Finite Group Databasebeta

← #412401

Subgroups — #412401

Order 512.

1880 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#466C211yesmaximal, characteristic
2C2×#2034C212yesmaximal, complemented
3C2×#1970C211yesmaximal, characteristic, complemented
4C2×#1970C211yesmaximal, characteristic, complemented
5C2×#1846C211yesmaximal, characteristic, complemented
6C2×#1808C211yesmaximal, characteristic, complemented
7C2×C8×C22⋊C4C211yesmaximal, characteristic, complemented
8#15219C212yesmaximal, complemented
9#11121C212yesmaximal, complemented
10#11051C212yesmaximal, complemented
11#10862C211yesmaximal, characteristic, complemented
12C23×D8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#121C2212yes—
15C2×#466—24no—
16C2×#466—24no—
17C2×#466—24no—
18C22×#107C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#106C2212yes—
22C22×#102C2212yes—
23C22×#102C2212yes—
24C22×#102C2212yes—
25C22×C22⋊C8C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#382C2212yes—
31C2×#380C2212yes—
32C2×C4×C22⋊C4C2211yescharacteristic
33#2034—28no—
34#1970—24no—
35#1970—24no—
36#1846—24no—
37#1808—24no—
38C8×C22⋊C4—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2211yescharacteristic
43#1440—24no—
44#1438—24no—
45C4⋊#95C2212yes—
46C23×D4—22no—
47C23×D4C2312yes—
48C24×C4C2311yescharacteristic
49C22×D8—22no—
50C22×D8—22no—
51C22×D8—22no—
52C22×D8—22no—
53C22×D8—22no—
54C22×D8—22no—
55C22×D8D414yes—
56C23×C8—22no—
57C23×C8—22no—
58C23×C8—22no—
59C23×C8—22no—
60C23×C8—22no—
61C23×C8—22no—
62C23×C8D414yes—
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C2×#121—24no—
68C2×#121—24no—
69C2×#121—24no—
70C2×#121—24no—
71C2×#121—42no—
72C2×#121—42no—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—24no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—24no—
80C22×C22⋊C4D412yes—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83#466—232no—
84C2×#107—22no—
85C2×#107—22no—
86C2×#107—22no—
87C2×#107—22no—
88C2×#107—22no—
89C2×#107—22no—
90C2×#107—22no—
91C2×#107—22no—
92C2×#107D414yes—
93C2×#106—24no—
94C2×#106—24no—
95C2×#106—42no—
96C2×#102—24no—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—24no—
102C2×#102—24no—
103C2×#102—24no—
104C2×#102—42no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8D414yes—
108C2×C4×C8—22no—
109C2×C4×C8—22no—
110C2×C4×C8—22no—
111C2×C4×C8—22no—
112C2×C4×C8—22no—
113C2×C4×C8D414yescomplemented
114#390—24no—
115#390—24no—
116#390—24no—
117#390—42no—
118#382—24no—
119#382—42no—
120#380—24no—
121#380—24no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C2×#95—24no—
125C2×#95—22no—
126C2×#95—24no—
127C2×#95D412yes—
128#340—24no—
129#340—24no—
130#336—22no—
131#336—22no—
132C25—22no—
133C25C2×D411yescharacteristic
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—44no—
148C22×D4—44no—
149C22×D4—44no—
150C22×D4—48no—
151C22×D4—84no—
152C22×D4C2×D414yes—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4C2×D412yes—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D412yes—
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4C2411yescharacteristic
174C2×D8—28no—
175C2×D8—28no—
176C2×D8—28no—
177C2×D8—44no—
178C2×D8—44no—
179C2×D8—44no—
180C2×D8—44no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—28no—
198C22×C8—28no—
199C22×C8—28no—
200C22×C8—44no—
201C22×C8—44no—
202C22×C8—44no—
203C22×C8—44no—
204C22×C8C2×D412yes—
205C22×C8C2×D412yes—
206C22×C8C2×D412yes—
207C22×C8C2×D412yes—
208C22×C8C2×D412yes—
209C22×C8C2×D412yes—
210C22×C8C2×D412yes—
211C22×C8C2×D412yes—
212C22×C8#4112yes—
213C22×C8#4112yes—
214C22×C8#4112yes—
215C22×C8#4112yes—
216#121—216no—
217#121—48no—
218#121—48no—
219#121—48no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4C2×D414yes—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—48no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—48no—
247C2×C22⋊C4—48no—
248C2×C22⋊C4—84no—
249C2×C22⋊C4—84no—
250C2×C22⋊C4—84no—
251C2×C22⋊C4—84no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42C2×D412yes—
263C2×C42#4112yes—
264#107—28no—
265#107—44no—
266#107—44no—
267#106—48no—
268#102—216no—
269#102—48no—
270#102—48no—
271C22⋊C8—28no—
272C4×C8—28no—
273C4×C8—44no—
274#95—24no—
275#95—48no—
276#95—44no—
277#95—84no—
278C24—21no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—21no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—24no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24—42no—
294C24—44no—
295C24—44no—
296C24—44no—
297C24—44no—
298C24#12712yes—
299C24#13514yes—
300C24C22×D411yescharacteristic
301C2×D4—24no—
302C2×D4—24no—
303C2×D4—24no—
304C2×D4—24no—
305C2×D4—24no—
306C2×D4—24no—
307C2×D4—416no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—816no—
331C2×D4—816no—
332C2×D4—88no—
333C2×D4—88no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4#12114yes—
