Finite Group Databasebeta

← C2×#27542 #6440964

Subgroups — C2×#27542 #6440964

Order 512.

2690 conjugacy classes of subgroups.

The subgroup lattice has 737 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2496C211yesmaximal, characteristic, complemented
2C2×#2339C211yesmaximal, characteristic
3#27542C2116yesmaximal, direct product factor
4C2×#2076C211yesmaximal, characteristic, complemented
5C2×#2077C211yesmaximal, characteristic, complemented
6C2×#2072C211yesmaximal, characteristic, complemented
7C2×#2066C211yesmaximal, characteristic, complemented
8C2×#1935C211yesmaximal, characteristic, complemented
9C2×#1928C211yesmaximal, characteristic, complemented
10C2×#1924C211yesmaximal, characteristic, complemented
11C2×#1864C211yesmaximal, characteristic, complemented
12C2×#1862C211yesmaximal, characteristic, complemented
13C2×#1802C211yesmaximal, characteristic, complemented
14C2×#1788C211yesmaximal, characteristic
15C2×#1788C211yesmaximal, characteristic
16C2×#1762C211yesmaximal, characteristic, complemented
17C23×#41C2211yescharacteristic
18C22×#126—21no—
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×C22⋊C8C2211yescharacteristic
33C22×C22⋊C8C2211yescharacteristic
34C22×#97C2211yescharacteristic
35#2496—24no—
36#2496C2218yescomplemented
37C2×#400—21no—
38C2×#397—21no—
39#2339—24no—
40#2339C2218yes—
41C2×#391C2211yescharacteristic
42C2×#391C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#387—21no—
46C2×#386—21no—
47C2×#386—21no—
48C2×#384C2211yescharacteristic
49C2×#381—21no—
50C2×#380C2211yescharacteristic
51C2×#380C2211yescharacteristic
52C2×#380C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4C2211yescharacteristic
55C22×#95—21no—
56#2077C2218yescomplemented
57#2076C2218yescomplemented
58#2072C2218yescomplemented
59#2066C2218yescomplemented
60#1935C2218yescomplemented
61#1928C2218yescomplemented
62#1924C2218yescomplemented
63#1864C2218yescomplemented
64#1862C2218yescomplemented
65#1802C2218yescomplemented
66#1788C2218yes—
67#1788C2218yes—
68#1762C2218yescomplemented
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C22×#41—24no—
78C22×#41—24no—
79C22×#41—22no—
80C22×#41—24no—
81C22×#41—24no—
82C22×#41—42no—
83C22×#41—42no—
84C22×#41C2314yes—
85C23×Q8C2311yescharacteristic
86C23×D4—21no—
87C23×D4C2311yescharacteristic
88C24×C4—21no—
89C24×C4—21no—
90C24×C4—21no—
91C24×C4C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C23×C8C2311yescharacteristic
95C23×C8C2311yescharacteristic
96C2×#126—28no—
97C2×#126—48no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C2×C4×D4—22no—
101C2×C4×D4—22no—
102C2×C4×D4—42no—
103C2×C4×D4—42no—
104C2×C4×D4C2314yes—
105C2×C4×D4C2314yes—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4D411yescharacteristic
122C22×C22⋊C4D411yescharacteristic
123C22×C42—21no—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C2×#107—42no—
127C2×#107C2314yes—
128C2×#106—42no—
129C2×#106C2314yes—
130C2×C4⋊C8—42no—
131C2×C4⋊C8C2314yes—
132C2×#103—24no—
133C2×#103—24no—
134C2×#103—42no—
135C2×#103—42no—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#102—24no—
141C2×#102—24no—
142C2×#102—24no—
143C2×#102—24no—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×C22⋊C8—42no—
149C2×C22⋊C8—42no—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2314yes—
152C2×#97—42no—
153C2×#97C2314yes—
154#400—28no—
155#397—28no—
156#391—22no—
157#391—22no—
158#391C2314yes—
159#391C2314yes—
160#389—22no—
161#389—22no—
162#389C2314yes—
163#389C2314yes—
164#387—28no—
165#386—28no—
166#386—28no—
167#384—22no—
168#384C2314yes—
169#381—28no—
170#380—22no—
171#380—22no—
172#380—22no—
173#380C2314yes—
