Finite Group Databasebeta

← C4×#1519 #182637

Subgroups — C4×#1519 #182637

Order 512.

1954 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2C2×C4×C22⋊C8C211yesmaximal, characteristic, complemented
3C2×#1519C212yesmaximal
4C4×#380C211yesmaximal, characteristic, complemented
5#8220C214yesmaximal, complemented
6C4×#327C214yesmaximal, complemented
7#8023C212yesmaximal, complemented
8C22×#121C2211yescharacteristic
9C22×C4×D4—21no—
10C22×C4×D4C411yescharacteristic, complemented
11C23×C42C2211yescharacteristic
12C22×C22⋊C8C2212yes—
13C22×C4×C8—21no—
14C4×#121—21no—
15C4×#121—21no—
16C4×#121—21no—
17C4×#121—21no—
18C4×#121C412yescomplemented
19C4×#121C412yescomplemented
20C4×#121C2212yes—
21C4×#121C2212yes—
22C2×#390C411yescharacteristic, complemented
23C2×#389C411yescharacteristic, complemented
24C2×#388—21no—
25C2×#386—21no—
26C2×#380C2212yes—
27C2×#380C2211yescharacteristic
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C411yescharacteristic, complemented
30C2×C4×C22⋊C4—21no—
31C2×C4×C22⋊C4—21no—
32C4×C22⋊C8—21no—
33C4×C22⋊C8—21no—
34C4×C22⋊C8C2212yescomplemented
35C4×C22⋊C8C2212yescomplemented
36C2×#336—21no—
37#1519C4116yesdirect product factor
38C2×#327C2218yes—
39C4⋊#95—21no—
40#1434C2212yescomplemented
41C22⋊#95—21no—
42C4×#95C2212yescomplemented
43#1294C2218yescomplemented
44C23×D4—21no—
45C23×D4C2×C411yescharacteristic
46C24×C4—21no—
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C24×C4C2311yescharacteristic
50C23×C8—22no—
51C2×#121—21no—
52C2×#121—21no—
53C2×#121—24no—
54C2×#121C2×C418yes—
55C2×#121C2×C412yes—
56C2×#121C2312yes—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4—44no—
64C2×C4×D4—44no—
65C2×C4×D4—44no—
66C2×C4×D4C2×C412yes—
67C2×C4×D4C2×C412yes—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C42—21no—
80C22×C42—22no—
81C22×C42—44no—
82C22×C42C2×C411yescharacteristic
83C22×C42C2×C411yescharacteristic
84C22×C42C2×C411yescharacteristic
85C22×C42C2×C411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94C2×C22⋊C8—22no—
95C2×C22⋊C8—24no—
96C2×C22⋊C8C2×C418yescomplemented
97C2×C22⋊C8C2314yes—
98C2×C4×C8—22no—
99C2×C4×C8—22no—
100C2×C4×C8—44no—
101#390—21no—
102#390—21no—
103#390C2×C412yes—
104#390C2×C412yes—
105#389—21no—
106#389—21no—
107#389C2×C412yes—
108#389C2×C412yes—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#386—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#380—21no—
118#380—21no—
119#380—24no—
120#380C2×C412yes—
121#380C2×C418yescomplemented
122#380C2312yes—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—44no—
126C4×C4⋊C4C2×C412yes—
127C4×C4⋊C4C2×C412yes—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—44no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95C2314yes—
140#336—22no—
141#336—22no—
142#327C2×C4132yescomplemented
143C25—21no—
144C25C2×D411yescharacteristic
145C22×D4—22no—
146C22×D4—28no—
147C22×D4—22no—
148C22×D4—21no—
149C22×D4—22no—
150C22×D4—42no—
151C22×D4—42no—
152C22×D4—42no—
153C22×D4—42no—
154C22×D4—44no—
155C22×D4C4214yescomplemented
156C22×D4C22×C412yes—
157C23×C4—21no—
158C23×C4—24no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—24no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—44no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C411yescharacteristic
181C23×C4D814yes—
182C23×C4SD1614yes—
183C23×C4C22×C411yescharacteristic
184C23×C4C22×C411yescharacteristic
185C23×C4C22×C414yes—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4#4114yes—
194C23×C4C2411yescharacteristic
195C22×C8—28no—
196C22×C8—24no—
197C22×C8—44no—
198C22×C8—44no—
199#121—24no—
200#121—28no—
201#121—28no—
202#121—24no—
203#121C4218yescomplemented
204#121C22×C418yes—
205C4×D4—44no—
206C4×D4—44no—
207C4×D4—44no—
208C4×D4—48no—
209C4×D4—48no—
210C4×D4—44no—
211C4×D4—44no—
212C4×D4—44no—
213C4×D4—44no—
214C4×D4—44no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4C4214yescomplemented
228C2×C4⋊C4C22×C412yes—
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C22×C412yes—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42—42no—
263C2×C42—42no—
264C2×C42—44no—
265C2×C42—44no—
266C2×C42—44no—
267C2×C42—44no—
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22⋊C411yescharacteristic
273C2×C42C22⋊C411yescharacteristic
274C2×C42C22⋊C411yescharacteristic
275C2×C42C22⋊C411yescharacteristic
276C2×C42C22×C411yescharacteristic
277C2×C42C22×C411yescharacteristic
278C2×C42C22×C411yescharacteristic
279C2×C42C22×C411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C22⋊C8—28no—
285C22⋊C8—24no—
286C22⋊C8C22×C418yescomplemented
287C4×C8—44no—
288C4×C8—44no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—24no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95C22×C418yescomplemented
300C24—22no—
301C24—24no—
302C24—41no—
303C24—41no—
304C24—41no—
305C24—41no—
306C24—42no—
307C24—42no—
308C24—42no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24C2×C22⋊C411yescharacteristic
313C24C2×C22⋊C411yescharacteristic
314C24C4×D414yes—
