Finite Group Databasebeta

← C2×#29396 #6461689

Subgroups — C2×#29396 #6461689

Order 512.

2034 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#125C211yesmaximal, characteristic, complemented
2C2×#2293C211yesmaximal, characteristic
3#29396C2116yesmaximal, direct product factor
4C2×#2085C211yesmaximal, characteristic, complemented
5C2×#2073C211yesmaximal, characteristic
6C2×#2059C211yesmaximal, characteristic, complemented
7C2×#2025C211yesmaximal, characteristic
8C2×#1926C211yesmaximal, characteristic
9C2×#1925C211yesmaximal, characteristic
10C2×#1921C211yesmaximal, characteristic, complemented
11C2×#1919C211yesmaximal, characteristic, complemented
12C2×#1919C211yesmaximal, characteristic, complemented
13C2×#1918C211yesmaximal, characteristic, complemented
14C2×#1864C211yesmaximal, characteristic
15C2×#1862C211yesmaximal, characteristic
16C2×#1839C211yesmaximal, characteristic, complemented
17C22×#125C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×#117—21no—
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C4×#125—24no—
29C4×#125C2218yescomplemented
30#2293—24no—
31#2293C2218yes—
32C2×#390C2211yescharacteristic
33C2×#390C2211yescharacteristic
34C2×#389C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#388—21no—
39C2×#388—21no—
40C2×#387—21no—
41C2×#387—21no—
42C2×#382C2211yescharacteristic
43C2×C4×C4⋊C4—21no—
44C2×C4×C4⋊C4—21no—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4—21no—
49C2×C43C2211yescharacteristic
50#2085C2218yescomplemented
51#2073C2218yes—
52#2059C2218yescomplemented
53#2025C2218yes—
54#1926C2218yes—
55#1925C2218yes—
56#1921C2218yescomplemented
57#1919C2218yescomplemented
58#1919C2218yescomplemented
59#1918C2218yescomplemented
60#1864C2218yes—
61#1862C2218yes—
62#1839C2218yescomplemented
63C2×#340C2211yescharacteristic
64C2×#340C2211yescharacteristic
65C2×#340C2211yescharacteristic
66C2×#340C2211yescharacteristic
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#336C2211yescharacteristic
72C2×#336C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×D4C2311yescharacteristic
78C24×C4—21no—
79C23×C8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C2×#125—24no—
84C2×#125—24no—
85C2×#125—22no—
86C2×#125C2314yes—
87C2×C4×D4—24no—
88C2×C4×D4—24no—
89C2×C4×D4—22no—
90C2×C4×D4C2314yes—
91C2×#117—28no—
92C2×#117—48no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4—21no—
107C22×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C42—21no—
112C22×C42—21no—
113C22×C42—21no—
114C22×C42D411yescharacteristic
115C22×C42D411yescharacteristic
116C22×C42D411yescharacteristic
117C22×C42D411yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C2×#107—24no—
122C2×#107—24no—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×#106—24no—
126C2×#106—24no—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×#102—24no—
130C2×#102—24no—
131C2×#102—24no—
132C2×#102—24no—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137#390—22no—
138#390—22no—
139#390C2314yes—
140#390C2314yes—
141#389—22no—
142#389—22no—
143#389—22no—
144#389—22no—
145#389C2314yes—
146#389C2314yes—
147#389C2314yes—
148#389C2314yes—
149#388—28no—
150#388—28no—
151#387—28no—
152#387—28no—
153#382—22no—
154#382C2314yes—
155C4×C4⋊C4—22no—
156C4×C4⋊C4—22no—
157C4×C4⋊C4—28no—
158C4×C4⋊C4—28no—
159C4×C4⋊C4—22no—
160C4×C4⋊C4C2314yes—
161C4×C4⋊C4C2314yes—
162C4×C4⋊C4C2314yes—
163C4×C22⋊C4—28no—
164C2×#95—21no—
165C2×#95—21no—
166C2×#95—21no—
167C2×#95—21no—
168C2×#95—21no—
169C2×#95—21no—
170C2×#95—21no—
171C2×#95—21no—
172C2×#95—21no—
173C43D414yes—
174C43C2314yes—
175#340C2314yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#336C2314yes—
184#336C2314yes—
185#336C2314yes—
186#336C2314yes—
187#336C2314yes—
188#336C2314yes—
189C25—21no—
190C22×D4—21no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4#4114yes—
197C22×D4C2412yes—
198C23×C4—28no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—24no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—21no—
206C23×C4—21no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2411yescharacteristic
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—22no—
228C22×C8C2412yes—
229C22×C8C2412yes—
230C22×C8C2412yes—
231C22×C8C2412yes—
232#125—216no—
233#125—28no—
234#125—28no—
235C4×D4—216no—
236C4×D4—28no—
237C4×D4—28no—
238#117—432no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4#4114yes—
270C2×C4⋊C4#4114yes—
271C2×C4⋊C4#4114yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—42no—
301C2×C42—42no—
302C2×C42—44no—
303C2×C42—44no—
304C2×C42—44no—
