Finite Group Databasebeta

← C2×#27206 #6451302

Subgroups — C2×#27206 #6451302

Order 512.

2352 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×#2447C211yesmaximal, characteristic, complemented
2C2×#2354C211yesmaximal, characteristic, complemented
3#27206C2116yesmaximal, direct product factor
4C2×#2093C211yesmaximal, characteristic, complemented
5C2×#2072C211yesmaximal, characteristic, complemented
6C2×#2067C211yesmaximal, characteristic, complemented
7C2×#2012C211yesmaximal, characteristic
8C2×#1983C211yesmaximal, characteristic, complemented
9C2×#1973C211yesmaximal, characteristic, complemented
10C2×#1928C211yesmaximal, characteristic, complemented
11C2×#1921C211yesmaximal, characteristic, complemented
12C2×#1917C211yesmaximal, characteristic, complemented
13C2×#1879C211yesmaximal, characteristic
14C2×#1879C211yesmaximal, characteristic
15C2×#1877C211yesmaximal, characteristic, complemented
16C2×#1762C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C23×D8C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×#117—21no—
23C22×#107C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#97C2211yescharacteristic
35#2447—24no—
36#2447C2218yescomplemented
37C2×#397—21no—
38C2×#397—21no—
39#2354—24no—
40#2354C2218yescomplemented
41C2×#391C2211yescharacteristic
42C2×#391C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#388—21no—
48C2×#388—21no—
49C2×#385—21no—
50C2×#384C2211yescharacteristic
51C2×#381—21no—
52C2×#380—21no—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56#2093C2218yescomplemented
57#2072C2218yescomplemented
58#2067C2218yescomplemented
59#2012C2218yes—
60#1983C2218yescomplemented
61#1973C2218yescomplemented
62#1928C2218yescomplemented
63#1921C2218yescomplemented
64#1917C2218yescomplemented
65#1879C2218yes—
66#1879C2218yes—
67#1877C2218yescomplemented
68#1762C2218yescomplemented
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C24×C4—21no—
82C22×SD16—24no—
83C22×SD16—24no—
84C22×SD16—42no—
85C22×SD16C2314yes—
86C22×D8—24no—
87C22×D8—24no—
88C22×D8—42no—
89C22×D8C2314yes—
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C2×#124—24no—
95C2×#124—22no—
96C2×#124—42no—
97C2×#124C2314yes—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C2×C4×D4—22no—
102C2×C4×D4—22no—
103C2×C4×D4—42no—
104C2×C4×D4C2314yes—
105C2×C4×D4C2314yes—
106C2×#117—28no—
107C2×#117—48no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4D411yescharacteristic
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
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⋊C4—21no—
128C22×C42—21no—
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C2×#107—24no—
134C2×#107—42no—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×C4⋊C8—42no—
138C2×C4⋊C8C2314yes—
139C2×#103—24no—
140C2×#103—42no—
141C2×#103—42no—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#102—24no—
146C2×#102—24no—
147C2×#102—42no—
148C2×#102—42no—
149C2×#102—42no—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#97—42no—
156C2×#97C2314yes—
157#397—28no—
158#397—28no—
159#391—22no—
160#391—22no—
161#391C2314yes—
162#391C2314yes—
163#390—22no—
164#390—22no—
165#390—22no—
166#390C2314yes—
167#390C2314yes—
168#390C2314yes—
169#389—22no—
170#389C2314yes—
171#388—28no—
172#388—28no—
173#385—28no—
174#384—22no—
175#384C2314yes—
176#381—28no—
177#380—28no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4D414yes—
181C4×C4⋊C4C2314yes—
182C4×C4⋊C4C2314yes—
183C4×C4⋊C4C2314yes—
184C2×#95—21no—
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#336C2314yes—
198#336C2314yes—
199#336C2314yes—
200#336C2314yes—
201C25—21no—
202C25—21no—
203C22×Q8—21no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8C2412yes—
211C22×D4—21no—
212C22×D4—21no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4#4114yes—
224C22×D4#4114yes—
225C22×D4C2412yes—
226C22×D4C2412yes—
227C23×C4—28no—
228C23×C4—21no—
229C23×C4—24no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—24no—
236C23×C4—44no—
237C23×C4—44no—
238C23×C4—44no—
239C23×C4—44no—
240C23×C4—82no—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2411yescharacteristic
248C2×SD16—28no—
249C2×SD16—28no—
250C2×SD16—44no—
251C2×SD16—416no—
252C2×D8—28no—
253C2×D8—28no—
254C2×D8—44no—
255C2×D8—416no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8#4114yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C22×C8C2412yes—
272#124—28no—
273#124—44no—
274#124—48no—
275C4×D4—216no—
276C4×D4—28no—
277C4×D4—28no—
278C4×D4—28no—
279C4×D4—44no—
280C4×D4—48no—
281#117—432no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4—82no—
304C2×C4⋊C4C2×D414yes—
305C2×C4⋊C4C2×D414yes—
306C2×C4⋊C4#4114yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
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—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—82no—
330C2×C22⋊C4—82no—
331C2×C22⋊C4—82no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—44no—
341C2×C42—44no—
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2412yes—
346C2×C42C2412yes—
347C2×C42C2412yes—
348#107—28no—
349#107—44no—
350C4⋊C8—44no—
351#103—28no—
352#103—44no—
