Finite Group Databasebeta

← C3×#10591216 #13158305

Subgroups — C3×#10591216 #13158305

Order 1728.

1018 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3752C211yesmaximal, characteristic
2C2×C3×#3724C211yesmaximal, characteristic, complemented
3C2×C3×#3722C212yesmaximal
4C32×C3⋊#95C212yesmaximal, complemented
5C3×#63579C211yesmaximal, characteristic, complemented
6C3×#4675—32nomaximal
7#10591216C311yesmaximal, characteristic, direct product factor
8C23×C3×C3⋊D3C2211yescharacteristic
9C22×C33×C4C2211yescharacteristic
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C32×Dic3C2212yes—
12C22×C32×Dic3C2212yes—
13C3×#3752—22no—
14C3×#3724—22no—
15C3×#3722—24no—
16C2×#3752C2×C311yescharacteristic
17C2×C3×(C2×D3)⋊C4—32no—
18C2×C3×(C2×D3)⋊C4—32no—
19C2×C3×(C2×D3)⋊C4—32no—
20C2×C3×(C2×D3)⋊C4—61no—
21C2×C3×Dic3⋊C4—32no—
22C2×#3724C2×C311yescharacteristic, complemented
23C2×#3722C2×C312yes—
24C32×#95—32no—
25C3×C3⋊#95—22no—
26C3×C3⋊#95—32no—
27C3×C3⋊#95D312yescomplemented
28C3×C3⋊#95C2×C312yescomplemented
29#63579C2×C311yescharacteristic, complemented
30C23×C33C2311yescharacteristic
31C22×C3×C3⋊D3—21no—
32C22×C3×C3⋊D3—21no—
33C22×C3×C3⋊D3—22no—
34C22×C3×C3⋊D3—22no—
35C22×C3×C3⋊D3D412yes—
36C2×C33×C4—21no—
37C2×C33×C4—22no—
38C2×C3×C3⋊Dic3—22no—
39C2×C3×C3⋊Dic3D412yes—
40C2×C32×Dic3—22no—
41C2×C32×Dic3—22no—
42C2×C32×Dic3—22no—
43C2×C32×Dic3—22no—
44C2×C32×Dic3—22no—
45C2×C32×Dic3—22no—
46C3×#397—91no—
47#4675—32no—
48C23×C3⋊D3C22×C311yescharacteristic
49C23×C3×D3—32no—
50C23×C3×D3—61no—
51C22×C32×C4—21no—
52C22×C32×C4—22no—
53C22×C32×C4—32no—
54C22×C32×C4—32no—
55C22×C32×C4—41no—
56C22×C32×C4C2×D312yes—
57C22×C32×C4C22×C311yescharacteristic
58C22×C3⋊Dic3C22×C311yescharacteristic
59C22×C3×Dic3—22no—
60C22×C3×Dic3—22no—
61C22×C3×Dic3—32no—
62C22×C3×Dic3—61no—
63C22×C3×Dic3C2×D312yes—
64C22×C3×Dic3C2×D312yes—
65C22×C3×Dic3C22×C312yes—
66C22×C3×Dic3C22×C312yes—
67#3752—22no—
68C3×(C2×D3)⋊C4—64no—
69C3×(C2×D3)⋊C4—64no—
70C3×(C2×D3)⋊C4—64no—
71C3×(C2×D3)⋊C4—64no—
72C3×Dic3⋊C4—64no—
73#3724—22no—
74#3722—24no—
75C22×C33C2×D411yescharacteristic
76C22×C33C2×D411yescharacteristic
77C22×C33#4111yescharacteristic
78C22×C33#4112yes—
79C22×C33#4112yes—
80C2×C3×C3⋊D3—22no—
81C2×C3×C3⋊D3—24no—
82C2×C3×C3⋊D3—41no—
83C2×C3×C3⋊D3—41no—
84C2×C3×C3⋊D3—42no—
85C33×C4—41no—
86C3×C3⋊Dic3—22no—
87C32×Dic3—42no—
88C32×Dic3—42no—
89C2×C3×C4⋊C4—91no—
90C2×C3×C22⋊C4—92no—
91C2×C3×C22⋊C4—91no—
92C2×(C2×D3)⋊C4—32no—
93C2×(C2×D3)⋊C4—32no—
94C2×(C2×D3)⋊C4—32no—
95C2×(C2×D3)⋊C4—61no—
96C2×Dic3⋊C4—32no—
97C3×#95—32no—
98C3×#95—32no—
99C3×#95—62no—
100C3×#95—91no—
101C3⋊#95—32no—
102C3⋊#95C3×D312yescomplemented
103C23×C32—21no—
104C23×C32—22no—
105C23×C32—41no—
106C23×C32C22×D312yes—
107C23×C32C23×C311yescharacteristic
108C22×C3⋊D3—21no—
109C22×C3⋊D3—21no—
