Finite Group Databasebeta

← C2×#27163 #6450809

Subgroups — C2×#27163 #6450809

Order 512.

2244 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2420C211yesmaximal, characteristic
2C2×#2344C211yesmaximal, characteristic, complemented
3#27163C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2076C211yesmaximal, characteristic, complemented
6C2×#2065C211yesmaximal, characteristic, complemented
7C2×#2034C211yesmaximal, characteristic
8C2×#1949C211yesmaximal, characteristic, complemented
9C2×#1941C211yesmaximal, characteristic, complemented
10C2×#1926C211yesmaximal, characteristic, complemented
11C2×#1923C211yesmaximal, characteristic, complemented
12C2×#1920C211yesmaximal, characteristic, complemented
13C2×#1874C211yesmaximal, characteristic
14C2×#1862C211yesmaximal, characteristic, complemented
15C2×#1853C211yesmaximal, characteristic, complemented
16C2×C4×#102C211yesmaximal, characteristic, complemented
17C22×#121C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×#117—21no—
20C23×C4⋊C4C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31C22×C22⋊C8C2211yescharacteristic
32C22×C4×C8C2211yescharacteristic
33#2420—24no—
34#2420C2218yes—
35C2×#397—21no—
36C2×#394—21no—
37#2344—24no—
38#2344C2218yescomplemented
39C2×#390C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#387—21no—
44C2×#387—21no—
45C2×#386—21no—
46C2×#382C2211yescharacteristic
47C2×#381—21no—
48C2×#381—21no—
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54#2101C2218yescomplemented
55#2076C2218yescomplemented
56#2065C2218yescomplemented
57#2034C2218yes—
58#1949C2218yescomplemented
59#1941C2218yescomplemented
60#1926C2218yescomplemented
61#1923C2218yescomplemented
62#1920C2218yescomplemented
63#1874C2218yes—
64#1862C2218yescomplemented
65#1853C2218yescomplemented
66C4×#102C2218yescomplemented
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×D4—21no—
78C23×D4C2311yescharacteristic
79C24×C4—21no—
80C24×C4—21no—
81C24×C4C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C2×#121—22no—
87C2×#121—24no—
88C2×#121—42no—
89C2×#121C2314yes—
90C2×C4×D4—24no—
91C2×C4×D4—22no—
92C2×C4×D4—42no—
93C2×C4×D4C2314yes—
94C2×#117—28no—
95C2×#117—48no—
96C22×C4⋊C4—22no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4D411yescharacteristic
118C22×C22⋊C4D411yescharacteristic
119C22×C42—21no—
120C22×C42—21no—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C2×#107—42no—
124C2×#107C2314yes—
125C2×#106—24no—
126C2×#106—24no—
127C2×#106—42no—
128C2×#106C2314yes—
129C2×#106C2314yes—
130C2×#106C2314yes—
131C2×C4⋊C8—24no—
132C2×C4⋊C8C2314yes—
133C2×#102—24no—
134C2×#102—24no—
135C2×#102—24no—
136C2×#102—24no—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×C22⋊C8—24no—
142C2×C22⋊C8—42no—
143C2×C22⋊C8C2314yes—
144C2×C22⋊C8C2314yes—
145C2×C4×C8—42no—
146C2×C4×C8C2314yes—
147#397—28no—
148#394—28no—
149#390—22no—
150#390C2314yes—
151#389—22no—
152#389—22no—
153#389—22no—
154#389C2314yes—
155#389C2314yes—
156#389C2314yes—
157#387—28no—
158#387—28no—
159#386—28no—
160#382—22no—
161#382C2314yes—
162#381—28no—
163#381—28no—
164#380—22no—
165#380—22no—
166#380C2314yes—
167#380C2314yes—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4C2314yes—
171C4×C4⋊C4C2314yes—
172C4×C22⋊C4D414yes—
173C4×C22⋊C4C2314yes—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95D411yescharacteristic
184C2×#95D411yescharacteristic
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#336C2314yes—
194#336C2314yes—
195C25—21no—
196C25C2×D411yescharacteristic
197C22×D4—24no—
198C22×D4—21no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—44no—
205C22×D4—44no—
206C22×D4—82no—
207C22×D4#4114yes—
208C22×D4C2412yes—
209C23×C4—28no—
210C23×C4—24no—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—24no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—82no—
230C23×C4C2×D412yes—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2412yes—
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#121—28no—
255#121—44no—
256#121—48no—
257C4×D4—28no—
258C4×D4—44no—
259C4×D4—48no—
260#117—432no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—216no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
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—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—82no—
291C2×C4⋊C4#4114yes—
292C2×C4⋊C4#4114yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—82no—
312C2×C22⋊C4—82no—
313C2×C22⋊C4—82no—
314C2×C22⋊C4C2×D414yes—
315C2×C22⋊C4C2×D414yes—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—44no—
323C2×C42—44no—
324C2×C42C2×D412yes—
325C2×C42C2×D412yes—
326C2×C42C2412yes—
327C2×C42C2412yes—
328#107—44no—
329#106—28no—
330#106—28no—
331#106—44no—
332C4⋊C8—28no—
333#102—28no—
334#102—28no—
335#102—28no—
336#102—28no—
337C22⋊C8—28no—
338C22⋊C8—44no—
339C4×C8—44no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95C2×D414yes—
