Finite Group Databasebeta

← C4×#5072 #10801357

Subgroups — C4×#5072 #10801357

Order 768.

1272 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#127C211yesmaximal, characteristic, complemented
2C2×#5072C211yesmaximal, characteristic
3C2×C4×#213C212yesmaximal, complemented
4C42×D12C211yesmaximal, characteristic, complemented
5#71761C211yesmaximal, characteristic, complemented
6C4×#832C212yesmaximal, complemented
7#69657C212yesmaximal, complemented
8#69636C212yesmaximal, complemented
9#69584C211yesmaximal, characteristic, complemented
10#69567C211yesmaximal, characteristic, complemented
11C4×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
12C4×#459—31nomaximal, Sylow
13C2×C3×#127C2211yescharacteristic
14C2×C3×C4×D4—21no—
15C2×C3×C4×D4—21no—
16C2×C3×C4×D4C2212yes—
17C22×#213C2212yes—
18C2×C4×D12C2211yescharacteristic
19C2×C4×D12C2211yescharacteristic
20C2×C4×D12C2211yescharacteristic
21C2×C42×D3—21no—
22C3×#390—21no—
23C3×#390—21no—
24C3×#390C2212yes—
25C3×#384C2211yescharacteristic
26C3×C43C2211yescharacteristic
27C2×#832C2212yes—
28C2×#832C2212yes—
29C2×#832C2212yes—
30#5072C418yesdirect product factor
31C4×#213—28no—
32C2×#809C2212yes—
33C2×#807C2212yes—
34C4×(C2×D3)⋊C4—21no—
35C4×(C2×D3)⋊C4—21no—
36C4×(C2×D3)⋊C4—21no—
37C4×C4⋊Dic3C2211yescharacteristic
38C4×C4⋊Dic3C2211yescharacteristic
39C4×C4⋊Dic3C2211yescharacteristic
40C2×C4⋊(C3⋊C8)C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×C4×C3⋊C8C2212yes—
43#4529C2212yes—
44#4529C2212yes—
45#4524C2212yes—
46C2×#459—31no—
47C2×C4×D8—32no—
48C4×#127D311yescharacteristic, complemented
49C42×D4—31no—
50#1982—31no—
51#1947—31no—
52#1925—32no—
53#1879—32no—
54#1802—31no—
55C4×C4⋊C8—31no—
56C4×#102—32no—
57C22×C3×D4—21no—
58C22×C3×D4—21no—
59C22×C3×D4C2312yes—
60C23×C3×C4—22no—
61C22×D12C2311yescharacteristic
62C22×C4×D3—21no—
63C22×C4×D3—21no—
64C22×C4×D3—21no—
65C3×#127—22no—
66C3×#127C2×C414yescomplemented
67C3×C4×D4—24no—
68C3×C4×D4—24no—
69C3×C4×D4—44no—
70C3×C4×D4—44no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C42D411yescharacteristic
78C2×C3×C42D411yescharacteristic
79C2×C3×C42D411yescharacteristic
80C2×C3×C42D411yescharacteristic
81C2×C3×C42C2311yescharacteristic
82C2×C3×C42C2311yescharacteristic
83C2×C3×C42C2311yescharacteristic
84C2×#213—24no—
85C2×#213—24no—
86C2×#213C2×C418yescomplemented
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×C4⋊Dic3C2311yescharacteristic
91C2×C4⋊Dic3C2311yescharacteristic
92C2×C4⋊Dic3C2311yescharacteristic
93C2×C4×Dic3—21no—
94C2×C4×Dic3—21no—
95C2×C4×Dic3—21no—
96C22×C3⋊C8C2312yes—
97C22×C3⋊C8C2312yes—
98C4×D12—22no—
99C4×D12—22no—
100C4×D12—22no—
101C4×D12—22no—
102C4×D12C2×C414yescomplemented
103C4×D12D414yescomplemented
104C42×D3—44no—
105#832—24no—
106#832—24no—
107#832C2×C418yescomplemented
108C3⋊#95—21no—
109C3⋊#95—21no—
110C3⋊#95—21no—
111#809—24no—
112#807—24no—
113C4⋊(C3⋊C8)—22no—
114C4⋊(C3⋊C8)C2×C414yescomplemented
115C4×C3⋊C8—24no—
116C22×D8—32no—
117C2×#127C2×D311yescharacteristic
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—31no—
121C2×C4×D4—31no—
122C2×C4×D4—31no—
123C2×C4×D4C2×D312yes—
124C22×C42—61no—
125#459—38no—
126C4×D8—68no—
