Finite Group Databasebeta

← #722633

Subgroups — #722633

Order 512.

3030 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#399C211yesmaximal, characteristic
2C2×#2345C211yesmaximal, characteristic, complemented
3#16531C214yesmaximal, complemented
4#15653C214yesmaximal, complemented
5#15649C214yesmaximal, complemented
6#14821C212yesmaximal, complemented
7#14793C214yesmaximal, complemented
8#14734C212yesmaximal
9#14338C211yesmaximal, characteristic
10#14149C211yesmaximal, characteristic, complemented
11#14039C212yesmaximal, complemented
12#13991C212yesmaximal
13#13965C211yesmaximal, characteristic, complemented
14#13924C211yesmaximal, characteristic, complemented
15#13775C211yesmaximal, characteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2212yes—
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2212yes—
20C2×#399—28no—
21C2×#399C2214yes—
22C2×#399C2212yes—
23C2×#399C2212yes—
24C2×#399C2214yes—
25C2×#398C2214yescomplemented
26#2345—28no—
27C2×#389C2212yescomplemented
28C2×#388C2214yes—
29C2×#387C2212yescomplemented
30C2×#387C2212yescomplemented
31C2×#387C2214yescomplemented
32C2×#385C2212yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#380C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×#380C2212yes—
38C2×#379C2211yescharacteristic
39C2×#379C2212yes—
40C2×#379C2211yescharacteristic
41C2×C4×C4⋊C4C2212yescomplemented
42C2×C4×C22⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2212yes—
44C2×C4×C22⋊C4C2211yescharacteristic
45C22×#95C2211yescharacteristic
46#1445C2212yescomplemented
47#1445C2212yescomplemented
48#1444C2214yescomplemented
49#1442C2214yescomplemented
50#1441C2214yescomplemented
51#1441C2214yescomplemented
52#1441C2214yescomplemented
53#1441C2214yescomplemented
54#1441C2214yescomplemented
55#1440C2214yes—
56#1439C2212yes—
57#1439C2212yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1437C2214yes—
64#1437C2214yes—
65#1437C2214yes—
66#1436C2214yescomplemented
67#1436C2214yescomplemented
68C4⋊#95C2212yescomplemented
69C4⋊#95C2214yescomplemented
70C4⋊#95C2212yescomplemented
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2211yescharacteristic
76C22⋊#95C2211yescharacteristic
77C26C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C22×C4⋊C4—24no—
83C22×C4⋊C4—24no—
84C22×C4⋊C4—28no—
85C22×C4⋊C4—48no—
86C22×C4⋊C4D418yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C22⋊C4—28no—
100C22×C22⋊C4—24no—
101C22×C22⋊C4—24no—
102C22×C22⋊C4—24no—
103C22×C22⋊C4—48no—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C42C2312yes—
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122#399—28no—
123#399—216no—
124#399—28no—
125#399—48no—
126#398—48no—
127#389—28no—
128#388—48no—
129#387—28no—
130#387—28no—
131#387—48no—
132#385—28no—
133#385—44no—
134#385—44no—
135#380—28no—
136#380—24no—
137#380—24no—
138#379—24no—
139#379—44no—
140#379Q818yes—
141C4×C4⋊C4—28no—
142C4×C22⋊C4—24no—
143C4×C22⋊C4—28no—
144C4×C22⋊C4—24no—
145C2×#95—24no—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2311yescharacteristic
150C2×#95C2311yescharacteristic
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C25—22no—
171C25—22no—
172C25—22no—
173C25—22no—
174C25—22no—
175C25—22no—
176C25—44no—
177C25—44no—
178C25C2×Q814yes—
179C25C2412yes—
180C25C2411yescharacteristic
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—28no—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4C2×D418yes—
196C23×C4C2×D414yes—
197C23×C4#4114yes—
198C23×C4#4114yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C2×C4⋊C4—232no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—432no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4C2×D414yes—
244C2×C4⋊C4C2×D418yes—
245C2×C4⋊C4C2×D414yes—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—432no—
277C2×C22⋊C4—416no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4C2×Q814yes—
287C2×C22⋊C4C2×Q814yes—
288C2×C22⋊C4C2×Q8116yes—
289C2×C22⋊C4#4114yes—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—44no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42—44no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—22no—
306#95—22no—
307#95—24no—
308#95—24no—
309#95—28no—
310#95—28no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322C24—24no—
323C24—24no—
324C24—28no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—24no—
330C24—24no—
331C24—22no—
332C24—22no—
333C24—24no—
334C24—24no—
335C24—28no—
336C24—28no—
337C24—416no—
338C24—48no—
339C24—48no—
340C24—416no—
341C24—44no—
342C24—44no—
343C24—44no—
344C24—44no—
345C24—84no—
