Finite Group Databasebeta

← C2×#27590 #6458784

Subgroups — C2×#27590 #6458784

Order 512.

2642 conjugacy classes of subgroups.

The subgroup lattice has 735 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2297C211yesmaximal, characteristic
2C2×#2283C211yesmaximal, characteristic, complemented
3#27590C2116yesmaximal, direct product factor
4C2×#2097C211yesmaximal, characteristic, complemented
5C2×#2092C211yesmaximal, characteristic, complemented
6C2×#2074C211yesmaximal, characteristic, complemented
7C2×#2073C211yesmaximal, characteristic, complemented
8C2×#1928C211yesmaximal, characteristic, complemented
9C2×#1926C211yesmaximal, characteristic, complemented
10C2×#1927C211yesmaximal, characteristic, complemented
11C2×#1925C211yesmaximal, characteristic, complemented
12C2×#1854C211yesmaximal, characteristic, complemented
13C2×#1854C211yesmaximal, characteristic, complemented
14C2×#1817C211yesmaximal, characteristic, complemented
15C2×#1788C211yesmaximal, characteristic
16C2×#1788C211yesmaximal, characteristic
17C23×#41C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C42C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×C22⋊C8C2211yescharacteristic
33C22×C22⋊C8C2211yescharacteristic
34#2297—24no—
35#2297C2218yes—
36C2×#391C2211yescharacteristic
37C2×#391C2211yescharacteristic
38C2×#390C2211yescharacteristic
39C2×#390C2211yescharacteristic
40C2×#386—21no—
41C2×#386—21no—
42#2283—24no—
43#2283C2218yescomplemented
44C2×#383—21no—
45C2×#382C2211yescharacteristic
46C2×#382C2211yescharacteristic
47C2×#381—21no—
48C2×#380C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4—21no—
53C2×C4×C22⋊C4—21no—
54C2×C4×C22⋊C4—21no—
55C2×#376—21no—
56#2097C2218yescomplemented
57#2092C2218yescomplemented
58#2074C2218yescomplemented
59#2073C2218yescomplemented
60#1928C2218yescomplemented
61#1927C2218yescomplemented
62#1926C2218yescomplemented
63#1925C2218yescomplemented
64#1854C2218yescomplemented
65#1854C2218yescomplemented
66#1817C2218yescomplemented
67#1788C2218yes—
68#1788C2218yes—
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
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×C4D411yescharacteristic
90C24×C4D411yescharacteristic
91C24×C4C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C23×C8C2311yescharacteristic
95C23×C8C2311yescharacteristic
96C2×#117—22no—
97C2×#117—42no—
98C2×#117—42no—
99C2×#117C2314yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4—21no—
107C22×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C42—24no—
114C22×C42—24no—
115C22×C42—21no—
116C22×C42—21no—
117C22×C42—21no—
118C22×C42D411yescharacteristic
119C22×C42D411yescharacteristic
120C22×C42D414yes—
121C22×C42C2314yes—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C2×#107—42no—
125C2×#107—42no—
126C2×#107C2314yes—
127C2×#107C2314yes—
128C2×#106—42no—
129C2×#106—42no—
130C2×#106C2314yes—
131C2×#106C2314yes—
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—42no—
143C2×#102—42no—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×C22⋊C8—24no—
149C2×C22⋊C8—24no—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2314yes—
152#391—22no—
153#391—22no—
154#391C2314yes—
155#391C2314yes—
156#390—22no—
157#390—22no—
158#390C2314yes—
159#390C2314yes—
160#386—28no—
161#386—28no—
162#383—28no—
163#382—22no—
164#382—22no—
165#382C2314yes—
166#382C2314yes—
167#381—28no—
168#380—22no—
169#380—22no—
170#380C2314yes—
171#380C2314yes—
172C4×C4⋊C4—22no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4C2314yes—
175C4×C4⋊C4C2314yes—
176C4×C22⋊C4—28no—
177C4×C22⋊C4—28no—
178C4×C22⋊C4—28no—
179#376—28no—
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—21no—
191C2×#95—21no—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2314yes—
