Finite Group Databasebeta

← C3×#10591242 #13158252

Subgroups — C3×#10591242 #13158252

Order 1728.

1090 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×Dic3C211yesmaximal, characteristic
2C2×C3×Dic3⋊Dic3C212yesmaximal
3C2×C3×Dic32C212yesmaximal
4C3×#63658C211yesmaximal, characteristic
5C32×C3⋊#95C211yesmaximal, characteristic
6C3×C4×C4⋊Dic3—31nomaximal
7C3×#4649—31nomaximal
8#10591242C311yesmaximal, characteristic, direct product factor
9C22×C33×C4C2211yescharacteristic
10C22×C3×C3⋊Dic3C2212yes—
11C22×C32×Dic3C2212yes—
12C22×C32×Dic3C2211yescharacteristic
13C22×C32×Dic3C2211yescharacteristic
14C32×C4×Dic3C414yescomplemented
15C3×Dic3⋊Dic3C418yescomplemented
16C3×Dic32—24no—
17C2×C32×C42—31no—
18C2×C3×C4⋊Dic3—32no—
19C2×C3×Dic3⋊C4—32no—
20C2×C3×C4×Dic3—21no—
21C2×C3×C4×Dic3—32no—
22C2×C3×C4×Dic3—32no—
23C2×C3×C4×Dic3D311yescharacteristic
24C2×C3×C4×Dic3C2×C311yescharacteristic
25C2×Dic3⋊Dic3C2×C312yes—
26C2×Dic32C2×C312yes—
27C32×#95—31no—
28#63658C2×C311yescharacteristic
29C3×C3⋊#95—21no—
30C3×C3⋊#95—31no—
31C3×C3⋊#95—31no—
32C3×C3⋊#95—61no—
33C3×C3⋊#95D311yescharacteristic
34C3×C3⋊#95C2×C311yescharacteristic
35C23×C33C2311yescharacteristic
36C2×C33×C4—22no—
37C2×C33×C4C2×C412yes—
38C2×C3×C3⋊Dic3—22no—
39C2×C3×C3⋊Dic3—22no—
40C2×C3×C3⋊Dic3C2×C414yes—
41C2×C32×Dic3—22no—
42C2×C32×Dic3—22no—
43C2×C32×Dic3C2×C414yes—
44C2×C32×Dic3C2×C414yes—
45C2×C32×Dic3C2×C412yes—
46C2×C32×Dic3C2×C412yes—
47C2×C32×Dic3D412yes—
48C2×C32×Dic3Q812yes—
49C3×C4×C4⋊C4—91no—
50C4×C4⋊Dic3—31no—
51#4649—31no—
52C22×C32×C4—21no—
53C22×C32×C4—21no—
54C22×C32×C4—21no—
55C22×C32×C4—31no—
56C22×C32×C4—31no—
57C22×C32×C4—32no—
58C22×C32×C4—41no—
59C22×C32×C4C2×D311yescharacteristic
60C22×C32×C4C2×D311yescharacteristic
61C22×C32×C4C22×C311yescharacteristic
62C22×C3⋊Dic3C22×C312yes—
63C22×C3×Dic3—21no—
64C22×C3×Dic3—21no—
65C22×C3×Dic3—22no—
66C22×C3×Dic3—32no—
67C22×C3×Dic3—32no—
68C22×C3×Dic3—62no—
69C22×C3×Dic3C2×D312yes—
70C22×C3×Dic3C2×D311yescharacteristic
71C22×C3×Dic3C2×D311yescharacteristic
72C22×C3×Dic3C22×C312yes—
73C22×C3×Dic3C22×C311yescharacteristic
74C22×C3×Dic3C22×C311yescharacteristic
75C32×C42—34no—
76C3×C4⋊Dic3—38no—
77C3×Dic3⋊C4—38no—
78C3×C4×Dic3—24no—
79C3×C4×Dic3—64no—
80C3×C4×Dic3—64no—
81C3×C4×Dic3Dic314yescomplemented
82C3×C4×Dic3C3×C414yescomplemented
83Dic3⋊Dic3C3×C418yescomplemented
84Dic32—24no—
85C22×C33C22×C411yescharacteristic
86C22×C33C22×C412yes—
87C22×C33C2×D411yescharacteristic
88C22×C33C2×Q811yescharacteristic
89C22×C33#4112yes—
90C33×C4—22no—
91C3×C3⋊Dic3—24no—
