Finite Group Databasebeta

← C3×#10596204 #13179484

Subgroups — C3×#10596204 #13179484

Order 1728.

1568 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C4⋊Q8C211yesmaximal, characteristic, complemented
2C3×#64156C212yesmaximal, complemented
3C3×#64155C212yesmaximal, complemented
4C3×#64107C212yesmaximal, complemented
5C3×#64106C211yesmaximal, characteristic, complemented
6C3×#64104C211yesmaximal, characteristic
7C3×#64103C212yesmaximal, complemented
8C3×#64102C211yesmaximal, characteristic, complemented
9C3×#64096C211yesmaximal, characteristic, complemented
10C3×#64084C211yesmaximal, characteristic, complemented
11C3×#64081C211yesmaximal, characteristic, complemented
12#10596204C311yesmaximal, characteristic, direct product factor
13C3×#5733—34nomaximal
14C2×C33×Q8C2212yes—
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4⋊(C3⋊D3)C2212yes—
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C3×C4×C3⋊D3C2212yes—
19C2×C3×#634C2211yescharacteristic
20C33×C4⋊C4C2211yescharacteristic
21C33×C4⋊C4C2211yescharacteristic
22C33×C4⋊C4C2212yes—
23C33×C42C2211yescharacteristic
24C3×#3752C2211yescharacteristic
25C3×#3752C2211yescharacteristic
26C3×#3752C2212yes—
27C3×#3752C2212yes—
28C3×#3752C2212yes—
29C3×#3752C2212yes—
30C3×#3752C2212yes—
31C3×C4⋊(C3⋊Dic3)C2212yes—
32C3×#3750C2211yescharacteristic
33C3×#3750C2211yescharacteristic
34C3×#3750C2212yes—
35C3×C4×C3⋊Dic3C2211yescharacteristic
36C3×C4×C3⋊Dic3C2212yes—
37C32×C4⋊Q8—24no—
38C32×C4⋊Q8D314yescomplemented
39C32×C4⋊Q8C2×C311yescharacteristic, complemented
40#64156C2×C312yescomplemented
41#64155C2×C312yescomplemented
42#64107C2×C312yescomplemented
43#64106C2×C311yescharacteristic, complemented
44#64104C2×C311yescharacteristic
45#64103C2×C312yescomplemented
46#64102C2×C311yescharacteristic, complemented
47#64096C2×C311yescharacteristic, complemented
48#64084C2×C311yescharacteristic, complemented
49#64081C2×C311yescharacteristic, complemented
50C3×#947—38no—
51C3×#946—38no—
52C3×#898—38no—
53C3×#897—34no—
54C3×#895—34no—
55C3×#894—38no—
56C3×#893—34no—
57C3×#887—34no—
58C3×#875—34no—
59C3×#872—34no—
60C22×C3×C3⋊D3C2311yescharacteristic
61C22×C3×C3⋊D3C2312yes—
62C33×Q8—42no—
63C2×C33×C4C2312yes—
64C2×C33×C4C2311yescharacteristic
65C2×C33×C4C2311yescharacteristic
66C2×C33×C4C2312yes—
67C2×C33×C4C2311yescharacteristic
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C2×C3×C3⋊Dic3C2312yes—
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C3×C4⋊(C3⋊D3)—22no—
