Finite Group Databasebeta

← #395912

Subgroups — #395912

Order 512.

1916 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#107C211yesmaximal, characteristic, complemented
2C2×#2067C212yesmaximal, complemented
3C2×#1893C211yesmaximal, characteristic
4C2×#1874C211yesmaximal, characteristic
5C2×#1850C211yesmaximal, characteristic, complemented
6#14137C211yesmaximal, characteristic
7#13907C211yesmaximal, characteristic, complemented
8#11232C212yesmaximal, complemented
9#11219C212yesmaximal, complemented
10#11123C212yesmaximal, complemented
11#10730C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×#121C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#107—28no—
17C22×#107—24no—
18C22×#107C2212yes—
19C22×#107C2212yes—
20C22×C4⋊C8C2212yes—
21C22×#102C2211yescharacteristic
22C22×#102C2212yes—
23C22×#102C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25C2×#390C2212yes—
26C2×#389C2212yes—
27C2×#386—21no—
28C2×#386—21no—
29C2×#380—21no—
30C2×#380C2211yescharacteristic
31C2×C4×C22⋊C4C2211yescharacteristic
32#2067—28no—
33#1893—24no—
34#1874—24no—
35#1850—24no—
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42#1441—22no—
43#1439—22no—
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
46C22⋊#95—21no—
47C23×D4—21no—
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C24×C4—21no—
51C24×C4C2311yescharacteristic
52C23×C8—22no—
53C23×C8—22no—
54C23×C8—24no—
55C23×C8—24no—
56C23×C8D414yes—
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C23×C8C2312yes—
60C2×#121—22no—
61C2×#121—22no—
62C2×#121—22no—
63C2×#121—22no—
64C2×#121—22no—
65C2×#121Q814yes—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—24no—
68C22×C4⋊C4—24no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—42no—
75C22×C4⋊C4—42no—
76C22×C4⋊C4D414yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—22no—
85C22×C22⋊C4—22no—
86C22×C22⋊C4D411yescharacteristic
87C22×C22⋊C4D411yescharacteristic
88C22×C42C2312yes—
89C2×#107—232no—
90C2×#107—24no—
91C2×#107—24no—
92C2×#107—24no—
93C2×#107—24no—
94C2×#107—24no—
95C2×#107D418yescomplemented
96C2×C4⋊C8—24no—
97C2×C4⋊C8—24no—
98C2×C4⋊C8—24no—
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—24no—
102C2×#102—24no—
103C2×#102—22no—
104C2×#102—22no—
105C2×#102—42no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109#390—24no—
110#390—42no—
111#389—24no—
112#389—42no—
113#386—44no—
114#386—44no—
115#380—22no—
116#380—22no—
117#380—44no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—21no—
125C2×#95—22no—
126C2×#95D412yes—
127#340—24no—
128#340—24no—
129#340—22no—
130#340—22no—
131#340—42no—
132#336—24no—
133C25—21no—
134C25C2×D411yescharacteristic
135C22×D4—24no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—22no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—22no—
142C22×D4—21no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—42no—
146C22×D4—42no—
147C22×D4—42no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4—42no—
151C22×D4C2×Q812yes—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—24no—
161C23×C4—22no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—82no—
170C23×C4—82no—
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4C2×Q812yes—
177C23×C4#4112yes—
178C23×C4#4112yes—
179C23×C4#4112yes—
180C23×C4C2411yescharacteristic
181C22×C8—216no—
182C22×C8—28no—
183C22×C8—28no—
184C22×C8—28no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—216no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—22no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8C2×D414yes—
201C22×C8C2×D414yes—
202C22×C8#4114yes—
203#121—28no—
204#121—28no—
205#121—44no—
206#121—44no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
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—44no—
232C2×C4⋊C4—48no—
233C2×C4⋊C4—48no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—82no—
241C2×C4⋊C4C2×D414yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D414yes—
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2×Q812yes—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—82no—
262C2×C22⋊C4—82no—
263C2×C22⋊C4D814yes—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270#107—216no—
271#107—216no—
272C4⋊C8—48no—
273#102—44no—
274#102—48no—
275#102—44no—
276C22⋊C8—44no—
277#95—24no—
278#95—44no—
279#95—44no—
280#95—42no—
281#95—44no—
282#95—44no—
283#95—82no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—41no—
290C24—41no—
291C24—42no—
292C24—42no—
293C24—41no—
294C24—41no—
295C24—41no—
296C24—41no—
297C24—42no—
298C24—42no—
299C24#12112yes—
300C24#12112yes—
301C24C22⋊Q812yes—
302C24C2×D812yes—
303C24#13512yes—
304C24AGL(1,ℤ/8)12yes—
305C24#13712yes—
306C24C22×D411yescharacteristic
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—24no—
310C2×D4—24no—
311C2×D4—48no—
312C2×D4—48no—
313C2×D4—48no—
314C2×D4—41no—
315C2×D4—42no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—42no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—84no—
