Finite Group Databasebeta

← C3×#10589617 #13152412

Subgroups — C3×#10589617 #13152412

Order 1728.

746 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C9×C4⋊C4C211yesmaximal, characteristic, complemented
2C3×#63492C211yesmaximal, characteristic
3C3×#63491C212yesmaximal, complemented
4C3×C4×#611C212yesmaximal, complemented
5C3×#63486C211yesmaximal, characteristic, complemented
6C3×#63460C212yesmaximal, complemented
7C3×#63458C212yesmaximal, complemented
8C3×#63456C212yesmaximal, complemented
9C3×#63451C212yesmaximal, complemented
10C3×#5515—31nomaximal
11#10589617C311yesmaximal, characteristic, direct product factor
12C22×C3×C4×C9C2211yescharacteristic
13C22×C3×C4×C9C2211yescharacteristic
14C22×C3×C4×C9C2211yescharacteristic
15C2×C3×#611C2212yes—
16C2×C3×D36C2211yescharacteristic
17C2×C3×C4×D9C2212yes—
18C2×C3×Dic18C2211yescharacteristic
19C3×C9×C4⋊C4C2212yes—
20C3×C9×C4⋊C4C2212yes—
21C3×#3676C2212yes—
22C3×(C2×D9)⋊C4C2212yes—
23C3×(C2×D9)⋊C4C2212yes—
24C3×(C2×D9)⋊C4C2212yes—
25C3×(C2×D9)⋊C4C2212yes—
26C3×C4⋊Dic9C2211yescharacteristic
27C3×C4⋊Dic9C2211yescharacteristic
28C3×Dic9⋊C4C2212yes—
29C3×Dic9⋊C4C2212yes—
30C3×Dic9⋊C4C2212yes—
31C3×Dic9⋊C4C2212yes—
32C3×C4×Dic9C2212yes—
33C2×C32×C4⋊C4D311yescharacteristic
34C3×#930—31no—
35C3×#929—32no—
36C3×C4×#67—32no—
37C3×#924—31no—
38C3×#898—32no—
39C3×#896—32no—
40C3×#894—32no—
41C3×#889—32no—
42C2×C9×C4⋊C4—21no—
43C2×C9×C4⋊C4C2×C311yescharacteristic, complemented
44#63492C2×C311yescharacteristic
45#63491C2×C312yescomplemented
46C4×#611C2×C312yescomplemented
47#63486C2×C311yescharacteristic, complemented
48#63460C2×C312yescomplemented
49#63458C2×C312yescomplemented
50#63456C2×C312yescomplemented
51#63451C2×C312yescomplemented
52C23×C3×C9C2311yescharacteristic
53C22×C3×D9C2312yes—
54C2×C3×C4×C9—22no—
55C2×C3×C4×C9—41no—
56C2×C3×C4×C9—41no—
57C2×C3×C4×C9C2311yescharacteristic
58C2×C3×C4×C9C2312yes—
59C2×C3×C4×C9C2311yescharacteristic
60C2×C3×C4×C9C2312yes—
61C3×#611—24no—
62C2×C3×Dic9C2312yes—
63C2×C3×Dic9C2312yes—
64C2×C3×Dic9C2312yes—
65C3×D36—41no—
66C3×C4×D9—42no—
67C3×Dic18—41no—
68C3×#539—91no—
69#5515—31no—
70C22×C32×C4C2×D311yescharacteristic
71C22×C32×C4C2×D311yescharacteristic
72C22×C32×C4C2×D311yescharacteristic
73C2×C3×#67—32no—
74C2×C3×D12—31no—
75C2×C3×C4×D3—32no—
76C2×C3×Dic6—31no—
77C32×C4⋊C4C2×D312yes—
78C32×C4⋊C4C2×D312yes—
79C3×#217—32no—
80C3×(C2×D3)⋊C4—32no—
81C3×(C2×D3)⋊C4—32no—
82C3×(C2×D3)⋊C4—32no—
83C3×(C2×D3)⋊C4—32no—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—31no—
86C3×Dic3⋊C4—32no—
87C3×Dic3⋊C4—32no—
88C3×Dic3⋊C4—32no—
89C3×Dic3⋊C4—32no—
90C3×C4×Dic3—32no—
91C22×C4×C9—21no—
92C22×C4×C9—21no—
