Finite Group Databasebeta

← C4×#1880 #354254

Subgroups — C4×#1880 #354254

Order 512.

2256 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×SD16C211yesmaximal, characteristic
2C2×C42×Q8C211yesmaximal, characteristic, complemented
3C2×#1880C212yesmaximal
4C2×C4×#103C211yesmaximal, characteristic, complemented
5C2×C4×#102C211yesmaximal, characteristic
6C4×#390C211yesmaximal, characteristic
7#10931C212yesmaximal
8#10908C212yesmaximal, complemented
9#10789C212yesmaximal, complemented
10C4×#340C211yesmaximal, characteristic, complemented
11C4×#336C211yesmaximal, characteristic, complemented
12C23×SD16C2211yescharacteristic
13C22×C4×Q8C2212yes—
14C22×C4×Q8C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C2×C4×SD16—24no—
17C2×C4×SD16—24no—
18C2×C4×SD16C418yescomplemented
19C22×#106C2211yescharacteristic
20C22×#103C2211yescharacteristic
21C22×#103C2212yes—
22C22×#102C2211yescharacteristic
23C22×#102C2212yes—
24C22×C4×C8C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C2×#390C2212yes—
27C42×Q8—24no—
28C42×Q8—24no—
29C2×C4×C4⋊C4—22no—
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4C2212yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4—21no—
35C2×C4×C22⋊C4—21no—
36C2×C43—21no—
37C2×C43C2211yescharacteristic
38#1880C4116yesdirect product factor
39C4×#103—24no—
40C4×#102—24no—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#336C2212yes—
46C2×#336C2212yes—
47#1437—22no—
48C4⋊#95C2212yes—
49C23×Q8C2311yescharacteristic
50C23×D4C2311yescharacteristic
51C24×C4—21no—
52C22×SD16—22no—
53C22×SD16—22no—
54C22×SD16—24no—
55C22×SD16C2×C418yes—
56C22×SD16C2×C414yes—
57C23×C8C2312yes—
58C23×C8C2311yescharacteristic
59C23×C8C2311yescharacteristic
60C2×C4×Q8—24no—
61C2×C4×Q8—24no—
62C2×C4×Q8—24no—
63C2×C4×Q8—24no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—24no—
67C2×C4×Q8—22no—
68C2×C4×Q8C2×C414yes—
69C2×C4×Q8C2×C418yescomplemented
70C2×C4×Q8D414yes—
71C2×C4×Q8D414yes—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4C2×C414yes—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—22no—
86C22×C22⋊C4—22no—
87C22×C42—22no—
88C22×C42—21no—
89C22×C42—22no—
90C22×C42—21no—
91C22×C42—21no—
92C22×C42D411yescharacteristic
93C22×C42D412yes—
94C22×C42D411yescharacteristic
95C22×C42C2312yes—
96C22×C42C2311yescharacteristic
97C4×SD16—232no—
98C2×#106—22no—
99C2×#106—22no—
100C2×#106C2×C414yes—
101C2×#103—22no—
102C2×#103—22no—
103C2×#103—24no—
104C2×#103—24no—
105C2×#103C2×C414yes—
106C2×#103C2×C418yescomplemented
107C2×#102—22no—
108C2×#102—22no—
109C2×#102—24no—
110C2×#102—24no—
111C2×#102C2×C418yes—
112C2×#102C2×C414yes—
113C2×C4×C8—22no—
114C2×C4×C8—22no—
115C2×C4×C8—22no—
116C2×C4×C8—22no—
117C2×C4×C8—22no—
118C2×C4×C8C2×C414yes—
119#390—24no—
120#390C2×C418yes—
121C4×C4⋊C4—28no—
122C4×C4⋊C4—216no—
123C4×C4⋊C4—24no—
124C4×C4⋊C4—24no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4D414yes—
129C4×C22⋊C4—44no—
130C4×C22⋊C4—44no—
131C2×#95—22no—
132C2×#95—21no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—21no—
136C43—28no—
137C43—22no—
138C43—22no—
139#340—24no—
140#340—24no—
141#340—24no—
142#340C2×C418yescomplemented
143#336—24no—
144#336C2×C418yescomplemented
145C25—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—22no—
149C22×Q8—21no—
150C22×Q8—22no—
151C22×Q8C22×C412yes—
152C22×Q8C22×C414yes—
153C22×Q8C2×D412yes—
154C22×Q8C2×D412yes—
155C22×Q8#4114yes—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—21no—
163C22×D4—22no—
164C22×D4C22×C412yes—
165C22×D4C22×C414yes—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—21no—
