Finite Group Databasebeta

← #950067

Subgroups — #950067

Order 512.

2234 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17251C214yesmaximal, complemented
2#17057C212yesmaximal, complemented
3#16987C212yesmaximal, complemented
4#16707C212yesmaximal, complemented
5#16706C212yesmaximal, complemented
6#15958C214yesmaximal
7#15223C212yesmaximal, complemented
8#15221C212yesmaximal, complemented
9#15220C212yesmaximal, complemented
10#14266C212yesmaximal, complemented
11#14130C212yesmaximal, complemented
12#13907C212yesmaximal, complemented
13#13872C211yesmaximal, characteristic
14#13872C211yesmaximal, characteristic
15#13847C211yesmaximal, characteristic
16C22×C4×D4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C2×#389C2212yes—
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#386C2212yes—
24C2×#386C2212yes—
25C2×#385C2214yes—
26C2×#382C2212yes—
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×#380C2211yescharacteristic
30C2×#380C2211yescharacteristic
31C2×#380C2214yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4C2212yes—
34C2×C4×C22⋊C4C2212yes—
35C2×#376C2211yescharacteristic
36#1445C2212yes—
37#1445C2214yes—
38#1445C2214yes—
39#1445C2214yes—
40#1445C2214yes—
41#1444C2214yes—
42#1444C2214yes—
43#1443C2214yes—
44#1443C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1441C2212yes—
50#1441C2212yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1439C2211yescharacteristic
56#1439C2211yescharacteristic
57#1439C2211yescharacteristic
58#1439C2211yescharacteristic
59#1439C2212yes—
60#1439C2212yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1438C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C23×D4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2312yes—
80C2×C4×D4—42no—
81C2×C4×D4—42no—
82C2×C4×D4Q818yes—
83C22×C4⋊C4—28no—
84C22×C4⋊C4—42no—
85C22×C4⋊C4—44no—
86C22×C4⋊C4—42no—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2311yescharacteristic
110C22×C42C2312yes—
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113#389—28no—
114#389—44no—
115#389—44no—
116#389—44no—
117#386—44no—
118#386—44no—
119#385—48no—
120#382—44no—
121#380—28no—
122#380—48no—
123#380—44no—
124#380—42no—
125#380—42no—
126C4×C4⋊C4—42no—
127C4×C22⋊C4—44no—
128C4×C22⋊C4—44no—
129#376—42no—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2311yescharacteristic
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2311yescharacteristic
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C25C2411yescharacteristic
159C25C2411yescharacteristic
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—41no—
165C22×D4—41no—
166C22×D4C2×Q814yes—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—28no—
178C23×C4—42no—
179C23×C4—44no—
180C23×C4—42no—
181C23×C4—44no—
182C23×C4—44no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4C2×Q814yes—
186C23×C4C2×Q814yes—
187C23×C4C2412yes—
188C23×C4C2414yes—
189C23×C4C2414yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2414yes—
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C23×C4C2411yescharacteristic
198C23×C4C2414yes—
199C23×C4C2411yescharacteristic
200C4×D4—416no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—416no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—416no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4C2×Q814yes—
237C2×C4⋊C4#4118yes—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4C2×Q814yes—
267C2×C22⋊C4C2×Q814yes—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42C2×Q814yes—
277#95—28no—
278#95—28no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—41no—
284#95—41no—
285#95—42no—
286#95—42no—
287#95—42no—
288#95—42no—
289#95—42no—
290#95—42no—
291#95—44no—
292#95—42no—
293#95—42no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—21no—
309C24—21no—
310C24—22no—
311C24—21no—
312C24—21no—
313C24—21no—
314C24—21no—
315C24—21no—
