Finite Group Databasebeta

← C3×#10596032 #13179312

Subgroups — C3×#10596032 #13179312

Order 1728.

1672 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C4×D4C211yesmaximal, characteristic, complemented
2C3×#64147C211yesmaximal, characteristic
3C3×#64145C211yesmaximal, characteristic, complemented
4C3×#64138C212yesmaximal, complemented
5C3×#64133C212yesmaximal, complemented
6C3×#64106C211yesmaximal, characteristic, complemented
7C3×#64097C211yesmaximal, characteristic, complemented
8C3×#64094C212yesmaximal, complemented
9C3×#64088C212yesmaximal, complemented
10C3×#64081C211yesmaximal, characteristic, complemented
11C3×C4×#634C211yesmaximal, characteristic, complemented
12#10596032C311yesmaximal, characteristic, direct product factor
13C3×#5561—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C22×C33×C4C2212yes—
16C2×C3×#638C2212yes—
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C3×#634C2211yescharacteristic
19C2×C3×#634C2212yes—
20C33×C4⋊C4C2211yescharacteristic
21C33×C22⋊C4C2212yes—
22C33×C42C2211yescharacteristic
23C3×#3757C2212yes—
24C3×#3757C2212yes—
25C3×#3757C2212yes—
26C3×#3752C2211yescharacteristic
27C3×#3752C2211yescharacteristic
28C3×#3752C2212yes—
29C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
30C3×C4⋊(C3⋊Dic3)C2212yes—
31C3×#3750C2212yes—
32C3×#3750C2211yescharacteristic
33C3×#3750C2211yescharacteristic
34C3×#3750C2212yes—
35C3×C4×C3⋊Dic3C2212yes—
36C3×C4×C3⋊Dic3C2211yescharacteristic
37C32×C4×D4—24no—
38C32×C4×D4D314yescomplemented
39C32×C4×D4C2×C311yescharacteristic, complemented
40#64147C2×C311yescharacteristic
41#64145C2×C311yescharacteristic, complemented
42#64138C2×C312yescomplemented
43#64133C2×C312yescomplemented
44#64106C2×C311yescharacteristic, complemented
45#64097C2×C311yescharacteristic, complemented
46#64094C2×C312yescomplemented
47#64088C2×C312yescomplemented
48#64081C2×C311yescharacteristic, complemented
49C4×#634C2×C311yescharacteristic, complemented
50C3×#938—34no—
51C3×#936—34no—
52C3×#929—38no—
53C3×#924—38no—
54C3×#897—34no—
55C3×#888—34no—
56C3×#885—38no—
57C3×#879—38no—
58C3×#872—34no—
59C3×C4×Dic6—34no—
60C23×C33C2312yes—
61C22×C3×C3⋊D3C2311yescharacteristic
62C33×D4—22no—
63C2×C33×C4—42no—
64C2×C33×C4C2311yescharacteristic
65C2×C33×C4C2311yescharacteristic
66C2×C33×C4C2312yes—
67C2×C33×C4C2311yescharacteristic
68C3×#638—42no—
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C2×C3×C3⋊Dic3C2312yes—
71C2×C3×C3⋊Dic3C2312yes—
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C2×C3×C3⋊Dic3C2311yescharacteristic
74C3×C4×C3⋊D3—22no—
75C3×#634—22no—
76C3×#634—42no—
77C3×#542—91no—
78#5561—34no—
79C2×C32×D4—24no—
80C2×C32×D4C2×D314yes—
81C2×C32×D4C22×C311yescharacteristic
82C22×C32×C4—28no—
83C22×C32×C4C2×D318yes—
84C22×C32×C4C22×C312yes—
85C2×#638C22×C312yes—
86C2×C4×C3⋊D3C22×C311yescharacteristic
87C2×#634C22×C311yescharacteristic
