Finite Group Databasebeta

← C3×#10589613 #13152408

Subgroups — C3×#10589613 #13152408

Order 1728.

776 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C9×C4⋊C4C211yesmaximal, characteristic, complemented
2C3×C4×#611C211yesmaximal, characteristic
3C3×C4×#611C212yesmaximal
4C3×#63486C211yesmaximal, characteristic, complemented
5C3×#63486C212yesmaximal, complemented
6C3×#63459C211yesmaximal, characteristic, complemented
7C3×#63458C212yesmaximal, complemented
8C3×C4⋊C4×D9C211yesmaximal, characteristic, complemented
9C3×#63451C212yesmaximal, complemented
10C3×#63450C211yesmaximal, characteristic, complemented
11C3×#63449C211yesmaximal, characteristic, complemented
12C3×#5511—31nomaximal
13#10589613C311yesmaximal, characteristic, direct product factor
14C22×C3×C4×C9C2211yescharacteristic
15C22×C3×C4×C9C2212yes—
16C2×C3×#611C2211yescharacteristic
17C2×C3×C4×D9C2211yescharacteristic
18C2×C3×C4×D9C2212yes—
19C2×C3×Dic18C2211yescharacteristic
20C2×C3×Dic18C2212yes—
21C3×C9×C4⋊C4C2211yescharacteristic
22C3×C9×C4⋊C4C2211yescharacteristic
23C3×C9×C4⋊C4C2212yes—
24C3×#3676C2211yescharacteristic
25C3×#3676C2212yes—
26C3×(C2×D9)⋊C4C2211yescharacteristic
27C3×(C2×D9)⋊C4C2212yes—
28C3×C4⋊Dic9C2211yescharacteristic
29C3×C4⋊Dic9C2212yes—
30C3×Dic9⋊C4C2212yes—
31C3×Dic9⋊C4C2212yes—
32C3×Dic9⋊C4C2212yes—
33C3×Dic9⋊C4C2212yes—
34C3×Dic9⋊C4C2211yescharacteristic
35C3×C4×Dic9C2211yescharacteristic
36C3×C4×Dic9C2212yes—
37C2×C32×C4⋊C4D311yescharacteristic
38C3×C4×#67—31no—
39C3×C4×#67—32no—
40C3×#924—32no—
41C3×#924—31no—
42C3×#897—31no—
43C3×#896—32no—
44C3×D3×C4⋊C4—31no—
45C3×#889—32no—
46C3×#888—31no—
47C3×#887—31no—
48C2×C9×C4⋊C4—21no—
49C2×C9×C4⋊C4C2×C311yescharacteristic, complemented
50C4×#611C2×C311yescharacteristic
51C4×#611C2×C312yes—
52#63486C2×C312yescomplemented
53#63486C2×C311yescharacteristic, complemented
54#63459C2×C311yescharacteristic, complemented
55#63458C2×C312yescomplemented
56C4⋊C4×D9C2×C311yescharacteristic, complemented
57#63451C2×C312yescomplemented
58#63450C2×C311yescharacteristic, complemented
59#63449C2×C311yescharacteristic, complemented
60C23×C3×C9C2311yescharacteristic
61C22×C3×D9C2311yescharacteristic
62C2×C3×C4×C9—22no—
63C2×C3×C4×C9—42no—
64C2×C3×C4×C9C2312yes—
65C2×C3×C4×C9C2312yes—
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3×C4×C9C2311yescharacteristic
68C3×#611Q814yescomplemented
69C2×C3×Dic9C2311yescharacteristic
70C2×C3×Dic9C2312yes—
71C2×C3×Dic9C2312yes—
72C2×C3×Dic9C2311yescharacteristic
