Finite Group Databasebeta

← C4×#5054 #10801416

Subgroups — C4×#5054 #10801416

Order 768.

1016 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#124C211yesmaximal, characteristic
2C2×#5054C211yesmaximal, characteristic
3C2×C4×#216C211yesmaximal, characteristic, complemented
4C2×C4×#214C211yesmaximal, characteristic
5C42×Dic6C211yesmaximal, characteristic, complemented
6#71721C211yesmaximal, characteristic
7C4×#835C211yesmaximal, characteristic, complemented
8C4×#832C211yesmaximal, characteristic
9#69690C211yesmaximal, characteristic, complemented
10#69666C211yesmaximal, characteristic, complemented
11#69660C211yesmaximal, characteristic
12#69636C211yesmaximal, characteristic
13#69559C211yesmaximal, characteristic, complemented
14#69540C211yesmaximal, characteristic, complemented
15C4×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
16C4×#464—31nomaximal, Sylow
17C2×C3×#124C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C22×#216C2211yescharacteristic
21C22×#214C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C3×#391C2211yescharacteristic
26C3×#390C2211yescharacteristic
27C3×#388—21no—
28C3×#388—21no—
29C3×#382C2211yescharacteristic
30C3×C4×C22⋊C4—21no—
31C3×C4×C22⋊C4—21no—
32C3×C43C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35C2×#835C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#832C2211yescharacteristic
38C2×#832C2211yescharacteristic
39#5054C418yesdirect product factor
40C4×#216—24no—
41C4×#214—24no—
42C2×#810C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C2×#807C2211yescharacteristic
46C4×C4⋊Dic3C2211yescharacteristic
47C4×C4⋊Dic3C2211yescharacteristic
48C4×C4⋊Dic3C2211yescharacteristic
49C4×Dic3⋊C4—21no—
50C4×Dic3⋊C4—21no—
51C4×Dic3⋊C4—21no—
52C42×Dic3—21no—
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54C2×C4⋊(C3⋊C8)C2211yescharacteristic
55C2×C4×C3⋊C8C2211yescharacteristic
56C2×C4×C3⋊C8C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62#4524C2211yescharacteristic
63C2×#464—31no—
64C2×C4×Q16—31no—
65C2×C4×SD16—31no—
66C4×#124D311yescharacteristic
67C42×Q8—31no—
68#1983—31no—
69#1949—31no—
70#1927—31no—
71#1925—31no—
72#1880—31no—
73#1878—31no—
74#1805—31no—
75C4×C4⋊C8—31no—
76C4×#103—31no—
77C4×#102—31no—
78C22×C3×Q8C2311yescharacteristic
79C22×C3×D4C2311yescharacteristic
80C23×C3×C4—21no—
81C22×Dic6C2311yescharacteristic
82C3×#124—22no—
83C3×#124C2×C414yes—
84C3×C4×Q8—22no—
85C3×C4×Q8—22no—
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C42—21no—
97C2×C3×C42D411yescharacteristic
98C2×C3×C42D411yescharacteristic
99C2×C3×C42D411yescharacteristic
100C2×C3×C42D411yescharacteristic
101C2×C3×C42C2311yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C2×C3×C42C2311yescharacteristic
104C2×#216—22no—
105C2×#216—22no—
106C2×#216C2×C414yescomplemented
107C2×#214—22no—
108C2×#214—22no—
109C2×#214C2×C414yes—
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3—21no—
120C2×C4×Dic3—21no—
121C2×C4×Dic3—21no—
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126C4×Dic6—22no—
127C4×Dic6—22no—
128C4×Dic6—22no—
129C4×Dic6—22no—
130C4×Dic6C2×C414yescomplemented
131C4×Dic6D414yes—
