Finite Group Databasebeta

← C2×#16821 #611591

Subgroups — C2×#16821 #611591

Order 512.

4426 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#398C211yesmaximal, characteristic, complemented
2C22×#397C211yesmaximal, characteristic
3C22×#396C212yesmaximal
4C22×#389C211yesmaximal, characteristic, complemented
5C22×#388C211yesmaximal, characteristic
6#16821C2116yesmaximal, direct product factor
7C2×#1442C212yesmaximal, complemented
8C2×#1441C211yesmaximal, characteristic, complemented
9C2×#1441C212yesmaximal, complemented
10C2×#1440C212yesmaximal
11C2×#1439C211yesmaximal, characteristic
12C2×C4⋊#95C211yesmaximal, characteristic, complemented
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2212yes—
15C23×C4⋊C4C2212yes—
16C23×C4⋊C4C2212yes—
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2212yes—
21C23×C22⋊C4C2212yes—
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24C23×C42C2211yescharacteristic
25C2×#398—28no—
26C2×#398C2218yescomplemented
27C2×#397—28no—
28C2×#397C2218yes—
29C2×#396—216no—
30C2×#396C22116yes—
31C2×#389—28no—
32C2×#389C2218yescomplemented
33C2×#388—28no—
34C2×#388C2218yes—
35C22×#95C2212yes—
36C22×#95C2211yescharacteristic
37C22×#95C2212yes—
38C22×#95C2212yes—
39C22×#95C2211yescharacteristic
40C22×#95C2212yes—
41C22×#95C2212yes—
42C22×#95C2211yescharacteristic
43C22×#95C2211yescharacteristic
44C22×#95C2212yes—
45C22×#95C2212yes—
46#1442C22116yescomplemented
47#1441C22116yescomplemented
48#1441C2218yescomplemented
49#1440C22116yes—
50#1439C2218yes—
51C4⋊#95C2218yescomplemented
52C26C2311yescharacteristic
53C24×C4C2312yes—
54C24×C4C2312yes—
55C24×C4C2311yescharacteristic
56C24×C4C2311yescharacteristic
57C24×C4C2311yescharacteristic
58C24×C4C2312yes—
59C24×C4C2311yescharacteristic
60C24×C4C2312yes—
61C24×C4C2311yescharacteristic
62C24×C4C2311yescharacteristic
63C22×C4⋊C4—24no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—28no—
66C22×C4⋊C4—28no—
67C22×C4⋊C4—28no—
68C22×C4⋊C4—28no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—24no—
71C22×C4⋊C4—24no—
72C22×C4⋊C4—28no—
73C22×C4⋊C4—28no—
74C22×C4⋊C4—24no—
75C22×C4⋊C4—28no—
76C22×C4⋊C4—44no—
77C22×C4⋊C4—44no—
78C22×C4⋊C4—44no—
79C22×C4⋊C4D418yes—
80C22×C4⋊C4C2318yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2318yes—
83C22×C4⋊C4C2318yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2314yes—
86C22×C22⋊C4—24no—
87C22×C22⋊C4—24no—
88C22×C22⋊C4—28no—
89C22×C22⋊C4—28no—
90C22×C22⋊C4—28no—
91C22×C22⋊C4—24no—
92C22×C22⋊C4—24no—
93C22×C22⋊C4—24no—
94C22×C22⋊C4—28no—
95C22×C22⋊C4—24no—
96C22×C22⋊C4—44no—
97C22×C22⋊C4—44no—
98C22×C22⋊C4—44no—
99C22×C22⋊C4D418yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2318yes—
102C22×C22⋊C4C2318yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C42—44no—
106C22×C42D418yes—
107C22×C42C2314yes—
108#398—232no—
109#397—232no—
110#396—264no—
111#389—232no—
112#388—232no—
113C2×#95—28no—
114C2×#95—28no—
115C2×#95—24no—
116C2×#95—24no—
117C2×#95—28no—
118C2×#95—24no—
119C2×#95—24no—
120C2×#95—28no—
121C2×#95—28no—
122C2×#95—44no—
123C2×#95—44no—
124C2×#95C2318yes—
125C2×#95C2314yes—
126C2×#95C2318yes—
127C2×#95C2314yes—
128C2×#95C2318yes—
129C2×#95C2318yes—
130C2×#95C2318yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2318yes—
134C2×#95C2318yes—
135C25—22no—
136C25—22no—
137C25—22no—
138C25—24no—
139C25—22no—
140C25—22no—
141C25—24no—
142C25—24no—
143C25—24no—
144C25—22no—
145C25C2×D414yes—
146C25C2412yes—
147C25C2411yescharacteristic
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—22no—