88C2×#634C22×C312yes—
89C2×C3×#67—38no—
90C2×C3×C4×D3—34no—
91C2×C3×Dic6—34no—
92C2×C3×Dic6—38no—
93C32×C4⋊C4—24no—
94C32×C4⋊C4C2×D314yes—
95C32×C4⋊C4C22×C311yescharacteristic
96C32×C22⋊C4—28no—
97C32×C22⋊C4C2×D318yes—
98C32×C22⋊C4C22×C312yes—
99C32×C42—24no—
100C32×C42C2×D314yes—
101C32×C42C22×C311yescharacteristic
102#3757C22×C312yes—
103#3757C22×C312yes—
104#3757C22×C312yes—
105#3752C22×C311yescharacteristic
106#3752C22×C311yescharacteristic
107#3752C22×C312yes—
108C4⋊(C3⋊Dic3)C22×C311yescharacteristic
109C4⋊(C3⋊Dic3)C22×C312yes—
110#3750C22×C312yes—
111#3750C22×C311yescharacteristic
112#3750C22×C311yescharacteristic
113#3750C22×C312yes—
114C4×C3⋊Dic3C22×C312yes—
115C4×C3⋊Dic3C22×C311yescharacteristic
116C3×#217—38no—
117C3×#217—38no—
118C3×#217—38no—
119C3×(C2×D3)⋊C4—34no—
120C3×(C2×D3)⋊C4—38no—
121C3×(C2×D3)⋊C4—34no—
122C3×C4⋊Dic3—38no—
123C3×C4⋊Dic3—34no—
124C3×Dic3⋊C4—34no—
125C3×Dic3⋊C4—38no—
126C3×Dic3⋊C4—34no—
127C3×Dic3⋊C4—38no—
128C3×C4×Dic3—38no—
129C3×C4×Dic3—34no—
130C22×C33—22no—
131C22×C33—22no—
132C22×C33—22no—
133C22×C33C2411yescharacteristic
134C2×C3×C3⋊D3—21no—
135C2×C3×C3⋊D3—21no—
136C2×C3×C3⋊D3—21no—
137C33×C4—21no—
138C33×C4—22no—
139C33×C4—21no—
140C33×C4#4112yes—
141C3×C3⋊Dic3—22no—
142C3×C3⋊Dic3—21no—
143C3×C3⋊Dic3—21no—
144C3×C3⋊Dic3—22no—
145C3×C3⋊Dic3—21no—
146C3×C4⋊Q8—91no—
147C3×#126—92no—
148C3×#124—91no—
149C3×#124—92no—
150C3×#123—92no—
151C3×C22⋊Q8—92no—
152C3×C22⋊Q8—91no—
153C3×#121—91no—
154C3×C4×Q8—91no—
155C3×C4×D4—24no—
156C3×C4×D4C3×D314yescomplemented
157C3×C4×D4C3⋊D311yescharacteristic, complemented
158C3×#117—91no—
159#938—34no—
160#936—34no—
161#929—38no—
162#924—38no—
163#897—34no—
164#888—34no—
165#885—38no—
166#879—38no—
167#872—34no—
168C4×Dic6—34no—
169C23×C32—28no—
170C23×C32C22×D318yes—
171C23×C32C23×C312yes—
172C22×C3⋊D3C23×C311yescharacteristic
173C22×C3×D3—34no—
174C32×D4—216no—
175C32×D4—28no—
176C32×D4—22no—
177C2×C32×C4—24no—
178C2×C32×C4—24no—
179C2×C32×C4—24no—
180C2×C32×C4—28no—
181C2×C32×C4—48no—
182C2×C32×C4—42no—
183C2×C32×C4—416no—
184C2×C32×C4C22×D314yes—
185C2×C32×C4C22×D314yes—
186C2×C32×C4C22×D318yes—
187C2×C32×C4C22×D314yes—
188C2×C32×C4C23×C311yescharacteristic
189C2×C32×C4C23×C311yescharacteristic
190C2×C32×C4C23×C312yes—
191C2×C32×C4C23×C311yescharacteristic
192#638—42no—
193C2×C3⋊Dic3C23×C311yescharacteristic
194C2×C3⋊Dic3C23×C312yes—
195C2×C3⋊Dic3C23×C312yes—
196C2×C3⋊Dic3C23×C311yescharacteristic
197C2×C3⋊Dic3C23×C311yescharacteristic
198C4×C3⋊D3—22no—
199#634—22no—
200#634—42no—
201C3×#67—128no—
202C2×C3×Dic3—34no—
203C2×C3×Dic3—38no—
204C2×C3×Dic3—38no—
205C2×C3×Dic3—34no—
206C2×C3×Dic3—34no—
207C3×C4×D3—68no—
208C3×Dic6—68no—
209C3×Dic6—128no—
210#542—91noSylow
211C2×C33—42no—
212C2×C33C2×#4111yescharacteristic
213C2×C33H(5,𝔽2)11yescharacteristic
214C2×C332−1+411yescharacteristic
215C3×C3⋊D3—41no—
