Finite Group Databasebeta

← C2×#28362 #6459727

Subgroups — C2×#28362 #6459727

Order 512.

3330 conjugacy classes of subgroups.

The subgroup lattice has 740 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#483C211yesmaximal, characteristic, complemented
2C22×#470C211yesmaximal, characteristic, complemented
3C22×#452C211yesmaximal, characteristic
4C22×#417C211yesmaximal, characteristic
5C2×#2502C211yesmaximal, characteristic, complemented
6C2×#2432C211yesmaximal, characteristic
7#28362C2116yesmaximal, direct product factor
8C2×#2096C211yesmaximal, characteristic, complemented
9C2×#2094C211yesmaximal, characteristic, complemented
10C2×#2016C211yesmaximal, characteristic, complemented
11C2×#2015C211yesmaximal, characteristic, complemented
12C2×#1971C211yesmaximal, characteristic, complemented
13C2×#1968C211yesmaximal, characteristic, complemented
14C2×#1959C211yesmaximal, characteristic, complemented
15C2×#1956C211yesmaximal, characteristic, complemented
16C2×#1753C211yesmaximal, characteristic, complemented
17C23×#41C2211yescharacteristic
18C23×Q16C2211yescharacteristic
19C23×Q16C2211yescharacteristic
20C23×SD16C2211yescharacteristic
21C23×M16C2211yescharacteristic
22C22×C4⋊Q8C2211yescharacteristic
23C22×#124C2211yescharacteristic
24C22×C22⋊Q8C2211yescharacteristic
25C22×C22⋊Q8C2211yescharacteristic
26C23×C4⋊C4C2211yescharacteristic
27C2×#483—28no—
28C2×#483C2218yescomplemented
29C2×#470—28no—
30C2×#470C2218yescomplemented
31C2×#452—28no—
32C2×#452C2218yes—
33C22×#106C2211yescharacteristic
34C22×#106C2211yescharacteristic
35C22×#106C2211yescharacteristic
36C2×#417—28no—
37C2×#417C2218yes—
38C22×#103C2211yescharacteristic
39C22×#103C2211yescharacteristic
40C22×#103C2211yescharacteristic
41C22×#103C2211yescharacteristic
42C22×#103C2211yescharacteristic
43C22×#103C2211yescharacteristic
44C22×#103C2211yescharacteristic
45C22×#102C2211yescharacteristic
46C22×#102C2211yescharacteristic
47C22×#102C2211yescharacteristic
48C22×C22⋊C8C2211yescharacteristic
49C22×#97C2211yescharacteristic
50C22×C4×C8C2211yescharacteristic
51#2502—24no—
52#2502C2218yescomplemented
53#2432—24no—
54#2432C2218yes—
55C2×#398—22no—
56C2×#394—22no—
57C2×#391C2211yescharacteristic
58C2×#391C2211yescharacteristic
59C2×#391C2211yescharacteristic
60C2×#390C2211yescharacteristic
61C2×#385—22no—
62C2×#385—22no—
63C2×#384C2211yescharacteristic
64C2×#384C2211yescharacteristic
65C2×C4×C22⋊C4C2211yescharacteristic
66#2096C2218yescomplemented
67#2094C2218yescomplemented
68#2016C2218yescomplemented
69#2015C2218yescomplemented
70#1971C2218yescomplemented
71#1968C2218yescomplemented
72#1959C2218yescomplemented
73#1956C2218yescomplemented
74#1753C2218yescomplemented
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C22×#41—22no—
78C22×#41—24no—
79C22×#41—24no—
80C22×#41—24no—
81C22×#41—24no—
