Finite Group Databasebeta

← C2×#26814 #6443773

Subgroups — C2×#26814 #6443773

Order 512.

2462 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#485C211yesmaximal, characteristic, complemented
2C22×#476C211yesmaximal, characteristic, complemented
3C2×#2492C211yesmaximal, characteristic, complemented
4C2×#2363C211yesmaximal, characteristic
5#26814C2116yesmaximal, direct product factor
6C2×#2068C211yesmaximal, characteristic, complemented
7C2×#2057C211yesmaximal, characteristic, complemented
8C2×#2033C211yesmaximal, characteristic
9C2×#2000C211yesmaximal, characteristic
10C2×#1973C211yesmaximal, characteristic, complemented
11C2×#1927C211yesmaximal, characteristic, complemented
12C2×#1925C211yesmaximal, characteristic, complemented
13C2×#1924C211yesmaximal, characteristic, complemented
14C2×#1877C211yesmaximal, characteristic, complemented
15C2×#1854C211yesmaximal, characteristic, complemented
16C2×#1776C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×#121C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C2×#485—28no—
24C2×#485C2218yescomplemented
25C2×#476—28no—
26C2×#476C2218yescomplemented
27C22×#107C2211yescharacteristic
28C22×#107C2211yescharacteristic
29C22×#107C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×#106C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×C22⋊C8C2211yescharacteristic
40C22×C22⋊C8C2211yescharacteristic
41C22×#97C2211yescharacteristic
42#2492—24no—
43#2492C2218yescomplemented
44C2×#400—21no—
45C2×#393—21no—
46#2363—24no—
47#2363C2218yes—
48C2×#391—21no—
49C2×#391C2211yescharacteristic
50C2×#391C2211yescharacteristic
51C2×#390—21no—
52C2×#390C2211yescharacteristic
53C2×#389C2211yescharacteristic
54C2×#388—21no—
55C2×#388—21no—
56C2×#386—21no—
57C2×#380C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59C2×C4×C4⋊C4C2211yescharacteristic
60C2×C4×C22⋊C4—21no—
61C2×C4×C22⋊C4C2211yescharacteristic
62#2068C2218yescomplemented
63#2057C2218yescomplemented
64#2033C2218yes—
65#2000C2218yes—
66#1973C2218yescomplemented
67#1927C2218yescomplemented
68#1925C2218yescomplemented
69#1924C2218yescomplemented
70#1877C2218yescomplemented
71#1854C2218yescomplemented
72#1776C2218yescomplemented
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8—21no—
80C23×Q8C2311yescharacteristic
81C23×D4—21no—
82C23×D4C2311yescharacteristic
83C24×C4—21no—
84C24×C4C2311yescharacteristic
85C22×SD16—24no—
86C22×SD16—24no—
87C22×SD16—42no—
88C22×SD16C2314yes—
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—22no—
95C2×C4⋊Q8—42no—
96C2×C4⋊Q8C2314yes—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—42no—
100C2×C22⋊Q8C2314yes—
101C2×#121—24no—
102C2×#121—22no—
103C2×#121—24no—
104C2×#121C2314yes—
105C2×C4×D4—24no—
106C2×C4×D4—22no—
107C2×C4×D4—42no—
108C2×C4×D4C2314yes—
109C2×#117—22no—
110C2×#117—24no—
111C2×#117—42no—
112C2×#117C2314yes—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4D411yescharacteristic
130C22×C22⋊C4D411yescharacteristic
131C22×C42—21no—
132C22×C42—21no—
133C22×C42—21no—
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#485—232no—
137#476—232no—
138C2×#107—24no—
139C2×#107—24no—
140C2×#107—42no—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#106—24no—
145C2×#106D418yescomplemented
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×C4⋊C8—24no—
149C2×C4⋊C8C2314yes—
150C2×#103—24no—
151C2×#103—24no—
152C2×#103—24no—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#102—24no—
157C2×#102—24no—
158C2×#102—24no—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×C22⋊C8—24no—
163C2×C22⋊C8—24no—
164C2×C22⋊C8C2314yes—
165C2×C22⋊C8C2314yes—
166C2×#97—24no—
167C2×#97C2314yes—
168#400—28no—
169#393—28no—
170#391—22no—
171#391—22no—
172#391—28no—
173#391C2314yes—
174#391C2314yes—
