Finite Group Databasebeta

← C3×#71851 #11792790

Subgroups — C3×#71851 #11792790

Order 1152.

1449 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#534C211yesmaximal, characteristic, complemented
2C2×C3×#937C212yesmaximal, complemented
3C2×C3×#834C212yesmaximal, complemented
4C3×#4955C211yesmaximal, characteristic, complemented
5C3×#4526C211yesmaximal, characteristic, complemented
6C3×#2021—31nomaximal
7#71851C311yesmaximal, characteristic, direct product factor
8C24×C3×D3C2211yescharacteristic
9C22×C32×D4C2212yes—
10C22×C3×#67—22no—
11C32×#123—22no—
12C32×#121—22no—
13C32×#120—22no—
14C2×C32×C22⋊C4C2211yescharacteristic
15C2×C3×#217C2211yescharacteristic
16C2×C3×#217C2212yes—
17C3×#937—24no—
18C3×#937—24no—
19C2×C3×(C2×D3)⋊C4—22no—
20C2×C3×(C2×D3)⋊C4—22no—
21C3×#834—24no—
22C3×#534—21no—
23C3×#534D311yescharacteristic, complemented
24C3×#534C2×C311yescharacteristic, complemented
25C2×C3×#120—32no—
26C2×#937C2×C312yescomplemented
27C2×C3×#99—32no—
28C3×#379—31no—
29C2×#834C2×C312yescomplemented
30#4955C2×C311yescharacteristic, complemented
31C3×#342—31no—
32#4526C2×C311yescharacteristic, complemented
33C24×C32D412yescomplemented
34C24×C32C2311yescharacteristic
35C23×C3×D3—21no—
36C23×C3×D3—22no—
37C23×C3×D3—22no—
38C23×C3×D3—21no—
39C23×C3×D3—44no—
40C23×C3×D3D412yescomplemented
41C2×C32×D4—22no—
42C2×C32×D4—22no—
43C2×C32×D4—22no—
44C2×C32×D4—22no—
45C2×C32×D4—22no—
46C2×C32×D4—22no—
47C2×C32×D4—22no—
48C22×C32×C4D412yescomplemented
49C22×C32×C4D412yescomplemented
50C2×C3×#67—28no—
51C2×C3×#67—48no—
52C22×C3×Dic3—22no—
53C22×C3×Dic3—21no—
54C32×C4⋊C4—22no—
55C32×C22⋊C4—24no—
56C32×C22⋊C4—22no—
57C3×#217—22no—
58C3×#217—22no—
59C3×#217D414yescomplemented
60C3×(C2×D3)⋊C4—28no—
61C3×(C2×D3)⋊C4—44no—
62#2021—31noSylow
63C25×C3—31no—
64C24×D3C22×C311yescharacteristic
65C22×C3×D4—22no—
66C22×C3×D4—62no—
67C22×C3×D4C2×D312yes—
68C22×C3×D4C22×C312yes—
69C22×#67—22no—
70C3×#123—24no—
71C3×#123—22no—
72C3×#123—22no—
73C3×#121—24no—
74C3×#121—22no—
75C3×#121—22no—
76C3×#120—24no—
77C3×#120—22no—
78C3×#120—22no—
79C3×#120—64no—
80C3×#120—64no—
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—32no—
83C2×C3×C22⋊C4—31no—
84C2×C3×C22⋊C4—62no—
85C2×C3×C22⋊C4—62no—
86C2×C3×C22⋊C4C2×D311yescharacteristic
87C2×C3×C22⋊C4C22×C311yescharacteristic
88C2×#217C22×C311yescharacteristic
89C2×#217C22×C312yes—
90#937—24no—
91#937—24no—
92C2×(C2×D3)⋊C4—22no—
93C2×(C2×D3)⋊C4—22no—
94C3×#99—64no—
95#834—24no—
96C23×C32—22no—
97C23×C32—22no—
98C23×C32—22no—
99C23×C32—42no—
100C23×C32—44no—
101C23×C32C2×D411yescharacteristic
102C23×C32C2×D412yes—
103C23×C32C2×D412yes—
104C23×C32C2×D411yescharacteristic
105C23×C32#4111yescharacteristic
106C22×C3×D3—24no—
107C22×C3×D3—28no—
108C22×C3×D3—24no—
109C22×C3×D3—22no—
110C22×C3×D3—44no—
111C22×C3×D3—48no—
112C22×C3×D3—44no—
113C22×C3×D3—42no—
114C22×C3×D3—44no—
115C22×C3×D3—41no—
116C22×C3×D3—42no—
117C22×C3×D3—41no—
118C32×D4—42no—
119C32×D4—42no—
120C32×D4—42no—
121C32×D4—42no—
122C32×D4—82no—
123C2×C32×C4—22no—
124C2×C32×C4—22no—
125C2×C32×C4—22no—
126C2×C32×C4—22no—
127C2×C32×C4—42no—
128C3×#67—416no—
129C2×C3×Dic3—24no—
130C2×C3×Dic3—44no—
131C2×C3×Dic3—42no—
132C2×C3×Dic3—41no—
133#534C3×D311yescharacteristic, complemented
134C2×#120—32no—
135C2×#99—32no—
136#379—31no—
137#342—31no—
138C24×C3—22no—
139C24×C3—21no—
140C24×C3—32no—
141C24×C3—61no—
142C24×C3—62no—
143C24×C3—62no—
144C24×C3—61no—
145C24×C3—124no—
146C24×C3#6712yescomplemented
147C24×C3C3×D412yescomplemented
148C24×C3C22×D311yescharacteristic