174#380C2314yes—
175#380C2314yes—
176C4×C4⋊C4—22no—
177C4×C4⋊C4C2314yes—
178C4×C22⋊C4D414yes—
179C4×C22⋊C4C2314yes—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—28no—
191C2×#95D411yescharacteristic
192C2×#95D411yescharacteristic
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201C25—21no—
202C25C2×D411yescharacteristic
203C2×#41—216no—
204C2×#41—28no—
205C2×#41—28no—
206C2×#41—28no—
207C2×#41—28no—
208C2×#41—44no—
209C2×#41—44no—
210C2×#41—48no—
211C2×#41—48no—
212C2×#41—416no—
213C2×#41—416no—
214C2×#41—416no—
215C2×#41—416no—
216C22×Q8—21no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8#4114yes—
223C22×Q8C2412yes—
224C22×D4—28no—
225C22×D4—24no—
226C22×D4—21no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—44no—
231C22×D4—44no—
232C22×D4#4114yes—
233C22×D4#4114yes—
234C22×D4#4114yes—
235C22×D4C2412yes—
236C23×C4—28no—
237C23×C4—21no—
238C23×C4—24no—
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—22no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—24no—
252C23×C4—24no—
253C23×C4—44no—
254C23×C4—44no—
255C23×C4—44no—
256C23×C4—44no—
257C23×C4—44no—
258C23×C4—44no—
259C23×C4—82no—
260C23×C4—82no—
261C23×C4C2×D412yes—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2412yes—
268C23×C4C2411yescharacteristic
269C22×C8—22no—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8—22no—
274C22×C8—22no—
275C22×C8—22no—
276C22×C8—22no—
277C22×C8—22no—
278C22×C8—22no—
279C22×C8—22no—
280C22×C8—22no—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C22×C8C2412yes—
284C22×C8C2412yes—
285#126—432no—
286C4×D4—28no—
287C4×D4—28no—
288C4×D4—44no—
289C4×D4—44no—
290C4×D4—48no—
291C4×D4—48no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—42no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—82no—
336C2×C22⋊C4C2×D414yes—
337C2×C22⋊C4C2×D414yes—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—24no—
343C2×C42—44no—
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2412yes—
347C2×C42C2412yes—
348#107—44no—
349#106—44no—
350C4⋊C8—44no—
351#103—28no—
352#103—28no—
353#103—44no—
354#103—44no—
355#102—28no—
356#102—28no—
357#102—28no—
358#102—28no—
359C22⋊C8—44no—
360C22⋊C8—44no—
361#97—44no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95C2×D414yes—
373#95C2×D414yes—
374C24—24no—
375C24—24no—
376C24—24no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24C22×D412yes—
390C24C22×D411yescharacteristic
391#41—416no—
392#41—416no—
393#41—416no—
394#41—464no—
395#41—416no—
396C2×Q8—24no—
397C2×Q8—22no—
398C2×Q8—28no—
399C2×Q8—28no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8C2×#4114yes—
409C2×D4—216no—
410C2×D4—24no—
411C2×D4—28no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—28no—
415C2×D4—28no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—48no—
420C2×D4—48no—
421C2×D4—432no—
422C2×D4#126116yescomplemented
423C2×D4C2×#4114yes—
424C2×D4C2×#4114yes—
425C2×D4C2×#4114yes—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—216no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—432no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—48no—
489C22×C4—84no—
490C22×C4—84no—
491C22×C4—816no—
492C22×C4—816no—
493C22×C4AGL(1,ℤ/8)14yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D411yescharacteristic
500C22×C4C2×#4112yes—
501C22×C4C2×#4112yes—
502C22×C4C2×#4112yes—
503C22×C4C2×#4112yes—
504C22×C4H(5,𝔽2)12yes—
505C22×C4H(5,𝔽2)12yes—
506C22×C4C2511yescharacteristic
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—44no—
520C2×C8—44no—
521C2×C8—44no—
522C2×C8—44no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—48no—
536C4⋊C4—48no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—48no—