110C22×C3⋊D3—22no—
111C22×C3⋊D3—22no—
112C22×C3⋊D3C3×D412yes—
113C22×C3×D3—34no—
114C22×C3×D3—62no—
115C22×C3×D3—62no—
116C22×C3×D3—62no—
117C22×C3×D3—62no—
118C22×C3×D3—62no—
119C22×C3×D3—62no—
120C22×C3×D3—62no—
121C22×C3×D3—62no—
122C22×C3×D3—62no—
123C22×C3×D3—122no—
124C22×C3×D3—122no—
125C2×C32×C4—22no—
126C2×C32×C4—21no—
127C2×C32×C4—22no—
128C2×C32×C4—24no—
129C2×C32×C4—22no—
130C2×C32×C4—22no—
131C2×C32×C4—22no—
132C2×C32×C4—42no—
133C2×C32×C4—42no—
134C2×C32×C4—42no—
135C2×C32×C4—42no—
136C2×C32×C4—44no—
137C2×C32×C4—62no—
138C2×C32×C4—62no—
139C2×C32×C4—62no—
140C2×C32×C4—62no—
141C2×C32×C4—62no—
142C2×C32×C4—62no—
143C2×C3⋊Dic3—22no—
144C2×C3⋊Dic3C3×D412yes—
145C2×C3×Dic3—24no—
146C2×C3×Dic3—24no—
147C2×C3×Dic3—22no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3—22no—
150C2×C3×Dic3—22no—
151C2×C3×Dic3—22no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3—22no—
155C2×C3×Dic3—22no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—22no—
159C2×C3×Dic3—34no—
160C2×C3×Dic3—42no—
161C2×C3×Dic3—42no—
162C2×C3×Dic3—42no—
163C2×C3×Dic3—42no—
164C2×C3×Dic3—62no—
165C2×C3×Dic3—62no—
166C2×C3×Dic3—62no—
167C2×C3×Dic3—122no—
168#397—91noSylow
169C2×C33#12111yescharacteristic
170C2×C33#12312yes—
171C2×C33#12311yescharacteristic
172C2×C33#12411yescharacteristic
173C2×C33#12612yes—
174C3×C3⋊D3—42no—
175C24×C3—91no—
176C23×D3—32no—
177C23×D3—61no—
178C22×C3×C4—22no—
179C22×C3×C4—21no—
180C22×C3×C4—32no—
181C22×C3×C4—32no—
182C22×C3×C4—32no—
183C22×C3×C4—32no—
184C22×C3×C4—41no—
185C22×C3×C4—62no—
186C22×C3×C4—62no—
187C22×C3×C4—91no—
188C22×C3×C4D3211yescharacteristic
189C22×C3×C4C2×C3×D312yes—
190C22×Dic3—32no—
191C22×Dic3—61no—
192C22×Dic3C2×C3×D312yes—
193C22×Dic3C2×C3×D312yes—
194C3×C4⋊C4—182no—
195C3×C22⋊C4—182no—
196C3×C22⋊C4—184no—
197(C2×D3)⋊C4—64no—
198(C2×D3)⋊C4—64no—
199(C2×D3)⋊C4—64no—
200(C2×D3)⋊C4—64no—
201Dic3⋊C4—64no—
202C22×C32—21no—
203C22×C32—22no—
204C22×C32—21no—
205C22×C32—22no—
206C22×C32—21no—
207C22×C32—22no—
208C22×C32—22no—
209C22×C32—22no—
210C22×C32—22no—
211C22×C32—22no—
212C22×C32—22no—
213C22×C32—22no—
214C22×C32—41no—
215C22×C32—41no—
216C22×C32—42no—
217C22×C32—41no—
218C22×C32—42no—
219C22×C32#23512yes—
220C22×C32#23512yes—
221C22×C32#23512yes—
222C22×C32D3×D412yes—
223C22×C32D3×D412yes—
224C22×C32#23712yes—
225C22×C32#23912yes—
226C22×C32C2×C3×D411yescharacteristic
227C22×C32C2×C3×D411yescharacteristic
228C22×C32C3×#4111yescharacteristic
229C22×C32C3×#4112yes—
230C22×C32C3×#4112yes—
231C2×C3⋊D3—22no—
232C2×C3⋊D3—24no—
233C2×C3⋊D3—41no—
234C2×C3⋊D3—41no—
235C2×C3⋊D3—42no—
236C2×C3×D3—64no—
237C2×C3×D3—64no—
238C2×C3×D3—64no—
239C2×C3×D3—64no—
240C2×C3×D3—124no—
241C2×C3×D3—124no—
242C2×C3×D3—122no—
243C2×C3×D3—122no—
244C2×C3×D3—122no—
245C2×C3×D3—122no—
246C2×C3×D3—122no—
247C2×C3×D3—122no—
