Finite Group Databasebeta

← #763435

Subgroups — #763435

Order 512.

3352 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2454C212yesmaximal, complemented
2C2×#2324C212yesmaximal, complemented
3#15776C214yesmaximal, complemented
4#15452C214yesmaximal, complemented
5#15376C214yesmaximal, complemented
6#15345C214yesmaximal, complemented
7#15216C211yesmaximal, characteristic, complemented
8#15126C212yesmaximal, complemented
9#15124C211yesmaximal, characteristic
10#14133C212yesmaximal, complemented
11#14025C212yesmaximal, complemented
12#14022C212yesmaximal, complemented
13#13844C211yesmaximal, characteristic, complemented
14C22×#127C2211yescharacteristic
15C22×#125C2211yescharacteristic
16C22×#121C2212yes—
17C22×C4×D4C2212yes—
18C22×#117C2211yescharacteristic
19#2454—216no—
20C2×#399C2214yes—
21C2×#396C2214yes—
22C2×#394C2214yes—
23C2×#392C2214yes—
24#2324—216no—
25C2×#390C2214yes—
26C2×#390C2212yes—
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#388C2214yes—
33C2×#388C2214yes—
34C2×#387C2214yes—
35C2×#386C2214yes—
36C2×#386C2214yes—
37C2×#386C2214yes—
38C2×#385C2214yes—
39C2×#385C2214yes—
40C2×#380C2214yes—
41C2×#380C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46#1444C2212yes—
47#1443C2214yes—
48#1442C2214yes—
49#1442C2212yes—
50#1442C2214yes—
51#1440C2214yes—
52#1440C2214yes—
53#1439C2212yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2212yes—
57#1439C2214yes—
58#1438C2214yes—
59#1438C2214yes—
60#1438C2214yes—
61#1438C2214yes—
62#1437C2214yes—
63#1436C2214yes—
64C4⋊#95C2211yescharacteristic
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2211yescharacteristic
68C4×#95C2212yes—
69C23×D4C2312yes—
70C23×D4C2314yes—
71C23×D4C2314yes—
72C24×C4C2311yescharacteristic
73C24×C4C2314yes—
74C2×#127—28no—
75C2×#127—42no—
76C2×#125—28no—
77C2×#125—42no—
78C2×#121—28no—
79C2×#121—28no—
80C2×#121—28no—
81C2×C4×D4—28no—
82C2×C4×D4—28no—
83C2×C4×D4—44no—
84C2×#117—28no—
85C2×#117—42no—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2311yescharacteristic
100C22×C22⋊C4C2311yescharacteristic
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2314yes—
115#399—216no—
116#396—216no—
117#394—48no—
118#392—48no—
119#390—28no—
120#390—216no—
121#390—28no—
122#390—28no—
123#390—28no—
124#389—28no—
125#389D4116yescomplemented
126#388—216no—
127#388—48no—
128#387—48no—
129#386—216no—
130#386—48no—
131#386—48no—
132#385—216no—
133#385—216no—
134#380—28no—
135#380—216no—
136C4×C4⋊C4—28no—
137C4×C4⋊C4—28no—
138C4×C22⋊C4—24no—
139C4×C22⋊C4—42no—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2311yescharacteristic
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2311yescharacteristic
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C25C2414yes—
160C25C2411yescharacteristic
161C22×D4—28no—
162C22×D4—28no—
163C22×D4—28no—
164C22×D4—28no—
165C22×D4—24no—
166C22×D4—24no—
167C22×D4—28no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—28no—
172C22×D4—24no—
173C22×D4—44no—
174C22×D4—44no—
175C22×D4—44no—
176C22×D4—44no—
177C22×D4—44no—
178C22×D4—44no—
179C22×D4—44no—
180C22×D4—44no—
181C22×D4#4118yes—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—28no—
187C23×C4—28no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4#4114yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2414yes—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205#127—416no—
206#125—416no—
207#121—264no—
208C4×D4—432no—
209#117—416no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4C2×D418yes—
235C2×C4⋊C4C2×D418yes—
236C2×C4⋊C4C2×D4116yes—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—41no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—28no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42C2×D418yes—
293#95—24no—
294#95—28no—
295#95—24no—
296#95—28no—
297#95—28no—
298#95—22no—
299#95—22no—
300#95—24no—
301#95—24no—
302#95—28no—
303#95—24no—
304#95—28no—
305#95—28no—
306#95—42no—
307#95—42no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95C2×D4116yes—
312C24—24no—
313C24—24no—
314C24—22no—
315C24—24no—
316C24—22no—
317C24—24no—
318C24—24no—
319C24—22no—
320C24—24no—
321C24—22no—
322C24—21no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24—21no—
327C24—24no—
328C24—22no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—22no—
333C24—24no—
334C24C2×#4118yes—
