Finite Group Databasebeta

← C3×#10591213 #13158075

Subgroups — C3×#10591213 #13158075

Order 1728.

1154 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#634C211yesmaximal, characteristic
2C2×C32×C4⋊Dic3C211yesmaximal, characteristic
3C2×C32×C4×Dic3C211yesmaximal, characteristic
4C3×#63579C214yesmaximal
5C3×#4973—31nomaximal
6C3×#4929—31nomaximal
7#10591213C311yesmaximal, characteristic, direct product factor
8C22×C33×C4C2211yescharacteristic
9C22×C3×C3⋊Dic3C2212yes—
10C2×C3×#634—22no—
11C2×C3×#634—22no—
12C2×C3×#634C414yescomplemented
13C22×C32×Dic3C2212yes—
14C22×C32×Dic3C2212yes—
15C32×C4⋊Dic3—22no—
16C32×C4×Dic3—22no—
17C22×#634C2×C311yescharacteristic
18C22×C3×Dic6—31no—
19C22×C3×Dic6—31no—
20C22×C3×Dic6—61no—
21C2×C32×C4⋊C4—31no—
22C2×C32×C42—31no—
23C2×C3×C4⋊Dic3—21no—
24C2×C3×C4⋊Dic3D311yescharacteristic
25C2×C3×C4⋊Dic3C2×C311yescharacteristic
26C2×C3×C4×Dic3—21no—
27C2×C3×C4×Dic3D311yescharacteristic
28C2×C3×C4×Dic3C2×C311yescharacteristic
29C3×C3⋊#95—34no—
30C3×C3⋊#95—34no—
31#63579C2×C314yes—
32C23×C33C2311yescharacteristic
33C2×C33×C4C2×C412yes—
34C2×C33×C4D412yes—
35C2×C33×C4D412yes—
36C2×C3×C3⋊Dic3—22no—
37C2×C3×C3⋊Dic3—22no—
38C2×C3×C3⋊Dic3C2×C414yes—
39C3×#634—28no—
40C2×C32×Dic3—22no—
41C2×C32×Dic3—22no—
42C2×C32×Dic3—22no—
43C2×C32×Dic3—24no—
44C2×C32×Dic3Q814yescomplemented
45C3×#391—91no—
46#4973—31no—
47#4929—31no—
48C22×C32×C4—21no—
49C22×C32×C4—21no—
50C22×C32×C4—21no—
51C22×C32×C4—32no—
52C22×C32×C4—32no—
53C22×C32×C4—41no—
54C22×C32×C4C2×D311yescharacteristic
55C22×C32×C4C2×D311yescharacteristic
56C22×C32×C4C22×C311yescharacteristic
57C22×C3⋊Dic3C22×C312yes—
58C2×#634—22no—
59C2×#634—22no—
60C2×#634C3×C414yescomplemented
61C22×C3×Dic3—22no—
62C22×C3×Dic3—22no—
63C22×C3×Dic3—32no—
64C22×C3×Dic3—32no—
65C22×C3×Dic3—62no—
66C22×C3×Dic3C2×D312yes—
67C22×C3×Dic3C2×D312yes—
68C22×C3×Dic3C22×C312yes—
69C22×C3×Dic3C22×C312yes—
70C2×C3×Dic6—34no—
71C2×C3×Dic6—34no—
72C2×C3×Dic6—64no—
73C2×C3×Dic6—64no—
74C2×C3×Dic6—64no—
75C2×C3×Dic6—62no—
76C2×C3×Dic6—62no—
77C2×C3×Dic6—62no—
78C2×C3×Dic6—62no—
79C32×C4⋊C4—62no—
80C32×C42—62no—
81C3×C4⋊Dic3—24no—
82C3×C4⋊Dic3—22no—
83C3×C4⋊Dic3—22no—
84C3×C4×Dic3—24no—
85C3×C4×Dic3—22no—
86C3×C4×Dic3—22no—
87C22×C33C22×C411yescharacteristic
88C22×C33C2×D411yescharacteristic
89C22×C33C2×D411yescharacteristic
90C22×C33C2×Q812yes—
91C22×C33#4112yes—
92C33×C4C22⋊C414yes—
93C3×C3⋊Dic3—24no—
94C32×Dic3—24no—
95C32×Dic3—42no—
96C22×C3×Q8—91no—
97C22×Dic6—31no—
98C22×Dic6—31no—
99C22×Dic6—61no—
100C2×C3×C4⋊C4—31no—
101C2×C3×C4⋊C4—31no—
102C2×C3×C4⋊C4—61no—
103C2×C3×C42—31no—
104C2×C3×C42—31no—
105C2×C3×C42—61no—
106C2×C4⋊Dic3C3×D311yescharacteristic
107C2×C4×Dic3C3×D311yescharacteristic
108C3×#95—94no—
109C3⋊#95—34no—
110C3⋊#95—34no—
111C23×C32—21no—
112C23×C32—21no—
113C23×C32—21no—
114C23×C32—41no—
115C23×C32C22×D311yescharacteristic
116C23×C32C22×D311yescharacteristic
117C23×C32C23×C311yescharacteristic
