Finite Group Databasebeta

← C3×#78640 #11864193

Subgroups — C3×#78640 #11864193

Order 1152.

1603 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#235C211yesmaximal, characteristic
2C2×C3×#924C211yesmaximal, characteristic, complemented
3C2×C3×#909C212yesmaximal, complemented
4C32×#407C211yesmaximal, characteristic, complemented
5C3×#5133C212yesmaximal, complemented
6C3×#4739C212yesmaximal, complemented
7C3×#4733C214yesmaximal, complemented
8C3×#4731C212yesmaximal, complemented
9C3×#3007—31nomaximal
10#78640C311yesmaximal, characteristic, direct product factor
11C23×C32×C4C2211yescharacteristic
12C22×C3×#67—21no—
13C2×C3×#235—21no—
14C2×C3×#235—21no—
15C2×C3×#235—28no—
16C2×C3×#235C2212yes—
17C2×C3×#235C2212yes—
18C22×C3×D12C2211yescharacteristic
19C22×C3×C4×D3—21no—
20C22×C3×Dic6C2211yescharacteristic
21C22×C3×Dic6C2211yescharacteristic, complemented
22C2×C32×M16C2212yescomplemented
23C2×C3×#217—21no—
24C2×C3×C4⋊Dic3C2211yescharacteristic, complemented
25C3×#924—21no—
26C3×#924—21no—
27C3×#924C2212yes—
28C3×#924C2212yescomplemented
29C2×C3×Dic3⋊C4—21no—
30C3×#909—28no—
31C2×C3×Dic12C2214yescomplemented
32C2×C3×#204C2214yescomplemented
33C32×C22⋊C8C2212yes—
34C32×C22⋊C8C2212yescomplemented
35C3×#821C2214yescomplemented
36C3×#816C2214yescomplemented
37C22×C3×#41—31no—
38C22×#235C2×C311yescharacteristic
39C2×C3×#137—32no—
40C2×C3×C22⋊Q8—31no—
41C2×#924C2×C311yescharacteristic, complemented
42C2×#909C2×C312yescomplemented
43C3×#452—34no—
44C3×#451—32no—
45C3×#450—32no—
46C3×#417—32no—
47C3×#407—21no—
48C3×#407D311yescharacteristic, complemented
49C3×#407C2×C311yescharacteristic, complemented
50#5133C2×C312yescomplemented
51#4739C2×C312yescomplemented
52#4733C2×C314yescomplemented
53#4731C2×C312yescomplemented
54C24×C32D411yescharacteristic
55C23×C3×D3—21no—
56C22×C32×C4—24no—
57C22×C32×C4D411yescharacteristic
58C22×C32×C4D411yescharacteristic
59C22×C32×C4D411yescharacteristic
60C22×C32×C4C2311yescharacteristic
61C22×C32×C4C2311yescharacteristic
62C22×C32×C4C2311yescharacteristic
63C2×C3×#67—28no—
64C2×C3×#67—22no—
65C2×C3×#67—22no—
66C22×C3×Dic3—21no—
67C22×C3×Dic3—21no—
68C22×C3×Dic3—21no—
69C3×#235—28no—
70C3×#235—28no—
71C3×#235—24no—
72C3×#235—24no—
73C3×#235—24no—
74C3×#235D418yes—
75C2×C3×D12—21no—
76C2×C3×D12—24no—
77C2×C3×D12C2312yes—
78C2×C3×C4×D3—28no—
79C2×C3×C4×D3—22no—
80C2×C3×C4×D3—22no—
81C2×C3×Dic6—21no—
82C2×C3×Dic6—24no—
83C2×C3×Dic6—24no—
84C2×C3×Dic6—21no—
85C2×C3×Dic6C2312yes—
86C2×C3×Dic6C2312yes—
87C32×M16—24no—