316C24—21no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24C22×Q812yes—
322C24C22×Q812yes—
323C24H(5,𝔽2)12yes—
324C24H(5,𝔽2)12yes—
325C242−1+412yes—
326C242−1+412yes—
327C24C2511yescharacteristic
328C2×D4—28no—
329C2×D4—28no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—44no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—24no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—216no—
362C22×C4—216no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—48no—
413C22×C4—416no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—48no—
421C22×C4—88no—
422C22×C4—84no—
423C22×C4—84no—
424C22×C4—88no—
425C22×C4—84no—
426C22×C4—84no—
427C22×C4—88no—
428C22×C4C22×Q812yes—
429C22×C4C22×Q812yes—
430C22×C4C22×Q812yes—
431C22×C4C22×Q812yes—
432C22×C4C22×Q812yes—
433C22×C4C2×#4118yes—
434C22×C4C2×#4114yes—
435C22×C4H(5,𝔽2)12yes—
436C22×C4H(5,𝔽2)14yes—
437C22×C4H(5,𝔽2)12yes—
438C4⋊C4—216no—
439C4⋊C4—44no—
440C4⋊C4—48no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—88no—
448C4⋊C4—88no—
449C4⋊C4—88no—
450C4⋊C4—88no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—416no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—48no—
458C22⋊C4—84no—
459C22⋊C4—84no—
460C22⋊C4—88no—
461C42—28no—
462C42—48no—
463C42—48no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—42no—
489C23—42no—
490C23—41no—
491C23—41no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23#54414yes—
509C23#54414yes—
510C23C23×Q811yescharacteristic
511C23C22×#4112yes—
512C23C2×H(5,𝔽2)11yescharacteristic
513C23C2×H(5,𝔽2)12yes—
514C23C2×H(5,𝔽2)11yescharacteristic
515C23C2×H(5,𝔽2)11yescharacteristic
516C23C2×H(5,𝔽2)11yescharacteristic
517C23C2×2−1+411yescharacteristic
518C23C2×2−1+411yescharacteristic
519C23#58512yes—
520C23#58512yes—
521D4—416no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—44no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—416no—
552C2×C4—416no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—816no—
603C2×C4—816no—
604C2×C4#54018yes—
605C2×C4#54314yes—
606C2×C4#55114yes—
607C2×C4#55418yes—
608C2×C4#56014yes—
609C22—24no—
610C22—24no—
611C22—28no—
612C22—28no—
613C22—44no—
614C22—44no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—44no—
620C22—44no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—44no—
628C22—44no—
629C22—42no—
630C22—42no—
631C22C2×#54012yes—
632C22C2×#54311yescharacteristic
633C22C2×#54411yescharacteristic
634C22C2×#54411yescharacteristic
635C22C2×#55111yescharacteristic
636C22C2×#55412yes—
637C22#350412yes—
638C22#351312yes—
639C22#351312yes—
640C22#351612yes—
641C22#351612yes—
642C22C2×#56011yescharacteristic
643C22#353611yescharacteristic
644C22#353611yescharacteristic
645C22#353711yescharacteristic
646C22#353811yescharacteristic
647C22#354011yescharacteristic
648C22#354011yescharacteristic
649C22#354112yes—
650C22#354112yes—
651C22#354212yes—
652C22#354212yes—
653C22#355112yes—
654C4—42no—
655C4—48no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—82no—
660C4—84no—
661C4—82no—
662C4—84no—
663C4—82no—
664C4—84no—
665C4—82no—
666C4#274818yes—
667C2—44no—
668C2—44no—
669C2#3742611yescharacteristic
670C2#3747711yescharacteristic
671C2#3810012yes—
672C2#3810112yes—
673C2#4254112yes—
674C2#4257012yes—
675C2#4538011yescharacteristic
676C2#4542011yescharacteristic
677C2C2×#274811yescharacteristic
678C2#5310311yescharacteristic
679C2#5313711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17251C2semidirect
2. #17057C2semidirect
3. #16987C2semidirect
4. #16707C2semidirect
5. #16706C2semidirect
7. #15223C2semidirect
8. #15221C2semidirect
9. #15220C2semidirect
10. #14266C2semidirect
11. #14130C2semidirect
12. #13907C2semidirect

Maximal subgroups

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

subgroupindex
#172512
#170572
#169872
#167072
#167062
#159582
#152232
#152212
#152202
#142662
#141302
#139072
#138722
#138472