171C23×C4—24no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4C2×D412yes—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D412yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2411yescharacteristic
183C2×SD16—28no—
184C2×SD16—28no—
185C2×SD16—216no—
186C2×SD16—216no—
187C2×SD16C42116yescomplemented
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8C22×C414yes—
202C22×C8C22×C412yes—
203C22×C8C22×C414yes—
204C22×C8C22×C412yes—
205C4×Q8—216no—
206C4×Q8—216no—
207C4×Q8—216no—
208C4×Q8—216no—
209C4×Q8—28no—
210C4×Q8—28no—
211C4×Q8—28no—
212C4×Q8C22⋊C4116yescomplemented
213C4×D4—28no—
214C4×D4—28no—
215C4×D4—44no—
216C4×D4—44no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4C22×C412yes—
240C2×C4⋊C4C22×C414yes—
241C2×C4⋊C4C22×C412yes—
242C2×C4⋊C4C22×C414yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2×D414yes—
246C2×C4⋊C4#4114yes—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C42—28no—
257C2×C42—28no—
258C2×C42—24no—
259C2×C42—28no—
260C2×C42—24no—
261C2×C42—22no—
262C2×C42—24no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—28no—
269C2×C42—24no—
270C2×C42—22no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—42no—
276C2×C42—44no—
277C2×C42—44no—
278C2×C42—42no—
279C2×C42—42no—
280C2×C42—42no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42SD1614yescomplemented
284C2×C42C22×C414yes—
285C2×C42C22×C412yes—
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42#4114yes—
290C2×C42#4112yes—
291C2×C42#4112yes—
292#106—28no—
293#103—28no—
294#103—216no—
295#102—28no—
296#102—216no—
297C4×C8—28no—
298C4×C8—44no—
299#95—28no—
300#95—24no—
301#95—44no—
302#95—42no—
303#95—44no—
304C24—24no—
305C24—22no—
306C24—41no—
307C24—41no—
308C24—42no—
309C24—41no—
310C24—42no—
311C24—41no—
312C24—41no—
313C24—41no—
314C24—42no—
315C24C22×D411yescharacteristic
316C2×Q8—22no—
317C2×Q8—24no—
318C2×Q8—22no—
319C2×Q8—24no—
320C2×Q8—22no—
321C2×Q8—24no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—24no—
325C2×Q8—24no—
326C2×Q8—24no—
327C2×Q8—24no—
328C2×Q8—24no—
329C2×Q8C2×C4214yes—
330C2×Q8C2×C22⋊C414yes—
331C2×Q8#11714yes—
332C2×Q8C4×D418yes—
333C2×Q8C4×D418yes—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—24no—
337C2×D4—22no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—22no—
341C2×D4—24no—
342C2×D4—22no—
343C2×D4—24no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4C2×C4214yes—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—84no—
399C22×C4—84no—
400C22×C4—84no—
401C22×C4—84no—
402C22×C4C4×D414yes—
403C22×C4C4×D412yes—
404C22×C4C4×D414yes—
405C22×C4C4×D412yes—
406C22×C4#12012yes—
407C22×C4#12012yes—
408C22×C4#12112yes—
409C22×C4#12112yes—
410C22×C4C2×SD1614yes—
411C22×C4C2×SD1612yes—
412C22×C4#13514yes—
413C22×C4#13512yes—
414C22×C4AGL(1,ℤ/8)12yes—
415C22×C4#13712yes—
416C22×C4C23×C411yescharacteristic
417C22×C4C23×C412yes—
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C2×#4112yes—
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4112yes—
424C22×C4C2×#4111yescharacteristic
425C22×C4C2×#4112yes—
426SD16—232no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—22no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—24no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8C2×C4214yes—
444C4⋊C4—232no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—216no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4C2×C22⋊C414yes—
461C4⋊C4C4×D418yes—
462C22⋊C4—416no—
463C22⋊C4—48no—
464C22⋊C4—416no—
465C42—232no—
466C42—24no—
467C42—24no—
468C42—24no—
469C42—24no—
470C42—216no—
471C42—28no—
472C42—48no—
473C42—416no—
474C42—44no—
475C42—48no—
476C42—84no—
477C42C2×C22⋊C414yes—
478C42C4×D414yes—
479C42#12114yes—
480C42#12314yes—
481C23—24no—
482C23—44no—
483C23—42no—
484C23—42no—
485C23—44no—