93C22×C4×C9—21no—
94C22×C4×C9C22×C311yescharacteristic
95C22×C4×C9C22×C311yescharacteristic
96C22×C4×C9C22×C311yescharacteristic
97C2×#611C22×C312yes—
98C2×D36C22×C311yescharacteristic
99C2×C4×D9C22×C312yes—
100C2×Dic18C22×C311yescharacteristic
101C9×C4⋊C4—22no—
102C9×C4⋊C4—22no—
103C9×C4⋊C4C22×C312yes—
104C9×C4⋊C4C22×C312yes—
105#3676C22×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×C311yescharacteristic
112Dic9⋊C4C22×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—22no—
125C3×C4×C9—21no—
126C3×C4×C9—22no—
127C3×C4×C9—21no—
128C3×Dic9—22no—
129C3×Dic9—22no—
130C3×Dic9—22no—
131C3×#126—92no—
132C3×#125—92no—
133C3×#123—92no—
134C3×#123—92no—
135C3×C22⋊Q8—91no—
136C3×C22⋊Q8—92no—
137C3×#121—91no—
138C3×C4×D4—92no—
139C2×C3×C4⋊C4—21no—
140C2×C3×C4⋊C4D911yescharacteristic, complemented
141C2×C3×C4⋊C4C3×D311yescharacteristic
142#930—31no—
143#929—32no—
144C4×#67—32no—
145#924—31no—
146#898—32no—
147#896—32no—
148#894—32no—
149#889—32no—
150C23×C32C22×D311yescharacteristic
151C22×C3×D3—32no—
152C2×C32×C4—22no—
153C2×C32×C4—41no—
154C2×C32×C4—41no—
155C2×C32×C4C22×D312yes—
156C2×C32×C4C22×D311yescharacteristic
157C2×C32×C4C22×D312yes—
158C2×C32×C4C22×D311yescharacteristic
159C3×#67—64no—
160C2×C3×Dic3—32no—
161C2×C3×Dic3—32no—
162C2×C3×Dic3—32no—
163C3×D12—121no—
164C3×C4×D3—122no—
165C3×Dic6—121no—
166C23×C9—21no—
167C23×C9C23×C311yescharacteristic
168C22×D9C23×C312yes—
169C2×C4×C9—24no—
170C2×C4×C9—22no—
171C2×C4×C9—22no—
172C2×C4×C9—22no—
173C2×C4×C9—21no—
174C2×C4×C9—21no—
175C2×C4×C9—41no—
176C2×C4×C9—41no—
177C2×C4×C9—42no—
178C2×C4×C9—42no—
179C2×C4×C9C23×C312yes—
180C2×C4×C9C23×C311yescharacteristic
181C2×C4×C9C23×C312yes—
182C2×C4×C9C23×C311yescharacteristic
183#611—24no—
184C2×Dic9C23×C312yes—
185C2×Dic9C23×C312yes—
186C2×Dic9C23×C312yes—
187D36—41no—
188C4×D9—42no—
189Dic18—41no—
190#539—91noSylow
191C2×C3×C9—22no—
192C2×C3×C9C2×#4111yescharacteristic
193C2×C3×C9H(5,𝔽2)11yescharacteristic
194C2×C3×C92−1+411yescharacteristic
195C3×D9—42no—
196C2×C3×Q8—91no—
197C2×C3×D4—92no—
198C2×C3×D4—91no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—92no—
203C22×C3×C4C2×D911yescharacteristic
204C22×C3×C4C2×D911yescharacteristic
205C22×C3×C4C2×D911yescharacteristic
206C22×C3×C4C2×C3×D311yescharacteristic
207C22×C3×C4C2×C3×D311yescharacteristic
208C22×C3×C4C2×C3×D311yescharacteristic
209C2×#67—32no—
210C2×D12—31no—
211C2×C4×D3—32no—
212C2×Dic6—31no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—92no—
216C3×C4⋊C4—92no—
217C3×C4⋊C4—92no—
218C3×C4⋊C4—92no—
219C3×C4⋊C4—91no—
220C3×C4⋊C4—91no—
221C3×C4⋊C4C2×D912yes—
222C3×C4⋊C4C2×D912yes—
223C3×C4⋊C4C2×C3×D312yes—
224C3×C4⋊C4C2×C3×D312yes—
