Finite Group Databasebeta

← C2×#27408 #6449355

Subgroups — C2×#27408 #6449355

Order 512.

2310 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2426C211yesmaximal, characteristic
2C2×#2344C211yesmaximal, characteristic, complemented
3#27408C2116yesmaximal, direct product factor
4C2×#2088C211yesmaximal, characteristic, complemented
5C2×#2042C211yesmaximal, characteristic, complemented
6C2×#2035C211yesmaximal, characteristic, complemented
7C2×#2033C211yesmaximal, characteristic
8C2×#1983C211yesmaximal, characteristic, complemented
9C2×#1936C211yesmaximal, characteristic, complemented
10C2×#1926C211yesmaximal, characteristic, complemented
11C2×#1924C211yesmaximal, characteristic, complemented
12C2×#1921C211yesmaximal, characteristic, complemented
13C2×#1874C211yesmaximal, characteristic
14C2×#1872C211yesmaximal, characteristic, complemented
15C2×#1863C211yesmaximal, characteristic, complemented
16C2×#1763C211yesmaximal, characteristic, complemented
17C22×#124C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×#117—21no—
22C23×C4⋊C4C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×C22⋊C8C2211yescharacteristic
35C22×C22⋊C8C2211yescharacteristic
36C22×#97C2211yescharacteristic
37#2426—24no—
38#2426C2218yes—
39C2×#396—21no—
40C2×#394—21no—
41#2344—24no—
42#2344C2218yescomplemented
43C2×#390—21no—
44C2×#390C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#388—21no—
49C2×#387—21no—
50C2×#387—21no—
51C2×#381—21no—
52C2×#380C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2211yescharacteristic
56#2088C2218yescomplemented
57#2042C2218yescomplemented
58#2035C2218yescomplemented
59#2033C2218yes—
60#1983C2218yescomplemented
61#1936C2218yescomplemented
62#1926C2218yescomplemented
63#1924C2218yescomplemented
64#1921C2218yescomplemented
65#1874C2218yes—
66#1872C2218yescomplemented
67#1863C2218yescomplemented
68#1763C2218yescomplemented
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×D4—21no—
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C24×C4C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C2×#124—24no—
87C2×#124—22no—
88C2×#124—42no—
89C2×#124C2314yes—
90C2×C22⋊Q8—22no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—42no—
93C2×C22⋊Q8C2314yes—
94C2×#121—22no—
95C2×#121—24no—
96C2×#121—42no—
97C2×#121C2314yes—
98C2×C4×Q8—22no—
99C2×C4×Q8—24no—
100C2×C4×Q8—42no—
101C2×C4×Q8C2314yes—
102C2×#117—28no—
103C2×#117—48no—
104C22×C4⋊C4—22no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4D411yescharacteristic
128C22×C22⋊C4D411yescharacteristic
129C22×C42—21no—
130C22×C42—21no—
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C2×#107—24no—
135C2×#107C2314yes—
136C2×#106—42no—
137C2×#106C2314yes—
138C2×C4⋊C8—42no—
139C2×C4⋊C8C2314yes—
140C2×#103—24no—
141C2×#103—24no—
142C2×#103—24no—
143C2×#103—42no—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#102—24no—
149C2×#102—24no—
150C2×#102—42no—
151C2×#102—42no—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×C22⋊C8—42no—
157C2×C22⋊C8—42no—
158C2×C22⋊C8C2314yes—
159C2×C22⋊C8C2314yes—
160C2×#97—42no—
161C2×#97C2314yes—
162#396—28no—
163#394—28no—
164#390—28no—
165#390—22no—
166#390C2314yes—
167#389—22no—
168#389—22no—
169#389—22no—
170#389C2314yes—
171#389C2314yes—
172#389C2314yes—
173#388—28no—
174#387—28no—
175#387—28no—
176#381—28no—
177#380—22no—
178#380C2314yes—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4C2314yes—
182C4×C4⋊C4C2314yes—
183C4×C22⋊C4D414yes—
184C4×C22⋊C4C2314yes—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95D411yescharacteristic
192C2×#95D411yescharacteristic
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201C25—21no—
202C25C2×D411yescharacteristic
203C22×Q8—21no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8#4114yes—
210C22×Q8C2412yes—
211C22×D4—24no—
212C22×D4—21no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—44no—
220C22×D4—44no—
221C22×D4—82no—
222C22×D4C2412yes—
223C23×C4—28no—
224C23×C4—24no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—22no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—44no—
239C23×C4—44no—
240C23×C4C2×D412yes—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2412yes—
247C23×C4C2411yescharacteristic
248C22×C8—22no—
249C22×C8—22no—
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×C8C2412yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C22×C8C2412yes—
264#124—28no—
265#124—44no—
266#124—48no—
267C22⋊Q8—28no—
268C22⋊Q8—44no—
269C22⋊Q8—48no—
270#121—28no—
271#121—44no—
272#121—48no—
273C4×Q8—28no—
274C4×Q8—44no—
275C4×Q8—48no—
276#117—432no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—216no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—82no—
307C2×C4⋊C4#4114yes—
308C2×C4⋊C4#4114yes—
309C2×C4⋊C4#4114yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—42no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—82no—
332C2×C22⋊C4C2×D414yes—
333C2×C22⋊C4C2×D414yes—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—44no—
