Finite Group Databasebeta

← C3×#10589772 #13153077

Subgroups — C3×#10589772 #13153077

Order 1728.

784 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×#124C211yesmaximal, characteristic, complemented
2C3×#63509C211yesmaximal, characteristic, complemented
3C3×#63506C211yesmaximal, characteristic, complemented
4C3×#63501C212yesmaximal, complemented
5C3×#63447C212yesmaximal, complemented
6C3×#63446C212yesmaximal, complemented
7C3×#611⋊C4C212yesmaximal, complemented
8C3×#63439C212yesmaximal, complemented
9C3×#63434C212yesmaximal, complemented
10C3×#5670—31nomaximal
11#10589772C311yesmaximal, characteristic, direct product factor
12C2×C3×Q8×C9C2211yescharacteristic
13C2×C3×D4×C9C2211yescharacteristic
14C2×C3×#611C2212yes—
15C2×C3×#611C2212yes—
16C22×C3×Dic9C2212yes—
17C2×C3×D36C2211yescharacteristic
18C2×C3×C4×D9C2212yes—
19C2×C3×Dic18C2211yescharacteristic
20C3×C9×C22⋊C4C2212yes—
21C3×C9×C22⋊C4C2212yes—
22C3×C42×C9C2211yescharacteristic
23C3×#3676C2212yes—
24C3×(C2×D9)⋊C4C2212yes—
25C3×(C2×D9)⋊C4C2212yes—
26C3×(C2×D9)⋊C4C2212yes—
27C3×Dic9⋊C4C2212yes—
28C3×Dic9⋊C4C2212yes—
29C3×Dic9⋊C4C2212yes—
30C3×C4×Dic9C2212yes—
31C3×C4×Dic9C2211yescharacteristic
32C3×C4×Dic9C2211yescharacteristic
33C32×#124D311yescharacteristic
34C3×#947—31no—
35C3×#944—31no—
36C3×#939—32no—
37C3×#885—32no—
38C3×#884—32no—
39C3×#67⋊C4—32no—
40C3×#877—32no—
41C3×#872—32no—
42C9×#124—21no—
43C9×#124C2×C311yescharacteristic, complemented
44#63509C2×C311yescharacteristic, complemented
45#63506C2×C311yescharacteristic, complemented
46#63501C2×C312yescomplemented
47#63447C2×C312yescomplemented
48#63446C2×C312yescomplemented
49#611⋊C4C2×C312yescomplemented
50#63439C2×C312yescomplemented
51#63434C2×C312yescomplemented
52C23×C3×C9C2312yes—
53C22×C3×D9C2312yes—
54C3×Q8×C9—41no—
55C3×D4×C9—41no—
56C2×C3×C4×C9C2311yescharacteristic
57C2×C3×C4×C9C2312yes—
58C2×C3×C4×C9C2311yescharacteristic
59C2×C3×C4×C9C2311yescharacteristic
60C3×#611—24no—
61C3×#611—42no—
62C2×C3×Dic9—24no—
63C2×C3×Dic9C2312yes—
64C2×C3×Dic9C2312yes—
65C2×C3×Dic9C2312yes—
66C3×D36—41no—
67C3×C4×D9—24no—
68C3×Dic18—41no—
69C3×#555—91no—
70#5670—31no—
71C2×C32×Q8C2×D311yescharacteristic
72C2×C32×D4C2×D311yescharacteristic
73C2×C3×#67—32no—
74C2×C3×#67—32no—
75C22×C3×Dic3—32no—
76C2×C3×D12—31no—
77C2×C3×C4×D3—32no—
78C2×C3×Dic6—31no—
79C32×C22⋊C4C2×D312yes—
80C32×C22⋊C4C2×D312yes—
81C32×C42C2×D311yescharacteristic
82C3×#217—32no—
83C3×(C2×D3)⋊C4—32no—
84C3×(C2×D3)⋊C4—32no—
85C3×(C2×D3)⋊C4—32no—
86C3×Dic3⋊C4—32no—
87C3×Dic3⋊C4—32no—
88C3×Dic3⋊C4—32no—
89C3×C4×Dic3—31no—
90C3×C4×Dic3—32no—
91C3×C4×Dic3—31no—
92C2×Q8×C9—21no—
93C2×Q8×C9C22×C311yescharacteristic
94C2×D4×C9—21no—
95C2×D4×C9C22×C311yescharacteristic