149C24×C3C23×C311yescharacteristic
150C23×D3—21no—
151C23×D3—22no—
152C23×D3—22no—
153C23×D3—21no—
154C23×D3—44no—
155C23×D3C3×D412yescomplemented
156C2×C3×D4—24no—
157C2×C3×D4—24no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—24no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—42no—
174C2×C3×D4—42no—
175C2×C3×D4—42no—
176C2×C3×D4—42no—
177C2×C3×D4—68no—
178C2×C3×D4—128no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—62no—
182C22×C3×C4—61no—
183C22×C3×C4#6712yescomplemented
184C22×C3×C4#6712yescomplemented
185C22×C3×C4C3×D412yescomplemented
186C22×C3×C4C3×D412yescomplemented
187C2×#67—28no—
188C2×#67—48no—
189C22×Dic3—22no—
190C22×Dic3—21no—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—22no—
194C3×C22⋊C4—24no—
195C3×C22⋊C4—28no—
196C3×C22⋊C4—24no—
197C3×C22⋊C4—24no—
198C3×C22⋊C4—22no—
199C3×C22⋊C4—22no—
200C3×C22⋊C4—34no—
201C3×C22⋊C4—68no—
202C3×C22⋊C4—62no—
203C3×C22⋊C4—62no—
204C3×C22⋊C4—124no—
205#217—22no—
206#217—22no—
207#217C3×D414yescomplemented
208(C2×D3)⋊C4—28no—
209(C2×D3)⋊C4—44no—
210C22×C32—22no—
211C22×C32—24no—
212C22×C32—42no—
213C22×C32—44no—
214C22×C32—41no—
215C22×C32—42no—
216C22×C32—41no—
217C22×C32—44no—
218C22×C32—82no—
219C22×C32#12012yes—
220C22×C32#12011yescharacteristic
221C22×C32#12111yescharacteristic
222C22×C32#12112yes—
223C22×C32#12711yescharacteristic
224C2×C3×D3—28no—
225C2×C3×D3—24no—
226C2×C3×D3—44no—
227C2×C3×D3—44no—
228C2×C3×D3—42no—
229C2×C3×D3—48no—
230C2×C3×D3—42no—
231C2×C3×D3—44no—
232C32×C4—42no—
233C32×C4—42no—
234C3×Dic3—44no—
235C3×Dic3—81no—
236C25—31no—
237C22×D4—62no—
238C22×D4C2×C3×D312yes—
239#123—22no—
240#121—22no—
241#120—22no—
242#120—64no—
243#120—64no—
244C2×C22⋊C4—32no—
245C2×C22⋊C4—31no—
246C2×C22⋊C4—62no—
247C2×C22⋊C4—62no—
248C2×C22⋊C4C2×C3×D311yescharacteristic
249#99—64no—
250C23×C3—24no—
251C23×C3—22no—
252C23×C3—22no—
253C23×C3—22no—
254C23×C3—22no—
255C23×C3—22no—
256C23×C3—22no—
257C23×C3—21no—
258C23×C3—22no—
259C23×C3—21no—
260C23×C3—22no—
261C23×C3—21no—
262C23×C3—42no—
263C23×C3—42no—
264C23×C3—42no—
265C23×C3—42no—
266C23×C3—44no—
267C23×C3—44no—
268C23×C3—64no—
269C23×C3—68no—
270C23×C3—64no—
271C23×C3—62no—
272C23×C3—82no—
273C23×C3—84no—
274C23×C3—122no—
275C23×C3—124no—
276C23×C3—124no—
277C23×C3—128no—
278C23×C3—124no—
279C23×C3—122no—
280C23×C3—121no—
281C23×C3—121no—
282C23×C3D3×D411yescharacteristic
283C23×C3D3×D412yes—
284C23×C3#23711yescharacteristic
285C23×C3C2×#6712yes—
286C23×C3C2×#6711yescharacteristic
287C23×C3C2×C3×D411yescharacteristic
288C23×C3C2×C3×D412yes—
289C23×C3C2×C3×D412yes—
290C23×C3C2×C3×D411yescharacteristic
291C23×C3C3×#4111yescharacteristic
292C22×D3—24no—
293C22×D3—28no—
294C22×D3—24no—
295C22×D3—22no—
296C22×D3—44no—
297C22×D3—48no—
298C22×D3—44no—
299C22×D3—42no—
300C22×D3—44no—
301C22×D3—41no—
302C22×D3—42no—
303C22×D3—41no—
304C3×D4—44no—
305C3×D4—44no—
306C3×D4—44no—
307C3×D4—44no—
308C3×D4—42no—
309C3×D4—42no—
310C3×D4—42no—
311C3×D4—42no—
312C3×D4—42no—
313C3×D4—42no—
314C3×D4—42no—
315C3×D4—42no—
316C3×D4—84no—
317C3×D4—82no—
318C3×D4—82no—
319C3×D4—1216no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—64no—
335C2×C3×C4—82no—
336C2×C3×C4—124no—
337C2×C3×C4—122no—
338C2×C3×C4—121no—
339#67—416no—
340C2×Dic3—24no—
341C2×Dic3—44no—
342C2×Dic3—42no—
343C2×Dic3—41no—
344C2×C32—22no—
345C2×C32—21no—
346C2×C32—21no—
347C2×C32—22no—
348C2×C32—82no—
349C2×C32#39211yescharacteristic
350C2×C32B(4,𝔽2)12yes—
351C3×D3—44no—
352C24—32no—
353C24—61no—
354C24—62no—
355C24—62no—
356C24—61no—
357C24—124no—
