no—
193C2×C4×C9—42no—
194C2×C4×C9—42no—
195C2×C4×C9C23×C312yes—
196C2×C4×C9C23×C311yescharacteristic
197C2×C4×C9C23×C311yescharacteristic
198#611—24no—
199#611—42no—
200#611—42no—
201C2×Dic9C23×C312yes—
202C2×Dic9C23×C312yes—
203C2×Dic9C23×C312yes—
204D36—41no—
205C4×D9—42no—
206Dic18—41no—
207#538—91noSylow
208C2×C3×C9—22no—
209C2×C3×C9—42no—
210C2×C3×C9C2×#4111yescharacteristic
211C2×C3×C9H(5,𝔽2)11yescharacteristic
212C2×C3×C9H(5,𝔽2)11yescharacteristic
213C3×D9—42no—
214C24×C3—21no—
215C24×C3C2×D911yescharacteristic
216C24×C3C2×C3×D311yescharacteristic
217C2×C3×Q8—91no—
218C2×C3×D4—92no—
219C2×C3×D4—92no—
220C2×C3×D4—92no—
221C2×C3×D4—91no—
222C22×C3×C4—21no—
223C22×C3×C4—21no—
224C22×C3×C4—92no—
225C22×C3×C4C2×D911yescharacteristic
226C22×C3×C4C2×D911yescharacteristic
227C22×C3×C4C2×C3×D311yescharacteristic
228C22×C3×C4C2×C3×D311yescharacteristic
229C2×#67—32no—
230C2×#67—32no—
231C2×#67—32no—
232C2×D12—31no—
233C2×C4×D3—32no—
234C2×Dic6—31no—
235C3×C4⋊C4—92no—
236C3×C4⋊C4—92no—
237C3×C4⋊C4—91no—
238C3×C4⋊C4—91no—
239C3×C22⋊C4—22no—
240C3×C22⋊C4—22no—
241C3×C22⋊C4—92no—
242C3×C22⋊C4—92no—
243C3×C22⋊C4—92no—
244C3×C22⋊C4—92no—
245C3×C22⋊C4—92no—
246C3×C22⋊C4C2×D912yes—
247C3×C22⋊C4C2×D912yes—
248C3×C22⋊C4C2×C3×D312yes—
249C3×C22⋊C4C2×C3×D312yes—
250C3×C42—92no—
251#217—32no—
252#217—32no—
253#217—32no—
254(C2×D3)⋊C4—32no—
255(C2×D3)⋊C4—32no—
256C4⋊Dic3—31no—
257C4⋊Dic3—31no—
258Dic3⋊C4—32no—
259Dic3⋊C4—32no—
260C4×Dic3—32no—
261C22×C32—21no—
262C22×C32—21no—
263C22×C32—22no—
264C22×C32—22no—
265C22×C32—22no—
266C22×C32—44no—
267C22×C32#23512yes—
268C22×C32C23×D311yescharacteristic
269C2×C3×D3—62no—
270C2×C3×D3—62no—
271C2×C3×D3—62no—
272C32×C4—21no—
273C32×C4—22no—
274C32×C4—21no—
275C3×Dic3—62no—
276C3×Dic3—62no—
277C3×Dic3—62no—
278C22×C9—22no—
279C22×C9—22no—
280C22×C9—21no—
281C22×C9—21no—
282C22×C9—24no—
283C22×C9—22no—
284C22×C9—22no—
285C22×C9—22no—
286C22×C9—21no—
287C22×C9—22no—
288C22×C9—42no—
289C22×C9—42no—
290C22×C9—48no—
291C22×C9—44no—
292C22×C9C3×#4112yes—
293C22×C9C24×C311yescharacteristic
294C2×D9—22no—
295C2×D9—22no—
296C2×D9—22no—
297C4×C9—21no—
298C4×C9—22no—
299C4×C9—21no—
300C4×C9—41no—
301C4×C9—42no—
302C4×C9—41no—
303Dic9—22no—
304Dic9—22no—
305Dic9—22no—
306#126—92no—
307#124—92no—
308#123—92no—
309C22⋊Q8—91no—
310#121—91no—
311#121—92no—
312#120—92no—
313C4×D4—92no—
314C2×C22⋊C4C3×D911yescharacteristic, complemented
315C3×C9#53811yescharacteristic, complemented, Sylow
316C23×C3—24no—
317C23×C3—22no—
318C23×C3—22no—
319C23×C3—21no—
320C23×C3—22no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3—42no—
326C23×C3—42no—