225C3×C22⋊C4—92no—
226C3×C22⋊C4—92no—
227C3×C22⋊C4—92no—
228C3×C22⋊C4—92no—
229C3×C22⋊C4—92no—
230C3×C42—92no—
231#217—32no—
232(C2×D3)⋊C4—32no—
233(C2×D3)⋊C4—32no—
234(C2×D3)⋊C4—32no—
235(C2×D3)⋊C4—32no—
236C4⋊Dic3—31no—
237C4⋊Dic3—31no—
238Dic3⋊C4—32no—
239Dic3⋊C4—32no—
240Dic3⋊C4—32no—
241Dic3⋊C4—32no—
242C4×Dic3—32no—
243C22×C32—21no—
244C22×C32—21no—
245C22×C32#23512yes—
246C22×C32C23×D311yescharacteristic
247C2×C3×D3—62no—
248C2×C3×D3—62no—
249C2×C3×D3—62no—
250C32×C4—22no—
251C32×C4—21no—
252C32×C4—22no—
253C32×C4—21no—
254C3×Dic3—62no—
255C3×Dic3—62no—
256C3×Dic3—62no—
257C22×C9—22no—
258C22×C9—22no—
259C22×C9—21no—
260C22×C9—21no—
261C22×C9—21no—
262C22×C9—22no—
263C22×C9C3×#4112yes—
264C22×C9C24×C311yescharacteristic
265C2×D9—22no—
266C2×D9—22no—
267C2×D9—22no—
268C4×C9—24no—
269C4×C9—22no—
270C4×C9—21no—
271C4×C9—22no—
272C4×C9—21no—
273C4×C9—41no—
274C4×C9—42no—
275C4×C9—41no—
276Dic9—22no—
277Dic9—22no—
278Dic9—22no—
279#126—92no—
280#125—92no—
281#123—92no—
282#123—92no—
283C22⋊Q8—91no—
284C22⋊Q8—92no—
285#121—91no—
286C4×D4—92no—
287C2×C4⋊C4C3×D911yescharacteristic, complemented
288C3×C9#53911yescharacteristic, complemented, Sylow
289C23×C3—21no—
290C23×C3—92no—
291C23×C3C22×D911yescharacteristic
292C23×C3C22×C3×D311yescharacteristic
293C22×D3—32no—
294C3×Q8—361no—
295C3×D4—184no—
296C3×D4—361no—
297C2×C3×C4—24no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—92no—
311C2×C3×C4—92no—
312C2×C3×C4—92no—
313C2×C3×C4—362no—
314C2×C3×C4C22×D912yes—
315C2×C3×C4C22×D912yes—
316C2×C3×C4C22×D911yescharacteristic
317C2×C3×C4C22×D911yescharacteristic
318C2×C3×C4C22×C3×D312yes—
319C2×C3×C4C22×C3×D311yescharacteristic
320C2×C3×C4C22×C3×D312yes—
321C2×C3×C4C22×C3×D311yescharacteristic
322#67—64no—
323C2×Dic3—32no—
324C2×Dic3—32no—
325C2×Dic3—32no—
326D12—121no—
327C4×D3—122no—
328Dic6—121no—
329C2×C32—22no—
330C2×C32C2×#23511yescharacteristic
331C2×C32#100411yescharacteristic
332C2×C32#100711yescharacteristic
333C3×D3—122no—
334C2×C9—24no—
335C2×C9—22no—
336C2×C9—21no—
337C2×C9—21no—
338C2×C9—21no—
339C2×C9C2×C3×#4111yescharacteristic
340C2×C9C3×H(5,𝔽2)11yescharacteristic
341C2×C9C3×2−1+411yescharacteristic
342D9—42no—
343C2×Q8—91no—
344C2×D4—92no—
345C2×D4—91no—
346C22×C4—92no—
347C22×C4C2×C3×D911yescharacteristic
348C22×C4C2×C3×D911yescharacteristic
349C22×C4C2×C3×D911yescharacteristic
350C4⋊C4—92no—
351C4⋊C4—91no—
352C4⋊C4—92no—
353C4⋊C4—92no—
354C4⋊C4—91no—
355C4⋊C4—92no—
356C4⋊C4C2×C3×D912yes—
357C4⋊C4C2×C3×D912yes—
358C22⋊C4—92no—
359C22⋊C4—92no—
360C22⋊C4—92no—
361C22⋊C4—92no—
362C22⋊C4—92no—
363C42—92no—
364C22×C3—22no—
365C22×C3—22no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—22no—