92C32×Dic3—24no—
93C32×Dic3C4214yescomplemented
94C32×Dic3C4⋊C414yescomplemented
95C2×C3×C4⋊C4—92no—
96C2×C3×C42—31no—
97C2×C3×C42—31no—
98C2×C3×C42—61no—
99C2×C3×C42—92no—
100C2×C4⋊Dic3—32no—
101C2×Dic3⋊C4—32no—
102C2×C4×Dic3—32no—
103C2×C4×Dic3—32no—
104C2×C4×Dic3C3×D311yescharacteristic
105C3×#95—31no—
106C3×#95—31no—
107C3×#95—61no—
108C3×#95—91no—
109C3⋊#95—31no—
110C3⋊#95—31no—
111C3⋊#95—61no—
112C3⋊#95C3×D311yescharacteristic
113C23×C32—21no—
114C23×C32—21no—
115C23×C32—21no—
116C23×C32—41no—
117C23×C32C22×D311yescharacteristic
118C23×C32C22×D311yescharacteristic
119C23×C32C23×C311yescharacteristic
120C2×C32×C4—22no—
121C2×C32×C4—24no—
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—32no—
129C2×C32×C4—32no—
130C2×C32×C4—32no—
131C2×C32×C4—32no—
132C2×C32×C4—34no—
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×C4C4×D312yes—
140C2×C32×C4C2×Dic312yes—
141C2×C32×C4C2×C3×C412yes—
142C2×C3⋊Dic3—22no—
143C2×C3⋊Dic3—22no—
144C2×C3⋊Dic3C2×C3×C414yes—
145C2×C3×Dic3—22no—
146C2×C3×Dic3—22no—
147C2×C3×Dic3—22no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3—22no—
150C2×C3×Dic3—24no—
151C2×C3×Dic3—24no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3—22no—
155C2×C3×Dic3—34no—
156C2×C3×Dic3—34no—
157C2×C3×Dic3—42no—
158C2×C3×Dic3—42no—
159C2×C3×Dic3—64no—
160C2×C3×Dic3—62no—
161C2×C3×Dic3—62no—
162C2×C3×Dic3—62no—
163C2×C3×Dic3—62no—
164C2×C3×Dic3—122no—
165C2×C3×Dic3—122no—
166C2×C3×Dic3Dic612yes—
167C2×C3×Dic3C4×D314yes—
168C2×C3×Dic3C4×D314yes—
169C2×C3×Dic3D1212yes—
170C2×C3×Dic3C2×Dic312yes—
171C2×C3×Dic3C2×Dic312yes—
172C2×C3×Dic3C2×C3×C414yes—
173C2×C3×Dic3C2×C3×C414yes—
174C2×C3×Dic3C2×C3×C412yes—
175C2×C3×Dic3C2×C3×C412yes—
176C2×C3×Dic3C3×D412yes—
177C2×C3×Dic3C3×Q812yes—
178C4×C4⋊C4—91noSylow
179C2×C33C2×C4211yescharacteristic
180C2×C33C2×C4⋊C411yescharacteristic
181C2×C33#11711yescharacteristic
182C2×C33C4×D412yes—
183C2×C33C4×Q812yes—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—31no—
188C22×C3×C4—31no—
189C22×C3×C4—31no—
190C22×C3×C4—31no—
191C22×C3×C4—32no—
192C22×C3×C4—32no—
193C22×C3×C4—41no—
194C22×C3×C4—61no—
195C22×C3×C4—61no—
196C22×C3×C4—62no—
197C22×C3×C4—92no—
198C22×C3×C4D3211yescharacteristic
199C22×C3×C4C2×C3×D311yescharacteristic
200C22×C3×C4C2×C3×D311yescharacteristic
201C22×Dic3—32no—
202C22×Dic3—32no—
203C22×Dic3—62no—
204C22×Dic3C2×C3×D312yes—
205C22×Dic3C2×C3×D311yescharacteristic
206C22×Dic3C2×C3×D311yescharacteristic
207C3×C4⋊C4—98no—
208C3×C42—34no—
209C3×C42—34no—
210C3×C42—64no—
211C3×C42—184no—
212C4⋊Dic3—38no—
