Finite Group Databasebeta

← C3×#10591063 #13158147

Subgroups — C3×#10591063 #13158147

Order 1728.

702 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×Dic3C211yesmaximal, characteristic
2C3×#63657C211yesmaximal, characteristic
3C32×#830C211yesmaximal, characteristic
4C3×Dic3⋊(C3⋊C8)C212yesmaximal, complemented
5C3×#63554C212yesmaximal, complemented
6C3×#4926—31nomaximal
7C3×#4719—31nomaximal
8#10591063C311yesmaximal, characteristic, direct product factor
9C22×C33×C4C2211yescharacteristic
10C22×C32×Dic3C411yescharacteristic
11C22×C32×Dic3C411yescharacteristic
12C2×C3×C3⋊(C3⋊C8)C2212yes—
13C32×C4×Dic3C411yescharacteristic
14C32×C4×Dic3C411yescharacteristic
15C32×C4×Dic3C2211yescharacteristic
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2212yes—
18C2×C32×C42—31no—
19C2×C3×C4×Dic3—21no—
20C2×C3×C4×Dic3D311yescharacteristic
21C2×C3×C4×Dic3C2×C311yescharacteristic
22C32×C22⋊C8—31no—
23#63657C2×C311yescharacteristic
24C3×#830—21no—
25C3×#830—31no—
26C3×#830—31no—
27C3×#830—61no—
28C3×#830D311yescharacteristic
29C3×#830C2×C311yescharacteristic
30C3×#815—32no—
31C3×Dic3⋊C8—32no—
32C3×C4⋊(C3⋊C8)—32no—
33C3×#803—32no—
34Dic3⋊(C3⋊C8)C2×C312yescomplemented
35#63554C2×C312yescomplemented
36C23×C33C2×C411yescharacteristic
37C2×C33×C4—22no—
38C2×C33×C4C2×C411yescharacteristic
39C2×C33×C4C2311yescharacteristic
40C2×C32×Dic3—22no—
41C2×C32×Dic3—22no—
42C2×C32×Dic3C2×C411yescharacteristic
43C2×C32×Dic3C2×C411yescharacteristic
44C2×C32×Dic3C2×C411yescharacteristic
45C2×C32×Dic3C2×C411yescharacteristic
46C3×C3⋊(C3⋊C8)—22no—
47C32×C3⋊C8—22no—
48C3×#432—91no—
49#4926—31no—
50#4719—31no—
51C22×C32×C4—21no—
52C22×C32×C4—21no—
53C22×C32×C4—21no—
54C22×C32×C4—31no—
55C22×C32×C4—31no—
56C22×C32×C4—41no—
57C22×C32×C4C2×D311yescharacteristic
58C22×C32×C4C2×D311yescharacteristic
59C22×C32×C4C22×C311yescharacteristic
60C22×C3×Dic3—21no—
61C22×C3×Dic3—21no—
62C22×C3×Dic3Dic311yescharacteristic
63C22×C3×Dic3Dic311yescharacteristic
64C22×C3×Dic3C3×C411yescharacteristic
65C22×C3×Dic3C3×C411yescharacteristic
66C2×C32×C8—32no—
67C32×C42—31no—
68C32×C42—31no—
69C32×C42—31no—
70C32×C42—31no—
71C2×C3⋊(C3⋊C8)C22×C312yes—
72C3×C4×Dic3—21no—
73C3×C4×Dic3—21no—
74C3×C4×Dic3—21no—
75C3×C4×Dic3—21no—
76C3×C4×Dic3Dic311yescharacteristic
77C3×C4×Dic3Dic311yescharacteristic
78C3×C4×Dic3C3×C411yescharacteristic
79C3×C4×Dic3C3×C411yescharacteristic
80C3×C4×Dic3C2×D311yescharacteristic
81C3×C4×Dic3C2×D311yescharacteristic
82C3×C4×Dic3C22×C311yescharacteristic
83C3×C4×Dic3C22×C311yescharacteristic
84C2×C3×C3⋊C8—22no—
85C2×C3×C3⋊C8—32no—
86C2×C3×C3⋊C8—32no—
87C2×C3×C3⋊C8—62no—
88C2×C3×C3⋊C8C2×D312yes—
89C2×C3×C3⋊C8C22×C312yes—
90C22×C33—21no—
91C22×C33—21no—
92C22×C33M1612yes—
93C22×C33C22×C411yescharacteristic
94C33×C4—21no—
95C33×C4#4112yes—
96C32×Dic3—21no—
97C32×Dic3—21no—
98C32×Dic3—21no—
99C32×Dic3M1612yes—
100C2×C3×C42—31no—
101C2×C3×C42—31no—
102C2×C3×C42—61no—
103C2×C4×Dic3C3×D311yescharacteristic
104C3×C4⋊C8—92no—
