Finite Group Databasebeta

← C3×#5657 #10595754

Subgroups — C3×#5657 #10595754

Order 576.

539 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#123C211yesmaximal, characteristic, complemented
2C3×#938C211yesmaximal, characteristic, complemented
3C3×#937C211yesmaximal, characteristic, complemented
4C3×#930C211yesmaximal, characteristic, complemented
5C3×C4×#67C211yesmaximal, characteristic, complemented
6C3×#898C211yesmaximal, characteristic, complemented
7C3×#897C211yesmaximal, characteristic, complemented
8C3×#894C211yesmaximal, characteristic, complemented
9C3×#893C211yesmaximal, characteristic, complemented
10C3×#885C211yesmaximal, characteristic, complemented
11C3×#885C211yesmaximal, characteristic, complemented
12C3×#884C211yesmaximal, characteristic, complemented
13C3×#882C211yesmaximal, characteristic, complemented
14C3×D3×C22⋊C4C211yesmaximal, characteristic, complemented
15C3×#879C211yesmaximal, characteristic, complemented
16C3×#538—31nomaximal
17#5657C311yesmaximal, characteristic, direct product factor
18C23×C3×D3C2211yescharacteristic
19C2×C32×D4C2211yescharacteristic
20C22×C32×C4C2211yescharacteristic, complemented
21C2×C3×#67C2211yescharacteristic
22C2×C3×#67C2211yescharacteristic
23C2×C3×#67C2211yescharacteristic
24C2×C3×#67C2211yescharacteristic, complemented
25C2×C3×D12C2211yescharacteristic
26C2×C3×D12C2211yescharacteristic
27C2×C3×C4×D3C2211yescharacteristic
28C2×C3×C4×D3C2211yescharacteristic
29C2×C3×C4×D3C2211yescharacteristic, complemented
30C2×C3×Dic6C2211yescharacteristic, complemented
31C32×C4⋊C4C2211yescharacteristic, complemented
32C32×C4⋊C4C2211yescharacteristic, complemented
33C32×C22⋊C4C2211yescharacteristic
34C32×C22⋊C4C2211yescharacteristic
35C32×C22⋊C4C2211yescharacteristic, complemented
36C3×#217C2211yescharacteristic
37C3×#217C2211yescharacteristic
38C3×#217C2211yescharacteristic, complemented
39C3×(C2×D3)⋊C4C2211yescharacteristic
40C3×(C2×D3)⋊C4C2211yescharacteristic
41C3×(C2×D3)⋊C4C2211yescharacteristic
42C3×(C2×D3)⋊C4C2211yescharacteristic
43C3×(C2×D3)⋊C4C2211yescharacteristic
44C3×(C2×D3)⋊C4C2211yescharacteristic
45C3×(C2×D3)⋊C4C2211yescharacteristic, complemented
46C3×(C2×D3)⋊C4C2211yescharacteristic, complemented
47C3×C4⋊Dic3C2211yescharacteristic, complemented
48C3×C4⋊Dic3C2211yescharacteristic, complemented
49C3×Dic3⋊C4C2211yescharacteristic, complemented
50C3×Dic3⋊C4C2211yescharacteristic, complemented
51C3×C4×Dic3C2211yescharacteristic, complemented
52C3×C4×Dic3C2211yescharacteristic, complemented
53C3×#126—31no—
54C3×#126—31no—
55C3×#124—31no—
56C3×#124—31no—
57C3×#123—21no—
58C3×#123—31no—
59C3×#123D311yescharacteristic, complemented
60C3×#123C2×C311yescharacteristic, complemented
61C3×C22⋊Q8—31no—
62C3×#121—31no—
63C3×#121—31no—
64C3×#121—31no—
65C3×#120—31no—
66C3×#120—31no—
67C3×C4×D4—31no—
68C3×C4×D4—31no—
69C2×C3×C22⋊C4—31no—
70#938C2×C311yescharacteristic, complemented
71#937C2×C311yescharacteristic, complemented
72#930C2×C311yescharacteristic, complemented
73C4×#67C2×C311yescharacteristic, complemented
74#898C2×C311yescharacteristic, complemented
75#897C2×C311yescharacteristic, complemented
76#894C2×C311yescharacteristic, complemented
