Finite Group Databasebeta

← C3×#10589745 #13152920

Subgroups — C3×#10589745 #13152920

Order 1728.

722 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×C22⋊Q8C211yesmaximal, characteristic, complemented
2C3×#63509C211yesmaximal, characteristic, complemented
3C3×Q8×Dic9C211yesmaximal, characteristic, complemented
4C3×#63492C211yesmaximal, characteristic
5C3×#63488C211yesmaximal, characteristic, complemented
6C3×#63460C212yesmaximal, complemented
7C3×#63459C212yesmaximal, complemented
8C3×#63455C211yesmaximal, characteristic, complemented
9C3×#63450C211yesmaximal, characteristic, complemented
10C3×#63447C212yesmaximal, complemented
11C3×#63445C212yesmaximal, complemented
12C3×#5643—31nomaximal
13#10589745C311yesmaximal, characteristic, direct product factor
14C2×C3×Q8×C9C2211yescharacteristic
15C22×C3×C4×C9C2211yescharacteristic
16C2×C3×#611C2212yes—
17C2×C3×D36C2211yescharacteristic
18C2×C3×C4×D9C2212yes—
19C2×C3×Dic18C2212yes—
20C3×C9×C4⋊C4C2211yescharacteristic
21C3×C9×C4⋊C4C2212yes—
22C3×C9×C22⋊C4C2212yes—
23C3×#3676C2211yescharacteristic
24C3×#3676C2211yescharacteristic
25C3×(C2×D9)⋊C4C2212yes—
26C3×(C2×D9)⋊C4C2212yes—
27C3×(C2×D9)⋊C4C2212yes—
28C3×(C2×D9)⋊C4C2212yes—
29C3×C4⋊Dic9C2211yescharacteristic
30C3×C4⋊Dic9C2211yescharacteristic
31C3×C4⋊Dic9C2211yescharacteristic
32C3×C4⋊Dic9C2212yes—
33C3×Dic9⋊C4C2212yes—
34C3×C4×Dic9C2211yescharacteristic
35C3×C4×Dic9C2211yescharacteristic
36C3×C4×Dic9C2212yes—
37C32×C22⋊Q8D311yescharacteristic
38C3×#947—31no—
39C3×Q8×Dic3—31no—
40C3×#930—31no—
41C3×#926—31no—
42C3×#898—32no—
43C3×#897—32no—
44C3×#893—31no—
45C3×#888—31no—
46C3×#885—32no—
47C3×#883—32no—
48C9×C22⋊Q8—21no—
49C9×C22⋊Q8C2×C311yescharacteristic, complemented
50#63509C2×C311yescharacteristic, complemented
51Q8×Dic9C2×C311yescharacteristic, complemented
52#63492C2×C311yescharacteristic
53#63488C2×C311yescharacteristic, complemented
54#63460C2×C312yescomplemented
55#63459C2×C312yescomplemented
56#63455C2×C311yescharacteristic, complemented
57#63450C2×C311yescharacteristic, complemented
58#63447C2×C312yescomplemented
59#63445C2×C312yescomplemented
60C23×C3×C9C2311yescharacteristic
61C22×C3×D9C2312yes—
62C3×Q8×C9—22no—
63C2×C3×C4×C9—22no—
64C2×C3×C4×C9C2312yes—
65C2×C3×C4×C9C2312yes—
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3×C4×C9C2311yescharacteristic
68C3×#611—42no—
69C2×C3×Dic9C2311yescharacteristic
70C2×C3×Dic9C2312yes—
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2311yescharacteristic