73C2×C3×Dic9C2311yescharacteristic
74C3×C4×D9—22no—
75C3×C4×D9—42no—
76C3×Dic18—22no—
77C3×Dic18—42no—
78C3×#554—91no—
79#5511—31no—
80C22×C32×C4C2×D311yescharacteristic
81C22×C32×C4C2×D312yes—
82C2×C3×#67—31no—
83C2×C3×C4×D3—31no—
84C2×C3×C4×D3—32no—
85C2×C3×Dic6—31no—
86C2×C3×Dic6—32no—
87C32×C4⋊C4C2×D311yescharacteristic
88C32×C4⋊C4C2×D311yescharacteristic
89C32×C4⋊C4C2×D312yes—
90C3×#217—32no—
91C3×#217—31no—
92C3×(C2×D3)⋊C4—31no—
93C3×(C2×D3)⋊C4—32no—
94C3×C4⋊Dic3—31no—
95C3×C4⋊Dic3—32no—
96C3×Dic3⋊C4—32no—
97C3×Dic3⋊C4—32no—
98C3×Dic3⋊C4—31no—
99C3×Dic3⋊C4—32no—
100C3×Dic3⋊C4—32no—
101C3×C4×Dic3—31no—
102C3×C4×Dic3—32no—
103C22×C4×C9—21no—
104C22×C4×C9—22no—
105C22×C4×C9C22×C311yescharacteristic
106C22×C4×C9C22×C312yes—
107C2×#611C22×C311yescharacteristic
108C2×C4×D9C22×C311yescharacteristic
109C2×C4×D9C22×C312yes—
110C2×Dic18C22×C311yescharacteristic
111C2×Dic18C22×C312yes—
112C9×C4⋊C4—21no—
113C9×C4⋊C4—21no—
114C9×C4⋊C4—22no—
115C9×C4⋊C4C22×C311yescharacteristic
116C9×C4⋊C4C22×C311yescharacteristic
117C9×C4⋊C4C22×C312yes—
118#3676C22×C312yes—
119#3676C22×C311yescharacteristic
120(C2×D9)⋊C4C22×C311yescharacteristic
121(C2×D9)⋊C4C22×C312yes—
122C4⋊Dic9C22×C311yescharacteristic
123C4⋊Dic9C22×C312yes—
124Dic9⋊C4C22×C312yes—
125Dic9⋊C4C22×C311yescharacteristic
126Dic9⋊C4C22×C312yes—
127Dic9⋊C4C22×C312yes—
128Dic9⋊C4C22×C312yes—
129C4×Dic9C22×C311yescharacteristic
130C4×Dic9C22×C312yes—
131C22×C3×C9—21no—
132C22×C3×C9—21no—
133C22×C3×C9C2×Q812yes—
134C22×C3×C9C2411yescharacteristic
135C2×C3×D9—21no—
136C2×C3×D9—21no—
137C2×C3×D9C2×Q812yes—
138C3×C4×C9—21no—
139C3×C4×C9—22no—
140C3×C4×C9—22no—
141C3×C4×C9#4112yes—
142C3×Dic9—22no—
143C3×Dic9—21no—
144C3×Dic9—21no—
145C3×Dic9—22no—
146C3×Dic9C2×Q812yes—
147C3×C4⋊Q8—91no—
148C3×#125—92no—
149C3×C22⋊Q8—92no—
150C3×C22⋊Q8—91no—
151C3×C22⋊Q8—92no—
152C3×C22⋊Q8—91no—
153C3×C4×Q8—91no—
154C3×C4×D4—92no—
155C3×C4×D4—91no—
156C2×C3×C4⋊C4—21no—
157C2×C3×C4⋊C4—91no—
158C2×C3×C4⋊C4D911yescharacteristic, complemented
159C2×C3×C4⋊C4C3×D311yescharacteristic
160C4×#67—31no—
161C4×#67—32no—
162#924—32no—
163#924—31no—
164#897—31no—
165#896—32no—
166D3×C4⋊C4—31no—
167#889—32no—
168#888—31no—
169#887—31no—
170C23×C32C22×D311yescharacteristic
171C22×C3×D3—31no—
172C2×C32×C4—22no—
173C2×C32×C4—42no—
174C2×C32×C4C22×D312yes—
175C2×C32×C4C22×D311yescharacteristic
176C2×C32×C4C22×D312yes—
177C2×C32×C4C22×D311yescharacteristic
178C3×#67—34no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3—31no—