159C23×C4—24no—
160C23×C4—22no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—22no—
166C23×C4—24no—
167C23×C4—22no—
168C23×C4—24no—
169C23×C4—22no—
170C23×C4—24no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—22no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4C2×D418yes—
199C23×C4C2×D414yes—
200C23×C4C2×D414yes—
201C23×C4C2×D414yes—
202C23×C4C2×D414yes—
203C23×C4C2×D414yes—
204C23×C4C2×D414yes—
205C23×C4#4118yes—
206C23×C4#4114yes—
207C23×C4#4114yes—
208C23×C4C2414yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2414yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C2×C4⋊C4—232no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—216no—
224C2×C4⋊C4—216no—
225C2×C4⋊C4—216no—
226C2×C4⋊C4—216no—
227C2×C4⋊C4—216no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—216no—
230C2×C4⋊C4—216no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—48no—
233C2×C4⋊C4—48no—
234C2×C4⋊C4—48no—
235C2×C4⋊C4—432no—
236C2×C4⋊C4—432no—
237C2×C4⋊C4—416no—
238C2×C4⋊C4—432no—
239C2×C4⋊C4—416no—
240C2×C4⋊C4C2×D4116yes—
241C2×C22⋊C4—232no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—216no—
245C2×C22⋊C4—216no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—216no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—216no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—48no—
253C2×C22⋊C4—48no—
254C2×C22⋊C4—48no—
255C2×C22⋊C4—432no—
256C2×C22⋊C4—432no—
257C2×C22⋊C4—416no—
258C2×C22⋊C4—416no—
259C2×C22⋊C4C2×D4116yes—
260C2×C42—416no—
261C2×C42—48no—
262C2×C42C2×D4116yes—
263#95—28no—
264#95—28no—
265#95—216no—
266#95—216no—
267#95—216no—
268#95—28no—
269#95—28no—
270#95—216no—
271#95—216no—
272#95—48no—
273#95—48no—
274C24—28no—
275C24—216no—
276C24—28no—
277C24—216no—
278C24—28no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—24no—
283C24—22no—
284C24—22no—
285C24—24no—
286C24—24no—
287C24—24no—
288C24—22no—
289C24—44no—
290C24—44no—
291C24—44no—
292C24—44no—
293C24—44no—
294C24—48no—
295C24—44no—
296C24—48no—
297C24—48no—
298C24—48no—
299C24—48no—
300C24—44no—
301C24—48no—
302C24—48no—
303C24—44no—
304C24—48no—
305C24—48no—
306C24—48no—
307C24C22×D414yes—
308C24C22×D412yes—
309C24C22×D412yes—
310C24C22×D412yes—
311C24C2×#4112yes—
312C24C2×#4112yes—
313C24C2×#4114yes—
314C24H(5,𝔽2)14yes—
315C24H(5,𝔽2)12yes—
316C242−1+414yes—
317C242−1+414yes—
318C242−1+412yes—
319C24C2511yescharacteristic
320C22×C4—28no—
321C22×C4—28no—
322C22×C4—28no—
323C22×C4—28no—
324C22×C4—216no—
325C22×C4—216no—
326C22×C4—216no—
327C22×C4—216no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—28no—
333C22×C4—216no—
334C22×C4—28no—
335C22×C4—28no—
336C22×C4—28no—
337C22×C4—216no—
338C22×C4—216no—
339C22×C4—28no—
340C22×C4—216no—
341C22×C4—28no—
342C22×C4—216no—
343C22×C4—216no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—44no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—44no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—48no—
411C22×C4—44no—
412C22×C4—48no—
413C22×C4—44no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4#121116yes—
429C22×C4#121116yes—
430C22×C4C22×D414yes—
431C22×C4C22×D418yes—
432C22×C4C22×D414yes—
433C22×C4C22×D414yes—
434C22×C4C22×D414yes—
435C22×C4C22×D414yes—
436C22×C4C22×D414yes—
437C22×C4C2×#4118yes—
438C22×C4C2×#4114yes—
439C22×C4C2×#4114yes—
440C4⋊C4—232no—
441C4⋊C4—432no—
442C4⋊C4—432no—
443C4⋊C4—416no—
444C4⋊C4—432no—
445C4⋊C4—416no—
446C22⋊C4—232no—
447C22⋊C4—432no—
448C22⋊C4—432no—
449C22⋊C4—416no—
450C22⋊C4—416no—
451C42—416no—
452C23—24no—
453C23—28no—
454C23—24no—
455C23—28no—