88C2×C32×C8C2314yes—
89C3×#217—22no—
90C3×#217—22no—
91C3×C4⋊Dic3—21no—
92C3×C4⋊Dic3C2312yes—
93C3×Dic3⋊C4—22no—
94C3×Dic3⋊C4—22no—
95C3×Dic12—28no—
96C3×#204—28no—
97#3007—31noSylow
98C2×C3×#41—32no—
99C2×C3×#41—32no—
100C2×C3×#41—68no—
101C2×C3×#41—61no—
102C2×C3×#41—61no—
103C22×C3×Q8—31no—
104C22×C3×Q8—31no—
105C22×C3×D4—31no—
106C22×C3×D4—61no—
107C23×C3×C4—21no—
108C23×C3×C4—61no—
109C23×C3×C4C2×D311yescharacteristic
110C23×C3×C4C22×C311yescharacteristic
111C22×#67—21no—
112C2×#235—21no—
113C2×#235—21no—
114C2×#235—28no—
115C2×#235C22×C312yes—
116C2×#235C22×C312yes—
117C22×D12C22×C311yescharacteristic
118C22×C4×D3—21no—
119C22×Dic6C22×C311yescharacteristic
120C22×Dic6C22×C311yescharacteristic, complemented
121C3×#137—68no—
122C2×C3×Q16—34no—
123C2×C3×SD16—34no—
124C2×C3×M16—22no—
125C2×C3×M16C2×D312yescomplemented
126C2×C3×M16C22×C312yescomplemented
127C3×C22⋊Q8—32no—
128C3×C22⋊Q8—32no—
129C3×C22⋊Q8—61no—
130C3×C22⋊Q8—61no—
131C2×C3×C4⋊C4—31no—
132C2×C3×C4⋊C4—61no—
133C2×C3×C22⋊C4—61no—
134C2×#217—21no—
135C2×C4⋊Dic3C22×C311yescharacteristic, complemented
136#924—21no—
137#924—21no—
138#924C22×C312yes—
139#924C22×C312yescomplemented
140C2×Dic3⋊C4—21no—
141#909—28no—
142C2×Dic12C22×C314yescomplemented
143C2×#204C22×C314yescomplemented
144C3×#103—34no—
145C3×#102—34no—
146C3×C22⋊C8—22no—
147C3×C22⋊C8—22no—
148C3×C22⋊C8C2×D312yes—
149C3×C22⋊C8C2×D312yescomplemented
150C3×C22⋊C8C22×C312yes—
151C3×C22⋊C8C22×C312yescomplemented
152#821C22×C314yescomplemented
153#816C22×C314yescomplemented
154C23×C32—22no—
155C23×C32—22no—
156C23×C32—41no—
157C23×C32C2×D411yescharacteristic
158C23×C32C2×D411yescharacteristic
159C23×C32C2×D411yescharacteristic
160C22×C3×D3—24no—
161C22×C3×D3—22no—
162C22×C3×D3—42no—
163C22×C3×D3—42no—
164C2×C32×C4—22no—
165C2×C32×C4—22no—
166C2×C32×C4—22no—
167C2×C32×C4—22no—
168C2×C32×C4—22no—
169C2×C32×C4C2×D411yescharacteristic
170C2×C32×C4C2×D411yescharacteristic
171C2×C32×C4C2×D411yescharacteristic
172C2×C32×C4C2×D414yes—
173C2×C32×C4C2411yescharacteristic
174C3×#67—28no—
175C3×#67—28no—
176C2×C3×Dic3—24no—
177C2×C3×Dic3—22no—
178C2×C3×Dic3—22no—
179C2×C3×Dic3—22no—
180C2×C3×Dic3—42no—
181C2×C3×Dic3—42no—
182C3×D12—24no—
183C3×D12—22no—
184C3×D12C2×D414yes—
185C3×C4×D3—28no—
186C3×C4×D3—28no—
187C3×Dic6—24no—
188C3×Dic6—22no—
189C3×Dic6—24no—
190C3×Dic6—42no—
191C3×Dic6C2×D414yes—
192C32×C8—24no—
193C22×#41—31no—
194C2×#137—32no—
195C2×C22⋊Q8—31no—
196#452—34no—
197#451—32no—
198#450—32no—
199#417—32no—
200#407C3×D311yescharacteristic, complemented
