Finite Group Databasebeta

← C3×#10595982 #13179262

Subgroups — C3×#10595982 #13179262

Order 1728.

1736 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×C33×C4⋊C4C211yesmaximal, characteristic, complemented
2C3×C4×#638C211yesmaximal, characteristic
3C3×C4×#638C212yesmaximal
4C3×#64133C212yesmaximal, complemented
5C3×#64133C211yesmaximal, characteristic, complemented
6C3×#64106C211yesmaximal, characteristic, complemented
7C3×#64105C212yesmaximal, complemented
8C3×C4⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
9C3×#64098C212yesmaximal, complemented
10C3×#64097C211yesmaximal, characteristic, complemented
11C3×#64096C211yesmaximal, characteristic, complemented
12#10595982C311yesmaximal, characteristic, direct product factor
13C3×#5511—34nomaximal
14C22×C33×C4C2211yescharacteristic
15C22×C33×C4C2212yes—
16C2×C3×#638C2211yescharacteristic
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C3×C4×C3⋊D3C2212yes—
19C2×C3×#634C2211yescharacteristic
20C2×C3×#634C2212yes—
21C33×C4⋊C4C2211yescharacteristic
22C33×C4⋊C4C2211yescharacteristic
23C33×C4⋊C4C2212yes—
24C3×#3757C2212yes—
25C3×#3757C2211yescharacteristic
26C3×#3752C2211yescharacteristic
27C3×#3752C2212yes—
28C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
29C3×C4⋊(C3⋊Dic3)C2212yes—
30C3×#3750C2212yes—
31C3×#3750C2212yes—
32C3×#3750C2211yescharacteristic
33C3×#3750C2212yes—
34C3×#3750C2212yes—
35C3×C4×C3⋊Dic3C2211yescharacteristic
36C3×C4×C3⋊Dic3C2212yes—
37C2×C32×C4⋊C4—24no—
38C2×C32×C4⋊C4D314yescomplemented
39C2×C32×C4⋊C4C2×C311yescharacteristic, complemented
40C4×#638C2×C311yescharacteristic
41C4×#638C2×C312yes—
42#64133C2×C312yescomplemented
43#64133C2×C311yescharacteristic, complemented
44#64106C2×C311yescharacteristic, complemented
45#64105C2×C312yescomplemented
46C4⋊C4×C3⋊D3C2×C311yescharacteristic, complemented
47#64098C2×C312yescomplemented
48#64097C2×C311yescharacteristic, complemented
49#64096C2×C311yescharacteristic, complemented
50C3×C4×#67—34no—
51C3×C4×#67—38no—
52C3×#924—38no—
53C3×#924—34no—
54C3×#897—34no—
55C3×#896—38no—
56C3×D3×C4⋊C4—34no—
57C3×#889—38no—
58C3×#888—34no—
59C3×#887—34no—
60C23×C33C2311yescharacteristic
61C22×C3×C3⋊D3C2311yescharacteristic
62C2×C33×C4—22no—
63C2×C33×C4—42no—
64C2×C33×C4C2312yes—
65C2×C33×C4C2311yescharacteristic
66C2×C33×C4C2312yes—
67C2×C33×C4C2311yescharacteristic
68C3×#638Q814yescomplemented
69C2×C3×C3⋊Dic3C2312yes—