358C24C3×#6712yescomplemented
359C24C22×C3×D311yescharacteristic
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—68no—
368C2×D4—128no—
369C22×C4—62no—
370C22×C4—61no—
371C22×C4C3×#6712yescomplemented
372C22×C4C3×#6712yescomplemented
373C4⋊C4—22no—
374C22⋊C4—24no—
375C22⋊C4—22no—
376C22⋊C4—34no—
377C22⋊C4—68no—
378C22⋊C4—62no—
379C22⋊C4—62no—
380C22⋊C4—124no—
381C22×C3—22no—
382C22×C3—24no—
383C22×C3—22no—
384C22×C3—24no—
385C22×C3—22no—
386C22×C3—21no—
387C22×C3—22no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—42no—
391C22×C3—44no—
392C22×C3—41no—
393C22×C3—42no—
394C22×C3—41no—
395C22×C3—41no—
396C22×C3—42no—
397C22×C3—41no—
398C22×C3—44no—
399C22×C3—44no—
400C22×C3—48no—
401C22×C3—44no—
402C22×C3—42no—
403C22×C3—44no—
404C22×C3—42no—
405C22×C3—44no—
406C22×C3—68no—
407C22×C3—64no—
408C22×C3—81no—
409C22×C3—82no—
410C22×C3—81no—
411C22×C3—84no—
412C22×C3—82no—
413C22×C3—82no—
414C22×C3—84no—
415C22×C3—124no—
416C22×C3—124no—
417C22×C3—122no—
418C22×C3—128no—
419C22×C3—122no—
420C22×C3—124no—
421C22×C3#93712yes—
422C22×C3#93811yescharacteristic
423C22×C3#93912yes—
424C22×C3#94011yescharacteristic
425C22×C3#95311yescharacteristic
426C22×C3C3×#12012yes—
427C22×C3C3×#12011yescharacteristic
428C22×C3C3×#12111yescharacteristic
429C22×C3C3×#12112yes—
430C22×C3C3×#12711yescharacteristic
431C2×D3—28no—
432C2×D3—24no—
433C2×D3—44no—
434C2×D3—44no—
435C2×D3—42no—
436C2×D3—48no—
437C2×D3—42no—
438C2×D3—44no—
439C3×C4—44no—
440C3×C4—44no—
441C3×C4—42no—
442C3×C4—42no—
443C3×C4—42no—
444C3×C4—42no—
445C3×C4—124no—
446C3×C4—241no—
447Dic3—44no—
448Dic3—81no—
449C32#202111yescharacteristic, complemented, Sylow
450C23—22no—
451C23—22no—
452C23—22no—
453C23—42no—
454C23—44no—
455C23—64no—
456C23—68no—
457C23—64no—
458C23—62no—
459C23—122no—
460C23—124no—
461C23—124no—
462C23—128no—
463C23—124no—
464C23—122no—
465C23—121no—
466C23—121no—
467C23C3×D3×D411yescharacteristic
468C23C3×D3×D412yes—
469C23C3×#23711yescharacteristic
470C23C2×C3×#6712yes—
471C23C2×C3×#6711yescharacteristic
472D4—42no—
473D4—42no—
474D4—42no—
475D4—42no—
476D4—82no—
477D4—1216no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—42no—
483C2×C4—64no—
484C2×C4—124no—
485C2×C4—122no—
486C2×C4—121no—
487C2×C3—22no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—22no—
491C2×C3—22no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—22no—
495C2×C3—21no—
496C2×C3—22no—
497C2×C3—42no—
498C2×C3—42no—
499C2×C3—41no—
500C2×C3—41no—
501C2×C3—84no—
502C2×C3—82no—
503C2×C3—82no—
504C2×C3—124no—
505C2×C3#522311yescharacteristic
506C2×C3#524412yes—
507C2×C3C3×#39211yescharacteristic
508C2×C3C3×B(4,𝔽2)12yes—
509D3—44no—
510C22—22no—
511C22—24no—
512C22—41no—
513C22—42no—
514C22—41no—
515C22—44no—
516C22—42no—
517C22—44no—
518C22—68no—
519C22—64no—
520C22—82no—
521C22—124no—
522C22—124no—
523C22—122no—
524C22—128no—
525C22—122no—
526C22—124no—
527C22C3×#93712yes—
528C22C3×#93811yescharacteristic
529C22C3×#93912yes—
530C22C3×#94011yescharacteristic
531C22C3×#95311yescharacteristic
532C4—42no—
533C4—42no—
534C4—124no—
535C4—241no—
536C3—21no—
537C3#7185111yescharacteristic, direct product factor
538C3C3×#202111yescharacteristic, complemented
539C2—22no—
540C2—21no—
541C2—21no—
542C2—22no—
543C2—82no—
544C2—124no—
545C2C3×#522311yescharacteristic
546C2C3×#524412yes—
subgroupG/H
centerC22×C3#953
commutator, cosocleC24×C3C23×C3
FrattiniC24C22×C3×D3
FittingC32×#534C2
socleC22×C32#120
hypercenterC3×#534D3
hypocenterC3C3×#2021
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ C3×#534

