Finite Group Databasebeta

← C2×#32126 #7507748

Subgroups — C2×#32126 #7507748

Order 512.

2226 conjugacy classes of subgroups.

The subgroup lattice has 729 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2492C211yesmaximal, characteristic, complemented
2C2×#2491C211yesmaximal, characteristic
3#32126C2116yesmaximal, direct product factor
4C2×#2077C211yesmaximal, characteristic, complemented
5C2×#2055C211yesmaximal, characteristic, complemented
6C2×#2055C211yesmaximal, characteristic, complemented
7C2×#2024C211yesmaximal, characteristic, complemented
8C2×#2023C211yesmaximal, characteristic, complemented
9C2×#2000C211yesmaximal, characteristic
10C2×#1988C211yesmaximal, characteristic, complemented
11C2×#1987C211yesmaximal, characteristic, complemented
12C2×#1926C211yesmaximal, characteristic, complemented
13C2×#1927C211yesmaximal, characteristic, complemented
14C2×#1895C211yesmaximal, characteristic
15C2×#1894C211yesmaximal, characteristic, complemented
16C2×#1851C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C23×M16C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×#124—21no—
22C22×C22⋊Q8C2211yescharacteristic
23C22×#121C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×C22⋊C8C2211yescharacteristic
34#2492—24no—
35#2492C2218yescomplemented
36#2491—24no—
37#2491C2218yes—
38C2×#400—21no—
39C2×#394—21no—
40C2×#393—21no—
41C2×#393—21no—
42C2×#391—21no—
43C2×#391C2211yescharacteristic
44C2×#391C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#388—21no—
48C2×#388—21no—
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54#2077C2218yescomplemented
55#2055C2218yescomplemented
56#2055C2218yescomplemented
57#2024C2218yescomplemented
58#2023C2218yescomplemented
59#2000C2218yes—
60#1988C2218yescomplemented
61#1987C2218yescomplemented
62#1927C2218yescomplemented
63#1926C2218yescomplemented
64#1895C2218yes—
65#1894C2218yescomplemented
66#1851C2218yescomplemented
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8—21no—
78C23×Q8—21no—
79C23×Q8C2311yescharacteristic
80C23×D4—21no—
81C23×D4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C22×SD16—42no—
84C22×SD16—42no—
85C22×SD16—42no—
86C22×SD16C2314yes—
87C22×M16—42no—
88C22×M16—42no—
89C22×M16—42no—
90C22×M16C2314yes—
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C23×C8C2311yescharacteristic
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—42no—
100C2×C4⋊Q8—42no—
101C2×C4⋊Q8C2314yes—
102C2×C4⋊Q8C2314yes—
103C2×#124—28no—
104C2×#124—48no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—42no—
108C2×C22⋊Q8C2314yes—
109C2×#121—22no—
110C2×#121—24no—
111C2×#121—42no—
112C2×#121C2314yes—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
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×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4D411yescharacteristic
131C22×C22⋊C4D411yescharacteristic
132C22×C42—21no—
133C22×C42—21no—
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C2×#106—42no—
137C2×#106C2314yes—
138C2×C4⋊C8—42no—
139C2×C4⋊C8—42no—
140C2×C4⋊C8C2314yes—
141C2×C4⋊C8C2314yes—
142C2×#103—24no—
143C2×#103—42no—
144C2×#103—42no—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#102—24no—
149C2×#102—42no—
150C2×#102—42no—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×C22⋊C8—42no—
155C2×C22⋊C8C2314yes—
156#400—28no—
157#394—28no—
158#393—28no—
159#393—28no—
160#391—22no—
161#391—22no—
162#391—28no—
163#391C2314yes—
164#391C2314yes—
165#390—22no—
166#390—22no—
167#390C2314yes—
168#390C2314yes—
169#388—28no—
170#388—28no—
171#380—22no—
172#380—22no—
173#380C2314yes—
174#380C2314yes—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4C2314yes—
178C4×C4⋊C4C2314yes—
179C4×C22⋊C4D414yes—
180C4×C22⋊C4C2314yes—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95D411yescharacteristic
190C2×#95D411yescharacteristic
191#340C2314yes—
192#340C2314yes—
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—24no—
205C22×Q8—24no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—44no—
213C22×Q8—44no—
214C22×Q8—44no—
215C22×Q8—44no—
216C22×Q8—82no—
217C22×Q8—82no—
218C22×Q8C2412yes—
219C22×D4—24no—
220C22×D4—21no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—44no—
228C22×D4—44no—
229C22×D4—82no—
230C22×D4C2412yes—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—22no—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4C2×D412yes—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2412yes—
251C23×C4C2411yescharacteristic
252C2×SD16—44no—
253C2×SD16—44no—
254C2×SD16—44no—
255C2×SD16—416no—
256C2×M16—44no—
257C2×M16—44no—
258C2×M16—44no—
259C2×M16—416no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C4⋊Q8—28no—
277C4⋊Q8—28no—
278C4⋊Q8—44no—
279C4⋊Q8—44no—
280C4⋊Q8—48no—
281C4⋊Q8—48no—
282#124—432no—
283C22⋊Q8—28no—
284C22⋊Q8—44no—
285C22⋊Q8—48no—
286#121—28no—
287#121—44no—
288#121—48no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4#4114yes—
312C2×C4⋊C4#4114yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—82no—
336C2×C22⋊C4—82no—
337C2×C22⋊C4C2×D414yes—