353#103—44no—
354#102—28no—
355#102—28no—
356#102—44no—
357#102—44no—
358#102—44no—
359#97—44no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95C2×D414yes—
368#95C2×D414yes—
369C24—24no—
370C24—24no—
371C24—22no—
372C24—22no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24C22×D411yescharacteristic
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×D4—24no—
400C2×D4—28no—
401C2×D4—24no—
402C2×D4—28no—
403C2×D4—28no—
404C2×D4—28no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—42no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—44no—
423C2×D4C2×#4114yes—
424C2×D4C2×#4114yes—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—216no—
428C22×C4—22no—
429C22×C4—24no—
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—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—432no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—48no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—84no—
473C22×C4—816no—
474C22×C4AGL(1,ℤ/8)14yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C2×#4112yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4H(5,𝔽2)12yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C4C2511yescharacteristic
489SD16—416no—
490D8—416no—
491C2×C8—28no—
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—22no—
503C2×C8—44no—
504C2×C8—44no—
505C2×C8—44no—
506C2×C8C2×#4114yes—
507C4⋊C4—24no—
508C4⋊C4—28no—
509C4⋊C4—22no—
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—44no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—48no—
524C4⋊C4—84no—
525C4⋊C4—84no—
526C4⋊C4C2×#4114yes—
527C22⋊C4—216no—
528C22⋊C4—48no—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—48no—
533C22⋊C4—48no—
534C22⋊C4—84no—
535C22⋊C4—84no—
536C22⋊C4—84no—
537C42—24no—
538C42—24no—
539C42—24no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—48no—
547C42—48no—
548C23—24no—
549C23—24no—
550C23—48no—
551C23—48no—
552C23—48no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—48no—
558C23—48no—
559C23—48no—
560C23—42no—
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—84no—
576C23#52512yes—
577C23#54612yes—
578C23#55012yes—
579C23C2×AGL(1,ℤ/8)12yes—
580C23#57512yes—
581C23#57612yes—
582C23#57812yes—
583C23C23×D411yescharacteristic
584Q8—42no—
585Q8—42no—
586Q8—42no—
587Q8—42no—
588Q8—84no—
589D4—24no—
590D4—24no—
591D4—24no—
592D4—24no—
593D4—42no—
594D4—42no—
595D4—42no—
596D4—42no—
597D4—48no—
598D4—48no—
599C2×C4—28no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
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—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—432no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—48no—
622C2×C4—44no—
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—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—816no—
643C2×C4—84no—
644C2×C4—84no—
645C2×C4—84no—
646C2×C4—84no—
647C2×C4—84no—
648C2×C4#52512yes—
649C2×C4#52914yes—
650C2×C4#54014yes—
651C2×C4#54612yes—
652C2×C4#55012yes—
653C2×C4#55114yes—
654C2×C4#55114yes—
655C2×C4C2×AGL(1,ℤ/8)14yes—
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—24no—
663C8—42no—
664C8—42no—
665C8—42no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—816no—
673C22—816no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22#247114yes—
683C22#311414yes—
684C22#312414yes—
685C22#331914yes—
686C22#333014yes—
687C22#339914yes—
688C22#340214yes—
689C22C2×#52511yescharacteristic
690C22C2×#54611yescharacteristic
691C22C2×#55011yescharacteristic
692C22C22×AGL(1,ℤ/8)11yescharacteristic
693C22C2×#57511yescharacteristic
694C22C2×#57611yescharacteristic
695C22C2×#57811yescharacteristic
696C4—24no—
697C4—44no—
698C4—42no—
699C4—42no—
700C4—42no—
701C4—42no—
702C4—42no—
703C4—82no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4#247114yes—
709C4C2×#52911yescharacteristic
710C4C2×#54011yescharacteristic
711C4C2×#55111yescharacteristic
712C4C2×#55111yescharacteristic
713C2—84no—
714C2—84no—
715C2#2720618yesdirect product factor
716C2C2×#247111yescharacteristic
717C2C2×#311411yescharacteristic
718C2C2×#312411yescharacteristic
719C2C2×#331911yescharacteristic
720C2C2×#333011yescharacteristic
721C2C2×#339911yescharacteristic
722C2C2×#340211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
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
1. C2×#2447C2semidirect
2. C2×#2354C2semidirect
3. #27206C2direct (N × H)
4. C2×#2093C2semidirect
5. C2×#2072C2semidirect
6. C2×#2067C2semidirect
8. C2×#1983C2semidirect
9. C2×#1973C2semidirect
10. C2×#1928C2semidirect
11. C2×#1921C2semidirect
12. C2×#1917C2semidirect
15. C2×#1877C2semidirect
16. C2×#1762C2semidirect
36. #2447C22semidirect
40. #2354C22semidirect
56. #2093C22semidirect
57. #2072C22semidirect
58. #2067C22semidirect
60. #1983C22semidirect
61. #1973C22semidirect
62. #1928C22semidirect
63. #1921C22semidirect
64. #1917C22semidirect
67. #1877C22semidirect
68. #1762C22semidirect
715. C2#27206direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24472
C2×#23542
#272062
C2×#20932
C2×#20722
C2×#20672
C2×#20122
C2×#19832
C2×#19732
C2×#19282
C2×#19212
C2×#19172
C2×#18792
C2×#18772
C2×#17622