213Dic3⋊C4—38no—
214C4×Dic3—64no—
215C4×Dic3—64no—
216C4×Dic3C3×Dic314yescomplemented
217C22×C32—21no—
218C22×C32—22no—
219C22×C32—21no—
220C22×C32—21no—
221C22×C32—22no—
222C22×C32—21no—
223C22×C32—22no—
224C22×C32—22no—
225C22×C32—21no—
226C22×C32—21no—
227C22×C32—22no—
228C22×C32—21no—
229C22×C32—22no—
230C22×C32—21no—
231C22×C32—21no—
232C22×C32—41no—
233C22×C32—42no—
234C22×C32—41no—
235C22×C32—41no—
236C22×C32—42no—
237C22×C32C2×Dic611yescharacteristic
238C22×C32C2×C4×D312yes—
239C22×C32C2×C4×D312yes—
240C22×C32C2×C4×D311yescharacteristic
241C22×C32C2×D1211yescharacteristic
242C22×C32#23512yes—
243C22×C32D3×D411yescharacteristic
244C22×C32#23712yes—
245C22×C32D3×Q811yescharacteristic
246C22×C32C22×Dic311yescharacteristic
247C22×C32C22×C3×C412yes—
248C22×C32C22×C3×C411yescharacteristic
249C22×C32C2×C3×D411yescharacteristic
250C22×C32C2×C3×Q811yescharacteristic
251C22×C32C3×#4112yes—
252C32×C4—22no—
253C32×C4—24no—
254C32×C4—22no—
255C32×C4—22no—
256C32×C4—34no—
257C32×C4—34no—
258C32×C4—42no—
259C32×C4—42no—
260C32×C4—44no—
261C32×C4—64no—
262C3⋊Dic3—24no—
263C3×Dic3—24no—
264C3×Dic3—24no—
265C3×Dic3—24no—
266C3×Dic3—24no—
267C3×Dic3—44no—
268C3×Dic3—64no—
269C3×Dic3—64no—
270C3×Dic3—124no—
271C3×Dic3C4×Dic314yescomplemented
272C3×Dic3C4⋊Dic314yescomplemented
273C3×Dic3C3×C4214yescomplemented
274C3×Dic3C3×C4⋊C414yescomplemented
275C2×C4⋊C4—92no—
276C2×C42—31no—
277C2×C42—92no—
278#95—31no—
279#95—91no—
280C33C4×C4⋊C411yescharacteristic, complemented, Sylow
281C23×C3—21no—
282C23×C3—21no—
283C23×C3—21no—
284C23×C3—41no—
285C23×C3C2×D3211yescharacteristic
286C23×C3C22×C3×D311yescharacteristic
287C23×C3C22×C3×D311yescharacteristic
288C2×C3×C4—24no—
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—32no—
296C2×C3×C4—34no—
297C2×C3×C4—32no—
298C2×C3×C4—32no—
299C2×C3×C4—32no—
300C2×C3×C4—34no—
301C2×C3×C4—32no—
302C2×C3×C4—32no—
303C2×C3×C4—32no—
304C2×C3×C4—32no—
305C2×C3×C4—34no—
306C2×C3×C4—34no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—44no—
310C2×C3×C4—42no—
311C2×C3×C4—62no—
312C2×C3×C4—62no—
313C2×C3×C4—62no—
314C2×C3×C4—64no—
315C2×C3×C4—62no—
316C2×C3×C4—64no—
317C2×C3×C4—62no—
318C2×C3×C4—62no—
319C2×C3×C4—62no—
320C2×C3×C4—62no—
321C2×C3×C4—94no—
322C2×C3×C4—122no—
323C2×C3×C4—122no—
324C2×C3×C4—182no—
325C2×C3×C4—182no—
326C2×C3×C4D3×Dic312yes—
327C2×C3×C4C3×C4×D312yes—
328C2×C3×C4C2×C3×Dic312yes—
329C2×Dic3—22no—
330C2×Dic3—22no—
331C2×Dic3—34no—
332C2×Dic3—34no—
333C2×Dic3—64no—
334C2×Dic3—62no—
335C2×Dic3—62no—
336C2×Dic3—62no—
337C2×Dic3—62no—
