Finite Group Databasebeta

← C2×#26184 #6438411

Subgroups — C2×#26184 #6438411

Order 512.

2612 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#463C211yesmaximal, characteristic
2C22×#440C211yesmaximal, characteristic, complemented
3C2×#2448C211yesmaximal, characteristic, complemented
4C2×#2349C211yesmaximal, characteristic, complemented
5#26184C2116yesmaximal, direct product factor
6C2×#2092C211yesmaximal, characteristic, complemented
7C2×#2086C211yesmaximal, characteristic, complemented
8C2×#2074C211yesmaximal, characteristic, complemented
9C2×#2042C211yesmaximal, characteristic, complemented
10C2×#1936C211yesmaximal, characteristic, complemented
11C2×#1917C211yesmaximal, characteristic, complemented
12C2×#1879C211yesmaximal, characteristic
13C2×#1879C211yesmaximal, characteristic
14C2×#1877C211yesmaximal, characteristic, complemented
15C2×#1865C211yesmaximal, characteristic, complemented
16C2×#1864C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C23×D8C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24C2×#463—28no—
25C2×#463C2218yes—
26C2×#440—28no—
27C2×#440C2218yescomplemented
28C22×#107C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×#97C2211yescharacteristic
40#2448—24no—
41#2448C2218yescomplemented
42C2×#399—21no—
43C2×#397—21no—
44#2349—24no—
45#2349C2218yescomplemented
46C2×#391C2211yescharacteristic
47C2×#390C2211yescharacteristic
48C2×#390C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#388—21no—
53C2×#388—21no—
54C2×#385—21no—
55C2×#385—21no—
56C2×#382C2211yescharacteristic
57C2×#380—21no—
58C2×C4×C4⋊C4C2211yescharacteristic
59C22×#95—21no—
60#2092C2218yescomplemented
61#2086C2218yescomplemented
62#2074C2218yescomplemented
63#2042C2218yescomplemented
64#1936C2218yescomplemented
65#1917C2218yescomplemented
66#1879C2218yes—
67#1879C2218yes—
68#1877C2218yescomplemented
69#1865C2218yescomplemented
70#1864C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C24×C4—21no—
83C24×C4—21no—
84C24×C4—21no—
85C22×SD16—24no—
86C22×SD16—24no—
87C22×SD16D418yes—
88C22×SD16C2314yes—
89C22×D8—24no—
90C22×D8—24no—
91C22×D8—24no—
92C22×D8C2314yes—
93C23×C8C2311yescharacteristic
94C23×C8C2311yescharacteristic
95C23×C8C2311yescharacteristic
96C23×C8C2311yescharacteristic
97C2×#124—24no—
98C2×#124—22no—
99C2×#124—42no—
100C2×#124C2314yes—
101C2×C4×Q8—22no—
102C2×C4×Q8—24no—
103C2×C4×Q8—42no—
104C2×C4×Q8C2314yes—
105C2×C4×D4—24no—
106C2×C4×D4—24no—
107C2×C4×D4—24no—
108C2×C4×D4—24no—
109C2×C4×D4—22no—
110C2×C4×D4—22no—
111C2×C4×D4—22no—
112C2×C4×D4—42no—
113C2×C4×D4—42no—
114C2×C4×D4C2314yes—
115C2×C4×D4C2314yes—
116C2×C4×D4C2314yes—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—21no—
135C22×C22⋊C4—21no—
136C22×C22⋊C4—21no—
137C22×C42—21no—
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141#463—232no—
142#440—232no—
143C2×#107—24no—
144C2×#107C2314yes—
145C2×#106—42no—
146C2×#106C2314yes—
147C2×C4⋊C8—24no—
148C2×C4⋊C8C2314yes—
149C2×#103—24no—
150C2×#103—24no—
151C2×#103—24no—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#102—24no—
156C2×#102—24no—
157C2×#102—24no—
158C2×#102—24no—
159C2×#102—42no—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#97—24no—
166C2×#97C2314yes—
167#399—28no—
168#397—28no—
169#391—22no—
170#391C2314yes—
171#390—22no—
172#390—22no—
173#390C2314yes—
174#390C2314yes—
175#389—22no—
176#389—22no—
177#389—22no—
178#389C2314yes—
179#389C2314yes—
180#389C2314yes—
181#388—28no—
182#388—28no—
183#385—28no—
184#385—28no—
185#382—22no—
186#382C2314yes—
187#380—28no—
188C4×C4⋊C4D414yes—
189C4×C4⋊C4C2314yes—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—28no—
198C2×#95D411yescharacteristic
199C2×#95D411yescharacteristic
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#336C2314yes—
205#336C2314yes—
206#336C2314yes—
207#336C2314yes—
