Finite Group Databasebeta

← C2×#32283 #7508744

Subgroups — C2×#32283 #7508744

Order 512.

1874 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2500C211yesmaximal, characteristic
2C2×#2497C211yesmaximal, characteristic, complemented
3#32283C2116yesmaximal, direct product factor
4C2×#2100C211yesmaximal, characteristic, complemented
5C2×#2094C211yesmaximal, characteristic, complemented
6C2×#2085C211yesmaximal, characteristic, complemented
7C2×#2067C211yesmaximal, characteristic, complemented
8C2×#2033C211yesmaximal, characteristic
9C2×#2033C211yesmaximal, characteristic
10C2×#1949C211yesmaximal, characteristic, complemented
11C2×#1934C211yesmaximal, characteristic, complemented
12C2×#1925C211yesmaximal, characteristic, complemented
13C2×#1923C211yesmaximal, characteristic, complemented
14C2×#1854C211yesmaximal, characteristic, complemented
15C2×#1854C211yesmaximal, characteristic, complemented
16C2×#1845C211yesmaximal, characteristic, complemented
17C22×#121C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31C22×C22⋊C8C2211yescharacteristic
32C22×C4×C8C2211yescharacteristic
33#2500—24no—
34#2500C2218yes—
35#2497—24no—
36#2497C2218yescomplemented
37C2×#400—21no—
38C2×#396—21no—
39C2×#394—21no—
40C2×#394—21no—
41C2×#390C2211yescharacteristic
42C2×#390C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#388—21no—
46C2×#388—21no—
47C2×#388—21no—
48C2×#387—21no—
49C2×#384C2211yescharacteristic
50C2×#384C2211yescharacteristic
51C2×#382C2211yescharacteristic
52C2×#380C2211yescharacteristic
53C2×#380C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2211yescharacteristic
56#2100C2218yescomplemented
57#2094C2218yescomplemented
58#2085C2218yescomplemented
59#2067C2218yescomplemented
60#2033C2218yes—
61#2033C2218yes—
62#1949C2218yescomplemented
63#1934C2218yescomplemented
64#1925C2218yescomplemented
65#1923C2218yescomplemented
66#1854C2218yescomplemented
67#1854C2218yescomplemented
68#1845C2218yescomplemented
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×D4—21no—
78C23×D4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C2×#121—22no—
85C2×#121—42no—
86C2×#121—42no—
87C2×#121C2314yes—
88C2×#117—22no—
89C2×#117—42no—
90C2×#117—42no—
91C2×#117C2314yes—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C22⋊C4—21no—
106C22×C22⋊C4—21no—
107C22×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4D411yescharacteristic
114C22×C22⋊C4D411yescharacteristic
115C22×C42—21no—
116C22×C42—21no—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C2×#107—42no—
120C2×#107—42no—
121C2×#107—42no—
122C2×#107—42no—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#107C2314yes—
127C2×#106—42no—
128C2×#106—42no—
129C2×#106C2314yes—
130C2×#106C2314yes—
131C2×C4⋊C8—24no—
132C2×C4⋊C8C2314yes—
133C2×#102—24no—
134C2×#102—24no—
135C2×#102—42no—
136C2×#102—42no—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×C22⋊C8—24no—
142C2×C22⋊C8—24no—
143C2×C22⋊C8C2314yes—
144C2×C22⋊C8C2314yes—
145C2×C4×C8—24no—
146C2×C4×C8C2314yes—
147#400—28no—
148#396—28no—
149#394—28no—
150#394—28no—
151#390—22no—
152#390—22no—
153#390C2314yes—
154#390C2314yes—
155#389—22no—
156#389—22no—
157#389C2314yes—
158#389C2314yes—
159#388—28no—
160#388—28no—
161#388—28no—
162#387—28no—
163#384—22no—
164#384—22no—
165#384C2314yes—
166#384C2314yes—
167#382—22no—
168#382C2314yes—
169#380—22no—
170#380—22no—
171#380C2314yes—
172#380C2314yes—
173C4×C4⋊C4—22no—
174C4×C4⋊C4C2314yes—
175C4×C22⋊C4D414yes—
176C4×C22⋊C4C2314yes—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95D411yescharacteristic
185C2×#95D411yescharacteristic
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#336C2314yes—
193#336C2314yes—
194C25—21no—
195C25C2×D411yescharacteristic
196C22×D4—24no—
197C22×D4—21no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—44no—
205C22×D4—82no—
206C22×D4—82no—
207C22×D4C2412yes—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4C2×D412yes—
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8#4114yes—
240C22×C8#4114yes—
241C22×C8C2412yes—
242C22×C8C2412yes—
243C22×C8C2412yes—
244C22×C8C2412yes—
245#121—44no—
246#121—44no—
247#121—48no—
248#117—44no—
249#117—44no—
250#117—48no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—82no—
280C2×C4⋊C4—82no—
281C2×C4⋊C4—82no—
282C2×C4⋊C4—82no—
283C2×C4⋊C4—82no—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—82no—
306C2×C22⋊C4—82no—
307C2×C22⋊C4—82no—
308C2×C22⋊C4—82no—
309C2×C22⋊C4C2×D414yes—
310C2×C22⋊C4C2×D414yes—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—44no—
318C2×C42—82no—
319C2×C42C2×D412yes—
320C2×C42C2×D412yes—
321C2×C42C2412yes—
322C2×C42C2412yes—
323#107—44no—
324#107—44no—
325#107—44no—