372C22×C3—182no—
373C22×C3—182no—
374C22×C3—182no—
375C22×C3#369712yes—
376C22×C3C23×D911yescharacteristic
377C22×C3C3×#23512yes—
378C22×C3C23×C3×D311yescharacteristic
379C2×D3—62no—
380C2×D3—62no—
381C2×D3—62no—
382C3×C4—24no—
383C3×C4—22no—
384C3×C4—21no—
385C3×C4—22no—
386C3×C4—21no—
387C3×C4—22no—
388C3×C4—22no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—42no—
392C3×C4—41no—
393C3×C4—41no—
394C3×C4—182no—
395C3×C4—182no—
396C3×C4—182no—
397Dic3—62no—
398Dic3—62no—
399Dic3—62no—
400C32#551511yescharacteristic
401C9—21no—
402C9C3×#53911yescharacteristic, complemented
403C23—92no—
404C23C22×C3×D911yescharacteristic
405Q8—361no—
406D4—184no—
407D4—361no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—92no—
412C2×C4—92no—
413C2×C4—92no—
414C2×C4—362no—
415C2×C4C22×C3×D912yes—
416C2×C4C22×C3×D912yes—
417C2×C4C22×C3×D911yescharacteristic
418C2×C4C22×C3×D911yescharacteristic
419C2×C3—24no—
420C2×C3—22no—
421C2×C3—22no—
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—362no—
426C2×C3C2×#369711yescharacteristic
427C2×C3#6371011yescharacteristic
428C2×C3#6371311yescharacteristic
429C2×C3C2×C3×#23511yescharacteristic
430C2×C3C3×#100411yescharacteristic
431C2×C3C3×#100711yescharacteristic
432D3—122no—
433C22—21no—
434C22—21no—
435C22—182no—
436C22—182no—
437C22—182no—
438C22C3×#369712yes—
439C22C23×C3×D911yescharacteristic
440C4—22no—
441C4—22no—
442C4—21no—
443C4—21no—
444C4—182no—
445C4—182no—
446C4—182no—
447C3—21no—
448C3#1058961711yescharacteristic, direct product factor
449C3C3×#551511yescharacteristic
450C2—22no—
451C2—362no—
452C2C2×C3×#369711yescharacteristic
453C2C3×#6371011yescharacteristic
454C2C3×#6371311yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC2×C3×C9×C4⋊C4C2
socleC22×C32C23×D3
prime radicalC3×#63492C2
hypercenterC2×C3×C4⋊C4D9
hypocenterC9C3×#539
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C9×C4⋊C4C2semidirect
3. C3×#63491C2semidirect
4. C3×C4×#611C2semidirect
5. C3×#63486C2semidirect
6. C3×#63460C2semidirect
7. C3×#63458C2semidirect
8. C3×#63456C2semidirect
9. C3×#63451C2semidirect
11. #10589617C3direct (N × H)
43. C2×C9×C4⋊C4C2×C3semidirect
45. #63491C2×C3semidirect
46. C4×#611C2×C3semidirect
47. #63486C2×C3semidirect
48. #63460C2×C3semidirect
49. #63458C2×C3semidirect
50. #63456C2×C3semidirect
51. #63451C2×C3semidirect
140. C2×C3×C4⋊C4D9semidirect
287. C2×C4⋊C4C3×D9semidirect
288. C3×C9#539semidirect
402. C9C3×#539semidirect
448. C3#10589617direct (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×C4⋊C42
C3×#634922
C3×#634912
C3×C4×#6112
C3×#634862
C3×#634602
C3×#634582
C3×#634562
C3×#634512
C3×#55153
#105896173