Finite Group Databasebeta

← C3×#5560 #10595657

Subgroups — C3×#5560 #10595657

Order 576.

547 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×C4×D4C211yesmaximal, characteristic, complemented
2C3×#939C211yesmaximal, characteristic, complemented
3C3×#937C211yesmaximal, characteristic, complemented
4C3×#930C211yesmaximal, characteristic, complemented
5C3×#929C211yesmaximal, characteristic, complemented
6C3×C4×#67C211yesmaximal, characteristic, complemented
7C2×C3×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
8C3×#898C211yesmaximal, characteristic, complemented
9C3×#894C211yesmaximal, characteristic, complemented
10C3×#885C211yesmaximal, characteristic, complemented
11C3×#884C211yesmaximal, characteristic, complemented
12C3×#882C211yesmaximal, characteristic, complemented
13C3×#878C211yesmaximal, characteristic, complemented
14C3×#876C211yesmaximal, characteristic, complemented
15C3×#875C211yesmaximal, characteristic, complemented
16C3×#538—31nomaximal
17#5560C311yesmaximal, characteristic, direct product factor
18C23×C3×D3C2211yescharacteristic
19C2×C32×D4C2211yescharacteristic
20C22×C32×C4C2211yescharacteristic
21C22×C32×C4C2211yescharacteristic, complemented
22C2×C3×#67C2211yescharacteristic
23C2×C3×#67C2211yescharacteristic
24C2×C3×#67C2211yescharacteristic
25C2×C3×#67C2211yescharacteristic, complemented
26C22×C3×Dic3C2211yescharacteristic
27C2×C3×D12C2211yescharacteristic
28C2×C3×D12C2211yescharacteristic
29C2×C3×C4×D3C2211yescharacteristic, complemented
30C2×C3×Dic6C2211yescharacteristic, complemented
31C32×C4⋊C4C2211yescharacteristic, complemented
32C32×C22⋊C4C2211yescharacteristic
33C32×C22⋊C4C2211yescharacteristic, complemented
34C32×C42C2211yescharacteristic, complemented
35C3×#217C2211yescharacteristic
36C3×#217C2211yescharacteristic, complemented
37C3×(C2×D3)⋊C4C2211yescharacteristic
38C3×(C2×D3)⋊C4C2211yescharacteristic
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×Dic3⋊C4C2211yescharacteristic, complemented
49C3×Dic3⋊C4C2211yescharacteristic, complemented
50C3×Dic3⋊C4C2211yescharacteristic, complemented
51C3×Dic3⋊C4C2211yescharacteristic, complemented
52C3×C4×Dic3C2211yescharacteristic, complemented
53C3×#126—31no—
54C3×#126—31no—
55C3×#124—31no—
56C3×#124—31no—
57C3×#123—31no—
58C3×#123—31no—
59C3×C22⋊Q8—31no—
60C3×#121—31no—
61C3×#121—31no—
62C3×#121—31no—
63C3×#120—31no—
64C3×#120—31no—
65C3×C4×D4—21no—
66C3×C4×D4—31no—
67C3×C4×D4D311yescharacteristic, complemented
68C3×C4×D4C2×C311yescharacteristic, complemented
69C2×C3×C22⋊C4—31no—
70#939C2×C311yescharacteristic, complemented
71#937C2×C311yescharacteristic, complemented
72#930C2×C311yescharacteristic, complemented
73#929C2×C311yescharacteristic, complemented
74C4×#67C2×C311yescharacteristic, complemented
75C2×(C2×D3)⋊C4C2×C311yescharacteristic, complemented
76#898C2×C311yescharacteristic, complemented
77#894C2×C311yescharacteristic, complemented
78#885C2×C311yescharacteristic, complemented
79#884C2×C311yescharacteristic, complemented
80#882C2×C311yescharacteristic, complemented
81#878C2×C311yescharacteristic, complemented
82#876C2×C311yescharacteristic, complemented
83#875C2×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—22no—
93C2×C32×C4—41no—
94C2×C32×C4—41no—
95C2×C32×C4C2311yescharacteristic
96C2×C32×C4C2311yescharacteristic
97C2×C32×C4C2311yescharacteristic
98C2×C32×C4C2311yescharacteristic
99C2×C32×C4C2311yescharacteristic
100C3×#67—22no—
101C3×#67—41no—
102C3×#67—41no—
103C3×#67—41no—
104C2×C3×Dic3—41no—
