Finite Group Databasebeta

← C3×#10589697 #13152759

Subgroups — C3×#10589697 #13152759

Order 1728.

886 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×#120C211yesmaximal, characteristic, complemented
2C3×#63515C212yesmaximal, complemented
3C3×#63502C211yesmaximal, characteristic, complemented
4C3×#63501C212yesmaximal, complemented
5C3×#63500C212yesmaximal, complemented
6C3×#63497C211yesmaximal, characteristic, complemented
7C3×#63446C212yesmaximal, complemented
8C3×#63445C212yesmaximal, complemented
9C3×#63441C212yesmaximal, complemented
10C3×#5595—31nomaximal
11#10589697C311yesmaximal, characteristic, direct product factor
12C24×C3×C9C2211yescharacteristic
13C2×C3×D4×C9C2211yescharacteristic
14C2×C3×D4×C9C2212yes—
15C2×C3×#611C2212yes—
16C2×C3×#611C2212yes—
17C2×C3×#611C2212yes—
18C2×C3×#611C2212yescomplemented
19C22×C3×Dic9C2211yescharacteristic, complemented
20C2×C3×D36C2211yescharacteristic, complemented
21C2×C3×C4×D9C2212yescomplemented
22C3×C9×C22⋊C4C2212yes—
23C3×C9×C22⋊C4C2211yescharacteristic
24C3×#3676C2212yes—
25C3×#3676C2212yes—
26C3×#3676C2212yes—
27C3×#3676C2211yescharacteristic, complemented
28C3×(C2×D9)⋊C4C2212yescomplemented
29C3×C4⋊Dic9C2212yescomplemented
30C3×Dic9⋊C4C2212yescomplemented
31C3×Dic9⋊C4C2212yescomplemented
32C3×C4×Dic9C2211yescharacteristic, complemented
33C32×#120D311yescharacteristic
34C3×#953—32no—
35C3×#940—31no—
36C3×#939—32no—
37C3×#938—32no—
38C3×#935—31no—
39C3×#884—32no—
40C3×#883—32no—
41C3×#879—32no—
42C9×#120—21no—
43C9×#120C2×C311yescharacteristic, complemented
44#63515C2×C312yescomplemented
45#63502C2×C311yescharacteristic, complemented
46#63501C2×C312yescomplemented
47#63500C2×C312yescomplemented
48#63497C2×C311yescharacteristic, complemented
49#63446C2×C312yescomplemented
50#63445C2×C312yescomplemented
51#63441C2×C312yescomplemented
52C23×C3×C9—42no—
53C23×C3×C9—41no—
54C23×C3×C9C2312yes—
55C23×C3×C9C2311yescharacteristic
56C23×C3×C9C2311yescharacteristic
57C22×C3×D9C2312yes—
58C3×D4×C9—42no—
59C3×D4×C9—41no—
60C2×C3×C4×C9C2312yes—
61C2×C3×C4×C9C2311yescharacteristic
62C3×#611—42no—
63C3×#611—42no—
64C3×#611—42no—
65C3×#611—42no—
66C2×C3×Dic9—41no—
67C2×C3×Dic9C2312yes—
68C2×C3×Dic9C2312yes—
69C2×C3×Dic9C2312yes—
70C3×D36—41no—
71C3×C4×D9—42no—
72C3×#560—91no—
73#5595—31no—
74C24×C32C2×D311yescharacteristic
75C2×C32×D4C2×D311yescharacteristic
76C2×C32×D4C2×D312yes—
77C2×C3×#67—32no—
78C2×C3×#67—32no—
79C2×C3×#67—32no—
80C2×C3×#67—32no—
81C22×C3×Dic3—31no—
82C2×C3×D12—31no—
83C2×C3×C4×D3—32no—
