Finite Group Databasebeta

← C2×#32556 #7510321

Subgroups — C2×#32556 #7510321

Order 512.

2008 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2C2×#2581C211yesmaximal, characteristic, complemented
3#32556C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2099C211yesmaximal, characteristic, complemented
6C2×#2087C211yesmaximal, characteristic, complemented
7C2×#2072C211yesmaximal, characteristic, complemented
8C2×#2073C211yesmaximal, characteristic
9C2×#2055C211yesmaximal, characteristic
10C2×#1985C211yesmaximal, characteristic, complemented
11C2×#1936C211yesmaximal, characteristic
12C2×#1935C211yesmaximal, characteristic, complemented
13C2×#1880C211yesmaximal, characteristic
14C2×#1877C211yesmaximal, characteristic
15C2×#1863C211yesmaximal, characteristic, complemented
16C2×#1862C211yesmaximal, characteristic
17C23×SD16C2211yescharacteristic
18C22×C22⋊Q8—21no—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#97C2211yescharacteristic
31#2581—24no—
32#2581C2218yescomplemented
33#2538—24no—
34#2538C2218yes—
35C2×#400—21no—
36C2×#396—21no—
37C2×#396—21no—
38C2×#395—21no—
39C2×#391C2211yescharacteristic
40C2×#391C2211yescharacteristic
41C2×#390C2211yescharacteristic
42C2×#390C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#387—21no—
48C2×#387—21no—
49C2×#384C2211yescharacteristic
50C2×#382C2211yescharacteristic
51C2×#380—21no—
52#2101C2218yescomplemented
53#2099C2218yescomplemented
54#2087C2218yescomplemented
55#2073C2218yes—
56#2072C2218yescomplemented
57#2055C2218yes—
58#1985C2218yescomplemented
59#1936C2218yes—
60#1935C2218yescomplemented
61#1880C2218yes—
62#1877C2218yes—
63#1863C2218yescomplemented
64#1862C2218yes—
65C2×#340C2211yescharacteristic
66C2×#340C2211yescharacteristic
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8D411yescharacteristic
78C23×Q8C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C22×SD16—24no—
82C22×SD16—24no—
83C22×SD16—42no—
84C22×SD16C2314yes—
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—48no—
91C2×C4×Q8—22no—
92C2×C4×Q8—42no—
93C2×C4×Q8—42no—
94C2×C4×Q8—42no—
95C2×C4×Q8—42no—
96C2×C4×Q8D414yes—
97C2×C4×Q8C2314yes—
98C2×C4×Q8C2314yes—
99C2×C4×D4—22no—
100C2×C4×D4—42no—
101C2×C4×D4—42no—
102C2×C4×D4C2314yes—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4D411yescharacteristic
111C22×C4⋊C4D411yescharacteristic
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C42—21no—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C2×#107—42no—
129C2×#107C2314yes—
130C2×C4⋊C8—42no—
131C2×C4⋊C8C2314yes—
132C2×#103—24no—
133C2×#103—24no—
134C2×#103—42no—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#102—24no—
139C2×#102—24no—
140C2×#102—42no—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#97—42no—
145C2×#97C2314yes—
146#400—28no—
147#396—28no—
148#396—28no—
149#395—28no—
150#391—22no—
151#391—22no—
152#391C2314yes—
153#391C2314yes—
154#390—22no—
155#390—22no—
156#390C2314yes—
157#390C2314yes—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389C2314yes—
163#389C2314yes—
164#389C2314yes—
165#389C2314yes—
166#387—28no—
167#387—28no—
168#384—22no—
169#384C2314yes—
170#382—22no—
171#382C2314yes—
172#380—28no—
173C2×#95—21no—
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—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#336C2314yes—
192#336C2314yes—
193#336C2314yes—
194#336C2314yes—
195C25—21no—
196C22×Q8—24no—
197C22×Q8—21no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—42no—
204C22×Q8—42no—
205C22×Q8C2×D414yes—
206C22×Q8#4114yes—
207C22×Q8C2412yes—
208C22×D4—21no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4#4114yes—
215C22×D4C2412yes—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—24no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2411yescharacteristic
234C2×SD16—28no—
235C2×SD16—28no—
236C2×SD16—44no—
237C2×SD16—416no—
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—22no—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C22⋊Q8—432no—
255C4×Q8—48no—
256C4×Q8—44no—
257C4×Q8—44no—
258C4×Q8—44no—
259C4×Q8—44no—
260C4×Q8—48no—
261C4×D4—44no—
262C4×D4—44no—
263C4×D4—48no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
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—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
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—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—82no—
299C2×C4⋊C4C2×D414yes—
300C2×C4⋊C4C2×D414yes—
301C2×C4⋊C4C2×D414yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—82no—
