Finite Group Databasebeta

← C3×#85297 #11922582

Subgroups — C3×#85297 #11922582

Order 1152.

1338 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 1338 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×S4C211yesmaximal, characteristic, complemented
2C2×C3×GL(2,ℤ/4)C211yesmaximal, characteristic, complemented
3C2×C3×C4⋊S4C212yesmaximal, complemented
4C3×A4×C22⋊C4C211yesmaximal, characteristic, complemented
5C3×(C2×S4)⋊C4C212yesmaximal, complemented
6#85297C311yesmaximal, characteristic, direct product factor
7C3×#2410—31nomaximal
8C23×C3×A4C2211yescharacteristic
9C22×C3×S4—22no—
10C22×C3×S4—24no—
11C22×C3×S4C2212yes—
12C22×C3×S4C2211yescharacteristic, complemented
13C2×C3×C4×A4C2212yescomplemented
14C3×GL(2,ℤ/4)—22no—
15C2×C3×SL(2,ℤ/4)C2211yescharacteristic, complemented
16C3×C4⋊S4—24no—
17C3×#882—41nomaximal
18C23×S4C2×C311yescharacteristic, complemented
19C23×C3×D4—31no—
20C2×GL(2,ℤ/4)C2×C311yescharacteristic, complemented
21C2×C4⋊S4C2×C312yescomplemented
22C2×C3×#121—32no—
23C2×C3×#120—31no—
24C2×C3×#120—31no—
25C22×C3×C22⋊C4D311yescharacteristic, complemented
26A4×C22⋊C4—21no—
27A4×C22⋊C4C2×C311yescharacteristic, complemented
28(C2×S4)⋊C4C2×C312yescomplemented
29C3×#394—32no—
30C3×#392—32no—
31C3×#386—31no—
32C3×#386—32no—
33C3×#385—31no—
34C3×#379—31no—
35C23×C3×D3—41no—
36C22×C3×A4—22no—
37C22×C3×A4D412yescomplemented
38C22×C3×A4C2311yescharacteristic
39C2×C3×S4—28no—
40C2×C3×S4—22no—
41C2×C3×S4—22no—
42C2×C3×S4—21no—
43C2×C3×S4—22no—
44C2×C3×S4D414yescomplemented
45C2×C3×#67—41no—
46C2×C3×D12—42no—
47C3×C4×A4—22no—
48C3×SL(2,ℤ/4)—21no—
49C32×C22⋊C4—41no—
50C3×(C2×D3)⋊C4—42no—
51#2410—31noSylow
52C25×C3—31no—
53C25×C3C2×D311yescharacteristic
54C23×A4—21no—
55C23×A4C22×C311yescharacteristic
56C22×S4—22no—
57C22×S4—24no—
58C22×S4C22×C312yes—
59C22×S4C22×C311yescharacteristic, complemented
60C22×C3×D4—32no—
61C22×C3×D4—32no—
62C22×C3×D4—31no—
63C22×C3×D4—31no—
64C22×C3×D4—32no—
65C22×C3×D4—31no—
66C22×C3×D4—64no—
67C22×C3×D4—64no—
68C22×C3×D4—62no—
69C22×C3×D4—62no—
70C23×C3×C4—31no—
71C23×C3×C4C2×D312yescomplemented
72C2×C4×A4—22no—
73C2×C4×A4C22×C312yescomplemented
74GL(2,ℤ/4)—22no—
75C2×SL(2,ℤ/4)C22×C311yescharacteristic, complemented
76C4⋊S4—24no—
77C3×#121—64no—
78C3×#121—64no—
79C3×#120—61no—
80C3×#120—62no—
81C3×#120—62no—
82C3×#120—62no—
83C3×#120—61no—
84C2×C3×C4⋊C4—32no—
85C2×C3×C22⋊C4—31no—
86C2×C3×C22⋊C4—31no—
87C2×C3×C22⋊C4—32no—
88C2×C3×C22⋊C4—31no—
89C2×C3×C22⋊C4—32no—
90C2×C3×C22⋊C4—32no—
91C2×C3×C22⋊C4—32no—
92C2×C3×C22⋊C4—31no—
93C2×C3×C22⋊C4—31no—
