Finite Group Databasebeta

← C3×#10589747 #13152922

Subgroups — C3×#10589747 #13152922

Order 1728.

728 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×C22⋊Q8C211yesmaximal, characteristic, complemented
2C3×#63508C212yesmaximal, complemented
3C3×#63492C211yesmaximal, characteristic
4C2×C3×Dic9⋊C4C211yesmaximal, characteristic, complemented
5C3×#63460C212yesmaximal, complemented
6C3×#63456C212yesmaximal, complemented
7C3×#63452C212yesmaximal, complemented
8C3×#63448C212yesmaximal, complemented
9C3×#611⋊C4C212yesmaximal, complemented
10C3×#5645—31nomaximal
11#10589747C311yesmaximal, characteristic, direct product factor
12C2×C3×Q8×C9C2211yescharacteristic
13C22×C3×C4×C9C2211yescharacteristic
14C2×C3×#611C2212yes—
15C22×C3×Dic9C2211yescharacteristic
16C22×C3×Dic9C2211yescharacteristic
17C2×C3×D36C2211yescharacteristic
18C2×C3×C4×D9C2212yes—
19C3×C9×C4⋊C4C2212yes—
20C3×C9×C4⋊C4C2211yescharacteristic
21C3×C9×C22⋊C4C2212yes—
22C3×(C2×D9)⋊C4C2212yes—
23C3×(C2×D9)⋊C4C2212yes—
24C3×(C2×D9)⋊C4C2212yes—
25C3×(C2×D9)⋊C4C2212yes—
26C3×C4⋊Dic9C2211yescharacteristic
27C3×C4⋊Dic9C2212yes—
28C3×C4⋊Dic9C2212yes—
29C3×Dic9⋊C4C2212yes—
30C3×Dic9⋊C4C2212yes—
31C3×Dic9⋊C4C2212yes—
32C3×C4×Dic9C2212yes—
33C32×C22⋊Q8D311yescharacteristic
34C3×#946—32no—
35C3×#930—31no—
36C2×C3×Dic3⋊C4—31no—
37C3×#898—32no—
38C3×#894—32no—
39C3×#890—32no—
40C3×#886—32no—
41C3×#67⋊C4—32no—
42C9×C22⋊Q8—21no—
43C9×C22⋊Q8C2×C311yescharacteristic, complemented
44#63508C2×C312yescomplemented
45#63492C2×C311yescharacteristic
46C2×Dic9⋊C4C2×C311yescharacteristic, complemented
47#63460C2×C312yescomplemented
48#63456C2×C312yescomplemented
49#63452C2×C312yescomplemented
50#63448C2×C312yescomplemented
51#611⋊C4C2×C312yescomplemented
52C23×C3×C9C2311yescharacteristic
53C22×C3×D9C2312yes—
54C3×Q8×C9—41no—
55C2×C3×C4×C9—41no—
56C2×C3×C4×C9C2312yes—
57C2×C3×C4×C9C2311yescharacteristic
58C2×C3×C4×C9C2312yes—
59C2×C3×C4×C9C2311yescharacteristic
60C3×#611—24no—
61C2×C3×Dic9—22no—
62C2×C3×Dic9—41no—
63C2×C3×Dic9C2312yes—
64C2×C3×Dic9C2312yes—
65C2×C3×Dic9C2312yes—
66C3×D36—41no—
67C3×C4×D9—42no—
68C3×#539—91no—
69#5645—31no—
70C2×C32×Q8C2×D311yescharacteristic
71C22×C32×C4C2×D311yescharacteristic
72C2×C3×#67—32no—
73C22×C3×Dic3—31no—
74C22×C3×Dic3—31no—
75C2×C3×D12—31no—
76C2×C3×C4×D3—32no—
77C32×C4⋊C4C2×D312yes—
78C32×C4⋊C4C2×D311yescharacteristic
79C32×C22⋊C4C2×D312yes—
80C3×(C2×D3)⋊C4—32no—
81C3×(C2×D3)⋊C4—32no—
82C3×(C2×D3)⋊C4—32no—
83C3×(C2×D3)⋊C4—32no—
84C3×C4⋊Dic3—32no—
85C3×C4⋊Dic3—32no—
86C3×C4⋊Dic3—31no—
87C3×Dic3⋊C4—32no—
88C3×Dic3⋊C4—32no—
89C3×Dic3⋊C4—32no—
90C3×C4×Dic3—32no—
91C2×Q8×C9—21no—
92C2×Q8×C9C22×C311yescharacteristic
93C22×C4×C9—21no—
94C22×C4×C9C22×C311yescharacteristic
95C2×#611C22×C312yes—
96C22×Dic9C22×C311yescharacteristic
