Finite Group Databasebeta

← C2×#32127 #7507753

Subgroups — C2×#32127 #7507753

Order 512.

2042 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2490C211yesmaximal, characteristic
2C2×#2489C211yesmaximal, characteristic, complemented
3#32127C2116yesmaximal, direct product factor
4C2×#2077C211yesmaximal, characteristic, complemented
5C2×#2055C211yesmaximal, characteristic, complemented
6C2×#2055C211yesmaximal, characteristic, complemented
7C2×#2026C211yesmaximal, characteristic, complemented
8C2×#2025C211yesmaximal, characteristic, complemented
9C2×#2000C211yesmaximal, characteristic
10C2×#1989C211yesmaximal, characteristic, complemented
11C2×#1989C211yesmaximal, 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×#125C2211yescharacteristic
20C22×#125C2211yescharacteristic
21C22×C22⋊Q8C2211yescharacteristic
22C22×#121C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×C22⋊C8C2211yescharacteristic
33#2490—24no—
34#2490C2218yes—
35#2489—24no—
36#2489C2218yescomplemented
37C2×#400—21no—
38C2×#397—21no—
39C2×#397—21no—
40C2×#394—21no—
41C2×#391C2211yescharacteristic
42C2×#391C2211yescharacteristic
43C2×#390—21no—
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#388—21no—
47C2×#388—21no—
48C2×#382—21no—
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54#2077C2218yescomplemented
55#2055C2218yescomplemented
56#2055C2218yescomplemented
57#2026C2218yescomplemented
58#2025C2218yescomplemented
59#2000C2218yes—
60#1989C2218yescomplemented
61#1989C2218yescomplemented
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×Q8C2311yescharacteristic
78C23×D4—21no—
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C22×SD16—42no—
82C22×SD16—42no—
83C22×SD16—42no—
84C22×SD16C2314yes—
85C22×M16—42no—
86C22×M16—42no—
87C22×M16—42no—
88C22×M16C2314yes—
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C2×#125—24no—
94C2×#125—24no—
95C2×#125—22no—
96C2×#125—22no—
97C2×#125—42no—
98C2×#125—42no—
99C2×#125C2314yes—
100C2×#125C2314yes—
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8—42no—
104C2×C22⋊Q8C2314yes—
105C2×#121—22no—
106C2×#121—24no—
107C2×#121—42no—
108C2×#121C2314yes—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
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⋊C4—21no—
128C22×C22⋊C4D411yescharacteristic
129C22×C22⋊C4D411yescharacteristic
130C22×C42—21no—
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C2×#106—42no—
135C2×#106C2314yes—
136C2×C4⋊C8—42no—
137C2×C4⋊C8—42no—
138C2×C4⋊C8C2314yes—
139C2×C4⋊C8C2314yes—
140C2×#103—24no—
141C2×#103—42no—
142C2×#103—42no—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#102—24no—
147C2×#102—42no—
148C2×#102—42no—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×C22⋊C8—42no—
153C2×C22⋊C8C2314yes—
154#400—28no—
155#397—28no—
156#397—28no—
157#394—28no—
158#391—22no—
159#391—22no—
160#391C2314yes—
161#391C2314yes—
162#390—28no—
163#390—22no—
164#390—22no—
165#390C2314yes—
166#390C2314yes—
167#388—28no—
168#388—28no—
169#382—28no—
170#380—22no—
171#380—22no—
172#380C2314yes—
173#380C2314yes—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4C2314yes—
177C4×C4⋊C4C2314yes—
178C4×C22⋊C4D414yes—
179C4×C22⋊C4C2314yes—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95D411yescharacteristic
189C2×#95D411yescharacteristic
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2314yes—
199#336C2314yes—
200C25—21no—
201C25C2×D411yescharacteristic
202C22×Q8—21no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8C2412yes—
210C22×D4—24no—
211C22×D4—21no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—44no—
219C22×D4—44no—
220C22×D4—82no—
221C22×D4C2412yes—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4C2×D412yes—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C2×SD16—44no—
244C2×SD16—44no—
245C2×SD16—44no—
246C2×SD16—416no—
247C2×M16—44no—
248C2×M16—44no—
249C2×M16—44no—
250C2×M16—416no—
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×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C22×C8C2412yes—
267#125—28no—
268#125—28no—
269#125—44no—
270#125—44no—
271#125—48no—
272#125—48no—
273C22⋊Q8—28no—
274C22⋊Q8—44no—
275C22⋊Q8—48no—
276#121—28no—
277#121—44no—
278#121—48no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
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—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4#4114yes—
306C2×C4⋊C4#4114yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
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—42no—
322C2×C22⋊C4—42no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—82no—
328C2×C22⋊C4—82no—
329C2×C22⋊C4—82no—
330C2×C22⋊C4—82no—
331C2×C22⋊C4C2×D414yes—
