Finite Group Databasebeta

← C3×#10589737 #13152915

Subgroups — C3×#10589737 #13152915

Order 1728.

758 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
1C3×C9×C22⋊Q8C211yesmaximal, characteristic, complemented
2C3×#63508C211yesmaximal, characteristic, complemented
3C3×Q8×Dic9C211yesmaximal, characteristic, complemented
4C3×C4×#611C211yesmaximal, characteristic
5C2×C3×C4⋊Dic9C211yesmaximal, characteristic, complemented
6C3×#63459C212yesmaximal, complemented
7C3×C4⋊C4×D9C211yesmaximal, characteristic, complemented
8C3×#63451C212yesmaximal, complemented
9C3×#63450C211yesmaximal, characteristic, complemented
10C3×#611⋊C4C212yesmaximal
11C3×#63440C212yesmaximal, complemented
12C3×#5635—31nomaximal
13#10589737C311yesmaximal, characteristic, direct product factor
14C2×C3×Q8×C9C2211yescharacteristic
15C22×C3×C4×C9C2211yescharacteristic
16C2×C3×#611C2211yescharacteristic
17C22×C3×Dic9C2212yes—
18C2×C3×C4×D9C2211yescharacteristic
19C2×C3×C4×D9C2212yes—
20C2×C3×Dic18C2212yes—
21C3×C9×C4⋊C4C2211yescharacteristic
22C3×C9×C4⋊C4C2212yes—
23C3×C9×C22⋊C4C2212yes—
24C3×#3676C2211yescharacteristic
25C3×(C2×D9)⋊C4C2211yescharacteristic
26C3×(C2×D9)⋊C4C2212yes—
27C3×C4⋊Dic9C2211yescharacteristic
28C3×C4⋊Dic9C2211yescharacteristic
29C3×C4⋊Dic9C2212yes—
30C3×C4⋊Dic9C2212yes—
31C3×Dic9⋊C4C2211yescharacteristic
32C3×Dic9⋊C4C2212yes—
33C3×Dic9⋊C4C2212yes—
34C3×Dic9⋊C4C2212yes—
35C3×C4×Dic9C2211yescharacteristic
36C3×C4×Dic9C2212yes—
37C32×C22⋊Q8D311yescharacteristic
38C3×#946—31no—
39C3×Q8×Dic3—31no—
40C3×C4×#67—31no—
41C2×C3×C4⋊Dic3—31no—
42C3×#897—32no—
43C3×D3×C4⋊C4—31no—
44C3×#889—32no—
45C3×#888—31no—
46C3×#67⋊C4—32no—
47C3×#878—32no—
48C9×C22⋊Q8—21no—
49C9×C22⋊Q8C2×C311yescharacteristic, complemented
50#63508C2×C311yescharacteristic, complemented
51Q8×Dic9C2×C311yescharacteristic, complemented
52C4×#611C2×C311yescharacteristic
53C2×C4⋊Dic9C2×C311yescharacteristic, complemented
54#63459C2×C312yescomplemented
55C4⋊C4×D9C2×C311yescharacteristic, complemented
56#63451C2×C312yescomplemented
57#63450C2×C311yescharacteristic, complemented
58#611⋊C4C2×C312yes—
59#63440C2×C312yescomplemented
60C23×C3×C9C2311yescharacteristic
61C22×C3×D9C2311yescharacteristic
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×#611Q814yescomplemented
69C2×C3×Dic9—42no—
70C2×C3×Dic9C2311yescharacteristic
71C2×C3×Dic9C2312yes—
72C2×C3×Dic9C2312yes—
73C2×C3×Dic9C2311yescharacteristic
74C2×C3×Dic9C2311yescharacteristic
75C3×C4×D9—22no—
76C3×C4×D9—42no—
77C3×Dic18—42no—
78C3×#554—91no—
79#5635—31no—
80C2×C32×Q8C2×D311yescharacteristic
81C22×C32×C4C2×D311yescharacteristic
82C2×C3×#67—31no—
83C22×C3×Dic3—32no—
84C2×C3×C4×D3—31no—
85C2×C3×C4×D3—32no—
86C2×C3×Dic6—32no—
87C32×C4⋊C4C2×D311yescharacteristic
88C32×C4⋊C4C2×D312yes—
