Finite Group Databasebeta

← C2×#26796 #6443756

Subgroups — C2×#26796 #6443756

Order 512.

2424 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#485C211yesmaximal, characteristic, complemented
2C22×#478C211yesmaximal, characteristic, complemented
3C2×#2489C211yesmaximal, characteristic, complemented
4C2×#2363C211yesmaximal, characteristic
5#26796C2116yesmaximal, direct product factor
6C2×#2066C211yesmaximal, characteristic, complemented
7C2×#2059C211yesmaximal, characteristic, complemented
8C2×#2034C211yesmaximal, characteristic
9C2×#2000C211yesmaximal, characteristic
10C2×#1968C211yesmaximal, characteristic, complemented
11C2×#1928C211yesmaximal, characteristic, complemented
12C2×#1925C211yesmaximal, characteristic, complemented
13C2×#1923C211yesmaximal, characteristic, complemented
14C2×#1877C211yesmaximal, characteristic, complemented
15C2×#1854C211yesmaximal, characteristic, complemented
16C2×C4×#106C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×#121C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C2×#485—28no—
24C2×#485C2218yescomplemented
25C2×#478—28no—
26C2×#478C2218yescomplemented
27C22×#107C2211yescharacteristic
28C22×#107C2211yescharacteristic
29C22×#106C2211yescharacteristic
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×C4×C8C2211yescharacteristic
42#2489—24no—
43#2489C2218yescomplemented
44C2×#400—21no—
45C2×#397—21no—
46#2363—24no—
47#2363C2218yes—
48C2×#391C2211yescharacteristic
49C2×#391C2211yescharacteristic
50C2×#390—21no—
51C2×#390C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#388—21no—
54C2×#388—21no—
55C2×#386—21no—
56C2×#382—21no—
57C2×#380C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59C2×C4×C4⋊C4C2211yescharacteristic
60C2×C4×C22⋊C4—21no—
61C2×C4×C22⋊C4C2211yescharacteristic
62#2066C2218yescomplemented
63#2059C2218yescomplemented
64#2034C2218yes—
65#2000C2218yes—
66#1968C2218yescomplemented
67#1928C2218yescomplemented
68#1925C2218yescomplemented
69#1923C2218yescomplemented
70#1877C2218yescomplemented
71#1854C2218yescomplemented
72C4×#106C2218yescomplemented
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4—21no—
81C23×D4C2311yescharacteristic
82C24×C4—21no—
83C24×C4C2311yescharacteristic
84C22×SD16—24no—
85C22×SD16—24no—
86C22×SD16—42no—
87C22×SD16C2314yes—
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C2×#125—24no—
93C2×#125—22no—
94C2×#125—42no—
95C2×#125C2314yes—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—42no—
99C2×C22⋊Q8C2314yes—
100C2×#121—24no—
101C2×#121—22no—
102C2×#121—24no—
103C2×#121C2314yes—
104C2×C4×D4—24no—
105C2×C4×D4—22no—
106C2×C4×D4—42no—
107C2×C4×D4C2314yes—
108C2×#117—22no—
109C2×#117—24no—
110C2×#117—42no—
111C2×#117C2314yes—
112C22×C4⋊C4—21no—
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#478—232no—
138C2×#107—24no—
139C2×#107—24no—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#106—24no—
143C2×#106—42no—
144C2×#106D418yescomplemented
145C2×#106C2314yes—
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×C4×C8—24no—
167C2×C4×C8C2314yes—
168#400—28no—
169#397—28no—
170#391—22no—
171#391—22no—
172#391C2314yes—
173#391C2314yes—
174#390—28no—
175#390—22no—
176#390C2314yes—
177#389—22no—
178#389C2314yes—
179#388—28no—
180#388—28no—
181#386—28no—
182#382—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—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8C2412yes—
218C22×D4—28no—
219C22×D4—24no—
220C22×D4—22no—
221C22×D4—21no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—82no—
227C22×D4—82no—
228C22×D4#4114yes—
229C22×D4C2412yes—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—22no—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—24no—
244C23×C4—44no—
245C23×C4—44no—
246C23×C4—82no—
247C23×C4C2×D412yes—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2412yes—
254C23×C4C2411yescharacteristic
255C2×SD16—28no—
256C2×SD16—28no—
257C2×SD16—44no—
258C2×SD16—416no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8C2×D414yes—
270C22×C8#4114yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275#125—28no—
276#125—44no—
277#125—48no—
278C22⋊Q8—28no—
279C22⋊Q8—44no—
280C22⋊Q8—48no—
281#121—216no—
282#121—28no—
283#121—28no—
284C4×D4—28no—
285C4×D4—44no—
286C4×D4—48no—
287#117—28no—
288#117—44no—
289#117—48no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
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—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—82no—
313C2×C4⋊C4C2×D414yes—
314C2×C4⋊C4C2×D414yes—
315C2×C4⋊C4#4114yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—28no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
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—42no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—82no—
338C2×C22⋊C4C2×D414yes—
339C2×C22⋊C4C2×D414yes—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42—44no—
348C2×C42—44no—
349C2×C42—82no—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2412yes—
353C2×C42C2412yes—
354#107—28no—
355#107—28no—
356#106—28no—