73C3×C4⋊(C3⋊D3)—42no—
74C3×C4×C3⋊D3—22no—
75C3×C4×C3⋊D3—42no—
76C3×#634—22no—
77C3×#542—91no—
78#5733—34no—
79C2×C32×Q8—28no—
80C2×C32×Q8C2×D318yes—
81C2×C32×Q8C22×C312yes—
82C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
83C2×C4⋊(C3⋊D3)C22×C312yes—
84C2×C4×C3⋊D3C22×C311yescharacteristic
85C2×C4×C3⋊D3C22×C312yes—
86C2×#634C22×C311yescharacteristic
87C2×C3×D12—34no—
88C2×C3×D12—38no—
89C2×C3×C4×D3—34no—
90C2×C3×C4×D3—38no—
91C2×C3×Dic6—34no—
92C32×C4⋊C4—24no—
93C32×C4⋊C4—24no—
94C32×C4⋊C4—28no—
95C32×C4⋊C4C2×D314yes—
96C32×C4⋊C4C2×D314yes—
97C32×C4⋊C4C2×D318yes—
98C32×C4⋊C4C22×C311yescharacteristic
99C32×C4⋊C4C22×C311yescharacteristic
100C32×C4⋊C4C22×C312yes—
101C32×C42—24no—
102C32×C42C2×D314yes—
103C32×C42C22×C311yescharacteristic
104#3752C22×C311yescharacteristic
105#3752C22×C311yescharacteristic
106#3752C22×C312yes—
107#3752C22×C312yes—
108#3752C22×C312yes—
109#3752C22×C312yes—
110#3752C22×C312yes—
111C4⋊(C3⋊Dic3)C22×C312yes—
112#3750C22×C311yescharacteristic
113#3750C22×C311yescharacteristic
114#3750C22×C312yes—
115C4×C3⋊Dic3C22×C311yescharacteristic
116C4×C3⋊Dic3C22×C312yes—
117C3×(C2×D3)⋊C4—38no—
118C3×(C2×D3)⋊C4—34no—
119C3×(C2×D3)⋊C4—38no—
120C3×(C2×D3)⋊C4—38no—
121C3×(C2×D3)⋊C4—38no—
122C3×(C2×D3)⋊C4—38no—
123C3×(C2×D3)⋊C4—34no—
124C3×C4⋊Dic3—38no—
125C3×Dic3⋊C4—38no—
126C3×Dic3⋊C4—34no—
127C3×Dic3⋊C4—34no—
128C3×C4×Dic3—34no—
129C3×C4×Dic3—38no—
130C22×C33C2411yescharacteristic
131C2×C3×C3⋊D3—21no—
132C2×C3×C3⋊D3—22no—
133C2×C3×C3⋊D3—22no—
134C2×C3×C3⋊D3—21no—
135C2×C3×C3⋊D3—21no—
136C2×C3×C3⋊D3—22no—
137C33×C4—21no—
138C33×C4—22no—
139C33×C4—22no—
140C33×C4—21no—
141C33×C4#4112yes—
142C3×C3⋊Dic3—22no—
143C3×C3⋊Dic3—21no—
144C3×C3⋊Dic3—21no—
145C3×C3⋊Dic3—21no—
146C3×C4⋊Q8—24no—
147C3×C4⋊Q8C3×D314yescomplemented
148C3×C4⋊Q8C3⋊D311yescharacteristic, complemented
149C3×#126—92no—
150C3×#124—92no—
151C3×#124—91no—
152C3×#123—92no—
153C3×C22⋊Q8—92no—
154C3×C22⋊Q8—91no—
155C3×#121—91no—
156C3×C4×Q8—91no—
157C3×C4×D4—91no—
158C3×#117—91no—
159#947—38no—
160#946—38no—
161#898—38no—
162#897—34no—
163#895—34no—
164#894—38no—
165#893—34no—
166#887—34no—
167#875—34no—
168#872—34no—
169C22×C3⋊D3C23×C311yescharacteristic
170C22×C3⋊D3C23×C312yes—
171C22×C3×D3—38no—
172C22×C3×D3—34no—
173C32×Q8—48no—
174C32×Q8—42no—
175C32×Q8—416no—
176C2×C32×C4—24no—
177C2×C32×C4—24no—
178C2×C32×C4—24no—
179C2×C32×C4—28no—
180C2×C32×C4—28no—
181C2×C32×C4C22×D318yes—
182C2×C32×C4C22×D314yes—
183C2×C32×C4C22×D314yes—
