Finite Group Databasebeta

← C3×#10589704 #13152795

Subgroups — C3×#10589704 #13152795

Order 1728.

748 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×#121C211yesmaximal, characteristic
2C3×#63497C212yesmaximal, complemented
3C3×#63497C212yesmaximal, complemented
4C3×D4×Dic9C212yesmaximal, complemented
5C3×#63486C211yesmaximal, characteristic, complemented
6C2×C3×Dic9⋊C4C211yesmaximal, characteristic, complemented
7C3×#63451C212yesmaximal, complemented
8C3×#63441C212yesmaximal, complemented
9C3×#63440C212yesmaximal, complemented
10C3×#5602—31nomaximal
11#10589704C311yesmaximal, characteristic, direct product factor
12C2×C3×D4×C9C2212yes—
13C2×C3×D4×C9C2211yescharacteristic
14C22×C3×C4×C9C2211yescharacteristic
15C22×C3×Dic9C2211yescharacteristic
16C22×C3×Dic9C2211yescharacteristic
17C22×C3×Dic9C2212yes—
18C2×C3×Dic18C2211yescharacteristic
19C3×C9×C4⋊C4C2211yescharacteristic
20C3×C9×C22⋊C4C2212yes—
21C3×#3676C2212yes—
22C3×#3676C2212yes—
23C3×#3676C2212yes—
24C3×#3676C2212yes—
25C3×C4⋊Dic9C2211yescharacteristic
26C3×C4⋊Dic9C2212yes—
27C3×Dic9⋊C4C2212yes—
28C3×Dic9⋊C4C2212yes—
29C3×Dic9⋊C4C2212yes—
30C3×Dic9⋊C4C2212yes—
31C3×Dic9⋊C4C2212yes—
32C3×C4×Dic9C2212yes—
33C32×#121D311yescharacteristic
34C3×#935—32no—
35C3×#935—32no—
36C3×D4×Dic3—32no—
37C3×#924—31no—
38C2×C3×Dic3⋊C4—31no—
39C3×#889—32no—
40C3×#879—32no—
41C3×#878—32no—
42C9×#121—21no—
43C9×#121C2×C311yescharacteristic
44#63497C2×C312yescomplemented
45#63497C2×C312yescomplemented
46D4×Dic9C2×C312yescomplemented
47#63486C2×C311yescharacteristic, complemented
48C2×Dic9⋊C4C2×C311yescharacteristic, complemented
49#63451C2×C312yescomplemented
50#63441C2×C312yescomplemented
51#63440C2×C312yescomplemented
52C23×C3×C9C2312yes—
53C23×C3×C9C2311yescharacteristic
54C3×D4×C9—24no—
55C3×D4×C9—41no—
56C2×C3×C4×C9—41no—
57C2×C3×C4×C9C2311yescharacteristic
58C2×C3×C4×C9C2312yes—
59C2×C3×C4×C9C2311yescharacteristic
60C2×C3×Dic9—22no—
61C2×C3×Dic9—42no—
62C2×C3×Dic9—41no—
63C2×C3×Dic9C2312yes—
64C2×C3×Dic9C2312yes—
65C2×C3×Dic9C2312yes—
66C2×C3×Dic9C2312yes—
67C3×Dic18—41no—
68C3×#539—91no—
69#5602—31no—
70C2×C32×D4C2×D312yes—
71C2×C32×D4C2×D311yescharacteristic
72C22×C32×C4C2×D311yescharacteristic
73C22×C3×Dic3—31no—
74C22×C3×Dic3—32no—
75C22×C3×Dic3—31no—
76C2×C3×Dic6—31no—
77C32×C4⋊C4C2×D311yescharacteristic
78C32×C22⋊C4C2×D312yes—
79C3×#217—32no—
80C3×#217—32no—
81C3×#217—32no—
82C3×#217—32no—
83C3×C4⋊Dic3—32no—
84C3×C4⋊Dic3—31no—
85C3×Dic3⋊C4—32no—
86C3×Dic3⋊C4—32no—
87C3×Dic3⋊C4—32no—
88C3×Dic3⋊C4—32no—
89C3×Dic3⋊C4—32no—
90C3×C4×Dic3—32no—
91C2×D4×C9—21no—
92C2×D4×C9—22no—
93C2×D4×C9C22×C312yes—
94C2×D4×C9C22×C311yescharacteristic
95C22×C4×C9—21no—
96C22×C4×C9C22×C311yescharacteristic