118C2×C32×C4—22no—
119C2×C32×C4—22no—
120C2×C32×C4—22no—
121C2×C32×C4—22no—
122C2×C32×C4—22no—
123C2×C32×C4—22no—
124C2×C32×C4—22no—
125C2×C32×C4—22no—
126C2×C32×C4—22no—
127C2×C32×C4—34no—
128C2×C32×C4—42no—
129C2×C32×C4—42no—
130C2×C32×C4—42no—
131C2×C32×C4—62no—
132C2×C32×C4—62no—
133C2×C32×C4—62no—
134C2×C32×C4—64no—
135C2×C32×C4C4×D312yes—
136C2×C32×C4C4×D312yes—
137C2×C32×C4D1212yes—
138C2×C32×C4D1212yes—
139C2×C32×C4#6712yes—
140C2×C32×C4#6712yes—
141C2×C32×C4C2×C3×C412yes—
142C2×C32×C4C3×D412yes—
143C2×C32×C4C3×D412yes—
144C2×C3⋊Dic3—22no—
145C2×C3⋊Dic3—22no—
146C2×C3⋊Dic3C2×C3×C414yes—
147#634—28no—
148C2×C3×Dic3—24no—
149C2×C3×Dic3—24no—
150C2×C3×Dic3—22no—
151C2×C3×Dic3—22no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3—24no—
155C2×C3×Dic3—24no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—24no—
159C2×C3×Dic3—24no—
160C2×C3×Dic3—34no—
161C2×C3×Dic3—34no—
162C2×C3×Dic3—44no—
163C2×C3×Dic3—64no—
164C2×C3×Dic3—64no—
165C2×C3×Dic3—64no—
166C2×C3×Dic3—62no—
167C2×C3×Dic3—62no—
168C2×C3×Dic3—62no—
169C2×C3×Dic3—62no—
170C2×C3×Dic3Dic614yescomplemented
171C2×C3×Dic3C3×Q814yescomplemented
172C3×Dic6—616no—
173C3×Dic6—68no—
174C3×Dic6—68no—
175#391—91noSylow
176C2×C33C2×C22⋊C411yescharacteristic
177C2×C33C4×Q812yes—
178C2×C33C22⋊Q812yes—
179C2×C33#12411yescharacteristic
180C2×C33C4⋊Q811yescharacteristic
181C2×C3×Q8—94no—
182C2×C3×Q8—182no—
183C2×C3×Q8—182no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—32no—
188C22×C3×C4—32no—
189C22×C3×C4—32no—
190C22×C3×C4—32no—
191C22×C3×C4—41no—
192C22×C3×C4—62no—
193C22×C3×C4—62no—
194C22×C3×C4—92no—
195C22×C3×C4D3211yescharacteristic
196C22×C3×C4C2×C3×D311yescharacteristic
197C22×C3×C4C2×C3×D311yescharacteristic
198C22×Dic3—32no—
199C22×Dic3—32no—
200C22×Dic3—62no—
201C22×Dic3C2×C3×D312yes—
202C22×Dic3C2×C3×D312yes—
203C2×Dic6—34no—
204C2×Dic6—34no—
205C2×Dic6—64no—
206C2×Dic6—64no—
207C2×Dic6—64no—
208C2×Dic6—62no—
209C2×Dic6—62no—
210C2×Dic6—62no—
211C2×Dic6—62no—
212C3×C4⋊C4—64no—
213C3×C4⋊C4—62no—
214C3×C4⋊C4—62no—
215C3×C42—64no—
216C3×C42—62no—
217C3×C42—62no—
218C4⋊Dic3—22no—
219C4×Dic3—22no—
220C22×C32—22no—
221C22×C32—22no—
222C22×C32—21no—
223C22×C32—21no—
224C22×C32—21no—
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—41no—
236C22×C32—41no—
237C22×C32—41no—
238C22×C32—42no—
239C22×C32—42no—
240C22×C32C2×Dic612yes—
241C22×C32C2×C4×D311yescharacteristic
242C22×C32C2×C4×D311yescharacteristic
243C22×C32C2×D1211yescharacteristic
244C22×C32C2×D1211yescharacteristic
245C22×C32#23512yes—
246C22×C32#23712yes—
247C22×C32D3×Q812yes—
248C22×C32C2×#6711yescharacteristic
249C22×C32C2×#6711yescharacteristic
250C22×C32C22×C3×C411yescharacteristic
251C22×C32C2×C3×D411yescharacteristic
252C22×C32C2×C3×D411yescharacteristic
253C22×C32C2×C3×Q812yes—
254C22×C32C3×#4112yes—
255C32×C4—24no—
256C32×C4—24no—
257C32×C4—24no—
258C32×C4—44no—
259C32×C4—64no—
260C32×C4—122no—
261C32×C4(C2×D3)⋊C414yes—