346C24—84no—
347C24—84no—
348C24—84no—
349C24—88no—
350C24—88no—
351C24C22×D412yes—
352C24C22×Q814yes—
353C24C22×Q812yes—
354C24C2×#4112yes—
355C24C2×#4112yes—
356C24C2×#4112yes—
357C24H(5,𝔽2)12yes—
358C242−1+412yes—
359C24C2511yescharacteristic
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—216no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—216no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—416no—
423C22×C4—416no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—44no—
428C22×C4—88no—
429C22×C4—88no—
430C22×C4—84no—
431C22×C4—84no—
432C22×C4#123116yes—
433C22×C4C22×D414yes—
434C22×C4C22×D414yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×Q814yes—
438C22×C4C22×Q814yes—
439C22×C4C2×#4114yes—
440C22×C4C2×#4112yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4H(5,𝔽2)18yes—
445C4⋊C4—28no—
446C4⋊C4—216no—
447C4⋊C4—28no—
448C4⋊C4—416no—
449C4⋊C4—416no—
450C4⋊C4—416no—
451C4⋊C4—48no—
452C4⋊C4—84no—
453C4⋊C4—84no—
454C4⋊C4—84no—
455C4⋊C4—88no—
456C4⋊C4—88no—
457C22⋊C4—28no—
458C22⋊C4—28no—
459C22⋊C4—28no—
460C22⋊C4—416no—
461C22⋊C4—416no—
462C22⋊C4—416no—
463C22⋊C4—48no—
464C22⋊C4—44no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—88no—
469C22⋊C4—88no—
470C22⋊C4—88no—
471C42—48no—
472C42—84no—
473C42—84no—
474C42—84no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—28no—
480C23—24no—
481C23—24no—
482C23—28no—
483C23—28no—
484C23—28no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—28no—
489C23—28no—
490C23—432no—
491C23—42no—
492C23—416no—
493C23—44no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—44no—
498C23—44no—
499C23—42no—
500C23—42no—
501C23—44no—
502C23—42no—
503C23—42no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—816no—
511C23—816no—
512C23—88no—
513C23—816no—
514C23—816no—
515C23—816no—
516C23—88no—
517C23C2×#12314yes—
518C23#53314yes—
519C23#54214yes—
520C23D4×Q814yes—
521C23#55014yes—
522C23#55214yes—
523C23#55414yes—
524C23C23×D411yescharacteristic
525C23C23×Q811yescharacteristic
526C23C22×#4111yescharacteristic
527C23C22×#4111yescharacteristic
528C23C22×#4111yescharacteristic
529C23C2×H(5,𝔽2)11yescharacteristic
530C23C2×H(5,𝔽2)14yes—
531C23C2×2−1+411yescharacteristic
532C23#58514yes—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—416no—
555C2×C4—432no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
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—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—816no—
601C2×C4—88no—
602C2×C4C2×#12314yes—
603C2×C4C2×#12314yes—
604C22—24no—
605C22—28no—
606C22—28no—
607C22—28no—
608C22—28no—
609C22—24no—
610C22—28no—
611C22—28no—
612C22—44no—
613C22—48no—
614C22—48no—
615C22—44no—
616C22—44no—
617C22—832no—
618C22—82no—
619C22—84no—
620C22—82no—
621C22—82no—
622C22—84no—
623C22—82no—
624C22#242318yes—
625C22C22×#12311yescharacteristic
626C22C2×#53311yescharacteristic
627C22#344912yes—
628C22C2×#54211yescharacteristic
629C22C2×D4×Q811yescharacteristic
630C22C2×#55011yescharacteristic
631C22C2×#55211yescharacteristic
632C22C2×#55411yescharacteristic
633C22#348412yes—
634C22#349112yes—
635C22#349112yes—
636C22#350114yes—
637C22#350812yes—
638C22#350812yes—
639C22#351314yes—
640C22#353414yes—
641C22#354114yes—
642C4—24no—
643C4—24no—
644C4—44no—
645C4—44no—
646C4—48no—
647C4—44no—
648C4—44no—
649C4—44no—
650C4—84no—
651C4—82no—
652C4—82no—
653C4—82no—
654C4—82no—
655C2—28no—
656C2—84no—
657C2C2×#242311yescharacteristic
658C2#3509711yescharacteristic
659C2#3555411yescharacteristic
660C2#3583912yes—
661C2#3912812yes—
662C2#3958812yes—
663C2#4139012yes—
664C2#4351212yes—
665C2#4408112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2345C2semidirect
3. #16531C2semidirect
4. #15653C2semidirect
5. #15649C2semidirect
6. #14821C2semidirect
7. #14793C2semidirect
10. #14149C2semidirect
11. #14039C2semidirect
13. #13965C2semidirect
14. #13924C2semidirect
25. C2×#398C22semidirect
27. C2×#389C22semidirect
29. C2×#387C22semidirect
30. C2×#387C22semidirect
31. C2×#387C22semidirect
41. C2×C4×C4⋊C4C22semidirect
46. #1445C22semidirect
47. #1445C22semidirect
48. #1444C22semidirect
49. #1442C22semidirect
50. #1441C22semidirect
51. #1441C22semidirect
52. #1441C22semidirect
53. #1441C22semidirect
54. #1441C22semidirect
66. #1436C22semidirect
67. #1436C22semidirect
68. C4⋊#95C22semidirect
69. C4⋊#95C22semidirect
70. C4⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C22×#3992
C2×#23452
#165312
#156532
#156492
#148212
#147932
#147342
#143382
#141492
#140392
#139912
#139652
#139242
#137752