216C2×C3×Q8—91no—
217C2×C3×Q8—92no—
218C2×C3×D4—24no—
219C2×C3×D4—92no—
220C2×C3×D4C2×C3×D314yes—
221C2×C3×D4C2×C3⋊D311yescharacteristic
222C22×C3×C4—28no—
223C22×C3×C4—91no—
224C22×C3×C4C2×C3×D318yes—
225C22×C3×C4C2×C3⋊D312yes—
226C2×#67—38no—
227C2×C4×D3—34no—
228C2×Dic6—34no—
229C2×Dic6—38no—
230C3×C4⋊C4—24no—
231C3×C4⋊C4—91no—
232C3×C4⋊C4—91no—
233C3×C4⋊C4—92no—
234C3×C4⋊C4—92no—
235C3×C4⋊C4—91no—
236C3×C4⋊C4—92no—
237C3×C4⋊C4C2×C3×D314yes—
238C3×C4⋊C4C2×C3⋊D311yescharacteristic
239C3×C22⋊C4—28no—
240C3×C22⋊C4—92no—
241C3×C22⋊C4—92no—
242C3×C22⋊C4—92no—
243C3×C22⋊C4—91no—
244C3×C22⋊C4—91no—
245C3×C22⋊C4—92no—
246C3×C22⋊C4C2×C3×D318yes—
247C3×C22⋊C4C2×C3⋊D312yes—
248C3×C42—24no—
249C3×C42—91no—
250C3×C42—92no—
251C3×C42C2×C3×D314yes—
252C3×C42C2×C3⋊D311yescharacteristic
253#217—38no—
254#217—38no—
255#217—38no—
256(C2×D3)⋊C4—34no—
257(C2×D3)⋊C4—34no—
258(C2×D3)⋊C4—38no—
259C4⋊Dic3—34no—
260C4⋊Dic3—38no—
261Dic3⋊C4—38no—
262Dic3⋊C4—34no—
263Dic3⋊C4—34no—
264Dic3⋊C4—38no—
265C4×Dic3—38no—
266C4×Dic3—34no—
267C22×C32—216no—
268C22×C32—28no—
269C22×C32—22no—
270C22×C32—22no—
271C22×C32—22no—
272C22×C32—24no—
273C22×C32—28no—
274C22×C32—28no—
275C22×C32—48no—
276C22×C32—48no—
277C22×C32C23×D314yes—
278C22×C32C24×C311yescharacteristic
279C2×C3⋊D3—21no—
280C2×C3⋊D3—21no—
281C2×C3⋊D3—21no—
282C2×C3×D3—64no—
283C2×C3×D3—64no—
284C2×C3×D3—64no—
285C32×C4—28no—
286C32×C4—24no—
287C32×C4—22no—
288C32×C4—21no—
289C32×C4—21no—
290C32×C4—28no—
291C32×C4—28no—
292C32×C4—24no—
293C32×C4—44no—
294C32×C4—48no—
295C32×C4#23518yes—
296C32×C4C3×#4112yes—
297C3⋊Dic3—22no—
298C3⋊Dic3—21no—
299C3⋊Dic3—21no—
300C3⋊Dic3—22no—
301C3⋊Dic3—21no—
302C3×Dic3—68no—
303C3×Dic3—64no—
304C3×Dic3—64no—
305C3×Dic3—68no—
306C3×Dic3—64no—
307C4⋊Q8—91no—
308#126—92no—
309#124—91no—
310#124—92no—
311#123—92no—
312C22⋊Q8—92no—
313C22⋊Q8—91no—
314#121—91no—
315C4×Q8—91no—
316C4×D4C3×C3⋊D311yescharacteristic, complemented
317#117—91no—
318C33#54211yescharacteristic, complemented, Sylow
319C23×C3—28no—
320C23×C3—91no—
321C23×C3C22×C3×D318yes—
322C23×C3C22×C3⋊D312yes—
323C22×D3—34no—
324C3×Q8—182no—
325C3×Q8—362no—
326C3×D4—216no—
327C3×D4—28no—
328C3×D4—22no—
329C3×D4—362no—
330C2×C3×C4—24no—
331C2×C3×C4—24no—
332C2×C3×C4—24no—
333C2×C3×C4—28no—
334C2×C3×C4—48no—
335C2×C3×C4—42no—
336C2×C3×C4—416no—
337C2×C3×C4—91no—
338C2×C3×C4—92no—
339C2×C3×C4—91no—
340C2×C3×C4—92no—
341C2×C3×C4—91no—
342C2×C3×C4—182no—
343C2×C3×C4C22×C3×D314yes—
344C2×C3×C4C22×C3×D314yes—
345C2×C3×C4C22×C3×D318yes—
346C2×C3×C4C22×C3×D314yes—
347C2×C3×C4C22×C3⋊D311yescharacteristic
348C2×C3×C4C22×C3⋊D311yescharacteristic
349C2×C3×C4C22×C3⋊D312yes—
350C2×C3×C4C22×C3⋊D311yescharacteristic
351#67—128no—
352C2×Dic3—34no—
353C2×Dic3—38no—
354C2×Dic3—38no—
355C2×Dic3—34no—
356C2×Dic3—34no—
357C4×D3—68no—
358Dic6—68no—
359Dic6—128no—
360C2×C32—24no—
361C2×C32—24no—