262C32×C4(C2×D3)⋊C414yes—
263C32×C4C3×C22⋊C414yes—
264C3⋊Dic3—24no—
265C3×Dic3—28no—
266C3×Dic3—24no—
267C3×Dic3—24no—
268C3×Dic3—44no—
269C3×Dic3—42no—
270C3×Dic3—42no—
271C3×Dic3—68no—
272C3×Dic3—64no—
273C3×Dic3—64no—
274C22×Q8—91no—
275C2×C4⋊C4—31no—
276C2×C42—31no—
277#95—94no—
278C33#39111yescharacteristic, complemented, Sylow
279C23×C3—21no—
280C23×C3—21no—
281C23×C3—21no—
282C23×C3—41no—
283C23×C3C2×D3211yescharacteristic
284C23×C3C22×C3×D311yescharacteristic
285C23×C3C22×C3×D311yescharacteristic
286C3×Q8—188no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—34no—
297C2×C3×C4—34no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—64no—
302C2×C3×C4—64no—
303C2×C3×C4—64no—
304C2×C3×C4—62no—
305C2×C3×C4—62no—
306C2×C3×C4—62no—
307C2×C3×C4—62no—
308C2×C3×C4—64no—
309C2×C3×C4—64no—
310C2×C3×C4—62no—
311C2×C3×C4—62no—
312C2×C3×C4—64no—
313C2×C3×C4—94no—
314C2×C3×C4—124no—
315C2×C3×C4—182no—
316C2×C3×C4—182no—
317C2×C3×C4#62412yes—
318C2×C3×C4#62612yes—
319C2×C3×C4#62612yes—
320C2×C3×C4C3×C4×D312yes—
321C2×C3×C4C3×C4×D312yes—
322C2×C3×C4C3×D1212yes—
323C2×C3×C4C3×D1212yes—
324C2×C3×C4C3×#6712yes—
325C2×C3×C4C3×#6712yes—
326C2×Dic3—22no—
327C2×Dic3—22no—
328C2×Dic3—24no—
329C2×Dic3—22no—
330C2×Dic3—34no—
331C2×Dic3—34no—
332C2×Dic3—64no—
333C2×Dic3—64no—
334C2×Dic3—64no—
335C2×Dic3—62no—
336C2×Dic3—62no—
337C2×Dic3—62no—
338C2×Dic3—62no—
339C2×Dic3C3×Dic614yescomplemented
340Dic6—616no—
341Dic6—68no—
342Dic6—68no—
343C2×C32—22no—
344C2×C32—22no—
345C2×C32—22no—
346C2×C32—21no—
347C2×C32—21no—
348C2×C32—21no—
349C2×C32—21no—
350C2×C32—22no—
351C2×C32—22no—
352C2×C32—21no—
353C2×C32—21no—
354C2×C32—21no—
355C2×C32—22no—
356C2×C32—21no—
357C2×C32—21no—
358C2×C32—41no—
359C2×C32—42no—
360C2×C32—42no—
361C2×C32—41no—
362C2×C32—41no—
363C2×C32C4×Dic612yes—
364C2×C32#86911yescharacteristic
365C2×C32#87511yescharacteristic
366C2×C32#88712yes—
367C2×C32#89712yes—
368C2×C32#92412yes—
369C2×C32C2×(C2×D3)⋊C411yescharacteristic
370C2×C32C2×(C2×D3)⋊C411yescharacteristic
371C2×C32#93611yescharacteristic
372C2×C32#94411yescharacteristic
373C2×C32C2×C3×C22⋊C411yescharacteristic
374C2×C32C3×C4×Q812yes—
375C2×C32C3×C22⋊Q812yes—
376C2×C32C3×#12411yescharacteristic
377C2×C32C3×C4⋊Q811yescharacteristic
378C2×Q8—94no—
379C2×Q8—182no—
380C2×Q8—182no—
381C22×C4—32no—
382C22×C4—32no—
383C22×C4—92no—
384C22×C4C3×D3211yescharacteristic
385C4⋊C4—62no—
386C42—62no—
387C22×C3—22no—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—22no—
392C22×C3—22no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—42no—
403C22×C3—42no—
404C22×C3—41no—
405C22×C3—41no—
406C22×C3—41no—
407C22×C3D3×Dic612yes—
408C22×C3#379512yes—
409C22×C3C2×#62411yescharacteristic
410C22×C3C2×#62611yescharacteristic
411C22×C3C2×#62611yescharacteristic
412C22×C3C2×C3×Dic612yes—
413C22×C3C2×C3×C4×D311yescharacteristic
414C22×C3C2×C3×C4×D311yescharacteristic
415C22×C3C2×C3×D1211yescharacteristic
