Finite Group Databasebeta

← C2×#31605 #7505234

Subgroups — C2×#31605 #7505234

Order 512.

1982 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×#2495C211yesmaximal, characteristic, complemented
2C2×#2420C211yesmaximal, characteristic
3#31605C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2088C211yesmaximal, characteristic, complemented
6C2×#2085C211yesmaximal, characteristic, complemented
7C2×#2077C211yesmaximal, characteristic, complemented
8C2×#2067C211yesmaximal, characteristic, complemented
9C2×#2034C211yesmaximal, characteristic
10C2×#1949C211yesmaximal, characteristic, complemented
11C2×#1926C211yesmaximal, characteristic, complemented
12C2×#1893C211yesmaximal, characteristic
13C2×#1862C211yesmaximal, characteristic, complemented
14C2×#1850C211yesmaximal, characteristic, complemented
15C2×#1808C211yesmaximal, characteristic, complemented
16C2×#1804C211yesmaximal, characteristic, complemented
17C24×C8C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31#2495—24no—
32#2495C2218yescomplemented
33#2420—24no—
34#2420C2218yes—
35C2×#400—21no—
36C2×#397—21no—
37C2×#397—21no—
38C2×#394—21no—
39C2×#390C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#388—21no—
44C2×#386—21no—
45C2×#383—21no—
46C2×#382C2211yescharacteristic
47C2×#382C2211yescharacteristic
48C2×#381—21no—
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×#380C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54#2101C2218yescomplemented
55#2088C2218yescomplemented
56#2085C2218yescomplemented
57#2077C2218yescomplemented
58#2067C2218yescomplemented
59#2034C2218yes—
60#1949C2218yescomplemented
61#1926C2218yescomplemented
62#1893C2218yes—
63#1862C2218yescomplemented
64#1850C2218yescomplemented
65#1808C2218yescomplemented
66#1804C2218yescomplemented
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×D4—21no—
78C23×D4C2311yescharacteristic
79C24×C4—21no—
80C24×C4C2311yescharacteristic
81C23×C8—24no—
82C23×C8—42no—
83C23×C8—42no—
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2314yes—
89C2×#121—22no—
90C2×#121—24no—
91C2×#121—42no—
92C2×#121C2314yes—
93C2×C4×D4—24no—
94C2×C4×D4—22no—
95C2×C4×D4—42no—
96C2×C4×D4C2314yes—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
114C22×C22⋊C4D411yescharacteristic
115C22×C22⋊C4D411yescharacteristic
116C22×C42—21no—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C2×#107—42no—
120C2×#107—42no—
121C2×#107C2314yes—
122C2×#107C2314yes—
123C2×#106—42no—
124C2×#106—42no—
125C2×#106C2314yes—
126C2×#106C2314yes—
127C2×C4⋊C8—24no—
128C2×C4⋊C8—42no—
129C2×C4⋊C8C2314yes—
130C2×C4⋊C8C2314yes—
131C2×#102—24no—
132C2×#102—24no—
133C2×#102—42no—
134C2×#102—42no—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×C22⋊C8—42no—
140C2×C22⋊C8C2314yes—
141#400—28no—
142#397—28no—
143#397—28no—
144#394—28no—
145#390—22no—
146#390C2314yes—
147#389—22no—
148#389—22no—
149#389—22no—
150#389C2314yes—
151#389C2314yes—
152#389C2314yes—
153#388—28no—
154#386—28no—
155#383—28no—
156#382—22no—
157#382—22no—
158#382C2314yes—
159#382C2314yes—
160#381—28no—
161#380—22no—
162#380—22no—
163#380—22no—
164#380C2314yes—
165#380C2314yes—
166#380C2314yes—
167C4×C4⋊C4—22no—
168C4×C4⋊C4C2314yes—
169C4×C22⋊C4D414yes—
170C4×C22⋊C4C2314yes—
171C2×#95—21no—
172C2×#95—21no—
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×#95D411yescharacteristic
183C2×#95D411yescharacteristic
184#340C2314yes—
185#340C2314yes—
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—44no—
204C22×D4—44no—
205C22×D4—82no—
206C22×D4#4114yes—
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×C4—24no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—82no—
225C23×C4C2×D412yes—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—28no—
245C22×C8—44no—
246C22×C8—44no—
247C22×C8—416no—
248C22×C8#4114yes—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253#121—28no—
254#121—44no—
255#121—48no—
256C4×D4—28no—
257C4×D4—44no—
258C4×D4—48no—
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—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—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—82no—
284C2×C4⋊C4—82no—
285C2×C4⋊C4—82no—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
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—42no—
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—44no—
317C2×C42C2×D412yes—
318C2×C42C2×D412yes—
319C2×C42C2412yes—
320C2×C42C2412yes—
321#107—44no—
322#107—44no—
323#106—44no—
324#106—44no—
325C4⋊C8—28no—
326C4⋊C8—44no—
327#102—28no—
328#102—28no—
329#102—44no—
330#102—44no—
331C22⋊C8—44no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95C2×D414yes—