73C2×C3×Dic9C2311yescharacteristic
74C3×D36—22no—
75C3×C4×D9—42no—
76C3×Dic18—42no—
77C3×#542—91no—
78#5643—31no—
79C2×C32×Q8C2×D311yescharacteristic
80C22×C32×C4C2×D311yescharacteristic
81C2×C3×#67—32no—
82C2×C3×D12—31no—
83C2×C3×C4×D3—32no—
84C2×C3×Dic6—32no—
85C32×C4⋊C4C2×D311yescharacteristic
86C32×C4⋊C4C2×D312yes—
87C32×C22⋊C4C2×D312yes—
88C3×#217—31no—
89C3×#217—31no—
90C3×(C2×D3)⋊C4—32no—
91C3×(C2×D3)⋊C4—32no—
92C3×(C2×D3)⋊C4—32no—
93C3×(C2×D3)⋊C4—32no—
94C3×C4⋊Dic3—32no—
95C3×C4⋊Dic3—31no—
96C3×C4⋊Dic3—31no—
97C3×C4⋊Dic3—31no—
98C3×Dic3⋊C4—32no—
99C3×C4×Dic3—32no—
100C3×C4×Dic3—31no—
101C3×C4×Dic3—31no—
102C2×Q8×C9—21no—
103C2×Q8×C9C22×C311yescharacteristic
104C22×C4×C9—21no—
105C22×C4×C9C22×C311yescharacteristic
106C2×#611C22×C312yes—
107C2×D36C22×C311yescharacteristic
108C2×C4×D9C22×C312yes—
109C2×Dic18C22×C312yes—
110C9×C4⋊C4—21no—
111C9×C4⋊C4—22no—
112C9×C4⋊C4C22×C311yescharacteristic
113C9×C4⋊C4C22×C312yes—
114C9×C22⋊C4—22no—
115C9×C22⋊C4C22×C312yes—
116#3676C22×C311yescharacteristic
117#3676C22×C311yescharacteristic
118(C2×D9)⋊C4C22×C312yes—
119(C2×D9)⋊C4C22×C312yes—
120(C2×D9)⋊C4C22×C312yes—
121(C2×D9)⋊C4C22×C312yes—
122C4⋊Dic9C22×C311yescharacteristic
123C4⋊Dic9C22×C311yescharacteristic
124C4⋊Dic9C22×C311yescharacteristic
125C4⋊Dic9C22×C312yes—
126Dic9⋊C4C22×C312yes—
127C4×Dic9C22×C311yescharacteristic
128C4×Dic9C22×C311yescharacteristic
129C4×Dic9C22×C312yes—
130C22×C3×C9—21no—
131C22×C3×C9—21no—
132C22×C3×C9—21no—
133C22×C3×C9C2411yescharacteristic
134C2×C3×D9—22no—
135C2×C3×D9—22no—
136C2×C3×D9—22no—
137C3×C4×C9—21no—
138C3×C4×C9—22no—
139C3×C4×C9—22no—
140C3×C4×C9#4112yes—
141C3×Dic9—21no—
142C3×Dic9—22no—
143C3×Dic9—21no—
144C3×Dic9—21no—
145C3×Dic9—21no—
146C3×C4⋊Q8—91no—
147C3×#126—92no—
148C3×#124—91no—
149C3×#124—92no—
150C3×#123—92no—
151C3×C22⋊Q8—21no—
152C3×C22⋊Q8—92no—
153C3×C22⋊Q8D911yescharacteristic, complemented
154C3×C22⋊Q8C3×D311yescharacteristic
155C3×#121—91no—
156C3×C4×Q8—91no—
157C3×C4×D4—91no—
158C3×#117—91no—
159#947—31no—
160Q8×Dic3—31no—
161#930—31no—
162#926—31no—
163#898—32no—
164#897—32no—
165#893—31no—
166#888—31no—
167#885—32no—
168#883—32no—
169C23×C32C22×D311yescharacteristic
170C22×C3×D3—32no—
171C32×Q8—22no—
172C2×C32×C4—22no—
173C2×C32×C4C22×D312yes—
174C2×C32×C4C22×D312yes—
175C2×C32×C4C22×D311yescharacteristic
176C2×C32×C4C22×D311yescharacteristic
177C3×#67—122no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3—31no—