338C2×Dic3—122no—
339C2×Dic3—122no—
340C2×Dic3C3×Dic612yes—
341C2×Dic3C3×C4×D314yes—
342C2×Dic3C3×C4×D314yes—
343C2×Dic3C3×D1212yes—
344C2×Dic3C2×C3×Dic312yes—
345C2×Dic3C2×C3×Dic312yes—
346C2×C32—21no—
347C2×C32—22no—
348C2×C32—22no—
349C2×C32—21no—
350C2×C32—21no—
351C2×C32—21no—
352C2×C32—21no—
353C2×C32—22no—
354C2×C32—21no—
355C2×C32—22no—
356C2×C32—21no—
357C2×C32—21no—
358C2×C32—22no—
359C2×C32—21no—
360C2×C32—22no—
361C2×C32—41no—
362C2×C32—41no—
363C2×C32—41no—
364C2×C32—42no—
365C2×C32—42no—
366C2×C32C4×Dic612yes—
367C2×C32C42×D311yescharacteristic
368C2×C32C4×D1212yes—
369C2×C32#87711yescharacteristic
370C2×C32#67⋊C412yes—
371C2×C32#88712yes—
372C2×C32D3×C4⋊C411yescharacteristic
373C2×C32C2×C4×Dic311yescharacteristic
374C2×C32C2×C4⋊Dic311yescharacteristic
375C2×C32#92611yescharacteristic
376C2×C32C2×C3×C4211yescharacteristic
377C2×C32C2×C3×C4⋊C411yescharacteristic
378C2×C32C3×#11711yescharacteristic
379C2×C32C3×C4×D412yes—
380C2×C32C3×C4×Q812yes—
381C22×C4—31no—
382C22×C4—31no—
383C22×C4—32no—
384C22×C4—92no—
385C22×C4C3×D3211yescharacteristic
386C4⋊C4—98no—
387C42—34no—
388C42—184no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—22no—
393C22×C3—22no—
394C22×C3—22no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—22no—
402C22×C3—22no—
403C22×C3—22no—
404C22×C3—41no—
405C22×C3—42no—
406C22×C3—41no—
407C22×C3—41no—
408C22×C3—42no—
409C22×C3D3×Dic611yescharacteristic
410C22×C3C4×D3212yes—
411C22×C3D3×D1211yescharacteristic
412C22×C3C2×D3×Dic311yescharacteristic
413C22×C3#380412yes—
414C22×C3C2×C3×Dic611yescharacteristic
415C22×C3C2×C3×C4×D312yes—
416C22×C3C2×C3×C4×D312yes—
417C22×C3C2×C3×C4×D311yescharacteristic
418C22×C3C2×C3×D1211yescharacteristic
419C22×C3C3×#23512yes—
420C22×C3C3×D3×D411yescharacteristic
421C22×C3C3×#23712yes—
422C22×C3C3×D3×Q811yescharacteristic
423C22×C3C22×C3×Dic311yescharacteristic
424C3×C4—24no—
425C3×C4—22no—
426C3×C4—22no—
427C3×C4—22no—
428C3×C4—34no—
429C3×C4—34no—
430C3×C4—34no—
431C3×C4—34no—
432C3×C4—42no—
433C3×C4—42no—
434C3×C4—44no—
435C3×C4—64no—
436C3×C4—64no—
437C3×C4—64no—
438C3×C4—64no—
439C3×C4—124no—
440C3×C4—184no—
441Dic3—24no—
442Dic3—64no—
443Dic3—64no—
444Dic3—124no—
445Dic3C3×C4×Dic314yescomplemented
446Dic3C3×C4⋊Dic314yescomplemented
447C32—21no—
448C32—21no—
449C32—21no—
450C32—41no—
451C32#464911yescharacteristic, complemented
452C32C4×C4⋊Dic311yescharacteristic, complemented
453C32C3×C4×C4⋊C411yescharacteristic, complemented
454C23C2×C3×D3211yescharacteristic
455C2×C4—22no—
456C2×C4—32no—
457C2×C4—34no—