77#893C2×C311yescharacteristic, complemented
78#885C2×C311yescharacteristic, complemented
79#885C2×C311yescharacteristic, complemented
80#884C2×C311yescharacteristic, complemented
81#882C2×C311yescharacteristic, complemented
82D3×C22⋊C4C2×C311yescharacteristic, complemented
83#879C2×C311yescharacteristic, complemented
84C23×C32C2311yescharacteristic
85C23×C32C2311yescharacteristic
86C22×C3×D3—22no—
87C22×C3×D3—41no—
88C22×C3×D3—41no—
89C22×C3×D3C2311yescharacteristic
90C22×C3×D3C2311yescharacteristic
91C22×C3×D3C2311yescharacteristic
92C32×D4—41no—
93C2×C32×C4—41no—
94C2×C32×C4C2311yescharacteristic
95C2×C32×C4C2311yescharacteristic
96C2×C32×C4C2311yescharacteristic
97C2×C32×C4C2311yescharacteristic
98C2×C32×C4C2311yescharacteristic
99C3×#67—22no—
100C3×#67—41no—
101C3×#67—41no—
102C3×#67—41no—
103C2×C3×Dic3C2311yescharacteristic
104C2×C3×Dic3C2311yescharacteristic
105C2×C3×Dic3C2311yescharacteristic
106C2×C3×Dic3C2311yescharacteristic
107C2×C3×Dic3C2311yescharacteristic
108C3×D12—22no—
109C3×D12—41no—
110C3×C4×D3—41no—
111C3×C4×D3—41no—
112C3×C4×D3—41no—
113C3×Dic6—41no—
114#538—31noSylow
115C24×C3—31no—
116C23×D3C22×C311yescharacteristic
117C2×C3×Q8—31no—
118C2×C3×D4—21no—
119C2×C3×D4—31no—
120C2×C3×D4—31no—
121C2×C3×D4—31no—
122C2×C3×D4—31no—
123C2×C3×D4—31no—
124C2×C3×D4—31no—
125C2×C3×D4C2×D311yescharacteristic
126C2×C3×D4C22×C311yescharacteristic
127C22×C3×C4—21no—
128C22×C3×C4—31no—
129C22×C3×C4—31no—
130C22×C3×C4—31no—
131C22×C3×C4C2×D311yescharacteristic, complemented
132C22×C3×C4C22×C311yescharacteristic, complemented
133C2×#67C22×C311yescharacteristic
134C2×#67C22×C311yescharacteristic
135C2×#67C22×C311yescharacteristic
136C2×#67C22×C311yescharacteristic, complemented
137C2×D12C22×C311yescharacteristic
138C2×D12C22×C311yescharacteristic
139C2×C4×D3C22×C311yescharacteristic
140C2×C4×D3C22×C311yescharacteristic
141C2×C4×D3C22×C311yescharacteristic, complemented
142C2×Dic6C22×C311yescharacteristic, complemented
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—31no—
146C3×C4⋊C4—31no—
147C3×C4⋊C4—31no—
148C3×C4⋊C4—31no—
149C3×C4⋊C4C2×D311yescharacteristic, complemented
150C3×C4⋊C4C2×D311yescharacteristic, complemented
151C3×C4⋊C4C22×C311yescharacteristic, complemented
152C3×C4⋊C4C22×C311yescharacteristic, complemented
153C3×C22⋊C4—21no—
154C3×C22⋊C4—21no—
155C3×C22⋊C4—21no—
156C3×C22⋊C4—31no—
157C3×C22⋊C4—31no—
158C3×C22⋊C4—31no—
159C3×C22⋊C4—31no—
160C3×C22⋊C4—31no—
161C3×C22⋊C4—31no—
162C3×C22⋊C4—31no—
163C3×C22⋊C4—31no—
164C3×C22⋊C4—31no—
165C3×C22⋊C4—31no—
166C3×C22⋊C4—31no—
167C3×C22⋊C4C2×D311yescharacteristic
168C3×C22⋊C4C2×D311yescharacteristic
169C3×C22⋊C4C2×D311yescharacteristic, complemented
170C3×C22⋊C4C22×C311yescharacteristic
171C3×C22⋊C4C22×C311yescharacteristic
172C3×C22⋊C4C22×C311yescharacteristic, complemented
173C3×C42—31no—
174C3×C42—31no—
175#217C22×C311yescharacteristic
176#217C22×C311yescharacteristic
177#217C22×C311yescharacteristic, complemented
178(C2×D3)⋊C4C22×C311yescharacteristic
179(C2×D3)⋊C4C22×C311yescharacteristic
180(C2×D3)⋊C4C22×C311yescharacteristic
181(C2×D3)⋊C4C22×C311yescharacteristic
182(C2×D3)⋊C4C22×C311yescharacteristic