182C2×C3×Dic3—31no—
183C3×D12—62no—
184C3×C4×D3—122no—
185C3×Dic6—122no—
186C23×C9—21no—
187C23×C9C23×C311yescharacteristic
188C22×D9C23×C312yes—
189Q8×C9—22no—
190Q8×C9—42no—
191C2×C4×C9—22no—
192C2×C4×C9—21no—
193C2×C4×C9—22no—
194C2×C4×C9—22no—
195C2×C4×C9—21no—
196C2×C4×C9—42no—
197C2×C4×C9C23×C312yes—
198C2×C4×C9C23×C312yes—
199C2×C4×C9C23×C311yescharacteristic
200C2×C4×C9C23×C311yescharacteristic
201#611—42no—
202C2×Dic9C23×C311yescharacteristic
203C2×Dic9C23×C311yescharacteristic
204C2×Dic9C23×C312yes—
205C2×Dic9C23×C311yescharacteristic
206C2×Dic9C23×C311yescharacteristic
207D36—22no—
208C4×D9—42no—
209Dic18—42no—
210#542—91noSylow
211C2×C3×C9—41no—
212C2×C3×C9C2×#4111yescharacteristic
213C2×C3×C9H(5,𝔽2)11yescharacteristic
214C2×C3×C92−1+411yescharacteristic
215C3×D9—42no—
216C2×C3×Q8—21no—
217C2×C3×Q8—92no—
218C2×C3×Q8C2×D911yescharacteristic
219C2×C3×Q8C2×C3×D311yescharacteristic
220C2×C3×D4—92no—
221C2×C3×D4—91no—
222C22×C3×C4—21no—
223C22×C3×C4—92no—
224C22×C3×C4C2×D911yescharacteristic
225C22×C3×C4C2×C3×D311yescharacteristic
226C2×#67—32no—
227C2×D12—31no—
228C2×C4×D3—32no—
229C2×Dic6—32no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—92no—
233C3×C4⋊C4—91no—
234C3×C4⋊C4—91no—
235C3×C4⋊C4—91no—
236C3×C4⋊C4—92no—
237C3×C4⋊C4C2×D911yescharacteristic
238C3×C4⋊C4C2×D912yes—
239C3×C4⋊C4C2×C3×D311yescharacteristic
240C3×C4⋊C4C2×C3×D312yes—
241C3×C22⋊C4—22no—
242C3×C22⋊C4—91no—
243C3×C22⋊C4—91no—
244C3×C22⋊C4—92no—
245C3×C22⋊C4—92no—
246C3×C22⋊C4—92no—
247C3×C22⋊C4—92no—
248C3×C22⋊C4C2×D912yes—
249C3×C22⋊C4C2×C3×D312yes—
250C3×C42—91no—
251C3×C42—91no—
252C3×C42—92no—
253#217—31no—
254#217—31no—
255(C2×D3)⋊C4—32no—
256(C2×D3)⋊C4—32no—
257(C2×D3)⋊C4—32no—
258(C2×D3)⋊C4—32no—
259C4⋊Dic3—32no—
260C4⋊Dic3—31no—
261C4⋊Dic3—31no—
262C4⋊Dic3—31no—
263Dic3⋊C4—32no—
264C4×Dic3—32no—
265C4×Dic3—31no—
266C4×Dic3—31no—
267C22×C32—21no—
268C22×C32—21no—
269C22×C32—21no—
270C22×C32C23×D311yescharacteristic
271C2×C3×D3—62no—
272C2×C3×D3—62no—
273C2×C3×D3—62no—
274C32×C4—21no—
275C32×C4—22no—
276C32×C4—22no—
277C32×C4#23712yes—
278C3×Dic3—61no—
279C3×Dic3—61no—
280C3×Dic3—61no—
281C3×Dic3—62no—
282C3×Dic3—61no—
283C22×C9—22no—
284C22×C9—21no—
285C22×C9—21no—
286C22×C9—21no—
287C22×C9—21no—
288C22×C9—41no—
289C22×C9—41no—
290C22×C9C24×C311yescharacteristic
291C2×D9—22no—
292C2×D9—22no—
293C2×D9—22no—
294C4×C9—21no—
295C4×C9—22no—
296C4×C9—22no—
297C4×C9—22no—
298C4×C9—42no—