89C32×C22⋊C4C2×D312yes—
90C3×#217—31no—
91C3×(C2×D3)⋊C4—32no—
92C3×(C2×D3)⋊C4—31no—
93C3×C4⋊Dic3—32no—
94C3×C4⋊Dic3—31no—
95C3×C4⋊Dic3—32no—
96C3×C4⋊Dic3—31no—
97C3×Dic3⋊C4—32no—
98C3×Dic3⋊C4—32no—
99C3×Dic3⋊C4—32no—
100C3×Dic3⋊C4—31no—
101C3×C4×Dic3—32no—
102C3×C4×Dic3—31no—
103C2×Q8×C9—21no—
104C2×Q8×C9C22×C311yescharacteristic
105C22×C4×C9—21no—
106C22×C4×C9C22×C311yescharacteristic
107C2×#611C22×C311yescharacteristic
108C22×Dic9C22×C312yes—
109C2×C4×D9C22×C311yescharacteristic
110C2×C4×D9C22×C312yes—
111C2×Dic18C22×C312yes—
112C9×C4⋊C4—21no—
113C9×C4⋊C4—22no—
114C9×C4⋊C4C22×C311yescharacteristic
115C9×C4⋊C4C22×C312yes—
116C9×C22⋊C4—22no—
117C9×C22⋊C4C22×C312yes—
118#3676C22×C311yescharacteristic
119(C2×D9)⋊C4C22×C311yescharacteristic
120(C2×D9)⋊C4C22×C312yes—
121C4⋊Dic9C22×C312yes—
122C4⋊Dic9C22×C311yescharacteristic
123C4⋊Dic9C22×C311yescharacteristic
124C4⋊Dic9C22×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—21no—
143C3×Dic9—21no—
144C3×Dic9—22no—
145C3×Dic9—22no—
146C3×Dic9C2×Q812yes—
147C3×C4⋊Q8—91no—
148C3×#125—92no—
149C3×C22⋊Q8—21no—
150C3×C22⋊Q8—91no—
151C3×C22⋊Q8—92no—
152C3×C22⋊Q8—92no—
153C3×C22⋊Q8D911yescharacteristic, complemented
154C3×C22⋊Q8C3×D311yescharacteristic
155C3×C4×Q8—91no—
156C3×C4×D4—91no—
157C3×C4×D4—92no—
158C2×C3×C4⋊C4—91no—
159C2×C3×C4⋊C4—91no—
160#946—31no—
161Q8×Dic3—31no—
162C4×#67—31no—
163C2×C4⋊Dic3—31no—
164#897—32no—
165D3×C4⋊C4—31no—
166#889—32no—
167#888—31no—
168#67⋊C4—32no—
169#878—32no—
170C23×C32C22×D311yescharacteristic
171C22×C3×D3—31no—
172C32×Q8—22no—
173C2×C32×C4—22no—
174C2×C32×C4C22×D312yes—
175C2×C32×C4C22×D312yes—
176C2×C32×C4C22×D311yescharacteristic
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—
184C2×C3×Dic3—122no—
185C3×C4×D3—62no—
186C3×C4×D3—122no—
187C3×Dic6—122no—
188C23×C9—21no—
189C23×C9C23×C311yescharacteristic
190C22×D9C23×C311yescharacteristic
191Q8×C9—22no—
192Q8×C9—42no—
193C2×C4×C9—22no—
194C2×C4×C9—21no—
195C2×C4×C9—22no—
196C2×C4×C9—22no—
197C2×C4×C9—21no—
198C2×C4×C9—42no—
199C2×C4×C9C23×C312yes—
200C2×C4×C9C23×C312yes—
201C2×C4×C9C23×C311yescharacteristic
202C2×C4×C9C23×C311yescharacteristic
203#611C3×Q814yescomplemented
204C2×Dic9—42no—
205C2×Dic9C23×C312yes—
206C2×Dic9C23×C311yescharacteristic
207C2×Dic9C23×C311yescharacteristic
208C2×Dic9C23×C312yes—
209C2×Dic9C23×C311yescharacteristic
210C4×D9—22no—
211C4×D9—42no—
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—21no—
220C2×C3×Q8—92no—
221C2×C3×Q8C2×D911yescharacteristic
222C2×C3×Q8C2×C3×D311yescharacteristic
223C2×C3×D4—91no—
224C22×C3×C4—21no—
225C22×C3×C4—92no—
226C22×C3×C4—91no—
227C22×C3×C4—92no—
228C22×C3×C4C2×D911yescharacteristic
229C22×C3×C4C2×C3×D311yescharacteristic
230C2×#67—31no—
231C22×Dic3—32no—
232C2×C4×D3—31no—
233C2×C4×D3—32no—