458C2×C4—32no—
459C2×C4—32no—
460C2×C4—32no—
461C2×C4—34no—
462C2×C4—62no—
463C2×C4—62no—
464C2×C4—94no—
465C2×C4—182no—
466C2×C4—182no—
467C2×C4C3×D3×Dic312yes—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—22no—
472C2×C3—22no—
473C2×C3—22no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—22no—
478C2×C3—22no—
479C2×C3—22no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—41no—
484C2×C3—42no—
485C2×C3—41no—
486C2×C3—42no—
487C2×C3—41no—
488C2×C3#6383712yes—
489C2×C3C4×D3×Dic311yescharacteristic
490C2×C3#6388012yes—
491C2×C3D3×C4⋊Dic311yescharacteristic
492C2×C3#6395511yescharacteristic
493C2×C3C3×C4×Dic612yes—
494C2×C3C3×C42×D311yescharacteristic
495C2×C3C3×C4×D1212yes—
496C2×C3C3×#87711yescharacteristic
497C2×C3C3×#67⋊C412yes—
498C2×C3C3×#88712yes—
499C2×C3C3×D3×C4⋊C411yescharacteristic
500C2×C3C2×C3×C4×Dic311yescharacteristic
501C2×C3C2×C3×C4⋊Dic311yescharacteristic
502C2×C3C3×#92611yescharacteristic
503C22C3×D3×Dic611yescharacteristic
504C22C3×C4×D3212yes—
505C22C3×D3×D1211yescharacteristic
506C22C2×C3×D3×Dic311yescharacteristic
507C22C3×#380412yes—
508C4—22no—
509C4—34no—
510C4—34no—
511C4—64no—
512C4—184no—
513C3—21no—
514C3—21no—
515C3—21no—
516C3—41no—
517C3#1059124211yescharacteristic, direct product factor
518C3C3×#464911yescharacteristic, complemented
519C3C3×C4×C4⋊Dic311yescharacteristic, complemented
520C2C3×#6383712yes—
521C2C3×C4×D3×Dic311yescharacteristic
522C2C3×#6388012yes—
523C2C3×D3×C4⋊Dic311yescharacteristic
524C2C3×#6395511yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutatorC2×C32C2×C3×C42
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C23
cosocleC23×C32C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×C4×C4⋊C4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
8. #10591242C3direct (N × H)
14. C32×C4×Dic3C4semidirect
15. C3×Dic3⋊Dic3C4semidirect
81. C3×C4×Dic3Dic3semidirect
82. C3×C4×Dic3C3×C4semidirect
83. Dic3⋊Dic3C3×C4semidirect
93. C32×Dic3C42semidirect
94. C32×Dic3C4⋊C4semidirect
216. C4×Dic3C3×Dic3semidirect
271. C3×Dic3C4×Dic3semidirect
272. C3×Dic3C4⋊Dic3semidirect
273. C3×Dic3C3×C42semidirect
274. C3×Dic3C3×C4⋊C4semidirect
280. C33C4×C4⋊C4semidirect
445. Dic3C3×C4×Dic3semidirect
446. Dic3C3×C4⋊Dic3semidirect
451. C32#4649semidirect
452. C32C4×C4⋊Dic3semidirect
453. C32C3×C4×C4⋊C4semidirect
517. C3#10591242direct (N × H)
518. C3C3×#4649semidirect
519. C3C3×C4×C4⋊Dic3semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4×Dic32
C2×C3×Dic3⋊Dic32
C2×C3×Dic322
C3×#636582
C32×C3⋊#952
C3×C4×C4⋊Dic33
C3×#46493
#105912423