Finite Group Databasebeta

← C3×#10589743 #13152897

Subgroups — C3×#10589743 #13152897

Order 1728.

710 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×C22⋊Q8C211yesmaximal, characteristic, complemented
2C3×#63508C211yesmaximal, characteristic, complemented
3C3×Q8×Dic9C211yesmaximal, characteristic, complemented
4C3×C4×#611C211yesmaximal, characteristic
5C3×#63488C211yesmaximal, characteristic, complemented
6C3×#63460C212yesmaximal, complemented
7C3×#63454C211yesmaximal, characteristic, complemented
8C3×#63450C211yesmaximal, characteristic, complemented
9C3×#63449C212yesmaximal, complemented
10C3×#63447C212yesmaximal
11C3×#63441C212yesmaximal, complemented
12C3×#5641—31nomaximal
13#10589743C311yesmaximal, characteristic, direct product factor
14C2×C3×Q8×C9C2211yescharacteristic
15C22×C3×C4×C9C2211yescharacteristic
16C2×C3×#611C2211yescharacteristic
17C2×C3×C4×D9C2211yescharacteristic
18C2×C3×Dic18C2212yes—
19C3×C9×C4⋊C4C2212yes—
20C3×C9×C4⋊C4C2211yescharacteristic
21C3×C9×C22⋊C4C2212yes—
22C3×#3676C2212yes—
23C3×#3676C2211yescharacteristic
24C3×(C2×D9)⋊C4C2211yescharacteristic
25C3×(C2×D9)⋊C4C2212yes—
26C3×(C2×D9)⋊C4C2212yes—
27C3×C4⋊Dic9C2211yescharacteristic
28C3×C4⋊Dic9C2211yescharacteristic
29C3×C4⋊Dic9C2212yes—
30C3×Dic9⋊C4C2211yescharacteristic
31C3×Dic9⋊C4C2212yes—
32C3×Dic9⋊C4C2212yes—
33C3×C4×Dic9C2212yes—
34C3×C4×Dic9C2211yescharacteristic
35C3×C4×Dic9C2212yes—
36C3×C4×Dic9C2212yes—
37C32×C22⋊Q8D311yescharacteristic
38C3×#946—31no—
39C3×Q8×Dic3—31no—
40C3×C4×#67—31no—
41C3×#926—31no—
42C3×#898—32no—
43C3×#892—31no—
44C3×#888—31no—
45C3×#887—32no—
46C3×#885—32no—
47C3×#879—32no—
48C9×C22⋊Q8—21no—
49C9×C22⋊Q8C2×C311yescharacteristic, complemented
50#63508C2×C311yescharacteristic, complemented
51Q8×Dic9C2×C311yescharacteristic, complemented
52C4×#611C2×C311yescharacteristic
53#63488C2×C311yescharacteristic, complemented
54#63460C2×C312yescomplemented
55#63454C2×C311yescharacteristic, complemented
56#63450C2×C311yescharacteristic, complemented
57#63449C2×C312yescomplemented
58#63447C2×C312yes—
59#63441C2×C312yescomplemented
60C23×C3×C9C2311yescharacteristic
61C22×C3×D9C2311yescharacteristic
62C3×Q8×C9—22no—
63C2×C3×C4×C9—22no—
64C2×C3×C4×C9C2312yes—
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3×C4×C9C2312yes—
67C2×C3×C4×C9C2311yescharacteristic
68C3×#611—22no—
69C2×C3×Dic9C2312yes—
70C2×C3×Dic9C2311yescharacteristic
71C2×C3×Dic9C2312yes—
72C2×C3×Dic9C2311yescharacteristic
73C2×C3×Dic9C2311yescharacteristic
74C3×C4×D9—22no—
75C3×Dic18—24no—
76C3×#556—91no—
77#5641—31no—
78C2×C32×Q8C2×D311yescharacteristic
79C22×C32×C4C2×D311yescharacteristic
80C2×C3×#67—31no—
81C2×C3×C4×D3—31no—
82C2×C3×Dic6—32no—
83C32×C4⋊C4C2×D312yes—
84C32×C4⋊C4C2×D311yescharacteristic
85C32×C22⋊C4C2×D312yes—
86C3×#217—31no—
87C3×#217—32no—
88C3×(C2×D3)⋊C4—32no—
89C3×(C2×D3)⋊C4—32no—
90C3×(C2×D3)⋊C4—31no—
91C3×C4⋊Dic3—32no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—31no—
94C3×Dic3⋊C4—32no—
95C3×Dic3⋊C4—32no—
96C3×Dic3⋊C4—31no—
97C3×C4×Dic3—32no—