416C22×C3C2×C3×D1211yescharacteristic
417C22×C3C3×#23512yes—
418C22×C3C3×#23712yes—
419C22×C3C3×D3×Q812yes—
420C22×C3C2×C3×#6711yescharacteristic
421C22×C3C2×C3×#6711yescharacteristic
422C3×C4—24no—
423C3×C4—24no—
424C3×C4—24no—
425C3×C4—44no—
426C3×C4—68no—
427C3×C4—64no—
428C3×C4—64no—
429C3×C4—124no—
430C3×C4—122no—
431C3×C4—122no—
432C3×C4—184no—
433C3×C4#372214yes—
434C3×C4C3×(C2×D3)⋊C414yes—
435C3×C4C3×(C2×D3)⋊C414yes—
436Dic3—24no—
437Dic3—42no—
438Dic3—68no—
439Dic3—64no—
440Dic3—64no—
441C32—21no—
442C32—21no—
443C32—21no—
444C32—41no—
445C32#492911yescharacteristic, complemented
446C32#497311yescharacteristic, complemented
447C32C3×#39111yescharacteristic, complemented
448C23C2×C3×D3211yescharacteristic
449Q8—188no—
450C2×C4—34no—
451C2×C4—62no—
452C2×C4—62no—
453C2×C4—64no—
454C2×C4—62no—
455C2×C4—94no—
456C2×C4—182no—
457C2×C4—182no—
458C2×C4C3×#62412yes—
459C2×C4C3×#62612yes—
460C2×C4C3×#62612yes—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—22no—
465C2×C3—22no—
466C2×C3—22no—
467C2×C3—22no—
468C2×C3—22no—
469C2×C3—22no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—41no—
477C2×C3—42no—
478C2×C3—42no—
479C2×C3—41no—
480C2×C3—41no—
481C2×C3#6384412yes—
482C2×C3#6385712yes—
483C2×C3#6386011yescharacteristic
484C2×C3#6386611yescharacteristic
485C2×C3C2×#372211yescharacteristic
486C2×C3C3×C4×Dic612yes—
487C2×C3C3×#86911yescharacteristic
488C2×C3C3×#87511yescharacteristic
489C2×C3C3×#88712yes—
490C2×C3C3×#89712yes—
491C2×C3C3×#92412yes—
492C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
493C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
494C2×C3C3×#93611yescharacteristic
495C2×C3C3×#94411yescharacteristic
496C22C3×D3×Dic612yes—
497C22C3×#379512yes—
498C22C2×C3×#62411yescharacteristic
499C22C2×C3×#62611yescharacteristic
500C22C2×C3×#62611yescharacteristic
501C4—64no—
502C4—122no—
503C4—184no—
504C4C3×#372214yes—
505C3—21no—
506C3—21no—
507C3—21no—
508C3—41no—
509C3#1059121311yescharacteristic, direct product factor
510C3C3×#492911yescharacteristic, complemented
511C3C3×#497311yescharacteristic, complemented
512C2C3×#6384412yes—
513C2C3×#6385712yes—
514C2C3×#6386011yescharacteristic
515C2C3×#6386611yescharacteristic
516C2C2×C3×#372211yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C23
cosocleC23×C32C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#391
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
7. #10591213C3direct (N × H)
12. C2×C3×#634C4semidirect
44. C2×C32×Dic3Q8semidirect
60. C2×#634C3×C4semidirect
170. C2×C3×Dic3Dic6semidirect
171. C2×C3×Dic3C3×Q8semidirect
278. C33#391semidirect
339. C2×Dic3C3×Dic6semidirect
445. C32#4929semidirect
446. C32#4973semidirect
447. C32C3×#391semidirect
509. C3#10591213direct (N × H)
510. C3C3×#4929semidirect
511. C3C3×#4973semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#6342
C2×C32×C4⋊Dic32
C2×C32×C4×Dic32
C3×#635792
C3×#49733
C3×#49293
#105912133