94C2×C3×C22⋊C4—32no—
95#882—41no—
96C3×#95—32no—
97C3×#95—32no—
98C23×C32—41no—
99C22×C3×D3—42no—
100C22×C3×D3—41no—
101C22×C3×D3—84no—
102C22×C3×D3—82no—
103C2×C3×A4—22no—
104C2×C3×A4C2×D411yescharacteristic
105C2×C3×A4C2×D412yes—
106C3×S4—24no—
107C3×S4—41no—
108C2×C32×C4—42no—
109C3×#67—82no—
110C2×C3×Dic3—41no—
111C3×D12—84no—
112C23×D4—31no—
113C2×#121—32no—
114C2×#120—31no—
115C2×#120—31no—
116C22×C22⋊C4C3×D311yescharacteristic, complemented
117#394—32no—
118#392—32no—
119#386—31no—
120#386—32no—
121#385—31no—
122#379—31no—
123C24×C3—22no—
124C24×C3—31no—
125C24×C3—31no—
126C24×C3—32no—
127C24×C3—32no—
128C24×C3—31no—
129C24×C3—62no—
130C24×C3—62no—
131C24×C3—64no—
132C24×C3—62no—
133C24×C3—61no—
134C24×C3—61no—
135C24×C3—122no—
136C24×C3D1212yescomplemented
137C24×C3C22×D311yescharacteristic
138C23×D3—41no—
139C22×A4—22no—
140C22×A4—22no—
141C22×A4—21no—
142C22×A4—42no—
143C22×A4C3×D412yescomplemented
144C22×A4C23×C311yescharacteristic
145C2×S4—28no—
146C2×S4—22no—
147C2×S4—22no—
148C2×S4—22no—
149C2×S4—21no—
150C2×S4C3×D414yescomplemented
151C2×C3×D4—34no—
152C2×C3×D4—34no—
153C2×C3×D4—64no—
154C2×C3×D4—62no—
155C2×C3×D4—68no—
156C2×C3×D4—62no—
157C2×C3×D4—62no—
158C2×C3×D4—64no—
159C2×C3×D4—64no—
160C2×C3×D4—68no—
161C2×C3×D4—616no—
162C2×C3×D4—62no—
163C2×C3×D4—62no—
164C2×C3×D4—61no—
165C2×C3×D4—62no—
166C2×C3×D4—62no—
167C2×C3×D4—61no—
168C2×C3×D4—64no—
169C2×C3×D4—62no—
170C2×C3×D4—62no—
171C2×C3×D4—61no—
172C2×C3×D4—62no—
173C2×C3×D4—61no—
174C2×C3×D4—62no—
175C2×C3×D4—122no—
176C2×C3×D4—122no—
177C22×C3×C4—22no—
178C22×C3×C4—32no—
179C22×C3×C4—32no—
180C22×C3×C4—32no—
181C22×C3×C4—31no—
182C22×C3×C4—31no—
183C22×C3×C4—31no—
184C22×C3×C4—64no—
185C22×C3×C4—62no—
186C22×C3×C4—62no—
187C2×#67—41no—
188C2×D12—42no—
189C4×A4—22no—
190C4×A4—42no—
191SL(2,ℤ/4)—21no—
192C3×C4⋊C4—64no—
193C3×C22⋊C4—31no—
194C3×C22⋊C4—32no—
195C3×C22⋊C4—41no—
196C3×C22⋊C4—61no—
197C3×C22⋊C4—62no—
198C3×C22⋊C4—64no—
199C3×C22⋊C4—61no—
200C3×C22⋊C4—62no—
201C3×C22⋊C4—62no—
202C3×C22⋊C4—64no—
203C3×C22⋊C4—61no—
204C3×C22⋊C4—62no—
205C3×C22⋊C4—81no—
206C3×C22⋊C4—122no—
207C3×C22⋊C4S411yescharacteristic, complemented
208(C2×D3)⋊C4—42no—
209C22×C32—42no—
210C22×C32—41no—
211C22×C32—82no—
212C2×C3×D3—44no—
213C2×C3×D3—82no—
214C2×C3×D3—82no—
215C2×C3×D3—88no—
216C2×C3×D3—82no—
217C2×C3×D3—81no—
218C3×A4#12011yescharacteristic, complemented
219C32×C4—82no—
220C3×Dic3—81no—
221C25—31no—
222C25C2×C3×D311yescharacteristic
223C22×D4—32no—
224C22×D4—31no—
225C22×D4—32no—
226C22×D4—31no—
227C22×D4—32no—