327C23×C3—92no—
328C23×C3C22×D912yes—
329C23×C3C22×D911yescharacteristic
330C23×C3C22×C3×D312yes—
331C23×C3C22×C3×D311yescharacteristic
332C22×D3—32no—
333C3×Q8—361no—
334C3×D4—184no—
335C3×D4—361no—
336C3×D4—362no—
337C3×D4—362no—
338C2×C3×C4—22no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—92no—
348C2×C3×C4—92no—
349C2×C3×C4—92no—
350C2×C3×C4—362no—
351C2×C3×C4C22×D912yes—
352C2×C3×C4C22×D911yescharacteristic
353C2×C3×C4C22×D911yescharacteristic
354C2×C3×C4C22×C3×D312yes—
355C2×C3×C4C22×C3×D311yescharacteristic
356C2×C3×C4C22×C3×D311yescharacteristic
357#67—64no—
358#67—122no—
359#67—122no—
360C2×Dic3—32no—
361C2×Dic3—32no—
362C2×Dic3—32no—
363D12—121no—
364C4×D3—122no—
365Dic6—121no—
366C2×C32—22no—
367C2×C32—42no—
368C2×C32C2×#23511yescharacteristic
369C2×C32#100411yescharacteristic
370C2×C32#100411yescharacteristic
371C3×D3—122no—
372C2×C9—24no—
373C2×C9—22no—
374C2×C9—21no—
375C2×C9—21no—
376C2×C9—21no—
377C2×C9—44no—
378C2×C9—42no—
379C2×C9C2×C3×#4111yescharacteristic
380C2×C9C3×H(5,𝔽2)11yescharacteristic
381C2×C9C3×H(5,𝔽2)11yescharacteristic
382D9—42no—
383C24C2×C3×D911yescharacteristic
384C2×Q8—91no—
385C2×D4—92no—
386C2×D4—92no—
387C2×D4—92no—
388C2×D4—91no—
389C22×C4—92no—
390C22×C4C2×C3×D911yescharacteristic
391C22×C4C2×C3×D911yescharacteristic
392C4⋊C4—92no—
393C4⋊C4—92no—
394C4⋊C4—91no—
395C4⋊C4—91no—
396C22⋊C4—92no—
397C22⋊C4—92no—
398C22⋊C4—92no—
399C22⋊C4—92no—
400C22⋊C4—92no—
401C22⋊C4C2×C3×D912yes—
402C22⋊C4C2×C3×D912yes—
403C42—92no—
404C22×C3—22no—
405C22×C3—22no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—24no—
411C22×C3—22no—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—22no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—21no—
418C22×C3—22no—
419C22×C3—42no—
420C22×C3—42no—
421C22×C3—44no—
422C22×C3—48no—
423C22×C3—44no—
424C22×C3—182no—
425C22×C3—182no—
426C22×C3—182no—
427C22×C3#369712yes—
428C22×C3C23×D911yescharacteristic
429C22×C3C3×#23512yes—
430C22×C3C23×C3×D311yescharacteristic
431C2×D3—62no—
432C2×D3—62no—
433C2×D3—62no—
434C3×C4—21no—
435C3×C4—22no—
436C3×C4—21no—
437C3×C4—22no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—42no—
441C3×C4—41no—
442C3×C4—41no—
443C3×C4—182no—
444C3×C4—182no—
445C3×C4—182no—
446Dic3—62no—
447Dic3—62no—
448Dic3—62no—
449C32#549411yescharacteristic
450C9—21no—
451C9C3×#53811yescharacteristic, complemented
452C23—22no—
453C23—41no—
454C23—41no—
455C23—92no—
456C23C22×C3×D912yes—
457C23C22×C3×D911yescharacteristic
458Q8—361no—
459D4—184no—
460D4—361no—
461D4—362no—
462D4—362no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—92no—
466C2×C4—92no—
467C2×C4—92no—
