Finite Group Databasebeta

← C3×#10589702 #13152791

Subgroups — C3×#10589702 #13152791

Order 1728.

796 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×#121C211yesmaximal, characteristic
2C3×#63498C211yesmaximal, characteristic, complemented
3C3×#63497C212yesmaximal, complemented
4C3×D4×Dic9C212yesmaximal, complemented
5C3×D4×Dic9C211yesmaximal, characteristic, complemented
6C3×#63486C211yesmaximal, characteristic, complemented
7C2×C3×C4×Dic9C211yesmaximal, characteristic, complemented
8C3×#63452C211yesmaximal, characteristic, complemented
9C3×#63449C211yesmaximal, characteristic, complemented
10C3×#63441C212yesmaximal, complemented
11C3×#63439C212yesmaximal, complemented
12C3×#5600—31nomaximal
13#10589702C311yesmaximal, characteristic, direct product factor
14C2×C3×D4×C9C2212yes—
15C2×C3×D4×C9C2211yescharacteristic
16C22×C3×C4×C9C2211yescharacteristic
17C22×C3×Dic9C2211yescharacteristic
18C22×C3×Dic9C2211yescharacteristic
19C22×C3×Dic9C2212yes—
20C2×C3×Dic18C2211yescharacteristic
21C3×C9×C4⋊C4C2211yescharacteristic
22C3×C9×C22⋊C4C2212yes—
23C3×#3676C2212yes—
24C3×#3676C2212yes—
25C3×#3676C2212yes—
26C3×#3676C2212yes—
27C3×C4⋊Dic9C2211yescharacteristic
28C3×C4⋊Dic9C2212yes—
29C3×Dic9⋊C4C2212yes—
30C3×Dic9⋊C4C2212yes—
31C3×Dic9⋊C4C2212yes—
32C3×C4×Dic9C2211yescharacteristic
33C3×C4×Dic9C2211yescharacteristic
34C3×C4×Dic9C2211yescharacteristic
35C3×C4×Dic9C2211yescharacteristic
36C3×C4×Dic9C2212yes—
37C32×#121D311yescharacteristic
38C3×#936—31no—
39C3×#935—32no—
40C3×D4×Dic3—32no—
41C3×D4×Dic3—31no—
42C3×#924—31no—
43C2×C3×C4×Dic3—31no—
44C3×#890—31no—
45C3×#887—31no—
46C3×#879—32no—
47C3×#877—32no—
48C9×#121—21no—
49C9×#121C2×C311yescharacteristic
50#63498C2×C311yescharacteristic, complemented
51#63497C2×C312yescomplemented
52D4×Dic9C2×C312yescomplemented
53D4×Dic9C2×C311yescharacteristic, complemented
54#63486C2×C311yescharacteristic, complemented
55C2×C4×Dic9C2×C311yescharacteristic, complemented
56#63452C2×C311yescharacteristic, complemented
57#63449C2×C311yescharacteristic, complemented
58#63441C2×C312yescomplemented
59#63439C2×C312yescomplemented
60C23×C3×C9C2312yes—
61C23×C3×C9C2311yescharacteristic
62C3×D4×C9—24no—
63C3×D4×C9—22no—
64C2×C3×C4×C9—22no—
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3×C4×C9C2312yes—
67C2×C3×C4×C9C2311yescharacteristic
68C2×C3×Dic9—22no—
69C2×C3×Dic9—22no—
70C2×C3×Dic9—24no—
71C2×C3×Dic9C2312yes—
72C2×C3×Dic9C2311yescharacteristic
73C2×C3×Dic9C2311yescharacteristic
