Finite Group Databasebeta

← C2×#28429 #6463151

Subgroups — C2×#28429 #6463151

Order 512.

2814 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#491C211yesmaximal, characteristic, complemented
2C22×#478C211yesmaximal, characteristic, complemented
3C22×#461C211yesmaximal, characteristic, complemented
4C22×#442C211yesmaximal, characteristic
5C2×#2547C211yesmaximal, characteristic, complemented
6C2×#2566C211yesmaximal, characteristic, complemented
7#28429C2116yesmaximal, direct product factor
8C2×#2097C211yesmaximal, characteristic, complemented
9C2×#2096C211yesmaximal, characteristic, complemented
10C2×#2012C211yesmaximal, characteristic
11C2×#2012C211yesmaximal, characteristic
12C2×#1973C211yesmaximal, characteristic, complemented
13C2×#1968C211yesmaximal, characteristic, complemented
14C2×#1959C211yesmaximal, characteristic, complemented
15C2×#1958C211yesmaximal, characteristic, complemented
16C2×#1771C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C23×SD16C2211yescharacteristic
19C23×D8C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×#125C2211yescharacteristic
22C22×#124C2211yescharacteristic
23C22×#124C2211yescharacteristic
24C22×C4×D4C2211yescharacteristic
25C22×C4×D4C2211yescharacteristic
26C2×#491—28no—
27C2×#491C2218yescomplemented
28C2×#478—28no—
29C2×#478C2218yescomplemented
30C2×#461—28no—
31C2×#461C2218yescomplemented
32C2×#442—28no—
33C2×#442C2218yes—
34C22×#107C2211yescharacteristic
35C22×#106C2211yescharacteristic
36C22×#106C2211yescharacteristic
37C22×C4⋊C8C2211yescharacteristic
38C22×#103C2211yescharacteristic
39C22×#103C2211yescharacteristic
40C22×#103C2211yescharacteristic
41C22×#103C2211yescharacteristic
42C22×#103C2211yescharacteristic
43C22×#103C2211yescharacteristic
44C22×#102C2211yescharacteristic
45C22×#102C2211yescharacteristic
46C22×#102C2211yescharacteristic
47C22×#102C2211yescharacteristic
48C22×#97C2211yescharacteristic
49C22×#97C2211yescharacteristic
50C22×C4×C8C2211yescharacteristic
51#2566—24no—
52#2566C2218yescomplemented
53#2547—24no—
54#2547C2218yescomplemented
55C2×#397—22no—
56C2×#396—22no—
57C2×#391C2211yescharacteristic
58C2×#391C2211yescharacteristic
59C2×#390C2211yescharacteristic
60C2×#390C2211yescharacteristic
61C2×#385—22no—
62C2×#384C2211yescharacteristic
63C2×#382C2211yescharacteristic
64C2×#380—22no—
65C2×C4×C4⋊C4C2211yescharacteristic
66#2097C2218yescomplemented
67#2096C2218yescomplemented
68#2012C2218yes—
69#2012C2218yes—
70#1973C2218yescomplemented
71#1968C2218yescomplemented
72#1959C2218yescomplemented
73#1958C2218yescomplemented
74#1771C2218yescomplemented
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C24×C4—21no—
82C24×C4—21no—
83C22×SD16—24no—
84C22×SD16—24no—
85C22×SD16—24no—
86C22×SD16—24no—
87C22×SD16—42no—
88C22×SD16—42no—
89C22×SD16C2314yes—
90C22×SD16C2314yes—
91C22×D8—24no—
92C22×D8—24no—
93C22×D8—24no—
94C22×D8C2314yes—
95C23×C8C2311yescharacteristic
96C23×C8C2311yescharacteristic
97C23×C8C2311yescharacteristic
98C23×C8C2311yescharacteristic
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—24no—
101C2×C4⋊Q8—24no—
102C2×C4⋊Q8C2314yes—
103C2×#125—22no—
104C2×#125—24no—
105C2×#125—24no—
106C2×#125C2314yes—
107C2×#124—24no—
108C2×#124—24no—
109C2×#124—22no—
110C2×#124—22no—
111C2×#124—42no—
112C2×#124—42no—
113C2×#124C2314yes—
114C2×#124C2314yes—
115C2×C4×D4—22no—
116C2×C4×D4—22no—
117C2×C4×D4—24no—
118C2×C4×D4—24no—
119C2×C4×D4—24no—
120C2×C4×D4—24no—
121C2×C4×D4C2314yes—
122C2×C4×D4C2314yes—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4D412yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C22⋊C4—21no—
134C22×C22⋊C4—22no—
135C22×C22⋊C4—21no—
136C22×C22⋊C4—22no—
137C22×C22⋊C4—22no—
138C22×C22⋊C4—22no—
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142#491—232no—
143#478—232no—
144#461—232no—