362C2×C32—24no—
363C2×C32—416no—
364C2×C32—48no—
365C2×C32—42no—
366C2×C32C2×#23514yes—
367C2×C32#100414yes—
368C2×C32#101014yes—
369C2×C32C2×C3×#4111yescharacteristic
370C2×C32C3×H(5,𝔽2)11yescharacteristic
371C2×C32C3×2−1+411yescharacteristic
372C3⋊D3—41no—
373C3×D3—124no—
374C2×Q8—91no—
375C2×Q8—92no—
376C2×D4—92no—
377C2×D4C2×C3×C3⋊D311yescharacteristic
378C22×C4—91no—
379C22×C4C2×C3×C3⋊D312yes—
380C4⋊C4—91no—
381C4⋊C4—91no—
382C4⋊C4—91no—
383C4⋊C4—92no—
384C4⋊C4—92no—
385C4⋊C4—92no—
386C4⋊C4C2×C3×C3⋊D311yescharacteristic
387C22⋊C4—92no—
388C22⋊C4—92no—
389C22⋊C4—92no—
390C22⋊C4—91no—
391C22⋊C4—91no—
392C22⋊C4—92no—
393C22⋊C4C2×C3×C3⋊D312yes—
394C42—91no—
395C42—92no—
396C42C2×C3×C3⋊D311yescharacteristic
397C22×C3—216no—
398C22×C3—28no—
399C22×C3—28no—
400C22×C3—28no—
401C22×C3—22no—
402C22×C3—22no—
403C22×C3—22no—
404C22×C3—24no—
405C22×C3—48no—
406C22×C3—48no—
407C22×C3—181no—
408C22×C3—181no—
409C22×C3—181no—
410C22×C3C23×C3×D314yes—
411C22×C3C23×C3⋊D311yescharacteristic
412C2×D3—64no—
413C2×D3—64no—
414C2×D3—64no—
415C3×C4—28no—
416C3×C4—28no—
417C3×C4—24no—
418C3×C4—24no—
419C3×C4—21no—
420C3×C4—22no—
421C3×C4—21no—
422C3×C4—28no—
423C3×C4—48no—
424C3×C4—44no—
425C3×C4—181no—
426C3×C4—182no—
427C3×C4—182no—
428C3×C4—181no—
429C3×C4—181no—
430C3×C4C3×#23518yes—
431C3×C4#382812yes—
432Dic3—68no—
433Dic3—64no—
434Dic3—64no—
435Dic3—68no—
436Dic3—64no—
437C32—24no—
438C32#556114yescomplemented
439C32C3×#54211yescharacteristic, complemented
440C23—91no—
441C23C22×C3×C3⋊D312yes—
442Q8—182no—
443Q8—362no—
444D4—22no—
445D4—362no—
446C2×C4—42no—
447C2×C4—91no—
448C2×C4—92no—
449C2×C4—91no—
450C2×C4—92no—
451C2×C4—91no—
452C2×C4—182no—
453C2×C4C22×C3×C3⋊D311yescharacteristic
454C2×C4C22×C3×C3⋊D311yescharacteristic
455C2×C4C22×C3×C3⋊D312yes—
456C2×C4C22×C3×C3⋊D311yescharacteristic
457C2×C3—24no—
458C2×C3—24no—
459C2×C3—24no—
460C2×C3—416no—
461C2×C3—48no—
462C2×C3—42no—
463C2×C3—361no—
464C2×C3C2×C3×#23514yes—
465C2×C3C3×#100414yes—
466C2×C3C3×#101014yes—
467C2×C3C2×#382811yescharacteristic
468C2×C3#6436111yescharacteristic
469C2×C3#6436711yescharacteristic
470D3—124no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—181no—
475C22—181no—
476C22—181no—
477C22C23×C3×C3⋊D311yescharacteristic
478C4—21no—
479C4—22no—
480C4—21no—
481C4—181no—
482C4—182no—
483C4—182no—
484C4—181no—
485C4—181no—
486C4C3×#382812yes—
487C3—24no—
488C3C3×#556114yescomplemented
489C3#1059603211yescharacteristic, direct product factor
490C2—42no—
491C2—361no—
492C2C2×C3×#382811yescharacteristic
493C2C3×#6436111yescharacteristic
494C2C3×#6436711yescharacteristic
subgroupG/H
centerC22×C3C23×C3⋊D3
commutator, cosocleC22×C32C24×C3
FrattiniC22C23×C3×C3⋊D3
FittingC33×C4×D4C2
socleC22×C33C24
prime radicalC3×#64147C2
hypercenterC3×C4×D4C3⋊D3
hypocenterC32C3×#542
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C4×D4

