Finite Group Databasebeta

← C2×#26332 #6448174

Subgroups — C2×#26332 #6448174

Order 512.

2114 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2548C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic, complemented
3#26332C2116yesmaximal, direct product factor
4C2×#2093C211yesmaximal, characteristic, complemented
5C2×#2092C211yesmaximal, characteristic, complemented
6C2×#2073C211yesmaximal, characteristic, complemented
7C2×#2071C211yesmaximal, characteristic, complemented
8C2×#1985C211yesmaximal, characteristic, complemented
9C2×#1982C211yesmaximal, characteristic
10C2×#1961C211yesmaximal, characteristic
11C2×#1949C211yesmaximal, characteristic, complemented
12C2×#1926C211yesmaximal, characteristic, complemented
13C2×#1926C211yesmaximal, characteristic, complemented
14C2×#1926C211yesmaximal, characteristic, complemented
15C2×#1925C211yesmaximal, characteristic, complemented
16C2×#1776C211yesmaximal, characteristic, complemented
17C22×#127C2211yescharacteristic
18C22×#124—21no—
19C22×C4×Q8C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
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#2548—24no—
35#2548C2218yes—
36C2×#394—21no—
37C2×#394—21no—
38#2355—24no—
39#2355C2218yescomplemented
40C2×#391—21no—
41C2×#390—21no—
42C2×#390C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#387—21no—
47C2×#387—21no—
48C2×#384C2211yescharacteristic
49C2×#382C2211yescharacteristic
50C2×#382C2211yescharacteristic
51C2×C4×C4⋊C4—21no—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56#2093C2218yescomplemented
57#2092C2218yescomplemented
58#2073C2218yescomplemented
59#2071C2218yescomplemented
60#1985C2218yescomplemented
61#1982C2218yes—
62#1961C2218yes—
63#1949C2218yescomplemented
64#1926C2218yescomplemented
65#1926C2218yescomplemented
66#1926C2218yescomplemented
67#1925C2218yescomplemented
68#1776C2218yescomplemented
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8D411yescharacteristic
78C23×D4—21no—
79C23×D4—21no—
80C23×D4C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C2×#127—22no—
87C2×#127—24no—
88C2×#127—24no—
89C2×#127C2314yes—
90C2×#124—28no—
91C2×#124—48no—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8D414yes—
95C2×C4×Q8C2314yes—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4D411yescharacteristic
99C22×C4⋊C4D411yescharacteristic
100C22×C4⋊C4D411yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C42—21no—
116C22×C42—21no—
117C22×C42—21no—
118C22×C42—21no—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C2×#107—24no—
123C2×#107—42no—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#106—24no—
127C2×#106—42no—
128C2×#106C2314yes—
129C2×#106C2314yes—
130C2×C4⋊C8—24no—
131C2×C4⋊C8C2314yes—
132C2×#102—24no—
133C2×#102—24no—
134C2×#102—24no—
135C2×#102—24no—
136C2×#102—42no—
137C2×#102—42no—
138C2×#102—42no—
139C2×#102—42no—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#97—24no—
149C2×#97C2314yes—
150#394—28no—
151#394—28no—
152#391—28no—
153#390—28no—
154#390—22no—
155#390—22no—
156#390—22no—
157#390—22no—
158#390C2314yes—
159#390C2314yes—
160#390C2314yes—
161#390C2314yes—
162#387—28no—
163#387—28no—
164#384—22no—
165#384C2314yes—
166#382—22no—
167#382—22no—
168#382C2314yes—
169#382C2314yes—
170C4×C4⋊C4—28no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4—22no—
175C4×C4⋊C4C2314yes—
176C4×C4⋊C4C2314yes—
177C4×C4⋊C4C2314yes—
178C4×C4⋊C4C2314yes—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#336C2314yes—
192#336C2314yes—
193#336C2314yes—
194#336C2314yes—
195C25—21no—
196C25—21no—
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—42no—
200C22×Q8C2×D414yes—
201C22×D4—24no—
202C22×D4—21no—
203C22×D4—21no—
204C22×D4—24no—
205C22×D4—22no—
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×D4C2412yes—
224C22×D4C2412yes—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2411yescharacteristic
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8#4114yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C22×C8C2412yes—
254#127—28no—
255#127—28no—
256#127—48no—
257#124—432no—
258C4×Q8—216no—
259C4×Q8—28no—
260C4×Q8—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—82no—
285C2×C4⋊C4C2×D414yes—
286C2×C4⋊C4C2×D414yes—
287C2×C4⋊C4C2×D414yes—
288C2×C4⋊C4#4114yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—82no—
308C2×C22⋊C4—82no—
309C2×C22⋊C4—82no—
310C2×C22⋊C4—82no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—44no—
320C2×C42—44no—
321C2×C42—82no—
322C2×C42—82no—
323C2×C42C2×D412yes—
324C2×C42C2×D412yes—
325C2×C42C2×D412yes—