82C22×#41—42no—
83C22×#41—42no—
84C22×#41C2314yes—
85C23×Q8C2311yescharacteristic
86C23×Q8C2311yescharacteristic
87C23×Q8C2311yescharacteristic
88C23×D4—21no—
89C23×D4C2311yescharacteristic
90C24×C4—21no—
91C24×C4—21no—
92C24×C4C2311yescharacteristic
93C22×Q16—24no—
94C22×Q16—24no—
95C22×Q16—24no—
96C22×Q16—42no—
97C22×Q16—42no—
98C22×Q16—42no—
99C22×Q16C2314yes—
100C22×Q16C2314yes—
101C22×SD16—24no—
102C22×SD16—24no—
103C22×SD16—42no—
104C22×SD16C2314yes—
105C22×M16—42no—
106C22×M16—42no—
107C22×M16D418yes—
108C22×M16C2314yes—
109C23×C8C2311yescharacteristic
110C23×C8C2311yescharacteristic
111C23×C8C2311yescharacteristic
112C23×C8C2311yescharacteristic
113C2×C4⋊Q8—24no—
114C2×C4⋊Q8—22no—
115C2×C4⋊Q8—42no—
116C2×C4⋊Q8C2314yes—
117C2×#124—24no—
118C2×#124—22no—
119C2×#124—42no—
120C2×#124C2314yes—
121C2×C22⋊Q8—22no—
122C2×C22⋊Q8—22no—
123C2×C22⋊Q8—24no—
124C2×C22⋊Q8—24no—
125C2×C22⋊Q8—24no—
126C2×C22⋊Q8—24no—
127C2×C22⋊Q8C2314yes—
128C2×C22⋊Q8C2314yes—
129C22×C4⋊C4—24no—
130C22×C4⋊C4—24no—
131C22×C4⋊C4—22no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4—22no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4—22no—
137C22×C4⋊C4—22no—
138C22×C4⋊C4C2314yes—
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2311yescharacteristic
141C22×C4⋊C4C2311yescharacteristic
142C22×C4⋊C4C2311yescharacteristic
143C22×C22⋊C4—21no—
144C22×C22⋊C4—21no—
145C22×C22⋊C4—22no—
146C22×C22⋊C4—22no—
147C22×C22⋊C4D412yes—
148C22×C42C2311yescharacteristic
149C22×C42C2311yescharacteristic
150#483—232no—
151#470—232no—
152#452—232no—
153C2×#106—24no—
154C2×#106—24no—
155C2×#106—24no—
156C2×#106C2314yes—
157C2×#106C2314yes—
158C2×#106C2314yes—
159#417—232no—
160C2×#103—24no—
161C2×#103—24no—
162C2×#103—24no—
163C2×#103—24no—
164C2×#103—24no—
165C2×#103—24no—
166C2×#103—24no—
167C2×#103C2314yes—
168C2×#103C2314yes—
169C2×#103C2314yes—
170C2×#103C2314yes—
171C2×#103C2314yes—
172C2×#103C2314yes—
173C2×#103C2314yes—
174C2×#102—24no—
175C2×#102—24no—
176C2×#102—24no—
177C2×#102C2314yes—
178C2×#102C2314yes—
179C2×#102C2314yes—
180C2×C22⋊C8D418yes—
181C2×C22⋊C8C2314yes—
182C2×#97—42no—
183C2×#97C2314yes—
184C2×C4×C8—42no—
185C2×C4×C8C2314yes—
186#398—216no—
187#394—216no—
188#391—22no—
189#391—22no—
190#391—22no—
191#391C2314yes—
192#391C2314yes—
193#391C2314yes—
194#390—22no—
195#390C2314yes—
196#385—216no—
197#385—216no—
198#384—22no—
199#384—22no—
200#384C2314yes—
201#384C2314yes—
202C4×C22⋊C4D414yes—
203C4×C22⋊C4C2314yes—
204C2×#95—22no—
205C2×#95—22no—
206C2×#95—22no—
207C2×#95D412yes—
208#336C2314yes—
209#336C2314yes—
210C25—21no—
211C25C2×D411yescharacteristic
212C2×#41—232no—
213C2×#41—28no—
214C2×#41—28no—
215C2×#41—28no—
216C2×#41—28no—
217C2×#41—44no—
218C2×#41—44no—
219C2×#41—48no—
220C2×#41—416no—
221C2×#41—48no—
222C2×#41—416no—
223C2×#41—416no—
224C2×#41—48no—
225C22×Q8—21no—