201C24×C3—21no—
202C24×C3—61no—
203C24×C3D1211yescharacteristic
204C24×C3C3×D411yescharacteristic
205C23×D3—21no—
206C3×#41—38no—
207C3×#41—64no—
208C3×#41—64no—
209C3×#41—68no—
210C3×#41—68no—
211C3×#41—64no—
212C2×C3×Q8—32no—
213C2×C3×Q8—32no—
214C2×C3×Q8—64no—
215C2×C3×Q8—61no—
216C2×C3×Q8—61no—
217C2×C3×Q8—64no—
218C2×C3×D4—32no—
219C2×C3×D4—64no—
220C2×C3×D4—68no—
221C2×C3×D4—62no—
222C2×C3×D4—62no—
223C2×C3×D4—61no—
224C22×C3×C4—24no—
225C22×C3×C4—24no—
226C22×C3×C4—21no—
227C22×C3×C4—21no—
228C22×C3×C4—21no—
229C22×C3×C4—21no—
230C22×C3×C4—21no—
231C22×C3×C4—21no—
232C22×C3×C4—44no—
233C22×C3×C4—68no—
234C22×C3×C4—61no—
235C22×C3×C4—62no—
236C22×C3×C4—62no—
237C22×C3×C4—61no—
238C22×C3×C4—61no—
239C22×C3×C4D1211yescharacteristic
240C22×C3×C4D1211yescharacteristic
241C22×C3×C4D1211yescharacteristic
242C22×C3×C4C3×D411yescharacteristic
243C22×C3×C4C3×D411yescharacteristic
244C22×C3×C4C3×D411yescharacteristic
245C22×C3×C4C22×D311yescharacteristic
246C22×C3×C4C22×D311yescharacteristic
247C22×C3×C4C22×D311yescharacteristic
248C22×C3×C4C23×C311yescharacteristic
249C22×C3×C4C23×C311yescharacteristic
250C22×C3×C4C23×C311yescharacteristic
251C2×#67—28no—
252C2×#67—22no—
253C2×#67—22no—
254C22×Dic3—21no—
255C22×Dic3—21no—
256C22×Dic3—21no—
257#235—28no—
258#235—28no—
259#235—24no—
260#235—24no—
261#235—24no—
262#235C3×D418yes—
263C2×D12—21no—
264C2×D12—24no—
265C2×D12C23×C312yes—
266C2×C4×D3—28no—
267C2×C4×D3—22no—
268C2×C4×D3—22no—
269C2×Dic6—21no—
270C2×Dic6—24no—
271C2×Dic6—24no—
272C2×Dic6—21no—
273C2×Dic6C23×C312yes—
274C2×Dic6C23×C312yes—
275C3×Q16—68no—
276C3×SD16—68no—
277C3×M16—24no—
278C3×M16—24no—
279C3×M16—44no—
280C2×C3×C8—24no—
281C2×C3×C8C22×D314yes—
282C2×C3×C8C23×C314yes—
283C3×C4⋊C4—32no—
284C3×C4⋊C4—62no—
285C3×C4⋊C4—62no—
286C3×C4⋊C4—61no—
287C3×C22⋊C4—62no—
288C3×C22⋊C4—62no—
289#217—22no—
290#217—22no—
291C4⋊Dic3—21no—
292C4⋊Dic3C23×C312yes—
293Dic3⋊C4—22no—
294Dic3⋊C4—22no—
295Dic12—28no—
296#204—28no—
297C22×C32—21no—
298C22×C32—21no—
299C22×C32—21no—
300C22×C32—21no—
301C22×C32—21no—
302C22×C32—21no—
303C22×C32—21no—
304C22×C32—44no—
305C22×C32#12012yes—
306C22×C32#13712yescomplemented
307C22×C32C22×D411yescharacteristic
308C2×C3×D3—24no—
309C2×C3×D3—48no—
310C2×C3×D3—42no—
311C2×C3×D3—42no—
312C32×C4—21no—
313C32×C4—21no—
314C32×C4#12012yes—
315C32×C4C22×D412yes—
316C3×Dic3—24no—
317C3×Dic3—42no—
318C3×Dic3—42no—
319C2×#41—32no—
320C2×#41—32no—
321C2×#41—68no—
322C2×#41—61no—
323C2×#41—61no—
324C22×Q8—31no—
325C22×Q8—31no—