96C2×#611C22×C312yes—
97C2×#611C22×C312yes—
98C22×Dic9C22×C312yes—
99C2×D36C22×C311yescharacteristic
100C2×C4×D9C22×C312yes—
101C2×Dic18C22×C311yescharacteristic
102C9×C22⋊C4—22no—
103C9×C22⋊C4—22no—
104C9×C22⋊C4C22×C312yes—
105C9×C22⋊C4C22×C312yes—
106C42×C9—21no—
107C42×C9C22×C311yescharacteristic
108#3676C22×C312yes—
109(C2×D9)⋊C4C22×C312yes—
110(C2×D9)⋊C4C22×C312yes—
111(C2×D9)⋊C4C22×C312yes—
112Dic9⋊C4C22×C312yes—
113Dic9⋊C4C22×C312yes—
114Dic9⋊C4C22×C312yes—
115C4×Dic9C22×C311yescharacteristic
116C4×Dic9C22×C311yescharacteristic
117C4×Dic9C22×C312yes—
118C22×C3×C9—22no—
119C22×C3×C9—22no—
120C22×C3×C9—22no—
121C22×C3×C9C2411yescharacteristic
122C2×C3×D9—22no—
123C2×C3×D9—22no—
124C2×C3×D9—22no—
125C3×C4×C9—22no—
126C3×C4×C9—21no—
127C3×C4×C9—21no—
128C3×C4×C9—21no—
129C3×Dic9—22no—
130C3×Dic9—22no—
131C3×Dic9#4114yes—
132C3×C4⋊Q8—91no—
133C3×#124—21no—
134C3×#124—91no—
135C3×#124—92no—
136C3×#124D911yescharacteristic, complemented
137C3×#124C3×D311yescharacteristic
138C3×#121—92no—
139C3×#121—92no—
140C3×C4×D4—92no—
141C3×#117—92no—
142C3×#117—92no—
143#947—31no—
144#944—31no—
145#939—32no—
146#885—32no—
147#884—32no—
148#67⋊C4—32no—
149#877—32no—
150#872—32no—
151C23×C32C22×D312yes—
152C22×C3×D3—32no—
153C32×Q8—41no—
154C32×D4—41no—
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C22×D312yes—
157C2×C32×C4C22×D311yescharacteristic
158C2×C32×C4C22×D311yescharacteristic
159C3×#67—64no—
160C3×#67—122no—
161C2×C3×Dic3—32no—
162C2×C3×Dic3—32no—
163C2×C3×Dic3—32no—
164C2×C3×Dic3—64no—
165C3×D12—121no—
166C3×C4×D3—64no—
167C3×Dic6—121no—
168C23×C9—22no—
169C23×C9C23×C312yes—
170C22×D9C23×C312yes—
171Q8×C9—41no—
172Q8×C9—42no—
173D4×C9—41no—
174D4×C9—42no—
175C2×C4×C9—21no—
176C2×C4×C9—22no—
177C2×C4×C9—21no—
178C2×C4×C9—21no—
179C2×C4×C9C23×C311yescharacteristic
180C2×C4×C9C23×C312yes—
181C2×C4×C9C23×C311yescharacteristic
182C2×C4×C9C23×C311yescharacteristic
183#611—24no—
184#611—42no—
185C2×Dic9—24no—
186C2×Dic9C23×C312yes—
187C2×Dic9C23×C312yes—
188C2×Dic9C23×C312yes—
189D36—41no—
190C4×D9—24no—
191Dic18—41no—
192#555—91noSylow
193C2×C3×C9—42no—
194C2×C3×C9C2×#4112yes—
195C2×C3×C9H(5,𝔽2)11yescharacteristic
196C3×D9—42no—
197C2×C3×Q8—21no—
198C2×C3×Q8—91no—
199C2×C3×Q8C2×D911yescharacteristic
200C2×C3×Q8C2×C3×D311yescharacteristic
201C2×C3×D4—21no—
202C2×C3×D4—92no—
203C2×C3×D4—92no—
204C2×C3×D4—91no—
205C2×C3×D4C2×D911yescharacteristic
206C2×C3×D4C2×C3×D311yescharacteristic
207C22×C3×C4—92no—
208C22×C3×C4—92no—
209C2×#67—32no—
210C2×#67—32no—
211C22×Dic3—32no—
212C2×D12—31no—
213C2×C4×D3—32no—
214C2×Dic6—31no—
215C3×C4⋊C4—92no—
216C3×C4⋊C4—92no—
217C3×C4⋊C4—92no—
218C3×C22⋊C4—22no—
219C3×C22⋊C4—22no—
220C3×C22⋊C4—92no—
221C3×C22⋊C4—92no—
