Finite Group Databasebeta

← C3×#5556 #10595653

Subgroups — C3×#5556 #10595653

Order 576.

543 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×C4×D4C211yesmaximal, characteristic, complemented
2C3×#937C211yesmaximal, characteristic
3C3×#935C211yesmaximal, characteristic, complemented
4C3×#929C211yesmaximal, characteristic, complemented
5C3×C4×#67C211yesmaximal, characteristic, complemented
6C2×C3×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
7C3×#924C211yesmaximal, characteristic, complemented
8C3×#898C211yesmaximal, characteristic, complemented
9C3×#896C211yesmaximal, characteristic, complemented
10C3×#885C211yesmaximal, characteristic, complemented
11C3×#883C211yesmaximal, characteristic, complemented
12C3×D3×C22⋊C4C211yesmaximal, characteristic, complemented
13C3×#877C211yesmaximal, characteristic, complemented
14C3×#876C211yesmaximal, characteristic, complemented
15C3×#872C211yesmaximal, characteristic, complemented
16C3×#551—31nomaximal
17#5556C311yesmaximal, characteristic, direct product factor
18C23×C3×D3C2211yescharacteristic
19C2×C32×D4C2211yescharacteristic
20C22×C32×C4C2211yescharacteristic
21C22×C32×C4C2211yescharacteristic, complemented
22C2×C3×#67C2211yescharacteristic
23C2×C3×#67C2211yescharacteristic
24C22×C3×Dic3C2211yescharacteristic
25C2×C3×C4×D3C2211yescharacteristic
26C2×C3×C4×D3C2211yescharacteristic
27C2×C3×Dic6C2211yescharacteristic, complemented
28C32×C4⋊C4C2211yescharacteristic, complemented
29C32×C22⋊C4C2211yescharacteristic
30C32×C22⋊C4C2211yescharacteristic, complemented
31C32×C42C2211yescharacteristic, complemented
32C3×#217C2211yescharacteristic
33C3×#217C2211yescharacteristic
34C3×#217C2211yescharacteristic, complemented
35C3×#217C2211yescharacteristic, complemented
36C3×(C2×D3)⋊C4C2211yescharacteristic
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×C4⋊Dic3C2211yescharacteristic, complemented
45C3×Dic3⋊C4C2211yescharacteristic, complemented
46C3×Dic3⋊C4C2211yescharacteristic, complemented
47C3×Dic3⋊C4C2211yescharacteristic, complemented
48C3×Dic3⋊C4C2211yescharacteristic, 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×#123—31no—
57C3×#123—31no—
58C3×#123—31no—
59C3×C22⋊Q8—31no—
60C3×C22⋊Q8—31no—
61C3×#120—31no—
62C3×C4×D4—21no—
63C3×C4×D4—31no—
64C3×C4×D4D311yescharacteristic, complemented
65C3×C4×D4C2×C311yescharacteristic, complemented
66C3×#117—31no—
67C3×#117—31no—
68C2×C3×C22⋊C4—31no—
69C2×C3×C22⋊C4—31no—
70#937C2×C311yescharacteristic
71#935C2×C311yescharacteristic, complemented
72#929C2×C311yescharacteristic, complemented
73C4×#67C2×C311yescharacteristic, complemented
74C2×(C2×D3)⋊C4C2×C311yescharacteristic, complemented
75#924C2×C311yescharacteristic, complemented
76#898C2×C311yescharacteristic, complemented
77#896C2×C311yescharacteristic, complemented