226C22×Q8—21no—
227C22×Q8—21no—
228C22×Q8—22no—
229C22×Q8—22no—
230C22×Q8—22no—
231C22×Q8—22no—
232C22×Q8—22no—
233C22×Q8—22no—
234C22×Q8—22no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8#4114yes—
246C22×Q8C2412yes—
247C22×Q8C2412yes—
248C22×Q8C2412yes—
249C22×D4—24no—
250C22×D4—22no—
251C22×D4—22no—
252C22×D4—21no—
253C22×D4—22no—
254C22×D4—22no—
255C22×D4—22no—
256C22×D4—44no—
257C22×D4—44no—
258C22×D4—44no—
259C22×D4#4114yes—
260C22×D4C2412yes—
261C23×C4—22no—
262C23×C4—22no—
263C23×C4—22no—
264C23×C4—22no—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—21no—
270C23×C4—21no—
271C23×C4—21no—
272C23×C4—24no—
273C23×C4—24no—
274C23×C4—44no—
275C23×C4—44no—
276C23×C4—44no—
277C23×C4—44no—
278C23×C4—84no—
279C23×C4C2×D412yes—
280C23×C4C2×D412yes—
281C23×C4C2×D411yescharacteristic
282C23×C4C2×D414yes—
283C23×C4C2×D414yes—
284C23×C4C2×D412yes—
285C23×C4C2412yes—
286C23×C4C2411yescharacteristic
287C2×Q16—28no—
288C2×Q16—28no—
289C2×Q16—28no—
290C2×Q16—44no—
291C2×Q16—44no—
292C2×Q16—44no—
293C2×Q16—416no—
294C2×Q16—416no—
295C2×SD16—28no—
296C2×SD16—28no—
297C2×SD16—44no—
298C2×SD16—416no—
299C2×M16—416no—
300C2×M16—44no—
301C2×M16—44no—
302C2×M16C2×D4116yes—
303C22×C8—22no—
304C22×C8—22no—
305C22×C8—22no—
306C22×C8—22no—
307C22×C8—22no—
308C22×C8—22no—
309C22×C8—22no—
310C22×C8—22no—
311C22×C8C2×D414yes—
312C22×C8C2×D414yes—
313C22×C8C2×D414yes—
314C22×C8C2×D414yes—
315C22×C8C2412yes—
316C22×C8C2412yes—
317C22×C8C2412yes—
318C22×C8C2412yes—
319C4⋊Q8—28no—
320C4⋊Q8—44no—
321C4⋊Q8—48no—
322#124—28no—
323#124—44no—
324#124—48no—
325C22⋊Q8—28no—
326C22⋊Q8—28no—
327C22⋊Q8—28no—
328C22⋊Q8—28no—
329C22⋊Q8—48no—
330C22⋊Q8—48no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—28no—
336C2×C4⋊C4—28no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—28no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—28no—
350C2×C4⋊C4—28no—
351C2×C4⋊C4—48no—
352C2×C4⋊C4—44no—
353C2×C4⋊C4—48no—
354C2×C4⋊C4—44no—
355C2×C4⋊C4—44no—
356C2×C4⋊C4—84no—
357C2×C4⋊C4—84no—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C22⋊C4—28no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—28no—
366C2×C22⋊C4—28no—
367C2×C22⋊C4—48no—
368C2×C22⋊C4—44no—
369C2×C22⋊C4—44no—
370C2×C22⋊C4—48no—
371C2×C22⋊C4C2×D418yes—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42C2×D412yes—
377C2×C42C2×D412yes—
378C2×C42C2412yes—
379C2×C42C2412yes—
380#106—28no—
381#106—28no—
382#106—28no—
383#103—28no—
384#103—28no—
385#103—28no—
386#103—28no—
387#103—28no—
388#103—28no—
389#103—28no—
390#102—28no—
391#102—28no—
392#102—28no—
393C22⋊C8C2×D4116yes—
394#97—44no—
395C4×C8—44no—
396#95—28no—
397#95—28no—
398#95—28no—
399#95C2×D418yes—
400C24—22no—
401C24—22no—
402C24—24no—
403C24—24no—
404C24—24no—
405C24—22no—
406C24—42no—
407C24—42no—
408C24—44no—
409C24—44no—
410C24#12714yes—
411C24C22×D412yes—