396C22×C4#12712yes—
397C22×C4#13514yes—
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4H(5,𝔽2)11yescharacteristic
409C22×C4H(5,𝔽2)11yescharacteristic
410C22×C42−1+411yescharacteristic
411C22×C42−1+411yescharacteristic
412D8—416no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—416no—
434C2×C8#12014yes—
435C2×C8#12114yes—
436C2×C8#12114yes—
437C2×C8#12114yes—
438C2×C8#12114yes—
439C2×C8#12314yes—
440C2×C8#12314yes—
441C2×C8#12414yes—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C22⋊C4—216no—
455C22⋊C4—416no—
456C22⋊C4—816no—
457C22⋊C4—816no—
458C22⋊C4—88no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—41no—
464C42—41no—
465C42—41no—
466C42—41no—
467C23—24no—
468C23—24no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—42no—
486C23—42no—
487C23—41no—
488C23—42no—
489C23—44no—
490C23—44no—
491C23—48no—
492C23—48no—
493C23—44no—
494C23—84no—
495C23—84no—
496C23—84no—
497C23—84no—
498C23—82no—
499C23—82no—
500C23—82no—
501C23—82no—
502C23—82no—
503C23—84no—
504C23—84no—
505C23—82no—
506C23—82no—
507C23—82no—
508C23—84no—
509C23—84no—
510C23—84no—
511C23—84no—
512C23#49514yes—
513C23C2×#12111yescharacteristic
514C23C2×#12711yescharacteristic
515C23#53211yescharacteristic
516C23D4211yescharacteristic
517C23D4211yescharacteristic
518C23#54711yescharacteristic
519C23#54711yescharacteristic
520C23C2×#13512yes—
521C23C2×#13512yes—
522C23#57612yes—
523C23#57612yes—
524D4—44no—
525D4—44no—
526D4—44no—
527D4—44no—
528D4—832no—
529D4—82no—
530D4—82no—
531D4—82no—
532D4—82no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—48no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—84no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—84no—
573C2×C4—82no—
574C2×C4—84no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4C2×#12011yescharacteristic
578C2×C4C2×#12111yescharacteristic
579C2×C4C2×#12111yescharacteristic
580C2×C4C2×#12111yescharacteristic
581C2×C4C2×#12111yescharacteristic
582C2×C4C2×#12311yescharacteristic
583C2×C4C2×#12311yescharacteristic
584C2×C4C2×#12411yescharacteristic
585C2×C4#53211yescharacteristic
586C2×C4#53211yescharacteristic
587C2×C4#53511yescharacteristic
588C2×C4#53611yescharacteristic
589C2×C4#53711yescharacteristic
590C2×C4#53711yescharacteristic
591C2×C4#54211yescharacteristic
592C2×C4#54211yescharacteristic
593C2×C4D4211yescharacteristic
594C2×C4D4211yescharacteristic
595C2×C4#54711yescharacteristic
596C2×C4#54711yescharacteristic
597C2×C4#54711yescharacteristic
598C2×C4#54711yescharacteristic
599C2×C4#54811yescharacteristic
600C2×C4#54811yescharacteristic
601C2×C4#55311yescharacteristic
602C2×C4#55311yescharacteristic
603C2×C4#55511yescharacteristic
604C2×C4#55511yescharacteristic
605C8—24no—
606C8—42no—
607C8—42no—
608C8#39418yes—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—42no—
613C22—42no—
614C22—44no—
615C22—44no—
616C22—42no—
617C22—42no—
618C22—44no—
619C22—44no—
620C22—84no—
621C22—88no—
622C22—84no—
623C22—84no—
624C22—88no—
625C22—162no—
626C22—162no—
627C22—164no—
628C22—162no—
629C22—162no—
630C22—164no—
631C22#241611yescharacteristic
632C22#242211yescharacteristic
633C22#243311yescharacteristic
634C22#249111yescharacteristic
635C22#250111yescharacteristic
636C22#253711yescharacteristic
637C22#285611yescharacteristic
638C22#305012yes—
639C22#305812yes—
640C22C2×#49511yescharacteristic
641C22#314711yescharacteristic
642C22#315511yescharacteristic
643C22#315511yescharacteristic
644C22#316812yes—
645C22#317612yes—
646C22#328012yes—
647C22#328012yes—
648C22#328012yes—
649C22#328712yes—
650C22#333912yes—
651C22#333912yes—
652C22#333912yes—
653C22#334112yes—
654C4—42no—
655C4—44no—
656C4—44no—
657C4—84no—
658C4—162no—
659C4—162no—
660C4C2×#39411yescharacteristic
661C4#241611yescharacteristic
662C4#242211yescharacteristic
663C4#243211yescharacteristic
664C4#249111yescharacteristic
665C4#250011yescharacteristic
666C4#253711yescharacteristic
667C4#262011yescharacteristic
668C2—44no—
669C2—164no—
670C2#1513011yescharacteristic
671C2#2477812yes—
672C2#2503212yes—
673C2#2577711yescharacteristic
674C2#2578111yescharacteristic
675C2#2779012yes—
676C2#2792811yescharacteristic
677C2#2793211yescharacteristic
678C2#2822512yes—
679C2#3317311yescharacteristic
680C2#3317911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#466C2
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2034C2semidirect
3. C2×#1970C2semidirect
4. C2×#1970C2semidirect
5. C2×#1846C2semidirect
6. C2×#1808C2semidirect
7. C2×C8×C22⋊C4C2semidirect
8. #15219C2semidirect
9. #11121C2semidirect
10. #11051C2semidirect
11. #10862C2semidirect
113. C2×C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
C22×#4662
C2×#20342
C2×#19702
C2×#18462
C2×#18082
C2×C8×C22⋊C42
#152192
#111212
#110512
#108622