127C2×#107—32no—
128C2×C4⋊C8—31no—
129C2×C4⋊C8—31no—
130C2×#102—32no—
131C2×#102—32no—
132C2×#102—32no—
133C2×#102—32no—
134C2×C4×C8—32no—
135#390—21no—
136#390—21no—
137#390C2×D312yes—
138#384C2×D311yescharacteristic
139C4×C4⋊C4—31no—
140C4×C4⋊C4—31no—
141C4×C4⋊C4—31no—
142C4×C22⋊C4—61no—
143C4×C22⋊C4—61no—
144C4×C22⋊C4—61no—
145C43C2×D311yescharacteristic
146#340—32no—
147#340—32no—
148#336—32no—
149C24×C3—22no—
150C23×D3—21no—
151C2×C3×D4—24no—
152C2×C3×D4—24no—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4—42no—
159C2×C3×D4—42no—
160C2×C3×D4—42no—
161C2×C3×D4—42no—
162C2×C3×D4C22×C414yes—
163C22×C3×C4—22no—
164C22×C3×C4—44no—
165C22×C3×C4—44no—
166C22×C3×C4—44no—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2411yescharacteristic
174C22×Dic3—21no—
175C22×Dic3—21no—
176C22×Dic3—21no—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12—21no—
180C2×D12C22×C412yes—
181C2×D12C2×D412yes—
182C2×D12#4112yes—
183C2×C4×D3—24no—
184C2×C4×D3—24no—
185C2×C4×D3—24no—
186C2×C4×D3—42no—
187C2×C4×D3—42no—
188C2×C4×D3—42no—
189C2×C4×D3—42no—
190C2×C4×D3—42no—
191C2×C4×D3—42no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—42no—
195C3×C4⋊C4—42no—
196C3×C22⋊C4—44no—
197C3×C22⋊C4—44no—
198C3×C22⋊C4—44no—
199C3×C42—22no—
200C3×C42—22no—
201C3×C42—21no—
202C3×C42—21no—
203C3×C42—41no—
204C3×C42D814yescomplemented
205C3×C42C22×C412yes—
206C3×C42C2×D412yes—
207C3×C42#4112yes—
208C3×C42#4112yes—
209#213—216no—
210(C2×D3)⋊C4—24no—
211(C2×D3)⋊C4—24no—
212(C2×D3)⋊C4—24no—
213C4⋊Dic3—21no—
214C4⋊Dic3—21no—
215C4⋊Dic3—21no—
216C4⋊Dic3C22×C412yes—
217C4⋊Dic3C2×D412yes—
218C4⋊Dic3#4112yes—
219C4×Dic3—42no—
220C4×Dic3—42no—
221C4×Dic3—42no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8C22×C414yes—
228C22×D4—21no—
229C22×D4—21no—
230C22×D4—31no—
231C22×D4C22×D312yes—
232C23×C4—22no—
233C23×C4—61no—
234C23×C4—61no—
235C23×C4—61no—
236C2×D8—38no—
237C2×D8—64no—
238C2×D8—64no—
239C22×C8—32no—
240C22×C8—32no—
241#127—22no—
242#127C4×D314yescomplemented
243C4×D4—24no—
244C4×D4—24no—
245C4×D4—34no—
246C4×D4—34no—
247C4×D4—44no—
248C4×D4—44no—
249C4×D4—62no—
250C4×D4—62no—
251C4×D4—62no—
252C4×D4—62no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—31no—
256C2×C4⋊C4—31no—
257C2×C4⋊C4—31no—
258C2×C4⋊C4C22×D312yes—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—61no—
263C2×C22⋊C4—61no—
264C2×C22⋊C4—61no—
265C2×C42—61no—
266C2×C42—61no—
267C2×C42—61no—
268C2×C42—124no—
269C2×C42#6711yescharacteristic
270C2×C42#6711yescharacteristic
271C2×C42#6711yescharacteristic
272C2×C42#6711yescharacteristic
273C2×C42C22×D311yescharacteristic
274C2×C42C22×D311yescharacteristic
275C2×C42C22×D311yescharacteristic
276#107—64no—
277C4⋊C8—34no—
278C4⋊C8—62no—
279#102—38no—
280#102—64no—
281#102—64no—
282#102—64no—
283C4×C8—64no—
284#95—61no—
285#95—61no—
286#95—61no—
287C23×C3—24no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3—42no—
294C23×C3C22×D411yescharacteristic
295C22×D3—22no—
296C22×D3—22no—
297C22×D3—22no—
298C22×D3—41no—
299C22×D3—41no—
300C22×D3—41no—
301C22×D3—41no—
302C3×D4—24no—
303C3×D4—24no—
304C3×D4—48no—
305C3×D4—42no—
306C3×D4—42no—
307C3×D4—48no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—88no—
