Finite Group Databasebeta

← C2×#26427 #6441916

Subgroups — C2×#26427 #6441916

Order 512.

2440 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2440C211yesmaximal, characteristic, complemented
2C2×#2363C211yesmaximal, characteristic
3#26427C2116yesmaximal, direct product factor
4C2×#2099C211yesmaximal, characteristic, complemented
5C2×#2088C211yesmaximal, characteristic, complemented
6C2×#2065C211yesmaximal, characteristic, complemented
7C2×#2023C211yesmaximal, characteristic, complemented
8C2×#1987C211yesmaximal, characteristic, complemented
9C2×#1921C211yesmaximal, characteristic, complemented
10C2×#1895C211yesmaximal, characteristic
11C2×#1874C211yesmaximal, characteristic
12C2×#1862C211yesmaximal, characteristic, complemented
13C2×#1854C211yesmaximal, characteristic, complemented
14C2×#1851C211yesmaximal, characteristic, complemented
15C2×#1802C211yesmaximal, characteristic, complemented
16C2×#1736C211yesmaximal, characteristic, complemented
17C23×M16C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8—21no—
20C22×#121C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×C22⋊C8C2211yescharacteristic
33#2440—24no—
34#2440C2218yescomplemented
35C2×#398—21no—
36C2×#395—21no—
37#2363—24no—
38#2363C2218yes—
39C2×#390—21no—
40C2×#390C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#388—21no—
45C2×#386—21no—
46C2×#385—21no—
47C2×#384C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4—21no—
51C2×C4×C22⋊C4C2211yescharacteristic
52#2099C2218yescomplemented
53#2088C2218yescomplemented
54#2065C2218yescomplemented
55#2023C2218yescomplemented
56#1987C2218yescomplemented
57#1921C2218yescomplemented
58#1895C2218yes—
59#1874C2218yes—
60#1862C2218yescomplemented
61#1854C2218yescomplemented
62#1851C2218yescomplemented
63#1802C2218yescomplemented
64#1736C2218yescomplemented
65C2×#340C2211yescharacteristic
66C2×#340C2211yescharacteristic
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8—21no—
78C23×D4—21no—
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C24×C4—21no—
82C24×C4C2311yescharacteristic
83C22×M16—24no—
84C22×M16—42no—
85C22×M16—42no—
86C22×M16C2314yes—
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—22no—
93C2×C4⋊Q8—42no—
94C2×C4⋊Q8C2314yes—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—48no—
97C2×#121—22no—
98C2×#121—24no—
99C2×#121Q818yescomplemented
100C2×#121C2314yes—
101C2×C4×D4—24no—
102C2×C4×D4—22no—
103C2×C4×D4—42no—
104C2×C4×D4C2314yes—
105C2×#117—22no—
106C2×#117—24no—
107C2×#117—42no—
108C2×#117C2314yes—
109C22×C4⋊C4—24no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—22no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—42no—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4D411yescharacteristic
134C22×C22⋊C4D411yescharacteristic
135C22×C42—21no—
136C22×C42—21no—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C2×#107—24no—
140C2×#107C2314yes—
141C2×#106—42no—
142C2×#106C2314yes—
143C2×C4⋊C8—42no—
144C2×C4⋊C8—42no—
145C2×C4⋊C8C2314yes—
146C2×C4⋊C8C2314yes—
147C2×#102—24no—
148C2×#102—42no—
149C2×#102—42no—
150C2×#102—42no—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×C22⋊C8—24no—
156C2×C22⋊C8C2314yes—
157#398—28no—
158#395—28no—
159#390—28no—
160#390—22no—
161#390C2314yes—
162#389—22no—
163#389—22no—
164#389—22no—
165#389C2314yes—
166#389C2314yes—
167#389C2314yes—
168#388—28no—
169#386—28no—
170#385—28no—
171#384—22no—
172#384C2314yes—
173#380—22no—