105C2×C3×Dic3C2311yescharacteristic
106C2×C3×Dic3C2311yescharacteristic
107C2×C3×Dic3C2311yescharacteristic
108C2×C3×Dic3C2311yescharacteristic
109C2×C3×Dic3C2311yescharacteristic
110C3×D12—41no—
111C3×D12—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—21no—
129C22×C3×C4—31no—
130C22×C3×C4—31no—
131C22×C3×C4C2×D311yescharacteristic
132C22×C3×C4C2×D311yescharacteristic, complemented
133C22×C3×C4C22×C311yescharacteristic
134C22×C3×C4C22×C311yescharacteristic, complemented
135C2×#67C22×C311yescharacteristic
136C2×#67C22×C311yescharacteristic
137C2×#67C22×C311yescharacteristic
138C2×#67C22×C311yescharacteristic, complemented
139C22×Dic3C22×C311yescharacteristic
140C2×D12C22×C311yescharacteristic
141C2×D12C22×C311yescharacteristic
142C2×C4×D3C22×C311yescharacteristic, complemented
143C2×Dic6C22×C311yescharacteristic, complemented
144C3×C4⋊C4—21no—
145C3×C4⋊C4—31no—
146C3×C4⋊C4—31no—
147C3×C4⋊C4—31no—
148C3×C4⋊C4—31no—
149C3×C4⋊C4—31no—
150C3×C4⋊C4C2×D311yescharacteristic, complemented
151C3×C4⋊C4C22×C311yescharacteristic, complemented
152C3×C22⋊C4—21no—
153C3×C22⋊C4—21no—
154C3×C22⋊C4—31no—
155C3×C22⋊C4—31no—
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⋊C4C2×D311yescharacteristic
167C3×C22⋊C4C2×D311yescharacteristic, complemented
168C3×C22⋊C4C22×C311yescharacteristic
169C3×C22⋊C4C22×C311yescharacteristic, complemented
170C3×C42—21no—
171C3×C42—31no—
172C3×C42C2×D311yescharacteristic, complemented
173C3×C42C22×C311yescharacteristic, complemented
174#217C22×C311yescharacteristic
175#217C22×C311yescharacteristic, complemented
176(C2×D3)⋊C4C22×C311yescharacteristic
177(C2×D3)⋊C4C22×C311yescharacteristic
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
187Dic3⋊C4C22×C311yescharacteristic, complemented
188Dic3⋊C4C22×C311yescharacteristic, complemented
189Dic3⋊C4C22×C311yescharacteristic, complemented
190Dic3⋊C4C22×C311yescharacteristic, complemented
191C4×Dic3C22×C311yescharacteristic, complemented
192C22×C32—21no—
193C22×C32—21no—
194C22×C32—21no—
195C22×C32—21no—
196C22×C32—21no—
197C22×C32#4112yes—
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—21no—
208C2×C3×D3—44no—
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#123—31no—
225C22⋊Q8—31no—
226#121—31no—
227#121—31no—
228#121—31no—
229#120—31no—
230#120—31no—
231C4×D4—31no—
232C4×D4C3×D311yescharacteristic, complemented
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—24no—
254C3×D4—22no—
255C3×D4—22no—
256C3×D4—62no—
257C3×D4—121no—
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—41no—
275C2×C3×C4—41no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—121no—
279C2×C3×C4—121no—
280C2×C3×C4C22×D311yescharacteristic
281C2×C3×C4C22×D311yescharacteristic
282C2×C3×C4C22×D311yescharacteristic
283C2×C3×C4C22×D311yescharacteristic
284C2×C3×C4C22×D311yescharacteristic
285C2×C3×C4C23×C311yescharacteristic
286C2×C3×C4C23×C311yescharacteristic
287C2×C3×C4C23×C311yescharacteristic
288C2×C3×C4C23×C311yescharacteristic
289C2×C3×C4C23×C311yescharacteristic
290#67—22no—
291#67—41no—
292#67—41no—
293#67—41no—
294C2×Dic3—41no—
295C2×Dic3C23×C311yescharacteristic
296C2×Dic3C23×C311yescharacteristic
297C2×Dic3C23×C311yescharacteristic
298C2×Dic3C23×C311yescharacteristic
299C2×Dic3C23×C311yescharacteristic
300D12—41no—
301D12—41no—
302C4×D3—41no—
303Dic6—41no—
304C2×C32—22no—
305C2×C32—41no—
306C2×C32C2×#4111yescharacteristic
307C2×C32H(5,𝔽2)11yescharacteristic
308C2×C32H(5,𝔽2)11yescharacteristic
309C3×D3—41no—
310C3×D3—41no—
311C3×D3—41no—
312C24—31no—