97C22×Dic9C22×C311yescharacteristic
98C22×Dic9C22×C311yescharacteristic
99C22×Dic9C22×C312yes—
100C2×Dic18C22×C311yescharacteristic
101C9×C4⋊C4—21no—
102C9×C4⋊C4C22×C311yescharacteristic
103C9×C22⋊C4—22no—
104C9×C22⋊C4C22×C312yes—
105#3676C22×C312yes—
106#3676C22×C312yes—
107#3676C22×C312yes—
108#3676C22×C312yes—
109C4⋊Dic9C22×C312yes—
110C4⋊Dic9C22×C311yescharacteristic
111Dic9⋊C4C22×C312yes—
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—22no—
120C22×C3×C9—22no—
121C22×C3×C9—22no—
122C22×C3×C9#4112yes—
123C22×C3×C9C2411yescharacteristic
124C3×C4×C9—22no—
125C3×C4×C9—21no—
126C3×C4×C9—21no—
127C3×Dic9—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—91no—
136C3×C22⋊Q8—92no—
137C3×#121—21no—
138C3×#121D911yescharacteristic
139C3×#121C3×D311yescharacteristic
140C3×C4×D4—92no—
141C2×C3×C4⋊C4—91no—
142#935—32no—
143#935—32no—
144D4×Dic3—32no—
145#924—31no—
146C2×Dic3⋊C4—31no—
147#889—32no—
148#879—32no—
149#878—32no—
150C23×C32C22×D312yes—
151C23×C32C22×D311yescharacteristic
152C32×D4—24no—
153C32×D4—41no—
154C2×C32×C4—41no—
155C2×C32×C4C22×D312yes—
156C2×C32×C4C22×D311yescharacteristic
157C2×C32×C4C22×D311yescharacteristic
158C2×C3×Dic3—32no—
159C2×C3×Dic3—32no—
160C2×C3×Dic3—32no—
161C2×C3×Dic3—32no—
162C2×C3×Dic3—62no—
163C2×C3×Dic3—122no—
164C2×C3×Dic3—121no—
165C3×Dic6—121no—
166C23×C9—21no—
167C23×C9—22no—
168C23×C9C23×C312yes—
169C23×C9C23×C311yescharacteristic
170D4×C9—24no—
171D4×C9—44no—
172D4×C9—41no—
173D4×C9—42no—
174C2×C4×C9—21no—
175C2×C4×C9—21no—
176C2×C4×C9—22no—
177C2×C4×C9—41no—
178C2×C4×C9—42no—
179C2×C4×C9C23×C312yes—
180C2×C4×C9C23×C311yescharacteristic
181C2×C4×C9C23×C311yescharacteristic
182C2×Dic9—22no—
183C2×Dic9—42no—
184C2×Dic9—41no—
185C2×Dic9C23×C312yes—
186C2×Dic9C23×C312yes—
187C2×Dic9C23×C312yes—
188C2×Dic9C23×C312yes—
189Dic18—41no—
190#539—91noSylow
191C2×C3×C9—22no—
192C2×C3×C9—42no—
193C2×C3×C9C2×#4111yescharacteristic
194C2×C3×C9H(5,𝔽2)11yescharacteristic
195C2×C3×C92−1+411yescharacteristic
196C2×C3×Q8—91no—
197C2×C3×D4—22no—
198C2×C3×D4—21no—
199C2×C3×D4C2×D912yes—
200C2×C3×D4C2×D911yescharacteristic
201C2×C3×D4C2×C3×D312yes—
202C2×C3×D4C2×C3×D311yescharacteristic
203C22×C3×C4—21no—
204C22×C3×C4—91no—
205C22×C3×C4—91no—
206C22×C3×C4—92no—
207C22×C3×C4C2×D911yescharacteristic
208C22×C3×C4C2×C3×D311yescharacteristic
209C22×Dic3—31no—
210C22×Dic3—32no—
211C22×Dic3—31no—
212C2×Dic6—31no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—92no—
215C3×C4⋊C4—92no—
216C3×C4⋊C4—92no—
217C3×C4⋊C4—92no—
218C3×C4⋊C4—92no—
219C3×C4⋊C4—92no—
220C3×C4⋊C4—91no—
221C3×C4⋊C4C2×D911yescharacteristic
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#217—32no—
232#217—32no—
233#217—32no—
234#217—32no—
235C4⋊Dic3—32no—