338C2×C22⋊C4C2×D414yes—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—44no—
346C2×C42—82no—
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2412yes—
350C2×C42C2412yes—
351#106—44no—
352C4⋊C8—44no—
353C4⋊C8—44no—
354#103—28no—
355#103—44no—
356#103—44no—
357#102—28no—
358#102—44no—
359#102—44no—
360C22⋊C8—44no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95C2×D414yes—
370#95C2×D414yes—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24C22×D412yes—
387C24C22×D411yescharacteristic
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—48no—
395C2×Q8—48no—
396C2×Q8—48no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—48no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—84no—
406C2×Q8—84no—
407C2×Q8—816no—
408C2×Q8—816no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—48no—
416C2×D4—48no—
417C2×D4—42no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—84no—
425C2×D4—816no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
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×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4AGL(1,ℤ/8)14yes—
481C22×C4#13714yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4H(5,𝔽2)12yes—
491C22×C4H(5,𝔽2)12yes—
492C22×C4H(5,𝔽2)12yes—
493C22×C42−1+412yes—
494C22×C4C2511yescharacteristic
495SD16—416no—
496M16—416no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—44no—
510C2×C8—44no—
511C2×C8—44no—
512C2×C8—44no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—48no—
526C4⋊C4—48no—
527C4⋊C4—48no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4C2×#4114yes—
533C4⋊C4C2×#4114yes—
534C22⋊C4—28no—
535C22⋊C4—28no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—48no—
539C22⋊C4—48no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C22⋊C4—84no—
543C22⋊C4—84no—
544C42—24no—
545C42—24no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—48no—
551C42—84no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—84no—
574C23—84no—
575C23—84no—
576C23—84no—
577C23—84no—
578C23—84no—
579C23—84no—
580C23#53712yes—
581C23#54612yes—
582C23#54612yes—
583C23C2×AGL(1,ℤ/8)12yes—
584C23C2×#13712yes—
585C23#57712yes—
586C23#57712yes—
587C23C23×D411yescharacteristic
588Q8—42no—
589Q8—42no—
590Q8—42no—
591Q8—42no—
592Q8—816no—
593Q8—816no—
594Q8—84no—
595D4—42no—
596D4—42no—
597D4—42no—
598D4—42no—
599D4—816no—
600D4—84no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—48no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—48no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—48no—
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—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—84no—
648C2×C4—84no—
649C2×C4—84no—
650C2×C4—84no—
651C2×C4—84no—
652C2×C4#53712yes—
653C2×C4#54214yes—
654C2×C4#54214yes—
655C2×C4#54612yes—
656C2×C4#54612yes—
657C2×C4#55914yes—
658C2×C4H(3,𝔽4)14yes—
659C2×C4C2×AGL(1,ℤ/8)14yes—
660C2×C4C2×#13714yes—
661C2×C4C22×#4111yescharacteristic
662C2×C4C22×#4111yescharacteristic
663C2×C4C2×H(5,𝔽2)11yescharacteristic
664C2×C4C2×H(5,𝔽2)11yescharacteristic
665C2×C4C2×H(5,𝔽2)11yescharacteristic
666C2×C4C2×2−1+411yescharacteristic
667C8—42no—
668C8—42no—
669C8—42no—
670C8—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—88no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—168no—
687C22#271214yes—
688C22#325414yes—
689C22#326314yes—
690C22#330614yes—
691C22#331614yes—
692C22#332414yes—
693C22#332914yes—
694C22C2×#53711yescharacteristic
695C22C2×#54611yescharacteristic
696C22C2×#54611yescharacteristic
697C22C22×AGL(1,ℤ/8)11yescharacteristic
698C22C22×#13711yescharacteristic
699C22C2×#57711yescharacteristic
700C22C2×#57711yescharacteristic
701C4—24no—
702C4—24no—
703C4—42no—
704C4—44no—
705C4—42no—
706C4—44no—
707C4—42no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4#271214yes—
714C4C2×#54211yescharacteristic
715C4C2×#54211yescharacteristic
716C4C2×#55911yescharacteristic
717C4C2×H(3,𝔽4)11yescharacteristic
718C2—82no—
719C2—162no—
720C2#3212618yesdirect product factor
721C2C2×#271211yescharacteristic
722C2C2×#325411yescharacteristic
723C2C2×#326311yescharacteristic
724C2C2×#330611yescharacteristic
725C2C2×#331611yescharacteristic
726C2C2×#332411yescharacteristic
727C2C2×#332911yescharacteristic
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
1. C2×#2492C2semidirect
3. #32126C2direct (N × H)
4. C2×#2077C2semidirect
5. C2×#2055C2semidirect
6. C2×#2055C2semidirect
7. C2×#2024C2semidirect
8. C2×#2023C2semidirect
10. C2×#1988C2semidirect
11. C2×#1987C2semidirect
12. C2×#1926C2semidirect
13. C2×#1927C2semidirect
15. C2×#1894C2semidirect
16. C2×#1851C2semidirect
35. #2492C22semidirect
54. #2077C22semidirect
55. #2055C22semidirect
56. #2055C22semidirect
57. #2024C22semidirect
58. #2023C22semidirect
60. #1988C22semidirect
61. #1987C22semidirect
62. #1927C22semidirect
63. #1926C22semidirect
65. #1894C22semidirect
66. #1851C22semidirect
720. C2#32126direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24922
C2×#24912
#321262
C2×#20772
C2×#20552
C2×#20242
C2×#20232
C2×#20002
C2×#19882
C2×#19872
C2×#19262
C2×#19272
C2×#18952
C2×#18942
C2×#18512