Complements

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

NH ≅ G/Ntype
1. C33×C4×D4C2semidirect
3. C3×#64145C2semidirect
4. C3×#64138C2semidirect
5. C3×#64133C2semidirect
6. C3×#64106C2semidirect
7. C3×#64097C2semidirect
8. C3×#64094C2semidirect
9. C3×#64088C2semidirect
10. C3×#64081C2semidirect
11. C3×C4×#634C2semidirect
12. #10596032C3direct (N × H)
38. C32×C4×D4D3semidirect
39. C32×C4×D4C2×C3semidirect
41. #64145C2×C3semidirect
42. #64138C2×C3semidirect
43. #64133C2×C3semidirect
44. #64106C2×C3semidirect
45. #64097C2×C3semidirect
46. #64094C2×C3semidirect
47. #64088C2×C3semidirect
48. #64081C2×C3semidirect
49. C4×#634C2×C3semidirect
156. C3×C4×D4C3×D3semidirect
157. C3×C4×D4C3⋊D3semidirect
316. C4×D4C3×C3⋊D3semidirect
318. C33#542semidirect
438. C32#5561semidirect
439. C32C3×#542semidirect
488. C3C3×#5561semidirect
489. C3#10596032direct (N × H)

Maximal subgroups

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

subgroupindex
C33×C4×D42
C3×#641472
C3×#641452
C3×#641382
C3×#641332
C3×#641062
C3×#640972
C3×#640942
C3×#640882
C3×#640812
C3×C4×#6342
#105960323
C3×#55613