236C4⋊Dic3—31no—
237Dic3⋊C4—32no—
238Dic3⋊C4—32no—
239Dic3⋊C4—32no—
240Dic3⋊C4—32no—
241Dic3⋊C4—32no—
242C4×Dic3—32no—
243C22×C32—21no—
244C22×C32—21no—
245C22×C32—22no—
246C22×C32—22no—
247C22×C32—22no—
248C22×C32#23712yes—
249C22×C32C23×D311yescharacteristic
250C32×C4—21no—
251C32×C4—22no—
252C32×C4—21no—
253C3×Dic3—62no—
254C3×Dic3—62no—
255C3×Dic3—62no—
256C3×Dic3—62no—
257C22×C9—21no—
258C22×C9—21no—
259C22×C9—22no—
260C22×C9—22no—
261C22×C9—22no—
262C22×C9—21no—
263C22×C9—22no—
264C22×C9—41no—
265C22×C9—41no—
266C22×C9—42no—
267C22×C9—42no—
268C22×C9—42no—
269C22×C9C3×#4112yes—
270C22×C9C24×C311yescharacteristic
271C4×C9—22no—
272C4×C9—22no—
273C4×C9—21no—
274C4×C9—21no—
275C4×C9—41no—
276C4×C9—42no—
277Dic9—22no—
278Dic9—22no—
279Dic9—22no—
280Dic9—22no—
281#126—92no—
282#125—92no—
283#123—92no—
284#123—92no—
285C22⋊Q8—91no—
286C22⋊Q8—92no—
287#121C3×D911yescharacteristic
288C4×D4—92no—
289C2×C4⋊C4—91no—
290C3×C9#53911yescharacteristic, complemented, Sylow
291C23×C3—21no—
292C23×C3—22no—
293C23×C3C22×D912yes—
294C23×C3C22×D911yescharacteristic
295C23×C3C22×C3×D312yes—
296C23×C3C22×C3×D311yescharacteristic
297C3×Q8—361no—
298C3×D4—24no—
299C3×D4—24no—
300C3×D4—44no—
301C3×D4—41no—
302C3×D4—41no—
303C3×D4—42no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—22no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—42no—
310C2×C3×C4—92no—
311C2×C3×C4—92no—
312C2×C3×C4—92no—
313C2×C3×C4—92no—
314C2×C3×C4—182no—
315C2×C3×C4—361no—
316C2×C3×C4—362no—
317C2×C3×C4C22×D912yes—
318C2×C3×C4C22×D911yescharacteristic
319C2×C3×C4C22×D911yescharacteristic
320C2×C3×C4C22×C3×D312yes—
321C2×C3×C4C22×C3×D311yescharacteristic
322C2×C3×C4C22×C3×D311yescharacteristic
323C2×Dic3—32no—
324C2×Dic3—32no—
325C2×Dic3—32no—
326C2×Dic3—32no—
327C2×Dic3—62no—
328C2×Dic3—122no—
329C2×Dic3—121no—
330Dic6—121no—
331C2×C32—22no—
332C2×C32—42no—
333C2×C32C2×#23711yescharacteristic
334C2×C32#100411yescharacteristic
335C2×C32#101011yescharacteristic
336C2×C9—22no—
337C2×C9—21no—
338C2×C9—21no—
339C2×C9—21no—
340C2×C9—42no—
341C2×C9—42no—
342C2×C9—82no—
343C2×C9C2×C3×#4111yescharacteristic
344C2×C9C3×H(5,𝔽2)11yescharacteristic
345C2×C9C3×2−1+411yescharacteristic
346C2×Q8—91no—
347C2×D4C2×C3×D912yes—
348C2×D4C2×C3×D911yescharacteristic
349C22×C4—91no—
350C22×C4—91no—
351C22×C4—92no—
352C22×C4C2×C3×D911yescharacteristic
353C4⋊C4—92no—
354C4⋊C4—92no—
355C4⋊C4—92no—
356C4⋊C4—92no—
357C4⋊C4—92no—
358C4⋊C4—91no—
359C4⋊C4—92no—
360C4⋊C4C2×C3×D911yescharacteristic
361C22⋊C4—92no—
362C22⋊C4—92no—
363C22⋊C4—92no—
364C22⋊C4—92no—
365C22⋊C4C2×C3×D912yes—
366C42—92no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—22no—
372C22×C3—22no—
373C22×C3—22no—
374C22×C3—22no—
375C22×C3—22no—