174#380C2314yes—
175C4×C4⋊C4—22no—
176C4×C4⋊C4C2314yes—
177C4×C22⋊C4—28no—
178C4×C22⋊C4D414yes—
179C4×C22⋊C4C2314yes—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95D411yescharacteristic
186C2×#95D411yescharacteristic
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#336C2314yes—
198#336C2314yes—
199C25—21no—
200C25C2×D411yescharacteristic
201C22×Q8—24no—
202C22×Q8—44no—
203C22×Q8—44no—
204C22×Q8—82no—
205C22×D4—28no—
206C22×D4—24no—
207C22×D4—21no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—82no—
214C22×D4—82no—
215C22×D4C2×Q814yes—
216C22×D4C2412yes—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4—82no—
235C23×C4—82no—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D412yes—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×Q814yes—
243C23×C4#4114yes—
244C23×C4C2412yes—
245C23×C4C2411yescharacteristic
246C2×M16—28no—
247C2×M16—44no—
248C2×M16—44no—
249C2×M16—416no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8C2412yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C4⋊Q8—28no—
267C4⋊Q8—44no—
268C4⋊Q8—48no—
269C22⋊Q8—432no—
270#121—216no—
271#121—28no—
272#121C2×Q8116yescomplemented
273C4×D4—28no—
274C4×D4—44no—
275C4×D4—48no—
276#117—28no—
277#117—44no—
278#117—48no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—48no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—82no—
313C2×C4⋊C4—82no—
314C2×C4⋊C4C2×Q814yes—
315C2×C4⋊C4#4114yes—
316C2×C4⋊C4#4114yes—
317C2×C4⋊C4C2412yes—
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—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—44no—
347C2×C42—82no—
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2412yes—
351C2×C42C2412yes—
352#107—28no—
353#106—44no—
354C4⋊C8—44no—
355C4⋊C8—44no—
356#102—28no—
357#102—44no—
358#102—44no—
359#102—44no—
360C22⋊C8—28no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95C2×D414yes—
367#95C2×D414yes—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—24no—
375C24—22no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24C22⋊Q814yes—
383C24C22×D412yes—
384C24C22×D411yescharacteristic
385C2×Q8—48no—
386C2×Q8—48no—
387C2×Q8—84no—
388C2×Q8—816no—
389C2×D4—216no—
390C2×D4—24no—
391C2×D4—28no—
392C2×D4—28no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—816no—
403C2×D4—84no—
404C2×D4—84no—
405C2×D4C22×Q814yes—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—48no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—48no—
460C22×C4—84no—
461C22×C4—816no—
462C22×C4—84no—
463C22×C4—816no—
464C22×C4C22⋊Q814yes—
465C22×C4C22⋊Q818yes—
466C22×C4AGL(1,ℤ/8)14yes—
467C22×C4#13714yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×Q814yes—
475C22×C4C22×Q812yes—
476C22×C4C2×#4114yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4H(5,𝔽2)12yes—
481C22×C4H(5,𝔽2)12yes—
482C22×C4C2511yescharacteristic
483M16—416no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—44no—
497C2×C8—44no—
498C2×C8—44no—
499C2×C8—44no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4—48no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—48no—
522C4⋊C4—48no—
523C4⋊C4—84no—
524C4⋊C4—84no—
525C4⋊C4C22×Q814yes—
526C4⋊C4C2×#4114yes—
527C4⋊C4C2×#4114yes—
528C22⋊C4—216no—
529C22⋊C4—28no—
530C22⋊C4—28no—
531C22⋊C4—48no—
532C22⋊C4—48no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—84no—
536C42—24no—
537C42—24no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—48no—
543C42—84no—
544C23—28no—
545C23—28no—