234C2×Dic6—32no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—22no—
237C3×C4⋊C4—92no—
238C3×C4⋊C4—91no—
239C3×C4⋊C4—92no—
240C3×C4⋊C4—92no—
241C3×C4⋊C4—91no—
242C3×C4⋊C4—92no—
243C3×C4⋊C4—91no—
244C3×C4⋊C4—92no—
245C3×C4⋊C4C2×D911yescharacteristic
246C3×C4⋊C4C2×D912yes—
247C3×C4⋊C4C2×C3×D311yescharacteristic
248C3×C4⋊C4C2×C3×D312yes—
249C3×C22⋊C4—22no—
250C3×C22⋊C4—91no—
251C3×C22⋊C4—91no—
252C3×C22⋊C4—92no—
253C3×C22⋊C4C2×D912yes—
254C3×C22⋊C4C2×C3×D312yes—
255C3×C42—92no—
256C3×C42—91no—
257#217—31no—
258(C2×D3)⋊C4—32no—
259(C2×D3)⋊C4—31no—
260C4⋊Dic3—32no—
261C4⋊Dic3—31no—
262C4⋊Dic3—32no—
263C4⋊Dic3—31no—
264Dic3⋊C4—32no—
265Dic3⋊C4—32no—
266Dic3⋊C4—32no—
267Dic3⋊C4—31no—
268C4×Dic3—32no—
269C4×Dic3—31no—
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#23712yes—
281C3×Dic3—32no—
282C3×Dic3—61no—
283C3×Dic3—61no—
284C3×Dic3—62no—
285C3×Dic3—62no—
286C22×C9—21no—
287C22×C9—21no—
288C22×C9—21no—
289C22×C9—22no—
290C22×C9—41no—
291C22×C9—41no—
292C22×C9C2×C3×Q812yes—
293C22×C9C24×C311yescharacteristic
294C2×D9—21no—
295C2×D9—21no—
296C2×D9C2×C3×Q812yes—
297C4×C9—21no—
298C4×C9—22no—
299C4×C9—22no—
300C4×C9—22no—
301C4×C9—42no—
302C4×C9—41no—
303C4×C9—42no—
304C4×C9C3×#4112yes—
305Dic9—21no—
306Dic9—21no—
307Dic9—22no—
308Dic9—22no—
309Dic9C2×C3×Q812yes—
310C4⋊Q8—91no—
311#125—92no—
312C22⋊Q8—91no—
313C22⋊Q8—92no—
314C22⋊Q8—92no—
315C22⋊Q8C3×D911yescharacteristic, complemented
316C4×Q8—91no—
317C4×D4—91no—
318C4×D4—92no—
319C2×C4⋊C4—91no—
320C2×C4⋊C4—91no—
321C3×C9#55411yescharacteristic, complemented, Sylow
322C23×C3—21no—
323C23×C3—91no—
324C23×C3C22×D911yescharacteristic
325C23×C3C22×C3×D311yescharacteristic
326C22×D3—31no—
327C3×Q8—22no—
328C3×Q8—22no—
329C3×Q8—42no—
330C3×Q8—362no—
331C3×D4—94no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—21no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—21no—
338C2×C3×C4—42no—
339C2×C3×C4—91no—
340C2×C3×C4—91no—
341C2×C3×C4—92no—
342C2×C3×C4—91no—
343C2×C3×C4—92no—
344C2×C3×C4—182no—
345C2×C3×C4—362no—
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×D312yes—
353C2×C3×C4C22×C3×D311yescharacteristic
354C2×C3×C4C22×C3×D311yescharacteristic
355#67—34no—
356C2×Dic3—31no—
357C2×Dic3—32no—
358C2×Dic3—31no—
359C2×Dic3—31no—
360C2×Dic3—32no—
361C2×Dic3—122no—
362C4×D3—62no—
363C4×D3—122no—
364Dic6—122no—
365C2×C32—22no—
366C2×C32C2×#23711yescharacteristic
367C2×C32C2×D3×Q811yescharacteristic
368C2×C32#100911yescharacteristic
369C3×D3—62no—
370C2×C9—22no—
371C2×C9—21no—
372C2×C9—21no—
373C2×C9—21no—
374C2×C9—42no—
375C2×C9C22×C3×Q811yescharacteristic
376C2×C9C2×C3×#4111yescharacteristic
377C2×C9C3×H(5,𝔽2)11yescharacteristic
378D9—22no—
379C2×Q8—92no—
380C2×Q8C2×C3×D911yescharacteristic
381C2×D4—91no—
382C22×C4—92no—
383C22×C4—91no—