132C3×#95—21no—
133C3×#95—21no—
134C3×#95—21no—
135C3×#95—21no—
136#835—22no—
137#835—22no—
138#835C2×C414yescomplemented
139#832—22no—
140#832—22no—
141#832C2×C414yes—
142C3⋊#95—21no—
143C3⋊#95—21no—
144C3⋊#95—21no—
145#810—22no—
146#810—22no—
147#808—22no—
148#807—22no—
149C4⋊(C3⋊C8)—22no—
150C4⋊(C3⋊C8)C2×C414yescomplemented
151C4×C3⋊C8—22no—
152C4×C3⋊C8—22no—
153C22×Q16—31no—
154C22×SD16—31no—
155C2×#124C2×D311yescharacteristic
156C2×C4×Q8—31no—
157C2×C4×Q8—31no—
158C2×C4×Q8—31no—
159C2×C4×Q8C2×D311yescharacteristic
160C2×C4×D4C2×D311yescharacteristic
161#464—38no—
162C4×Q16—64no—
163C4×SD16—64no—
164C2×#107—31no—
165C2×#106—31no—
166C2×C4⋊C8—31no—
167C2×C4⋊C8—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#103—31no—
171C2×#103—31no—
172C2×#103—31no—
173C2×#102—31no—
174C2×#102—31no—
175C2×#102—31no—
176C2×C4×C8—31no—
177C2×C4×C8—31no—
178#391C2×D311yescharacteristic
179#390C2×D311yescharacteristic
180#388—21no—
181#388—21no—
182#382C2×D311yescharacteristic
183C4×C4⋊C4—31no—
184C4×C4⋊C4—31no—
185C4×C4⋊C4—31no—
186C4×C4⋊C4—61no—
187C4×C4⋊C4—61no—
188C4×C4⋊C4—61no—
189C4×C22⋊C4—21no—
190C4×C22⋊C4—21no—
191C43—61no—
192C43C2×D311yescharacteristic
193#340—31no—
194#340—31no—
195#340—31no—
196#340—31no—
197#336—31no—
198#336—31no—
199C24×C3—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8C22×C412yes—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4C22×C412yes—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—42no—
216C22×C3×C4—42no—
217C22×C3×C4—42no—
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2411yescharacteristic
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6C22×C412yes—
233C2×Dic6C2×D412yes—
234C2×Dic6#4112yes—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—42no—
240C3×C22⋊C4—24no—
241C3×C22⋊C4—24no—
242C3×C22⋊C4—42no—
243C3×C22⋊C4—42no—
244C3×C22⋊C4—42no—
245C3×C42—22no—
246C3×C42—22no—
247C3×C42—22no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42—41no—
251C3×C42—42no—
252C3×C42C22×C412yes—
253C3×C42C2×D412yes—
254C3×C42#4112yes—
255C3×C42#4112yes—
256#216—28no—
257#214—28no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3C22×C412yes—
262C4⋊Dic3C2×D412yes—
263C4⋊Dic3#4112yes—
264Dic3⋊C4—24no—
265Dic3⋊C4—24no—
266Dic3⋊C4—24no—
267C4×Dic3—24no—
268C4×Dic3—24no—
269C4×Dic3—24no—
270C4×Dic3—42no—
271C4×Dic3—42no—
272C4×Dic3—42no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8C22×C412yes—
284C2×C3⋊C8C22×C412yes—
285C22×Q8—31no—
286C22×Q8C22×D311yescharacteristic
287C22×D4C22×D311yescharacteristic
288C23×C4—21no—
289C2×Q16—34no—
290C2×Q16—62no—
291C2×Q16—62no—
292C2×SD16—34no—
293C2×SD16—62no—
294C2×SD16—62no—
295C22×C8—31no—
296C22×C8—31no—
297C22×C8—31no—
298C22×C8—31no—
299#124—22no—
300#124C4×D314yes—
301C4×Q8—22no—
302C4×Q8—22no—
303C4×Q8—34no—
304C4×Q8—34no—
305C4×Q8—62no—
306C4×Q8—62no—
307C4×Q8—62no—
308C4×Q8—62no—
309C4×D4—22no—
310C4×D4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4C22×D311yescharacteristic
319C2×C4⋊C4C22×D311yescharacteristic
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C42—21no—
326C2×C42—61no—
327C2×C42—61no—
328C2×C42—61no—
329C2×C42—61no—