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C2×C3×C3⋊Dic3C2312yes—
73C2×C3×C3⋊Dic3C2311yescharacteristic
74C3×C4×C3⋊D3—22no—
75C3×C4×C3⋊D3—42no—
76C3×#634—22no—
77C3×#634—42no—
78C3×#554—91no—
79#5511—34no—
80C22×C32×C4—24no—
81C22×C32×C4—28no—
82C22×C32×C4C2×D314yes—
83C22×C32×C4C2×D318yes—
84C22×C32×C4C22×C311yescharacteristic
85C22×C32×C4C22×C312yes—
86C2×#638C22×C311yescharacteristic
87C2×C4×C3⋊D3C22×C311yescharacteristic
88C2×C4×C3⋊D3C22×C312yes—
89C2×#634C22×C311yescharacteristic
90C2×#634C22×C312yes—
91C2×C3×#67—34no—
92C2×C3×C4×D3—34no—
93C2×C3×C4×D3—38no—
94C2×C3×Dic6—34no—
95C2×C3×Dic6—38no—
96C32×C4⋊C4—24no—
97C32×C4⋊C4—24no—
98C32×C4⋊C4—28no—
99C32×C4⋊C4C2×D314yes—
100C32×C4⋊C4C2×D314yes—
101C32×C4⋊C4C2×D318yes—
102C32×C4⋊C4C22×C311yescharacteristic
103C32×C4⋊C4C22×C311yescharacteristic
104C32×C4⋊C4C22×C312yes—
105#3757C22×C312yes—
106#3757C22×C311yescharacteristic
107#3752C22×C311yescharacteristic
108#3752C22×C312yes—
109C4⋊(C3⋊Dic3)C22×C311yescharacteristic
110C4⋊(C3⋊Dic3)C22×C312yes—
111#3750C22×C312yes—
112#3750C22×C312yes—
113#3750C22×C311yescharacteristic
114#3750C22×C312yes—
115#3750C22×C312yes—
116C4×C3⋊Dic3C22×C311yescharacteristic
117C4×C3⋊Dic3C22×C312yes—
118C3×#217—38no—
119C3×#217—34no—
120C3×(C2×D3)⋊C4—34no—
121C3×(C2×D3)⋊C4—38no—
122C3×C4⋊Dic3—34no—
123C3×C4⋊Dic3—38no—
124C3×Dic3⋊C4—38no—
125C3×Dic3⋊C4—38no—
126C3×Dic3⋊C4—34no—
127C3×Dic3⋊C4—38no—
128C3×Dic3⋊C4—38no—
129C3×C4×Dic3—34no—
130C3×C4×Dic3—38no—
131C22×C33—21no—
132C22×C33—21no—
133C22×C33C2×Q812yes—
134C22×C33C2411yescharacteristic
135C2×C3×C3⋊D3—21no—
136C2×C3×C3⋊D3—21no—
137C2×C3×C3⋊D3C2×Q812yes—
138C33×C4—21no—
139C33×C4—22no—
140C33×C4—22no—
141C33×C4#4112yes—
142C3×C3⋊Dic3—22no—
143C3×C3⋊Dic3—21no—
144C3×C3⋊Dic3—21no—
145C3×C3⋊Dic3—22no—
146C3×C3⋊Dic3C2×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—24no—
157C2×C3×C4⋊C4—91no—
158C2×C3×C4⋊C4C3×D314yescomplemented
159C2×C3×C4⋊C4C3⋊D311yescharacteristic, complemented
160C4×#67—34no—
161C4×#67—38no—
162#924—38no—
163#924—34no—
164#897—34no—
165#896—38no—
166D3×C4⋊C4—34no—
167#889—38no—
168#888—34no—
169#887—34no—
170C23×C32—24no—
171C23×C32C22×D314yes—
172C23×C32C23×C311yescharacteristic
173C22×C3⋊D3C23×C311yescharacteristic
174C22×C3×D3—34no—
175C2×C32×C4—216no—
176C2×C32×C4—28no—
177C2×C32×C4—22no—
178C2×C32×C4—24no—
179C2×C32×C4—24no—
180C2×C32×C4—28no—
181C2×C32×C4—28no—
182C2×C32×C4—48no—
183C2×C32×C4—42no—
184C2×C32×C4—416no—