222C3×C22⋊C4—92no—
223C3×C22⋊C4—92no—
224C3×C22⋊C4C2×D912yes—
225C3×C22⋊C4C2×D912yes—
226C3×C22⋊C4C2×C3×D312yes—
227C3×C22⋊C4C2×C3×D312yes—
228C3×C42—21no—
229C3×C42—91no—
230C3×C42—92no—
231C3×C42—91no—
232C3×C42C2×D911yescharacteristic
233C3×C42C2×C3×D311yescharacteristic
234#217—32no—
235(C2×D3)⋊C4—32no—
236(C2×D3)⋊C4—32no—
237(C2×D3)⋊C4—32no—
238Dic3⋊C4—32no—
239Dic3⋊C4—32no—
240Dic3⋊C4—32no—
241C4×Dic3—32no—
242C4×Dic3—31no—
243C4×Dic3—31no—
244C22×C32—22no—
245C22×C32—22no—
246C22×C32—22no—
247C22×C32C23×D311yescharacteristic
248C2×C3×D3—62no—
249C2×C3×D3—62no—
250C2×C3×D3—62no—
251C32×C4—21no—
252C32×C4—21no—
253C32×C4—21no—
254C32×C4—22no—
255C3×Dic3—34no—
256C3×Dic3—62no—
257C3×Dic3—62no—
258C22×C9—22no—
259C22×C9—22no—
260C22×C9—22no—
261C22×C9—21no—
262C22×C9—42no—
263C22×C9—42no—
264C22×C9—42no—
265C22×C9C24×C311yescharacteristic
266C2×D9—22no—
267C2×D9—22no—
268C2×D9—22no—
269C4×C9—22no—
270C4×C9—22no—
271C4×C9—21no—
272C4×C9—21no—
273C4×C9—21no—
274C4×C9—42no—
275C4×C9—41no—
276C4×C9—41no—
277Dic9—22no—
278Dic9—22no—
279Dic9C3×#4114yes—
280C4⋊Q8—91no—
281#124—91no—
282#124—92no—
283#124C3×D911yescharacteristic, complemented
284#121—92no—
285#121—92no—
286C4×D4—92no—
287#117—92no—
288#117—92no—
289C3×C9#55511yescharacteristic, complemented, Sylow
290C23×C3—22no—
291C23×C3—92no—
292C23×C3C22×D912yes—
293C23×C3C22×C3×D312yes—
294C22×D3—32no—
295C3×Q8—41no—
296C3×Q8—41no—
297C3×Q8—42no—
298C3×Q8—361no—
299C3×D4—41no—
300C3×D4—41no—
301C3×D4—42no—
302C3×D4—184no—
303C3×D4—361no—
304C3×D4—362no—
305C2×C3×C4—21no—
306C2×C3×C4—22no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—92no—
310C2×C3×C4—92no—
311C2×C3×C4—92no—
312C2×C3×C4—184no—
313C2×C3×C4—184no—
314C2×C3×C4C22×D912yes—
315C2×C3×C4C22×D911yescharacteristic
316C2×C3×C4C22×D911yescharacteristic
317C2×C3×C4C22×D911yescharacteristic
318C2×C3×C4C22×C3×D311yescharacteristic
319C2×C3×C4C22×C3×D312yes—
320C2×C3×C4C22×C3×D311yescharacteristic
321C2×C3×C4C22×C3×D311yescharacteristic
322#67—64no—
323#67—122no—
324C2×Dic3—32no—
325C2×Dic3—32no—
326C2×Dic3—32no—
327C2×Dic3—64no—
328D12—121no—
329C4×D3—64no—
330Dic6—121no—
331C2×C32—42no—
332C2×C32#100411yescharacteristic
333C2×C32D3×#4112yes—
334C3×D3—122no—
335C2×C9—21no—
336C2×C9—22no—
337C2×C9—42no—
338C2×C9—82no—
339C2×C9C2×C3×#4112yes—
340C2×C9C3×H(5,𝔽2)11yescharacteristic
341D9—42no—
342C2×Q8—91no—
343C2×Q8C2×C3×D911yescharacteristic
344C2×D4—92no—
345C2×D4—92no—
346C2×D4—91no—
347C2×D4C2×C3×D911yescharacteristic
348C22×C4—92no—
349C22×C4—92no—
350C4⋊C4—92no—
351C4⋊C4—92no—
352C4⋊C4—92no—
353C22⋊C4—92no—
354C22⋊C4—92no—
355C22⋊C4—92no—
356C22⋊C4—92no—
357C22⋊C4C2×C3×D912yes—
358C22⋊C4C2×C3×D912yes—
359C42—91no—
360C42—92no—
361C42—91no—
