Finite Group Databasebeta

← C2×#28428 #6463158

Subgroups — C2×#28428 #6463158

Order 512.

2900 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#489C211yesmaximal, characteristic, complemented
2C22×#474C211yesmaximal, characteristic, complemented
3C22×#464C211yesmaximal, characteristic, complemented
4C22×#440C211yesmaximal, characteristic, complemented
5C2×#2567C211yesmaximal, characteristic, complemented
6C2×#2544C211yesmaximal, characteristic
7#28428C2116yesmaximal, direct product factor
8C2×#2095C211yesmaximal, characteristic, complemented
9C2×#2093C211yesmaximal, characteristic, complemented
10C2×#2015C211yesmaximal, characteristic, complemented
11C2×#2014C211yesmaximal, characteristic
12C2×#1973C211yesmaximal, characteristic, complemented
13C2×#1969C211yesmaximal, characteristic, complemented
14C2×#1959C211yesmaximal, characteristic, complemented
15C2×#1957C211yesmaximal, characteristic
16C2×#1771C211yesmaximal, characteristic, complemented
17C23×Q16C2211yescharacteristic
18C23×SD16C2211yescharacteristic
19C23×SD16C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×#127C2211yescharacteristic
22C22×#124C2211yescharacteristic
23C22×#124C2211yescharacteristic
24C22×C4×Q8C2211yescharacteristic
25C22×C4×D4C2211yescharacteristic
26C2×#489—28no—
27C2×#489C2218yescomplemented
28C2×#474—28no—
29C2×#474C2218yescomplemented
30C2×#464—28no—
31C2×#464C2218yescomplemented
32C2×#440—28no—
33C2×#440C2218yescomplemented
34C22×#107C2211yescharacteristic
35C22×#107C2211yescharacteristic
36C22×#107C2211yescharacteristic
37C22×C4⋊C8C2211yescharacteristic
38C22×#103C2211yescharacteristic
39C22×#103C2211yescharacteristic
40C22×#103C2211yescharacteristic
41C22×#103C2211yescharacteristic
42C22×#102C2211yescharacteristic
43C22×#102C2211yescharacteristic
44C22×#102C2211yescharacteristic
45C22×#102C2211yescharacteristic
46C22×#102C2211yescharacteristic
47C22×#102C2211yescharacteristic
48C22×#97C2211yescharacteristic
49C22×#97C2211yescharacteristic
50C22×C4×C8C2211yescharacteristic
51#2567—24no—
52#2567C2218yescomplemented
53#2544—24no—
54#2544C2218yes—
55C2×#395—22no—
56C2×#394—22no—
57C2×#391C2211yescharacteristic
58C2×#391C2211yescharacteristic
59C2×#391C2211yescharacteristic
60C2×#390C2211yescharacteristic
61C2×#389—22no—
62C2×#386—22no—
63C2×#384C2211yescharacteristic
64C2×#384C2211yescharacteristic
65C2×C4×C4⋊C4C2211yescharacteristic
66#2095C2218yescomplemented
67#2093C2218yescomplemented
68#2015C2218yescomplemented
69#2014C2218yes—
70#1973C2218yescomplemented
71#1969C2218yescomplemented
72#1959C2218yescomplemented
73#1957C2218yes—
74#1771C2218yescomplemented
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8—21no—
78C23×Q8C2311yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4—22no—
81C23×D4C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C24×C4—21no—
84C22×Q16—24no—
85C22×Q16—24no—
86C22×Q16—42no—
87C22×Q16C2314yes—
88C22×SD16—24no—
89C22×SD16—24no—
90C22×SD16—24no—
91C22×SD16—24no—
92C22×SD16—24no—
93C22×SD16—42no—
94C22×SD16C2314yes—
95C22×SD16C2314yes—
96C23×C8C2311yescharacteristic
97C23×C8C2311yescharacteristic
98C23×C8C2311yescharacteristic
99C23×C8C2311yescharacteristic
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—24no—
102C2×C4⋊Q8—24no—
103C2×C4⋊Q8C2314yes—
104C2×#127—24no—
105C2×#127—22no—
106C2×#127—42no—
107C2×#127C2314yes—
108C2×#124—24no—
109C2×#124—22no—
110C2×#124—22no—
111C2×#124—24no—
112C2×#124—24no—
113C2×#124—42no—
114C2×#124C2314yes—
115C2×#124C2314yes—
116C2×C4×Q8—22no—
117C2×C4×Q8—24no—
118C2×C4×Q8—24no—
119C2×C4×Q8C2314yes—
120C2×C4×D4—22no—
121C2×C4×D4—24no—
122C2×C4×D4—24no—
123C2×C4×D4C2314yes—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—22no—
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—21no—
135C22×C22⋊C4—22no—
136C22×C22⋊C4—21no—
137C22×C22⋊C4—22no—
138C22×C42—21no—
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142#489—232no—
143#474—232no—
144#464—232no—
145#440—232no—
146C2×#107—24no—
147C2×#107—24no—
148C2×#107—24no—
149C2×#107C2314yes—
150C2×#107C2314yes—
151C2×#107C2314yes—
152C2×C4⋊C8D418yescomplemented