185C2×C32×C4C22×D318yes—
186C2×C32×C4C22×D314yes—
187C2×C32×C4C22×D318yes—
188C2×C32×C4C22×D314yes—
189C2×C32×C4C23×C312yes—
190C2×C32×C4C23×C311yescharacteristic
191C2×C32×C4C23×C312yes—
192C2×C32×C4C23×C311yescharacteristic
193#638C3×Q814yescomplemented
194C2×C3⋊Dic3C23×C312yes—
195C2×C3⋊Dic3C23×C311yescharacteristic
196C2×C3⋊Dic3C23×C311yescharacteristic
197C2×C3⋊Dic3C23×C312yes—
198C2×C3⋊Dic3C23×C311yescharacteristic
199C4×C3⋊D3—22no—
200C4×C3⋊D3—42no—
201#634—22no—
202#634—42no—
203C3×#67—316no—
204C2×C3×Dic3—34no—
205C2×C3×Dic3—38no—
206C2×C3×Dic3—34no—
207C2×C3×Dic3—34no—
208C2×C3×Dic3—38no—
209C3×C4×D3—68no—
210C3×C4×D3—128no—
211C3×Dic6—68no—
212C3×Dic6—128no—
213#554—91noSylow
214C2×C33—22no—
215C2×C33C22×Q811yescharacteristic
216C2×C33C2×#4111yescharacteristic
217C2×C33H(5,𝔽2)11yescharacteristic
218C3×C3⋊D3—22no—
219C2×C3×Q8—91no—
220C2×C3×Q8—92no—
221C2×C3×D4—91no—
222C22×C3×C4—24no—
223C22×C3×C4—28no—
224C22×C3×C4—91no—
225C22×C3×C4—92no—
226C22×C3×C4C2×C3×D314yes—
227C22×C3×C4C2×C3×D318yes—
228C22×C3×C4C2×C3⋊D311yescharacteristic
229C22×C3×C4C2×C3⋊D312yes—
230C2×#67—34no—
231C2×C4×D3—34no—
232C2×C4×D3—38no—
233C2×Dic6—34no—
234C2×Dic6—38no—
235C3×C4⋊C4—24no—
236C3×C4⋊C4—28no—
237C3×C4⋊C4—24no—
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×C3×D314yes—
246C3×C4⋊C4C2×C3×D314yes—
247C3×C4⋊C4C2×C3×D318yes—
248C3×C4⋊C4C2×C3⋊D311yescharacteristic
249C3×C4⋊C4C2×C3⋊D312yes—
250C3×C4⋊C4C2×C3⋊D311yescharacteristic
251C3×C22⋊C4—92no—
252C3×C22⋊C4—91no—
253C3×C22⋊C4—92no—
254C3×C22⋊C4—91no—
255C3×C42—92no—
256C3×C42—91no—
257#217—38no—
258#217—34no—
259(C2×D3)⋊C4—34no—
260(C2×D3)⋊C4—38no—
261C4⋊Dic3—34no—
262C4⋊Dic3—38no—
263Dic3⋊C4—38no—
264Dic3⋊C4—38no—
265Dic3⋊C4—34no—
266Dic3⋊C4—38no—
267Dic3⋊C4—38no—
268C4×Dic3—34no—
269C4×Dic3—38no—
270C22×C32—28no—
271C22×C32—28no—
272C22×C32—24no—
273C22×C32—24no—
274C22×C32—21no—
275C22×C32—21no—
276C22×C32—24no—
277C22×C32—28no—
278C22×C32D3×Q818yes—
279C22×C32C2×C3×Q812yes—
280C22×C32C23×D314yes—
281C22×C32C24×C311yescharacteristic
282C2×C3⋊D3—21no—
283C2×C3⋊D3—21no—
284C2×C3⋊D3C2×C3×Q812yes—
285C2×C3×D3—38no—
286C2×C3×D3—64no—
287C2×C3×D3—64no—
288C32×C4—28no—
289C32×C4—24no—
290C32×C4—21no—
291C32×C4—22no—
292C32×C4—22no—
293C32×C4—28no—
294C32×C4—28no—
295C32×C4—28no—
296C32×C4—48no—
297C32×C4—48no—
298C32×C4#23518yes—
299C32×C4C3×#4112yes—
300C3⋊Dic3—22no—
301C3⋊Dic3—21no—
302C3⋊Dic3—21no—
303C3⋊Dic3—22no—