145#442—232no—
146C2×#107—24no—
147C2×#107C2314yes—
148C2×#106—24no—
149C2×#106—24no—
150C2×#106C2314yes—
151C2×#106C2314yes—
152C2×C4⋊C8D418yes—
153C2×C4⋊C8C2314yes—
154C2×#103—24no—
155C2×#103—24no—
156C2×#103—24no—
157C2×#103—24no—
158C2×#103—24no—
159C2×#103—24no—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#102—24no—
167C2×#102—24no—
168C2×#102—24no—
169C2×#102—24no—
170C2×#102C2314yes—
171C2×#102C2314yes—
172C2×#102C2314yes—
173C2×#102C2314yes—
174C2×#97—42no—
175C2×#97D418yes—
176C2×#97C2314yes—
177C2×#97C2314yes—
178C2×C4×C8—42no—
179C2×C4×C8C2314yes—
180#397—216no—
181#396—216no—
182#391—22no—
183#391—22no—
184#391C2314yes—
185#391C2314yes—
186#390—22no—
187#390—22no—
188#390C2314yes—
189#390C2314yes—
190#385—216no—
191#384—22no—
192#384C2314yes—
193#382—22no—
194#382C2314yes—
195#380—216no—
196C4×C4⋊C4D414yes—
197C4×C4⋊C4C2314yes—
198C2×#95—22no—
199C2×#95—22no—
200C2×#95—22no—
201C2×#95D412yes—
202#336C2314yes—
203#336C2314yes—
204C25—21no—
205C25—21no—
206C22×Q8—21no—
207C22×Q8—21no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8C2412yes—
221C22×Q8C2412yes—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—21no—
227C22×D4—21no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4#4114yes—
235C22×D4#4114yes—
236C22×D4C2412yes—
237C22×D4C2412yes—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—24no—
245C23×C4—24no—
246C23×C4—48no—
247C23×C4—44no—
248C23×C4—44no—
249C23×C4—84no—
250C23×C4C2×D412yes—
251C23×C4C2×D412yes—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2411yescharacteristic
255C2×SD16—28no—
256C2×SD16—28no—
257C2×SD16—28no—
258C2×SD16—28no—
259C2×SD16—44no—
260C2×SD16—44no—
261C2×SD16—416no—
262C2×SD16—416no—
263C2×D8—232no—
264C2×D8—28no—
265C2×D8—28no—
266C2×D8—28no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8—22no—
274C22×C8—22no—
275C22×C8C2×D414yes—
276C22×C8C2×D414yes—
277C22×C8C2×D414yes—
278C22×C8C2×D414yes—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C4⋊Q8—28no—
284C4⋊Q8—28no—
285C4⋊Q8—48no—
286#125—28no—
287#125—28no—
288#125—48no—
289#124—28no—
290#124—28no—
291#124—44no—
292#124—44no—
293#124—48no—
294#124—48no—
295C4×D4—28no—
296C4×D4—28no—
297C4×D4—28no—
298C4×D4—28no—
299C4×D4—48no—
300C4×D4—48no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—28no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4—84no—
324C2×C4⋊C4C2×D418yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—28no—
332C2×C22⋊C4—28no—
333C2×C22⋊C4—28no—
334C2×C22⋊C4—28no—
335C2×C22⋊C4—48no—
336C2×C22⋊C4—48no—
337C2×C22⋊C4—48no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—84no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2×D414yes—
349C2×C42C2×D414yes—
350C2×C42C2412yes—
351C2×C42C2412yes—
352C2×C42C2412yes—
353#107—28no—
354#106—28no—
355#106—28no—
356C4⋊C8C2×D4116yes—
357#103—28no—
358#103—28no—
359#103—28no—
360#103—28no—
361#103—28no—
362#103—28no—
363#102—28no—
364#102—28no—
365#102—28no—
366#102—28no—
367#97—44no—
368#97C2×D4116yes—
369C4×C8—44no—
370#95—28no—
371#95—28no—
372#95—28no—
373#95C2×D418yes—
374C24—24no—
375C24—24no—
376C24—22no—
377C24—22no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24—44no—
383C24—44no—
384C24—44no—
385C24—44no—
386C24C22×D411yescharacteristic
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×D4—28no—
414C2×D4—28no—
415C2×D4—28no—
416C2×D4—24no—
417C2×D4—28no—
418C2×D4—24no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—42no—
435C2×D4—44no—
436C2×D4—42no—
437C2×D4C2×#4114yes—
438C2×D4C2×#4114yes—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—416no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—88no—
471C22×C4—816no—
472C22×C4—816no—
473C22×C4#12714yes—
474C22×C4#12714yes—
475C22×C4C22×D414yes—
