Finite Group Databasebeta

← C2×#33081 #6629581

Subgroups — C2×#33081 #6629581

Order 512.

2232 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2860C211yesmaximal, characteristic
2C2×C42×Q8C211yesmaximal, characteristic, complemented
3#33081C2116yesmaximal, direct product factor
4C2×#1926C211yesmaximal, characteristic
5C2×#1925C211yesmaximal, characteristic
6C2×#1925C211yesmaximal, characteristic
7C2×#1921C211yesmaximal, characteristic, complemented
8C2×#1920C211yesmaximal, characteristic, complemented
9C2×#1918C211yesmaximal, characteristic, complemented
10C2×#1880C211yesmaximal, characteristic
11C2×#1880C211yesmaximal, characteristic
12C2×#1880C211yesmaximal, characteristic
13C2×#1865C211yesmaximal, characteristic, complemented
14C2×#1863C211yesmaximal, characteristic, complemented
15C2×#1863C211yesmaximal, characteristic, complemented
16C2×#1839C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C22×#126—21no—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31#2860—24no—
32#2860C2218yes—
33C2×#390C2211yescharacteristic
34C2×#390C2211yescharacteristic
35C2×#390C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#388—21no—
40C2×#388—21no—
41C2×#388—21no—
42C42×Q8—24no—
43C42×Q8C2218yescomplemented
44C2×C4×C4⋊C4—21no—
45C2×C4×C4⋊C4—21no—
46C2×C4×C4⋊C4—21no—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C43—21no—
51C2×C43C2211yescharacteristic
52#1926C2218yes—
53#1925C2218yes—
54#1925C2218yes—
55#1921C2218yescomplemented
56#1920C2218yescomplemented
57#1918C2218yescomplemented
58#1880C2218yes—
59#1880C2218yes—
60#1880C2218yes—
61#1865C2218yescomplemented
62#1863C2218yescomplemented
63#1863C2218yescomplemented
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×Q8C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C22×SD16—24no—
80C22×SD16—24no—
81C22×SD16—24no—
82C22×SD16C2314yes—
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C2×#126—28no—
88C2×#126—48no—
89C2×C4×Q8—24no—
90C2×C4×Q8—24no—
91C2×C4×Q8—24no—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8—24no—
95C2×C4×Q8—22no—
96C2×C4×Q8—22no—
97C2×C4×Q8—22no—
98C2×C4×Q8C2314yes—
99C2×C4×Q8C2314yes—
100C2×C4×Q8C2314yes—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
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×C42—21no—
120C22×C42—21no—
121C22×C42—21no—
122C22×C42—21no—
123C22×C42—21no—
124C22×C42—21no—
125C22×C42D411yescharacteristic
126C22×C42D411yescharacteristic
127C22×C42D411yescharacteristic
128C22×C42D411yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C2×#107—24no—
133C2×#107—24no—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#106—24no—
137C2×#106C2314yes—
138C2×#103—24no—
139C2×#103—24no—
140C2×#103—24no—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#102—24no—
145C2×#102—24no—
146C2×#102—24no—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150#390—22no—
151#390—22no—
152#390—22no—
153#390C2314yes—
154#390C2314yes—
155#390C2314yes—
156#389—22no—
157#389—22no—
158#389—22no—
159#389C2314yes—
160#389C2314yes—
161#389C2314yes—
162#388—28no—
163#388—28no—
164#388—28no—
165C4×C4⋊C4—28no—
166C4×C4⋊C4—28no—
167C4×C4⋊C4—28no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4C2314yes—
172C4×C4⋊C4C2314yes—
173C4×C4⋊C4C2314yes—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C43—28no—
181C43D414yes—
182C43C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#336C2314yes—
190#336C2314yes—
191#336C2314yes—
192#336C2314yes—
193#336C2314yes—
194#336C2314yes—
195C25—21no—
196C22×Q8—21no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8#4114yes—
201C22×Q8#4114yes—
202C22×Q8#4114yes—
203C22×Q8C2412yes—
204C22×D4—21no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4C2412yes—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2411yescharacteristic
226C2×SD16—28no—
227C2×SD16—28no—
228C2×SD16—28no—
229C2×SD16—416no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8C2412yes—
243C22×C8C2412yes—
244C22×C8C2412yes—
245C22×C8C2412yes—
246#126—432no—
247C4×Q8—216no—
248C4×Q8—216no—
249C4×Q8—216no—
250C4×Q8—28no—
251C4×Q8—28no—
252C4×Q8—28no—
253C4×Q8—28no—
254C4×Q8—28no—
255C4×Q8—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4#4114yes—
284C2×C4⋊C4#4114yes—
285C2×C4⋊C4#4114yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C42—28no—
305C2×C42—28no—
306C2×C42—28no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—42no—
323C2×C42—44no—
324C2×C42—44no—
325C2×C42—44no—
326C2×C42C2×D412yes—
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2×D414yes—