304C3⋊Dic3C2×C3×Q812yes—
305C3×Dic3—38no—
306C3×Dic3—68no—
307C3×Dic3—64no—
308C3×Dic3—64no—
309C3×Dic3—68no—
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×C3⋊D311yescharacteristic, complemented
321C33#55411yescharacteristic, complemented, Sylow
322C23×C3—24no—
323C23×C3—91no—
324C23×C3C22×C3×D314yes—
325C23×C3C22×C3⋊D311yescharacteristic
326C22×D3—34no—
327C3×Q8—182no—
328C3×Q8—362no—
329C3×D4—94no—
330C2×C3×C4—216no—
331C2×C3×C4—28no—
332C2×C3×C4—22no—
333C2×C3×C4—24no—
334C2×C3×C4—24no—
335C2×C3×C4—28no—
336C2×C3×C4—28no—
337C2×C3×C4—48no—
338C2×C3×C4—42no—
339C2×C3×C4—416no—
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×C3×D318yes—
348C2×C3×C4C22×C3×D314yes—
349C2×C3×C4C22×C3×D318yes—
350C2×C3×C4C22×C3×D314yes—
351C2×C3×C4C22×C3⋊D312yes—
352C2×C3×C4C22×C3⋊D311yescharacteristic
353C2×C3×C4C22×C3⋊D312yes—
354C2×C3×C4C22×C3⋊D311yescharacteristic
355#67—316no—
356C2×Dic3—34no—
357C2×Dic3—38no—
358C2×Dic3—34no—
359C2×Dic3—34no—
360C2×Dic3—38no—
361C4×D3—68no—
362C4×D3—128no—
363Dic6—68no—
364Dic6—128no—
365C2×C32—216no—
366C2×C32—28no—
367C2×C32—22no—
368C2×C32—24no—
369C2×C32—24no—
370C2×C32—24no—
371C2×C32C2×#23514yes—
372C2×C32#100414yes—
373C2×C32C2×D3×Q814yes—
374C2×C32C22×C3×Q811yescharacteristic
375C2×C32C2×C3×#4111yescharacteristic
376C2×C32C3×H(5,𝔽2)11yescharacteristic
377C3⋊D3—22no—
378C3×D3—68no—
379C2×Q8—91no—
380C2×Q8—92no—
381C2×D4—91no—
382C22×C4—91no—
383C22×C4—92no—
384C22×C4C2×C3×C3⋊D311yescharacteristic
385C22×C4C2×C3×C3⋊D312yes—
386C4⋊C4—92no—
387C4⋊C4—91no—
388C4⋊C4—91no—
389C4⋊C4—92no—
390C4⋊C4—92no—
391C4⋊C4—92no—
392C4⋊C4—92no—
393C4⋊C4C2×C3×C3⋊D311yescharacteristic
394C4⋊C4C2×C3×C3⋊D312yes—
395C4⋊C4C2×C3×C3⋊D311yescharacteristic
396C22⋊C4—92no—
397C22⋊C4—91no—
398C22⋊C4—92no—
399C22⋊C4—91no—
400C42—92no—
401C42—91no—
402C22×C3—28no—
403C22×C3—28no—
404C22×C3—24no—
405C22×C3—24no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—24no—
409C22×C3—28no—
410C22×C3—92no—
411C22×C3—181no—
412C22×C3—181no—
413C22×C3C3×D3×Q818yes—
414C22×C3Q8×C3⋊D312yes—
415C22×C3C23×C3×D314yes—
416C22×C3C23×C3⋊D311yescharacteristic
417C2×D3—38no—
418C2×D3—64no—
419C2×D3—64no—
420C3×C4—28no—
421C3×C4—24no—
422C3×C4—28no—
423C3×C4—28no—
424C3×C4—21no—
425C3×C4—22no—
426C3×C4—22no—
427C3×C4—28no—
428C3×C4—48no—
429C3×C4—48no—
430C3×C4—92no—
431C3×C4—181no—
432C3×C4—182no—
433C3×C4—182no—
434C3×C4—181no—
435C3×C4C3×#23518yes—
436C3×C4#382812yes—
