Finite Group Databasebeta

← C2×#24941 #6441497

Subgroups — C2×#24941 #6441497

Order 512.

2304 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic
2C2×#2352C211yesmaximal, characteristic
3#24941C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2085C211yesmaximal, characteristic, complemented
6C2×#2076C211yesmaximal, characteristic, complemented
7C2×#2077C211yesmaximal, characteristic, complemented
8C2×#1921C211yesmaximal, characteristic, complemented
9C2×#1920C211yesmaximal, characteristic, complemented
10C2×#1869C211yesmaximal, characteristic
11C2×#1854C211yesmaximal, characteristic, complemented
12C2×#1853C211yesmaximal, characteristic, complemented
13C2×#1825C211yesmaximal, characteristic, complemented
14C2×#1823C211yesmaximal, characteristic, complemented
15C2×#1808C211yesmaximal, characteristic, complemented
16C22×#340C211yesmaximal, characteristic, complemented
17C24×C8C2211yescharacteristic
18C22×#123—21no—
19C22×C4×D4C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C22×C22⋊C8C2211yescharacteristic
28C22×C22⋊C8C2211yescharacteristic
29C22×C22⋊C8C2211yescharacteristic
30#2486—24no—
31#2486C2218yes—
32C2×#398—21no—
33C2×#397—21no—
34#2352—24no—
35#2352C2218yes—
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#388—21no—
39C2×#387—21no—
40C2×#386—21no—
41C2×#382C2211yescharacteristic
42C2×#381—21no—
43C2×#380C2211yescharacteristic
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4—21no—
50#2101C2218yescomplemented
51#2085C2218yescomplemented
52#2077C2218yescomplemented
53#2076C2218yescomplemented
54#1921C2218yescomplemented
55#1920C2218yescomplemented
56#1869C2218yes—
57#1854C2218yescomplemented
58#1853C2218yescomplemented
59#1825C2218yescomplemented
60#1823C2218yescomplemented
61#1808C2218yescomplemented
62C2×#340C2211yescharacteristic
63C2×#340C2211yescharacteristic
64C2×#340C2211yescharacteristic
65C2×#340C2211yescharacteristic
66C2×#340C2211yescharacteristic
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2218yescomplemented
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×D4D411yescharacteristic
78C24×C4—21no—
79C24×C4—21no—
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C23×C8—24no—
83C23×C8—24no—
84C23×C8—42no—
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2314yes—
89C23×C8C2311yescharacteristic
90C2×#123—28no—
91C2×#123—48no—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×C4×D4D414yes—
95C2×C4×D4C2314yes—
96C2×#117—22no—
97C2×#117—24no—
98C2×#117—42no—
99C2×#117C2314yes—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—22no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—22no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—42no—
110C22×C4⋊C4D411yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4D411yescharacteristic
128C22×C22⋊C4D411yescharacteristic
129C22×C42—21no—
130C22×C42—21no—
131C22×C42C2311yescharacteristic
132C2×#107—24no—
133C2×#107—42no—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#106—24no—
137C2×#106—42no—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×C22⋊C8—24no—
141C2×C22⋊C8—24no—
142C2×C22⋊C8—42no—
143C2×C22⋊C8C2314yes—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2314yes—
146#398—28no—
147#397—28no—
148#389—22no—
149#389—22no—
150#389C2314yes—
151#389C2314yes—
152#388—28no—
153#387—28no—
154#386—28no—
155#382—22no—
156#382C2314yes—
157#381—28no—
158#380—22no—
159#380—22no—
160#380—22no—
161#380—22no—
162#380—22no—
163#380C2314yes—
164#380C2314yes—
165#380C2314yes—
166#380C2314yes—
167#380C2314yes—
168C4×C4⋊C4—22no—
169C4×C4⋊C4C2314yes—
170C4×C22⋊C4—28no—
171C2×#95—21no—
172C2×#95—21no—
173C2×#95—21no—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#336C2314yes—
194#336C2314yes—
195C25C2×D411yescharacteristic
196C25C2×D411yescharacteristic
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4C2×D414yes—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—21no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—24no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—82no—
227C23×C4C2×D412yes—
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D412yes—
232C23×C4C2×D411yescharacteristic
233C23×C4#4114yes—
234C23×C4#4114yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—28no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—28no—
251C22×C8—44no—
252C22×C8—416no—
253C22×C8#4114yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258#123—432no—
259C4×D4—216no—
260C4×D4—28no—
261C4×D4—28no—
262#117—28no—
263#117—44no—
264#117—48no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
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—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—48no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—48no—
303C2×C4⋊C4C2×D414yes—