313C2×Q8—31no—
314C2×D4—31no—
315C2×D4—31no—
316C2×D4—31no—
317C2×D4—31no—
318C2×D4—31no—
319C2×D4—31no—
320C2×D4C2×C3×D311yescharacteristic
321C22×C4—31no—
322C22×C4—31no—
323C22×C4C2×C3×D311yescharacteristic
324C22×C4C2×C3×D311yescharacteristic, complemented
325C4⋊C4—31no—
326C4⋊C4—31no—
327C4⋊C4—31no—
328C4⋊C4—31no—
329C4⋊C4—31no—
330C4⋊C4C2×C3×D311yescharacteristic, complemented
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⋊C4—31no—
342C22⋊C4—31no—
343C22⋊C4C2×C3×D311yescharacteristic
344C22⋊C4C2×C3×D311yescharacteristic, complemented
345C42—31no—
346C42C2×C3×D311yescharacteristic, complemented
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—22no—
352C22×C3—21no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—22no—
360C22×C3—41no—
361C22×C3—41no—
362C22×C3—41no—
363C22×C3—41no—
364C22×C3—61no—
365C22×C3—61no—
366C22×C3—61no—
367C22×C3—61no—
368C22×C3—61no—
369C22×C3—61no—
370C22×C3—61no—
371C22×C3—61no—
372C22×C3—61no—
373C22×C3—124no—
374C22×C3#23512yes—
375C22×C3C3×#4112yes—
376C22×C3C23×D311yescharacteristic
377C22×C3C24×C311yescharacteristic
378C2×D3—21no—
379C2×D3—21no—
380C2×D3—21no—
381C2×D3—21no—
382C2×D3—21no—
383C2×D3—21no—
384C2×D3—21no—
385C2×D3—21no—
386C2×D3—21no—
387C2×D3—44no—
388C3×C4—22no—
389C3×C4—22no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—21no—
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—41no—
401C3×C4—41no—
402C3×C4—41no—
403C3×C4—61no—
404C3×C4—61no—
405C3×C4—61no—
406C3×C4—61no—
407C3×C4—61no—
408Dic3—21no—
409Dic3—21no—
410Dic3—21no—
411Dic3—21no—
412Dic3—21no—
413C32#53811yescharacteristic, complemented, Sylow
414C23—31no—
415C23—31no—
416C23—31no—
417C23—62no—
418C23—121no—
419C23—121no—
420C23C22×C3×D311yescharacteristic
421C23C22×C3×D311yescharacteristic
422Q8—121no—
423D4—22no—
424D4—62no—
425D4—121no—
426D4—121no—
427D4—121no—
428D4—121no—
429D4—121no—
430C2×C4—31no—
431C2×C4—31no—
432C2×C4—31no—
433C2×C4—31no—
434C2×C4—31no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—121no—
438C2×C4—121no—
439C2×C4C22×C3×D311yescharacteristic
440C2×C4C22×C3×D311yescharacteristic
441C2×C4C22×C3×D311yescharacteristic
442C2×C4C22×C3×D311yescharacteristic
443C2×C4C22×C3×D311yescharacteristic
444C2×C3—22no—
445C2×C3—22no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—42no—
450C2×C3—42no—
451C2×C3—41no—
452C2×C3—41no—
453C2×C3—121no—
454C2×C3—121no—
455C2×C3—121no—
456C2×C3C2×#23511yescharacteristic
457C2×C3#100411yescharacteristic
458C2×C3#100911yescharacteristic
459C2×C3C2×C3×#4111yescharacteristic
460C2×C3C3×H(5,𝔽2)11yescharacteristic
461C2×C3C3×H(5,𝔽2)11yescharacteristic
462D3—41no—
463D3—41no—
464D3—41no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—61no—
471C22—61no—
472C22—61no—
473C22—61no—
474C22—61no—
475C22—61no—
476C22—61no—
477C22—61no—
478C22—61no—
479C22—124no—
480C22C3×#23512yes—
481C22C23×C3×D311yescharacteristic
482C4—21no—
483C4—21no—
484C4—21no—
485C4—21no—
486C4—21no—
487C4—61no—
488C4—61no—
489C4—61no—
490C4—61no—
491C4—61no—
492C3—21no—
493C3#556011yescharacteristic, direct product factor
494C3C3×#53811yescharacteristic, complemented
495C2—22no—
496C2—41no—
497C2—121no—
498C2—121no—
499C2—121no—
500C2C2×C3×#23511yescharacteristic
501C2C3×#100411yescharacteristic
502C2C3×#100911yescharacteristic
subgroupG/H
centerC22×C3C23×D3
commutator, cosocleC22×C3C24×C3
FrattiniC22C23×C3×D3
FittingC32×C4×D4C2
socleC22×C32C24
hypercenterC3×C4×D4D3
hypocenterC3C3×#538
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C4×D4

