Finite Group Databasebeta

← C2×#26705 #6447876

Subgroups — C2×#26705 #6447876

Order 512.

2352 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2445C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic, complemented
3#26705C2116yesmaximal, direct product factor
4C2×#2088C211yesmaximal, characteristic, complemented
5C2×#2041C211yesmaximal, characteristic
6C2×#2023C211yesmaximal, characteristic, complemented
7C2×#2023C211yesmaximal, characteristic, complemented
8C2×#1982C211yesmaximal, characteristic
9C2×#1936C211yesmaximal, characteristic, complemented
10C2×#1926C211yesmaximal, characteristic, complemented
11C2×#1926C211yesmaximal, characteristic, complemented
12C2×#1926C211yesmaximal, characteristic, complemented
13C2×#1920C211yesmaximal, characteristic, complemented
14C2×#1917C211yesmaximal, characteristic, complemented
15C2×#1864C211yesmaximal, characteristic, complemented
16C2×#1762C211yesmaximal, characteristic, complemented
17C22×C4⋊Q8C2211yescharacteristic
18C22×#127C2211yescharacteristic
19C22×#123—21no—
20C22×C4×Q8—21no—
21C22×C4×D4C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#97C2211yescharacteristic
34#2445—24no—
35#2445C2218yes—
36C2×#394—21no—
37C2×#394—21no—
38#2365—24no—
39#2365C2218yescomplemented
40C2×#391—21no—
41C2×#390C2211yescharacteristic
42C2×#390C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#387—21no—
48C2×#387—21no—
49C2×#386—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54#2088C2218yescomplemented
55#2041C2218yes—
56#2023C2218yescomplemented
57#2023C2218yescomplemented
58#1982C2218yes—
59#1936C2218yescomplemented
60#1926C2218yescomplemented
61#1926C2218yescomplemented
62#1926C2218yescomplemented
63#1920C2218yescomplemented
64#1917C2218yescomplemented
65#1864C2218yescomplemented
66#1762C2218yescomplemented
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8—21no—
78C23×D4—21no—
79C23×D4—21no—
80C23×D4C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C24×C4—21no—
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C2×C4⋊Q8—24no—
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—22no—
90C2×C4⋊Q8C2314yes—
91C2×#127—24no—
92C2×#127—22no—
93C2×#127—42no—
94C2×#127C2314yes—
95C2×#123—28no—
96C2×#123—48no—
97C2×C4×Q8—28no—
98C2×C4×Q8—48no—
99C2×C4×D4—24no—
100C2×C4×D4—22no—
101C2×C4×D4—42no—
102C2×C4×D4C2314yes—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4D411yescharacteristic
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C42—21no—
123C22×C42—21no—
124C22×C42—21no—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C2×#106—24no—
129C2×#106—42no—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×C4⋊C8—42no—
133C2×C4⋊C8C2314yes—
134C2×#102—24no—
135C2×#102—24no—
136C2×#102—24no—
137C2×#102—24no—
138C2×#102—24no—
139C2×#102—24no—
140C2×#102—24no—
141C2×#102—42no—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#97—42no—
151C2×#97C2314yes—
152#394—28no—
153#394—28no—
154#391—28no—
155#390—22no—
156#390—22no—
157#390—22no—
158#390C2314yes—
159#390C2314yes—
160#390C2314yes—
161#389—22no—
162#389—22no—
163#389—22no—
164#389C2314yes—
165#389C2314yes—
166#389C2314yes—
167#387—28no—
168#387—28no—
169#386—28no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4D414yes—
174C4×C4⋊C4C2314yes—
175C4×C4⋊C4C2314yes—
176C4×C4⋊C4C2314yes—
177C4×C4⋊C4C2314yes—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95D411yescharacteristic
185C2×#95D411yescharacteristic
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#336C2314yes—
193#336C2314yes—
194#336C2314yes—
195#336C2314yes—
196C25—21no—
197C25—21no—
198C22×Q8—28no—
199C22×Q8—24no—
200C22×Q8—44no—
201C22×Q8—44no—
202C22×D4—21no—
203C22×D4—21no—
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—44no—
218C22×D4—44no—
219C22×D4—44no—
220C22×D4—44no—
221C22×D4—82no—
222C22×D4—82no—
223C22×D4#4114yes—
224C22×D4C2412yes—
225C22×D4C2412yes—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—24no—
233C23×C4—44no—
234C23×C4—44no—
235C23×C4—82no—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2411yescharacteristic
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C4⋊Q8—216no—
260C4⋊Q8—28no—
261C4⋊Q8—28no—
262#127—28no—
263#127—44no—
264#127—48no—
265#123—432no—
266C4×Q8—432no—
267C4×D4—28no—
268C4×D4—44no—
269C4×D4—48no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4C2×D414yes—
296C2×C4⋊C4C2×D414yes—
297C2×C4⋊C4#4114yes—
298C2×C4⋊C4#4114yes—
299C2×C4⋊C4#4114yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—82no—
318C2×C22⋊C4—82no—
319C2×C22⋊C4—82no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—44no—
330C2×C42—44no—
331C2×C42—44no—
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2412yes—
336C2×C42C2412yes—
337C2×C42C2412yes—
338#106—28no—
339#106—44no—
340C4⋊C8—44no—
341#102—28no—
342#102—28no—