304C2×C4⋊C4#4114yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—82no—
327C2×C22⋊C4C2×D414yes—
328C2×C22⋊C4C2×D414yes—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—44no—
334C2×C42—82no—
335C2×C42C2×D412yes—
336C2×C42C2412yes—
337#107—28no—
338#107—44no—
339#106—28no—
340#106—44no—
341C22⋊C8—28no—
342C22⋊C8—28no—
343C22⋊C8—44no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24#12314yes—
366C24#13514yes—
367C24AGL(1,ℤ/8)14yes—
368C24C22×D412yes—
369C24C22×D412yes—
370C24C22×D411yescharacteristic
371C2×D4—28no—
372C2×D4—28no—
373C2×D4—28no—
374C2×D4—416no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
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—48no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—48no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—48no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—84no—
447C22×C4—816no—
448C22×C4—816no—
449C22×C4#12314yes—
450C22×C4#12318yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C2×#4114yes—
457C22×C4C2×#4114yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C42−1+412yes—
464C22×C4C2511yescharacteristic
465C2×C8—28no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—416no—
478C2×C8—44no—
479C2×C8—44no—
480C2×C8—44no—
481C2×C8C2×#4114yes—
482C4⋊C4—28no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
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—48no—
502C4⋊C4—48no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4C2×#4114yes—
511C22⋊C4—28no—
512C22⋊C4—28no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—84no—
519C42—24no—
520C42—22no—
521C42—22no—
522C42—48no—
523C42—84no—
524C23—28no—
525C23—28no—
526C23—28no—
527C23—28no—
528C23—22no—
529C23—22no—
530C23—28no—
531C23—28no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23#482116yes—
549C23C2×#12314yes—
550C23C2×#12312yes—
551C23#54612yes—
552C23#54712yes—
553C23C2×#13514yes—
554C23C2×#13512yes—
555C23C2×AGL(1,ℤ/8)14yes—
556C23C2×AGL(1,ℤ/8)12yes—
557C23#57712yes—
558C23#57812yes—
559C23C23×D411yescharacteristic
560D4—416no—
561C2×C4—28no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—216no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—48no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—48no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—816no—
612C2×C4—816no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4—84no—
620C2×C4C2×#12314yes—
621C2×C4C2×#12312yes—
622C2×C4#52914yes—
623C2×C4#53914yes—
624C2×C4#54612yes—
625C2×C4#54712yes—
626C2×C4#55114yes—
627C2×C4#55214yes—
628C2×C4C22×#4111yescharacteristic
629C2×C4C22×#4111yescharacteristic
630C2×C4C22×#4111yescharacteristic
631C2×C4C22×#4111yescharacteristic
632C2×C4C2×H(5,𝔽2)11yescharacteristic
633C2×C4C2×2−1+411yescharacteristic
634C8—24no—
635C8—42no—
636C8—42no—
637C8—42no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—24no—
642C22—232no—
643C22—24no—
644C22—24no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—88no—
653C22#241914yes—
654C22C2×#48218yes—
655C22C2×#48214yes—
656C22#309714yes—
657C22#329614yes—
658C22#330814yes—
659C22#333614yes—
660C22#334514yes—
661C22C22×#12311yescharacteristic
662C22C2×#54611yescharacteristic
663C22C2×#54711yescharacteristic
664C22C22×#13511yescharacteristic
665C22C22×AGL(1,ℤ/8)11yescharacteristic
666C22C2×#57711yescharacteristic
667C22C2×#57811yescharacteristic
668C4—24no—
669C4—42no—
670C4—42no—
671C4—42no—
672C4—44no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4#241914yes—
681C4C2×#52911yescharacteristic
682C4C2×#53911yescharacteristic
683C4C2×#55111yescharacteristic
684C4C2×#55211yescharacteristic
685C2—28no—
686C2—82no—
687C2#2494118yesdirect product factor
688C2C2×#241911yescharacteristic
689C2C22×#48211yescharacteristic
690C2C2×#309711yescharacteristic
691C2C2×#329611yescharacteristic
692C2C2×#330811yescharacteristic
693C2C2×#333611yescharacteristic
694C2C2×#334511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #24941C2direct (N × H)
4. C2×#2101C2semidirect
5. C2×#2085C2semidirect
6. C2×#2076C2semidirect
7. C2×#2077C2semidirect
8. C2×#1921C2semidirect
9. C2×#1920C2semidirect
11. C2×#1854C2semidirect
12. C2×#1853C2semidirect
13. C2×#1825C2semidirect
14. C2×#1823C2semidirect
15. C2×#1808C2semidirect
16. C22×#340C2semidirect
50. #2101C22semidirect
51. #2085C22semidirect
52. #2077C22semidirect
53. #2076C22semidirect
54. #1921C22semidirect
55. #1920C22semidirect
57. #1854C22semidirect
58. #1853C22semidirect
59. #1825C22semidirect
60. #1823C22semidirect
61. #1808C22semidirect
74. C2×#340C22semidirect
687. C2#24941direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24862
C2×#23522
#249412
C2×#21012
C2×#20852
C2×#20762
C2×#20772
C2×#19212
C2×#19202
C2×#18692
C2×#18542
C2×#18532
C2×#18252
C2×#18232
C2×#18082
C22×#3402