321C2×C3×C4C4×D412yes—
322C2×C3×C4C4×D412yes—
323C2×C3×C4#12112yes—
324C2×C3×C4#12112yes—
325C2×C3×C4C2×D812yes—
326C2×C3×C4C2×D812yes—
327C2×C3×C4C2×D812yes—
328C2×C3×C4#13512yes—
329C2×C3×C4AGL(1,ℤ/8)12yes—
330C2×C3×C4C23×C411yescharacteristic
331C2×C3×C4C22×D411yescharacteristic
332C2×C3×C4C2×#4111yescharacteristic
333C2×C3×C4C2×#4111yescharacteristic
334C2×C3×C4C2×#4111yescharacteristic
335C2×C3×C4C2×#4111yescharacteristic
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—41no—
340C2×Dic3—41no—
341C2×Dic3—41no—
342C2×Dic3—41no—
343C2×Dic3—41no—
344C2×Dic3—41no—
345D12—22no—
346D12—22no—
347D12—22no—
348D12C4×D414yescomplemented
349C4×D3—48no—
350C4×D3—48no—
351C4×D3—48no—
352C3⋊C8—24no—
353C3⋊C8—42no—
354C24—22no—
355C24—61no—
356C2×D4—24no—
357C2×D4—24no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—32no—
364C2×D4—32no—
365C2×D4—32no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—61no—
371C2×D4—61no—
372C2×D4—61no—
373C2×D4C2×C4×D314yes—
374C22×C4—22no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—64no—
379C22×C4—61no—
380C22×C4—61no—
381C22×C4—61no—
382C22×C4—64no—
383C22×C4—64no—
384C22×C4—122no—
385C22×C4—122no—
386C22×C4—122no—
387C22×C4—122no—
388C22×C4—122no—
389C22×C4—122no—
390C22×C4C2×#6711yescharacteristic
391C22×C4C2×#6711yescharacteristic
392C22×C4C2×#6711yescharacteristic
393C22×C4C2×#6711yescharacteristic
394C22×C4C2×#6711yescharacteristic
395C22×C4C2×#6711yescharacteristic
396C22×C4C23×D311yescharacteristic
397D8—616no—
398C2×C8—34no—
399C2×C8—62no—
400C2×C8—62no—
401C2×C8—62no—
402C2×C8—62no—
403C2×C8—62no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—32no—
407C4⋊C4—32no—
408C4⋊C4—32no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—61no—
412C4⋊C4—61no—
413C4⋊C4—61no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—64no—
418C22⋊C4—64no—
419C22⋊C4—64no—
420C42—22no—
421C42—22no—
422C42—21no—
423C42—21no—
424C42—41no—
425C42—122no—
426C42—122no—
427C42—122no—
428C42#21314yescomplemented
429C42C2×C4×D312yes—
430C42#23512yes—
431C42D3×D412yes—
432C42#23712yes—
433C22×C3—44no—
434C22×C3—44no—
435C22×C3—44no—
436C22×C3—82no—
437C22×C3—82no—
438C22×C3—82no—
439C22×C3—82no—
440C22×C3C2×C4×D411yescharacteristic
441C22×C3C2×#12111yescharacteristic
442C22×C3#52511yescharacteristic
443C22×C3C22×D811yescharacteristic
444C22×C3C2×#13511yescharacteristic
445C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
446C22×C3#57511yescharacteristic
447C2×D3—24no—
448C2×D3—42no—
449C2×D3—42no—
450C2×D3—42no—
451C2×D3—42no—
452C2×D3—42no—
453C2×D3—42no—
454C3×C4—22no—
455C3×C4—22no—
456C3×C4—22no—
457C3×C4—41no—
458C3×C4—82no—
459C3×C4C4×D814yescomplemented
460C3×C4#45914yescomplemented
461C3×C4C2×C4×D411yescharacteristic
462C3×C4C4×#4111yescharacteristic
463C3×C4#52911yescharacteristic
464C3×C4#53211yescharacteristic
465Dic3—42no—
466Dic3—42no—
467Dic3—42no—
468C23—24no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—62no—
476C23—62no—
477C23—62no—
478C23—121no—
479C23—121no—
480C23—121no—
481C23—121no—
482C23C22×#6711yescharacteristic
483D4—24no—
484D4—24no—
485D4—34no—
486D4—48no—
487D4—48no—
488D4—42no—