Complements

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

NH ≅ G/Ntype
1. C32×C4×D4C2semidirect
2. C3×#939C2semidirect
3. C3×#937C2semidirect
4. C3×#930C2semidirect
5. C3×#929C2semidirect
6. C3×C4×#67C2semidirect
7. C2×C3×(C2×D3)⋊C4C2semidirect
8. C3×#898C2semidirect
9. C3×#894C2semidirect
10. C3×#885C2semidirect
11. C3×#884C2semidirect
12. C3×#882C2semidirect
13. C3×#878C2semidirect
14. C3×#876C2semidirect
15. C3×#875C2semidirect
17. #5560C3direct (N × H)
21. C22×C32×C4C22semidirect
25. C2×C3×#67C22semidirect
29. C2×C3×C4×D3C22semidirect
30. C2×C3×Dic6C22semidirect
31. C32×C4⋊C4C22semidirect
33. C32×C22⋊C4C22semidirect
34. C32×C42C22semidirect
36. C3×#217C22semidirect
45. C3×(C2×D3)⋊C4C22semidirect
46. C3×(C2×D3)⋊C4C22semidirect
47. C3×C4⋊Dic3C22semidirect
48. C3×Dic3⋊C4C22semidirect
49. C3×Dic3⋊C4C22semidirect
50. C3×Dic3⋊C4C22semidirect
51. C3×Dic3⋊C4C22semidirect
52. C3×C4×Dic3C22semidirect
67. C3×C4×D4D3semidirect
68. C3×C4×D4C2×C3semidirect
70. #939C2×C3semidirect
71. #937C2×C3semidirect
72. #930C2×C3semidirect
73. #929C2×C3semidirect
74. C4×#67C2×C3semidirect
75. C2×(C2×D3)⋊C4C2×C3semidirect
76. #898C2×C3semidirect
77. #894C2×C3semidirect
78. #885C2×C3semidirect
79. #884C2×C3semidirect
80. #882C2×C3semidirect
81. #878C2×C3semidirect
82. #876C2×C3semidirect
83. #875C2×C3semidirect
132. C22×C3×C4C2×D3semidirect
134. C22×C3×C4C22×C3semidirect
138. C2×#67C22×C3semidirect
142. C2×C4×D3C22×C3semidirect
143. C2×Dic6C22×C3semidirect
150. C3×C4⋊C4C2×D3semidirect
151. C3×C4⋊C4C22×C3semidirect
167. C3×C22⋊C4C2×D3semidirect
169. C3×C22⋊C4C22×C3semidirect
172. C3×C42C2×D3semidirect
173. C3×C42C22×C3semidirect
175. #217C22×C3semidirect
184. (C2×D3)⋊C4C22×C3semidirect
185. (C2×D3)⋊C4C22×C3semidirect
186. C4⋊Dic3C22×C3semidirect
187. Dic3⋊C4C22×C3semidirect
188. Dic3⋊C4C22×C3semidirect
189. Dic3⋊C4C22×C3semidirect
190. Dic3⋊C4C22×C3semidirect
191. C4×Dic3C22×C3semidirect
232. C4×D4C3×D3semidirect
324. C22×C4C2×C3×D3semidirect
330. C4⋊C4C2×C3×D3semidirect
344. C22⋊C4C2×C3×D3semidirect
346. C42C2×C3×D3semidirect
413. C32#538semidirect
493. C3#5560direct (N × H)
494. C3C3×#538semidirect

Maximal subgroups

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

subgroupindex
C32×C4×D42
C3×#9392
C3×#9372
C3×#9302
C3×#9292
C3×C4×#672
C2×C3×(C2×D3)⋊C42
C3×#8982
C3×#8942
C3×#8852
C3×#8842
C3×#8822
C3×#8782
C3×#8762
C3×#8752
C3×#5383
#55603