305C2×C42C2×D414yes—
306C2×C42C2×D412yes—
307C2×C42C2×D412yes—
308C2×C42C2×D412yes—
309C2×C42C2×D414yes—
310C2×C42C2×D414yes—
311C2×C42C2×D414yes—
312C2×C42C2412yes—
313C2×C42C2412yes—
314C2×C42C2412yes—
315#107—28no—
316#107—28no—
317#106—28no—
318#106—28no—
319#102—28no—
320#102—28no—
321#102—28no—
322#102—28no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332C24—24no—
333C24—22no—
334C24—42no—
335C24—42no—
336C24—42no—
337C24—42no—
338C24—42no—
339C24—42no—
340C24C22×D411yescharacteristic
341C2×D4—24no—
342C2×D4—28no—
343C2×D4—28no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4C2×#4114yes—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—216no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—48no—
382C22×C4—432no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—48no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4#13514yes—
403C22×C4AGL(1,ℤ/8)14yes—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D412yes—
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D411yescharacteristic
411C22×C4C2×#4112yes—
412C22×C4C2×#4112yes—
413C22×C4C2×#4112yes—
414C22×C4C2×#4112yes—
415C22×C42−1+412yes—
416C22×C42−1+412yes—
417C22×C4C2511yescharacteristic
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—44no—
431C2×C8—44no—
432C2×C8—44no—
433C2×C8—44no—
434C4⋊C4—216no—
435C4⋊C4—216no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4C2×#4114yes—
459C4⋊C4C2×#4114yes—
460C4⋊C4C2×#4114yes—
461C22⋊C4—216no—
462C22⋊C4—48no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C42—24no—
467C42—24no—
468C42—24no—
469C42—28no—
470C42—28no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—44no—
478C42—44no—
479C42—48no—
480C42—48no—
481C42—48no—
482C23—24no—
483C23—48no—
484C23—48no—
485C23—48no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—84no—
493C23—84no—
494C23—84no—
495C23—84no—
496C23#52512yes—
497C23#52512yes—
498C23#53612yes—
499C23C2×#13512yes—
500C23C2×AGL(1,ℤ/8)12yes—
501C23#57512yes—
502C23#57512yes—
503C23C23×D411yescharacteristic
504D4—24no—
505D4—24no—
506D4—48no—
507D4—42no—
508D4—42no—
509C2×C4—28no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—28no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—432no—
531C2×C4—44no—
532C2×C4—48no—
533C2×C4—48no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—48no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—84no—
557C2×C4—84no—
558C2×C4—84no—
559C2×C4—84no—
560C2×C4#52512yes—
561C2×C4#52512yes—
562C2×C4#53612yes—
563C2×C4#55214yes—
564C2×C4#55214yes—
565C2×C4#55614yes—
566C2×C4#55614yes—
567C2×C4C2×#13514yes—
568C2×C4C2×AGL(1,ℤ/8)14yes—
569C2×C4C22×#4111yescharacteristic
570C2×C4C22×#4111yescharacteristic
571C2×C4C22×#4111yescharacteristic
572C2×C4C22×#4111yescharacteristic
573C2×C4C2×2−1+411yescharacteristic
574C2×C4C2×2−1+411yescharacteristic
575C8—42no—
576C8—42no—
577C8—42no—
578C8—42no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—816no—
587C22#259914yes—
588C22#310214yes—
589C22#310514yes—
590C22#311114yes—
591C22#311514yes—
592C22#323314yes—
593C22#323414yes—
594C22C2×#52511yescharacteristic
595C22C2×#52511yescharacteristic
596C22C2×#53611yescharacteristic
597C22C22×#13511yescharacteristic
598C22C22×AGL(1,ℤ/8)11yescharacteristic
599C22C2×#57511yescharacteristic
600C22C2×#57511yescharacteristic
601C4—24no—
602C4—24no—
603C4—42no—
604C4—44no—
605C4—44no—
606C4—42no—
607C4—44no—
608C4—42no—
609C4—42no—
610C4—82no—
611C4—82no—
612C4—82no—
613C4—82no—
614C4#259914yes—
615C4C2×#55211yescharacteristic
616C4C2×#55211yescharacteristic
617C4C2×#55611yescharacteristic
618C4C2×#55611yescharacteristic
619C2—84no—
620C2#2939618yesdirect product factor
621C2C2×#259911yescharacteristic
622C2C2×#310211yescharacteristic
623C2C2×#310511yescharacteristic
624C2C2×#311511yescharacteristic
625C2C2×#311111yescharacteristic
626C2C2×#323311yescharacteristic
627C2C2×#323411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×C4×#125C2semidirect
3. #29396C2direct (N × H)
4. C2×#2085C2semidirect
6. C2×#2059C2semidirect
10. C2×#1921C2semidirect
11. C2×#1919C2semidirect
12. C2×#1919C2semidirect
13. C2×#1918C2semidirect
16. C2×#1839C2semidirect
29. C4×#125C22semidirect
50. #2085C22semidirect
52. #2059C22semidirect
56. #1921C22semidirect
57. #1919C22semidirect
58. #1919C22semidirect
59. #1918C22semidirect
62. #1839C22semidirect
620. C2#29396direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#1252
C2×#22932
#293962
C2×#20852
C2×#20732
C2×#20592
C2×#20252
C2×#19262
C2×#19252
C2×#19212
C2×#19192
C2×#19182
C2×#18642
C2×#18622
C2×#18392