197#336C2314yes—
198#336C2314yes—
199#336C2314yes—
200C25—21no—
201C25C2×D411yescharacteristic
202C2×#41—216no—
203C2×#41—28no—
204C2×#41—28no—
205C2×#41—28no—
206C2×#41—28no—
207C2×#41—44no—
208C2×#41—44no—
209C2×#41—416no—
210C2×#41—416no—
211C2×#41—416no—
212C2×#41—416no—
213C2×#41—48no—
214C2×#41—48no—
215C22×Q8—21no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8#4114yes—
222C22×Q8C2412yes—
223C22×D4—28no—
224C22×D4—24no—
225C22×D4—21no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—44no—
232C22×D4—44no—
233C22×D4#4114yes—
234C22×D4C2412yes—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—28no—
240C23×C4—24no—
241C23×C4—24no—
242C23×C4—21no—
243C23×C4—24no—
244C23×C4—24no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—42no—
252C23×C4—42no—
253C23×C4—42no—
254C23×C4—44no—
255C23×C4—44no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D414yes—
260C23×C4C2×D412yes—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D414yes—
264C23×C4#4114yes—
265C23×C4#4114yes—
266C23×C4C2412yes—
267C23×C4C2411yescharacteristic
268C22×C8—22no—
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×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C22×C8C2412yes—
284#117—44no—
285#117—44no—
286#117—48no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
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—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—216no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—28no—
331C2×C42—28no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—44no—
338C2×C42—44no—
339C2×C42—44no—
340C2×C42C2×D414yes—
341C2×C42C2×D414yes—
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2412yes—
345C2×C42C2412yes—
346#107—44no—
347#107—44no—
348#106—44no—
349#106—44no—
350#103—28no—
351#103—28no—
352#103—44no—
353#103—44no—
354#102—28no—
355#102—28no—
356#102—44no—
357#102—44no—
358C22⋊C8—28no—
359C22⋊C8—28no—
360#95—24no—
361#95—24no—
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—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—24no—
379C24—22no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24#12314yes—
387C24C22×D412yes—
388C24C22×D411yescharacteristic
389#41—416no—
390#41—416no—
391#41—416no—
392#41—416no—
393#41—464no—
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—28no—
397C2×Q8—28no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8C2×#4114yes—
407C2×D4—216no—
408C2×D4—24no—
409C2×D4—28no—
410C2×D4—28no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—48no—
417C2×D4—432no—
418C2×D4—48no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4C2×#4114yes—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—28no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—216no—
439C22×C4—28no—
440C22×C4—28no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—28no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—416no—
463C22×C4—48no—
464C22×C4—432no—
465C22×C4—416no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—48no—
469C22×C4—42no—
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#12314yes—
488C22×C4#12318yes—
489C22×C4#13514yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D411yescharacteristic
496C22×C4C2×#4114yes—
497C22×C4C2×#4114yes—
498C22×C4C2×#4112yes—
499C22×C4C2×#4112yes—
500C22×C4C2×#4112yes—
501C22×C4C2×#4112yes—
502C22×C4H(5,𝔽2)12yes—
503C22×C4H(5,𝔽2)12yes—
504C22×C4C2511yescharacteristic
505C2×C8—22no—
506C2×C8—22no—
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—44no—
518C2×C8—44no—
519C2×C8—44no—
520C2×C8—44no—
521C4⋊C4—22no—
522C4⋊C4—22no—
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—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C22⋊C4—48no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C22⋊C4—48no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C22⋊C4—48no—
