Finite Group Databasebeta

← C3×#80028 #11870696

Subgroups — C3×#80028 #11870696

Order 1152.

935 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q8×Dic3C211yesmaximal, characteristic, complemented
2C3×#5596C211yesmaximal, characteristic
3C32×#448C211yesmaximal, characteristic, complemented
4C3×#5227C211yesmaximal, characteristic, complemented
5C3×#5163C212yesmaximal, complemented
6C3×Dic3×SD16C212yesmaximal, complemented
7C3×#5034C211yesmaximal, characteristic, complemented
8C3×#4791C212yesmaximal, complemented
9C3×#4788C212yesmaximal, complemented
10C3×#4735C211yesmaximal, characteristic, complemented
11C3×#4718C211yesmaximal, characteristic, complemented
12C3×#3103—31nomaximal
13#80028C311yesmaximal, characteristic, direct product factor
14C22×C32×Q8C2211yescharacteristic
15C2×C3×#237—21no—
16C2×C32×SD16C2212yes—
17C32×#121C2211yescharacteristic
18C3×Q8×Dic3—21no—
19C3×Q8×Dic3—21no—
20C3×Q8×Dic3C2212yes—
21C3×Q8×Dic3C2212yes—
22C2×C3×#215C2212yes—
23C3×#940C2211yescharacteristic
24C3×#939—21no—
25C3×D4×Dic3C2211yescharacteristic
26C3×#930C2211yescharacteristic
27C2×C3×C4⋊Dic3—21no—
28C2×C3×C4⋊Dic3C2211yescharacteristic
29C2×C3×C4×Dic3—21no—
30C2×C3×C4×Dic3C2211yescharacteristic
31C3×#893C2211yescharacteristic
32C3×#888C2211yescharacteristic
33C3×#885—21no—
34C3×#67⋊C4—21no—
35C32×#103C2212yes—
36C32×C22⋊C8C2211yescharacteristic
37C3×#835C2212yes—
38C3×#832C2212yes—
39C3×#830C2211yescharacteristic
40C3×#821C2212yes—
41C3×#817C2212yes—
42C3×Dic3⋊C8C2212yes—
43C3×C8×Dic3C2212yes—
44C3×#808C2212yes—
45C3×#532—31no—
46C2×C3×C4×Q8—31no—
47C2×Q8×Dic3C2×C311yescharacteristic, complemented
48#5596C2×C311yescharacteristic
49C3×#481—31no—
50C3×#481—31no—
51C3×#475—32no—
52C3×#460—32no—
53C3×#448—21no—
54C3×#448—31no—
55C3×#448D311yescharacteristic, complemented
56C3×#448C2×C311yescharacteristic, complemented
57C3×C4×SD16—32no—
58C3×C4×SD16—32no—
59C3×#431—31no—
60#5227C2×C311yescharacteristic, complemented
61#5163C2×C312yescomplemented
62Dic3×SD16C2×C312yescomplemented
63#5034C2×C311yescharacteristic, complemented
64#4791C2×C312yescomplemented
65#4788C2×C312yescomplemented
66#4735C2×C311yescharacteristic, complemented
67#4718C2×C311yescharacteristic, complemented
68C2×C32×Q8—24no—
69C2×C32×Q8—21no—
70C2×C32×Q8C2312yes—
71C2×C32×D4—21no—
72C2×C32×D4C2311yescharacteristic
73C22×C32×C4—21no—
74C22×C32×C4C2311yescharacteristic
75C2×C3×#67—21no—
76C2×C3×#67—21no—
77C2×C3×#67—21no—
78C22×C3×Dic3—21no—
79C22×C3×Dic3—21no—
80C22×C3×Dic3D411yescharacteristic
81C22×C3×Dic3D411yescharacteristic
82C3×#237—44no—
83C2×C3×D12C2311yescharacteristic
84C2×C3×C4×D3—21no—
85C2×C3×Dic6—21no—
86C32×SD16—24no—
87C2×C32×C8C2312yes—
88C32×C4⋊C4C2311yescharacteristic
89C32×C22⋊C4—21no—
90C3×#217—21no—
91C3×#215—24no—
92C3×(C2×D3)⋊C4—21no—
93C3×(C2×D3)⋊C4—21no—
94C3×C4⋊Dic3—21no—
95C3×C4⋊Dic3—22no—
96C3×C4⋊Dic3—22no—