456C23—24no—
457C23—416no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—44no—
465C23—432no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—432no—
471C23—42no—
472C23—44no—
473C23—44no—
474C23—42no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—416no—
479C23—416no—
480C23—88no—
481C23—88no—
482C23—816no—
483C23—816no—
484C23—88no—
485C23—88no—
486C23C2×#12114yes—
487C23C2×#12114yes—
488C23#53214yes—
489C23#53414yes—
490C23#53614yes—
491C23#53618yes—
492C23#53614yes—
493C23#53718yes—
494C23#53714yes—
495C23#53918yes—
496C23#53918yes—
497C23#54118yes—
498C23#54118yes—
499C23D4214yes—
500C23#54618yes—
501C23#54714yes—
502C23#54714yes—
503C23#54818yes—
504C23#55318yes—
505C23#55514yes—
506C23#55618yes—
507C23#56218yes—
508C23#56318yes—
509C23C23×D411yescharacteristic
510C23C23×D411yescharacteristic
511C23C23×D411yescharacteristic
512C23C22×#4111yescharacteristic
513C23C22×#4111yescharacteristic
514C23C22×#4112yes—
515C23C2×H(5,𝔽2)12yes—
516C23C2×H(5,𝔽2)11yescharacteristic
517C23C2×2−1+412yes—
518C23C2×2−1+412yes—
519C23C2×2−1+411yescharacteristic
520C2×C4—232no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—232no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—432no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—416no—
558C2×C4—432no—
559C2×C4—416no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—88no—
585C2×C4—816no—
586C2×C4—816no—
587C2×C4—88no—
588C2×C4C2×#12118yes—
589C2×C4C2×#12118yes—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—48no—
594C22—48no—
595C22—84no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—832no—
600C22—84no—
601C22—82no—
602C22#253718yes—
603C22#254318yes—
604C22#254518yes—
605C22#261818yes—
606C22#262118yes—
607C22#266418yes—
608C22#2681116yes—
609C22#2689116yes—
610C22#2725116yes—
611C22#2783116yes—
612C22#281418yes—
613C22C22×#12111yescharacteristic
614C22C22×#12111yescharacteristic
615C22C2×#53211yescharacteristic
616C22C2×#53411yescharacteristic
617C22C2×#53611yescharacteristic
618C22C2×#53611yescharacteristic
619C22C2×#53612yes—
620C22C2×#53711yescharacteristic
621C22C2×#53712yes—
622C22C2×#53912yes—
623C22C2×#53912yes—
624C22C2×#54112yes—
625C22C2×#54112yes—
626C22C2×D4211yescharacteristic
627C22C2×#54612yes—
628C22C2×#54711yescharacteristic
629C22C2×#54711yescharacteristic
630C22C2×#54812yes—
631C22C2×#55312yes—
632C22C2×#55511yescharacteristic
633C22C2×#55612yes—
634C22C2×#56212yes—
635C22C2×#56312yes—
636C4—28no—
637C4—28no—
638C4—44no—
639C4—44no—
640C4—48no—
641C4—48no—
642C4—84no—
643C4—82no—
644C4—84no—
645C4—82no—
646C2—84no—
647C2#16821116yesdirect product factor
648C2C2×#261811yescharacteristic
649C2C2×#266411yescharacteristic
650C2C2×#262111yescharacteristic
651C2C2×#254311yescharacteristic
652C2C2×#254511yescharacteristic
653C2C2×#253711yescharacteristic
654C2C2×#281411yescharacteristic
655C2C2×#268112yes—
656C2C2×#268912yes—
657C2C2×#272512yes—
658C2C2×#278312yes—
subgroupG/H
center, socleC25C24
commutator, Frattini, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#398C2semidirect
4. C22×#389C2semidirect
6. #16821C2direct (N × H)
7. C2×#1442C2semidirect
8. C2×#1441C2semidirect
9. C2×#1441C2semidirect
12. C2×C4⋊#95C2semidirect
26. C2×#398C22semidirect
32. C2×#389C22semidirect
46. #1442C22semidirect
47. #1441C22semidirect
48. #1441C22semidirect
51. C4⋊#95C22semidirect
647. C2#16821direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#3982
C22×#3972
C22×#3962
C22×#3892
C22×#3882
#168212
C2×#14422
C2×#14412
C2×#14402
C2×#14392
C2×C4⋊#952