Finite Group Databasebeta

← C2×#33080 #6629577

Subgroups — C2×#33080 #6629577

Order 512.

2664 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2860C211yesmaximal, characteristic, complemented
2C2×C42×D4C211yesmaximal, characteristic, complemented
3#33080C2116yesmaximal, direct product factor
4C2×#1926C211yesmaximal, characteristic, complemented
5C2×#1926C211yesmaximal, characteristic, complemented
6C2×#1925C211yesmaximal, characteristic, complemented
7C2×#1921C211yesmaximal, characteristic, complemented
8C2×#1919C211yesmaximal, characteristic, complemented
9C2×#1920C211yesmaximal, characteristic, complemented
10C2×#1879C211yesmaximal, characteristic
11C2×#1879C211yesmaximal, characteristic
12C2×#1879C211yesmaximal, characteristic
13C2×#1864C211yesmaximal, characteristic, complemented
14C2×#1864C211yesmaximal, characteristic, complemented
15C2×#1862C211yesmaximal, characteristic, complemented
16C2×#1839C211yesmaximal, characteristic, complemented
17C23×D8C2211yescharacteristic
18C22×#126—21no—
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C23×C42—21no—
23C22×#107C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32#2860—24no—
33#2860C2218yescomplemented
34C2×#390C2211yescharacteristic
35C2×#390C2211yescharacteristic
36C2×#390C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#388—21no—
41C2×#388—21no—
42C2×#388—21no—
43C42×D4—24no—
44C42×D4C2218yescomplemented
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4—21no—
49C2×C4×C22⋊C4—21no—
50C2×C4×C22⋊C4—21no—
51C2×C43C2211yescharacteristic
52#1926C2218yescomplemented
53#1926C2218yescomplemented
54#1925C2218yescomplemented
55#1921C2218yescomplemented
56#1920C2218yescomplemented
57#1919C2218yescomplemented
58#1879C2218yes—
59#1879C2218yes—
60#1879C2218yes—
61#1864C2218yescomplemented
62#1864C2218yescomplemented
63#1862C2218yescomplemented
64#1839C2218yescomplemented
65C2×#340C2211yescharacteristic
66C2×#340C2211yescharacteristic
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#336C2211yescharacteristic
72C2×#336C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×D4C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C24×C4—21no—
80C24×C4—21no—
81C24×C4—21no—
82C22×D8—24no—
83C22×D8—24no—
84C22×D8—24no—
85C22×D8C2314yes—
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C2×#126—28no—
91C2×#126—48no—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×C4×D4—24no—
95C2×C4×D4—24no—
96C2×C4×D4—24no—
97C2×C4×D4—24no—
98C2×C4×D4—22no—
99C2×C4×D4—22no—
100C2×C4×D4—22no—
101C2×C4×D4C2314yes—
102C2×C4×D4C2314yes—
103C2×C4×D4C2314yes—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C42—21no—
123C22×C42—21no—
124C22×C42—21no—
125C22×C42—28no—
126C22×C42—48no—
127C22×C42D411yescharacteristic
128C22×C42D411yescharacteristic
129C22×C42D411yescharacteristic
130C22×C42D411yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C2×#107—24no—
135C2×#107C2314yes—
136C2×#106—24no—
137C2×#106—24no—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×#102—24no—
141C2×#102—24no—
142C2×#102—24no—
143C2×#102—24no—
144C2×#102—24no—
145C2×#102—24no—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152#390—22no—
153#390—22no—
154#390—22no—
155#390C2314yes—
156#390C2314yes—
157#390C2314yes—
158#389—22no—
159#389—22no—
160#389—22no—
161#389C2314yes—
162#389C2314yes—
163#389C2314yes—
164#388—28no—
165#388—28no—
166#388—28no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4C2314yes—
171C4×C4⋊C4C2314yes—
172C4×C4⋊C4C2314yes—
173C4×C22⋊C4—28no—
174C4×C22⋊C4—28no—
175C4×C22⋊C4—28no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C43D414yes—
183C43C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#336C2314yes—
191#336C2314yes—
192#336C2314yes—
193#336C2314yes—
194#336C2314yes—
195#336C2314yes—
196C25—21no—
197C25—21no—
198C22×D4—21no—
199C22×D4—21no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4#4114yes—
210C22×D4#4114yes—
211C22×D4#4114yes—
212C22×D4C2412yes—
213C22×D4C2412yes—
214C23×C4—28no—
215C23×C4—28no—
216C23×C4—28no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2411yescharacteristic
239C2×D8—28no—
240C2×D8—28no—
241C2×D8—28no—
242C2×D8—416no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C22×C8C2412yes—
259#126—432no—
260C4×D4—216no—
261C4×D4—216no—
262C4×D4—216no—
263C4×D4—28no—
264C4×D4—28no—
265C4×D4—28no—
266C4×D4—28no—
267C4×D4—28no—
268C4×D4—28no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4#4114yes—
291C2×C4⋊C4#4114yes—
292C2×C4⋊C4#4114yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—432no—
330C2×C42—42no—
331C2×C42—44no—
332C2×C42—44no—
333C2×C42—44no—
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D414yes—
338C2×C42C2×D414yes—
339C2×C42C2×D414yes—
340C2×C42C2×D414yes—
341C2×C42C2412yes—
342C2×C42C2412yes—
343C2×C42C2412yes—