183(C2×D3)⋊C4C22×C311yescharacteristic
184(C2×D3)⋊C4C22×C311yescharacteristic, complemented
185(C2×D3)⋊C4C22×C311yescharacteristic, complemented
186C4⋊Dic3C22×C311yescharacteristic, complemented
187C4⋊Dic3C22×C311yescharacteristic, complemented
188Dic3⋊C4C22×C311yescharacteristic, complemented
189Dic3⋊C4C22×C311yescharacteristic, complemented
190C4×Dic3C22×C311yescharacteristic, complemented
191C4×Dic3C22×C311yescharacteristic, complemented
192C22×C32—21no—
193C22×C32—21no—
194C22×C32—21no—
195C22×C32—21no—
196C22×C32—21no—
197C22×C32—21no—
198C22×C32C2411yescharacteristic
199C2×C3×D3—21no—
200C2×C3×D3—21no—
201C2×C3×D3—21no—
202C2×C3×D3—21no—
203C2×C3×D3—21no—
204C2×C3×D3—21no—
205C2×C3×D3—21no—
206C2×C3×D3—21no—
207C2×C3×D3—44no—
208C2×C3×D3#4112yes—
209C32×C4—21no—
210C32×C4—21no—
211C32×C4—21no—
212C32×C4—21no—
213C32×C4—21no—
214C3×Dic3—21no—
215C3×Dic3—21no—
216C3×Dic3—21no—
217C3×Dic3—21no—
218C3×Dic3—21no—
219#126—31no—
220#126—31no—
221#124—31no—
222#124—31no—
223#123—31no—
224#123C3×D311yescharacteristic, complemented
225C22⋊Q8—31no—
226#121—31no—
227#121—31no—
228#121—31no—
229#120—31no—
230#120—31no—
231C4×D4—31no—
232C4×D4—31no—
233C2×C22⋊C4—31no—
234C23×C3—21no—
235C23×C3—21no—
236C23×C3—31no—
237C23×C3—31no—
238C23×C3—31no—
239C23×C3—62no—
240C23×C3—121no—
241C23×C3—121no—
242C23×C3C22×D311yescharacteristic
243C23×C3C22×D311yescharacteristic
244C23×C3C23×C311yescharacteristic
245C23×C3C23×C311yescharacteristic
246C22×D3—22no—
247C22×D3—41no—
248C22×D3—41no—
249C22×D3C23×C311yescharacteristic
250C22×D3C23×C311yescharacteristic
251C22×D3C23×C311yescharacteristic
252C3×Q8—121no—
253C3×D4—41no—
254C3×D4—41no—
255C3×D4—42no—
256C3×D4—62no—
257C3×D4—62no—
258C3×D4—121no—
259C3×D4—121no—
260C3×D4—121no—
261C3×D4—121no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—31no—
268C2×C3×C4—31no—
269C2×C3×C4—31no—
270C2×C3×C4—31no—
271C2×C3×C4—31no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—42no—
275C2×C3×C4—121no—
276C2×C3×C4—121no—
277C2×C3×C4—121no—
278C2×C3×C4C22×D311yescharacteristic
279C2×C3×C4C22×D311yescharacteristic
280C2×C3×C4C22×D311yescharacteristic
281C2×C3×C4C22×D311yescharacteristic
282C2×C3×C4C22×D311yescharacteristic
283C2×C3×C4C23×C311yescharacteristic
284C2×C3×C4C23×C311yescharacteristic
285C2×C3×C4C23×C311yescharacteristic
286C2×C3×C4C23×C311yescharacteristic
287C2×C3×C4C23×C311yescharacteristic
288#67—22no—
289#67—41no—
290#67—41no—
291#67—41no—
292C2×Dic3C23×C311yescharacteristic
293C2×Dic3C23×C311yescharacteristic
294C2×Dic3C23×C311yescharacteristic
295C2×Dic3C23×C311yescharacteristic
296C2×Dic3C23×C311yescharacteristic
297D12—22no—
298D12—41no—
299C4×D3—41no—
300C4×D3—41no—
301C4×D3—41no—
302Dic6—41no—
303C2×C32—41no—
304C2×C32—41no—
305C2×C32C2×#4111yescharacteristic
306C2×C32H(5,𝔽2)11yescharacteristic
307C2×C32H(5,𝔽2)11yescharacteristic
308C3×D3—22no—
309C3×D3—41no—
310C3×D3—41no—
311C24—31no—
312C2×Q8—31no—
313C2×D4—31no—
314C2×D4—31no—
315C2×D4—31no—
316C2×D4—31no—
317C2×D4—31no—
318C2×D4—31no—
319C2×D4C2×C3×D311yescharacteristic
320C22×C4—31no—
321C22×C4—31no—
322C22×C4—31no—
323C22×C4C2×C3×D311yescharacteristic, complemented
324C4⋊C4—31no—
325C4⋊C4—31no—
326C4⋊C4—31no—