317C2×C22⋊C4—82no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—44no—
326C2×C42C2×D412yes—
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2412yes—
330C2×C42C2412yes—
331C2×C42C2412yes—
332#107—44no—
333C4⋊C8—44no—
334#103—28no—
335#103—28no—
336#103—44no—
337#102—28no—
338#102—28no—
339#102—44no—
340#97—44no—
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#95—24no—
350#95—24no—
351C24—24no—
352C24—22no—
353C24—42no—
354C24—42no—
355C24—42no—
356C24—42no—
357C24—42no—
358C24—42no—
359C24C22×D411yescharacteristic
360C2×Q8—24no—
361C2×Q8—28no—
362C2×Q8—22no—
363C2×Q8—28no—
364C2×Q8—28no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—44no—
370C2×Q8—44no—
371C2×Q8—416no—
372C2×Q8—42no—
373C2×Q8—44no—
374C2×Q8—42no—
375C2×Q8—44no—
376C2×Q8C2×#4114yes—
377C2×D4—28no—
378C2×D4—24no—
379C2×D4—28no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—42no—
386C2×D4—44no—
387C2×D4—42no—
388C2×D4—44no—
389C2×D4C2×#4114yes—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
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—48no—
416C22×C4—48no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—48no—
438C22×C4—816no—
439C22×C4AGL(1,ℤ/8)14yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C22×C4H(5,𝔽2)12yes—
450C22×C4H(5,𝔽2)12yes—
451C22×C42−1+412yes—
452C22×C42−1+412yes—
453C22×C4C2511yescharacteristic
454SD16—416no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—44no—
468C2×C8—44no—
469C2×C8—44no—
470C2×C8—44no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—84no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—84no—
503C22⋊C4—84no—
504C42—24no—
505C42—24no—
506C42—24no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—48no—
514C23—24no—
515C23—48no—
516C23—48no—
517C23—42no—
518C23—42no—
519C23—48no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—84no—
525C23—84no—
526C23—84no—
527C23—84no—
528C23#53412yes—
529C23#54812yes—
530C23#54812yes—
531C23C2×AGL(1,ℤ/8)12yes—
532C23#57512yes—
533C23#57712yes—
534C23#57812yes—
535C23C23×D411yescharacteristic
536Q8—24no—
537Q8—24no—
538Q8—416no—
539Q8—42no—
540Q8—48no—
541Q8—42no—
542D4—24no—
543D4—24no—
544D4—42no—
545D4—48no—
546D4—42no—
547C2×C4—22no—
548C2×C4—28no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—48no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—816no—
593C2×C4—84no—
594C2×C4—84no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4#53412yes—
600C2×C4#53914yes—
601C2×C4#53914yes—
602C2×C4#54812yes—
603C2×C4#54812yes—
604C2×C4#55114yes—
605C2×C4#56314yes—
606C2×C4C2×AGL(1,ℤ/8)14yes—
607C2×C4C22×#4111yescharacteristic
608C2×C4C22×#4111yescharacteristic
609C2×C4C2×H(5,𝔽2)11yescharacteristic
610C2×C4C2×H(5,𝔽2)11yescharacteristic
611C2×C4C2×2−1+411yescharacteristic
612C2×C4C2×2−1+411yescharacteristic
613C8—42no—
614C8—42no—
615C8—42no—
616C8—42no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—816no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22#272814yes—
626C22#319414yes—
627C22#320114yes—
628C22#336714yes—
629C22#336814yes—
630C22#337614yes—
631C22#337714yes—
632C22C2×#53411yescharacteristic
633C22C2×#54811yescharacteristic
634C22C2×#54811yescharacteristic
635C22C22×AGL(1,ℤ/8)11yescharacteristic
636C22C2×#57511yescharacteristic
637C22C2×#57711yescharacteristic
638C22C2×#57811yescharacteristic
639C4—24no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—44no—
646C4—82no—
647C4—82no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4—82no—
652C4#272814yes—
653C4C2×#53911yescharacteristic
654C4C2×#53911yescharacteristic
655C4C2×#55111yescharacteristic
656C4C2×#56311yescharacteristic
657C2—84no—
658C2#3255618yesdirect product factor
659C2C2×#272811yescharacteristic
660C2C2×#319411yescharacteristic
661C2C2×#320111yescharacteristic
662C2C2×#336711yescharacteristic
663C2C2×#336811yescharacteristic
664C2C2×#337611yescharacteristic
665C2C2×#337711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2581C2semidirect
3. #32556C2direct (N × H)
4. C2×#2101C2semidirect
5. C2×#2099C2semidirect
6. C2×#2087C2semidirect
7. C2×#2072C2semidirect
10. C2×#1985C2semidirect
12. C2×#1935C2semidirect
15. C2×#1863C2semidirect
32. #2581C22semidirect
52. #2101C22semidirect
53. #2099C22semidirect
54. #2087C22semidirect
56. #2072C22semidirect
58. #1985C22semidirect
60. #1935C22semidirect
63. #1863C22semidirect
658. C2#32556direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#25812
#325562
C2×#21012
C2×#20992
C2×#20872
C2×#20722
C2×#20732
C2×#20552
C2×#19852
C2×#19362
C2×#19352
C2×#18802
C2×#18772
C2×#18632
C2×#18622