412C24C22×D411yescharacteristic
413#41—232no—
414#41—416no—
415#41—416no—
416#41—416no—
417#41—464no—
418C2×Q8—24no—
419C2×Q8—22no—
420C2×Q8—28no—
421C2×Q8—28no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—44no—
442C2×Q8—42no—
443C2×Q8—44no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—44no—
449C2×Q8—44no—
450C2×Q8—44no—
451C2×Q8—44no—
452C2×Q8—44no—
453C2×Q8—44no—
454C2×Q8—44no—
455C2×Q8—44no—
456C2×Q8C2×#4114yes—
457C2×D4—28no—
458C2×D4—28no—
459C2×D4—24no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—48no—
466C2×D4—48no—
467C2×D4—432no—
468C2×D4—48no—
469C2×D4—42no—
470C2×D4—44no—
471C2×D4—44no—
472C2×D4—42no—
473C2×D4C2×#4114yes—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—28no—
479C22×C4—28no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—28no—
483C22×C4—28no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—48no—
500C22×C4—48no—
501C22×C4—432no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—48no—
505C22×C4—42no—
506C22×C4—42no—
507C22×C4—42no—
508C22×C4—42no—
509C22×C4—48no—
510C22×C4—42no—
511C22×C4—42no—
512C22×C4—42no—
513C22×C4—42no—
514C22×C4—42no—
515C22×C4—42no—
516C22×C4—44no—
517C22×C4—44no—
518C22×C4—44no—
519C22×C4—44no—
520C22×C4—88no—
521C22×C4—816no—
522C22×C4#12714yes—
523C22×C4#12718yes—
524C22×C4#13718yes—
525C22×C4C22×D414yes—
526C22×C4C22×D412yes—
527C22×C4C22×D414yes—
528C22×C4C22×D414yes—
529C22×C4C22×D414yes—
530C22×C4C22×D412yes—
531C22×C4C22×D411yescharacteristic
532C22×C4C22×D412yes—
533C22×C4C2×#4112yes—
534C22×C4C2×#4112yes—
535C22×C4H(5,𝔽2)12yes—
536C22×C4H(5,𝔽2)12yes—
537C22×C4C2511yescharacteristic
538Q16—416no—
539Q16—416no—
540SD16—416no—
541M16—416no—
542C2×C8—28no—
543C2×C8—28no—
544C2×C8—28no—
545C2×C8—28no—
546C2×C8—22no—
547C2×C8—22no—
548C2×C8—22no—
549C2×C8—22no—
550C2×C8—22no—
551C2×C8—22no—
552C2×C8—22no—
553C2×C8—22no—
554C2×C8C22×D414yes—
555C2×C8C22×D414yes—
556C2×C8C22×D414yes—
557C2×C8C22×D414yes—
558C4⋊C4—22no—
559C4⋊C4—22no—
560C4⋊C4—22no—
561C4⋊C4—22no—
562C4⋊C4—22no—
563C4⋊C4—22no—
564C4⋊C4—22no—
565C4⋊C4—22no—
566C4⋊C4—416no—
567C4⋊C4—48no—
568C4⋊C4—42no—
569C4⋊C4—42no—
570C4⋊C4—42no—
571C4⋊C4—42no—
572C4⋊C4—48no—
573C4⋊C4—48no—
574C4⋊C4—88no—
575C4⋊C4—88no—
576C22⋊C4—216no—
577C22⋊C4—416no—
578C22⋊C4—48no—
579C22⋊C4—48no—
580C22⋊C4—416no—
581C42—24no—
582C42—24no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C23—28no—
588C23—28no—
589C23—22no—
590C23—22no—
591C23—28no—
592C23—24no—
593C23—24no—
594C23—24no—
595C23—48no—
596C23—48no—
597C23—416no—
598C23—42no—
599C23—42no—
600C23—48no—
601C23—44no—
602C23—44no—
603C23—84no—
604C23—84no—
605C23—88no—
606C23C2×#12714yes—
607C23C2×#12712yes—
608C23#54612yes—
609C23#54612yes—
610C23C2×#13714yes—
611C23#57714yes—
612C23C23×D411yescharacteristic
613Q8—24no—
614Q8—24no—
615Q8—42no—
616Q8—42no—
617Q8—48no—