299C4×C9—41no—
300C4×C9—42no—
301C4×C9C3×#4112yes—
302Dic9—21no—
303Dic9—22no—
304Dic9—21no—
305Dic9—21no—
306Dic9—21no—
307C4⋊Q8—91no—
308#126—92no—
309#124—91no—
310#124—92no—
311#123—92no—
312C22⋊Q8—92no—
313C22⋊Q8C3×D911yescharacteristic, complemented
314#121—91no—
315C4×Q8—91no—
316C4×D4—91no—
317#117—91no—
318C3×C9#54211yescharacteristic, complemented, Sylow
319C23×C3—21no—
320C23×C3—92no—
321C23×C3C22×D911yescharacteristic
322C23×C3C22×C3×D311yescharacteristic
323C22×D3—32no—
324C3×Q8—22no—
325C3×Q8—22no—
326C3×Q8—42no—
327C3×Q8—362no—
328C3×D4—182no—
329C3×D4—362no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—21no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—21no—
336C2×C3×C4—42no—
337C2×C3×C4—91no—
338C2×C3×C4—91no—
339C2×C3×C4—91no—
340C2×C3×C4—91no—
341C2×C3×C4—92no—
342C2×C3×C4—362no—
343C2×C3×C4C22×D912yes—
344C2×C3×C4C22×D911yescharacteristic
345C2×C3×C4C22×D912yes—
346C2×C3×C4C22×D911yescharacteristic
347C2×C3×C4C22×C3×D312yes—
348C2×C3×C4C22×C3×D312yes—
349C2×C3×C4C22×C3×D311yescharacteristic
350C2×C3×C4C22×C3×D311yescharacteristic
351#67—122no—
352C2×Dic3—31no—
353C2×Dic3—31no—
354C2×Dic3—32no—
355C2×Dic3—31no—
356C2×Dic3—31no—
357D12—62no—
358C4×D3—122no—
359Dic6—122no—
360C2×C32—41no—
361C2×C32C2×#23711yescharacteristic
362C2×C32#100711yescharacteristic
363C2×C32#100911yescharacteristic
364C3×D3—122no—
365C2×C9—21no—
366C2×C9—21no—
367C2×C9—21no—
368C2×C9—42no—
369C2×C9—41no—
370C2×C9C2×C3×#4111yescharacteristic
371C2×C9C3×H(5,𝔽2)11yescharacteristic
372C2×C9C3×2−1+411yescharacteristic
373D9—42no—
374C2×Q8—92no—
375C2×Q8C2×C3×D911yescharacteristic
376C2×D4—92no—
377C2×D4—91no—
378C22×C4—92no—
379C22×C4C2×C3×D911yescharacteristic
380C4⋊C4—91no—
381C4⋊C4—91no—
382C4⋊C4—91no—
383C4⋊C4—92no—
384C4⋊C4—92no—
385C4⋊C4C2×C3×D911yescharacteristic
386C4⋊C4C2×C3×D912yes—
387C22⋊C4—91no—
388C22⋊C4—91no—
389C22⋊C4—92no—
390C22⋊C4—92no—
391C22⋊C4—92no—
392C22⋊C4—92no—
393C22⋊C4C2×C3×D912yes—
394C42—91no—
395C42—91no—
396C42—92no—
397C22×C3—22no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—41no—
406C22×C3—41no—
407C22×C3—182no—
408C22×C3—182no—
409C22×C3—182no—
410C22×C3C23×D911yescharacteristic
411C22×C3C23×C3×D311yescharacteristic
412C2×D3—62no—
413C2×D3—62no—
414C2×D3—62no—
415C3×C4—21no—
416C3×C4—22no—
417C3×C4—22no—
418C3×C4—21no—
419C3×C4—22no—
420C3×C4—22no—
421C3×C4—22no—
422C3×C4—41no—
423C3×C4—42no—
424C3×C4—42no—
425C3×C4—181no—
426C3×C4—181no—