98C3×C4×Dic3—32no—
99C3×C4×Dic3—32no—
100C3×C4×Dic3—31no—
101C2×Q8×C9—21no—
102C2×Q8×C9C22×C311yescharacteristic
103C22×C4×C9—21no—
104C22×C4×C9C22×C311yescharacteristic
105C2×#611C22×C311yescharacteristic
106C2×C4×D9C22×C311yescharacteristic
107C2×Dic18C22×C312yes—
108C9×C4⋊C4—21no—
109C9×C4⋊C4—22no—
110C9×C4⋊C4C22×C312yes—
111C9×C4⋊C4C22×C311yescharacteristic
112C9×C22⋊C4—22no—
113C9×C22⋊C4C22×C312yes—
114#3676C22×C312yes—
115#3676C22×C311yescharacteristic
116(C2×D9)⋊C4C22×C311yescharacteristic
117(C2×D9)⋊C4C22×C312yes—
118(C2×D9)⋊C4C22×C312yes—
119C4⋊Dic9C22×C311yescharacteristic
120C4⋊Dic9C22×C311yescharacteristic
121C4⋊Dic9C22×C312yes—
122Dic9⋊C4C22×C311yescharacteristic
123Dic9⋊C4C22×C312yes—
124Dic9⋊C4C22×C312yes—
125C4×Dic9C22×C312yes—
126C4×Dic9C22×C311yescharacteristic
127C4×Dic9C22×C312yes—
128C4×Dic9C22×C312yes—
129C22×C3×C9—21no—
130C22×C3×C9—21no—
131C22×C3×C9—21no—
132C22×C3×C9C2411yescharacteristic
133C2×C3×D9—21no—
134C2×C3×D9—21no—
135C2×C3×D9—21no—
136C3×C4×C9—22no—
137C3×C4×C9—21no—
138C3×C4×C9—22no—
139C3×C4×C9#4112yes—
140C3×Dic9—22no—
141C3×Dic9—21no—
142C3×Dic9—22no—
143C3×Dic9—21no—
144C3×Dic9#4112yes—
145C3×C4⋊Q8—91no—
146C3×#126—92no—
147C3×#126—92no—
148C3×#124—92no—
149C3×C22⋊Q8—21no—
150C3×C22⋊Q8—91no—
151C3×C22⋊Q8D911yescharacteristic, complemented
152C3×C22⋊Q8C3×D311yescharacteristic
153C3×C4×Q8—91no—
154C3×C4×Q8—92no—
155C3×C4×D4—91no—
156C3×#117—91no—
157C3×#117—91no—
158#946—31no—
159Q8×Dic3—31no—
160C4×#67—31no—
161#926—31no—
162#898—32no—
163#892—31no—
164#888—31no—
165#887—32no—
166#885—32no—
167#879—32no—
168C23×C32C22×D311yescharacteristic
169C22×C3×D3—31no—
170C32×Q8—22no—
171C2×C32×C4—22no—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D312yes—
174C2×C32×C4C22×D312yes—
175C2×C32×C4C22×D311yescharacteristic
176C3×#67—62no—
177C2×C3×Dic3—31no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3—31no—
181C2×C3×Dic3—32no—
182C3×C4×D3—62no—
183C3×Dic6—64no—
184C23×C9—21no—
185C23×C9C23×C311yescharacteristic
186C22×D9C23×C311yescharacteristic
187Q8×C9—22no—
188Q8×C9—42no—
189C2×C4×C9—22no—
190C2×C4×C9—21no—
191C2×C4×C9—22no—
192C2×C4×C9—22no—
193C2×C4×C9—21no—
194C2×C4×C9—42no—
195C2×C4×C9C23×C311yescharacteristic
196C2×C4×C9C23×C312yes—
197C2×C4×C9C23×C312yes—
198C2×C4×C9C23×C311yescharacteristic
199#611—22no—
200C2×Dic9C23×C312yes—
201C2×Dic9C23×C311yescharacteristic
202C2×Dic9C23×C311yescharacteristic
203C2×Dic9C23×C312yes—
204C2×Dic9C23×C311yescharacteristic
205C4×D9—22no—
206Dic18—24no—
207#556—91noSylow
208C2×C3×C9—41no—
209C2×C3×C9C2×#4111yescharacteristic
210C2×C3×C9C2×#4111yescharacteristic
211C2×C3×C92−1+411yescharacteristic
212C3×D9—41no—
213C2×C3×Q8—21no—
214C2×C3×Q8—92no—
215C2×C3×Q8C2×D911yescharacteristic
216C2×C3×Q8C2×C3×D311yescharacteristic
217C2×C3×D4—91no—
218C22×C3×C4—21no—
219C22×C3×C4—91no—
220C22×C3×C4C2×D911yescharacteristic
221C22×C3×C4C2×C3×D311yescharacteristic
222C2×#67—31no—
223C2×C4×D3—31no—
224C2×Dic6—32no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—91no—