153C2×C4⋊C8C2314yes—
154C2×#103—24no—
155C2×#103—24no—
156C2×#103—24no—
157C2×#103—24no—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#102—24no—
163C2×#102—24no—
164C2×#102—24no—
165C2×#102—24no—
166C2×#102—24no—
167C2×#102—24no—
168C2×#102C2314yes—
169C2×#102C2314yes—
170C2×#102C2314yes—
171C2×#102C2314yes—
172C2×#102C2314yes—
173C2×#102C2314yes—
174C2×#97—42no—
175C2×#97D418yescomplemented
176C2×#97C2314yes—
177C2×#97C2314yes—
178C2×C4×C8—42no—
179C2×C4×C8C2314yes—
180#395—216no—
181#394—216no—
182#391—22no—
183#391—22no—
184#391—22no—
185#391C2314yes—
186#391C2314yes—
187#391C2314yes—
188#390—22no—
189#390C2314yes—
190#389—216no—
191#386—216no—
192#384—22no—
193#384—22no—
194#384C2314yes—
195#384C2314yes—
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—24no—
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×Q8—44no—
221C22×Q8—84no—
222C22×Q8#4114yes—
223C22×Q8C2412yes—
224C22×Q8C2412yes—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—21no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—21no—
231C22×D4—28no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4—22no—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4—48no—
240C22×D4—48no—
241C22×D4—84no—
242C22×D4#4114yes—
243C22×D4C2412yes—
244C22×D4C2412yes—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—24no—
252C23×C4—44no—
253C23×C4—84no—
254C23×C4C2×D412yes—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D412yes—
258C23×C4C2411yescharacteristic
259C2×Q16—28no—
260C2×Q16—28no—
261C2×Q16—44no—
262C2×Q16—416no—
263C2×SD16—232no—
264C2×SD16—28no—
265C2×SD16—28no—
266C2×SD16—28no—
267C2×SD16—28no—
268C2×SD16—28no—
269C2×SD16—44no—
270C2×SD16—416no—
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×C8C2×D414yes—
280C22×C8C2×D414yes—
281C22×C8C2×D414yes—
282C22×C8C2×D414yes—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C4⋊Q8—28no—
288C4⋊Q8—28no—
289C4⋊Q8—48no—
290#127—28no—
291#127—44no—
292#127—48no—
293#124—28no—
294#124—28no—
295#124—28no—
296#124—44no—
297#124—48no—
298#124—48no—
299C4×Q8—28no—
300C4×Q8—28no—
301C4×Q8—48no—
302C4×D4—28no—
303C4×D4—28no—
304C4×D4—48no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—28no—
321C2×C4⋊C4—28no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—84no—
325C2×C4⋊C4—84no—
326C2×C4⋊C4C2×D418yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—28no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—28no—
336C2×C22⋊C4—48no—
337C2×C22⋊C4—44no—
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×C42—24no—
346C2×C42—44no—
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2×D414yes—
351C2×C42C2×D414yes—
352C2×C42C2412yes—
353C2×C42C2412yes—
354C2×C42C2412yes—
355#107—28no—
356#107—28no—
357#107—28no—
358C4⋊C8C2×D4116yescomplemented
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#102—28no—
368#102—28no—
369#97—44no—
370#97C2×D4116yescomplemented
371C4×C8—44no—
372#95—28no—
373#95—28no—
374#95—28no—
375#95C2×D418yes—
376C24—24no—
377C24—22no—
378C24—22no—
379C24—42no—
380C24—42no—
381C24—44no—
382C24—44no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24—44no—
387C24—44no—
388C24C22×D411yescharacteristic
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—28no—
392C2×Q8—28no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—48no—
406C2×Q8—44no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—44no—
410C2×Q8—42no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—88no—
416C2×Q8—816no—
417C2×Q8C2×#4114yes—
418C2×D4—28no—
419C2×D4—24no—
420C2×D4—28no—
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—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—416no—
433C2×D4—44no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—44no—
439C2×D4—44no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—44no—
443C2×D4—42no—
444C2×D4—416no—
445C2×D4—88no—
446C2×D4—832no—
447C2×D4C2×#4114yes—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—48no—
476C22×C4—88no—
477C22×C4—816no—
478C22×C4#12714yes—
479C22×C4#12714yes—
480C22×C4#13718yes—
481C22×C4C22×D414yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D414yes—