326#107—44no—
327#106—44no—
328#106—44no—
329C4⋊C8—28no—
330#102—28no—
331#102—28no—
332#102—44no—
333#102—44no—
334C22⋊C8—28no—
335C22⋊C8—28no—
336C4×C8—28no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95C2×D414yes—
345#95C2×D414yes—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—42no—
355C24—42no—
356C24—42no—
357C24—42no—
358C24—42no—
359C24—42no—
360C24—42no—
361C24C22×D412yes—
362C24C22×D411yescharacteristic
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—48no—
370C2×D4—42no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—84no—
378C2×D4—84no—
379C2×D4—816no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
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—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
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#13514yes—
435C22×C4#13714yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4H(5,𝔽2)12yes—
445C22×C4H(5,𝔽2)12yes—
446C22×C42−1+412yes—
447C22×C42−1+412yes—
448C22×C4C2511yescharacteristic
449C2×C8—28no—
450C2×C8—28no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—44no—
462C2×C8—44no—
463C2×C8C2×#4114yes—
464C2×C8C2×#4114yes—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
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—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—48no—
487C4⋊C4—84no—
488C4⋊C4—84no—
489C4⋊C4—84no—
490C4⋊C4—84no—
491C4⋊C4—84no—
492C22⋊C4—28no—
493C22⋊C4—28no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C22⋊C4—84no—
499C22⋊C4—84no—
500C22⋊C4—84no—
501C22⋊C4—84no—
502C42—24no—
503C42—24no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—48no—
509C42—84no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—84no—
532C23—84no—
533C23—84no—
534C23—84no—
535C23—84no—
536C23—84no—
537C23—84no—
538C23#53512yes—
539C23#54712yes—
540C23#54712yes—
541C23C2×#13512yes—
542C23C2×#13712yes—
543C23#57612yes—
544C23#57612yes—
545C23C23×D411yescharacteristic
546D4—42no—
547D4—42no—
548D4—42no—
549D4—42no—
550D4—816no—
551D4—84no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—22no—
555C2×C4—22no—
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—22no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
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—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4#53512yes—
604C2×C4#53914yes—
605C2×C4#54214yes—
606C2×C4#54712yes—
607C2×C4#54712yes—
608C2×C4#55314yes—
609C2×C4#56314yes—
610C2×C4C2×#13514yes—
611C2×C4C2×#13714yes—
612C2×C4C22×#4111yescharacteristic
613C2×C4C22×#4111yescharacteristic
614C2×C4C2×H(5,𝔽2)11yescharacteristic
615C2×C4C2×H(5,𝔽2)11yescharacteristic
616C2×C4C2×2−1+411yescharacteristic
617C2×C4C2×2−1+411yescharacteristic
618C8—24no—
619C8—24no—
620C8—42no—
621C8—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—88no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—168no—
638C22#271814yes—
639C22#321014yes—
640C22#322014yes—
641C22#333914yes—
642C22#334114yes—
643C22#334614yes—
644C22#335714yes—
645C22C2×#53511yescharacteristic
646C22C2×#54711yescharacteristic
647C22C2×#54711yescharacteristic
648C22C22×#13511yescharacteristic
649C22C22×#13711yescharacteristic
650C22C2×#57611yescharacteristic
651C22C2×#57611yescharacteristic
652C4—24no—
653C4—24no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—82no—
658C4—82no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4#271814yes—
665C4C2×#53911yescharacteristic
666C4C2×#54211yescharacteristic
667C4C2×#55311yescharacteristic
668C4C2×#56311yescharacteristic
669C2—82no—
670C2—162no—
671C2#3228318yesdirect product factor
672C2C2×#271811yescharacteristic
673C2C2×#321011yescharacteristic
674C2C2×#322011yescharacteristic
675C2C2×#333911yescharacteristic
676C2C2×#334111yescharacteristic
677C2C2×#334611yescharacteristic
678C2C2×#335711yescharacteristic
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×#2497C2semidirect
3. #32283C2direct (N × H)
4. C2×#2100C2semidirect
5. C2×#2094C2semidirect
6. C2×#2085C2semidirect
7. C2×#2067C2semidirect
10. C2×#1949C2semidirect
11. C2×#1934C2semidirect
12. C2×#1925C2semidirect
13. C2×#1923C2semidirect
14. C2×#1854C2semidirect
15. C2×#1854C2semidirect
16. C2×#1845C2semidirect
36. #2497C22semidirect
56. #2100C22semidirect
57. #2094C22semidirect
58. #2085C22semidirect
59. #2067C22semidirect
62. #1949C22semidirect
63. #1934C22semidirect
64. #1925C22semidirect
65. #1923C22semidirect
66. #1854C22semidirect
67. #1854C22semidirect
68. #1845C22semidirect
671. C2#32283direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25002
C2×#24972
#322832
C2×#21002
C2×#20942
C2×#20852
C2×#20672
C2×#20332
C2×#19492
C2×#19342
C2×#19252
C2×#19232
C2×#18542
C2×#18452