228C22×D4—31no—
229C22×D4—64no—
230C22×D4—64no—
231C22×D4—62no—
232C22×D4—62no—
233C23×C4—31no—
234C23×C4C2×C3×D312yescomplemented
235#121—64no—
236#121—64no—
237#120—61no—
238#120—62no—
239#120—62no—
240#120—62no—
241#120—61no—
242C2×C4⋊C4—32no—
243C2×C22⋊C4—31no—
244C2×C22⋊C4—31no—
245C2×C22⋊C4—32no—
246C2×C22⋊C4—31no—
247C2×C22⋊C4—32no—
248C2×C22⋊C4—31no—
249C2×C22⋊C4—32no—
250C2×C22⋊C4—32no—
251C2×C22⋊C4—31no—
252C2×C22⋊C4—32no—
253#95—32no—
254#95—32no—
255C23×C3—22no—
256C23×C3—31no—
257C23×C3—31no—
258C23×C3—32no—
259C23×C3—32no—
260C23×C3—32no—
261C23×C3—32no—
262C23×C3—31no—
263C23×C3—34no—
264C23×C3—41no—
265C23×C3—62no—
266C23×C3—62no—
267C23×C3—61no—
268C23×C3—62no—
269C23×C3—64no—
270C23×C3—62no—
271C23×C3—62no—
272C23×C3—62no—
273C23×C3—62no—
274C23×C3—62no—
275C23×C3—62no—
276C23×C3—62no—
277C23×C3—62no—
278C23×C3—61no—
279C23×C3—61no—
280C23×C3—62no—
281C23×C3—68no—
282C23×C3—62no—
283C23×C3—61no—
284C23×C3—81no—
285C23×C3—124no—
286C23×C3—122no—
287C23×C3—122no—
288C23×C3—128no—
289C23×C3—124no—
290C23×C3—124no—
291C23×C3—122no—
292C23×C3C2×D1211yescharacteristic
293C23×C3D3×D412yes—
294C23×C3C2×S411yescharacteristic
295C22×D3—42no—
296C22×D3—41no—
297C22×D3—84no—
298C22×D3—82no—
299C2×A4—22no—
300C2×A4—21no—
301C2×A4—22no—
302C2×A4—42no—
303C2×A4C2×C3×D411yescharacteristic
304C2×A4C2×C3×D412yes—
305S4—24no—
306S4—41no—
307C3×D4—64no—
308C3×D4—64no—
309C3×D4—64no—
310C3×D4—616no—
311C3×D4—64no—
312C3×D4—124no—
313C3×D4—124no—
314C3×D4—121no—
315C3×D4—122no—
316C3×D4—122no—
317C3×D4—124no—
318C3×D4—121no—
319C3×D4—122no—
320C2×C3×C4—34no—
321C2×C3×C4—32no—
322C2×C3×C4—42no—
323C2×C3×C4—62no—
324C2×C3×C4—62no—
325C2×C3×C4—62no—
326C2×C3×C4—62no—
327C2×C3×C4—68no—
328C2×C3×C4—62no—
329C2×C3×C4—62no—
330C2×C3×C4—61no—
331C2×C3×C4—61no—
332C2×C3×C4—61no—
333C2×C3×C4—62no—
334C2×C3×C4—62no—
335C2×C3×C4—61no—
336C2×C3×C4—61no—
337C2×C3×C4—82no—
338C2×C3×C4C2×S412yescomplemented
339#67—82no—
340C2×Dic3—41no—
341D12—84no—
342C2×C32—41no—
343C2×C32—42no—
344C2×C32—82no—
345C3×D3—84no—
346C3×D3—161no—
347C24—22no—
348C24—31no—
349C24—31no—
350C24—32no—
351C24—32no—
352C24—31no—
353C24—62no—
354C24—62no—
355C24—64no—
356C24—61no—
357C24—62no—
358C24—61no—
359C24—122no—
360C24C3×D1212yescomplemented
361C24C22×C3×D311yescharacteristic
362C2×D4—34no—
363C2×D4—34no—
364C2×D4—64no—
365C2×D4—62no—
366C2×D4—68no—
367C2×D4—62no—
368C2×D4—62no—
369C2×D4—64no—
370C2×D4—64no—
371C2×D4—68no—
372C2×D4—616no—
373C2×D4—62no—
374C2×D4—62no—
375C2×D4—61no—
376C2×D4—62no—
377C2×D4—62no—
378C2×D4—61no—