329C2×D4—84no—
330C2×D4—84no—
331C2×D4—84no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
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—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—48no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—48no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—48no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—84no—
396C22×C4—88no—
397C22×C4—88no—
398C22×C4—88no—
399C22×C4—88no—
400C22×C4—84no—
401C22×C4—84no—
402C22×C4#12114yes—
403C22×C4#12112yes—
404C22×C4#12112yes—
405C22×C4C22⋊Q814yes—
406C22×C4C22⋊Q814yes—
407C22×C4C22⋊Q812yes—
408C22×C4#12614yes—
409C22×C4C2×D814yes—
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D412yes—
412C22×C4C22×D411yescharacteristic
413C22×C4C22×Q811yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4112yes—
416C22×C4C2×#4111yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4H(5,𝔽2)12yes—
419C22×C42−1+412yes—
420C2×C8—232no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—42no—
431C2×C8—41no—
432C2×C8—42no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8#12118yes—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—48no—
447C4⋊C4—48no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—88no—
458C4⋊C4#12014yes—
459C22⋊C4—28no—
460C22⋊C4—44no—
461C22⋊C4—48no—
462C22⋊C4—88no—
463C22⋊C4—88no—
464C22⋊C4—84no—
465C42—42no—
466C42—42no—
467C42—42no—
468C42—42no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—42no—
486C23—44no—
487C23—42no—
488C23—44no—
489C23—42no—
490C23—82no—
491C23—82no—
492C23—82no—
493C23—82no—
494C23—81no—
495C23—81no—
496C23—82no—
497C23—82no—
498C23—82no—
499C23—82no—
500C23—81no—
501C23—82no—
502C23—81no—
503C23—81no—
504C23—81no—
505C23—81no—
506C23—81no—
507C23—82no—
508C23#39614yes—
509C23#45914yes—
510C23#46414yes—
511C23#46614yes—
512C23#46814yes—
513C23#47414yes—
514C23#47814yes—
515C23C2×#12111yescharacteristic
516C23C2×#12111yescharacteristic
517C23C2×C22⋊Q811yescharacteristic
518C23D4212yes—
519C23#54712yes—
520C23C22×D811yescharacteristic
521C23C2×#13511yescharacteristic
522C23C2×AGL(1,ℤ/8)11yescharacteristic
523C23C2×#13711yescharacteristic
524C23#57612yes—
525C23#57812yes—
526D4—42no—
527D4—42no—
528D4—44no—
529D4—816no—
530D4—81no—
531D4—81no—
532D4—82no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—816no—
572C2×C4—816no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—82no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4#39614yes—
591C2×C4C2×#12011yescharacteristic
592C2×C4C2×#12111yescharacteristic
593C2×C4C2×#12112yes—
594C2×C4C2×C22⋊Q811yescharacteristic
595C2×C4C2×C22⋊Q811yescharacteristic
596C2×C4#52511yescharacteristic
597C2×C4#52512yes—
598C2×C4C2×#12611yescharacteristic
599C2×C4#53411yescharacteristic
600C2×C4#53611yescharacteristic
601C2×C4#53712yes—
602C2×C4#53912yes—
603C2×C4#54612yes—
604C2×C4#54812yes—
605C2×C4D4×Q812yes—
606C2×C4#55212yes—
607C2×C4#55312yes—
608C2×C4#55412yes—
609C8—28no—
610C8—81no—
611C8—81no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—28no—
619C22—28no—
620C22—28no—
621C22—28no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—84no—
626C22—84no—
627C22—161no—
628C22—161no—
629C22—161no—
630C22—161no—
631C22—162no—
632C22—162no—
633C22#206718yescomplemented
634C22C2×#39611yescharacteristic
635C22#241011yescharacteristic
636C22#241311yescharacteristic
637C22#241612yes—
638C22#244912yes—
639C22C2×#45911yescharacteristic
640C22C2×#46411yescharacteristic
641C22#304111yescharacteristic
642C22#304411yescharacteristic
643C22C2×#46611yescharacteristic
644C22C2×#46811yescharacteristic
645C22#306112yes—
646C22C2×#47411yescharacteristic
647C22C2×#47811yescharacteristic
648C22#308312yes—
649C22#326412yes—
650C22#327012yes—
651C22D4×D812yes—
652C22#328512yes—
653C22#333412yes—
654C22#333912yes—
655C22#334312yes—
656C22#334612yes—
657C4—24no—
658C4—44no—
659C4—44no—
660C4—84no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—162no—
665C4#240911yescharacteristic
666C4#241712yes—
667C4#248112yes—
668C4#257911yescharacteristic
669C4#261811yescharacteristic
670C4#284711yescharacteristic
671C2—28no—
672C2—162no—
673C2#1472911yescharacteristic
674C2C2×#206711yescharacteristic
675C2#2361111yescharacteristic
676C2#2435411yescharacteristic
677C2#2436111yescharacteristic
678C2#2457611yescharacteristic
679C2#2458311yescharacteristic
680C2#2478012yes—
681C2#2478712yes—
682C2#2633612yes—
683C2#2635012yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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
1. C23×#107C2semidirect
2. C2×#2067C2semidirect
5. C2×#1850C2semidirect
7. #13907C2semidirect
8. #11232C2semidirect
9. #11219C2semidirect
10. #11123C2semidirect
11. #10730C2semidirect
95. C2×#107D4semidirect
633. C22#2067semidirect

Maximal subgroups

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

subgroupindex
C23×#1072
C2×#20672
C2×#18932
C2×#18742
C2×#18502
#141372
#139072
#112322
#112192
#111232
#107302