184C2×C32×C4C22×D318yes—
185C2×C32×C4C22×D314yes—
186C2×C32×C4C23×C312yes—
187C2×C32×C4C23×C311yescharacteristic
188C2×C32×C4C23×C311yescharacteristic
189C2×C32×C4C23×C312yes—
190C2×C32×C4C23×C311yescharacteristic
191C2×C3⋊Dic3C23×C311yescharacteristic
192C2×C3⋊Dic3C23×C311yescharacteristic
193C2×C3⋊Dic3C23×C312yes—
194C2×C3⋊Dic3C23×C311yescharacteristic
195C4⋊(C3⋊D3)—22no—
196C4⋊(C3⋊D3)—42no—
197C4×C3⋊D3—22no—
198C4×C3⋊D3—42no—
199#634—22no—
200C2×C3×Dic3—34no—
201C2×C3×Dic3—34no—
202C2×C3×Dic3—38no—
203C2×C3×Dic3—34no—
204C3×D12—68no—
205C3×D12—128no—
206C3×C4×D3—68no—
207C3×C4×D3—128no—
208C3×Dic6—68no—
209#542—91noSylow
210C2×C33C2×#4111yescharacteristic
211C2×C33H(5,𝔽2)11yescharacteristic
212C2×C332−1+411yescharacteristic
213C3×C3⋊D3—41no—
214C3×C3⋊D3—42no—
215C2×C3×Q8—28no—
216C2×C3×Q8—91no—
217C2×C3×Q8C2×C3×D318yes—
218C2×C3×Q8C2×C3⋊D312yes—
219C2×C3×D4—91no—
220C2×C3×D4—92no—
221C22×C3×C4—91no—
222C22×C3×C4—92no—
223C2×D12—34no—
224C2×D12—38no—
225C2×C4×D3—34no—
226C2×C4×D3—38no—
227C2×Dic6—34no—
228C3×C4⋊C4—24no—
229C3×C4⋊C4—24no—
230C3×C4⋊C4—28no—
231C3×C4⋊C4—91no—
232C3×C4⋊C4—91no—
233C3×C4⋊C4—92no—
234C3×C4⋊C4—92no—
235C3×C4⋊C4C2×C3×D314yes—
236C3×C4⋊C4C2×C3×D314yes—
237C3×C4⋊C4C2×C3×D318yes—
238C3×C4⋊C4C2×C3⋊D311yescharacteristic
239C3×C4⋊C4C2×C3⋊D311yescharacteristic
240C3×C4⋊C4C2×C3⋊D312yes—
241C3×C22⋊C4—91no—
242C3×C22⋊C4—91no—
243C3×C22⋊C4—92no—
244C3×C22⋊C4—92no—
245C3×C22⋊C4—92no—
246C3×C22⋊C4—92no—
247C3×C22⋊C4—92no—
248C3×C42—24no—
249C3×C42—91no—
250C3×C42—92no—
251C3×C42C2×C3×D314yes—
252C3×C42C2×C3⋊D311yescharacteristic
253(C2×D3)⋊C4—34no—
254(C2×D3)⋊C4—34no—
255(C2×D3)⋊C4—38no—
256(C2×D3)⋊C4—38no—
257(C2×D3)⋊C4—38no—
258(C2×D3)⋊C4—38no—
259(C2×D3)⋊C4—38no—
260C4⋊Dic3—38no—
261Dic3⋊C4—34no—
262Dic3⋊C4—34no—
263Dic3⋊C4—38no—
264C4×Dic3—34no—
265C4×Dic3—38no—
266C22×C32—24no—
267C22×C32C23×D314yes—
268C22×C32C24×C311yescharacteristic
269C2×C3⋊D3—21no—
270C2×C3⋊D3—22no—
271C2×C3⋊D3—22no—
272C2×C3⋊D3—21no—
273C2×C3⋊D3—21no—
274C2×C3⋊D3—22no—
275C2×C3×D3—68no—
276C2×C3×D3—64no—
277C2×C3×D3—64no—
278C2×C3×D3—68no—
279C2×C3×D3—68no—
280C2×C3×D3—64no—
281C32×C4—28no—
282C32×C4—28no—
283C32×C4—24no—
284C32×C4—24no—
285C32×C4—22no—
286C32×C4—22no—
287C32×C4—21no—
288C32×C4—21no—
289C32×C4—28no—
290C32×C4—28no—
291C32×C4—28no—
292C32×C4—48no—
293C32×C4—48no—
294C32×C4#23918yes—
295C32×C4C3×#4112yes—
296C3⋊Dic3—22no—
297C3⋊Dic3—21no—
298C3⋊Dic3—21no—
299C3⋊Dic3—21no—
300C3×Dic3—64no—