332C2×C22⋊C4C2×D414yes—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—44no—
340C2×C42—82no—
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2412yes—
344C2×C42C2412yes—
345#106—44no—
346C4⋊C8—44no—
347C4⋊C8—44no—
348#103—28no—
349#103—44no—
350#103—44no—
351#102—28no—
352#102—44no—
353#102—44no—
354C22⋊C8—44no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95C2×D414yes—
364#95C2×D414yes—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24C22×D412yes—
381C24C22×D411yescharacteristic
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—48no—
402C2×D4—48no—
403C2×D4—42no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—84no—
411C2×D4—816no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
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—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
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×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4H(5,𝔽2)12yes—
477C22×C4H(5,𝔽2)12yes—
478C22×C4H(5,𝔽2)12yes—
479C22×C42−1+412yes—
480C22×C4C2511yescharacteristic
481SD16—416no—
482M16—416no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—44no—
496C2×C8—44no—
497C2×C8—44no—
498C2×C8—44no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—48no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4C2×#4114yes—
521C4⋊C4C2×#4114yes—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—84no—
529C22⋊C4—84no—
530C22⋊C4—84no—
531C22⋊C4—84no—
532C42—24no—
533C42—24no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—48no—
539C42—84no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—84no—
562C23—84no—
563C23—84no—
564C23—84no—
565C23—84no—
566C23—84no—
567C23—84no—
568C23#53712yes—
569C23#54612yes—
570C23#54612yes—
571C23#57512yes—
572C23#57512yes—
573C23#57712yes—
574C23#57712yes—
575C23C23×D411yescharacteristic
576Q8—42no—
577Q8—42no—
578Q8—42no—
579Q8—42no—
580Q8—84no—
581D4—42no—
582D4—42no—
583D4—42no—
584D4—42no—
585D4—816no—
586D4—84no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—48no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—48no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—48no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
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—84no—
636C2×C4—84no—
637C2×C4—84no—
638C2×C4—84no—
639C2×C4—84no—
640C2×C4#53712yes—
641C2×C4#54214yes—
642C2×C4#54214yes—
643C2×C4#54612yes—
644C2×C4#54612yes—
645C2×C4#55914yes—
646C2×C4H(3,𝔽4)14yes—
647C2×C4C22×#4111yescharacteristic
648C2×C4C22×#4111yescharacteristic
649C2×C4C2×H(5,𝔽2)11yescharacteristic
650C2×C4C2×H(5,𝔽2)11yescharacteristic
651C2×C4C2×H(5,𝔽2)11yescharacteristic
652C2×C4C2×2−1+411yescharacteristic
653C8—42no—
654C8—42no—
655C8—42no—
656C8—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—88no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—168no—
673C22#271214yes—
674C22#325614yes—
675C22#326014yes—
676C22#330714yes—
677C22#331714yes—
678C22#332314yes—
679C22#332614yes—
680C22C2×#53711yescharacteristic
681C22C2×#54611yescharacteristic
682C22C2×#54611yescharacteristic
683C22C2×#57511yescharacteristic
684C22C2×#57511yescharacteristic
685C22C2×#57711yescharacteristic
686C22C2×#57711yescharacteristic
687C4—42no—
688C4—42no—
689C4—44no—
690C4—42no—
691C4—42no—
692C4—44no—
693C4—42no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4#271214yes—
700C4C2×#54211yescharacteristic
701C4C2×#54211yescharacteristic
702C4C2×#55911yescharacteristic
703C4C2×H(3,𝔽4)11yescharacteristic
704C2—82no—
705C2—162no—
706C2#3212718yesdirect product factor
707C2C2×#271211yescharacteristic
708C2C2×#325611yescharacteristic
709C2C2×#326011yescharacteristic
710C2C2×#330711yescharacteristic
711C2C2×#331711yescharacteristic
712C2C2×#332311yescharacteristic
713C2C2×#332611yescharacteristic
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×#2489C2semidirect
3. #32127C2direct (N × H)
4. C2×#2077C2semidirect
5. C2×#2055C2semidirect
6. C2×#2055C2semidirect
7. C2×#2026C2semidirect
8. C2×#2025C2semidirect
10. C2×#1989C2semidirect
11. C2×#1989C2semidirect
12. C2×#1926C2semidirect
13. C2×#1927C2semidirect
15. C2×#1894C2semidirect
16. C2×#1851C2semidirect
36. #2489C22semidirect
54. #2077C22semidirect
55. #2055C22semidirect
56. #2055C22semidirect
57. #2026C22semidirect
58. #2025C22semidirect
60. #1989C22semidirect
61. #1989C22semidirect
62. #1927C22semidirect
63. #1926C22semidirect
65. #1894C22semidirect
66. #1851C22semidirect
706. C2#32127direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24902
C2×#24892
#321272
C2×#20772
C2×#20552
C2×#20262
C2×#20252
C2×#20002
C2×#19892
C2×#19262
C2×#19272
C2×#18952
C2×#18942
C2×#18512