84C24×C9—21no—
85C24×C9C22×C311yescharacteristic
86C32×C22⋊C4C2×D312yes—
87C32×C22⋊C4C2×D311yescharacteristic
88C3×#217—32no—
89C3×#217—32no—
90C3×#217—32no—
91C3×#217—31no—
92C3×(C2×D3)⋊C4—32no—
93C3×C4⋊Dic3—32no—
94C3×Dic3⋊C4—32no—
95C3×Dic3⋊C4—32no—
96C3×C4×Dic3—31no—
97C2×D4×C9—21no—
98C2×D4×C9—22no—
99C2×D4×C9C22×C311yescharacteristic
100C2×D4×C9C22×C312yes—
101C2×#611C22×C312yes—
102C2×#611C22×C312yes—
103C2×#611C22×C312yes—
104C2×#611C22×C312yescomplemented
105C22×Dic9C22×C311yescharacteristic, complemented
106C2×D36C22×C311yescharacteristic, complemented
107C2×C4×D9C22×C312yescomplemented
108C9×C22⋊C4—22no—
109C9×C22⋊C4—21no—
110C9×C22⋊C4C22×C312yes—
111C9×C22⋊C4C22×C311yescharacteristic
112#3676C22×C312yes—
113#3676C22×C312yes—
114#3676C22×C312yes—
115#3676C22×C311yescharacteristic, complemented
116(C2×D9)⋊C4C22×C312yescomplemented
117C4⋊Dic9C22×C312yescomplemented
118Dic9⋊C4C22×C312yescomplemented
119Dic9⋊C4C22×C312yescomplemented
120C4×Dic9C22×C311yescharacteristic, complemented
121C22×C3×C9—22no—
122C22×C3×C9—21no—
123C22×C3×C9—21no—
124C22×C3×C9—22no—
125C22×C3×C9—22no—
126C22×C3×C9—22no—
127C22×C3×C9—22no—
128C22×C3×C9—44no—
129C22×C3×C9C2411yescharacteristic
130C2×C3×D9—22no—
131C2×C3×D9—22no—
132C2×C3×D9—22no—
133C3×C4×C9—22no—
134C3×C4×C9—21no—
135C3×Dic9—22no—
136C3×Dic9—22no—
137C3×Dic9—22no—
138C3×#127—91no—
139C3×#126—92no—
140C3×#123—91no—
141C3×#123—92no—
142C3×#121—92no—
143C3×#121—92no—
144C3×#121—92no—
145C3×#120—21no—
146C3×#120—92no—
147C3×#120D911yescharacteristic, complemented
148C3×#120C3×D311yescharacteristic
149#953—32no—
150#940—31no—
151#939—32no—
152#938—32no—
153#935—31no—
154#884—32no—
155#883—32no—
156#879—32no—
157C23×C32—42no—
158C23×C32—41no—
159C23×C32C22×D312yes—
160C23×C32C22×D311yescharacteristic
161C23×C32C22×D311yescharacteristic
162C22×C3×D3—32no—
163C32×D4—42no—
164C32×D4—41no—
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C22×D312yes—
167C3×#67—122no—
168C3×#67—122no—
169C3×#67—122no—
170C3×#67—122no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3—32no—
174C2×C3×Dic3—121no—
175C3×D12—121no—
176C3×C4×D3—122no—
177C23×C9—22no—
178C23×C9—21no—
179C23×C9—21no—
180C23×C9—42no—
181C23×C9—41no—
182C23×C9—44no—
183C23×C9—42no—
184C23×C9C23×C311yescharacteristic
185C23×C9C23×C312yes—
186C23×C9C23×C311yescharacteristic
187C22×D9C23×C312yes—
188D4×C9—42no—
189D4×C9—41no—
190D4×C9—44no—
191D4×C9—42no—
192C2×C4×C9—21no—
193C2×C4×C9—22no—
194C2×C4×C9C23×C311yescharacteristic
195C2×C4×C9C23×C312yes—
196#611—42no—