182C2×C3×Dic3—31no—
183C2×C3×Dic3—32no—
184C3×C4×D3—62no—
185C3×C4×D3—122no—
186C3×Dic6—62no—
187C3×Dic6—122no—
188C23×C9—21no—
189C23×C9C23×C311yescharacteristic
190C22×D9C23×C311yescharacteristic
191C2×C4×C9—24no—
192C2×C4×C9—22no—
193C2×C4×C9—21no—
194C2×C4×C9—21no—
195C2×C4×C9—22no—
196C2×C4×C9—22no—
197C2×C4×C9—42no—
198C2×C4×C9—44no—
199C2×C4×C9C23×C312yes—
200C2×C4×C9C23×C311yescharacteristic
201C2×C4×C9C23×C312yes—
202C2×C4×C9C23×C311yescharacteristic
203#611C3×Q814yescomplemented
204C2×Dic9C23×C312yes—
205C2×Dic9C23×C311yescharacteristic
206C2×Dic9C23×C311yescharacteristic
207C2×Dic9C23×C312yes—
208C2×Dic9C23×C311yescharacteristic
209C4×D9—22no—
210C4×D9—42no—
211Dic18—22no—
212Dic18—42no—
213#554—91noSylow
214C2×C3×C9—22no—
215C2×C3×C9C22×Q811yescharacteristic
216C2×C3×C9C2×#4111yescharacteristic
217C2×C3×C9H(5,𝔽2)11yescharacteristic
218C3×D9—22no—
219C2×C3×Q8—91no—
220C2×C3×Q8—92no—
221C2×C3×D4—91no—
222C22×C3×C4—21no—
223C22×C3×C4—22no—
224C22×C3×C4—91no—
225C22×C3×C4—92no—
226C22×C3×C4C2×D911yescharacteristic
227C22×C3×C4C2×D912yes—
228C22×C3×C4C2×C3×D311yescharacteristic
229C22×C3×C4C2×C3×D312yes—
230C2×#67—31no—
231C2×C4×D3—31no—
232C2×C4×D3—32no—
233C2×Dic6—31no—
234C2×Dic6—32no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—22no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—92no—
239C3×C4⋊C4—91no—
240C3×C4⋊C4—92no—
241C3×C4⋊C4—92no—
242C3×C4⋊C4—92no—
243C3×C4⋊C4—91no—
244C3×C4⋊C4—92no—
245C3×C4⋊C4C2×D911yescharacteristic
246C3×C4⋊C4C2×D912yes—
247C3×C4⋊C4C2×D911yescharacteristic
248C3×C4⋊C4C2×C3×D311yescharacteristic
249C3×C4⋊C4C2×C3×D311yescharacteristic
250C3×C4⋊C4C2×C3×D312yes—
251C3×C22⋊C4—92no—
252C3×C22⋊C4—91no—
253C3×C22⋊C4—92no—
254C3×C22⋊C4—91no—
255C3×C42—92no—
256C3×C42—91no—
257#217—32no—
258#217—31no—
259(C2×D3)⋊C4—31no—
260(C2×D3)⋊C4—32no—
261C4⋊Dic3—31no—
262C4⋊Dic3—32no—
263Dic3⋊C4—32no—
264Dic3⋊C4—32no—
265Dic3⋊C4—31no—
266Dic3⋊C4—32no—
267Dic3⋊C4—32no—
268C4×Dic3—31no—
269C4×Dic3—32no—
270C22×C32—21no—
271C22×C32—21no—
272C22×C32D3×Q812yes—
273C22×C32C23×D311yescharacteristic
274C2×C3×D3—32no—
275C2×C3×D3—61no—
276C2×C3×D3—61no—
277C32×C4—21no—
278C32×C4—22no—
279C32×C4—22no—
280C32×C4#23512yes—
281C3×Dic3—32no—
282C3×Dic3—62no—
283C3×Dic3—61no—
284C3×Dic3—61no—
285C3×Dic3—62no—
286C22×C9—22no—
287C22×C9—22no—
288C22×C9—21no—
289C22×C9—21no—
290C22×C9—21no—
291C22×C9—22no—
292C22×C9C2×C3×Q812yes—
293C22×C9C24×C311yescharacteristic
294C2×D9—21no—
295C2×D9—21no—
296C2×D9C2×C3×Q812yes—