74C2×C3×Dic9C2311yescharacteristic
75C2×C3×Dic9C2312yes—
76C2×C3×Dic9C2311yescharacteristic
77C3×Dic18—22no—
78C3×#529—91no—
79#5600—31no—
80C2×C32×D4C2×D312yes—
81C2×C32×D4C2×D311yescharacteristic
82C22×C32×C4C2×D311yescharacteristic
83C22×C3×Dic3—32no—
84C22×C3×Dic3—31no—
85C22×C3×Dic3—31no—
86C2×C3×Dic6—31no—
87C32×C4⋊C4C2×D311yescharacteristic
88C32×C22⋊C4C2×D312yes—
89C3×#217—32no—
90C3×#217—32no—
91C3×#217—32no—
92C3×#217—32no—
93C3×C4⋊Dic3—32no—
94C3×C4⋊Dic3—31no—
95C3×Dic3⋊C4—32no—
96C3×Dic3⋊C4—32no—
97C3×Dic3⋊C4—32no—
98C3×C4×Dic3—31no—
99C3×C4×Dic3—32no—
100C3×C4×Dic3—31no—
101C3×C4×Dic3—31no—
102C3×C4×Dic3—31no—
103C2×D4×C9—21no—
104C2×D4×C9—22no—
105C2×D4×C9C22×C312yes—
106C2×D4×C9C22×C311yescharacteristic
107C22×C4×C9—21no—
108C22×C4×C9C22×C311yescharacteristic
109C22×Dic9C22×C312yes—
110C22×Dic9C22×C311yescharacteristic
111C22×Dic9C22×C311yescharacteristic
112C2×Dic18C22×C311yescharacteristic
113C9×C4⋊C4—21no—
114C9×C4⋊C4C22×C311yescharacteristic
115C9×C22⋊C4—22no—
116C9×C22⋊C4C22×C312yes—
117#3676C22×C312yes—
118#3676C22×C312yes—
119#3676C22×C312yes—
120#3676C22×C312yes—
121C4⋊Dic9C22×C312yes—
122C4⋊Dic9C22×C311yescharacteristic
123Dic9⋊C4C22×C312yes—
124Dic9⋊C4C22×C312yes—
125Dic9⋊C4C22×C312yes—
126C4×Dic9C22×C312yes—
127C4×Dic9C22×C311yescharacteristic
128C4×Dic9C22×C311yescharacteristic
129C4×Dic9C22×C311yescharacteristic
130C4×Dic9C22×C311yescharacteristic
131C22×C3×C9—21no—
132C22×C3×C9—21no—
133C22×C3×C9—22no—
134C22×C3×C9—22no—
135C22×C3×C9—22no—
136C22×C3×C9#4112yes—
137C22×C3×C9C2411yescharacteristic
138C3×C4×C9—21no—
139C3×C4×C9—22no—
140C3×C4×C9#4112yes—
141C3×Dic9—22no—
142C3×Dic9—21no—
143C3×Dic9—21no—
144C3×Dic9—21no—
145C3×Dic9—22no—
146C3×Dic9#4112yes—
147C3×#126—92no—
148C3×#125—91no—
149C3×#124—91no—
150C3×#123—92no—
151C3×C22⋊Q8—91no—
152C3×#121—21no—
153C3×#121D911yescharacteristic
154C3×#121C3×D311yescharacteristic
155C3×C4×Q8—91no—
156C3×C4×D4—91no—
157C3×C4×D4—92no—
158C3×#117—92no—
159C2×C3×C42—91no—
160#936—31no—
161#935—32no—
162D4×Dic3—32no—
163D4×Dic3—31no—
164#924—31no—
165C2×C4×Dic3—31no—
166#890—31no—
167#887—31no—
168#879—32no—
169#877—32no—
170C23×C32C22×D312yes—
171C23×C32C22×D311yescharacteristic
172C32×D4—24no—
173C32×D4—22no—
174C2×C32×C4—22no—
175C2×C32×C4C22×D311yescharacteristic
176C2×C32×C4C22×D312yes—
177C2×C32×C4C22×D311yescharacteristic
178C2×C3×Dic3—31no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3—31no—
181C2×C3×Dic3—31no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—32no—