362C42C2×C3×D911yescharacteristic
363C22×C3—22no—
364C22×C3—22no—
365C22×C3—22no—
366C22×C3—22no—
367C22×C3—22no—
368C22×C3—22no—
369C22×C3—21no—
370C22×C3—42no—
371C22×C3—42no—
372C22×C3—42no—
373C22×C3—182no—
374C22×C3—182no—
375C22×C3—182no—
376C22×C3C23×D911yescharacteristic
377C22×C3C23×C3×D311yescharacteristic
378C2×D3—62no—
379C2×D3—62no—
380C2×D3—62no—
381C3×C4—22no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—22no—
386C3×C4—21no—
387C3×C4—21no—
388C3×C4—21no—
389C3×C4—22no—
390C3×C4—41no—
391C3×C4—41no—
392C3×C4—42no—
393C3×C4—94no—
394C3×C4—182no—
395C3×C4—182no—
396Dic3—34no—
397Dic3—62no—
398Dic3—62no—
399C32#567011yescharacteristic
400C9—21no—
401C9C3×#55511yescharacteristic, complemented
402C23—92no—
403C23C22×C3×D912yes—
404Q8—41no—
405Q8—361no—
406D4—41no—
407D4—184no—
408D4—361no—
409D4—362no—
410C2×C4—92no—
411C2×C4—92no—
412C2×C4—92no—
413C2×C4—184no—
414C2×C4—184no—
415C2×C4C22×C3×D912yes—
416C2×C4C22×C3×D911yescharacteristic
417C2×C4C22×C3×D911yescharacteristic
418C2×C4C22×C3×D911yescharacteristic
419C2×C3—21no—
420C2×C3—22no—
421C2×C3—42no—
422C2×C3—42no—
423C2×C3—82no—
424C2×C3—362no—
425C2×C3#6371011yescharacteristic
426C2×C3#41×D912yes—
427C2×C3C3×#100411yescharacteristic
428C2×C3C3×D3×#4112yes—
429D3—122no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—182no—
434C22—182no—
435C22—182no—
436C22C23×C3×D911yescharacteristic
437C4—21no—
438C4—21no—
439C4—21no—
440C4—22no—
441C4—94no—
442C4—182no—
443C4—182no—
444C3—21no—
445C3#1058977211yescharacteristic, direct product factor
446C3C3×#567011yescharacteristic
447C2—42no—
448C2—362no—
449C2C3×#6371011yescharacteristic
450C2C3×#41×D912yes—
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×#124C2
socleC22×C32C23×D3
hypercenterC3×#124D9
hypocenterC9C3×#555
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×#124

Complements

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

NH ≅ G/Ntype
1. C3×C9×#124C2semidirect
2. C3×#63509C2semidirect
3. C3×#63506C2semidirect
4. C3×#63501C2semidirect
5. C3×#63447C2semidirect
6. C3×#63446C2semidirect
7. C3×#611⋊C4C2semidirect
8. C3×#63439C2semidirect
9. C3×#63434C2semidirect
11. #10589772C3direct (N × H)
43. C9×#124C2×C3semidirect
44. #63509C2×C3semidirect
45. #63506C2×C3semidirect
46. #63501C2×C3semidirect
47. #63447C2×C3semidirect
48. #63446C2×C3semidirect
49. #611⋊C4C2×C3semidirect
50. #63439C2×C3semidirect
51. #63434C2×C3semidirect
136. C3×#124D9semidirect
283. #124C3×D9semidirect
289. C3×C9#555semidirect
401. C9C3×#555semidirect
445. C3#10589772direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C9×#1242
C3×#635092
C3×#635062
C3×#635012
C3×#634472
C3×#634462
C3×#611⋊C42
C3×#634392
C3×#634342
C3×#56703
#105897723