427C3×C4—181no—
428C3×C4—182no—
429C3×C4—181no—
430C3×C4#369912yes—
431C3×C4C3×#23712yes—
432Dic3—61no—
433Dic3—61no—
434Dic3—61no—
435Dic3—62no—
436Dic3—61no—
437C32#564311yescharacteristic
438C9—21no—
439C9C3×#54211yescharacteristic, complemented
440C23—92no—
441C23C22×C3×D911yescharacteristic
442Q8—22no—
443Q8—362no—
444D4—182no—
445D4—362no—
446C2×C4—22no—
447C2×C4—91no—
448C2×C4—91no—
449C2×C4—91no—
450C2×C4—91no—
451C2×C4—92no—
452C2×C4—362no—
453C2×C4C22×C3×D912yes—
454C2×C4C22×C3×D911yescharacteristic
455C2×C4C22×C3×D912yes—
456C2×C4C22×C3×D911yescharacteristic
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—42no—
461C2×C3—41no—
462C2×C3—41no—
463C2×C3—362no—
464C2×C3C2×#369911yescharacteristic
465C2×C3#6371311yescharacteristic
466C2×C3#6371511yescharacteristic
467C2×C3C2×C3×#23711yescharacteristic
468C2×C3C3×#100711yescharacteristic
469C2×C3C3×#100911yescharacteristic
470D3—122no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—182no—
475C22—182no—
476C22—182no—
477C22C23×C3×D911yescharacteristic
478C4—21no—
479C4—22no—
480C4—22no—
481C4—181no—
482C4—181no—
483C4—181no—
484C4—182no—
485C4—181no—
486C4C3×#369912yes—
487C3—21no—
488C3#1058974511yescharacteristic, direct product factor
489C3C3×#564311yescharacteristic
490C2—41no—
491C2—362no—
492C2C2×C3×#369911yescharacteristic
493C2C3×#6371311yescharacteristic
494C2C3×#6371511yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×C22⋊Q8C2
socleC22×C32C23×D3
prime radicalC3×#63492C2
hypercenterC3×C22⋊Q8D9
hypocenterC9C3×#542
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×C9×C22⋊Q8C2semidirect
2. C3×#63509C2semidirect
3. C3×Q8×Dic9C2semidirect
5. C3×#63488C2semidirect
6. C3×#63460C2semidirect
7. C3×#63459C2semidirect
8. C3×#63455C2semidirect
9. C3×#63450C2semidirect
10. C3×#63447C2semidirect
11. C3×#63445C2semidirect
13. #10589745C3direct (N × H)
49. C9×C22⋊Q8C2×C3semidirect
50. #63509C2×C3semidirect
51. Q8×Dic9C2×C3semidirect
53. #63488C2×C3semidirect
54. #63460C2×C3semidirect
55. #63459C2×C3semidirect
56. #63455C2×C3semidirect
57. #63450C2×C3semidirect
58. #63447C2×C3semidirect
59. #63445C2×C3semidirect
153. C3×C22⋊Q8D9semidirect
313. C22⋊Q8C3×D9semidirect
318. C3×C9#542semidirect
439. C9C3×#542semidirect
488. C3#10589745direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C9×C22⋊Q82
C3×#635092
C3×Q8×Dic92
C3×#634922
C3×#634882
C3×#634602
C3×#634592
C3×#634552
C3×#634502
C3×#634472
C3×#634452
C3×#56433
#105897453