486C23—42no—
487C23—42no—
488C23—44no—
489C23—44no—
490C23—42no—
491C23—42no—
492C23—44no—
493C23—44no—
494C23—82no—
495C23—82no—
496C23—82no—
497C23—82no—
498C23—81no—
499C23—81no—
500C23—82no—
501C23—81no—
502C23—81no—
503C23—82no—
504C23C2×C4×D411yescharacteristic
505C23C2×C4×D412yes—
506C23C2×#12011yescharacteristic
507C23C2×#12111yescharacteristic
508C23#52512yes—
509C23C22×SD1611yescharacteristic
510C23C2×#13511yescharacteristic
511C23C2×#13512yes—
512C23C2×AGL(1,ℤ/8)11yescharacteristic
513C23C2×#13711yescharacteristic
514C23#57512yes—
515Q8—24no—
516Q8—24no—
517Q8—24no—
518Q8—28no—
519Q8—28no—
520Q8C4×C22⋊C418yescomplemented
521D4—24no—
522D4—24no—
523D4—42no—
524D4—42no—
525D4—44no—
526D4—44no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—82no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—82no—
575C2×C4—816no—
576C2×C4#38614yes—
577C2×C4#38614yes—
578C2×C4C4×SD1614yes—
579C2×C4C4×SD1618yescomplemented
580C2×C4#44014yes—
581C2×C4#44814yes—
582C2×C4#44914yes—
583C2×C4#46014yes—
584C2×C4#46414yes—
585C2×C4C22×C4211yescharacteristic
586C2×C4C22×C22⋊C411yescharacteristic
587C2×C4C2×#11711yescharacteristic
588C2×C4C2×C4×D412yes—
589C2×C4C2×C4×D412yes—
590C2×C4C2×C4×D411yescharacteristic
591C2×C4C2×C4×D412yes—
592C2×C4C4×#4111yescharacteristic
593C2×C4C4×#4112yes—
594C2×C4C4×#4111yescharacteristic
595C2×C4C4×#4112yes—
596C2×C4C2×#12111yescharacteristic
597C2×C4C2×#12311yescharacteristic
598C2×C4#52512yes—
599C2×C4#52512yes—
600C2×C4#52912yes—
601C2×C4#52912yes—
602C2×C4#53212yes—
603C8—24no—
604C8—42no—
605C8—44no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—84no—
614C22—84no—
615C22—84no—
616C22—84no—
617C22C42×D411yescharacteristic
618C22C2×#38611yescharacteristic
619C22#229011yescharacteristic
620C22C4×#12012yes—
621C22C4×#12112yes—
622C22C2×C4×SD1611yescharacteristic
623C22C2×C4×SD1612yes—
624C22C4×#13511yescharacteristic
625C22C4×#13512yes—
626C22C2×#44011yescharacteristic
627C22#294311yescharacteristic
628C22C4×AGL(1,ℤ/8)12yes—
629C22C4×#13712yes—
630C22C2×#44811yescharacteristic
631C22C2×#44911yescharacteristic
632C22#300212yes—
633C22C2×#46011yescharacteristic
634C22C2×#46411yescharacteristic
635C22#303512yes—
636C22#310212yes—
637C22#310312yes—
638C22#310612yes—
639C22#310812yes—
640C4—24no—
641C4—28no—
642C4—28no—
643C4—44no—
644C4—44no—
645C4—48no—
646C4—82no—
647C4—84no—
648C4#188018yesdirect product factor
649C4C2×C4×C22⋊C411yescharacteristic
650C4C42×D411yescharacteristic
651C4#229011yescharacteristic
652C4#229711yescharacteristic
653C4C4×#12112yes—
654C4C4×#12312yes—
655C2—84no—
656C2C4×#38611yescharacteristic
657C2C42×SD1611yescharacteristic
658C2C4×#44011yescharacteristic
659C2C2×#188011yescharacteristic
660C2#2092411yescharacteristic
661C2#2111511yescharacteristic
662C2#2112011yescharacteristic
663C2C4×#44812yes—
664C2C4×#44912yes—
665C2C4×#46012yes—
666C2C4×#46412yes—
subgroupG/H
centerC23×C4C2×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC24C22×D4
prime radicalC23×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C42×Q8C2semidirect
4. C2×C4×#103C2semidirect
8. #10908C2semidirect
9. #10789C2semidirect
10. C4×#340C2semidirect
11. C4×#336C2semidirect
18. C2×C4×SD16C4semidirect
38. #1880C4direct (N × H)
69. C2×C4×Q8C2×C4semidirect
106. C2×#103C2×C4semidirect
142. #340C2×C4semidirect
144. #336C2×C4semidirect
187. C2×SD16C42semidirect
212. C4×Q8C22⋊C4semidirect
283. C2×C42SD16semidirect
520. Q8C4×C22⋊C4semidirect
579. C2×C4C4×SD16semidirect
648. C4#1880direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×SD162
C2×C42×Q82
C2×#18802
C2×C4×#1032
C2×C4×#1022
C4×#3902
#109312
#109082
#107892
C4×#3402
C4×#3362