327C4⋊C4—31no—
328C4⋊C4C2×C3×D311yescharacteristic, complemented
329C4⋊C4C2×C3×D311yescharacteristic, complemented
330C22⋊C4—31no—
331C22⋊C4—31no—
332C22⋊C4—31no—
333C22⋊C4—31no—
334C22⋊C4—31no—
335C22⋊C4—31no—
336C22⋊C4—31no—
337C22⋊C4—31no—
338C22⋊C4—31no—
339C22⋊C4—31no—
340C22⋊C4—31no—
341C22⋊C4C2×C3×D311yescharacteristic
342C22⋊C4C2×C3×D311yescharacteristic
343C22⋊C4C2×C3×D311yescharacteristic, complemented
344C42—31no—
345C42—31no—
346C22×C3—22no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—32no—
361C22×C3—41no—
362C22×C3—41no—
363C22×C3—41no—
364C22×C3—41no—
365C22×C3—41no—
366C22×C3—61no—
367C22×C3—61no—
368C22×C3—61no—
369C22×C3—61no—
370C22×C3—61no—
371C22×C3—61no—
372C22×C3—61no—
373C22×C3—61no—
374C22×C3—124no—
375C22×C3C23×D311yescharacteristic
376C22×C3C24×C311yescharacteristic
377C2×D3—21no—
378C2×D3—21no—
379C2×D3—21no—
380C2×D3—21no—
381C2×D3—21no—
382C2×D3—21no—
383C2×D3—21no—
384C2×D3—21no—
385C2×D3—44no—
386C2×D3C3×#4112yes—
387C3×C4—21no—
388C3×C4—21no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—41no—
398C3×C4—41no—
399C3×C4—41no—
400C3×C4—41no—
401C3×C4—41no—
402C3×C4—61no—
403C3×C4—61no—
404C3×C4—61no—
405C3×C4—61no—
406C3×C4—61no—
407Dic3—21no—
408Dic3—21no—
409Dic3—21no—
410Dic3—21no—
411Dic3—21no—
412C32#53811yescharacteristic, complemented, Sylow
413C23—31no—
414C23—31no—
415C23—31no—
416C23—62no—
417C23—121no—
418C23—121no—
419C23C22×C3×D311yescharacteristic
420C23C22×C3×D311yescharacteristic
421Q8—121no—
422D4—41no—
423D4—62no—
424D4—62no—
425D4—121no—
426D4—121no—
427D4—121no—
428D4—121no—
429C2×C4—31no—
430C2×C4—31no—
431C2×C4—31no—
432C2×C4—31no—
433C2×C4—31no—
434C2×C4—41no—
435C2×C4—121no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4C22×C3×D311yescharacteristic
439C2×C4C22×C3×D311yescharacteristic
440C2×C4C22×C3×D311yescharacteristic
441C2×C4C22×C3×D311yescharacteristic
442C2×C4C22×C3×D311yescharacteristic
443C2×C3—21no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—42no—
447C2×C3—41no—
448C2×C3—41no—
449C2×C3—41no—
450C2×C3—41no—
451C2×C3—62no—
452C2×C3—81no—
453C2×C3—121no—
454C2×C3—121no—
455C2×C3#100411yescharacteristic
456C2×C3D3×#4111yescharacteristic
457C2×C3#100911yescharacteristic
458C2×C3C2×C3×#4111yescharacteristic
459C2×C3C3×H(5,𝔽2)11yescharacteristic
460C2×C3C3×H(5,𝔽2)11yescharacteristic
461D3—22no—
462D3—41no—
463D3—41no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—32no—
471C22—61no—
472C22—61no—
473C22—61no—
474C22—61no—
475C22—61no—
476C22—61no—
477C22—61no—
478C22—61no—
479C22—124no—
480C22C23×C3×D311yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—21no—
486C4—61no—
487C4—61no—
488C4—61no—
489C4—61no—
490C4—61no—
491C3—21no—
492C3#565711yescharacteristic, direct product factor
493C3C3×#53811yescharacteristic, complemented
494C2—41no—
495C2—41no—
496C2—62no—
497C2—121no—
498C2—121no—
499C2C3×#100411yescharacteristic
500C2C3×D3×#4111yescharacteristic
501C2C3×#100911yescharacteristic
subgroupG/H
centerC22×C3C23×D3
commutator, cosocleC22×C3C24×C3
FrattiniC22C23×C3×D3
FittingC32×#123C2
socleC22×C32C24
hypercenterC3×#123D3
hypocenterC3C3×#538
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×#123