618Q8—42no—
619Q8—42no—
620Q8—42no—
621Q8—42no—
622Q8—42no—
623Q8—42no—
624Q8—42no—
625Q8—42no—
626Q8—84no—
627Q8—84no—
628D4—24no—
629D4—24no—
630D4—432no—
631D4—42no—
632D4—42no—
633D4—48no—
634C2×C4—24no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—28no—
638C2×C4—216no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—216no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—24no—
649C2×C4—432no—
650C2×C4—48no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—48no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—816no—
669C2×C4—84no—
670C2×C4—84no—
671C2×C4—84no—
672C2×C4—84no—
673C2×C4—84no—
674C2×C4—84no—
675C2×C4#52514yes—
676C2×C4C2×#12714yes—
677C2×C4C2×#12712yes—
678C2×C4#53414yes—
679C2×C4#54614yes—
680C2×C4#54614yes—
681C2×C4#54612yes—
682C2×C4#54612yes—
683C2×C4C2×#13718yes—
684C2×C4C23×D411yescharacteristic
685C2×C4C23×D411yescharacteristic
686C2×C4C22×#4111yescharacteristic
687C2×C4C22×#4111yescharacteristic
688C2×C4C2×H(5,𝔽2)11yescharacteristic
689C2×C4C2×H(5,𝔽2)11yescharacteristic
690C8—24no—
691C8—24no—
692C8—24no—
693C8—24no—
694C22—24no—
695C22—24no—
696C22—24no—
697C22—416no—
698C22—44no—
699C22—44no—
700C22—44no—
701C22—48no—
702C22—816no—
703C22—84no—
704C22—82no—
705C22—82no—
706C22#253914yes—
707C22#315214yes—
708C22#315414yes—
709C22#331618yes—
710C22#333118yes—
711C22C22×#12711yescharacteristic
712C22C2×#54611yescharacteristic
713C22C2×#54611yescharacteristic
714C22C22×#13712yes—
715C22C2×#57712yes—
716C4—24no—
717C4—28no—
718C4—44no—
719C4—44no—
720C4—82no—
721C4—82no—
722C4—82no—
723C4—82no—
724C4—82no—
725C4—82no—
726C4#253914yes—
727C4C2×#52511yescharacteristic
728C4C2×#53411yescharacteristic
729C4C2×#54611yescharacteristic
730C4C2×#54611yescharacteristic
731C2—44no—
732C2—84no—
733C2#2836218yesdirect product factor
734C2C2×#253911yescharacteristic
735C2C2×#315211yescharacteristic
736C2C2×#315411yescharacteristic
737C2C2×#331612yes—
738C2C2×#333112yes—
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. C22×#483C2semidirect
2. C22×#470C2semidirect
5. C2×#2502C2semidirect
7. #28362C2direct (N × H)
8. C2×#2096C2semidirect
9. C2×#2094C2semidirect
10. C2×#2016C2semidirect
11. C2×#2015C2semidirect
12. C2×#1971C2semidirect
13. C2×#1968C2semidirect
14. C2×#1959C2semidirect
15. C2×#1956C2semidirect
16. C2×#1753C2semidirect
28. C2×#483C22semidirect
30. C2×#470C22semidirect
52. #2502C22semidirect
66. #2096C22semidirect
67. #2094C22semidirect
68. #2016C22semidirect
69. #2015C22semidirect
70. #1971C22semidirect
71. #1968C22semidirect
72. #1959C22semidirect
73. #1956C22semidirect
74. #1753C22semidirect
733. C2#28362direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4832
C22×#4702
C22×#4522
C22×#4172
C2×#25022
C2×#24322
#283622
C2×#20962
C2×#20942
C2×#20162
C2×#20152
C2×#19712
C2×#19682
C2×#19592
C2×#19562
C2×#17532