330C2×C42C2×D414yes—
331C2×C42C2×D414yes—
332C2×C42C2×D414yes—
333C2×C42C2412yes—
334C2×C42C2412yes—
335C2×C42C2412yes—
336#107—28no—
337#107—28no—
338#106—28no—
339#103—28no—
340#103—28no—
341#103—28no—
342#102—28no—
343#102—28no—
344#102—28no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351C24—22no—
352C24—42no—
353C24—42no—
354C24—42no—
355C24—42no—
356C24—42no—
357C24—42no—
358C24—42no—
359C24C22×D411yescharacteristic
360C2×Q8—24no—
361C2×Q8—24no—
362C2×Q8—24no—
363C2×Q8—28no—
364C2×Q8—28no—
365C2×Q8—28no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8#126116yescomplemented
370C2×Q8C2×#4114yes—
371C2×Q8C2×#4114yes—
372C2×Q8C2×#4114yes—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—42no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—44no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4#13514yes—
430C22×C4AGL(1,ℤ/8)14yes—
431C22×C4#13714yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C2×#4112yes—
440C22×C4C2×#4112yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4C2511yescharacteristic
446SD16—416no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—44no—
460C2×C8—44no—
461C2×C8—44no—
462C2×C8—44no—
463C4⋊C4—216no—
464C4⋊C4—216no—
465C4⋊C4—216no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—48no—
484C4⋊C4C2×#4114yes—
485C4⋊C4C2×#4114yes—
486C4⋊C4C2×#4114yes—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C42—216no—
494C42—216no—
495C42—216no—
496C42—24no—
497C42—24no—
498C42—24no—
499C42—28no—
500C42—28no—
501C42—28no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—44no—
509C42—48no—
510C42—48no—
511C42—48no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—84no—
520C23—84no—
521C23—84no—
522C23—84no—
523C23—84no—
524C23—84no—
525C23—84no—
526C23#52512yes—
527C23#52512yes—
528C23#52512yes—
529C23C2×#13512yes—
530C23C2×AGL(1,ℤ/8)12yes—
531C23C2×#13712yes—
532C23#57512yes—
533C23C23×D411yescharacteristic
534Q8—216no—
535Q8—24no—
536Q8—24no—
537Q8—24no—
538Q8C2×#12618yescomplemented
539D4—42no—
540D4—42no—
541D4—42no—
542D4—42no—
543D4—84no—
544C2×C4—216no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—44no—
570C2×C4—48no—
571C2×C4—44no—
572C2×C4—48no—
573C2×C4—44no—
574C2×C4—48no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—84no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4#52512yes—
594C2×C4#52512yes—
595C2×C4#52512yes—
596C2×C4C2×#12614yes—
597C2×C4#52914yes—
598C2×C4#52914yes—
599C2×C4#52914yes—
600C2×C4C2×#13514yes—
601C2×C4C2×AGL(1,ℤ/8)14yes—
602C2×C4C2×#13714yes—
603C2×C4C22×#4111yescharacteristic
604C2×C4C22×#4111yescharacteristic
605C2×C4C22×#4111yescharacteristic
606C2×C4C22×#4111yescharacteristic
607C2×C4C22×#4111yescharacteristic
608C2×C4C22×#4111yescharacteristic
609C8—42no—
610C8—42no—
611C8—42no—
612C8—42no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—168no—
621C22#285214yes—
622C22#310214yes—
623C22#310614yes—
624C22#310814yes—
625C22#311214yes—
626C22#311514yes—
627C22#311714yes—
628C22C2×#52511yescharacteristic
629C22C2×#52511yescharacteristic
630C22C2×#52511yescharacteristic
631C22C22×#13511yescharacteristic
632C22C22×AGL(1,ℤ/8)11yescharacteristic
633C22C22×#13711yescharacteristic
634C22C2×#57511yescharacteristic
635C4—28no—
636C4—24no—
637C4—24no—
638C4—24no—
639C4—42no—
640C4—44no—
641C4—42no—
642C4—44no—
643C4—42no—
644C4—44no—
645C4—82no—
646C4—82no—
647C4—82no—
648C4#285214yes—
649C4C22×#12611yescharacteristic
650C4C2×#52911yescharacteristic
651C4C2×#52911yescharacteristic
652C4C2×#52911yescharacteristic
653C2—162no—
654C2#3308118yesdirect product factor
655C2C2×#285211yescharacteristic
656C2C2×#310211yescharacteristic
657C2C2×#310611yescharacteristic
658C2C2×#310811yescharacteristic
659C2C2×#311511yescharacteristic
660C2C2×#311711yescharacteristic
661C2C2×#311211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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
2. C2×C42×Q8C2semidirect
3. #33081C2direct (N × H)
7. C2×#1921C2semidirect
8. C2×#1920C2semidirect
9. C2×#1918C2semidirect
13. C2×#1865C2semidirect
14. C2×#1863C2semidirect
15. C2×#1863C2semidirect
16. C2×#1839C2semidirect
43. C42×Q8C22semidirect
55. #1921C22semidirect
56. #1920C22semidirect
57. #1918C22semidirect
61. #1865C22semidirect
62. #1863C22semidirect
63. #1863C22semidirect
64. #1839C22semidirect
369. C2×Q8#126semidirect
538. Q8C2×#126semidirect
654. C2#33081direct (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×Q82
#330812
C2×#19262
C2×#19252
C2×#19212
C2×#19202
C2×#19182
C2×#18802
C2×#18652
C2×#18632
C2×#18392