326C22×D4—31no—
327C22×D4—61no—
328C23×C4—61no—
329C23×C4C2×C3×D311yescharacteristic
330#137—68no—
331C2×Q16—34no—
332C2×SD16—34no—
333C2×M16C2×C3×D312yescomplemented
334C22⋊Q8—32no—
335C22⋊Q8—32no—
336C22⋊Q8—61no—
337C22⋊Q8—61no—
338C2×C4⋊C4—31no—
339C2×C4⋊C4—61no—
340C2×C22⋊C4—61no—
341#103—34no—
342#102—34no—
343C22⋊C8C2×C3×D312yes—
344C22⋊C8C2×C3×D312yescomplemented
345C23×C3—22no—
346C23×C3—22no—
347C23×C3—22no—
348C23×C3—22no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—42no—
353C23×C3—42no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—42no—
357C23×C3—64no—
358C23×C3—62no—
359C23×C3—122no—
360C23×C3—122no—
361C23×C3C2×D1211yescharacteristic
362C23×C3C2×D1211yescharacteristic
363C23×C3C2×D1211yescharacteristic
364C23×C3C2×C3×D411yescharacteristic
365C23×C3C2×C3×D411yescharacteristic
366C23×C3C2×C3×D411yescharacteristic
367C22×D3—24no—
368C22×D3—22no—
369C22×D3—42no—
370C22×D3—42no—
371C3×Q8—34no—
372C3×Q8—64no—
373C3×Q8—62no—
374C3×Q8—64no—
375C3×Q8—122no—
376C3×D4—34no—
377C3×D4—64no—
378C3×D4—68no—
379C3×D4—68no—
380C3×D4—62no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—24no—
396C2×C3×C4—42no—
397C2×C3×C4—42no—
398C2×C3×C4—42no—
399C2×C3×C4—42no—
400C2×C3×C4—42no—
401C2×C3×C4—68no—
402C2×C3×C4—64no—
403C2×C3×C4—62no—
404C2×C3×C4—68no—
405C2×C3×C4—62no—
406C2×C3×C4—62no—
407C2×C3×C4—122no—
408C2×C3×C4—122no—
409C2×C3×C4C2×D1211yescharacteristic
410C2×C3×C4C2×D1211yescharacteristic
411C2×C3×C4C2×D1211yescharacteristic
412C2×C3×C4D3×D414yes—
413C2×C3×C4C2×C3×D411yescharacteristic
414C2×C3×C4C2×C3×D411yescharacteristic
415C2×C3×C4C2×C3×D411yescharacteristic
416C2×C3×C4C2×C3×D414yes—
417C2×C3×C4C23×D311yescharacteristic
418C2×C3×C4C24×C311yescharacteristic
419#67—28no—
420#67—28no—
421C2×Dic3—24no—
422C2×Dic3—22no—
423C2×Dic3—22no—
424C2×Dic3—22no—
425C2×Dic3—42no—
426C2×Dic3—42no—
427D12—24no—
428D12—22no—
429D12C2×C3×D414yes—
430C4×D3—28no—
431C4×D3—28no—
432Dic6—24no—
433Dic6—22no—
434Dic6—24no—
435Dic6—42no—
436Dic6C2×C3×D414yes—
437C3×C8—24no—
438C3×C8—24no—
439C3×C8—44no—
440C2×C32—22no—
441C2×C32—22no—
442C2×C32—41no—
443C2×C32C2×#12011yescharacteristic
444C2×C32C2×#13711yescharacteristic
445C2×C32#57511yescharacteristic
446C3×D3—44no—
447C24—61no—
448C24C3×D1211yescharacteristic
449#41—38no—
450#41—64no—
451#41—64no—
452#41—68no—
453#41—68no—
454#41—64no—
455C2×Q8—32no—
456C2×Q8—32no—
457C2×Q8—64no—
458C2×Q8—61no—
459C2×Q8—61no—
460C2×Q8—64no—
461C2×D4—32no—
462C2×D4—64no—
463C2×D4—68no—
464C2×D4—62no—
465C2×D4—62no—