379C2×D4—64no—
380C2×D4—62no—
381C2×D4—62no—
382C2×D4—61no—
383C2×D4—61no—
384C2×D4—62no—
385C2×D4—62no—
386C2×D4—122no—
387C2×D4—122no—
388C22×C4—22no—
389C22×C4—32no—
390C22×C4—32no—
391C22×C4—32no—
392C22×C4—31no—
393C22×C4—31no—
394C22×C4—31no—
395C22×C4—64no—
396C22×C4—62no—
397C22×C4—62no—
398C4⋊C4—64no—
399C22⋊C4—31no—
400C22⋊C4—32no—
401C22⋊C4—61no—
402C22⋊C4—62no—
403C22⋊C4—64no—
404C22⋊C4—61no—
405C22⋊C4—62no—
406C22⋊C4—62no—
407C22⋊C4—64no—
408C22⋊C4—61no—
409C22⋊C4—62no—
410C22⋊C4—122no—
411C22⋊C4C3×S411yescharacteristic, complemented
412C22×C3—22no—
413C22×C3—31no—
414C22×C3—32no—
415C22×C3—31no—
416C22×C3—31no—
417C22×C3—32no—
418C22×C3—32no—
419C22×C3—32no—
420C22×C3—41no—
421C22×C3—42no—
422C22×C3—62no—
423C22×C3—62no—
424C22×C3—64no—
425C22×C3—61no—
426C22×C3—62no—
427C22×C3—62no—
428C22×C3—64no—
429C22×C3—64no—
430C22×C3—62no—
431C22×C3—64no—
432C22×C3—82no—
433C22×C3—81no—
434C22×C3—82no—
435C22×C3—128no—
436C22×C3—122no—
437C22×C3—122no—
438C22×C3—122no—
439C22×C3—121no—
440C22×C3—121no—
441C22×C3—122no—
442C22×C3—128no—
443C22×C3—122no—
444C22×C3—122no—
445C22×C3—121no—
446C22×C3—122no—
447C22×C3—162no—
448C22×C3#88211yescharacteristic, complemented
449C22×C3C4⋊S412yescomplemented
450C22×C3C22×S411yescharacteristic
451C2×D3—44no—
452C2×D3—82no—
453C2×D3—82no—
454C2×D3—88no—
455C2×D3—82no—
456C2×D3—81no—
457A4—21no—
458A4C3×#12011yescharacteristic, complemented
459C3×C4—22no—
460C3×C4—64no—
461C3×C4—62no—
462C3×C4—82no—
463C3×C4—121no—
464C3×C4—121no—
465C3×C4—121no—
466C3×C4—162no—
467Dic3—81no—
468C32—41noSylow
469C23—22no—
470C23—31no—
471C23—31no—
472C23—32no—
473C23—32no—
474C23—31no—
475C23—32no—
476C23—32no—
477C23—34no—
478C23—62no—
479C23—62no—
480C23—61no—
481C23—62no—
482C23—64no—
483C23—62no—
484C23—62no—
485C23—62no—
486C23—62no—
487C23—62no—
488C23—62no—
489C23—62no—
490C23—62no—
491C23—61no—
492C23—61no—
493C23—62no—
494C23—68no—
495C23—62no—
496C23—61no—
497C23—124no—
498C23—122no—
499C23—122no—
500C23—128no—
501C23—124no—
502C23—124no—
503C23—122no—
504C23C2×C3×D1211yescharacteristic
505C23C3×D3×D412yes—
506C23C2×C3×S411yescharacteristic
507D4—64no—
508D4—64no—
509D4—64no—
510D4—616no—
511D4—64no—
512D4—124no—
513D4—124no—
514D4—121no—
515D4—122no—
516D4—122no—
517D4—124no—
518D4—121no—
519D4—122no—
520C2×C4—34no—
521C2×C4—32no—
522C2×C4—62no—
523C2×C4—62no—
524C2×C4—62no—
525C2×C4—62no—
526C2×C4—68no—
527C2×C4—62no—
528C2×C4—62no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4C2×C3×S412yescomplemented
537C2×C3—22no—
538C2×C3—31no—