Complements

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

NH ≅ G/Ntype
1. C32×#123C2semidirect
2. C3×#938C2semidirect
3. C3×#937C2semidirect
4. C3×#930C2semidirect
5. C3×C4×#67C2semidirect
6. C3×#898C2semidirect
7. C3×#897C2semidirect
8. C3×#894C2semidirect
9. C3×#893C2semidirect
10. C3×#885C2semidirect
11. C3×#885C2semidirect
12. C3×#884C2semidirect
13. C3×#882C2semidirect
14. C3×D3×C22⋊C4C2semidirect
15. C3×#879C2semidirect
17. #5657C3direct (N × H)
20. C22×C32×C4C22semidirect
24. C2×C3×#67C22semidirect
29. C2×C3×C4×D3C22semidirect
30. C2×C3×Dic6C22semidirect
31. C32×C4⋊C4C22semidirect
32. C32×C4⋊C4C22semidirect
35. C32×C22⋊C4C22semidirect
38. C3×#217C22semidirect
45. C3×(C2×D3)⋊C4C22semidirect
46. C3×(C2×D3)⋊C4C22semidirect
47. C3×C4⋊Dic3C22semidirect
48. C3×C4⋊Dic3C22semidirect
49. C3×Dic3⋊C4C22semidirect
50. C3×Dic3⋊C4C22semidirect
51. C3×C4×Dic3C22semidirect
52. C3×C4×Dic3C22semidirect
59. C3×#123D3semidirect
60. C3×#123C2×C3semidirect
70. #938C2×C3semidirect
71. #937C2×C3semidirect
72. #930C2×C3semidirect
73. C4×#67C2×C3semidirect
74. #898C2×C3semidirect
75. #897C2×C3semidirect
76. #894C2×C3semidirect
77. #893C2×C3semidirect
78. #885C2×C3semidirect
79. #885C2×C3semidirect
80. #884C2×C3semidirect
81. #882C2×C3semidirect
82. D3×C22⋊C4C2×C3semidirect
83. #879C2×C3semidirect
131. C22×C3×C4C2×D3semidirect
132. C22×C3×C4C22×C3semidirect
136. C2×#67C22×C3semidirect
141. C2×C4×D3C22×C3semidirect
142. C2×Dic6C22×C3semidirect
149. C3×C4⋊C4C2×D3semidirect
150. C3×C4⋊C4C2×D3semidirect
151. C3×C4⋊C4C22×C3semidirect
152. C3×C4⋊C4C22×C3semidirect
169. C3×C22⋊C4C2×D3semidirect
172. C3×C22⋊C4C22×C3semidirect
177. #217C22×C3semidirect
184. (C2×D3)⋊C4C22×C3semidirect
185. (C2×D3)⋊C4C22×C3semidirect
186. C4⋊Dic3C22×C3semidirect
187. C4⋊Dic3C22×C3semidirect
188. Dic3⋊C4C22×C3semidirect
189. Dic3⋊C4C22×C3semidirect
190. C4×Dic3C22×C3semidirect
191. C4×Dic3C22×C3semidirect
224. #123C3×D3semidirect
323. C22×C4C2×C3×D3semidirect
328. C4⋊C4C2×C3×D3semidirect
329. C4⋊C4C2×C3×D3semidirect
343. C22⋊C4C2×C3×D3semidirect
412. C32#538semidirect
492. C3#5657direct (N × H)
493. C3C3×#538semidirect

Maximal subgroups

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

subgroupindex
C32×#1232
C3×#9382
C3×#9372
C3×#9302
C3×C4×#672
C3×#8982
C3×#8972
C3×#8942
C3×#8932
C3×#8852
C3×#8842
C3×#8822
C3×D3×C22⋊C42
C3×#8792
C3×#5383
#56573