344#107—28no—
345#106—28no—
346#106—28no—
347#102—28no—
348#102—28no—
349#102—28no—
350#102—28no—
351#102—28no—
352#102—28no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359C24—24no—
360C24—24no—
361C24—24no—
362C24—22no—
363C24—22no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24C22×D411yescharacteristic
376C2×D4—24no—
377C2×D4—24no—
378C2×D4—24no—
379C2×D4—28no—
380C2×D4—28no—
381C2×D4—28no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4#126116yescomplemented
399C2×D4C2×#4114yes—
400C2×D4C2×#4114yes—
401C2×D4C2×#4114yes—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—216no—
409C22×C4—216no—
410C22×C4—216no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—432no—
432C22×C4—432no—
433C22×C4—432no—
434C22×C4—48no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4#13514yes—
454C22×C4AGL(1,ℤ/8)14yes—
455C22×C4AGL(1,ℤ/8)14yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C2×#4112yes—
464C22×C4C2×#4112yes—
465C22×C4C2×#4112yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4C2511yescharacteristic
470D8—416no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—44no—
484C2×C8—44no—
485C2×C8—44no—
486C2×C8—44no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—48no—
505C4⋊C4C2×#4114yes—
506C4⋊C4C2×#4114yes—
507C4⋊C4C2×#4114yes—
508C22⋊C4—216no—
509C22⋊C4—216no—
510C22⋊C4—216no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C42—24no—
518C42—24no—
519C42—24no—
520C42—28no—
521C42—28no—
522C42—28no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—44no—
530C42—48no—
531C42—48no—
532C42—48no—
533C23—216no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—48no—
538C23—48no—
539C23—48no—
540C23—48no—
541C23—48no—
542C23—48no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—84no—
555C23—84no—
556C23—84no—
557C23—84no—
558C23—84no—
559C23—84no—
560C23—84no—
561C23#52512yes—
562C23#52512yes—
563C23#52512yes—
564C23C2×#13512yes—
565C23C2×AGL(1,ℤ/8)12yes—
566C23C2×AGL(1,ℤ/8)12yes—
567C23#57512yes—
568C23C23×D411yescharacteristic
569D4—216no—
570D4—24no—
571D4—24no—
572D4—24no—
573D4—42no—
574D4—42no—
575D4—42no—
576D4—42no—
577D4—84no—
578D4C2×#12618yescomplemented
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—432no—
601C2×C4—432no—
602C2×C4—432no—
603C2×C4—44no—
604C2×C4—48no—
605C2×C4—44no—
606C2×C4—48no—
607C2×C4—48no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4#52512yes—
628C2×C4#52512yes—
629C2×C4#52512yes—
630C2×C4C2×#12614yes—
631C2×C4#52914yes—
632C2×C4#52914yes—
633C2×C4#52914yes—
634C2×C4C2×#13514yes—
635C2×C4C2×AGL(1,ℤ/8)14yes—
636C2×C4C2×AGL(1,ℤ/8)14yes—
637C2×C4C22×#4111yescharacteristic
638C2×C4C22×#4111yescharacteristic
639C2×C4C22×#4111yescharacteristic
640C2×C4C22×#4111yescharacteristic
641C2×C4C22×#4111yescharacteristic
642C2×C4C22×#4111yescharacteristic
643C8—42no—
644C8—42no—
645C8—42no—
646C8—42no—
647C22—28no—
648C22—432no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—168no—
663C22#285214yes—
664C22#310214yes—
665C22#310514yes—
666C22#310614yes—
667C22#310914yes—
668C22#311414yes—
669C22#311514yes—
670C22C2×#52511yescharacteristic
671C22C2×#52511yescharacteristic
672C22C2×#52511yescharacteristic
673C22C22×#13511yescharacteristic
674C22C22×AGL(1,ℤ/8)11yescharacteristic
675C22C22×AGL(1,ℤ/8)11yescharacteristic
676C22C2×#57511yescharacteristic
677C4—24no—
678C4—24no—
679C4—24no—
680C4—42no—
681C4—44no—
682C4—42no—
683C4—44no—
684C4—42no—
685C4—44no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4#285214yes—
690C4C22×#12611yescharacteristic
691C4C2×#52911yescharacteristic
692C4C2×#52911yescharacteristic
693C4C2×#52911yescharacteristic
694C2—48no—
695C2—162no—
696C2#3308018yesdirect product factor
697C2C2×#285211yescharacteristic
698C2C2×#310211yescharacteristic
699C2C2×#310511yescharacteristic
700C2C2×#310611yescharacteristic
701C2C2×#311411yescharacteristic
702C2C2×#311511yescharacteristic
703C2C2×#310911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2860C2semidirect
2. C2×C42×D4C2semidirect
3. #33080C2direct (N × H)
4. C2×#1926C2semidirect
5. C2×#1926C2semidirect
6. C2×#1925C2semidirect
7. C2×#1921C2semidirect
8. C2×#1919C2semidirect
9. C2×#1920C2semidirect
13. C2×#1864C2semidirect
14. C2×#1864C2semidirect
15. C2×#1862C2semidirect
16. C2×#1839C2semidirect
33. #2860C22semidirect
44. C42×D4C22semidirect
52. #1926C22semidirect
53. #1926C22semidirect
54. #1925C22semidirect
55. #1921C22semidirect
56. #1920C22semidirect
57. #1919C22semidirect
61. #1864C22semidirect
62. #1864C22semidirect
63. #1862C22semidirect
64. #1839C22semidirect
398. C2×D4#126semidirect
578. D4C2×#126semidirect
696. C2#33080direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#28602
C2×C42×D42
#330802
C2×#19262
C2×#19252
C2×#19212
C2×#19192
C2×#19202
C2×#18792
C2×#18642
C2×#18622
C2×#18392