476C22×C4C22×D414yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D412yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4112yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C42−1+412yes—
486C22×C4C2511yescharacteristic
487SD16—416no—
488SD16—416no—
489D8—232no—
490C2×C8—28no—
491C2×C8—28no—
492C2×C8—28no—
493C2×C8—28no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8C22×D414yes—
503C2×C8C22×D414yes—
504C2×C8C22×D414yes—
505C2×C8C22×D414yes—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—48no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—48no—
520C4⋊C4—48no—
521C4⋊C4—48no—
522C4⋊C4—48no—
523C4⋊C4—88no—
524C22⋊C4—416no—
525C22⋊C4—416no—
526C22⋊C4—416no—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C22⋊C4—88no—
530C42—24no—
531C42—24no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42#12718yes—
537C42C22×D414yes—
538C42C22×D414yes—
539C23—24no—
540C23—24no—
541C23—416no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—416no—
547C23—48no—
548C23—48no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—88no—
554C23—84no—
555C23—84no—
556C23—84no—
557C23—84no—
558C23—88no—
559C23C2×#12712yes—
560C23#54612yes—
561C23#54812yes—
562C23#57514yes—
563C23#57714yes—
564C23C23×D411yescharacteristic
565Q8—42no—
566Q8—42no—
567Q8—42no—
568Q8—42no—
569Q8—42no—
570Q8—42no—
571Q8—42no—
572Q8—42no—
573Q8—84no—
574Q8—84no—
575D4—24no—
576D4—24no—
577D4—24no—
578D4—24no—
579D4—42no—
580D4—42no—
581D4—42no—
582D4—42no—
583D4—48no—
584D4—48no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—48no—
596C2×C4—48no—
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—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
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#52514yes—
620C2×C4C2×#12714yes—
621C2×C4C2×#12714yes—
622C2×C4C2×#12712yes—
623C2×C4#53714yes—
624C2×C4#54614yes—
625C2×C4#54612yes—
626C2×C4#54814yes—
627C2×C4#54812yes—
628C2×C4C23×D411yescharacteristic
629C2×C4C23×D411yescharacteristic
630C2×C4C22×#4111yescharacteristic
631C2×C4C22×#4111yescharacteristic
632C2×C4C2×H(5,𝔽2)11yescharacteristic
633C2×C4C2×2−1+411yescharacteristic
634C8—24no—
635C8—24no—
636C8—24no—
637C8—24no—
638C22—44no—
639C22—44no—
640C22—48no—
641C22—48no—
642C22—816no—
643C22—816no—
644C22—84no—
645C22—84no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22#254014yes—
651C22#315314yes—
652C22#315414yes—
653C22#330718yes—
654C22#337618yes—
655C22C22×#12711yescharacteristic
656C22C2×#54611yescharacteristic
657C22C2×#54811yescharacteristic
658C22C2×#57512yes—
659C22C2×#57712yes—
660C4—24no—
661C4—24no—
662C4—44no—
663C4—44no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4#254014yes—
671C4C2×#52511yescharacteristic
672C4C2×#53711yescharacteristic
673C4C2×#54611yescharacteristic
674C4C2×#54811yescharacteristic
675C2—84no—
676C2—84no—
677C2#2842918yesdirect product factor
678C2C2×#254011yescharacteristic
679C2C2×#315311yescharacteristic
680C2C2×#315411yescharacteristic
681C2C2×#330712yes—
682C2C2×#337612yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#491C2semidirect
2. C22×#478C2semidirect
3. C22×#461C2semidirect
5. C2×#2547C2semidirect
6. C2×#2566C2semidirect
7. #28429C2direct (N × H)
8. C2×#2097C2semidirect
9. C2×#2096C2semidirect
12. C2×#1973C2semidirect
13. C2×#1968C2semidirect
14. C2×#1959C2semidirect
15. C2×#1958C2semidirect
16. C2×#1771C2semidirect
27. C2×#491C22semidirect
29. C2×#478C22semidirect
31. C2×#461C22semidirect
52. #2566C22semidirect
54. #2547C22semidirect
66. #2097C22semidirect
67. #2096C22semidirect
70. #1973C22semidirect
71. #1968C22semidirect
72. #1959C22semidirect
73. #1958C22semidirect
74. #1771C22semidirect
677. C2#28429direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4912
C22×#4782
C22×#4612
C22×#4422
C2×#25472
C2×#25662
#284292
C2×#20972
C2×#20962
C2×#20122
C2×#19732
C2×#19682
C2×#19592
C2×#19582
C2×#17712