466C2×D4—61no—
467C22×C4—24no—
468C22×C4—68no—
469C22×C4—61no—
470C22×C4—62no—
471C22×C4—62no—
472C22×C4—61no—
473C22×C4—61no—
474C22×C4C3×D1211yescharacteristic
475C22×C4C3×D1211yescharacteristic
476C22×C4C3×D1211yescharacteristic
477C22×C4C22×C3×D311yescharacteristic
478C22×C4C22×C3×D311yescharacteristic
479C22×C4C22×C3×D311yescharacteristic
480Q16—68no—
481SD16—68no—
482M16—24no—
483C2×C8C22×C3×D314yes—
484C4⋊C4—32no—
485C4⋊C4—61no—
486C4⋊C4—62no—
487C4⋊C4—62no—
488C22⋊C4—62no—
489C22⋊C4—62no—
490C22×C3—21no—
491C22×C3—21no—
492C22×C3—21no—
493C22×C3—21no—
494C22×C3—21no—
495C22×C3—21no—
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—22no—
499C22×C3—21no—
500C22×C3—21no—
501C22×C3—21no—
502C22×C3—21no—
503C22×C3—21no—
504C22×C3—21no—
505C22×C3—22no—
506C22×C3—22no—
507C22×C3—21no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3—41no—
511C22×C3—41no—
512C22×C3—41no—
513C22×C3—41no—
514C22×C3—44no—
515C22×C3—44no—
516C22×C3—48no—
517C22×C3—64no—
518C22×C3—128no—
519C22×C3—122no—
520C22×C3—122no—
521C22×C3#88212yes—
522C22×C3#90912yescomplemented
523C22×C3C3×#12012yes—
524C22×C3C3×#13712yescomplemented
525C22×C3C22×D1211yescharacteristic
526C22×C3C22×C3×D411yescharacteristic
527C2×D3—24no—
528C2×D3—48no—
529C2×D3—42no—
530C2×D3—42no—
531C3×C4—21no—
532C3×C4—21no—
533C3×C4—21no—
534C3×C4—21no—
535C3×C4—22no—
536C3×C4—22no—
537C3×C4—41no—
538C3×C4—41no—
539C3×C4—64no—
540C3×C4—122no—
541C3×C4—122no—
542C3×C4#88212yes—
543C3×C4C3×#12012yes—
544C3×C4C2×D3×D412yes—
545C3×C4C22×C3×D412yes—
546Dic3—24no—
547Dic3—42no—
548Dic3—42no—
549C32#300711yescharacteristic, complemented, Sylow
550C23—22no—
551C23—22no—
552C23—41no—
553C23—64no—
554C23—62no—
555C23—122no—
556C23—122no—
557C23C2×C3×D1211yescharacteristic
558C23C2×C3×D1211yescharacteristic
559C23C2×C3×D1211yescharacteristic
560Q8—34no—
561Q8—64no—
562Q8—62no—
563Q8—64no—
564Q8—122no—
565D4—34no—
566D4—64no—
567D4—68no—
568D4—68no—
569D4—62no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—68no—
576C2×C4—64no—
577C2×C4—62no—
578C2×C4—68no—
579C2×C4—62no—
580C2×C4—62no—
581C2×C4—122no—
582C2×C4—122no—
583C2×C4C2×C3×D1211yescharacteristic
584C2×C4C2×C3×D1211yescharacteristic
585C2×C4C2×C3×D1211yescharacteristic
586C2×C4C3×D3×D414yes—
587C2×C4C23×C3×D311yescharacteristic
588C8—24no—
589C2×C3—22no—
590C2×C3—22no—
591C2×C3—22no—
592C2×C3—22no—
593C2×C3—21no—
594C2×C3—21no—
595C2×C3—21no—
596C2×C3—42no—
597C2×C3—42no—
598C2×C3—42no—
599C2×C3—41no—
600C2×C3—41no—
601C2×C3—124no—