97C3×C4⋊Dic3C2311yescharacteristic
98C3×C4⋊Dic3C2312yes—
99C3×Dic3⋊C4—21no—
100C3×C4×Dic3—22no—
101C3×C4×Dic3—22no—
102C3×C4×Dic3D411yescharacteristic
103C3×C4×Dic3D411yescharacteristic
104C3×C4×Dic3C2311yescharacteristic
105C3×C4×Dic3C2311yescharacteristic
106C2×C3×C3⋊C8C2312yes—
107#3103—31noSylow
108C2×C3×#41—61no—
109C22×C3×Q8—21no—
110C22×C3×Q8C2×D311yescharacteristic
111C22×C3×Q8C22×C311yescharacteristic
112C2×#237—21no—
113C2×C3×SD16—22no—
114C2×C3×SD16—32no—
115C2×C3×SD16C2×D312yes—
116C2×C3×SD16C22×C312yes—
117C3×C4⋊Q8—31no—
118C3×#127—31no—
119C3×#124—61no—
120C3×#121—21no—
121C3×#121—31no—
122C3×#121—61no—
123C3×#121C2×D311yescharacteristic
124C3×#121C22×C311yescharacteristic
125C3×C4×Q8—32no—
126C3×C4×Q8—32no—
127C3×C4×Q8—61no—
128C3×C4×Q8—61no—
129C3×C4×D4—31no—
130C3×C4×D4—31no—
131C3×C4×D4—61no—
132C2×C3×C4⋊C4—31no—
133C2×C3×C4⋊C4—61no—
134C2×C3×C42—31no—
135C2×C3×C42—61no—
136Q8×Dic3—21no—
137Q8×Dic3—21no—
138Q8×Dic3C22×C312yes—
139Q8×Dic3C22×C312yes—
140C2×#215C22×C312yes—
141#940C22×C311yescharacteristic
142#939—21no—
143D4×Dic3C22×C311yescharacteristic
144#930C22×C311yescharacteristic
145C2×C4⋊Dic3—21no—
146C2×C4⋊Dic3C22×C311yescharacteristic
147C2×C4×Dic3—21no—
148C2×C4×Dic3C22×C311yescharacteristic
149#893C22×C311yescharacteristic
150#888C22×C311yescharacteristic
151#885—21no—
152#67⋊C4—21no—
153C3×#106—32no—
154C3×#106—32no—
155C3×C4⋊C8—32no—
156C3×#103—22no—
157C3×#103—32no—
158C3×#103C2×D312yes—
159C3×#103C22×C312yes—
160C3×#102—32no—
161C3×#102—32no—
162C3×C22⋊C8—21no—
163C3×C22⋊C8—31no—
164C3×C22⋊C8C2×D311yescharacteristic
165C3×C22⋊C8C22×C311yescharacteristic
166C3×C4×C8—32no—
167#835C22×C312yes—
168#832C22×C312yes—
169#830C22×C311yescharacteristic
170#821C22×C312yes—
171#817C22×C312yes—
172Dic3⋊C8C22×C312yes—
173C8×Dic3C22×C312yes—
174#808C22×C312yes—
175C23×C32—21no—
176C23×C32C2×D411yescharacteristic
177C22×C3×D3—21no—
178C32×Q8—24no—
179C32×Q8—22no—
180C32×Q8#4114yescomplemented
181C32×D4—22no—
182C32×D4—42no—
183C2×C32×C4—24no—
184C2×C32×C4—22no—
185C2×C32×C4—22no—
186C2×C32×C4—21no—
187C2×C32×C4C2×D411yescharacteristic
188C2×C32×C4C2411yescharacteristic
189C3×#67—42no—
190C3×#67—42no—
191C3×#67—42no—
192C2×C3×Dic3—22no—
193C2×C3×Dic3—21no—
194C2×C3×Dic3—41no—
195C2×C3×Dic3—42no—
196C2×C3×Dic3—42no—
197C2×C3×Dic3SD1614yescomplemented
198C2×C3×Dic3C2×D411yescharacteristic
199C2×C3×Dic3C2×D411yescharacteristic
200C2×C3×Dic3C2×D411yescharacteristic
201C2×C3×Dic3C2×D411yescharacteristic
202C3×D12—22no—
203C3×C4×D3—42no—
204C3×Dic6—42no—
205C32×C8—22no—
206C3×C3⋊C8—22no—
207#532—31no—
208C2×C4×Q8—31no—
209#481—31no—
210#481—31no—
211#475—32no—
212#460—32no—
213#448—31no—
214#448C3×D311yescharacteristic, complemented
215C4×SD16—32no—
216C4×SD16—32no—
217#431—31no—
218C3×#41—124no—
219C2×C3×Q8—24no—
220C2×C3×Q8—24no—
221C2×C3×Q8—21no—
222C2×C3×Q8—21no—
223C2×C3×Q8—22no—
224C2×C3×Q8—41no—