301C3×Dic3—68no—
302C3×Dic3—64no—
303C3×Dic3—64no—
304C4⋊Q8C3×C3⋊D311yescharacteristic, complemented
305#126—92no—
306#124—92no—
307#124—91no—
308#123—92no—
309C22⋊Q8—92no—
310C22⋊Q8—91no—
311#121—91no—
312C4×Q8—91no—
313C4×D4—91no—
314#117—91no—
315C33#54211yescharacteristic, complemented, Sylow
316C23×C3—92no—
317C23×C3—91no—
318C22×D3—38no—
319C22×D3—34no—
320C3×Q8—48no—
321C3×Q8—42no—
322C3×Q8—416no—
323C3×Q8—182no—
324C3×D4—182no—
325C3×D4—362no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—28no—
330C2×C3×C4—28no—
331C2×C3×C4—91no—
332C2×C3×C4—92no—
333C2×C3×C4—91no—
334C2×C3×C4—91no—
335C2×C3×C4—182no—
336C2×C3×C4—362no—
337C2×C3×C4C22×C3×D318yes—
338C2×C3×C4C22×C3×D314yes—
339C2×C3×C4C22×C3×D314yes—
340C2×C3×C4C22×C3×D318yes—
341C2×C3×C4C22×C3×D314yes—
342C2×C3×C4C22×C3⋊D312yes—
343C2×C3×C4C22×C3⋊D311yescharacteristic
344C2×C3×C4C22×C3⋊D311yescharacteristic
345C2×C3×C4C22×C3⋊D312yes—
346C2×C3×C4C22×C3⋊D311yescharacteristic
347C2×Dic3—34no—
348C2×Dic3—34no—
349C2×Dic3—38no—
350C2×Dic3—34no—
351D12—68no—
352D12—128no—
353C4×D3—68no—
354C4×D3—128no—
355Dic6—68no—
356C2×C32—24no—
357C2×C32—24no—
358C2×C32—24no—
359C2×C32#100414yes—
360C2×C32C2×#23914yes—
361C2×C32#100714yes—
362C2×C32C2×C3×#4111yescharacteristic
363C2×C32C3×H(5,𝔽2)11yescharacteristic
364C2×C32C3×2−1+411yescharacteristic
365C3⋊D3—41no—
366C3⋊D3—42no—
367C3×D3—128no—
368C3×D3—124no—
369C2×Q8—91no—
370C2×Q8C2×C3×C3⋊D312yes—
371C2×D4—91no—
372C2×D4—92no—
373C22×C4—91no—
374C22×C4—92no—
375C4⋊C4—91no—
376C4⋊C4—91no—
377C4⋊C4—92no—
378C4⋊C4—92no—
379C4⋊C4C2×C3×C3⋊D311yescharacteristic
380C4⋊C4C2×C3×C3⋊D311yescharacteristic
381C4⋊C4C2×C3×C3⋊D312yes—
382C22⋊C4—91no—
383C22⋊C4—91no—
384C22⋊C4—92no—
385C22⋊C4—92no—
386C22⋊C4—92no—
387C22⋊C4—92no—
388C22⋊C4—92no—
389C42—91no—
390C42—92no—
391C42C2×C3×C3⋊D311yescharacteristic
392C22×C3—24no—
393C22×C3—182no—
394C22×C3—181no—
395C22×C3—181no—
396C22×C3—182no—
397C22×C3—182no—
398C22×C3—181no—
399C22×C3C23×C3×D314yes—
400C22×C3C23×C3⋊D311yescharacteristic
401C2×D3—68no—
402C2×D3—64no—
403C2×D3—64no—
404C2×D3—68no—
405C2×D3—68no—
406C2×D3—64no—
407C3×C4—28no—
408C3×C4—28no—
409C3×C4—28no—
410C3×C4—28no—
411C3×C4—24no—
412C3×C4—24no—
413C3×C4—22no—
414C3×C4—22no—
415C3×C4—21no—
416C3×C4—21no—
417C3×C4—28no—
418C3×C4—48no—
419C3×C4—48no—
420C3×C4—181no—
421C3×C4—181no—
422C3×C4—182no—
423C3×C4—181no—
424C3×C4C3×#23918yes—
425C3×C4#383212yes—
426Dic3—64no—
427Dic3—68no—
428Dic3—64no—
429Dic3—64no—