248C32×C4—44no—
249C32×C4—42no—
250C32×C4—44no—
251C32×C4—42no—
252C32×C4—41no—
253C32×C4—122no—
254C32×C4—122no—
255C3⋊Dic3—22no—
256C3×Dic3—44no—
257C3×Dic3—44no—
258C3×Dic3—42no—
259C3×Dic3—42no—
260C3×Dic3—42no—
261C3×Dic3—42no—
262C3×Dic3—64no—
263C3×Dic3—122no—
264C2×C4⋊C4—91no—
265C2×C22⋊C4—92no—
266C2×C22⋊C4—91no—
267#95—32no—
268#95—91no—
269C33#39711yescharacteristic, complemented, Sylow
270C23×C3—22no—
271C23×C3—21no—
272C23×C3—41no—
273C23×C3—92no—
274C23×C3—181no—
275C23×C3—182no—
276C23×C3—182no—
277C23×C3—181no—
278C23×C3C2×D3211yescharacteristic
279C23×C3C22×C3×D312yes—
280C22×D3—34no—
281C22×D3—62no—
282C22×D3—62no—
283C22×D3—62no—
284C22×D3—62no—
285C22×D3—62no—
286C22×D3—62no—
287C22×D3—62no—
288C22×D3—62no—
289C22×D3—62no—
290C22×D3—122no—
291C22×D3—122no—
292C2×C3×C4—22no—
293C2×C3×C4—24no—
294C2×C3×C4—22no—
295C2×C3×C4—21no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—44no—
303C2×C3×C4—42no—
304C2×C3×C4—64no—
305C2×C3×C4—64no—
306C2×C3×C4—62no—
307C2×C3×C4—62no—
308C2×C3×C4—62no—
309C2×C3×C4—62no—
310C2×C3×C4—62no—
311C2×C3×C4—62no—
312C2×C3×C4—62no—
313C2×C3×C4—62no—
314C2×C3×C4—62no—
315C2×C3×C4—62no—
316C2×C3×C4—62no—
317C2×C3×C4—62no—
318C2×C3×C4—92no—
319C2×C3×C4—122no—
320C2×C3×C4—122no—
321C2×C3×C4—122no—
322C2×C3×C4—122no—
323C2×C3×C4—182no—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—22no—
330C2×Dic3—34no—
331C2×Dic3—62no—
332C2×Dic3—62no—
333C2×Dic3—62no—
334C2×Dic3—122no—
335C2×C32—22no—
336C2×C32—22no—
337C2×C32—22no—
338C2×C32—22no—
339C2×C32—22no—
340C2×C32—22no—
341C2×C32—22no—
342C2×C32—22no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—22no—
347C2×C32—41no—
348C2×C32—42no—
349C2×C32—41no—
350C2×C32—41no—
351C2×C32—42no—
352C2×C32#87612yes—
353C2×C32#88312yes—
354C2×C32#88412yes—
355C2×C32#88512yes—
356C2×C32#89412yes—
357C2×C32#89412yes—
358C2×C32#89812yes—
359C2×C32C3×#12111yescharacteristic
360C2×C32C3×#12312yes—
361C2×C32C3×#12311yescharacteristic
362C2×C32C3×#12411yescharacteristic
363C2×C32C3×#12612yes—
364C3⋊D3—42no—
365C3×D3—124no—
366C3×D3—124no—
367C24—91no—
368C22×C4—32no—
369C22×C4—32no—
370C22×C4—91no—
371C22×C4C3×D3211yescharacteristic
372C4⋊C4—182no—
373C22⋊C4—182no—
374C22⋊C4—184no—
375C22×C3—22no—
376C22×C3—21no—
377C22×C3—22no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3—22no—
381C22×C3—21no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—22no—
385C22×C3—21no—
386C22×C3—22no—
387C22×C3—41no—
388C22×C3—42no—
389C22×C3—41no—
390C22×C3—42no—
391C22×C3—41no—
392C22×C3—184no—
393C22×C3—182no—
394C22×C3—362no—
395C22×C3—361no—
396C22×C3—361no—
397C22×C3#379811yescharacteristic
398C22×C3#379912yes—
399C22×C3#380211yescharacteristic
400C22×C3#380412yes—
401C22×C3#381111yescharacteristic
402C22×C3C3×#23512yes—
403C22×C3C3×#23512yes—
404C22×C3C3×#23512yes—