197#611—42no—
198#611—42no—
199#611—42no—
200C2×Dic9—41no—
201C2×Dic9C23×C312yes—
202C2×Dic9C23×C312yes—
203C2×Dic9C23×C312yes—
204D36—41no—
205C4×D9—42no—
206#560—91noSylow
207C2×C3×C9—42no—
208C2×C3×C9—41no—
209C2×C3×C9—41no—
210C2×C3×C9H(5,𝔽2)12yes—
211C2×C3×C9H(5,𝔽2)11yescharacteristic
212C3×D9—42no—
213C24×C3—21no—
214C24×C3C2×D911yescharacteristic
215C24×C3C2×C3×D311yescharacteristic
216C2×C3×D4—22no—
217C2×C3×D4—21no—
218C2×C3×D4—92no—
219C2×C3×D4—92no—
220C2×C3×D4—92no—
221C2×C3×D4—92no—
222C2×C3×D4—91no—
223C2×C3×D4C2×D912yes—
224C2×C3×D4C2×D911yescharacteristic
225C2×C3×D4C2×C3×D311yescharacteristic
226C2×C3×D4C2×C3×D312yes—
227C22×C3×C4—91no—
228C22×C3×C4—92no—
229C2×#67—32no—
230C2×#67—32no—
231C2×#67—32no—
232C2×#67—32no—
233C22×Dic3—31no—
234C2×D12—31no—
235C2×C4×D3—32no—
236C3×C4⋊C4—92no—
237C3×C4⋊C4—92no—
238C3×C4⋊C4—92no—
239C3×C22⋊C4—22no—
240C3×C22⋊C4—21no—
241C3×C22⋊C4—92no—
242C3×C22⋊C4—92no—
243C3×C22⋊C4—92no—
244C3×C22⋊C4—91no—
245C3×C22⋊C4—92no—
246C3×C22⋊C4C2×D912yes—
247C3×C22⋊C4C2×D911yescharacteristic
248C3×C22⋊C4C2×C3×D312yes—
249C3×C22⋊C4C2×C3×D311yescharacteristic
250C3×C42—91no—
251#217—32no—
252#217—32no—
253#217—32no—
254#217—31no—
255(C2×D3)⋊C4—32no—
256C4⋊Dic3—32no—
257Dic3⋊C4—32no—
258Dic3⋊C4—32no—
259C4×Dic3—31no—
260C22×C32—22no—
261C22×C32—21no—
262C22×C32—21no—
263C22×C32—22no—
264C22×C32—22no—
265C22×C32—22no—
266C22×C32—22no—
267C22×C32—44no—
268C22×C32C23×D311yescharacteristic
269C2×C3×D3—62no—
270C2×C3×D3—62no—
271C2×C3×D3—62no—
272C32×C4—22no—
273C32×C4—21no—
274C3×Dic3—62no—
275C3×Dic3—62no—
276C3×Dic3—62no—
277C22×C9—24no—
278C22×C9—22no—
279C22×C9—21no—
280C22×C9—22no—
281C22×C9—22no—
282C22×C9—22no—
283C22×C9—22no—
284C22×C9—22no—
285C22×C9—21no—
286C22×C9—21no—
287C22×C9—42no—
288C22×C9—42no—
289C22×C9—42no—
290C22×C9—41no—
291C22×C9—42no—
292C22×C9—48no—
293C22×C9—44no—
294C22×C9C24×C311yescharacteristic
295C2×D9—22no—
296C2×D9—22no—
297C2×D9—22no—
298C4×C9—22no—
299C4×C9—21no—
300C4×C9—42no—
301C4×C9—41no—
302Dic9—22no—
303Dic9—22no—
304Dic9—22no—
305#127—91no—
306#126—92no—
307#123—91no—
308#123—92no—
309#121—92no—
310#121—92no—
311#121—92no—
312#120—92no—
313#120C3×D911yescharacteristic, complemented
314C3×C9#56011yescharacteristic, complemented, Sylow
315C23×C3—22no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—42no—
319C23×C3—41no—
320C23×C3—42no—
321C23×C3—41no—
322C23×C3—44no—
323C23×C3—42no—
324C23×C3—92no—
325C23×C3C22×D912yes—
326C23×C3C22×D911yescharacteristic
327C23×C3C22×D911yescharacteristic