357#106—44no—
358#106C2×D4116yescomplemented
359C4⋊C8—28no—
360#103—28no—
361#103—28no—
362#103—28no—
363#102—28no—
364#102—28no—
365#102—28no—
366C22⋊C8—28no—
367C22⋊C8—28no—
368C4×C8—28no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95C2×D414yes—
378#95C2×D414yes—
379C24—24no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24C22×D412yes—
395C24C22×D411yescharacteristic
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×D4—216no—
410C2×D4—24no—
411C2×D4—28no—
412C2×D4—28no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—84no—
423C2×D4—84no—
424C2×D4—816no—
425C2×D4C2×#4114yes—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
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—84no—
484C22×C4—816no—
485C22×C4#13514yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D412yes—
493C22×C4C2×#4112yes—
494C22×C4C2×#4112yes—
495C22×C4C2×#4112yes—
496C22×C4H(5,𝔽2)12yes—
497C22×C42−1+412yes—
498C22×C4C2511yescharacteristic
499SD16—416no—
500C2×C8—28no—
501C2×C8—28no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—44no—
513C2×C8—44no—
514C2×C8C22×D414yes—
515C2×C8C2×#4114yes—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—48no—
529C4⋊C4—48no—
530C4⋊C4—48no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—84no—
535C4⋊C4C22×D414yes—
536C4⋊C4C22×D414yes—
537C4⋊C4C2×#4114yes—
538C22⋊C4—216no—
539C22⋊C4—28no—
540C22⋊C4—44no—
541C22⋊C4—48no—
542C22⋊C4—48no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C22⋊C4—48no—
546C22⋊C4—84no—
547C42—24no—
548C42—24no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—48no—
554C42—48no—
555C42—84no—
556C23—24no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—48no—
568C23—44no—
569C23—48no—
570C23—42no—
571C23—42no—
572C23—44no—
573C23—48no—
574C23—44no—
575C23—44no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—84no—
581C23—84no—
582C23—84no—
583C23—84no—
584C23#52512yes—
585C23D4212yes—
586C23#54712yes—
587C23C2×#13512yes—
588C23#57512yes—
589C23#57712yes—
590C23#57712yes—
591C23C23×D411yescharacteristic
592Q8—42no—
593Q8—42no—
594Q8—42no—
595Q8—42no—
596Q8—84no—
597D4—24no—
598D4—24no—
599D4—42no—
600D4—42no—
601D4—48no—
602D4—816no—
603C2×C4—28no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—44no—
622C2×C4—48no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—48no—
626C2×C4—44no—
627C2×C4—48no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—48no—
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—816no—
652C2×C4—84no—
653C2×C4—84no—
654C2×C4—84no—
655C2×C4—84no—
656C2×C4#52512yes—
657C2×C4#53214yes—
658C2×C4#54214yes—
659C2×C4D4212yes—
660C2×C4#54712yes—
661C2×C4#54814yes—
662C2×C4#55114yes—
663C2×C4C2×#13514yes—
664C2×C4C23×D411yescharacteristic
665C2×C4C22×#4111yescharacteristic
666C2×C4C22×#4111yescharacteristic
667C2×C4C22×#4111yescharacteristic
668C2×C4C2×H(5,𝔽2)11yescharacteristic
669C2×C4C2×2−1+411yescharacteristic
670C8—24no—
671C8—24no—
672C8—42no—
673C8—42no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—44no—
679C22—42no—
680C22—44no—
681C22—42no—
682C22—42no—
683C22—44no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—88no—
689C22—816no—
690C22#246114yes—
691C22#310214yes—
692C22#312314yes—
693C22#327514yes—
694C22#328414yes—
695C22#333714yes—
696C22#334814yes—
697C22C2×#52511yescharacteristic
698C22C2×D4211yescharacteristic
699C22C2×#54711yescharacteristic
700C22C22×#13511yescharacteristic
701C22C2×#57511yescharacteristic
702C22C2×#57711yescharacteristic
703C22C2×#57711yescharacteristic
704C4—24no—
705C4—42no—
706C4—42no—
707C4—44no—
708C4—42no—
709C4—42no—
710C4—44no—
711C4—44no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4#246114yes—
717C4C2×#53211yescharacteristic
718C4C2×#54211yescharacteristic
719C4C2×#54811yescharacteristic
720C4C2×#55111yescharacteristic
721C2—82no—
722C2—84no—
723C2#2679618yesdirect product factor
724C2C2×#246111yescharacteristic
725C2C2×#310211yescharacteristic
726C2C2×#312311yescharacteristic
727C2C2×#327511yescharacteristic
728C2C2×#328411yescharacteristic
729C2C2×#333711yescharacteristic
730C2C2×#334811yescharacteristic
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×#478C2semidirect
3. C2×#2489C2semidirect
5. #26796C2direct (N × H)
6. C2×#2066C2semidirect
7. C2×#2059C2semidirect
10. C2×#1968C2semidirect
11. C2×#1928C2semidirect
12. C2×#1925C2semidirect
13. C2×#1923C2semidirect
14. C2×#1877C2semidirect
15. C2×#1854C2semidirect
16. C2×C4×#106C2semidirect
24. C2×#485C22semidirect
26. C2×#478C22semidirect
43. #2489C22semidirect
62. #2066C22semidirect
63. #2059C22semidirect
66. #1968C22semidirect
67. #1928C22semidirect
68. #1925C22semidirect
69. #1923C22semidirect
70. #1877C22semidirect
71. #1854C22semidirect
72. C4×#106C22semidirect
144. C2×#106D4semidirect
358. #106C2×D4semidirect
723. C2#26796direct (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×#4782
C2×#24892
C2×#23632
#267962
C2×#20662
C2×#20592
C2×#20342
C2×#20002
C2×#19682
C2×#19282
C2×#19252
C2×#19232
C2×#18772
C2×#18542
C2×C4×#1062