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D412yes—
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4H(5,𝔽2)12yes—
491C22×C42−1+412yes—
492C22×C4C2511yescharacteristic
493Q16—416no—
494SD16—232no—
495SD16—416no—
496C2×C8—28no—
497C2×C8—28no—
498C2×C8—28no—
499C2×C8—28no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8C22×D414yes—
509C2×C8C22×D414yes—
510C2×C8C22×D414yes—
511C2×C8C22×D414yes—
512C4⋊C4—216no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—48no—
526C4⋊C4—48no—
527C4⋊C4—88no—
528C4⋊C4—88no—
529C22⋊C4—416no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—48no—
533C22⋊C4—88no—
534C42—24no—
535C42—24no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—48no—
541C42#12718yes—
542C42C22×D414yes—
543C42C22×D414yes—
544C23—24no—
545C23—42no—
546C23—42no—
547C23—48no—
548C23—416no—
549C23—44no—
550C23—44no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—44no—
555C23—44no—
556C23—88no—
557C23—84no—
558C23—84no—
559C23—88no—
560C23—84no—
561C23—84no—
562C23—84no—
563C23—88no—
564C23C2×#12712yes—
565C23#54612yes—
566C23#54812yes—
567C23C2×#13714yes—
568C23#57614yes—
569C23C23×D411yescharacteristic
570Q8—24no—
571Q8—24no—
572Q8—42no—
573Q8—42no—
574Q8—48no—
575Q8—42no—
576Q8—42no—
577Q8—42no—
578Q8—42no—
579Q8—816no—
580Q8—84no—
581D4—24no—
582D4—24no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—42no—
587D4—42no—
588D4—42no—
589D4—48no—
590D4—832no—
591D4—84no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—24no—
595C2×C4—216no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—48no—
606C2×C4—48no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—816no—
619C2×C4—84no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4#52514yes—
625C2×C4C2×#12714yes—
626C2×C4C2×#12714yes—
627C2×C4C2×#12712yes—
628C2×C4#53714yes—
629C2×C4#54614yes—
630C2×C4#54612yes—
631C2×C4#54814yes—
632C2×C4#54812yes—
633C2×C4C2×#13718yes—
634C2×C4C23×D411yescharacteristic
635C2×C4C23×D411yescharacteristic
636C2×C4C22×#4111yescharacteristic
637C2×C4C22×#4111yescharacteristic
638C2×C4C2×H(5,𝔽2)11yescharacteristic
639C2×C4C2×2−1+411yescharacteristic
640C8—24no—
641C8—24no—
642C8—24no—
643C8—24no—
644C22—44no—
645C22—48no—
646C22—816no—
647C22—84no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—84no—
652C22—84no—
653C22—82no—
654C22—82no—
655C22—168no—
656C22#254014yes—
657C22#315214yes—
658C22#315514yes—
659C22#331418yes—
660C22#337518yes—
661C22C22×#12711yescharacteristic
662C22C2×#54611yescharacteristic
663C22C2×#54811yescharacteristic
664C22C22×#13712yes—
665C22C2×#57612yes—
666C4—28no—
667C4—24no—
668C4—24no—
669C4—44no—
670C4—44no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4#254014yes—
677C4C2×#52511yescharacteristic
678C4C2×#53711yescharacteristic
679C4C2×#54611yescharacteristic
680C4C2×#54811yescharacteristic
681C2—84no—
682C2—162no—
683C2#2842818yesdirect product factor
684C2C2×#254011yescharacteristic
685C2C2×#315211yescharacteristic
686C2C2×#315511yescharacteristic
687C2C2×#331412yes—
688C2C2×#337512yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
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×#489C2semidirect
2. C22×#474C2semidirect
3. C22×#464C2semidirect
4. C22×#440C2semidirect
5. C2×#2567C2semidirect
7. #28428C2direct (N × H)
8. C2×#2095C2semidirect
9. C2×#2093C2semidirect
10. C2×#2015C2semidirect
12. C2×#1973C2semidirect
13. C2×#1969C2semidirect
14. C2×#1959C2semidirect
16. C2×#1771C2semidirect
27. C2×#489C22semidirect
29. C2×#474C22semidirect
31. C2×#464C22semidirect
33. C2×#440C22semidirect
52. #2567C22semidirect
66. #2095C22semidirect
67. #2093C22semidirect
68. #2015C22semidirect
70. #1973C22semidirect
71. #1969C22semidirect
72. #1959C22semidirect
74. #1771C22semidirect
152. C2×C4⋊C8D4semidirect
175. C2×#97D4semidirect
358. C4⋊C8C2×D4semidirect
370. #97C2×D4semidirect
683. C2#28428direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4892
C22×#4742
C22×#4642
C22×#4402
C2×#25672
C2×#25442
#284282
C2×#20952
C2×#20932
C2×#20152
C2×#20142
C2×#19732
C2×#19692
C2×#19592
C2×#19572
C2×#17712