78#885C2×C311yescharacteristic, complemented
79#883C2×C311yescharacteristic, complemented
80D3×C22⋊C4C2×C311yescharacteristic, complemented
81#877C2×C311yescharacteristic, complemented
82#876C2×C311yescharacteristic, complemented
83#872C2×C311yescharacteristic, complemented
84C23×C32C2311yescharacteristic
85C23×C32C2311yescharacteristic
86C22×C3×D3—22no—
87C22×C3×D3—22no—
88C22×C3×D3—41no—
89C22×C3×D3C2311yescharacteristic
90C22×C3×D3C2311yescharacteristic
91C32×D4—22no—
92C2×C32×C4—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—
101C2×C3×Dic3—22no—
102C2×C3×Dic3C2311yescharacteristic
103C2×C3×Dic3C2311yescharacteristic
104C2×C3×Dic3C2311yescharacteristic
105C2×C3×Dic3C2311yescharacteristic
106C2×C3×Dic3C2311yescharacteristic
107C2×C3×Dic3C2311yescharacteristic
108C3×C4×D3—22no—
109C3×C4×D3—41no—
110C3×Dic6—41no—
111#551—31noSylow
112C24×C3—31no—
113C23×D3C22×C311yescharacteristic
114C2×C3×Q8—31no—
115C2×C3×D4—21no—
116C2×C3×D4—31no—
117C2×C3×D4—31no—
118C2×C3×D4C2×D311yescharacteristic
119C2×C3×D4C22×C311yescharacteristic
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—31no—
123C22×C3×C4—31no—
124C22×C3×C4—31no—
125C22×C3×C4C2×D311yescharacteristic
126C22×C3×C4C2×D311yescharacteristic, complemented
127C22×C3×C4C22×C311yescharacteristic
128C22×C3×C4C22×C311yescharacteristic, complemented
129C2×#67C22×C311yescharacteristic
130C2×#67C22×C311yescharacteristic
131C22×Dic3C22×C311yescharacteristic
132C2×C4×D3C22×C311yescharacteristic
133C2×C4×D3C22×C311yescharacteristic
134C2×Dic6C22×C311yescharacteristic, complemented
135C3×C4⋊C4—21no—
136C3×C4⋊C4—31no—
137C3×C4⋊C4—31no—
138C3×C4⋊C4—31no—
139C3×C4⋊C4—31no—
140C3×C4⋊C4—31no—
141C3×C4⋊C4—31no—
142C3×C4⋊C4—31no—
143C3×C4⋊C4C2×D311yescharacteristic, complemented
144C3×C4⋊C4C22×C311yescharacteristic, complemented
145C3×C22⋊C4—21no—
146C3×C22⋊C4—21no—
147C3×C22⋊C4—31no—
148C3×C22⋊C4—31no—
149C3×C22⋊C4—31no—
150C3×C22⋊C4—31no—
151C3×C22⋊C4—31no—
152C3×C22⋊C4—31no—
153C3×C22⋊C4—31no—
154C3×C22⋊C4—31no—
155C3×C22⋊C4—31no—
156C3×C22⋊C4—31no—
157C3×C22⋊C4—31no—
158C3×C22⋊C4—31no—
159C3×C22⋊C4C2×D311yescharacteristic
160C3×C22⋊C4C2×D311yescharacteristic, complemented
161C3×C22⋊C4C22×C311yescharacteristic
162C3×C22⋊C4C22×C311yescharacteristic, complemented
163C3×C42—21no—
164C3×C42—31no—
165C3×C42—31no—
166C3×C42C2×D311yescharacteristic, complemented
167C3×C42C22×C311yescharacteristic, complemented
168#217C22×C311yescharacteristic
169#217C22×C311yescharacteristic
170#217C22×C311yescharacteristic, complemented
171#217C22×C311yescharacteristic, complemented
172(C2×D3)⋊C4C22×C311yescharacteristic
173(C2×D3)⋊C4C22×C311yescharacteristic
174(C2×D3)⋊C4C22×C311yescharacteristic
175(C2×D3)⋊C4C22×C311yescharacteristic
176(C2×D3)⋊C4C22×C311yescharacteristic
177(C2×D3)⋊C4C22×C311yescharacteristic
178(C2×D3)⋊C4C22×C311yescharacteristic