297C4×C9—22no—
298C4×C9—21no—
299C4×C9—22no—
300C4×C9—22no—
301C4×C9—22no—
302C4×C9—42no—
303C4×C9—42no—
304C4×C9C3×#4112yes—
305Dic9—22no—
306Dic9—21no—
307Dic9—21no—
308Dic9—22no—
309Dic9C2×C3×Q812yes—
310C4⋊Q8—91no—
311#125—92no—
312C22⋊Q8—92no—
313C22⋊Q8—91no—
314C22⋊Q8—92no—
315C22⋊Q8—91no—
316C4×Q8—91no—
317C4×D4—92no—
318C4×D4—91no—
319C2×C4⋊C4—91no—
320C2×C4⋊C4C3×D911yescharacteristic, complemented
321C3×C9#55411yescharacteristic, complemented, Sylow
322C23×C3—21no—
323C23×C3—91no—
324C23×C3C22×D911yescharacteristic
325C23×C3C22×C3×D311yescharacteristic
326C22×D3—31no—
327C3×Q8—182no—
328C3×Q8—362no—
329C3×D4—94no—
330C2×C3×C4—24no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—44no—
340C2×C3×C4—91no—
341C2×C3×C4—92no—
342C2×C3×C4—91no—
343C2×C3×C4—91no—
344C2×C3×C4—92no—
345C2×C3×C4—182no—
346C2×C3×C4—362no—
347C2×C3×C4C22×D912yes—
348C2×C3×C4C22×D911yescharacteristic
349C2×C3×C4C22×D912yes—
350C2×C3×C4C22×D911yescharacteristic
351C2×C3×C4C22×C3×D312yes—
352C2×C3×C4C22×C3×D311yescharacteristic
353C2×C3×C4C22×C3×D312yes—
354C2×C3×C4C22×C3×D311yescharacteristic
355#67—34no—
356C2×Dic3—31no—
357C2×Dic3—32no—
358C2×Dic3—31no—
359C2×Dic3—31no—
360C2×Dic3—32no—
361C4×D3—62no—
362C4×D3—122no—
363Dic6—62no—
364Dic6—122no—
365C2×C32—22no—
366C2×C32C2×#23511yescharacteristic
367C2×C32#100411yescharacteristic
368C2×C32C2×D3×Q811yescharacteristic
369C3×D3—62no—
370C2×C9—24no—
371C2×C9—22no—
372C2×C9—21no—
373C2×C9—21no—
374C2×C9—21no—
375C2×C9C22×C3×Q811yescharacteristic
376C2×C9C2×C3×#4111yescharacteristic
377C2×C9C3×H(5,𝔽2)11yescharacteristic
378D9—22no—
379C2×Q8—91no—
380C2×Q8—92no—
381C2×D4—91no—
382C22×C4—91no—
383C22×C4—92no—
384C22×C4C2×C3×D911yescharacteristic
385C22×C4C2×C3×D912yes—
386C4⋊C4—92no—
387C4⋊C4—91no—
388C4⋊C4—91no—
389C4⋊C4—92no—
390C4⋊C4—92no—
391C4⋊C4—92no—
392C4⋊C4—92no—
393C4⋊C4C2×C3×D911yescharacteristic
394C4⋊C4C2×C3×D912yes—
395C4⋊C4C2×C3×D911yescharacteristic
396C22⋊C4—92no—
397C22⋊C4—91no—
398C22⋊C4—92no—
399C22⋊C4—91no—
400C42—92no—
401C42—91no—
402C22×C3—22no—
403C22×C3—22no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—22no—
410C22×C3—92no—
411C22×C3—181no—
412C22×C3—181no—
413C22×C3Q8×D912yes—
414C22×C3C23×D911yescharacteristic
415C22×C3C3×D3×Q812yes—
416C22×C3C23×C3×D311yescharacteristic
417C2×D3—32no—
418C2×D3—61no—
419C2×D3—61no—
420C3×C4—22no—
421C3×C4—21no—
422C3×C4—22no—
423C3×C4—22no—
424C3×C4—21no—
425C3×C4—22no—
426C3×C4—22no—
427C3×C4—22no—
428C3×C4—42no—
429C3×C4—42no—
430C3×C4—92no—