175#390—28no—
176#390—22no—
177#390C2314yes—
178#389—22no—
179#389C2314yes—
180#388—28no—
181#388—28no—
182#386—28no—
183#380—22no—
184#380C2314yes—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4C2314yes—
188C4×C4⋊C4C2314yes—
189C4×C22⋊C4—28no—
190C4×C22⋊C4D414yes—
191C4×C22⋊C4C2314yes—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95D411yescharacteristic
201C2×#95D411yescharacteristic
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#336C2314yes—
207#336C2314yes—
208C25—21no—
209C25C2×D411yescharacteristic
210C22×Q8—21no—
211C22×Q8—24no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—44no—
219C22×Q8—44no—
220C22×Q8—82no—
221C22×Q8C2412yes—
222C22×D4—28no—
223C22×D4—24no—
224C22×D4—22no—
225C22×D4—21no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—82no—
231C22×D4—82no—
232C22×D4#4114yes—
233C22×D4C2412yes—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—24no—
248C23×C4—44no—
249C23×C4—44no—
250C23×C4—82no—
251C23×C4C2×D412yes—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2412yes—
258C23×C4C2411yescharacteristic
259C2×SD16—28no—
260C2×SD16—28no—
261C2×SD16—44no—
262C2×SD16—416no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8C2×D414yes—
274C22×C8#4114yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C4⋊Q8—28no—
280C4⋊Q8—44no—
281C4⋊Q8—48no—
282C22⋊Q8—28no—
283C22⋊Q8—44no—
284C22⋊Q8—48no—
285#121—216no—
286#121—28no—
287#121—28no—
288C4×D4—28no—
289C4×D4—44no—
290C4×D4—48no—
291#117—28no—
292#117—44no—
293#117—48no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
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—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—82no—
315C2×C4⋊C4C2×D414yes—
316C2×C4⋊C4C2×D414yes—
317C2×C4⋊C4#4114yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C22⋊C4—28no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—42no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—82no—
339C2×C22⋊C4—82no—
340C2×C22⋊C4C2×D414yes—
341C2×C22⋊C4C2×D414yes—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—24no—
347C2×C42—24no—
348C2×C42—24no—
349C2×C42—44no—
350C2×C42—82no—
351C2×C42—82no—
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2412yes—
355C2×C42C2412yes—
356#107—28no—
357#107—28no—
358#107—44no—
359#106—28no—
360#106C2×D4116yescomplemented
361C4⋊C8—28no—
362#103—28no—
363#103—28no—
364#103—28no—
365#102—28no—
366#102—28no—
367#102—28no—
368C22⋊C8—28no—
369C22⋊C8—28no—
370#97—28no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95C2×D414yes—
380#95C2×D414yes—
381C24—24no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24C22×D412yes—
397C24C22×D411yescharacteristic
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—48no—
405C2×Q8—48no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—84no—
414C2×Q8—816no—
415C2×D4—216no—
416C2×D4—24no—
417C2×D4—28no—
418C2×D4—28no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—84no—
429C2×D4—84no—
430C2×D4—816no—
431C2×D4C2×#4114yes—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—48no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
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—816no—
491C22×C4#13714yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D412yes—
499C22×C4C2×#4112yes—
500C22×C4C2×#4112yes—
501C22×C4C2×#4112yes—
502C22×C4H(5,𝔽2)12yes—
503C22×C42−1+412yes—
504C22×C4C2511yescharacteristic
505SD16—416no—
506C2×C8—28no—
507C2×C8—28no—
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—44no—
519C2×C8—44no—
520C2×C8C22×D414yes—
521C2×C8C2×#4114yes—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—24no—
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—48no—
535C4⋊C4—48no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—84no—
540C4⋊C4C22×D414yes—
541C4⋊C4C22×D414yes—