344#95C2×D414yes—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—24no—
352C24—22no—
353C24—42no—
354C24—42no—
355C24—42no—
356C24—42no—
357C24—42no—
358C24—42no—
359C24#13514yes—
360C24C22×D412yes—
361C24C22×D411yescharacteristic
362C2×D4—28no—
363C2×D4—24no—
364C2×D4—28no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—48no—
371C2×D4—48no—
372C2×D4—42no—
373C2×D4—44no—
374C2×D4—42no—
375C2×D4—44no—
376C2×D4—84no—
377C2×D4—816no—
378C2×D4C2×#4114yes—
379C22×C4—22no—
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—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
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—22no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
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—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—84no—
438C22×C4—816no—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C2×#4112yes—
446C22×C4C2×#4112yes—
447C22×C4H(5,𝔽2)12yes—
448C22×C4H(5,𝔽2)12yes—
449C22×C42−1+412yes—
450C22×C42−1+412yes—
451C22×C4C2511yescharacteristic
452C2×C8—28no—
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—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—416no—
465C2×C8—44no—
466C2×C8—44no—
467C2×C8—44no—
468C2×C8C2×#4114yes—
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—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—84no—
491C4⋊C4—84no—
492C4⋊C4—84no—
493C22⋊C4—28no—
494C22⋊C4—44no—
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—
509C23—28no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—28no—
514C23—28no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—24no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—42no—
524C23—48no—
525C23—42no—
526C23—48no—
527C23—48no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—84no—
533C23—84no—
534C23—84no—
535C23—84no—
536C23#53712yes—
537C23#54612yes—
538C23#54712yes—
539C23C2×#13514yes—
540C23C2×#13512yes—
541C23#57512yes—
542C23#57612yes—
543C23#57712yes—
544C23C23×D411yescharacteristic
545D4—24no—
546D4—24no—
547D4—42no—
548D4—48no—
549D4—42no—
550D4—816no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
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—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
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—42no—
597C2×C4—816no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4#53712yes—
606C2×C4#53814yes—
607C2×C4#53914yes—
608C2×C4#54612yes—
609C2×C4#54712yes—
610C2×C4#55214yes—
611C2×C4#56314yes—
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—42no—
620C8—42no—
621C8—42no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—42no—
626C22—416no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—82no—
634C22—82no—
635C22—816no—
636C22—82no—
637C22—82no—
638C22#269514yes—
639C22#325014yes—
640C22#325714yes—
641C22#329614yes—
642C22#331914yes—
643C22#333914yes—
644C22#334814yes—
645C22C2×#53711yescharacteristic
646C22C2×#54611yescharacteristic
647C22C2×#54711yescharacteristic
648C22C22×#13511yescharacteristic
649C22C2×#57511yescharacteristic
650C22C2×#57611yescharacteristic
651C22C2×#57711yescharacteristic
652C4—42no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—82no—
658C4—82no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4#269514yes—
665C4C2×#53811yescharacteristic
666C4C2×#53911yescharacteristic
667C4C2×#55211yescharacteristic
668C4C2×#56311yescharacteristic
669C2—44no—
670C2—84no—
671C2#3160518yesdirect product factor
672C2C2×#269511yescharacteristic
673C2C2×#325011yescharacteristic
674C2C2×#325711yescharacteristic
675C2C2×#329611yescharacteristic
676C2C2×#331911yescharacteristic
677C2C2×#333911yescharacteristic
678C2C2×#334811yescharacteristic
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×#2495C2semidirect
3. #31605C2direct (N × H)
4. C2×#2101C2semidirect
5. C2×#2088C2semidirect
6. C2×#2085C2semidirect
7. C2×#2077C2semidirect
8. C2×#2067C2semidirect
10. C2×#1949C2semidirect
11. C2×#1926C2semidirect
13. C2×#1862C2semidirect
14. C2×#1850C2semidirect
15. C2×#1808C2semidirect
16. C2×#1804C2semidirect
32. #2495C22semidirect
54. #2101C22semidirect
55. #2088C22semidirect
56. #2085C22semidirect
57. #2077C22semidirect
58. #2067C22semidirect
60. #1949C22semidirect
61. #1926C22semidirect
63. #1862C22semidirect
64. #1850C22semidirect
65. #1808C22semidirect
66. #1804C22semidirect
671. C2#31605direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24952
C2×#24202
#316052
C2×#21012
C2×#20882
C2×#20852
C2×#20772
C2×#20672
C2×#20342
C2×#19492
C2×#19262
C2×#18932
C2×#18622
C2×#18502
C2×#18082
C2×#18042