184C2×C3×Dic3—62no—
185C2×C3×Dic3—64no—
186C2×C3×Dic3—62no—
187C3×Dic6—62no—
188C23×C9—21no—
189C23×C9—22no—
190C23×C9C23×C312yes—
191C23×C9C23×C311yescharacteristic
192D4×C9—24no—
193D4×C9—22no—
194D4×C9—42no—
195D4×C9—44no—
196C2×C4×C9—22no—
197C2×C4×C9—21no—
198C2×C4×C9—21no—
199C2×C4×C9—22no—
200C2×C4×C9—42no—
201C2×C4×C9C23×C311yescharacteristic
202C2×C4×C9C23×C312yes—
203C2×C4×C9C23×C311yescharacteristic
204C2×Dic9—22no—
205C2×Dic9—22no—
206C2×Dic9—24no—
207C2×Dic9C23×C311yescharacteristic
208C2×Dic9C23×C311yescharacteristic
209C2×Dic9C23×C311yescharacteristic
210C2×Dic9C23×C312yes—
211C2×Dic9C23×C312yes—
212C2×Dic9C23×C311yescharacteristic
213Dic18—22no—
214#529—91noSylow
215C2×C3×C9—22no—
216C2×C3×C9—42no—
217C2×C3×C9C2×#4111yescharacteristic
218C2×C3×C9C2×#4111yescharacteristic
219C2×C3×C9C2×#4111yescharacteristic
220C2×C3×Q8—91no—
221C2×C3×D4—22no—
222C2×C3×D4—21no—
223C2×C3×D4C2×D912yes—
224C2×C3×D4C2×D911yescharacteristic
225C2×C3×D4C2×C3×D312yes—
226C2×C3×D4C2×C3×D311yescharacteristic
227C22×C3×C4—21no—
228C22×C3×C4—91no—
229C22×C3×C4—91no—
230C22×C3×C4—92no—
231C22×C3×C4C2×D911yescharacteristic
232C22×C3×C4C2×C3×D311yescharacteristic
233C22×Dic3—32no—
234C22×Dic3—31no—
235C22×Dic3—31no—
236C2×Dic6—31no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—92no—
239C3×C4⋊C4—92no—
240C3×C4⋊C4—92no—
241C3×C4⋊C4—91no—
242C3×C4⋊C4—92no—
243C3×C4⋊C4C2×D911yescharacteristic
244C3×C4⋊C4C2×C3×D311yescharacteristic
245C3×C22⋊C4—22no—
246C3×C22⋊C4—92no—
247C3×C22⋊C4—92no—
248C3×C22⋊C4—92no—
249C3×C22⋊C4—92no—
250C3×C22⋊C4C2×D912yes—
251C3×C22⋊C4C2×C3×D312yes—
252C3×C42—91no—
253C3×C42—91no—
254C3×C42—91no—
255C3×C42—91no—
256C3×C42—92no—
257#217—32no—
258#217—32no—
259#217—32no—
260#217—32no—
261C4⋊Dic3—32no—
262C4⋊Dic3—31no—
263Dic3⋊C4—32no—
264Dic3⋊C4—32no—
265Dic3⋊C4—32no—
266C4×Dic3—31no—
267C4×Dic3—32no—
268C4×Dic3—31no—
269C4×Dic3—31no—
270C4×Dic3—31no—
271C22×C32—21no—
272C22×C32—21no—
273C22×C32—22no—
274C22×C32—22no—
275C22×C32—22no—
276C22×C32#23712yes—
277C22×C32C23×D311yescharacteristic
278C32×C4—21no—
279C32×C4—22no—
280C32×C4#23712yes—
281C3×Dic3—32no—
282C3×Dic3—62no—
283C3×Dic3—61no—
284C3×Dic3—61no—
285C3×Dic3—61no—
286C3×Dic3—62no—
287C22×C9—21no—
288C22×C9—21no—
289C22×C9—22no—
290C22×C9—22no—
291C22×C9—22no—
292C22×C9—21no—
293C22×C9—22no—
294C22×C9—41no—
295C22×C9—41no—
296C22×C9—42no—
297C22×C9—42no—
298C22×C9—42no—
299C22×C9C3×#4112yes—
300C22×C9C24×C311yescharacteristic
301C4×C9—21no—