542C4⋊C4C2×#4114yes—
543C22⋊C4—216no—
544C22⋊C4—28no—
545C22⋊C4—44no—
546C22⋊C4—48no—
547C22⋊C4—48no—
548C22⋊C4—48no—
549C22⋊C4—48no—
550C22⋊C4—84no—
551C22⋊C4—84no—
552C42—24no—
553C42—24no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—48no—
559C42—84no—
560C42—84no—
561C23—24no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—48no—
573C23—44no—
574C23—48no—
575C23—42no—
576C23—42no—
577C23—44no—
578C23—48no—
579C23—44no—
580C23—44no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—84no—
586C23—84no—
587C23—84no—
588C23—84no—
589C23#52512yes—
590C23D4212yes—
591C23#54712yes—
592C23C2×#13712yes—
593C23#57512yes—
594C23#57612yes—
595C23#57712yes—
596C23C23×D411yescharacteristic
597Q8—42no—
598Q8—42no—
599Q8—42no—
600Q8—42no—
601Q8—816no—
602Q8—84no—
603D4—24no—
604D4—24no—
605D4—42no—
606D4—42no—
607D4—48no—
608D4—816no—
609C2×C4—22no—
610C2×C4—28no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—48no—
630C2×C4—44no—
631C2×C4—48no—
632C2×C4—44no—
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—48no—
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—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—816no—
658C2×C4—84no—
659C2×C4—84no—
660C2×C4—84no—
661C2×C4—84no—
662C2×C4#52512yes—
663C2×C4#53214yes—
664C2×C4#54214yes—
665C2×C4D4212yes—
666C2×C4#54712yes—
667C2×C4#54814yes—
668C2×C4#55114yes—
669C2×C4C2×#13714yes—
670C2×C4C23×D411yescharacteristic
671C2×C4C22×#4111yescharacteristic
672C2×C4C22×#4111yescharacteristic
673C2×C4C22×#4111yescharacteristic
674C2×C4C2×H(5,𝔽2)11yescharacteristic
675C2×C4C2×2−1+411yescharacteristic
676C8—24no—
677C8—24no—
678C8—42no—
679C8—42no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—44no—
686C22—42no—
687C22—44no—
688C22—42no—
689C22—44no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—88no—
695C22—816no—
696C22#246114yes—
697C22#311614yes—
698C22#311814yes—
699C22#326614yes—
700C22#326814yes—
701C22#335014yes—
702C22#335714yes—
703C22C2×#52511yescharacteristic
704C22C2×D4211yescharacteristic
705C22C2×#54711yescharacteristic
706C22C22×#13711yescharacteristic
707C22C2×#57511yescharacteristic
708C22C2×#57611yescharacteristic
709C22C2×#57711yescharacteristic
710C4—24no—
711C4—42no—
712C4—42no—
713C4—44no—
714C4—42no—
715C4—44no—
716C4—42no—
717C4—44no—
718C4—82no—
719C4—82no—
720C4—82no—
721C4—82no—
722C4#246114yes—
723C4C2×#53211yescharacteristic
724C4C2×#54211yescharacteristic
725C4C2×#54811yescharacteristic
726C4C2×#55111yescharacteristic
727C2—82no—
728C2—84no—
729C2#2681418yesdirect product factor
730C2C2×#246111yescharacteristic
731C2C2×#311611yescharacteristic
732C2C2×#311811yescharacteristic
733C2C2×#326611yescharacteristic
734C2C2×#326811yescharacteristic
735C2C2×#335011yescharacteristic
736C2C2×#335711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
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×#485C2semidirect
2. C22×#476C2semidirect
3. C2×#2492C2semidirect
5. #26814C2direct (N × H)
6. C2×#2068C2semidirect
7. C2×#2057C2semidirect
10. C2×#1973C2semidirect
11. C2×#1927C2semidirect
12. C2×#1925C2semidirect
13. C2×#1924C2semidirect
14. C2×#1877C2semidirect
15. C2×#1854C2semidirect
16. C2×#1776C2semidirect
24. C2×#485C22semidirect
26. C2×#476C22semidirect
43. #2492C22semidirect
62. #2068C22semidirect
63. #2057C22semidirect
66. #1973C22semidirect
67. #1927C22semidirect
68. #1925C22semidirect
69. #1924C22semidirect
70. #1877C22semidirect
71. #1854C22semidirect
72. #1776C22semidirect
145. C2×#106D4semidirect
360. #106C2×D4semidirect
729. C2#26814direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4852
C22×#4762
C2×#24922
C2×#23632
#268142
C2×#20682
C2×#20572
C2×#20332
C2×#20002
C2×#19732
C2×#19272
C2×#19252
C2×#19242
C2×#18772
C2×#18542
C2×#17762