225C2×C3×Q8—44no—
226C2×C3×Q8—61no—
227C2×C3×Q8C22×D312yes—
228C2×C3×Q8C23×C312yes—
229C2×C3×D4—21no—
230C2×C3×D4—21no—
231C2×C3×D4—21no—
232C2×C3×D4—31no—
233C2×C3×D4—41no—
234C2×C3×D4—61no—
235C2×C3×D4—61no—
236C2×C3×D4—61no—
237C2×C3×D4C22×D311yescharacteristic
238C2×C3×D4C23×C311yescharacteristic
239C22×C3×C4—21no—
240C22×C3×C4—21no—
241C22×C3×C4—21no—
242C22×C3×C4—31no—
243C22×C3×C4—31no—
244C22×C3×C4—41no—
245C22×C3×C4—61no—
246C22×C3×C4—61no—
247C22×C3×C4—61no—
248C22×C3×C4C22×D311yescharacteristic
249C22×C3×C4C23×C311yescharacteristic
250C2×#67—21no—
251C2×#67—21no—
252C2×#67—21no—
253C22×Dic3—21no—
254C22×Dic3—21no—
255C22×Dic3C3×D411yescharacteristic
256C22×Dic3C3×D411yescharacteristic
257#237—44no—
258C2×D12C23×C311yescharacteristic
259C2×C4×D3—21no—
260C2×Dic6—21no—
261C3×SD16—24no—
262C3×SD16—24no—
263C3×SD16—44no—
264C3×SD16—64no—
265C2×C3×C8—22no—
266C2×C3×C8—32no—
267C2×C3×C8C22×D312yes—
268C2×C3×C8C23×C312yes—
269C3×C4⋊C4—21no—
270C3×C4⋊C4—32no—
271C3×C4⋊C4—31no—
272C3×C4⋊C4—61no—
273C3×C4⋊C4—62no—
274C3×C4⋊C4—62no—
275C3×C4⋊C4—61no—
276C3×C4⋊C4C22×D311yescharacteristic
277C3×C4⋊C4C23×C311yescharacteristic
278C3×C22⋊C4—21no—
279C3×C22⋊C4—21no—
280C3×C22⋊C4—41no—
281C3×C22⋊C4—61no—
282C3×C22⋊C4—61no—
283C3×C22⋊C4—61no—
284C3×C42—31no—
285C3×C42—31no—
286C3×C42—31no—
287C3×C42—31no—
288C3×C42—62no—
289C3×C42—62no—
290#217—21no—
291#215—24no—
292(C2×D3)⋊C4—21no—
293(C2×D3)⋊C4—21no—
294C4⋊Dic3—21no—
295C4⋊Dic3—22no—
296C4⋊Dic3—22no—
297C4⋊Dic3C23×C312yes—
298C4⋊Dic3C23×C311yescharacteristic
299Dic3⋊C4—21no—
300C4×Dic3—22no—
301C4×Dic3—22no—
302C4×Dic3C3×D411yescharacteristic
303C4×Dic3C3×D411yescharacteristic
304C4×Dic3C23×C311yescharacteristic
305C4×Dic3C23×C311yescharacteristic
306C2×C3⋊C8C23×C312yes—
307C22×C32—21no—
308C22×C32—21no—
309C22×C32—41no—
310C22×C32—41no—
311C22×C32—41no—
312C22×C32C2×SD1612yes—
313C22×C32C22×D411yescharacteristic
314C2×C3×D3—41no—
315C2×C3×D3—41no—
316C2×C3×D3—41no—
317C32×C4—24no—
318C32×C4—21no—
319C32×C4—41no—
320C32×C4C2×#4112yes—
321C3×Dic3—21no—
322C3×Dic3—21no—
323C3×Dic3—41no—
324C3×Dic3—42no—
325C3×Dic3C2×SD1612yes—
326C3×Dic3C2×SD1612yes—
327C2×#41—61no—
328C22×Q8C2×C3×D311yescharacteristic
329C2×SD16—32no—
330C2×SD16C2×C3×D312yes—
331C4⋊Q8—31no—
332#127—31no—
333#124—61no—
334#121—31no—
335#121—61no—
336#121C2×C3×D311yescharacteristic
337C4×Q8—32no—
338C4×Q8—32no—
339C4×Q8—61no—
340C4×Q8—61no—
341C4×D4—31no—
342C4×D4—31no—
343C4×D4—61no—
344C2×C4⋊C4—31no—
345C2×C4⋊C4—61no—
346C2×C42—31no—
347C2×C42—61no—
348#106—32no—
349#106—32no—
350C4⋊C8—32no—
351#103—32no—
352#103C2×C3×D312yes—
353#102—32no—
354#102—32no—
355C22⋊C8—31no—
356C22⋊C8C2×C3×D311yescharacteristic
357C4×C8—32no—
358C23×C3—21no—
359C23×C3—21no—
360C23×C3—21no—
361C23×C3—41no—
362C23×C3—61no—
363C23×C3D3×D411yescharacteristic
364C23×C3C2×C3×D411yescharacteristic
365C22×D3—21no—