302C4×C9—22no—
303C4×C9—22no—
304C4×C9—41no—
305C4×C9—42no—
306C4×C9C3×#4112yes—
307Dic9—22no—
308Dic9—21no—
309Dic9—21no—
310Dic9—21no—
311Dic9—22no—
312Dic9C3×#4112yes—
313#126—92no—
314#125—91no—
315#124—91no—
316#123—92no—
317C22⋊Q8—91no—
318#121C3×D911yescharacteristic
319C4×Q8—91no—
320C4×D4—91no—
321C4×D4—92no—
322#117—92no—
323C2×C42—91no—
324C3×C9#52911yescharacteristic, complemented, Sylow
325C23×C3—21no—
326C23×C3—22no—
327C23×C3C22×D912yes—
328C23×C3C22×D911yescharacteristic
329C23×C3C22×C3×D312yes—
330C23×C3C22×C3×D311yescharacteristic
331C3×Q8—182no—
332C3×D4—24no—
333C3×D4—24no—
334C3×D4—22no—
335C3×D4—22no—
336C3×D4—42no—
337C3×D4—44no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—22no—
343C2×C3×C4—42no—
344C2×C3×C4—91no—
345C2×C3×C4—91no—
346C2×C3×C4—91no—
347C2×C3×C4—91no—
348C2×C3×C4—92no—
349C2×C3×C4—92no—
350C2×C3×C4—182no—
351C2×C3×C4—184no—
352C2×C3×C4—182no—
353C2×C3×C4C22×D911yescharacteristic
354C2×C3×C4C22×D912yes—
355C2×C3×C4C22×D911yescharacteristic
356C2×C3×C4C22×C3×D311yescharacteristic
357C2×C3×C4C22×C3×D312yes—
358C2×C3×C4C22×C3×D311yescharacteristic
359C2×Dic3—31no—
360C2×Dic3—31no—
361C2×Dic3—31no—
362C2×Dic3—31no—
363C2×Dic3—32no—
364C2×Dic3—32no—
365C2×Dic3—62no—
366C2×Dic3—64no—
367C2×Dic3—62no—
368Dic6—62no—
369C2×C32—22no—
370C2×C32—42no—
371C2×C32C2×#23711yescharacteristic
372C2×C32C2×#23711yescharacteristic
373C2×C32D3×#4111yescharacteristic
374C2×C9—22no—
375C2×C9—21no—
376C2×C9—21no—
377C2×C9—21no—
378C2×C9—42no—
379C2×C9—42no—
380C2×C9—82no—
381C2×C9C2×C3×#4111yescharacteristic
382C2×C9C2×C3×#4111yescharacteristic
383C2×C9C2×C3×#4111yescharacteristic
384C2×Q8—91no—
385C2×D4C2×C3×D912yes—
386C2×D4C2×C3×D911yescharacteristic
387C22×C4—91no—
388C22×C4—91no—
389C22×C4—92no—
390C22×C4C2×C3×D911yescharacteristic
391C4⋊C4—92no—
392C4⋊C4—92no—
393C4⋊C4—91no—
394C4⋊C4—92no—
395C4⋊C4—92no—
396C4⋊C4C2×C3×D911yescharacteristic
397C22⋊C4—92no—
398C22⋊C4—92no—
399C22⋊C4—92no—
400C22⋊C4—92no—
401C22⋊C4C2×C3×D912yes—
402C42—91no—
403C42—91no—
404C42—91no—
405C42—91no—
406C42—92no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—22no—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—22no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—21no—
418C22×C3—22no—
419C22×C3—41no—
420C22×C3—41no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3#369912yes—
425C22×C3C23×D911yescharacteristic
426C22×C3C3×#23712yes—
427C22×C3C23×C3×D311yescharacteristic
428C3×C4—21no—
429C3×C4—22no—