105C3×C22⋊C8—31no—
106C3×C22⋊C8—31no—
107C3×C22⋊C8—61no—
108C3×C22⋊C8—91no—
109C3×#97—92no—
110#830—31no—
111#830—31no—
112#830—61no—
113#830C3×D311yescharacteristic
114#815—32no—
115Dic3⋊C8—32no—
116C4⋊(C3⋊C8)—32no—
117#803—32no—
118C23×C32—21no—
119C23×C32—21no—
120C23×C32—21no—
121C23×C32—41no—
122C23×C32C4×D311yescharacteristic
123C23×C32C2×Dic311yescharacteristic
124C23×C32C2×C3×C411yescharacteristic
125C2×C32×C4—22no—
126C2×C32×C4—24no—
127C2×C32×C4—22no—
128C2×C32×C4—22no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—31no—
136C2×C32×C4—31no—
137C2×C32×C4—31no—
138C2×C32×C4—31no—
139C2×C32×C4—41no—
140C2×C32×C4—41no—
141C2×C32×C4—42no—
142C2×C32×C4—42no—
143C2×C32×C4—44no—
144C2×C32×C4—62no—
145C2×C32×C4—62no—
146C2×C32×C4C4×D311yescharacteristic
147C2×C32×C4C2×Dic311yescharacteristic
148C2×C32×C4C2×C3×C411yescharacteristic
149C2×C32×C4C22×D311yescharacteristic
150C2×C32×C4C22×D311yescharacteristic
151C2×C32×C4C23×C311yescharacteristic
152C2×C3×Dic3—24no—
153C2×C3×Dic3—24no—
154C2×C3×Dic3—22no—
155C2×C3×Dic3—22no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—21no—
159C2×C3×Dic3—21no—
160C2×C3×Dic3—21no—
161C2×C3×Dic3—21no—
162C2×C3×Dic3C2×Dic311yescharacteristic
163C2×C3×Dic3C2×Dic311yescharacteristic
164C2×C3×Dic3C2×Dic311yescharacteristic
165C2×C3×Dic3C2×Dic311yescharacteristic
166C2×C3×Dic3C2×C3×C411yescharacteristic
167C2×C3×Dic3C2×C3×C411yescharacteristic
168C2×C3×Dic3C2×C3×C411yescharacteristic
169C2×C3×Dic3C2×C3×C411yescharacteristic
170C32×C8—62no—
171C3⋊(C3⋊C8)—22no—
172C3×C3⋊C8—22no—
173C3×C3⋊C8—22no—
174C3×C3⋊C8—42no—
175C3×C3⋊C8—62no—
176C3×C3⋊C8—62no—
177C3×C3⋊C8—122no—
178#432—91noSylow
179C2×C33—22no—
180C2×C33#11711yescharacteristic
181C2×C33C2×M1611yescharacteristic
182C2×C33C2×M1611yescharacteristic
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—31no—
187C22×C3×C4—31no—
188C22×C3×C4—31no—
189C22×C3×C4—31no—
190C22×C3×C4—41no—
191C22×C3×C4—61no—
192C22×C3×C4—61no—
193C22×C3×C4D3211yescharacteristic
194C22×C3×C4C2×C3×D311yescharacteristic
195C22×C3×C4C2×C3×D311yescharacteristic
196C22×Dic3C3×Dic311yescharacteristic
197C22×Dic3C3×Dic311yescharacteristic
198C2×C3×C8—32no—
199C2×C3×C8—32no—
200C2×C3×C8—62no—
201C2×C3×C8—92no—
202C3×C42—31no—
203C3×C42—31no—
204C3×C42—31no—
205C3×C42—31no—
206C3×C42—31no—
207C3×C42—31no—
208C3×C42—31no—
209C3×C42—31no—
210C3×C42—61no—
211C3×C42—61no—
212C3×C42—61no—
213C3×C42—61no—
214C4×Dic3C3×Dic311yescharacteristic
215C4×Dic3C3×Dic311yescharacteristic
216C4×Dic3C2×C3×D311yescharacteristic
217C4×Dic3C2×C3×D311yescharacteristic
218C2×C3⋊C8—32no—
219C2×C3⋊C8—32no—
220C2×C3⋊C8—62no—
221C2×C3⋊C8C2×C3×D312yes—
222C22×C32—21no—
223C22×C32—21no—
224C22×C32—22no—
225C22×C32—22no—
226C22×C32—21no—
227C22×C32—21no—
228C22×C32—21no—
229C22×C32—21no—
230C22×C32—22no—
231C22×C32—22no—
232C22×C32—22no—
233C22×C32—21no—
234C22×C32—21no—
235C22×C32—21no—
236C22×C32—42no—
237C22×C32—41no—
238C22×C32—41no—
239C22×C32—41no—
240C22×C32—41no—