343#102—28no—
344#102—28no—
345#102—28no—
346#102—28no—
347#102—28no—
348#102—44no—
349#97—44no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95C2×D414yes—
357#95C2×D414yes—
358C24—24no—
359C24—22no—
360C24—22no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24C22×D411yescharacteristic
375C2×Q8—216no—
376C2×Q8—48no—
377C2×Q8—432no—
378C2×Q8—48no—
379C2×D4—28no—
380C2×D4—24no—
381C2×D4—28no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—48no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—48no—
408C2×D4—84no—
409C2×D4—84no—
410C2×D4—816no—
411C2×D4—816no—
412C2×D4C2×#4114yes—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
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—42no—
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—84no—
456C22×C4—816no—
457C22×C4AGL(1,ℤ/8)14yes—
458C22×C4AGL(1,ℤ/8)14yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4H(5,𝔽2)12yes—
471C22×C42−1+412yes—
472C22×C4C2511yescharacteristic
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—44no—
486C2×C8—44no—
487C2×C8—44no—
488C2×C8—44no—
489C4⋊C4—216no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—28no—
494C4⋊C4—28no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—48no—
510C4⋊C4C2×#4114yes—
511C4⋊C4C2×#4114yes—
512C4⋊C4C2×#4114yes—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—84no—
518C22⋊C4—84no—
519C22⋊C4—84no—
520C42—24no—
521C42—24no—
522C42—24no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—48no—
530C42—48no—
531C42—48no—
532C23—24no—
533C23—42no—
534C23—42no—
535C23—48no—
536C23—48no—
537C23—42no—
538C23—42no—
539C23—48no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—84no—
550C23—84no—
551C23—84no—
552C23—84no—
553C23—84no—
554C23—84no—
555C23—84no—
556C23—84no—
557C23—84no—
558C23—84no—
559C23—84no—
560C23#52512yes—
561C23#54612yes—
562C23#54812yes—
563C23C2×AGL(1,ℤ/8)12yes—
564C23C2×AGL(1,ℤ/8)12yes—
565C23#57712yes—
566C23#57712yes—
567C23C23×D411yescharacteristic
568Q8—432no—
569D4—24no—
570D4—24no—
571D4—42no—
572D4—42no—
573D4—42no—
574D4—42no—
575D4—42no—
576D4—42no—
577D4—48no—
578D4—816no—
579D4—816no—
580D4—84no—
581C2×C4—28no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—28no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—48no—
603C2×C4—48no—
604C2×C4—48no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—816no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4#52512yes—
629C2×C4#52914yes—
630C2×C4#54214yes—
631C2×C4#54612yes—
632C2×C4#54812yes—
633C2×C4#55214yes—
634C2×C4#55914yes—
635C2×C4C2×AGL(1,ℤ/8)14yes—
636C2×C4C2×AGL(1,ℤ/8)14yes—
637C2×C4C22×#4111yescharacteristic
638C2×C4C22×#4111yescharacteristic
639C2×C4C22×#4111yescharacteristic
640C2×C4C22×#4111yescharacteristic
641C2×C4C2×H(5,𝔽2)11yescharacteristic
642C2×C4C2×2−1+411yescharacteristic
643C8—42no—
644C8—42no—
645C8—42no—
646C8—42no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—816no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—168no—
663C22#245814yes—
664C22#310914yes—
665C22#312614yes—
666C22#330614yes—
667C22#332414yes—
668C22#336814yes—
669C22#337314yes—
670C22C2×#52511yescharacteristic
671C22C2×#54611yescharacteristic
672C22C2×#54811yescharacteristic
673C22C22×AGL(1,ℤ/8)11yescharacteristic
674C22C22×AGL(1,ℤ/8)11yescharacteristic
675C22C2×#57711yescharacteristic
676C22C2×#57711yescharacteristic
677C4—24no—
678C4—24no—
679C4—44no—
680C4—44no—
681C4—44no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4#245814yes—
690C4C2×#52911yescharacteristic
691C4C2×#54211yescharacteristic
692C4C2×#55211yescharacteristic
693C4C2×#55911yescharacteristic
694C2—84no—
695C2—162no—
696C2#2670518yesdirect product factor
697C2C2×#245811yescharacteristic
698C2C2×#310911yescharacteristic
699C2C2×#312611yescharacteristic
700C2C2×#330611yescharacteristic
701C2C2×#332411yescharacteristic
702C2C2×#337311yescharacteristic
703C2C2×#336811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
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. C2×#2365C2semidirect
3. #26705C2direct (N × H)
4. C2×#2088C2semidirect
6. C2×#2023C2semidirect
7. C2×#2023C2semidirect
9. C2×#1936C2semidirect
10. C2×#1926C2semidirect
11. C2×#1926C2semidirect
12. C2×#1926C2semidirect
13. C2×#1920C2semidirect
14. C2×#1917C2semidirect
15. C2×#1864C2semidirect
16. C2×#1762C2semidirect
39. #2365C22semidirect
54. #2088C22semidirect
56. #2023C22semidirect
57. #2023C22semidirect
59. #1936C22semidirect
60. #1926C22semidirect
61. #1926C22semidirect
62. #1926C22semidirect
63. #1920C22semidirect
64. #1917C22semidirect
65. #1864C22semidirect
66. #1762C22semidirect
696. C2#26705direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24452
C2×#23652
#267052
C2×#20882
C2×#20412
C2×#20232
C2×#19822
C2×#19362
C2×#19262
C2×#19202
C2×#19172
C2×#18642
C2×#17622