335C24C2×#4112yes—
336C24C2511yescharacteristic
337C2×D4—232no—
338C2×D4—216no—
339C2×D4—216no—
340C2×D4—216no—
341C2×D4—416no—
342C2×D4—416no—
343C2×D4—416no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—48no—
347C2×D4—48no—
348C2×D4—88no—
349C2×D4—88no—
350C2×D4—88no—
351C2×D4—88no—
352C2×D4—88no—
353C2×D4—88no—
354C2×D4—88no—
355C2×D4—88no—
356C2×D4—88no—
357C2×D4—88no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—216no—
399C22×C4—28no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—416no—
422C22×C4—416no—
423C22×C4—44no—
424C22×C4—48no—
425C22×C4—44no—
426C22×C4—88no—
427C22×C4—88no—
428C22×C4—88no—
429C22×C4—88no—
430C22×C4—88no—
431C22×C4C22×D418yes—
432C22×C4C22×D414yes—
433C22×C4C22×D418yes—
434C22×C4C22×D414yes—
435C22×C4C22×D414yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4114yes—
439C22×C4H(5,𝔽2)14yes—
440C22×C4H(5,𝔽2)14yes—
441C22×C4H(5,𝔽2)12yes—
442C22×C42−1+412yes—
443C4⋊C4—216no—
444C4⋊C4—216no—
445C4⋊C4—416no—
446C4⋊C4—48no—
447C4⋊C4—48no—
448C4⋊C4—416no—
449C4⋊C4—48no—
450C4⋊C4—48no—
451C4⋊C4—88no—
452C22⋊C4—232no—
453C22⋊C4—416no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—416no—
457C22⋊C4—48no—
458C22⋊C4—416no—
459C22⋊C4—48no—
460C22⋊C4—88no—
461C22⋊C4—88no—
462C22⋊C4—88no—
463C22⋊C4—88no—
464C22⋊C4—88no—
465C22⋊C4—88no—
466C22⋊C4—88no—
467C42—216no—
468C42—48no—
469C42—44no—
470C42—44no—
471C42—88no—
472C23—28no—
473C23—24no—
474C23—22no—
475C23—28no—
476C23—28no—
477C23—24no—
478C23—22no—
479C23—22no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—28no—
484C23—44no—
485C23—44no—
486C23—41no—
487C23—44no—
488C23—42no—
489C23—44no—
490C23—41no—
491C23—44no—
492C23—42no—
493C23—44no—
494C23—44no—
495C23—42no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—42no—
502C23—44no—
503C23—44no—
504C23—42no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—42no—
509C23—44no—
510C23—42no—
511C23—44no—
512C23—42no—
513C23—44no—
514C23—42no—
515C23—42no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—42no—
521C23—41no—
522C23—44no—
523C23—44no—
524C23—42no—
525C23—44no—
526C23—44no—
527C23—41no—
528C23C23×D412yes—
529C23C22×#4111yescharacteristic
530C23C22×#4112yes—
531C23C2×H(5,𝔽2)11yescharacteristic
532C23C2×H(5,𝔽2)12yes—
533C23C2×H(5,𝔽2)12yes—
534C23C2×2−1+411yescharacteristic
535C23#58512yes—
536C23#58512yes—
537D4—432no—
538D4—832no—
539D4—832no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—416no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—832no—
610C2×C4#52518yes—
611C2×C4#53914yes—
612C2×C4D4214yes—
613C2×C4#54818yes—
614C2×C4#55114yes—
615C22—48no—
616C22—48no—
617C22—48no—
618C22—42no—
619C22—48no—
620C22—42no—
621C22—48no—
622C22—44no—
623C22—42no—
624C22—44no—
625C22—48no—
626C22—48no—
627C22—82no—
628C22—82no—
629C22—84no—
630C22—84no—
631C22—82no—
632C22—84no—
633C22—84no—
634C22—84no—
635C22—84no—
636C22—84no—
637C22—82no—
638C22—84no—
639C22C2×#52512yes—
640C22C2×#53911yescharacteristic
641C22C2×D4211yescharacteristic
642C22C2×#54812yes—
643C22C2×#55111yescharacteristic
644C22#348712yes—
645C22#348812yes—
646C22#349412yes—
647C22#349612yes—
648C22#349812yes—
649C22#350811yescharacteristic
650C22#351211yescharacteristic
651C22#351312yes—
652C22#351312yes—
653C22#351912yes—
654C22#352112yes—
655C22#354212yes—
656C22#354612yes—
657C22#354811yescharacteristic
658C22#355412yes—
659C22#355711yescharacteristic
660C4—44no—
661C4—44no—
662C4—48no—
663C4—44no—
664C4—42no—
665C4—44no—
666C4—48no—
667C4—82no—
668C4—82no—
669C4—84no—
670C4#246318yes—
671C2—88no—
672C2—82no—
673C2C2×#246311yescharacteristic
674C2#3552512yes—
675C2#3553812yes—
676C2#3728611yescharacteristic
677C2#3731811yescharacteristic
678C2#4488612yes—
679C2#4503412yes—
680C2#4506112yes—
681C2#4511112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#15124C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2454C2semidirect
2. C2×#2324C2semidirect
3. #15776C2semidirect
4. #15452C2semidirect
5. #15376C2semidirect
6. #15345C2semidirect
7. #15216C2semidirect
8. #15126C2semidirect
10. #14133C2semidirect
11. #14025C2semidirect
12. #14022C2semidirect
13. #13844C2semidirect
125. #389D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24542
C2×#23242
#157762
#154522
#153762
#153452
#152162
#151262
#151242
#141332
#140252
#140222
#138442