376C22×C3—22no—
377C22×C3—21no—
378C22×C3—22no—
379C22×C3—41no—
380C22×C3—41no—
381C22×C3—42no—
382C22×C3—42no—
383C22×C3—42no—
384C22×C3#369912yes—
385C22×C3C23×D911yescharacteristic
386C22×C3C3×#23712yes—
387C22×C3C23×C3×D311yescharacteristic
388C3×C4—22no—
389C3×C4—22no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—22no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—41no—
396C3×C4—42no—
397C3×C4—182no—
398C3×C4—182no—
399C3×C4—182no—
400C3×C4—182no—
401Dic3—62no—
402Dic3—62no—
403Dic3—62no—
404Dic3—62no—
405C32#560211yescharacteristic
406C9—21no—
407C9C3×#53911yescharacteristic, complemented
408C23C22×C3×D912yes—
409C23C22×C3×D911yescharacteristic
410Q8—361no—
411D4—24no—
412D4—41no—
413C2×C4—41no—
414C2×C4—92no—
415C2×C4—92no—
416C2×C4—92no—
417C2×C4—92no—
418C2×C4—182no—
419C2×C4—361no—
420C2×C4—362no—
421C2×C4C22×C3×D912yes—
422C2×C4C22×C3×D911yescharacteristic
423C2×C4C22×C3×D911yescharacteristic
424C2×C3—22no—
425C2×C3—22no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—21no—
429C2×C3—42no—
430C2×C3—42no—
431C2×C3—42no—
432C2×C3—82no—
433C2×C3C2×#369911yescharacteristic
434C2×C3#6371011yescharacteristic
435C2×C3#6371611yescharacteristic
436C2×C3C2×C3×#23711yescharacteristic
437C2×C3C3×#100411yescharacteristic
438C2×C3C3×#101011yescharacteristic
439C22—21no—
440C22—21no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22C3×#369912yes—
445C22C23×C3×D911yescharacteristic
446C4—22no—
447C4—21no—
448C4—21no—
449C4—182no—
450C4—182no—
451C4—182no—
452C4—182no—
453C3—21no—
454C3#1058970411yescharacteristic, direct product factor
455C3C3×#560211yescharacteristic
456C2—22no—
457C2—42no—
458C2C2×C3×#369911yescharacteristic
459C2C3×#6371011yescharacteristic
460C2C3×#6371611yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×#121C2
socleC22×C32C23×D3
prime radicalC32×#121D3
hypercenterC3×#121D9
hypocenterC9C3×#539
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×#121

Complements

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

NH ≅ G/Ntype
2. C3×#63497C2semidirect
3. C3×#63497C2semidirect
4. C3×D4×Dic9C2semidirect
5. C3×#63486C2semidirect
6. C2×C3×Dic9⋊C4C2semidirect
7. C3×#63451C2semidirect
8. C3×#63441C2semidirect
9. C3×#63440C2semidirect
11. #10589704C3direct (N × H)
44. #63497C2×C3semidirect
45. #63497C2×C3semidirect
46. D4×Dic9C2×C3semidirect
47. #63486C2×C3semidirect
48. C2×Dic9⋊C4C2×C3semidirect
49. #63451C2×C3semidirect
50. #63441C2×C3semidirect
51. #63440C2×C3semidirect
290. C3×C9#539semidirect
407. C9C3×#539semidirect
454. C3#10589704direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C9×#1212
C3×#634972
C3×D4×Dic92
C3×#634862
C2×C3×Dic9⋊C42
C3×#634512
C3×#634412
C3×#634402
C3×#56023
#105897043