208C25—21no—
209C25—21no—
210C22×Q8—21no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8C2×D414yes—
217C22×Q8C2412yes—
218C22×D4—21no—
219C22×D4—22no—
220C22×D4—21no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4C2×D414yes—
230C22×D4#4114yes—
231C22×D4#4114yes—
232C22×D4C2412yes—
233C22×D4C2412yes—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—28no—
237C23×C4—24no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—24no—
243C23×C4—24no—
244C23×C4—44no—
245C23×C4—44no—
246C23×C4—44no—
247C23×C4—44no—
248C23×C4—82no—
249C23×C4—82no—
250C23×C4—82no—
251C23×C4—82no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2411yescharacteristic
259C2×SD16—232no—
260C2×SD16—28no—
261C2×SD16—28no—
262C2×SD16C2×D4116yes—
263C2×D8—28no—
264C2×D8—28no—
265C2×D8—28no—
266C2×D8—416no—
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×C8—22no—
276C22×C8—22no—
277C22×C8—22no—
278C22×C8C2×D414yes—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2412yes—
283#124—28no—
284#124—44no—
285#124—48no—
286C4×Q8—28no—
287C4×Q8—44no—
288C4×Q8—48no—
289C4×D4—216no—
290C4×D4—28no—
291C4×D4—28no—
292C4×D4—28no—
293C4×D4—28no—
294C4×D4—44no—
295C4×D4—44no—
296C4×D4—48no—
297C4×D4—48no—
298C2×C4⋊C4—21no—
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—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4—42no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—82no—
320C2×C4⋊C4—82no—
321C2×C4⋊C4—82no—
322C2×C4⋊C4C2×D414yes—
323C2×C4⋊C4C2×D414yes—
324C2×C4⋊C4#4114yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C22⋊C4—28no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4—44no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4—44no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4—82no—
347C2×C22⋊C4—82no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—24no—
355C2×C42—44no—
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2×D412yes—
359C2×C42C2412yes—
360C2×C42C2412yes—
361C2×C42C2412yes—
362#107—28no—
363#106—44no—
364C4⋊C8—28no—
365#103—28no—
366#103—28no—
367#103—28no—
368#102—28no—
369#102—28no—
370#102—28no—
371#102—28no—
372#102—44no—
373#97—28no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95C2×D414yes—
382#95C2×D414yes—
383C24—24no—
384C24—24no—
385C24—24no—
386C24—22no—
387C24—22no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24C22×D411yescharacteristic
400C2×Q8—24no—
401C2×Q8—22no—
402C2×Q8—28no—
403C2×Q8—28no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—42no—
409C2×Q8—44no—
410C2×Q8—42no—
411C2×Q8—44no—
412C2×Q8C22×D414yes—
413C2×D4—24no—
414C2×D4—28no—
415C2×D4—24no—
416C2×D4—28no—
417C2×D4—24no—
418C2×D4—28no—
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—44no—
429C2×D4—42no—
430C2×D4—44no—
431C2×D4—42no—
432C2×D4—44no—
433C2×D4—42no—
434C2×D4—44no—
435C2×D4#123116yes—
436C2×D4C22×D414yes—
437C2×D4C2×#4114yes—
438C2×D4C2×#4114yes—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
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—22no—
460C22×C4—22no—
461C22×C4—216no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—42no—
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—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—84no—
487C22×C4—84no—
488C22×C4—84no—
489C22×C4—84no—
490C22×C4—816no—
491C22×C4—816no—
492C22×C4—816no—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D411yescharacteristic
501C22×C4C2×#4112yes—
502C22×C4C2×#4112yes—
503C22×C4C2×#4112yes—
504C22×C4H(5,𝔽2)12yes—
505C22×C42−1+412yes—
506C22×C4C2511yescharacteristic
507SD16—232no—
508D8—416no—
509C2×C8—28no—
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—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—44no—
522C2×C8—44no—
523C2×C8—44no—
524C2×C8C22×D414yes—
525C4⋊C4—24no—