330C2×C42—61no—
331C2×C42—61no—
332C2×C42#6711yescharacteristic
333C2×C42#6711yescharacteristic
334C2×C42#6711yescharacteristic
335C2×C42#6711yescharacteristic
336C2×C42C22×D311yescharacteristic
337C2×C42C22×D311yescharacteristic
338C2×C42C22×D311yescharacteristic
339#107—62no—
340#106—62no—
341C4⋊C8—34no—
342C4⋊C8—62no—
343#103—34no—
344#103—62no—
345#103—62no—
346#103—62no—
347#103—62no—
348#102—34no—
349#102—62no—
350#102—62no—
351C4×C8—62no—
352C4×C8—62no—
353#95—21no—
354#95—21no—
355#95—21no—
356#95—21no—
357#95—61no—
358#95—61no—
359#95—61no—
360C23×C3—22no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3C22×D411yescharacteristic
368C3×Q8—22no—
369C3×Q8—22no—
370C3×Q8—41no—
371C3×Q8—41no—
372C3×D4—22no—
373C3×D4—22no—
374C3×D4—41no—
375C3×D4—41no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—22no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—84no—
397C2×C3×C4C4×D412yes—
398C2×C3×C4C4×D412yes—
399C2×C3×C4#12112yes—
400C2×C3×C4#12112yes—
401C2×C3×C4#13512yes—
402C2×C3×C4#13512yes—
403C2×C3×C4#13712yes—
404C2×C3×C4C23×C411yescharacteristic
405C2×C3×C4C22×D411yescharacteristic
406C2×C3×C4C2×#4111yescharacteristic
407C2×C3×C4C2×#4111yescharacteristic
408C2×C3×C4C2×#4111yescharacteristic
409C2×C3×C4C2×#4111yescharacteristic
410C2×Dic3—22no—
411C2×Dic3—22no—
412C2×Dic3—22no—
413C2×Dic3—22no—
414C2×Dic3—22no—
415C2×Dic3—22no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422Dic6—22no—
423Dic6—22no—
424Dic6—22no—
425Dic6C4×D414yes—
426C3⋊C8—22no—
427C3⋊C8—22no—
428C3⋊C8—41no—
429C3⋊C8—41no—
430C24—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—32no—
437C2×Q8—32no—
438C2×Q8—32no—
439C2×Q8—61no—
440C2×Q8—61no—
441C2×Q8—61no—
442C2×Q8C2×C4×D312yes—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4C2×C4×D312yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4C2×#6711yescharacteristic
460C22×C4C2×#6711yescharacteristic
461C22×C4C2×#6711yescharacteristic
462C22×C4C2×#6711yescharacteristic
463C22×C4C2×#6711yescharacteristic
464C22×C4C2×#6711yescharacteristic
465C22×C4C23×D311yescharacteristic
466Q16—68no—
467SD16—68no—
468C2×C8—32no—
469C2×C8—32no—
470C2×C8—61no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8—61no—
479C2×C8—61no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—32no—
485C4⋊C4—32no—
486C4⋊C4—32no—
487C4⋊C4—42no—
488C4⋊C4—64no—
489C4⋊C4—64no—
490C4⋊C4—64no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—21no—
503C42—21no—
504C42—41no—
505C42—42no—
506C42—64no—
507C42—64no—
508C42—64no—
509C42—122no—
510C42—122no—
511C42—122no—
512C42C2×C4×D312yes—
513C42#23512yes—
514C42D3×D412yes—
515C42#23712yes—
516C22×C3—42no—
517C22×C3—42no—
518C22×C3—42no—
519C22×C3—81no—
520C22×C3—81no—
521C22×C3—81no—
522C22×C3—81no—
523C22×C3C2×C4×D411yescharacteristic
524C22×C3C2×#12111yescharacteristic
525C22×C3#52511yescharacteristic
526C22×C3C2×#13511yescharacteristic
527C22×C3C2×#13511yescharacteristic
528C22×C3C2×#13711yescharacteristic
529C22×C3#57511yescharacteristic
530C3×C4—22no—
531C3×C4—22no—
532C3×C4—22no—
533C3×C4—22no—
534C3×C4—41no—
535C3×C4—42no—
536C3×C4—81no—
537C3×C4—81no—