430C3×C4—21no—
431C3×C4—22no—
432C3×C4—22no—
433C3×C4—41no—
434C3×C4—42no—
435C3×C4—92no—
436C3×C4—181no—
437C3×C4—181no—
438C3×C4—182no—
439C3×C4—181no—
440C3×C4—182no—
441C3×C4#369912yes—
442C3×C4C3×#23712yes—
443Dic3—32no—
444Dic3—62no—
445Dic3—61no—
446Dic3—61no—
447Dic3—61no—
448Dic3—62no—
449C32#560011yescharacteristic
450C9—21no—
451C9C3×#52911yescharacteristic, complemented
452C23C22×C3×D912yes—
453C23C22×C3×D911yescharacteristic
454Q8—182no—
455D4—24no—
456D4—22no—
457C2×C4—22no—
458C2×C4—91no—
459C2×C4—91no—
460C2×C4—91no—
461C2×C4—91no—
462C2×C4—92no—
463C2×C4—92no—
464C2×C4—182no—
465C2×C4—184no—
466C2×C4—182no—
467C2×C4C22×C3×D911yescharacteristic
468C2×C4C22×C3×D912yes—
469C2×C4C22×C3×D911yescharacteristic
470C2×C3—22no—
471C2×C3—22no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—42no—
476C2×C3—42no—
477C2×C3—42no—
478C2×C3—82no—
479C2×C3C2×#369911yescharacteristic
480C2×C3C2×#369911yescharacteristic
481C2×C3#41×D911yescharacteristic
482C2×C3C2×C3×#23711yescharacteristic
483C2×C3C2×C3×#23711yescharacteristic
484C2×C3C3×D3×#4111yescharacteristic
485C22—21no—
486C22—21no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22C3×#369912yes—
491C22C23×C3×D911yescharacteristic
492C4—21no—
493C4—22no—
494C4—92no—
495C4—181no—
496C4—181no—
497C4—182no—
498C4—181no—
499C4—182no—
500C4C3×#369912yes—
501C3—21no—
502C3#1058970211yescharacteristic, direct product factor
503C3C3×#560011yescharacteristic
504C2—22no—
505C2—42no—
506C2C2×C3×#369911yescharacteristic
507C2C2×C3×#369911yescharacteristic
508C2C3×#41×D911yescharacteristic
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×#529
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×#63498C2semidirect
3. C3×#63497C2semidirect
4. C3×D4×Dic9C2semidirect
5. C3×D4×Dic9C2semidirect
6. C3×#63486C2semidirect
7. C2×C3×C4×Dic9C2semidirect
8. C3×#63452C2semidirect
9. C3×#63449C2semidirect
10. C3×#63441C2semidirect
11. C3×#63439C2semidirect
13. #10589702C3direct (N × H)
50. #63498C2×C3semidirect
51. #63497C2×C3semidirect
52. D4×Dic9C2×C3semidirect
53. D4×Dic9C2×C3semidirect
54. #63486C2×C3semidirect
55. C2×C4×Dic9C2×C3semidirect
56. #63452C2×C3semidirect
57. #63449C2×C3semidirect
58. #63441C2×C3semidirect
59. #63439C2×C3semidirect
324. C3×C9#529semidirect
451. C9C3×#529semidirect
502. C3#10589702direct (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×#634982
C3×#634972
C3×D4×Dic92
C3×#634862
C2×C3×C4×Dic92
C3×#634522
C3×#634492
C3×#634412
C3×#634392
C3×#56003
#105897023