546C42—28no—
547C42—24no—
548C42—28no—
549C42—24no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—48no—
555C42—48no—
556C42—48no—
557C23—28no—
558C23—28no—
559C23—22no—
560C23—22no—
561C23—28no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—24no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—48no—
572C23—48no—
573C23—42no—
574C23—42no—
575C23—48no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—84no—
581C23—84no—
582C23—84no—
583C23—84no—
584C23C2×#12314yes—
585C23C2×#12312yes—
586C23#52512yes—
587C23#53512yes—
588C23C2×#13512yes—
589C23#57512yes—
590C23#57512yes—
591C23#57512yes—
592C23C23×D411yescharacteristic
593Q8—24no—
594Q8—24no—
595Q8—42no—
596Q8—42no—
597Q8—48no—
598D4—24no—
599D4—24no—
600D4—432no—
601D4—42no—
602D4—42no—
603D4—48no—
604C2×C4—28no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—28no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—216no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—416no—
626C2×C4—432no—
627C2×C4—416no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—48no—
634C2×C4—42no—
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—84no—
653C2×C4—84no—
654C2×C4—84no—
655C2×C4—84no—
656C2×C4—84no—
657C2×C4—84no—
658C2×C4C2×#12314yes—
659C2×C4C2×#12312yes—
660C2×C4#52512yes—
661C2×C4#53512yes—
662C2×C4#55114yes—
663C2×C4#55114yes—
664C2×C4#55114yes—
665C2×C4#55114yes—
666C2×C4C2×#13514yes—
667C2×C4C22×#4111yescharacteristic
668C2×C4C22×#4111yescharacteristic
669C2×C4C22×#4111yescharacteristic
670C2×C4C22×#4111yescharacteristic
671C2×C4C2×H(5,𝔽2)11yescharacteristic
672C2×C4C2×H(5,𝔽2)11yescharacteristic
673C8—42no—
674C8—42no—
675C8—42no—
676C8—42no—
677C22—24no—
678C22—24no—
679C22—24no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—416no—
685C22—44no—
686C22—44no—
687C22—44no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—816no—
693C22#248214yes—
694C22#309114yes—
695C22#309314yes—
696C22#310214yes—
697C22#311114yes—
698C22#321214yes—
699C22#321914yes—
700C22C22×#12311yescharacteristic
701C22C2×#52511yescharacteristic
702C22C2×#53511yescharacteristic
703C22C22×#13511yescharacteristic
704C22C2×#57511yescharacteristic
705C22C2×#57511yescharacteristic
706C22C2×#57511yescharacteristic
707C4—24no—
708C4—24no—
709C4—42no—
710C4—42no—
711C4—42no—
712C4—44no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4#248214yes—
720C4C2×#55111yescharacteristic
721C4C2×#55111yescharacteristic
722C4C2×#55111yescharacteristic
723C4C2×#55111yescharacteristic
724C2—44no—
725C2—84no—
726C2#2759018yesdirect product factor
727C2C2×#248211yescharacteristic
728C2C2×#309111yescharacteristic
729C2C2×#309311yescharacteristic
730C2C2×#310211yescharacteristic
731C2C2×#311111yescharacteristic
732C2C2×#321211yescharacteristic
733C2C2×#321911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2283C2semidirect
3. #27590C2direct (N × H)
4. C2×#2097C2semidirect
5. C2×#2092C2semidirect
6. C2×#2074C2semidirect
7. C2×#2073C2semidirect
8. C2×#1928C2semidirect
9. C2×#1926C2semidirect
10. C2×#1927C2semidirect
11. C2×#1925C2semidirect
12. C2×#1854C2semidirect
13. C2×#1854C2semidirect
14. C2×#1817C2semidirect
43. #2283C22semidirect
56. #2097C22semidirect
57. #2092C22semidirect
58. #2074C22semidirect
59. #2073C22semidirect
60. #1928C22semidirect
61. #1927C22semidirect
62. #1926C22semidirect
63. #1925C22semidirect
64. #1854C22semidirect
65. #1854C22semidirect
66. #1817C22semidirect
726. C2#27590direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#22972
C2×#22832
#275902
C2×#20972
C2×#20922
C2×#20742
C2×#20732
C2×#19282
C2×#19262
C2×#19272
C2×#19252
C2×#18542
C2×#18172
C2×#17882