539C2×C3—31no—
540C2×C3—32no—
541C2×C3—41no—
542C2×C3—42no—
543C2×C3—62no—
544C2×C3—82no—
545C2×C3—81no—
546C2×C3—82no—
547C2×C3—124no—
548C2×C3—162no—
549C2×C3—241no—
550C2×C3C2×C4⋊S411yescharacteristic
551C2×C3D4×S412yes—
552D3—84no—
553D3—161no—
554C22—22no—
555C22—31no—
556C22—32no—
557C22—31no—
558C22—31no—
559C22—32no—
560C22—32no—
561C22—32no—
562C22—62no—
563C22—62no—
564C22—64no—
565C22—61no—
566C22—62no—
567C22—62no—
568C22—64no—
569C22—64no—
570C22—62no—
571C22—64no—
572C22—128no—
573C22—122no—
574C22—122no—
575C22—122no—
576C22—121no—
577C22—121no—
578C22—122no—
579C22—128no—
580C22—122no—
581C22—122no—
582C22—121no—
583C22—122no—
584C22C3×#88211yescharacteristic, complemented
585C22C3×C4⋊S412yescomplemented
586C22C22×C3×S411yescharacteristic
587C4—22no—
588C4—64no—
589C4—62no—
590C4—121no—
591C4—121no—
592C4—121no—
593C3—41no—
594C3—81no—
595C3#8529711yescharacteristic, direct product factor
596C2—22no—
597C2—31no—
598C2—31no—
599C2—32no—
600C2—62no—
601C2—124no—
602C2—241no—
603C2C2×C3×C4⋊S411yescharacteristic
604C2C3×D4×S412yes—
subgroupG/H
centerC22×C3C22×S4
commutator, cosocleC22×A4C23×C3
FrattiniC22C22×C3×S4
FittingC22×C3×C22⋊C4D3
socleC24×C3C22×D3
hypercenterC3×C22⋊C4S4
hypocenterA4C3×#120
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22×C3 ◁ C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C23×C3×S4C2semidirect
2. C2×C3×GL(2,ℤ/4)C2semidirect
3. C2×C3×C4⋊S4C2semidirect
4. C3×A4×C22⋊C4C2semidirect
5. C3×(C2×S4)⋊C4C2semidirect
6. #85297C3direct (N × H)
12. C22×C3×S4C22semidirect
13. C2×C3×C4×A4C22semidirect
15. C2×C3×SL(2,ℤ/4)C22semidirect
18. C23×S4C2×C3semidirect
20. C2×GL(2,ℤ/4)C2×C3semidirect
21. C2×C4⋊S4C2×C3semidirect
25. C22×C3×C22⋊C4D3semidirect
27. A4×C22⋊C4C2×C3semidirect
28. (C2×S4)⋊C4C2×C3semidirect
37. C22×C3×A4D4semidirect
44. C2×C3×S4D4semidirect
59. C22×S4C22×C3semidirect
71. C23×C3×C4C2×D3semidirect
73. C2×C4×A4C22×C3semidirect
75. C2×SL(2,ℤ/4)C22×C3semidirect
116. C22×C22⋊C4C3×D3semidirect
136. C24×C3D12semidirect
143. C22×A4C3×D4semidirect
150. C2×S4C3×D4semidirect
207. C3×C22⋊C4S4semidirect
218. C3×A4#120semidirect
234. C23×C4C2×C3×D3semidirect
338. C2×C3×C4C2×S4semidirect
360. C24C3×D12semidirect
411. C22⋊C4C3×S4semidirect
448. C22×C3#882semidirect
449. C22×C3C4⋊S4semidirect
458. A4C3×#120semidirect
536. C2×C4C2×C3×S4semidirect
584. C22C3×#882semidirect
585. C22C3×C4⋊S4semidirect
595. C3#85297direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×S42
C2×C3×GL(2,ℤ/4)2
C2×C3×C4⋊S42
C3×A4×C22⋊C42
C3×(C2×S4)⋊C42
#852973
C3×#24103
C3×#8824