241C22×C32—41no—
242C22×C32—42no—
243C22×C32—42no—
244C22×C32ΓL(1,𝔽25)12yes—
245C22×C32#20812yes—
246C22×C32C3×M1612yes—
247C22×C32C2×C4×D311yescharacteristic
248C22×C32C22×Dic311yescharacteristic
249C22×C32C22×C3×C411yescharacteristic
250C32×C4—21no—
251C32×C4—22no—
252C32×C4—21no—
253C32×C4—21no—
254C32×C4—22no—
255C32×C4—22no—
256C32×C4—22no—
257C32×C4—32no—
258C32×C4—42no—
259C32×C4—41no—
260C32×C4—41no—
261C32×C4—42no—
262C32×C4—61no—
263C32×C4—61no—
264C32×C4—61no—
265C32×C4#23512yes—
266C32×C4#23712yes—
267C32×C4C3×#4112yes—
268C3×Dic3—22no—
269C3×Dic3—22no—
270C3×Dic3—22no—
271C3×Dic3—21no—
272C3×Dic3—21no—
273C3×Dic3—21no—
274C3×Dic3—21no—
275C3×Dic3—21no—
276C3×Dic3—21no—
277C3×Dic3—22no—
278C3×Dic3#20812yes—
279C3×Dic3C3×M1612yes—
280C2×C42—31no—
281C4⋊C8—92no—
282C22⋊C8—31no—
283C22⋊C8—91no—
284#97—92no—
285C33#43211yescharacteristic, complemented, Sylow
286C23×C3—21no—
287C23×C3—21no—
288C23×C3—21no—
289C23×C3—41no—
290C23×C3D3×Dic311yescharacteristic
291C23×C3C3×C4×D311yescharacteristic
292C23×C3C2×C3×Dic311yescharacteristic
293C2×C3×C4—24no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—31no—
304C2×C3×C4—31no—
305C2×C3×C4—31no—
306C2×C3×C4—31no—
307C2×C3×C4—31no—
308C2×C3×C4—31no—
309C2×C3×C4—31no—
310C2×C3×C4—31no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—44no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—61no—
317C2×C3×C4—61no—
318C2×C3×C4—61no—
319C2×C3×C4—61no—
320C2×C3×C4—62no—
321C2×C3×C4—62no—
322C2×C3×C4—64no—
323C2×C3×C4—64no—
324C2×C3×C4—62no—
325C2×C3×C4—62no—
326C2×C3×C4D3×Dic311yescharacteristic
327C2×C3×C4C3×C4×D311yescharacteristic
328C2×C3×C4C2×C3×Dic311yescharacteristic
329C2×C3×C4C2×D3211yescharacteristic
330C2×C3×C4C22×C3×D311yescharacteristic
331C2×C3×C4C22×C3×D311yescharacteristic
332C2×Dic3—22no—
333C2×Dic3—22no—
334C2×Dic3C2×C3×Dic311yescharacteristic
335C2×Dic3C2×C3×Dic311yescharacteristic
336C2×Dic3C2×C3×Dic311yescharacteristic
337C2×Dic3C2×C3×Dic311yescharacteristic
338C3×C8—62no—
339C3×C8—62no—
340C3×C8—122no—
341C3×C8—182no—
342C3⋊C8—22no—
343C3⋊C8—62no—
344C3⋊C8—62no—
345C3⋊C8—122no—
346C2×C32—22no—
347C2×C32—24no—
348C2×C32—22no—
349C2×C32—22no—
350C2×C32—21no—
351C2×C32—21no—
352C2×C32—21no—
353C2×C32—21no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32—21no—
357C2×C32—21no—
358C2×C32—21no—
359C2×C32—41no—
360C2×C32—41no—
361C2×C32—41no—
362C2×C32—42no—
363C2×C32—42no—
364C2×C32—44no—
365C2×C32#87711yescharacteristic
366C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
367C2×C32D3×M1611yescharacteristic
368C2×C32C2×#20811yescharacteristic
369C2×C32C2×#20811yescharacteristic
370C2×C32#92611yescharacteristic
371C2×C32C3×#11711yescharacteristic
372C2×C32C2×C3×M1611yescharacteristic
373C2×C32C2×C3×M1611yescharacteristic
374C22×C4—31no—
375C22×C4—31no—
376C22×C4C3×D3211yescharacteristic
377C2×C8—32no—
378C2×C8—92no—
379C42—31no—
380C42—31no—
381C42—31no—
382C42—31no—
383C22×C3—22no—