328C23×C3C22×C3×D312yes—
329C23×C3C22×C3×D311yescharacteristic
330C23×C3C22×C3×D311yescharacteristic
331C22×D3—32no—
332C3×D4—42no—
333C3×D4—41no—
334C3×D4—42no—
335C3×D4—41no—
336C3×D4—44no—
337C3×D4—42no—
338C3×D4—361no—
339C3×D4—362no—
340C3×D4—362no—
341C3×D4—362no—
342C3×D4—362no—
343C2×C3×C4—21no—
344C2×C3×C4—22no—
345C2×C3×C4—92no—
346C2×C3×C4—92no—
347C2×C3×C4—92no—
348C2×C3×C4—362no—
349C2×C3×C4—361no—
350C2×C3×C4C22×D911yescharacteristic
351C2×C3×C4C22×D912yes—
352C2×C3×C4C22×C3×D311yescharacteristic
353C2×C3×C4C22×C3×D312yes—
354#67—122no—
355#67—122no—
356#67—122no—
357#67—122no—
358C2×Dic3—32no—
359C2×Dic3—32no—
360C2×Dic3—32no—
361C2×Dic3—121no—
362D12—121no—
363C4×D3—122no—
364C2×C32—42no—
365C2×C32—41no—
366C2×C32—41no—
367C2×C32#100412yes—
368C2×C32#100411yescharacteristic
369C3×D3—122no—
370C2×C9—22no—
371C2×C9—21no—
372C2×C9—44no—
373C2×C9—42no—
374C2×C9—41no—
375C2×C9—42no—
376C2×C9—41no—
377C2×C9—81no—
378C2×C9C3×H(5,𝔽2)12yes—
379C2×C9C3×H(5,𝔽2)11yescharacteristic
380D9—42no—
381C24C2×C3×D911yescharacteristic
382C2×D4—92no—
383C2×D4—92no—
384C2×D4—92no—
385C2×D4—92no—
386C2×D4—91no—
387C2×D4C2×C3×D912yes—
388C2×D4C2×C3×D911yescharacteristic
389C22×C4—91no—
390C22×C4—92no—
391C4⋊C4—92no—
392C4⋊C4—92no—
393C4⋊C4—92no—
394C22⋊C4—92no—
395C22⋊C4—92no—
396C22⋊C4—92no—
397C22⋊C4—91no—
398C22⋊C4—92no—
399C22⋊C4C2×C3×D912yes—
400C22⋊C4C2×C3×D911yescharacteristic
401C42—91no—
402C22×C3—24no—
403C22×C3—22no—
404C22×C3—21no—
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—22no—
408C22×C3—22no—
409C22×C3—22no—
410C22×C3—21no—
411C22×C3—21no—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—21no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—21no—
419C22×C3—42no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—41no—
423C22×C3—42no—
424C22×C3—44no—
425C22×C3—48no—
426C22×C3—44no—
427C22×C3—182no—
428C22×C3—182no—
429C22×C3—182no—
430C22×C3C23×D911yescharacteristic
431C22×C3C23×C3×D311yescharacteristic
432C2×D3—62no—
433C2×D3—62no—
434C2×D3—62no—
435C3×C4—22no—
436C3×C4—21no—
437C3×C4—22no—
438C3×C4—21no—
439C3×C4—42no—
440C3×C4—41no—
441C3×C4—182no—
442C3×C4—182no—
443C3×C4—182no—
444Dic3—62no—
445Dic3—62no—
446Dic3—62no—
447C32#559511yescharacteristic
448C9—21no—
449C9C3×#56011yescharacteristic, complemented
450C23—42no—
451C23—41no—
452C23—92no—
453C23C22×C3×D912yes—
454C23C22×C3×D911yescharacteristic
455C23C22×C3×D911yescharacteristic
456D4—42no—
457D4—41no—
458D4—361no—
459D4—362no—
460D4—362no—
461D4—362no—
462D4—362no—