179(C2×D3)⋊C4C22×C311yescharacteristic
180C4⋊Dic3C22×C311yescharacteristic, complemented
181Dic3⋊C4C22×C311yescharacteristic, complemented
182Dic3⋊C4C22×C311yescharacteristic, complemented
183Dic3⋊C4C22×C311yescharacteristic, complemented
184Dic3⋊C4C22×C311yescharacteristic, complemented
185Dic3⋊C4C22×C311yescharacteristic, complemented
186Dic3⋊C4C22×C311yescharacteristic, complemented
187C4×Dic3C22×C311yescharacteristic, complemented
188C4×Dic3C22×C311yescharacteristic, complemented
189C22×C32—21no—
190C22×C32—21no—
191C22×C32—21no—
192C22×C32—21no—
193C22×C32—21no—
194C22×C32#4112yes—
195C22×C32C2411yescharacteristic
196C2×C3×D3—21no—
197C2×C3×D3—21no—
198C2×C3×D3—21no—
199C2×C3×D3—21no—
200C2×C3×D3—21no—
201C2×C3×D3—44no—
202C2×C3×D3#4112yes—
203C32×C4—21no—
204C32×C4—21no—
205C32×C4—21no—
206C32×C4—21no—
207C32×C4—21no—
208C3×Dic3—21no—
209C3×Dic3—21no—
210C3×Dic3—21no—
211C3×Dic3—21no—
212C3×Dic3—21no—
213C3×Dic3—21no—
214#126—31no—
215#126—31no—
216#124—31no—
217#123—31no—
218#123—31no—
219#123—31no—
220C22⋊Q8—31no—
221C22⋊Q8—31no—
222#120—31no—
223C4×D4—31no—
224C4×D4C3×D311yescharacteristic, complemented
225#117—31no—
226#117—31no—
227C2×C22⋊C4—31no—
228C2×C22⋊C4—31no—
229C23×C3—21no—
230C23×C3—21no—
231C23×C3—31no—
232C23×C3—31no—
233C23×C3—62no—
234C23×C3—62no—
235C23×C3—121no—
236C23×C3C22×D311yescharacteristic
237C23×C3C22×D311yescharacteristic
238C23×C3C23×C311yescharacteristic
239C23×C3C23×C311yescharacteristic
240C22×D3—22no—
241C22×D3—22no—
242C22×D3—41no—
243C22×D3C23×C311yescharacteristic
244C22×D3C23×C311yescharacteristic
245C3×Q8—121no—
246C3×D4—24no—
247C3×D4—22no—
248C3×D4—22no—
249C3×D4—62no—
250C3×D4—121no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—31no—
257C2×C3×C4—31no—
258C2×C3×C4—31no—
259C2×C3×C4—31no—
260C2×C3×C4—31no—
261C2×C3×C4—31no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—62no—
269C2×C3×C4—62no—
270C2×C3×C4—121no—
271C2×C3×C4C22×D311yescharacteristic
272C2×C3×C4C22×D311yescharacteristic
273C2×C3×C4C22×D311yescharacteristic
274C2×C3×C4C22×D311yescharacteristic
275C2×C3×C4C22×D311yescharacteristic
276C2×C3×C4C23×C311yescharacteristic
277C2×C3×C4C23×C311yescharacteristic
278C2×C3×C4C23×C311yescharacteristic
279C2×C3×C4C23×C311yescharacteristic
280C2×C3×C4C23×C311yescharacteristic
281#67—22no—
282#67—41no—
283C2×Dic3—22no—
284C2×Dic3C23×C311yescharacteristic
285C2×Dic3C23×C311yescharacteristic
286C2×Dic3C23×C311yescharacteristic
287C2×Dic3C23×C311yescharacteristic
288C2×Dic3C23×C311yescharacteristic
289C2×Dic3C23×C311yescharacteristic
290C4×D3—22no—
291C4×D3—41no—
292Dic6—41no—
293C2×C32—22no—
294C2×C32—41no—
295C2×C32C2×#4111yescharacteristic
296C2×C32C2×#4111yescharacteristic
297C2×C32H(5,𝔽2)11yescharacteristic
298C3×D3—22no—
299C3×D3—41no—
300C24—31no—
301C2×Q8—31no—