384C22×C4—92no—
385C22×C4C2×C3×D911yescharacteristic
386C4⋊C4—91no—
387C4⋊C4—92no—
388C4⋊C4—92no—
389C4⋊C4—91no—
390C4⋊C4—91no—
391C4⋊C4—92no—
392C4⋊C4—92no—
393C4⋊C4—92no—
394C4⋊C4C2×C3×D911yescharacteristic
395C4⋊C4C2×C3×D912yes—
396C22⋊C4—91no—
397C22⋊C4—91no—
398C22⋊C4—92no—
399C22⋊C4C2×C3×D912yes—
400C42—92no—
401C42—91no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—22no—
408C22×C3—41no—
409C22×C3—41no—
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—21no—
421C3×C4—22no—
422C3×C4—22no—
423C3×C4—21no—
424C3×C4—22no—
425C3×C4—22no—
426C3×C4—22no—
427C3×C4—41no—
428C3×C4—42no—
429C3×C4—42no—
430C3×C4—92no—
431C3×C4—181no—
432C3×C4—182no—
433C3×C4—182no—
434C3×C4—181no—
435C3×C4#369912yes—
436C3×C4C3×#23712yes—
437Dic3—32no—
438Dic3—61no—
439Dic3—61no—
440Dic3—62no—
441Dic3—62no—
442C32#563511yescharacteristic
443C9—21no—
444C9C3×#55411yescharacteristic, complemented
445C23—91no—
446C23C22×C3×D911yescharacteristic
447Q8—22no—
448Q8—362no—
449D4—94no—
450C2×C4—22no—
451C2×C4—91no—
452C2×C4—91no—
453C2×C4—92no—
454C2×C4—91no—
455C2×C4—92no—
456C2×C4—182no—
457C2×C4—362no—
458C2×C4—362no—
459C2×C4C22×C3×D912yes—
460C2×C4C22×C3×D911yescharacteristic
461C2×C4C22×C3×D912yes—
462C2×C4C22×C3×D911yescharacteristic
463C2×C3—22no—
464C2×C3—22no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—42no—
469C2×C3—182no—
470C2×C3C2×#369911yescharacteristic
471C2×C3C2×Q8×D911yescharacteristic
472C2×C3#6371511yescharacteristic
473C2×C3C2×C3×#23711yescharacteristic
474C2×C3C2×C3×D3×Q811yescharacteristic
475C2×C3C3×#100911yescharacteristic
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×#369912yes—
493C3—21no—
494C3#1058973711yescharacteristic, direct product factor
495C3C3×#563511yescharacteristic
496C2—22no—
497C2—182no—
498C2C2×C3×#369911yescharacteristic
499C2C2×C3×Q8×D911yescharacteristic
500C2C3×#6371511yescharacteristic
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×#554
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. C2×C3×C4⋊Dic9C2semidirect
6. C3×#63459C2semidirect
7. C3×C4⋊C4×D9C2semidirect
8. C3×#63451C2semidirect
9. C3×#63450C2semidirect
11. C3×#63440C2semidirect
13. #10589737C3direct (N × H)
49. C9×C22⋊Q8C2×C3semidirect
50. #63508C2×C3semidirect
51. Q8×Dic9C2×C3semidirect
53. C2×C4⋊Dic9C2×C3semidirect
54. #63459C2×C3semidirect
55. C4⋊C4×D9C2×C3semidirect
56. #63451C2×C3semidirect
57. #63450C2×C3semidirect
59. #63440C2×C3semidirect
68. C3×#611Q8semidirect
153. C3×C22⋊Q8D9semidirect
203. #611C3×Q8semidirect
315. C22⋊Q8C3×D9semidirect
321. C3×C9#554semidirect
444. C9C3×#554semidirect
494. C3#10589737direct (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
C2×C3×C4⋊Dic92
C3×#634592
C3×C4⋊C4×D92
C3×#634512
C3×#634502
C3×#611⋊C42
C3×#634402
C3×#56353
#105897373