228C3×C4⋊C4—92no—
229C3×C4⋊C4—92no—
230C3×C4⋊C4—91no—
231C3×C4⋊C4—91no—
232C3×C4⋊C4—92no—
233C3×C4⋊C4C2×D911yescharacteristic
234C3×C4⋊C4C2×D912yes—
235C3×C4⋊C4C2×C3×D312yes—
236C3×C4⋊C4C2×C3×D311yescharacteristic
237C3×C22⋊C4—22no—
238C3×C22⋊C4—92no—
239C3×C22⋊C4—91no—
240C3×C22⋊C4—92no—
241C3×C22⋊C4—91no—
242C3×C22⋊C4—92no—
243C3×C22⋊C4C2×D912yes—
244C3×C22⋊C4C2×C3×D312yes—
245C3×C42—92no—
246C3×C42—92no—
247C3×C42—91no—
248C3×C42—92no—
249#217—31no—
250#217—32no—
251(C2×D3)⋊C4—32no—
252(C2×D3)⋊C4—32no—
253(C2×D3)⋊C4—31no—
254C4⋊Dic3—32no—
255C4⋊Dic3—31no—
256C4⋊Dic3—31no—
257Dic3⋊C4—32no—
258Dic3⋊C4—32no—
259Dic3⋊C4—31no—
260C4×Dic3—32no—
261C4×Dic3—32no—
262C4×Dic3—32no—
263C4×Dic3—31no—
264C22×C32—21no—
265C22×C32—21no—
266C22×C32—21no—
267C22×C32C23×D311yescharacteristic
268C2×C3×D3—61no—
269C2×C3×D3—61no—
270C2×C3×D3—61no—
271C32×C4—21no—
272C32×C4—22no—
273C32×C4—22no—
274C32×C4#23712yes—
275C3×Dic3—32no—
276C3×Dic3—61no—
277C3×Dic3—62no—
278C3×Dic3—61no—
279C3×Dic3—62no—
280C22×C9—22no—
281C22×C9—21no—
282C22×C9—21no—
283C22×C9—21no—
284C22×C9—21no—
285C22×C9—41no—
286C22×C9—41no—
287C22×C9C24×C311yescharacteristic
288C2×D9—21no—
289C2×D9—21no—
290C2×D9—21no—
291C4×C9—22no—
292C4×C9—21no—
293C4×C9—22no—
294C4×C9—22no—
295C4×C9—42no—
296C4×C9—41no—
297C4×C9—42no—
298C4×C9C3×#4112yes—
299Dic9—22no—
300Dic9—21no—
301Dic9—22no—
302Dic9—21no—
303Dic9C3×#4112yes—
304C4⋊Q8—91no—
305#126—92no—
306#126—92no—
307#124—92no—
308C22⋊Q8—91no—
309C22⋊Q8C3×D911yescharacteristic, complemented
310C4×Q8—91no—
311C4×Q8—92no—
312C4×D4—91no—
313#117—91no—
314#117—91no—
315C3×C9#55611yescharacteristic, complemented, Sylow
316C23×C3—21no—
317C23×C3—91no—
318C23×C3C22×D911yescharacteristic
319C23×C3C22×C3×D311yescharacteristic
320C22×D3—31no—
321C3×Q8—22no—
322C3×Q8—22no—
323C3×Q8—42no—
324C3×Q8—184no—
325C3×D4—182no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—21no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—21no—
332C2×C3×C4—42no—
333C2×C3×C4—91no—
334C2×C3×C4—92no—
335C2×C3×C4—91no—
336C2×C3×C4—91no—
337C2×C3×C4—92no—
338C2×C3×C4—182no—
339C2×C3×C4C22×D911yescharacteristic
340C2×C3×C4C22×D912yes—
341C2×C3×C4C22×D912yes—
342C2×C3×C4C22×D911yescharacteristic
343C2×C3×C4C22×C3×D311yescharacteristic
344C2×C3×C4C22×C3×D312yes—
345C2×C3×C4C22×C3×D312yes—
346C2×C3×C4C22×C3×D311yescharacteristic
347#67—62no—
348C2×Dic3—31no—
349C2×Dic3—32no—
350C2×Dic3—31no—
351C2×Dic3—31no—
352C2×Dic3—32no—
353C4×D3—62no—
354Dic6—64no—
355C2×C32—41no—
356C2×C32C2×#23711yescharacteristic
357C2×C32#100711yescharacteristic
358C2×C32D3×#4111yescharacteristic
359C3×D3—121no—
360C2×C9—21no—
361C2×C9—21no—
362C2×C9—21no—
363C2×C9—42no—
364C2×C9—41no—
365C2×C9C2×C3×#4111yescharacteristic
366C2×C9C2×C3×#4111yescharacteristic
367C2×C9C3×2−1+411yescharacteristic
368D9—41no—
369C2×Q8—92no—
370C2×Q8C2×C3×D911yescharacteristic
371C2×D4—91no—
372C22×C4—91no—
373C22×C4C2×C3×D911yescharacteristic
374C4⋊C4—91no—
375C4⋊C4—91no—
376C4⋊C4—91no—
377C4⋊C4—92no—
378C4⋊C4—92no—
379C4⋊C4—92no—