302C2×D4—31no—
303C2×D4—31no—
304C2×D4C2×C3×D311yescharacteristic
305C22×C4—31no—
306C22×C4—31no—
307C22×C4—31no—
308C22×C4C2×C3×D311yescharacteristic
309C22×C4C2×C3×D311yescharacteristic, complemented
310C4⋊C4—31no—
311C4⋊C4—31no—
312C4⋊C4—31no—
313C4⋊C4—31no—
314C4⋊C4—31no—
315C4⋊C4—31no—
316C4⋊C4—31no—
317C4⋊C4C2×C3×D311yescharacteristic, complemented
318C22⋊C4—31no—
319C22⋊C4—31no—
320C22⋊C4—31no—
321C22⋊C4—31no—
322C22⋊C4—31no—
323C22⋊C4—31no—
324C22⋊C4—31no—
325C22⋊C4—31no—
326C22⋊C4—31no—
327C22⋊C4—31no—
328C22⋊C4—31no—
329C22⋊C4—31no—
330C22⋊C4C2×C3×D311yescharacteristic
331C22⋊C4C2×C3×D311yescharacteristic, complemented
332C42—31no—
333C42—31no—
334C42C2×C3×D311yescharacteristic, complemented
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—22no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—22no—
348C22×C3—32no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—41no—
352C22×C3—41no—
353C22×C3—61no—
354C22×C3—61no—
355C22×C3—61no—
356C22×C3—61no—
357C22×C3—61no—
358C22×C3—124no—
359C22×C3#23512yes—
360C22×C3C3×#4112yes—
361C22×C3C23×D311yescharacteristic
362C22×C3C24×C311yescharacteristic
363C2×D3—21no—
364C2×D3—21no—
365C2×D3—21no—
366C2×D3—21no—
367C2×D3—21no—
368C2×D3—44no—
369C2×D3C3×#4112yes—
370C3×C4—22no—
371C3×C4—22no—
372C3×C4—21no—
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—41no—
383C3×C4—41no—
384C3×C4—41no—
385C3×C4—61no—
386C3×C4—61no—
387C3×C4—61no—
388C3×C4—61no—
389C3×C4—61no—
390C3×C4—61no—
391Dic3—21no—
392Dic3—21no—
393Dic3—21no—
394Dic3—21no—
395Dic3—21no—
396Dic3—21no—
397C32#55111yescharacteristic, complemented, Sylow
398C23—31no—
399C23—31no—
400C23—62no—
401C23—62no—
402C23—121no—
403C23C22×C3×D311yescharacteristic
404C23C22×C3×D311yescharacteristic
405Q8—121no—
406D4—22no—
407D4—62no—
408D4—121no—
409C2×C4—31no—
410C2×C4—31no—
411C2×C4—31no—
412C2×C4—31no—
413C2×C4—31no—
414C2×C4—31no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—62no—
418C2×C4—62no—
419C2×C4—121no—
420C2×C4C22×C3×D311yescharacteristic
421C2×C4C22×C3×D311yescharacteristic
422C2×C4C22×C3×D311yescharacteristic
423C2×C4C22×C3×D311yescharacteristic
424C2×C4C22×C3×D311yescharacteristic
425C2×C3—22no—
426C2×C3—22no—
427C2×C3—21no—
428C2×C3—21no—
429C2×C3—21no—
430C2×C3—42no—
431C2×C3—42no—
432C2×C3—41no—
433C2×C3—41no—
434C2×C3—62no—
435C2×C3—121no—
436C2×C3C2×#23511yescharacteristic
437C2×C3#100411yescharacteristic
438C2×C3D3×#4111yescharacteristic
439C2×C3C2×C3×#4111yescharacteristic
440C2×C3C2×C3×#4111yescharacteristic
441C2×C3C3×H(5,𝔽2)11yescharacteristic
442D3—22no—
443D3—41no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—32no—
450C22—61no—
451C22—61no—
452C22—61no—
453C22—61no—
454C22—61no—
455C22—124no—
456C22C3×#23512yes—
457C22C23×C3×D311yescharacteristic
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—61no—