97C22×Dic9C22×C311yescharacteristic
98C2×D36C22×C311yescharacteristic
99C2×C4×D9C22×C312yes—
100C9×C4⋊C4—21no—
101C9×C4⋊C4—22no—
102C9×C4⋊C4C22×C312yes—
103C9×C4⋊C4C22×C311yescharacteristic
104C9×C22⋊C4—22no—
105C9×C22⋊C4C22×C312yes—
106(C2×D9)⋊C4C22×C312yes—
107(C2×D9)⋊C4C22×C312yes—
108(C2×D9)⋊C4C22×C312yes—
109(C2×D9)⋊C4C22×C312yes—
110C4⋊Dic9C22×C311yescharacteristic
111C4⋊Dic9C22×C312yes—
112C4⋊Dic9C22×C312yes—
113Dic9⋊C4C22×C312yes—
114Dic9⋊C4C22×C312yes—
115Dic9⋊C4C22×C312yes—
116C4×Dic9C22×C312yes—
117C22×C3×C9—21no—
118C22×C3×C9—21no—
119C22×C3×C9#4112yes—
120C22×C3×C9C2411yescharacteristic
121C2×C3×D9—22no—
122C2×C3×D9—22no—
123C2×C3×D9—22no—
124C3×C4×C9—21no—
125C3×C4×C9—22no—
126C3×C4×C9—21no—
127C3×C4×C9—22no—
128C3×Dic9—22no—
129C3×Dic9—22no—
130C3×Dic9—22no—
131C3×#126—92no—
132C3×#125—92no—
133C3×#123—92no—
134C3×#123—92no—
135C3×C22⋊Q8—21no—
136C3×C22⋊Q8—92no—
137C3×C22⋊Q8D911yescharacteristic, complemented
138C3×C22⋊Q8C3×D311yescharacteristic
139C3×#121—91no—
140C3×C4×D4—92no—
141C2×C3×C4⋊C4—91no—
142#946—32no—
143#930—31no—
144C2×Dic3⋊C4—31no—
145#898—32no—
146#894—32no—
147#890—32no—
148#886—32no—
149#67⋊C4—32no—
150C23×C32C22×D311yescharacteristic
151C22×C3×D3—32no—
152C32×Q8—41no—
153C2×C32×C4—41no—
154C2×C32×C4C22×D312yes—
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C22×D311yescharacteristic
157C2×C32×C4C22×D312yes—
158C3×#67—64no—
159C2×C3×Dic3—32no—
160C2×C3×Dic3—32no—
161C2×C3×Dic3—32no—
162C2×C3×Dic3—62no—
163C2×C3×Dic3—121no—
164C3×D12—121no—
165C3×C4×D3—122no—
166C23×C9—21no—
167C23×C9C23×C311yescharacteristic
168C22×D9C23×C312yes—
169Q8×C9—41no—
170Q8×C9—42no—
171C2×C4×C9—21no—
172C2×C4×C9—22no—
173C2×C4×C9—22no—
174C2×C4×C9—21no—
175C2×C4×C9—41no—
176C2×C4×C9—42no—
177C2×C4×C9C23×C312yes—
178C2×C4×C9C23×C311yescharacteristic
179C2×C4×C9C23×C311yescharacteristic
180C2×C4×C9C23×C312yes—
181#611—24no—
182C2×Dic9—22no—
183C2×Dic9—41no—
184C2×Dic9C23×C312yes—
185C2×Dic9C23×C312yes—
186C2×Dic9C23×C312yes—
187D36—41no—
188C4×D9—42no—
189#539—91noSylow
190C2×C3×C9—22no—
191C2×C3×C9C2×#4111yescharacteristic
192C2×C3×C9H(5,𝔽2)11yescharacteristic
193C2×C3×C92−1+411yescharacteristic
194C3×D9—42no—
195C2×C3×Q8—21no—
196C2×C3×Q8C2×D911yescharacteristic
197C2×C3×Q8C2×C3×D311yescharacteristic
198C2×C3×D4—92no—
199C2×C3×D4—91no—
200C22×C3×C4—21no—
201C22×C3×C4—91no—
202C22×C3×C4—91no—
203C22×C3×C4—92no—
204C22×C3×C4C2×D911yescharacteristic
205C22×C3×C4C2×C3×D311yescharacteristic
206C2×#67—32no—
207C22×Dic3—31no—
208C22×Dic3—31no—
209C2×D12—31no—
210C2×C4×D3—32no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—92no—
214C3×C4⋊C4—92no—
215C3×C4⋊C4—92no—
216C3×C4⋊C4—91no—
217C3×C4⋊C4—92no—
218C3×C4⋊C4—92no—
219C3×C4⋊C4C2×D912yes—
220C3×C4⋊C4C2×D911yescharacteristic
221C3×C4⋊C4C2×C3×D312yes—