430C32—24no—
431C32#573314yescomplemented
432C32C3×#54211yescharacteristic, complemented
433C23—92no—
434C23—91no—
435Q8—42no—
436Q8—182no—
437D4—182no—
438D4—362no—
439C2×C4—91no—
440C2×C4—92no—
441C2×C4—91no—
442C2×C4—91no—
443C2×C4—182no—
444C2×C4—362no—
445C2×C4C22×C3×C3⋊D312yes—
446C2×C4C22×C3×C3⋊D311yescharacteristic
447C2×C4C22×C3×C3⋊D311yescharacteristic
448C2×C4C22×C3×C3⋊D312yes—
449C2×C4C22×C3×C3⋊D311yescharacteristic
450C2×C3—24no—
451C2×C3—24no—
452C2×C3—24no—
453C2×C3—362no—
454C2×C3—361no—
455C2×C3C3×#100414yes—
456C2×C3C2×C3×#23914yes—
457C2×C3C3×#100714yes—
458C2×C3#6436111yescharacteristic
459C2×C3C2×#383211yescharacteristic
460C2×C3#6436411yescharacteristic
461D3—128no—
462D3—124no—
463C22—182no—
464C22—181no—
465C22—181no—
466C22—182no—
467C22—182no—
468C22—181no—
469C22C23×C3×C3⋊D311yescharacteristic
470C4—22no—
471C4—22no—
472C4—21no—
473C4—21no—
474C4—181no—
475C4—181no—
476C4—182no—
477C4—181no—
478C4C3×#383212yes—
479C3—24no—
480C3C3×#573314yescomplemented
481C3#1059620411yescharacteristic, direct product factor
482C2—362no—
483C2—361no—
484C2C3×#6436111yescharacteristic
485C2C2×C3×#383211yescharacteristic
486C2C3×#6436411yescharacteristic
subgroupG/H
centerC22×C3C23×C3⋊D3
commutator, cosocleC22×C32C24×C3
FrattiniC22C23×C3×C3⋊D3
FittingC33×C4⋊Q8C2
socleC22×C33C24
prime radicalC3×#64104C2
hypercenterC3×C4⋊Q8C3⋊D3
hypocenterC32C3×#542
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C33×C4⋊Q8C2semidirect
2. C3×#64156C2semidirect
3. C3×#64155C2semidirect
4. C3×#64107C2semidirect
5. C3×#64106C2semidirect
7. C3×#64103C2semidirect
8. C3×#64102C2semidirect
9. C3×#64096C2semidirect
10. C3×#64084C2semidirect
11. C3×#64081C2semidirect
12. #10596204C3direct (N × H)
38. C32×C4⋊Q8D3semidirect
39. C32×C4⋊Q8C2×C3semidirect
40. #64156C2×C3semidirect
41. #64155C2×C3semidirect
42. #64107C2×C3semidirect
43. #64106C2×C3semidirect
45. #64103C2×C3semidirect
46. #64102C2×C3semidirect
47. #64096C2×C3semidirect
48. #64084C2×C3semidirect
49. #64081C2×C3semidirect
147. C3×C4⋊Q8C3×D3semidirect
148. C3×C4⋊Q8C3⋊D3semidirect
304. C4⋊Q8C3×C3⋊D3semidirect
315. C33#542semidirect
431. C32#5733semidirect
432. C32C3×#542semidirect
480. C3C3×#5733semidirect
481. C3#10596204direct (N × H)

Maximal subgroups

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

subgroupindex
C33×C4⋊Q82
C3×#641562
C3×#641552
C3×#641072
C3×#641062
C3×#641042
C3×#641032
C3×#641022
C3×#640962
C3×#640842
C3×#640812
#105962043
C3×#57333