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—22no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—48no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—84no—
543C4⋊C4—84no—
544C4⋊C4—84no—
545C4⋊C4C2×#4114yes—
546C22⋊C4—216no—
547C22⋊C4—48no—
548C22⋊C4—48no—
549C22⋊C4—48no—
550C22⋊C4—48no—
551C22⋊C4—48no—
552C22⋊C4—48no—
553C22⋊C4—84no—
554C22⋊C4—84no—
555C42—24no—
556C42—24no—
557C42—24no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—48no—
565C23—24no—
566C23—24no—
567C23—24no—
568C23—216no—
569C23—48no—
570C23—48no—
571C23—48no—
572C23—48no—
573C23—48no—
574C23—48no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—84no—
587C23—84no—
588C23—84no—
589C23—84no—
590C23—84no—
591C23—84no—
592C23—84no—
593C23#52512yes—
594C23D4212yes—
595C23#54812yes—
596C23#57512yes—
597C23#57512yes—
598C23#57612yes—
599C23#57712yes—
600C23C23×D411yescharacteristic
601Q8—24no—
602Q8—24no—
603Q8—42no—
604Q8—48no—
605Q8—42no—
606D4—24no—
607D4—24no—
608D4—216no—
609D4—24no—
610D4—42no—
611D4—42no—
612D4—42no—
613D4—42no—
614D4—84no—
615D4C2×#12318yes—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—48no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—48no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
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—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—816no—
661C2×C4—816no—
662C2×C4—816no—
663C2×C4—84no—
664C2×C4—84no—
665C2×C4—84no—
666C2×C4—84no—
667C2×C4C2×#12314yes—
668C2×C4#52512yes—
669C2×C4#53914yes—
670C2×C4D4212yes—
671C2×C4#54714yes—
672C2×C4#54812yes—
673C2×C4#55314yes—
674C2×C4C23×D411yescharacteristic
675C2×C4C22×#4111yescharacteristic
676C2×C4C22×#4111yescharacteristic
677C2×C4C22×#4111yescharacteristic
678C2×C4C2×H(5,𝔽2)11yescharacteristic
679C2×C4C2×2−1+411yescharacteristic
680C8—24no—
681C8—42no—
682C8—42no—
683C8—42no—
684C22—28no—
685C22—432no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22—44no—
690C22—44no—
691C22—44no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—168no—
700C22#244314yes—
701C22#311114yes—
702C22#311814yes—
703C22#326814yes—
704C22#327514yes—
705C22#337014yes—
706C22#337614yes—
707C22C2×#52511yescharacteristic
708C22C2×D4211yescharacteristic
709C22C2×#54811yescharacteristic
710C22C2×#57511yescharacteristic
711C22C2×#57511yescharacteristic
712C22C2×#57611yescharacteristic
713C22C2×#57711yescharacteristic
714C4—44no—
715C4—42no—
716C4—42no—
717C4—42no—
718C4—42no—
719C4—42no—
720C4—42no—
721C4—44no—
722C4—82no—
723C4—82no—
724C4—82no—
725C4—82no—
726C4#244314yes—
727C4C22×#12311yescharacteristic
728C4C2×#53911yescharacteristic
729C4C2×#54711yescharacteristic
730C4C2×#55311yescharacteristic
731C2—48no—
732C2—162no—
733C2#2618418yesdirect product factor
734C2C2×#244311yescharacteristic
735C2C2×#311111yescharacteristic
736C2C2×#311811yescharacteristic
737C2C2×#326811yescharacteristic
738C2C2×#327511yescharacteristic
739C2C2×#337011yescharacteristic
740C2C2×#337611yescharacteristic
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
2. C22×#440C2semidirect
3. C2×#2448C2semidirect
4. C2×#2349C2semidirect
5. #26184C2direct (N × H)
6. C2×#2092C2semidirect
7. C2×#2086C2semidirect
8. C2×#2074C2semidirect
9. C2×#2042C2semidirect
10. C2×#1936C2semidirect
11. C2×#1917C2semidirect
14. C2×#1877C2semidirect
15. C2×#1865C2semidirect
16. C2×#1864C2semidirect
27. C2×#440C22semidirect
41. #2448C22semidirect
45. #2349C22semidirect
60. #2092C22semidirect
61. #2086C22semidirect
62. #2074C22semidirect
63. #2042C22semidirect
64. #1936C22semidirect
65. #1917C22semidirect
68. #1877C22semidirect
69. #1865C22semidirect
70. #1864C22semidirect
733. C2#26184direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4632
C22×#4402
C2×#24482
C2×#23492
#261842
C2×#20922
C2×#20862
C2×#20742
C2×#20422
C2×#19362
C2×#19172
C2×#18792
C2×#18772
C2×#18652
C2×#18642