489D4—42no—
490D4—62no—
491D4—62no—
492D4—62no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—62no—
508C2×C4—88no—
509C2×C4—128no—
510C2×C4—121no—
511C2×C4—121no—
512C2×C4—121no—
513C2×C4—121no—
514C2×C4—121no—
515C2×C4—121no—
516C2×C4—128no—
517C2×C4—128no—
518C2×C4C4×#6712yes—
519C2×C4C4×#6712yes—
520C2×C4C2×#21312yes—
521C2×C4C2×#21312yes—
522C2×C4C2×#21312yes—
523C2×C4#93212yes—
524C2×C4#93812yes—
525C2×C4#93812yes—
526C2×C4#95012yes—
527C2×C4C22×C4×D311yescharacteristic
528C2×C4C2×#23511yescharacteristic
529C2×C4C2×D3×D411yescharacteristic
530C2×C4C2×#23711yescharacteristic
531C2×C4D3×#4111yescharacteristic
532C2×C4D3×#4111yescharacteristic
533C8—64no—
534C8—122no—
535C2×C3—84no—
536C2×C3C4×#12111yescharacteristic
537C2×C3C2×C4×D811yescharacteristic
538C2×C3C4×AGL(1,ℤ/8)11yescharacteristic
539C2×C3C2×#45911yescharacteristic
540C2×C3#303511yescharacteristic
541C2×C3#310011yescharacteristic
542C2×C3#310611yescharacteristic
543D3—44no—
544C22—44no—
545C22—44no—
546C22—44no—
547C22—64no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—122no—
553C22—122no—
554C22—122no—
555C22—122no—
556C22—122no—
557C22—122no—
558C22C2×C4×#6711yescharacteristic
559C22C22×#21311yescharacteristic
560C22C2×#93211yescharacteristic
561C22C2×#93811yescharacteristic
562C22C2×#95011yescharacteristic
563C22#582211yescharacteristic
564C22#582811yescharacteristic
565C4—22no—
566C4—22no—
567C4—22no—
568C4—41no—
569C4—82no—
570C4—122no—
571C4—122no—
572C4—122no—
573C4C4×#21314yescomplemented
574C4#507214yesdirect product factor
575C4C4×#23711yescharacteristic
576C4C4×D3×D411yescharacteristic
577C4#554811yescharacteristic
578C4#555711yescharacteristic
579C3C4×#45911yescharacteristic, complemented, Sylow
580C2—84no—
581C2—124no—
582C2C2×C4×#21311yescharacteristic
583C2C4×#93211yescharacteristic
584C2C4×#93811yescharacteristic
585C2C2×#507211yescharacteristic
586C2#8329611yescharacteristic
587C2#8329811yescharacteristic
588C2#8329911yescharacteristic
subgroupG/H
centerC22×C4C2×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×C4×#127C2
socleC23×C3C22×D4
prime radicalC2×#5072C2
cosocleC22×C3×C4C24
hypercenterC4×#127D3
hypocenterC3C4×#459
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C43 ◁ C4×#127

Complements

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

NH ≅ G/Ntype
1. C3×C4×#127C2semidirect
3. C2×C4×#213C2semidirect
4. C42×D12C2semidirect
5. #71761C2semidirect
6. C4×#832C2semidirect
7. #69657C2semidirect
8. #69636C2semidirect
9. #69584C2semidirect
10. #69567C2semidirect
11. C4×C4⋊(C3⋊C8)C2semidirect
30. #5072C4direct (N × H)
48. C4×#127D3semidirect
66. C3×#127C2×C4semidirect
86. C2×#213C2×C4semidirect
102. C4×D12C2×C4semidirect
103. C4×D12D4semidirect
107. #832C2×C4semidirect
114. C4⋊(C3⋊C8)C2×C4semidirect
204. C3×C42D8semidirect
242. #127C4×D3semidirect
348. D12C4×D4semidirect
428. C42#213semidirect
459. C3×C4C4×D8semidirect
460. C3×C4#459semidirect
573. C4C4×#213semidirect
574. C4#5072direct (N × H)
579. C3C4×#459semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#1272
C2×#50722
C2×C4×#2132
C42×D122
#717612
C4×#8322
#696572
#696362
#695842
#695672
C4×C4⋊(C3⋊C8)2
C4×#4593