546C23—22no—
547C23—22no—
548C23—28no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—24no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—48no—
559C23—48no—
560C23—42no—
561C23—42no—
562C23—48no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—84no—
568C23—84no—
569C23—84no—
570C23—84no—
571C23C2×C22⋊Q814yes—
572C23C2×C22⋊Q812yes—
573C23#54612yes—
574C23#54612yes—
575C23C2×AGL(1,ℤ/8)12yes—
576C23C2×#13712yes—
577C23#57612yes—
578C23#57712yes—
579C23C23×D411yescharacteristic
580Q8—816no—
581D4—24no—
582D4—24no—
583D4—42no—
584D4—42no—
585D4—48no—
586D4—816no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—216no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—48no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—48no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—816no—
634C2×C4—816no—
635C2×C4—84no—
636C2×C4—84no—
637C2×C4—84no—
638C2×C4—84no—
639C2×C4C2×C22⋊Q814yes—
640C2×C4C2×C22⋊Q812yes—
641C2×C4#52914yes—
642C2×C4#53814yes—
643C2×C4#54612yes—
644C2×C4#54612yes—
645C2×C4#55414yes—
646C2×C4#55414yes—
647C2×C4C2×AGL(1,ℤ/8)14yes—
648C2×C4C2×#13714yes—
649C2×C4C23×Q811yescharacteristic
650C2×C4C22×#4111yescharacteristic
651C2×C4C22×#4111yescharacteristic
652C2×C4C22×#4111yescharacteristic
653C2×C4C2×H(5,𝔽2)11yescharacteristic
654C2×C4C2×H(5,𝔽2)11yescharacteristic
655C8—42no—
656C8—42no—
657C8—42no—
658C8—42no—
659C22—24no—
660C22—24no—
661C22—24no—
662C22—416no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—816no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22#245014yes—
676C22#307914yes—
677C22#308114yes—
678C22#330614yes—
679C22#332014yes—
680C22#332514yes—
681C22#333314yes—
682C22C22×C22⋊Q811yescharacteristic
683C22C2×#54611yescharacteristic
684C22C2×#54611yescharacteristic
685C22C22×AGL(1,ℤ/8)11yescharacteristic
686C22C22×#13711yescharacteristic
687C22C2×#57611yescharacteristic
688C22C2×#57711yescharacteristic
689C4—24no—
690C4—24no—
691C4—24no—
692C4—44no—
693C4—44no—
694C4—42no—
695C4—44no—
696C4—42no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4#245014yes—
702C4C2×#52911yescharacteristic
703C4C2×#53811yescharacteristic
704C4C2×#55411yescharacteristic
705C4C2×#55411yescharacteristic
706C2—44no—
707C2—84no—
708C2#2642718yesdirect product factor
709C2C2×#245011yescharacteristic
710C2C2×#307911yescharacteristic
711C2C2×#308111yescharacteristic
712C2C2×#330611yescharacteristic
713C2C2×#332011yescharacteristic
714C2C2×#332511yescharacteristic
715C2C2×#333311yescharacteristic
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. C2×#2440C2semidirect
3. #26427C2direct (N × H)
4. C2×#2099C2semidirect
5. C2×#2088C2semidirect
6. C2×#2065C2semidirect
7. C2×#2023C2semidirect
8. C2×#1987C2semidirect
9. C2×#1921C2semidirect
12. C2×#1862C2semidirect
13. C2×#1854C2semidirect
14. C2×#1851C2semidirect
15. C2×#1802C2semidirect
16. C2×#1736C2semidirect
34. #2440C22semidirect
52. #2099C22semidirect
53. #2088C22semidirect
54. #2065C22semidirect
55. #2023C22semidirect
56. #1987C22semidirect
57. #1921C22semidirect
60. #1862C22semidirect
61. #1854C22semidirect
62. #1851C22semidirect
63. #1802C22semidirect
64. #1736C22semidirect
99. C2×#121Q8semidirect
272. #121C2×Q8semidirect
708. C2#26427direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24402
C2×#23632
#264272
C2×#20992
C2×#20882
C2×#20652
C2×#20232
C2×#19872
C2×#19212
C2×#18952
C2×#18742
C2×#18622
C2×#18542
C2×#18512
C2×#18022
C2×#17362