437Dic3—38no—
438Dic3—68no—
439Dic3—64no—
440Dic3—64no—
441Dic3—68no—
442C32—24no—
443C32#551114yescomplemented
444C32C3×#55411yescharacteristic, complemented
445C23—91no—
446C23C22×C3×C3⋊D311yescharacteristic
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×C3⋊D312yes—
460C2×C4C22×C3×C3⋊D311yescharacteristic
461C2×C4C22×C3×C3⋊D312yes—
462C2×C4C22×C3×C3⋊D311yescharacteristic
463C2×C3—216no—
464C2×C3—28no—
465C2×C3—22no—
466C2×C3—24no—
467C2×C3—24no—
468C2×C3—24no—
469C2×C3—182no—
470C2×C3C2×C3×#23514yes—
471C2×C3C3×#100414yes—
472C2×C3C2×C3×D3×Q814yes—
473C2×C3C2×#382811yescharacteristic
474C2×C3#6436111yescharacteristic
475C2×C3C2×Q8×C3⋊D311yescharacteristic
476D3—68no—
477C22—21no—
478C22—21no—
479C22—92no—
480C22—181no—
481C22—181no—
482C22C3×Q8×C3⋊D312yes—
483C22C23×C3×C3⋊D311yescharacteristic
484C4—21no—
485C4—22no—
486C4—22no—
487C4—92no—
488C4—181no—
489C4—182no—
490C4—182no—
491C4—181no—
492C4C3×#382812yes—
493C3—24no—
494C3C3×#551114yescomplemented
495C3#1059598211yescharacteristic, direct product factor
496C2—22no—
497C2—182no—
498C2C2×C3×#382811yescharacteristic
499C2C3×#6436111yescharacteristic
500C2C2×C3×Q8×C3⋊D311yescharacteristic
subgroupG/H
centerC22×C3C23×C3⋊D3
commutator, cosocleC22×C32C24×C3
FrattiniC22C23×C3×C3⋊D3
FittingC2×C33×C4⋊C4C2
socleC22×C33C24
prime radicalC2×C3×#638C22
hypercenterC2×C3×C4⋊C4C3⋊D3
hypocenterC32C3×#554
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C33×C4⋊C4C2semidirect
4. C3×#64133C2semidirect
5. C3×#64133C2semidirect
6. C3×#64106C2semidirect
7. C3×#64105C2semidirect
8. C3×C4⋊C4×C3⋊D3C2semidirect
9. C3×#64098C2semidirect
10. C3×#64097C2semidirect
11. C3×#64096C2semidirect
12. #10595982C3direct (N × H)
38. C2×C32×C4⋊C4D3semidirect
39. C2×C32×C4⋊C4C2×C3semidirect
42. #64133C2×C3semidirect
43. #64133C2×C3semidirect
44. #64106C2×C3semidirect
45. #64105C2×C3semidirect
46. C4⋊C4×C3⋊D3C2×C3semidirect
47. #64098C2×C3semidirect
48. #64097C2×C3semidirect
49. #64096C2×C3semidirect
68. C3×#638Q8semidirect
158. C2×C3×C4⋊C4C3×D3semidirect
159. C2×C3×C4⋊C4C3⋊D3semidirect
193. #638C3×Q8semidirect
320. C2×C4⋊C4C3×C3⋊D3semidirect
321. C33#554semidirect
443. C32#5511semidirect
444. C32C3×#554semidirect
494. C3C3×#5511semidirect
495. C3#10595982direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C33×C4⋊C42
C3×C4×#6382
C3×#641332
C3×#641062
C3×#641052
C3×C4⋊C4×C3⋊D32
C3×#640982
C3×#640972
C3×#640962
#105959823
C3×#55113