384C22×C3—22no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—22no—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3—41no—
398C22×C3—41no—
399C22×C3—41no—
400C22×C3—41no—
401C22×C3—42no—
402C22×C3—42no—
403C22×C3—41no—
404C22×C3—42no—
405C22×C3#371112yes—
406C22×C3C3×ΓL(1,𝔽25)12yes—
407C22×C3C3×#20812yes—
408C22×C3C2×D3×Dic311yescharacteristic
409C22×C3C2×C3×C4×D311yescharacteristic
410C22×C3C22×C3×Dic311yescharacteristic
411C3×C4—22no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—22no—
418C3×C4—32no—
419C3×C4—32no—
420C3×C4—41no—
421C3×C4—41no—
422C3×C4—42no—
423C3×C4—42no—
424C3×C4—62no—
425C3×C4—62no—
426C3×C4—62no—
427C3×C4—62no—
428C3×C4—61no—
429C3×C4—61no—
430C3×C4—61no—
431C3×C4—61no—
432C3×C4—61no—
433C3×C4—61no—
434C3×C4#380412yes—
435C3×C4C3×#23512yes—
436C3×C4C3×#23712yes—
437Dic3—21no—
438Dic3—21no—
439Dic3—21no—
440Dic3C3×#20812yes—
441C32—21no—
442C32—21no—
443C32—21no—
444C32—41no—
445C32#471911yescharacteristic, complemented
446C32#492611yescharacteristic, complemented
447C32C3×#43211yescharacteristic, complemented
448C23C3×D3×Dic311yescharacteristic
449C2×C4—22no—
450C2×C4—31no—
451C2×C4—31no—
452C2×C4—31no—
453C2×C4—31no—
454C2×C4—62no—
455C2×C4—62no—
456C2×C4C3×D3×Dic311yescharacteristic
457C2×C4C2×C3×D3211yescharacteristic
458C8—62no—
459C8—182no—
460C2×C3—24no—
461C2×C3—22no—
462C2×C3—22no—
463C2×C3—22no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—42no—
474C2×C3—42no—
475C2×C3—44no—
476C2×C3—41no—
477C2×C3—41no—
478C2×C3—41no—
479C2×C3D3×#20811yescharacteristic
480C2×C3C2×#371111yescharacteristic
481C2×C3#6395511yescharacteristic
482C2×C3C3×#87711yescharacteristic
483C2×C3C2×C3×ΓL(1,𝔽25)11yescharacteristic
484C2×C3C3×D3×M1611yescharacteristic
485C2×C3C2×C3×#20811yescharacteristic
486C2×C3C2×C3×#20811yescharacteristic
487C2×C3C3×#92611yescharacteristic
488C22—21no—
489C22—21no—
490C22C3×#371112yes—
491C22C2×C3×D3×Dic311yescharacteristic
492C4—21no—
493C4—32no—
494C4—61no—
495C4—61no—
496C4—61no—
497C4C3×#380412yes—
498C3—21no—
499C3—21no—
500C3—21no—
501C3—41no—
502C3#1059106311yescharacteristic, direct product factor
503C3C3×#471911yescharacteristic, complemented
504C3C3×#492611yescharacteristic, complemented
505C2—22no—
506C2C3×D3×#20811yescharacteristic
507C2C2×C3×#371111yescharacteristic
508C2C3×#6395511yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C22×C4
prime radicalC23×C33C2×C4
cosocleC2×C32×C4C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#432
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
4. C3×Dic3⋊(C3⋊C8)C2semidirect
5. C3×#63554C2semidirect
8. #10591063C3direct (N × H)
34. Dic3⋊(C3⋊C8)C2×C3semidirect
35. #63554C2×C3semidirect
285. C33#432semidirect
445. C32#4719semidirect
446. C32#4926semidirect
447. C32C3×#432semidirect
502. C3#10591063direct (N × H)
503. C3C3×#4719semidirect
504. C3C3×#4926semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4×Dic32
C3×#636572
C32×#8302
C3×Dic3⋊(C3⋊C8)2
C3×#635542
C3×#49263
C3×#47193
#105910633