405C22×C3C3×D3×D412yes—
406C22×C3C3×D3×D412yes—
407C22×C3C3×#23712yes—
408C22×C3C3×#23912yes—
409C2×D3—64no—
410C2×D3—64no—
411C2×D3—64no—
412C2×D3—64no—
413C2×D3—124no—
414C2×D3—122no—
415C2×D3—124no—
416C2×D3—122no—
417C2×D3—122no—
418C2×D3—122no—
419C2×D3—122no—
420C2×D3—122no—
421C3×C4—42no—
422C3×C4—44no—
423C3×C4—44no—
424C3×C4—42no—
425C3×C4—41no—
426C3×C4—124no—
427C3×C4—124no—
428C3×C4—122no—
429C3×C4—122no—
430C3×C4—122no—
431C3×C4—122no—
432C3×C4—182no—
433Dic3—42no—
434Dic3—42no—
435Dic3—64no—
436Dic3—122no—
437C32—22no—
438C32—21no—
439C32—41no—
440C32#467512yescomplemented
441C32C3×#39711yescharacteristic, complemented
442C23—92no—
443C23—181no—
444C23—182no—
445C23—182no—
446C23—181no—
447C23C2×C3×D3211yescharacteristic
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—62no—
451C2×C4—62no—
452C2×C4—62no—
453C2×C4—62no—
454C2×C4—62no—
455C2×C4—62no—
456C2×C4—92no—
457C2×C4—182no—
458C2×C3—21no—
459C2×C3—22no—
460C2×C3—22no—
461C2×C3—22no—
462C2×C3—22no—
463C2×C3—22no—
464C2×C3—22no—
465C2×C3—22no—
466C2×C3—21no—
467C2×C3—22no—
468C2×C3—22no—
469C2×C3—21no—
470C2×C3—41no—
471C2×C3—42no—
472C2×C3—42no—
473C2×C3—41no—
474C2×C3—41no—
475C2×C3—362no—
476C2×C3#6385412yes—
477C2×C3#6386812yes—
478C2×C3#6389711yescharacteristic
479C2×C3#6391211yescharacteristic
480C2×C3#6395311yescharacteristic
481C2×C3C3×#87612yes—
482C2×C3C3×#88312yes—
483C2×C3C3×#88412yes—
484C2×C3C3×#88512yes—
485C2×C3C3×#89412yes—
486C2×C3C3×#89412yes—
487C2×C3C3×#89812yes—
488D3—124no—
489D3—124no—
490C22—184no—
491C22—182no—
492C22—361no—
493C22—362no—
494C22—361no—
495C22C3×#379811yescharacteristic
496C22C3×#379912yes—
497C22C3×#380211yescharacteristic
498C22C3×#380412yes—
499C22C3×#381111yescharacteristic
500C4—41no—
501C4—122no—
502C4—122no—
503C4—182no—
504C3—21no—
505C3—22no—
506C3—41no—
507C3#1059121611yescharacteristic, direct product factor
508C3C3×#467512yescomplemented
509C2—362no—
510C2C3×#6385412yes—
511C2C3×#6386812yes—
512C2C3×#6389711yescharacteristic
513C2C3×#6391211yescharacteristic
514C2C3×#6395311yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutator, cosocleC23×C32C23×C3
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC23×C3×C3⋊D3C22
hypercenterC22×C3×C4D32
hypocenterC32C3×#397
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3724C2semidirect
4. C32×C3⋊#95C2semidirect
5. C3×#63579C2semidirect
7. #10591216C3direct (N × H)
22. C2×#3724C2×C3semidirect
27. C3×C3⋊#95D3semidirect
28. C3×C3⋊#95C2×C3semidirect
29. #63579C2×C3semidirect
102. C3⋊#95C3×D3semidirect
269. C33#397semidirect
440. C32#4675semidirect
441. C32C3×#397semidirect
507. C3#10591216direct (N × H)
508. C3C3×#4675semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37522
C2×C3×#37242
C2×C3×#37222
C32×C3⋊#952
C3×#635792
C3×#46753
#105912163