315C24C2×D812yes—
316C24C2×SD1612yes—
317C24#13514yes—
318C24C22×D411yescharacteristic
319C2×D4—28no—
320C2×D4—28no—
321C2×D4—22no—
322C2×D4—24no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—48no—
326C2×D4—48no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—416no—
333C2×D4—416no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4C2×C4214yes—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—28no—
366C22×C4—48no—
367C22×C4—48no—
368C22×C4—48no—
369C22×C4—48no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—416no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—44no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—816no—
410C22×C4#10214yes—
411C22×C4#10214yes—
412C22×C4#10214yes—
413C22×C4#10214yes—
414C22×C4C2×C4212yes—
415C22×C4C2×C4212yes—
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4#11712yes—
427C22×C4#11712yes—
428C22×C4C4×D414yes—
429C22×C4C4×D414yes—
430C22×C4C4×D414yes—
431C22×C4C2×D812yes—
432C22×C4C2×D812yes—
433C22×C4C2×SD1612yes—
434C22×C4C2×SD1612yes—
435C22×C4C23×C411yescharacteristic
436C22×C4C23×C412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C2×#4112yes—
439C2×C8—28no—
440C2×C8—44no—
441C2×C8—44no—
442C2×C8—416no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—416no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4C2×C4214yes—
453C22⋊C4—28no—
454C22⋊C4—28no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—44no—
460C22⋊C4—416no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—41no—
475C42—41no—
476C42—41no—
477C42—41no—
478C42—41no—
479C42—41no—
480C42—44no—
481C42—44no—
482C42—88no—
483C42#9914yes—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—24no—
497C23—44no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—44no—
502C23—42no—
503C23—42no—
504C23—44no—
505C23—48no—
506C23—48no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—48no—
514C23—84no—
515C23—84no—
516C23—84no—
517C23—84no—
518C23C4×C22⋊C412yes—
519C23C4×C22⋊C412yes—
520C23C2×#10212yes—
521C23C2×#10212yes—
522C23#41612yes—
523C23#41612yes—
524C23C4×D818yes—
525C23C4×SD1618yes—
526C23C22×C22⋊C411yescharacteristic
527C23C2×C4×D412yes—
528C23C22×D811yescharacteristic
529C23C22×SD1611yescharacteristic
530C23C2×#13512yes—
531D4—44no—
532D4—44no—
533D4—48no—
534D4—48no—
535D4—416no—
536D4—48no—
537C2×C4—28no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—24no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—48no—
586C2×C4—48no—
587C2×C4—48no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—48no—
595C2×C4—88no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4—88no—
603C2×C4C4×C22⋊C412yes—
604C2×C4C4×C22⋊C412yes—
605C2×C4C2×#9912yes—
606C2×C4C2×#9912yes—
607C2×C4C2×#9912yes—
608C2×C4#41012yes—
609C2×C4C2×#10212yes—
610C2×C4C2×#10212yes—
611C2×C4C2×#10212yes—
612C2×C4C2×#10212yes—
613C2×C4#42112yes—
614C2×C4#42112yes—
615C2×C4C22×C4211yescharacteristic
616C2×C4C2×#11711yescharacteristic
617C2×C4C2×C4×D412yes—
618C8—48no—
619C22—21no—
620C22—22no—
621C22—21no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—21no—
630C22—22no—
631C22—21no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—48no—
640C22—48no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—816no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22C4×#10214yes—
650C22C4×#10214yes—
651C22C2×C4×C22⋊C411yescharacteristic
652C22C22×#9911yescharacteristic
653C22C2×#41011yescharacteristic
654C22C22×#10211yescharacteristic
655C22C2×#41611yescharacteristic
656C22C2×#42111yescharacteristic
657C22C2×#42111yescharacteristic
658C22C2×C4×D812yes—
659C22C2×C4×SD1612yes—
660C4—22no—
661C4—22no—
662C4—22no—
663C4—22no—
664C4—22no—
665C4—22no—
666C4—21no—
667C4—22no—
668C4—21no—
669C4—41no—
670C4—41no—
671C4—44no—
672C4—88no—
673C4—82no—
674C4—82no—
675C4#151914yesdirect product factor
676C4C4×#9914yes—
677C2—22no—
678C2—24no—
679C2—22no—
680C2—24no—
681C2—84no—
682C2C2×#151911yescharacteristic
683C2#1756711yescharacteristic
684C2#1757211yescharacteristic
685C2#1757411yescharacteristic
686C2C2×C4×#9911yescharacteristic
687C2C4×#42111yescharacteristic
688C2C2×C4×#10211yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC22×#121C22
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊C8C2semidirect
4. C4×#380C2semidirect
5. #8220C2semidirect
6. C4×#327C2semidirect
7. #8023C2semidirect
10. C22×C4×D4C4semidirect
18. C4×#121C4semidirect
19. C4×#121C4semidirect
22. C2×#390C4semidirect
23. C2×#389C4semidirect
29. C2×C4×C4⋊C4C4semidirect
34. C4×C22⋊C8C22semidirect
35. C4×C22⋊C8C22semidirect
37. #1519C4direct (N × H)
40. #1434C22semidirect
42. C4×#95C22semidirect
43. #1294C22semidirect
96. C2×C22⋊C8C2×C4semidirect
121. #380C2×C4semidirect
142. #327C2×C4semidirect
155. C22×D4C42semidirect
203. #121C42semidirect
227. C2×C4⋊C4C42semidirect
286. C22⋊C8C22×C4semidirect
299. #95C22×C4semidirect
675. C4#1519direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#1212
C2×C4×C22⋊C82
C2×#15192
C4×#3802
#82202
C4×#3272
#80232