366C3×Q8—24no—
367C3×Q8—24no—
368C3×Q8—22no—
369C3×Q8—22no—
370C3×Q8—24no—
371C3×Q8—42no—
372C3×Q8—44no—
373C3×Q8—122no—
374C3×Q8#23714yescomplemented
375C3×Q8C3×#4114yescomplemented
376C3×D4—22no—
377C3×D4—22no—
378C3×D4—42no—
379C3×D4—42no—
380C3×D4—42no—
381C3×D4—62no—
382C3×D4—82no—
383C3×D4—122no—
384C3×D4—122no—
385C3×D4—122no—
386C2×C3×C4—24no—
387C2×C3×C4—24no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—34no—
397C2×C3×C4—31no—
398C2×C3×C4—31no—
399C2×C3×C4—31no—
400C2×C3×C4—31no—
401C2×C3×C4—42no—
402C2×C3×C4—42no—
403C2×C3×C4—44no—
404C2×C3×C4—41no—
405C2×C3×C4—62no—
406C2×C3×C4—61no—
407C2×C3×C4—122no—
408C2×C3×C4—122no—
409C2×C3×C4—121no—
410C2×C3×C4—122no—
411C2×C3×C4D3×D411yescharacteristic
412C2×C3×C4C2×C3×D411yescharacteristic
413C2×C3×C4C23×D311yescharacteristic
414C2×C3×C4C24×C311yescharacteristic
415#67—42no—
416#67—42no—
417#67—42no—
418C2×Dic3—22no—
419C2×Dic3—21no—
420C2×Dic3—41no—
421C2×Dic3—42no—
422C2×Dic3—42no—
423C2×Dic3C3×SD1614yescomplemented
424C2×Dic3C2×C3×D411yescharacteristic
425C2×Dic3C2×C3×D411yescharacteristic
426C2×Dic3C2×C3×D411yescharacteristic
427C2×Dic3C2×C3×D411yescharacteristic
428D12—22no—
429C4×D3—42no—
430Dic6—42no—
431C3×C8—22no—
432C3×C8—22no—
433C3×C8—42no—
434C3×C8—62no—
435C3⋊C8—22no—
436C2×C32—22no—
437C2×C32—81no—
438C2×C32#52511yescharacteristic
439C2×C32C22×SD1611yescharacteristic
440C2×C32#57511yescharacteristic
441C3×D3—81no—
442#41—124no—
443C2×Q8—24no—
444C2×Q8—21no—
445C2×Q8—61no—
446C2×Q8C22×C3×D312yes—
447C2×D4—21no—
448C2×D4—31no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4C22×C3×D311yescharacteristic
453C22×C4—21no—
454C22×C4—31no—
455C22×C4—31no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4C22×C3×D311yescharacteristic
460SD16—24no—
461SD16—64no—
462C2×C8—32no—
463C2×C8C22×C3×D312yes—
464C4⋊C4—32no—
465C4⋊C4—31no—
466C4⋊C4—62no—
467C4⋊C4—62no—
468C4⋊C4—61no—
469C4⋊C4—61no—
470C4⋊C4C22×C3×D311yescharacteristic
471C22⋊C4—21no—
472C22⋊C4—61no—
473C22⋊C4—61no—
474C22⋊C4—61no—
475C42—31no—
476C42—31no—
477C42—31no—
478C42—31no—
479C42—62no—
480C42—62no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3—22no—
486C22×C3—21no—
487C22×C3—41no—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3—41no—
491C22×C3—41no—
492C22×C3—41no—
493C22×C3—41no—
494C22×C3—41no—
495C22×C3—81no—
496C22×C3—81no—
497C22×C3—81no—
498C22×C3—121no—
499C22×C3—121no—
500C22×C3—121no—
501C22×C3D3×SD1612yes—
502C22×C3C2×C3×SD1612yes—
503C22×C3C2×D3×D411yescharacteristic
504C22×C3C22×C3×D411yescharacteristic
505C2×D3—41no—
506C2×D3—41no—
507C2×D3—41no—
508C3×C4—24no—
509C3×C4—21no—
510C3×C4—24no—
511C3×C4—21no—
512C3×C4—22no—
513C3×C4—32no—
514C3×C4—32no—
515C3×C4—41no—
516C3×C4—44no—
517C3×C4—41no—
518C3×C4—41no—
519C3×C4—61no—
520C3×C4—61no—
521C3×C4—81no—
522C3×C4—121no—
523C3×C4—122no—
524C3×C4C2×#23712yes—
525C3×C4C2×C3×#4112yes—
526Dic3—21no—