602C2×C3C2×#88211yescharacteristic
603C2×C3C2×#90911yescharacteristic
604C2×C3#574811yescharacteristic
605C2×C3C2×C3×#12011yescharacteristic
606C2×C3C2×C3×#13711yescharacteristic
607C2×C3C3×#57511yescharacteristic
608D3—44no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—44no—
617C22—64no—
618C22—128no—
619C22—122no—
620C22—122no—
621C22C3×#88212yes—
622C22C3×#90912yescomplemented
623C22C22×C3×D1211yescharacteristic
624C4—21no—
625C4—21no—
626C4—64no—
627C4—122no—
628C4—122no—
629C4C3×#88212yes—
630C4C2×C3×D3×D412yes—
631C3—21no—
632C3#7864011yescharacteristic, direct product factor
633C3C3×#300711yescharacteristic, complemented
634C2—22no—
635C2—22no—
636C2—41no—
637C2—124no—
638C2C2×C3×#88211yescharacteristic
639C2C2×C3×#90911yescharacteristic
640C2C3×#574811yescharacteristic
subgroupG/H
centerC22×C3C22×D12
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#407C2
socleC22×C32C22×D4
prime radicalC22×C3×#235C2
hypercenterC3×#407D3
hypocenterC3C3×#3007
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ C3×#407

Complements

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

NH ≅ G/Ntype
2. C2×C3×#924C2semidirect
3. C2×C3×#909C2semidirect
4. C32×#407C2semidirect
5. C3×#5133C2semidirect
6. C3×#4739C2semidirect
7. C3×#4733C2semidirect
8. C3×#4731C2semidirect
10. #78640C3direct (N × H)
21. C22×C3×Dic6C22semidirect
22. C2×C32×M16C22semidirect
24. C2×C3×C4⋊Dic3C22semidirect
28. C3×#924C22semidirect
31. C2×C3×Dic12C22semidirect
32. C2×C3×#204C22semidirect
34. C32×C22⋊C8C22semidirect
35. C3×#821C22semidirect
36. C3×#816C22semidirect
41. C2×#924C2×C3semidirect
42. C2×#909C2×C3semidirect
48. C3×#407D3semidirect
49. C3×#407C2×C3semidirect
50. #5133C2×C3semidirect
51. #4739C2×C3semidirect
52. #4733C2×C3semidirect
53. #4731C2×C3semidirect
120. C22×Dic6C22×C3semidirect
125. C2×C3×M16C2×D3semidirect
126. C2×C3×M16C22×C3semidirect
135. C2×C4⋊Dic3C22×C3semidirect
139. #924C22×C3semidirect
142. C2×Dic12C22×C3semidirect
143. C2×#204C22×C3semidirect
149. C3×C22⋊C8C2×D3semidirect
151. C3×C22⋊C8C22×C3semidirect
152. #821C22×C3semidirect
153. #816C22×C3semidirect
200. #407C3×D3semidirect
306. C22×C32#137semidirect
333. C2×M16C2×C3×D3semidirect
344. C22⋊C8C2×C3×D3semidirect
522. C22×C3#909semidirect
524. C22×C3C3×#137semidirect
549. C32#3007semidirect
622. C22C3×#909semidirect
632. C3#78640direct (N × H)
633. C3C3×#3007semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#2352
C2×C3×#9242
C2×C3×#9092
C32×#4072
C3×#51332
C3×#47392
C3×#47332
C3×#47312
C3×#30073
#786403