538C3×C4#46414yescomplemented
539C3×C4C2×C4×D411yescharacteristic
540C3×C4C4×#4111yescharacteristic
541C3×C4#52911yescharacteristic
542C3×C4#53211yescharacteristic
543Dic3—24no—
544Dic3—42no—
545Dic3—42no—
546Dic3—42no—
547C23—22no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23C22×#6711yescharacteristic
555Q8—22no—
556Q8—22no—
557Q8—34no—
558Q8—41no—
559Q8—41no—
560Q8—62no—
561Q8—62no—
562Q8—62no—
563D4—22no—
564D4—22no—
565D4—41no—
566D4—41no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—62no—
590C2×C4—62no—
591C2×C4—62no—
592C2×C4—62no—
593C2×C4—84no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4C4×#6712yes—
601C2×C4C4×#6712yes—
602C2×C4#93812yes—
603C2×C4#93812yes—
604C2×C4#95012yes—
605C2×C4#95012yes—
606C2×C4#95112yes—
607C2×C4C22×C4×D311yescharacteristic
608C2×C4C2×#23511yescharacteristic
609C2×C4C2×D3×D411yescharacteristic
610C2×C4C2×#23711yescharacteristic
611C2×C4D3×#4111yescharacteristic
612C2×C4D3×#4111yescharacteristic
613C8—62no—
614C8—62no—
615C8—121no—
616C8—121no—
617C2×C3—82no—
618C2×C3C4×#12111yescharacteristic
619C2×C3C4×#13511yescharacteristic
620C2×C3C4×#13711yescharacteristic
621C2×C3C2×#46411yescharacteristic
622C2×C3#303511yescharacteristic
623C2×C3#310211yescharacteristic
624C2×C3#310811yescharacteristic
625C22—42no—
626C22—42no—
627C22—42no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22C2×C4×#6711yescharacteristic
633C22C2×#93811yescharacteristic
634C22C2×#95011yescharacteristic
635C22C2×#95011yescharacteristic
636C22#582211yescharacteristic
637C22C2×#95111yescharacteristic
638C22#582811yescharacteristic
639C4—22no—
640C4—22no—
641C4—22no—
642C4—22no—
643C4—41no—
644C4—42no—
645C4—64no—
646C4—81no—
647C4—81no—
648C4—122no—
649C4—122no—
650C4—122no—
651C4#505414yesdirect product factor
652C4C4×#23711yescharacteristic
653C4C4×D3×D411yescharacteristic
654C4#554811yescharacteristic
655C4#555711yescharacteristic
656C3C4×#46411yescharacteristic, complemented, Sylow
657C2—82no—
658C2C4×#93811yescharacteristic
659C2C4×#95011yescharacteristic
660C2C4×#95111yescharacteristic
661C2C2×#505411yescharacteristic
662C2#8314011yescharacteristic
663C2#8315211yescharacteristic
664C2#8329611yescharacteristic
subgroupG/H
centerC22×C4C2×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×C4×#124C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
cosocleC22×C3×C4C24
hypercenterC4×#124D3
hypocenterC3C4×#464
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C43 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
3. C2×C4×#216C2semidirect
5. C42×Dic6C2semidirect
7. C4×#835C2semidirect
9. #69690C2semidirect
10. #69666C2semidirect
13. #69559C2semidirect
14. #69540C2semidirect
15. C4×C4⋊(C3⋊C8)C2semidirect
39. #5054C4direct (N × H)
106. C2×#216C2×C4semidirect
130. C4×Dic6C2×C4semidirect
138. #835C2×C4semidirect
150. C4⋊(C3⋊C8)C2×C4semidirect
538. C3×C4#464semidirect
651. C4#5054direct (N × H)
656. C3C4×#464semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#1242
C2×#50542
C2×C4×#2162
C2×C4×#2142
C42×Dic62
#717212
C4×#8352
C4×#8322
#696902
#696662
#696602
#696362
#695592
#695402
C4×C4⋊(C3⋊C8)2
C4×#4643