463C2×C4—92no—
464C2×C4—92no—
465C2×C4—92no—
466C2×C4—362no—
467C2×C4—361no—
468C2×C4C22×C3×D911yescharacteristic
469C2×C4C22×C3×D912yes—
470C2×C3—22no—
471C2×C3—21no—
472C2×C3—44no—
473C2×C3—42no—
474C2×C3—42no—
475C2×C3—41no—
476C2×C3—41no—
477C2×C3—42no—
478C2×C3—41no—
479C2×C3—41no—
480C2×C3—81no—
481C2×C3—362no—
482C2×C3#6371012yes—
483C2×C3#6371011yescharacteristic
484C2×C3C3×#100412yes—
485C2×C3C3×#100411yescharacteristic
486D3—122no—
487C22—21no—
488C22—22no—
489C22—22no—
490C22—21no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—44no—
495C22—182no—
496C22—182no—
497C22—182no—
498C22C23×C3×D911yescharacteristic
499C4—22no—
500C4—21no—
501C4—182no—
502C4—182no—
503C4—182no—
504C3—21no—
505C3#1058969711yescharacteristic, direct product factor
506C3C3×#559511yescharacteristic
507C2—42no—
508C2—41no—
509C2—41no—
510C2—362no—
511C2C3×#6371012yes—
512C2C3×#6371011yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×#120C2
socleC22×C32C23×D3
hypercenterC3×#120D9
hypocenterC9C3×#560
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×#120

Complements

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

NH ≅ G/Ntype
1. C3×C9×#120C2semidirect
2. C3×#63515C2semidirect
3. C3×#63502C2semidirect
4. C3×#63501C2semidirect
5. C3×#63500C2semidirect
6. C3×#63497C2semidirect
7. C3×#63446C2semidirect
8. C3×#63445C2semidirect
9. C3×#63441C2semidirect
11. #10589697C3direct (N × H)
18. C2×C3×#611C22semidirect
19. C22×C3×Dic9C22semidirect
20. C2×C3×D36C22semidirect
21. C2×C3×C4×D9C22semidirect
27. C3×#3676C22semidirect
28. C3×(C2×D9)⋊C4C22semidirect
29. C3×C4⋊Dic9C22semidirect
30. C3×Dic9⋊C4C22semidirect
31. C3×Dic9⋊C4C22semidirect
32. C3×C4×Dic9C22semidirect
43. C9×#120C2×C3semidirect
44. #63515C2×C3semidirect
45. #63502C2×C3semidirect
46. #63501C2×C3semidirect
47. #63500C2×C3semidirect
48. #63497C2×C3semidirect
49. #63446C2×C3semidirect
50. #63445C2×C3semidirect
51. #63441C2×C3semidirect
104. C2×#611C22×C3semidirect
105. C22×Dic9C22×C3semidirect
106. C2×D36C22×C3semidirect
107. C2×C4×D9C22×C3semidirect
115. #3676C22×C3semidirect
116. (C2×D9)⋊C4C22×C3semidirect
117. C4⋊Dic9C22×C3semidirect
118. Dic9⋊C4C22×C3semidirect
119. Dic9⋊C4C22×C3semidirect
120. C4×Dic9C22×C3semidirect
147. C3×#120D9semidirect
313. #120C3×D9semidirect
314. C3×C9#560semidirect
449. C9C3×#560semidirect
505. C3#10589697direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C9×#1202
C3×#635152
C3×#635022
C3×#635012
C3×#635002
C3×#634972
C3×#634462
C3×#634452
C3×#634412
C3×#55953
#105896973