464C4—61no—
465C4—61no—
466C4—61no—
467C4—61no—
468C4—61no—
469C3—21no—
470C3#555611yescharacteristic, direct product factor
471C3C3×#55111yescharacteristic, complemented
472C2—22no—
473C2—41no—
474C2—62no—
475C2—121no—
476C2C2×C3×#23511yescharacteristic
477C2C3×#100411yescharacteristic
478C2C3×D3×#4111yescharacteristic
subgroupG/H
centerC22×C3C23×D3
commutator, cosocleC22×C3C24×C3
FrattiniC22C23×C3×D3
FittingC32×C4×D4C2
socleC22×C32C24
prime radicalC3×#937C2
hypercenterC3×C4×D4D3
hypocenterC3C3×#551
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
3. C3×#935C2semidirect
4. C3×#929C2semidirect
5. C3×C4×#67C2semidirect
6. C2×C3×(C2×D3)⋊C4C2semidirect
7. C3×#924C2semidirect
8. C3×#898C2semidirect
9. C3×#896C2semidirect
10. C3×#885C2semidirect
11. C3×#883C2semidirect
12. C3×D3×C22⋊C4C2semidirect
13. C3×#877C2semidirect
14. C3×#876C2semidirect
15. C3×#872C2semidirect
17. #5556C3direct (N × H)
21. C22×C32×C4C22semidirect
27. C2×C3×Dic6C22semidirect
28. C32×C4⋊C4C22semidirect
30. C32×C22⋊C4C22semidirect
31. C32×C42C22semidirect
34. C3×#217C22semidirect
35. C3×#217C22semidirect
44. C3×C4⋊Dic3C22semidirect
45. C3×Dic3⋊C4C22semidirect
46. C3×Dic3⋊C4C22semidirect
47. C3×Dic3⋊C4C22semidirect
48. C3×Dic3⋊C4C22semidirect
49. C3×Dic3⋊C4C22semidirect
50. C3×Dic3⋊C4C22semidirect
51. C3×C4×Dic3C22semidirect
52. C3×C4×Dic3C22semidirect
64. C3×C4×D4D3semidirect
65. C3×C4×D4C2×C3semidirect
71. #935C2×C3semidirect
72. #929C2×C3semidirect
73. C4×#67C2×C3semidirect
74. C2×(C2×D3)⋊C4C2×C3semidirect
75. #924C2×C3semidirect
76. #898C2×C3semidirect
77. #896C2×C3semidirect
78. #885C2×C3semidirect
79. #883C2×C3semidirect
80. D3×C22⋊C4C2×C3semidirect
81. #877C2×C3semidirect
82. #876C2×C3semidirect
83. #872C2×C3semidirect
126. C22×C3×C4C2×D3semidirect
128. C22×C3×C4C22×C3semidirect
134. C2×Dic6C22×C3semidirect
143. C3×C4⋊C4C2×D3semidirect
144. C3×C4⋊C4C22×C3semidirect
160. C3×C22⋊C4C2×D3semidirect
162. C3×C22⋊C4C22×C3semidirect
166. C3×C42C2×D3semidirect
167. C3×C42C22×C3semidirect
170. #217C22×C3semidirect
171. #217C22×C3semidirect
180. C4⋊Dic3C22×C3semidirect
181. Dic3⋊C4C22×C3semidirect
182. Dic3⋊C4C22×C3semidirect
183. Dic3⋊C4C22×C3semidirect
184. Dic3⋊C4C22×C3semidirect
185. Dic3⋊C4C22×C3semidirect
186. Dic3⋊C4C22×C3semidirect
187. C4×Dic3C22×C3semidirect
188. C4×Dic3C22×C3semidirect
224. C4×D4C3×D3semidirect
309. C22×C4C2×C3×D3semidirect
317. C4⋊C4C2×C3×D3semidirect
331. C22⋊C4C2×C3×D3semidirect
334. C42C2×C3×D3semidirect
397. C32#551semidirect
470. C3#5556direct (N × H)
471. C3C3×#551semidirect

Maximal subgroups

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

subgroupindex
C32×C4×D42
C3×#9372
C3×#9352
C3×#9292
C3×C4×#672
C2×C3×(C2×D3)⋊C42
C3×#9242
C3×#8982
C3×#8962
C3×#8852
C3×#8832
C3×D3×C22⋊C42
C3×#8772
C3×#8762
C3×#8722
C3×#5513
#55563