468C2×C4—362no—
469C2×C4C22×C3×D912yes—
470C2×C4C22×C3×D911yescharacteristic
471C2×C4C22×C3×D911yescharacteristic
472C2×C3—24no—
473C2×C3—22no—
474C2×C3—22no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—44no—
479C2×C3—42no—
480C2×C3—42no—
481C2×C3—362no—
482C2×C3C2×#369711yescharacteristic
483C2×C3#6371011yescharacteristic
484C2×C3#6371011yescharacteristic
485C2×C3C2×C3×#23511yescharacteristic
486C2×C3C3×#100411yescharacteristic
487C2×C3C3×#100411yescharacteristic
488D3—122no—
489C22—21no—
490C22—21no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—44no—
495C22—182no—
496C22—182no—
497C22—182no—
498C22C3×#369712yes—
499C22C23×C3×D911yescharacteristic
500C4—22no—
501C4—21no—
502C4—21no—
503C4—182no—
504C4—182no—
505C4—182no—
506C3—21no—
507C3#1058959611yescharacteristic, direct product factor
508C3C3×#549411yescharacteristic
509C2—22no—
510C2—42no—
511C2—362no—
512C2C2×C3×#369711yescharacteristic
513C2C3×#6371011yescharacteristic
514C2C3×#6371011yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC2×C3×C9×C22⋊C4C2
socleC22×C32C23×D3
hypercenterC2×C3×C22⋊C4D9
hypocenterC9C3×#538
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C2×C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C9×C22⋊C4C2semidirect
2. C3×#63515C2semidirect
3. C3×#63492C2semidirect
4. C3×C4×#611C2semidirect
5. C3×#63486C2semidirect
6. C3×#63447C2semidirect
7. C3×#63446C2semidirect
8. C3×#63445C2semidirect
9. C3×#63441C2semidirect
11. #10589596C3direct (N × H)
18. C2×C3×D36C22semidirect
19. C2×C3×C4×D9C22semidirect
20. C2×C3×Dic18C22semidirect
26. C3×(C2×D9)⋊C4C22semidirect
27. C3×(C2×D9)⋊C4C22semidirect
28. C3×C4⋊Dic9C22semidirect
29. C3×C4⋊Dic9C22semidirect
30. C3×Dic9⋊C4C22semidirect
31. C3×Dic9⋊C4C22semidirect
32. C3×C4×Dic9C22semidirect
43. C2×C9×C22⋊C4C2×C3semidirect
44. #63515C2×C3semidirect
45. #63492C2×C3semidirect
46. C4×#611C2×C3semidirect
47. #63486C2×C3semidirect
48. #63447C2×C3semidirect
49. #63446C2×C3semidirect
50. #63445C2×C3semidirect
51. #63441C2×C3semidirect
104. C2×D36C22×C3semidirect
105. C2×C4×D9C22×C3semidirect
106. C2×Dic18C22×C3semidirect
114. (C2×D9)⋊C4C22×C3semidirect
115. (C2×D9)⋊C4C22×C3semidirect
116. C4⋊Dic9C22×C3semidirect
117. C4⋊Dic9C22×C3semidirect
118. Dic9⋊C4C22×C3semidirect
119. Dic9⋊C4C22×C3semidirect
120. C4×Dic9C22×C3semidirect
147. C2×C3×C22⋊C4D9semidirect
314. C2×C22⋊C4C3×D9semidirect
315. C3×C9#538semidirect
451. C9C3×#538semidirect
507. C3#10589596direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C9×C22⋊C42
C3×#635152
C3×#634922
C3×C4×#6112
C3×#634862
C3×#634472
C3×#634462
C3×#634452
C3×#634412
C3×#54943
#105895963