527Dic3—21no—
528Dic3—41no—
529Dic3—42no—
530Dic3C2×C3×SD1612yes—
531Dic3C2×C3×SD1612yes—
532C32#310311yescharacteristic, complemented, Sylow
533C23—21no—
534C23—61no—
535C23C3×D3×D411yescharacteristic
536Q8—24no—
537Q8—22no—
538Q8—122no—
539Q8C3×#23714yescomplemented
540D4—22no—
541D4—42no—
542D4—62no—
543D4—122no—
544D4—122no—
545D4—122no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—34no—
551C2×C4—31no—
552C2×C4—31no—
553C2×C4—31no—
554C2×C4—31no—
555C2×C4—62no—
556C2×C4—61no—
557C2×C4—122no—
558C2×C4—122no—
559C2×C4—121no—
560C2×C4—122no—
561C2×C4C3×D3×D411yescharacteristic
562C2×C4C23×C3×D311yescharacteristic
563C8—22no—
564C8—62no—
565C2×C3—22no—
566C2×C3—22no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—42no—
571C2×C3—81no—
572C2×C3—81no—
573C2×C3—161no—
574C2×C3—241no—
575C2×C3#558611yescharacteristic
576C2×C3C2×D3×SD1611yescharacteristic
577C2×C3#577711yescharacteristic
578C2×C3C3×#52511yescharacteristic
579C2×C3C22×C3×SD1611yescharacteristic
580C2×C3C3×#57511yescharacteristic
581D3—81no—
582C22—21no—
583C22—21no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—121no—
588C22—121no—
589C22—121no—
590C22C3×D3×SD1612yes—
591C22C2×C3×D3×D411yescharacteristic
592C4—24no—
593C4—21no—
594C4—32no—
595C4—32no—
596C4—41no—
597C4—61no—
598C4—61no—
599C4—121no—
600C4—122no—
601C4C2×C3×#23712yes—
602C3—21no—
603C3#8002811yescharacteristic, direct product factor
604C3C3×#310311yescharacteristic, complemented
605C2—22no—
606C2—81no—
607C2—241no—
608C2C3×#558611yescharacteristic
609C2C2×C3×D3×SD1611yescharacteristic
610C2C3×#577711yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#448C2
socleC22×C32C22×D4
prime radicalC3×#5596C2
hypercenterC3×#448D3
hypocenterC3C3×#3103
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#448

Complements

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

NH ≅ G/Ntype
1. C2×C3×Q8×Dic3C2semidirect
3. C32×#448C2semidirect
4. C3×#5227C2semidirect
5. C3×#5163C2semidirect
6. C3×Dic3×SD16C2semidirect
7. C3×#5034C2semidirect
8. C3×#4791C2semidirect
9. C3×#4788C2semidirect
10. C3×#4735C2semidirect
11. C3×#4718C2semidirect
13. #80028C3direct (N × H)
47. C2×Q8×Dic3C2×C3semidirect
55. C3×#448D3semidirect
56. C3×#448C2×C3semidirect
60. #5227C2×C3semidirect
61. #5163C2×C3semidirect
62. Dic3×SD16C2×C3semidirect
63. #5034C2×C3semidirect
64. #4791C2×C3semidirect
65. #4788C2×C3semidirect
66. #4735C2×C3semidirect
67. #4718C2×C3semidirect
180. C32×Q8#41semidirect
197. C2×C3×Dic3SD16semidirect
214. #448C3×D3semidirect
374. C3×Q8#237semidirect
375. C3×Q8C3×#41semidirect
423. C2×Dic3C3×SD16semidirect
532. C32#3103semidirect
539. Q8C3×#237semidirect
603. C3#80028direct (N × H)
604. C3C3×#3103semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×Dic32
C3×#55962
C32×#4482
C3×#52272
C3×#51632
C3×Dic3×SD162
C3×#50342
C3×#47912
C3×#47882
C3×#47352
C3×#47182
C3×#31033
#800283