431C3×C4—181no—
432C3×C4—182no—
433C3×C4—182no—
434C3×C4—181no—
435C3×C4#369712yes—
436C3×C4C3×#23512yes—
437Dic3—32no—
438Dic3—62no—
439Dic3—61no—
440Dic3—61no—
441Dic3—62no—
442C32#551111yescharacteristic
443C9—21no—
444C9C3×#55411yescharacteristic, complemented
445C23—91no—
446C23C22×C3×D911yescharacteristic
447Q8—182no—
448Q8—362no—
449D4—94no—
450C2×C4—22no—
451C2×C4—42no—
452C2×C4—91no—
453C2×C4—92no—
454C2×C4—91no—
455C2×C4—91no—
456C2×C4—92no—
457C2×C4—182no—
458C2×C4—362no—
459C2×C4C22×C3×D912yes—
460C2×C4C22×C3×D911yescharacteristic
461C2×C4C22×C3×D912yes—
462C2×C4C22×C3×D911yescharacteristic
463C2×C3—24no—
464C2×C3—22no—
465C2×C3—22no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—182no—
470C2×C3C2×#369711yescharacteristic
471C2×C3#6371011yescharacteristic
472C2×C3C2×Q8×D911yescharacteristic
473C2×C3C2×C3×#23511yescharacteristic
474C2×C3C3×#100411yescharacteristic
475C2×C3C2×C3×D3×Q811yescharacteristic
476D3—62no—
477C22—21no—
478C22—21no—
479C22—92no—
480C22—181no—
481C22—181no—
482C22C3×Q8×D912yes—
483C22C23×C3×D911yescharacteristic
484C4—21no—
485C4—22no—
486C4—22no—
487C4—92no—
488C4—181no—
489C4—182no—
490C4—182no—
491C4—181no—
492C4C3×#369712yes—
493C3—21no—
494C3#1058961311yescharacteristic, direct product factor
495C3C3×#551111yescharacteristic
496C2—22no—
497C2—182no—
498C2C2×C3×#369711yescharacteristic
499C2C3×#6371011yescharacteristic
500C2C2×C3×Q8×D911yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC2×C3×C9×C4⋊C4C2
socleC22×C32C23×D3
prime radicalC2×C3×#611C22
hypercenterC2×C3×C4⋊C4D9
hypocenterC9C3×#554
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C9×C4⋊C4C2semidirect
4. C3×#63486C2semidirect
5. C3×#63486C2semidirect
6. C3×#63459C2semidirect
7. C3×#63458C2semidirect
8. C3×C4⋊C4×D9C2semidirect
9. C3×#63451C2semidirect
10. C3×#63450C2semidirect
11. C3×#63449C2semidirect
13. #10589613C3direct (N × H)
49. C2×C9×C4⋊C4C2×C3semidirect
52. #63486C2×C3semidirect
53. #63486C2×C3semidirect
54. #63459C2×C3semidirect
55. #63458C2×C3semidirect
56. C4⋊C4×D9C2×C3semidirect
57. #63451C2×C3semidirect
58. #63450C2×C3semidirect
59. #63449C2×C3semidirect
68. C3×#611Q8semidirect
158. C2×C3×C4⋊C4D9semidirect
203. #611C3×Q8semidirect
320. C2×C4⋊C4C3×D9semidirect
321. C3×C9#554semidirect
444. C9C3×#554semidirect
494. C3#10589613direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C9×C4⋊C42
C3×C4×#6112
C3×#634862
C3×#634592
C3×#634582
C3×C4⋊C4×D92
C3×#634512
C3×#634502
C3×#634492
C3×#55113
#105896133