350#95C2×D414yes—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—24no—
358C24—22no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24#13514yes—
366C24C22×D412yes—
367C24C22×D411yescharacteristic
368C2×D4—28no—
369C2×D4—24no—
370C2×D4—28no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—48no—
377C2×D4—48no—
378C2×D4—42no—
379C2×D4—44no—
380C2×D4—42no—
381C2×D4—44no—
382C2×D4—84no—
383C2×D4—816no—
384C2×D4C2×#4114yes—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—216no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
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—22no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—432no—
421C22×C4—48no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
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—48no—
446C22×C4—84no—
447C22×C4—816no—
448C22×C4AGL(1,ℤ/8)14yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4H(5,𝔽2)12yes—
460C22×C42−1+412yes—
461C22×C4C2511yescharacteristic
462C2×C8—28no—
463C2×C8—22no—
464C2×C8—22no—
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—44no—
475C2×C8—44no—
476C2×C8—44no—
477C2×C8C2×#4114yes—
478C4⋊C4—216no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
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—48no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—48no—
500C4⋊C4—84no—
501C4⋊C4C2×#4114yes—
502C4⋊C4C2×#4114yes—
503C22⋊C4—28no—
504C22⋊C4—28no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—48no—
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—
519C23—28no—
520C23—28no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—28no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—24no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—42no—
534C23—42no—
535C23—48no—
536C23—48no—
537C23—48no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—84no—
543C23—84no—
544C23—84no—
545C23—84no—
546C23#52512yes—
547C23#54612yes—
548C23#54712yes—
549C23C2×#13514yes—
550C23C2×#13512yes—
551C23C2×AGL(1,ℤ/8)12yes—
552C23#57712yes—
553C23#57712yes—
554C23C23×D411yescharacteristic
555D4—24no—
556D4—24no—
557D4—42no—
558D4—42no—
559D4—48no—
560D4—816no—
561C2×C4—22no—
562C2×C4—28no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—432no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—48no—
582C2×C4—48no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—48no—
594C2×C4—42no—
595C2×C4—42no—
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—816no—
610C2×C4—84no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4#52512yes—
616C2×C4#52914yes—
617C2×C4#54214yes—
618C2×C4#54612yes—
619C2×C4#54712yes—
620C2×C4#55214yes—
621C2×C4#55314yes—
622C2×C4C2×AGL(1,ℤ/8)14yes—
623C2×C4C22×#4111yescharacteristic
624C2×C4C22×#4111yescharacteristic
625C2×C4C22×#4111yescharacteristic
626C2×C4C22×#4111yescharacteristic
627C2×C4C2×H(5,𝔽2)11yescharacteristic
628C2×C4C2×2−1+411yescharacteristic
629C8—24no—
630C8—42no—
631C8—42no—
632C8—42no—
633C22—24no—
634C22—24no—
635C22—24no—
636C22—416no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—816no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22#247014yes—
650C22#310514yes—
651C22#312614yes—
652C22#329614yes—
653C22#330614yes—
654C22#333714yes—
655C22#335414yes—
656C22C2×#52511yescharacteristic
657C22C2×#54611yescharacteristic
658C22C2×#54711yescharacteristic
659C22C22×#13511yescharacteristic
660C22C22×AGL(1,ℤ/8)11yescharacteristic
661C22C2×#57711yescharacteristic
662C22C2×#57711yescharacteristic
663C4—24no—
664C4—42no—
665C4—42no—
666C4—44no—
667C4—44no—
668C4—42no—
669C4—42no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4#247014yes—
676C4C2×#52911yescharacteristic
677C4C2×#54211yescharacteristic
678C4C2×#55211yescharacteristic
679C4C2×#55311yescharacteristic
680C2—44no—
681C2—84no—
682C2#2716318yesdirect product factor
683C2C2×#247011yescharacteristic
684C2C2×#310511yescharacteristic
685C2C2×#312611yescharacteristic
686C2C2×#329611yescharacteristic
687C2C2×#330611yescharacteristic
688C2C2×#333711yescharacteristic
689C2C2×#335411yescharacteristic
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
2. C2×#2344C2semidirect
3. #27163C2direct (N × H)
4. C2×#2101C2semidirect
5. C2×#2076C2semidirect
6. C2×#2065C2semidirect
8. C2×#1949C2semidirect
9. C2×#1941C2semidirect
10. C2×#1926C2semidirect
11. C2×#1923C2semidirect
12. C2×#1920C2semidirect
14. C2×#1862C2semidirect
15. C2×#1853C2semidirect
16. C2×C4×#102C2semidirect
38. #2344C22semidirect
54. #2101C22semidirect
55. #2076C22semidirect
56. #2065C22semidirect
58. #1949C22semidirect
59. #1941C22semidirect
60. #1926C22semidirect
61. #1923C22semidirect
62. #1920C22semidirect
64. #1862C22semidirect
65. #1853C22semidirect
66. C4×#102C22semidirect
682. C2#27163direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24202
C2×#23442
#271632
C2×#21012
C2×#20762
C2×#20652
C2×#20342
C2×#19492
C2×#19412
C2×#19262
C2×#19232
C2×#19202
C2×#18742
C2×#18622
C2×#18532
C2×C4×#1022