222C3×C4⋊C4C2×C3×D311yescharacteristic
223C3×C22⋊C4—22no—
224C3×C22⋊C4—92no—
225C3×C22⋊C4—92no—
226C3×C22⋊C4—92no—
227C3×C22⋊C4—92no—
228C3×C22⋊C4C2×D912yes—
229C3×C22⋊C4C2×C3×D312yes—
230C3×C42—92no—
231(C2×D3)⋊C4—32no—
232(C2×D3)⋊C4—32no—
233(C2×D3)⋊C4—32no—
234(C2×D3)⋊C4—32no—
235C4⋊Dic3—32no—
236C4⋊Dic3—32no—
237C4⋊Dic3—31no—
238Dic3⋊C4—32no—
239Dic3⋊C4—32no—
240Dic3⋊C4—32no—
241C4×Dic3—32no—
242C22×C32—21no—
243C22×C32—21no—
244C22×C32#23912yes—
245C22×C32C23×D311yescharacteristic
246C2×C3×D3—62no—
247C2×C3×D3—62no—
248C2×C3×D3—62no—
249C32×C4—21no—
250C32×C4—22no—
251C32×C4—21no—
252C32×C4—22no—
253C3×Dic3—62no—
254C3×Dic3—62no—
255C3×Dic3—62no—
256C22×C9—21no—
257C22×C9—21no—
258C22×C9—21no—
259C22×C9—22no—
260C22×C9—41no—
261C22×C9—41no—
262C22×C9C3×#4112yes—
263C22×C9C24×C311yescharacteristic
264C2×D9—22no—
265C2×D9—22no—
266C2×D9—22no—
267C4×C9—22no—
268C4×C9—22no—
269C4×C9—21no—
270C4×C9—21no—
271C4×C9—22no—
272C4×C9—42no—
273C4×C9—41no—
274C4×C9—42no—
275Dic9—22no—
276Dic9—22no—
277Dic9—22no—
278#126—92no—
279#125—92no—
280#123—92no—
281#123—92no—
282C22⋊Q8—92no—
283C22⋊Q8C3×D911yescharacteristic, complemented
284#121—91no—
285C4×D4—92no—
286C2×C4⋊C4—91no—
287C3×C9#53911yescharacteristic, complemented, Sylow
288C23×C3—21no—
289C23×C3—92no—
290C23×C3C22×D911yescharacteristic
291C23×C3C22×C3×D311yescharacteristic
292C22×D3—32no—
293C3×Q8—41no—
294C3×Q8—41no—
295C3×Q8—42no—
296C3×D4—184no—
297C3×D4—361no—
298C2×C3×C4—21no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—21no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—42no—
305C2×C3×C4—92no—
306C2×C3×C4—92no—
307C2×C3×C4—92no—
308C2×C3×C4—182no—
309C2×C3×C4—362no—
310C2×C3×C4—361no—
311C2×C3×C4C22×D912yes—
312C2×C3×C4C22×D911yescharacteristic
313C2×C3×C4C22×D912yes—
314C2×C3×C4C22×D911yescharacteristic
315C2×C3×C4C22×C3×D312yes—
316C2×C3×C4C22×C3×D311yescharacteristic
317C2×C3×C4C22×C3×D311yescharacteristic
318C2×C3×C4C22×C3×D312yes—
319#67—64no—
320C2×Dic3—32no—
321C2×Dic3—32no—
322C2×Dic3—32no—
323C2×Dic3—62no—
324C2×Dic3—121no—
325D12—121no—
326C4×D3—122no—
327C2×C32—22no—
328C2×C32#100411yescharacteristic
329C2×C32C2×#23911yescharacteristic
330C2×C32#101011yescharacteristic
331C3×D3—122no—
332C2×C9—22no—
333C2×C9—21no—
334C2×C9—21no—
335C2×C9—21no—
336C2×C9—42no—
337C2×C9C2×C3×#4111yescharacteristic
338C2×C9C3×H(5,𝔽2)11yescharacteristic
339C2×C9C3×2−1+411yescharacteristic
340D9—42no—
341C2×Q8C2×C3×D911yescharacteristic
342C2×D4—92no—
343C2×D4—91no—
344C22×C4—91no—
345C22×C4—91no—
346C22×C4—92no—
347C22×C4C2×C3×D911yescharacteristic
348C4⋊C4—92no—
349C4⋊C4—92no—
350C4⋊C4—92no—
351C4⋊C4—91no—
352C4⋊C4—92no—
353C4⋊C4—92no—
354C4⋊C4C2×C3×D912yes—
355C4⋊C4C2×C3×D911yescharacteristic
356C22⋊C4—92no—
357C22⋊C4—92no—
358C22⋊C4—92no—
359C22⋊C4—92no—