380C4⋊C4C2×C3×D911yescharacteristic
381C4⋊C4C2×C3×D912yes—
382C22⋊C4—92no—
383C22⋊C4—91no—
384C22⋊C4—92no—
385C22⋊C4—91no—
386C22⋊C4—92no—
387C22⋊C4C2×C3×D912yes—
388C42—92no—
389C42—92no—
390C42—91no—
391C42—92no—
392C22×C3—22no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—41no—
401C22×C3—41no—
402C22×C3—181no—
403C22×C3—181no—
404C22×C3—181no—
405C22×C3C23×D911yescharacteristic
406C22×C3C23×C3×D311yescharacteristic
407C2×D3—61no—
408C2×D3—61no—
409C2×D3—61no—
410C3×C4—21no—
411C3×C4—22no—
412C3×C4—22no—
413C3×C4—21no—
414C3×C4—22no—
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—41no—
418C3×C4—42no—
419C3×C4—42no—
420C3×C4—92no—
421C3×C4—181no—
422C3×C4—182no—
423C3×C4—181no—
424C3×C4—182no—
425C3×C4#369912yes—
426C3×C4C3×#23712yes—
427Dic3—32no—
428Dic3—61no—
429Dic3—62no—
430Dic3—61no—
431Dic3—62no—
432C32#564111yescharacteristic
433C9—21no—
434C9C3×#55611yescharacteristic, complemented
435C23—91no—
436C23C22×C3×D911yescharacteristic
437Q8—22no—
438Q8—184no—
439D4—182no—
440C2×C4—22no—
441C2×C4—91no—
442C2×C4—92no—
443C2×C4—91no—
444C2×C4—91no—
445C2×C4—92no—
446C2×C4—182no—
447C2×C4C22×C3×D911yescharacteristic
448C2×C4C22×C3×D912yes—
449C2×C4C22×C3×D912yes—
450C2×C4C22×C3×D911yescharacteristic
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—42no—
455C2×C3—41no—
456C2×C3—41no—
457C2×C3—361no—
458C2×C3C2×#369911yescharacteristic
459C2×C3#6371311yescharacteristic
460C2×C3#41×D911yescharacteristic
461C2×C3C2×C3×#23711yescharacteristic
462C2×C3C3×#100711yescharacteristic
463C2×C3C3×D3×#4111yescharacteristic
464D3—121no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—181no—
469C22—181no—
470C22—181no—
471C22C23×C3×D911yescharacteristic
472C4—21no—
473C4—22no—
474C4—22no—
475C4—92no—
476C4—181no—
477C4—182no—
478C4—181no—
479C4—182no—
480C4C3×#369912yes—
481C3—21no—
482C3#1058974311yescharacteristic, direct product factor
483C3C3×#564111yescharacteristic
484C2—41no—
485C2—361no—
486C2C2×C3×#369911yescharacteristic
487C2C3×#6371311yescharacteristic
488C2C3×#41×D911yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×C22⋊Q8C2
socleC22×C32C23×D3
prime radicalC2×C3×#611C22
hypercenterC3×C22⋊Q8D9
hypocenterC9C3×#556
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×#63508C2semidirect
3. C3×Q8×Dic9C2semidirect
5. C3×#63488C2semidirect
6. C3×#63460C2semidirect
7. C3×#63454C2semidirect
8. C3×#63450C2semidirect
9. C3×#63449C2semidirect
11. C3×#63441C2semidirect
13. #10589743C3direct (N × H)
49. C9×C22⋊Q8C2×C3semidirect
50. #63508C2×C3semidirect
51. Q8×Dic9C2×C3semidirect
53. #63488C2×C3semidirect
54. #63460C2×C3semidirect
55. #63454C2×C3semidirect
56. #63450C2×C3semidirect
57. #63449C2×C3semidirect
59. #63441C2×C3semidirect
151. C3×C22⋊Q8D9semidirect
309. C22⋊Q8C3×D9semidirect
315. C3×C9#556semidirect
434. C9C3×#556semidirect
482. C3#10589743direct (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×#635082
C3×Q8×Dic92
C3×C4×#6112
C3×#634882
C3×#634602
C3×#634542
C3×#634502
C3×#634492
C3×#634472
C3×#634412
C3×#56413
#105897433