544C4⋊C4—48no—
545C22⋊C4—28no—
546C22⋊C4—44no—
547C22⋊C4—48no—
548C22⋊C4—48no—
549C22⋊C4—48no—
550C22⋊C4—48no—
551C22⋊C4—84no—
552C42—24no—
553C42—24no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—48no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—216no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—48no—
571C23—48no—
572C23—48no—
573C23—48no—
574C23—48no—
575C23—48no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23#52512yes—
588C23#54612yes—
589C23#54612yes—
590C23C2×AGL(1,ℤ/8)12yes—
591C23#57512yes—
592C23#57712yes—
593C23#57812yes—
594C23C23×D411yescharacteristic
595Q8—24no—
596Q8—24no—
597Q8—42no—
598Q8—42no—
599Q8—48no—
600D4—24no—
601D4—24no—
602D4—24no—
603D4—216no—
604D4—432no—
605D4C2×#12618yescomplemented
606C2×C4—22no—
607C2×C4—28no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—432no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—48no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—48no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—816no—
654C2×C4—816no—
655C2×C4—84no—
656C2×C4—84no—
657C2×C4—84no—
658C2×C4—84no—
659C2×C4—84no—
660C2×C4—84no—
661C2×C4#52512yes—
662C2×C4C2×#12614yes—
663C2×C4#53814yes—
664C2×C4#54612yes—
665C2×C4#54612yes—
666C2×C4#55114yes—
667C2×C4#55114yes—
668C2×C4C2×AGL(1,ℤ/8)14yes—
669C2×C4C22×#4111yescharacteristic
670C2×C4C22×#4111yescharacteristic
671C2×C4C22×#4111yescharacteristic
672C2×C4C22×#4111yescharacteristic
673C2×C4C2×H(5,𝔽2)11yescharacteristic
674C2×C4C2×H(5,𝔽2)11yescharacteristic
675C8—42no—
676C8—42no—
677C8—42no—
678C8—42no—
679C22—28no—
680C22—44no—
681C22—44no—
682C22—432no—
683C22—44no—
684C22—44no—
685C22—44no—
686C22—44no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22—42no—
693C22—42no—
694C22—88no—
695C22#248014yes—
696C22#311514yes—
697C22#312014yes—
698C22#330614yes—
699C22#331114yes—
700C22#331514yes—
701C22#333014yes—
702C22C2×#52511yescharacteristic
703C22C2×#54611yescharacteristic
704C22C2×#54611yescharacteristic
705C22C22×AGL(1,ℤ/8)11yescharacteristic
706C22C2×#57511yescharacteristic
707C22C2×#57711yescharacteristic
708C22C2×#57811yescharacteristic
709C4—24no—
710C4—42no—
711C4—44no—
712C4—42no—
713C4—42no—
714C4—42no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4—82no—
720C4—82no—
721C4#248014yes—
722C4C22×#12611yescharacteristic
723C4C2×#53811yescharacteristic
724C4C2×#55111yescharacteristic
725C4C2×#55111yescharacteristic
726C2—48no—
727C2—82no—
728C2#2754218yesdirect product factor
729C2C2×#248011yescharacteristic
730C2C2×#311511yescharacteristic
731C2C2×#312011yescharacteristic
732C2C2×#330611yescharacteristic
733C2C2×#331511yescharacteristic
734C2C2×#331111yescharacteristic
735C2C2×#333011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2496C2semidirect
3. #27542C2direct (N × H)
4. C2×#2076C2semidirect
5. C2×#2077C2semidirect
6. C2×#2072C2semidirect
7. C2×#2066C2semidirect
8. C2×#1935C2semidirect
9. C2×#1928C2semidirect
10. C2×#1924C2semidirect
11. C2×#1864C2semidirect
12. C2×#1862C2semidirect
13. C2×#1802C2semidirect
16. C2×#1762C2semidirect
36. #2496C22semidirect
56. #2077C22semidirect
57. #2076C22semidirect
58. #2072C22semidirect
59. #2066C22semidirect
60. #1935C22semidirect
61. #1928C22semidirect
62. #1924C22semidirect
63. #1864C22semidirect
64. #1862C22semidirect
65. #1802C22semidirect
68. #1762C22semidirect
422. C2×D4#126semidirect
605. D4C2×#126semidirect
728. C2#27542direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24962
C2×#23392
#275422
C2×#20762
C2×#20772
C2×#20722
C2×#20662
C2×#19352
C2×#19282
C2×#19242
C2×#18642
C2×#18622
C2×#18022
C2×#17882
C2×#17622