360C22⋊C4C2×C3×D912yes—
361C42—92no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—22no—
368C22×C3—41no—
369C22×C3—41no—
370C22×C3—182no—
371C22×C3—182no—
372C22×C3—182no—
373C22×C3#370112yes—
374C22×C3C23×D911yescharacteristic
375C22×C3C3×#23912yes—
376C22×C3C23×C3×D311yescharacteristic
377C2×D3—62no—
378C2×D3—62no—
379C2×D3—62no—
380C3×C4—22no—
381C3×C4—22no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—22no—
385C3×C4—21no—
386C3×C4—21no—
387C3×C4—22no—
388C3×C4—22no—
389C3×C4—41no—
390C3×C4—42no—
391C3×C4—42no—
392C3×C4—182no—
393C3×C4—182no—
394C3×C4—182no—
395Dic3—62no—
396Dic3—62no—
397Dic3—62no—
398C32#564511yescharacteristic
399C9—21no—
400C9C3×#53911yescharacteristic, complemented
401C23—92no—
402C23C22×C3×D911yescharacteristic
403Q8—41no—
404D4—184no—
405D4—361no—
406C2×C4—41no—
407C2×C4—92no—
408C2×C4—92no—
409C2×C4—92no—
410C2×C4—182no—
411C2×C4—362no—
412C2×C4—361no—
413C2×C4C22×C3×D912yes—
414C2×C4C22×C3×D911yescharacteristic
415C2×C4C22×C3×D912yes—
416C2×C4C22×C3×D911yescharacteristic
417C2×C3—22no—
418C2×C3—22no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—21no—
422C2×C3—42no—
423C2×C3—362no—
424C2×C3#6371011yescharacteristic
425C2×C3C2×#370111yescharacteristic
426C2×C3#6371611yescharacteristic
427C2×C3C3×#100411yescharacteristic
428C2×C3C2×C3×#23911yescharacteristic
429C2×C3C3×#101011yescharacteristic
430D3—122no—
431C22—21no—
432C22—21no—
433C22—182no—
434C22—182no—
435C22—182no—
436C22C3×#370112yes—
437C22C23×C3×D911yescharacteristic
438C4—21no—
439C4—21no—
440C4—22no—
441C4—22no—
442C4—182no—
443C4—182no—
444C4—182no—
445C3—21no—
446C3#1058974711yescharacteristic, direct product factor
447C3C3×#564511yescharacteristic
448C2—22no—
449C2—362no—
450C2C3×#6371011yescharacteristic
451C2C2×C3×#370111yescharacteristic
452C2C3×#6371611yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×C22⋊Q8C2
socleC22×C32C23×D3
prime radicalC3×#63492C2
hypercenterC3×C22⋊Q8D9
hypocenterC9C3×#539
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×C9×C22⋊Q8C2semidirect
2. C3×#63508C2semidirect
4. C2×C3×Dic9⋊C4C2semidirect
5. C3×#63460C2semidirect
6. C3×#63456C2semidirect
7. C3×#63452C2semidirect
8. C3×#63448C2semidirect
9. C3×#611⋊C4C2semidirect
11. #10589747C3direct (N × H)
43. C9×C22⋊Q8C2×C3semidirect
44. #63508C2×C3semidirect
46. C2×Dic9⋊C4C2×C3semidirect
47. #63460C2×C3semidirect
48. #63456C2×C3semidirect
49. #63452C2×C3semidirect
50. #63448C2×C3semidirect
51. #611⋊C4C2×C3semidirect
137. C3×C22⋊Q8D9semidirect
283. C22⋊Q8C3×D9semidirect
287. C3×C9#539semidirect
400. C9C3×#539semidirect
446. C3#10589747direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C9×C22⋊Q82
C3×#635082
C3×#634922
C2×C3×Dic9⋊C42
C3×#634602
C3×#634562
C3×#634522
C3×#634482
C3×#611⋊C42
C3×#56453
#105897473