Complements

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

NH ≅ G/Ntype
1. C32×#534C2semidirect
2. C2×C3×#937C2semidirect
3. C2×C3×#834C2semidirect
4. C3×#4955C2semidirect
5. C3×#4526C2semidirect
7. #71851C3direct (N × H)
23. C3×#534D3semidirect
24. C3×#534C2×C3semidirect
26. C2×#937C2×C3semidirect
29. C2×#834C2×C3semidirect
30. #4955C2×C3semidirect
32. #4526C2×C3semidirect
33. C24×C32D4semidirect
40. C23×C3×D3D4semidirect
48. C22×C32×C4D4semidirect
49. C22×C32×C4D4semidirect
59. C3×#217D4semidirect
133. #534C3×D3semidirect
146. C24×C3#67semidirect
147. C24×C3C3×D4semidirect
155. C23×D3C3×D4semidirect
183. C22×C3×C4#67semidirect
184. C22×C3×C4#67semidirect
185. C22×C3×C4C3×D4semidirect
186. C22×C3×C4C3×D4semidirect
207. #217C3×D4semidirect
358. C24C3×#67semidirect
371. C22×C4C3×#67semidirect
372. C22×C4C3×#67semidirect
449. C32#2021semidirect
537. C3#71851direct (N × H)
538. C3C3×#2021semidirect

Maximal subgroups

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

subgroupindex
C32×#5342
C2×C3×#9372
C2×C3×#8342
C3×#49552
C3×#45262
C3×#20213
#718513