342C2×C42—44no—
343C2×C42—44no—
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2412yes—
347C2×C42C2412yes—
348#107—28no—
349#106—44no—
350C4⋊C8—44no—
351#103—28no—
352#103—28no—
353#103—28no—
354#103—44no—
355#102—28no—
356#102—28no—
357#102—44no—
358#102—44no—
359C22⋊C8—44no—
360C22⋊C8—44no—
361#97—44no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95C2×D414yes—
369#95C2×D414yes—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24AGL(1,ℤ/8)14yes—
385C24C22×D412yes—
386C24C22×D411yescharacteristic
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—28no—
390C2×Q8—28no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8—42no—
398C2×Q8—44no—
399C2×Q8C2×#4114yes—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—48no—
407C2×D4—48no—
408C2×D4—42no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—84no—
416C2×D4—816no—
417C22×C4—216no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
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—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—432no—
454C22×C4—48no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—48no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C4C2511yescharacteristic
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—44no—
503C2×C8—44no—
504C2×C8—44no—
505C2×C8—44no—
506C4⋊C4—216no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—48no—
520C4⋊C4—48no—
521C4⋊C4—48no—
522C4⋊C4—48no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—84no—
527C4⋊C4C2×#4114yes—
528C4⋊C4C2×#4114yes—
529C4⋊C4C2×#4114yes—
530C22⋊C4—28no—
531C22⋊C4—44no—
532C22⋊C4—48no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—84no—
539C42—24no—
540C42—24no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—48no—
546C42—48no—
547C42—48no—
548C23—28no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—28no—
553C23—28no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—84no—
569C23—84no—
570C23—84no—
571C23—84no—
572C23—84no—
573C23—84no—
574C23—84no—
575C23#52512yes—
576C23#54612yes—
577C23#54612yes—
578C23C2×AGL(1,ℤ/8)14yes—
579C23C2×AGL(1,ℤ/8)12yes—
580C23#57512yes—
581C23#57712yes—
582C23#57812yes—
583C23C23×D411yescharacteristic
584Q8—24no—
585Q8—24no—
586Q8—42no—
587Q8—42no—
588Q8—48no—
589D4—42no—
590D4—42no—
591D4—42no—
592D4—42no—
593D4—816no—
594D4—84no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—432no—
615C2×C4—48no—
616C2×C4—48no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—48no—
621C2×C4—44no—
622C2×C4—48no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—48no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—84no—
647C2×C4—84no—
648C2×C4—84no—
649C2×C4#52512yes—
650C2×C4#52914yes—
651C2×C4#54014yes—
652C2×C4#54612yes—
653C2×C4#54612yes—
654C2×C4#55314yes—
655C2×C4#55514yes—
656C2×C4C22×#4111yescharacteristic
657C2×C4C22×#4111yescharacteristic
658C2×C4C22×#4111yescharacteristic
659C2×C4C22×#4111yescharacteristic
660C2×C4C2×H(5,𝔽2)11yescharacteristic
661C2×C4C2×H(5,𝔽2)11yescharacteristic
662C8—42no—
663C8—42no—
664C8—42no—
665C8—42no—
666C22—24no—
667C22—24no—
668C22—24no—
669C22—416no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—168no—
682C22#247614yes—
683C22#311414yes—
684C22#312114yes—
685C22#330614yes—
686C22#330814yes—
687C22#331714yes—
688C22#332714yes—
689C22C2×#52511yescharacteristic
690C22C2×#54611yescharacteristic
691C22C2×#54611yescharacteristic
692C22C22×AGL(1,ℤ/8)11yescharacteristic
693C22C2×#57511yescharacteristic
694C22C2×#57711yescharacteristic
695C22C2×#57811yescharacteristic
696C4—44no—
697C4—44no—
698C4—42no—
699C4—44no—
700C4—42no—
701C4—44no—
702C4—42no—
703C4—42no—
704C4—42no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4#247614yes—
709C4C2×#52911yescharacteristic
710C4C2×#54011yescharacteristic
711C4C2×#55311yescharacteristic
712C4C2×#55511yescharacteristic
713C2—44no—
714C2—162no—
715C2#2740818yesdirect product factor
716C2C2×#247611yescharacteristic
717C2C2×#311411yescharacteristic
718C2C2×#312111yescharacteristic
719C2C2×#330611yescharacteristic
720C2C2×#330811yescharacteristic
721C2C2×#331711yescharacteristic
722C2C2×#332711yescharacteristic
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. #27408C2direct (N × H)
4. C2×#2088C2semidirect
5. C2×#2042C2semidirect
6. C2×#2035C2semidirect
8. C2×#1983C2semidirect
9. C2×#1936C2semidirect
10. C2×#1926C2semidirect
11. C2×#1924C2semidirect
12. C2×#1921C2semidirect
14. C2×#1872C2semidirect
15. C2×#1863C2semidirect
16. C2×#1763C2semidirect
42. #2344C22semidirect
56. #2088C22semidirect
57. #2042C22semidirect
58. #2035C22semidirect
60. #1983C22semidirect
61. #1936C22semidirect
62. #1926C22semidirect
63. #1924C22semidirect
64. #1921C22semidirect
66. #1872C22semidirect
67. #1863C22semidirect
68. #1763C22semidirect
715. C2#27408direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24262
C2×#23442
#274082
C2×#20882
C2×#20422
C2×#20352
C2×#20332
C2×#19832
C2×#19362
C2×#19262
C2×#19242
C2×#19212
C2×#18742
C2×#18722
C2×#18632
C2×#17632