326C2×C42#4114yes—
327C2×C42#4114yes—
328C2×C42C2412yes—
329C2×C42C2412yes—
330C2×C42C2412yes—
331#107—28no—
332#107—44no—
333#106—28no—
334#106—44no—
335C4⋊C8—28no—
336#102—28no—
337#102—28no—
338#102—28no—
339#102—28no—
340#102—44no—
341#102—44no—
342#102—44no—
343#102—44no—
344#97—28no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353C24—22no—
354C24—22no—
355C24—42no—
356C24—42no—
357C24—42no—
358C24—42no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24C22×D411yescharacteristic
370C2×Q8—28no—
371C2×Q8—28no—
372C2×Q8—44no—
373C2×Q8—416no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—48no—
387C2×D4—48no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—84no—
405C2×D4—84no—
406C2×D4—816no—
407C2×D4—816no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4#12314yes—
448C22×C4#12314yes—
449C22×C4AGL(1,ℤ/8)14yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4H(5,𝔽2)12yes—
462C22×C42−1+412yes—
463C22×C4C2511yescharacteristic
464C2×C8—28no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—44no—
477C2×C8—44no—
478C2×C8—44no—
479C2×C8C2×#4114yes—
480C4⋊C4—28no—
481C4⋊C4—28no—
482C4⋊C4—28no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—48no—
501C4⋊C4—84no—
502C4⋊C4C2×#4114yes—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—84no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C42—24no—
512C42—24no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—48no—
518C42—48no—
519C42—84no—
520C42—84no—
521C42#12318yes—
522C42C2×#4114yes—
523C42C2×#4114yes—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—84no—
539C23—84no—
540C23—84no—
541C23—84no—
542C23—84no—
543C23—84no—
544C23—84no—
545C23—84no—
546C23—84no—
547C23—84no—
548C23—84no—
549C23—84no—
550C23—84no—
551C23—84no—
552C23C2×#12312yes—
553C23#54812yes—
554C23#55012yes—
555C23C2×AGL(1,ℤ/8)12yes—
556C23#57512yes—
557C23#57612yes—
558C23#57712yes—
559C23C23×D411yescharacteristic
560Q8—416no—
561D4—42no—
562D4—42no—
563D4—42no—
564D4—42no—
565D4—42no—
566D4—42no—
567D4—42no—
568D4—42no—
569D4—816no—
570D4—816no—
571D4—84no—
572D4—84no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—48no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4C2×#12314yes—
619C2×C4C2×#12314yes—
620C2×C4C2×#12312yes—
621C2×C4#52914yes—
622C2×C4#54214yes—
623C2×C4#54812yes—
624C2×C4#55012yes—
625C2×C4#55614yes—
626C2×C4#55914yes—
627C2×C4C2×AGL(1,ℤ/8)14yes—
628C2×C4C22×#4111yescharacteristic
629C2×C4C22×#4111yescharacteristic
630C2×C4C22×#4111yescharacteristic
631C2×C4C22×#4111yescharacteristic
632C2×C4C2×H(5,𝔽2)11yescharacteristic
633C2×C4C2×2−1+411yescharacteristic
634C8—24no—
635C8—42no—
636C8—42no—
637C8—42no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—168no—
653C22—168no—
654C22#244814yes—
655C22#309414yes—
656C22#309614yes—
657C22#336814yes—
658C22#337014yes—
659C22#339914yes—
660C22#340114yes—
661C22C22×#12311yescharacteristic
662C22C2×#54811yescharacteristic
663C22C2×#55011yescharacteristic
664C22C22×AGL(1,ℤ/8)11yescharacteristic
665C22C2×#57511yescharacteristic
666C22C2×#57611yescharacteristic
667C22C2×#57711yescharacteristic
668C4—24no—
669C4—24no—
670C4—24no—
671C4—42no—
672C4—42no—
673C4—42no—
674C4—44no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4#244814yes—
681C4C2×#52911yescharacteristic
682C4C2×#54211yescharacteristic
683C4C2×#55611yescharacteristic
684C4C2×#55911yescharacteristic
685C2—162no—
686C2—162no—
687C2#2633218yesdirect product factor
688C2C2×#244811yescharacteristic
689C2C2×#309411yescharacteristic
690C2C2×#309611yescharacteristic
691C2C2×#336811yescharacteristic
692C2C2×#337011yescharacteristic
693C2C2×#339911yescharacteristic
694C2C2×#340111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2355C2semidirect
3. #26332C2direct (N × H)
4. C2×#2093C2semidirect
5. C2×#2092C2semidirect
6. C2×#2073C2semidirect
7. C2×#2071C2semidirect
8. C2×#1985C2semidirect
11. C2×#1949C2semidirect
12. C2×#1926C2semidirect
13. C2×#1926C2semidirect
14. C2×#1926C2semidirect
15. C2×#1925C2semidirect
16. C2×#1776C2semidirect
39. #2355C22semidirect
56. #2093C22semidirect
57. #2092C22semidirect
58. #2073C22semidirect
59. #2071C22semidirect
60. #1985C22semidirect
63. #1949C22semidirect
64. #1926C22semidirect
65. #1926C22semidirect
66. #1926C22semidirect
67. #1925C22semidirect
68. #1776C22semidirect
687. C2#26332direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25482
C2×#23552
#263322
C2×#20932
C2×#20922
C2×#20732
C2×#20712
C2×#19852
C2×#19822
C2×#19612
C2×#19492
C2×#19262
C2×#19252
C2×#17762