Finite Group Databasebeta

← #536189

Subgroups — #536189

Order 512.

3224 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×#399C211yesmaximal, characteristic
2#14618C212yesmaximal, complemented
3#14592C212yesmaximal, complemented
4#14578C214yesmaximal, complemented
5#14240C214yesmaximal, complemented
6#14215C212yesmaximal, complemented
7#14194C212yesmaximal, complemented
8#13903C211yesmaximal, characteristic
9#13882C212yesmaximal, complemented
10#13874C212yesmaximal, complemented
11#13873C212yesmaximal
12#13869C211yesmaximal, characteristic
13C22⋊C4×C4⋊C4C212yesmaximal, complemented
14C22⋊C42C212yesmaximal
15#13776C212yesmaximal, complemented
16C23×C4⋊C4C2212yes—
17C23×C4⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2212yes—
21C2×#399C4116yescomplemented
22C2×#399C2211yescharacteristic
23C2×#399C2211yescharacteristic
24C2×#399C2212yes—
25C2×#399C2211yescharacteristic
26C2×#399C2211yescharacteristic
27C2×#399C2212yes—
28C2×#388C2214yes—
29C2×#387C2214yescomplemented
30C2×#381C2212yes—
31C2×#381C2212yes—
32C2×#380C2212yes—
33C2×#380C2211yescharacteristic
34C2×#379C2212yes—
35C2×#379C2211yescharacteristic
36C2×C4×C4⋊C4C2214yescomplemented
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C2×C4×C22⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2214yes—
42C22×#95C2211yescharacteristic
43#1445C2214yescomplemented
44#1443C2214yescomplemented
45#1442C2214yescomplemented
46#1441C2214yescomplemented
47#1440C2212yes—
48#1440C2214yes—
49#1440C2212yes—
50#1439C2212yes—
51#1439C2214yes—
52#1439C2212yes—
53#1437C2214yes—
54#1437C2212yes—
55#1437C2214yes—
56#1437C2212yes—
57#1437C2214yes—
58#1436C2214yescomplemented
59#1436C2214yescomplemented
60#1436C2214yescomplemented
61#1436C2214yescomplemented
62#1436C2214yescomplemented
63#1436C2214yescomplemented
64#1436C2214yescomplemented
65#1436C2214yescomplemented
66C4⋊#95C2214yescomplemented
67#1434C2214yescomplemented
68C22⋊#95C2212yes—
69C22⋊#95C2214yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C26C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2312yes—
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C22×C4⋊C4—48no—
83C22×C4⋊C4—42no—
84C22×C4⋊C4—42no—
85C22×C4⋊C4C2×C4116yescomplemented
86C22×C4⋊C4C2×C418yescomplemented
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C22⋊C4—28no—
97C22×C22⋊C4—48no—
98C22×C22⋊C4—42no—
99C22×C22⋊C4—42no—
100C22×C22⋊C4—42no—
101C22×C22⋊C4C2×C4116yes—
102C22×C22⋊C4C2×C418yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C42C2314yes—
121C22×C42C2314yes—
122C22×C42C2314yes—
123C22×C42C2312yes—
124#399C2×C418yes—
125#399C2×C4116yes—
126#399C2×C418yes—
127#399C2×C418yescomplemented
128#399C2×C4116yescomplemented
129#399C2×C418yescomplemented
130#388—48no—
131#387—48no—
132#381—28no—
133#381—28no—
134#380—28no—
135#380—42no—
136#379—28no—
137#379—42no—
138C4×C4⋊C4—48no—
139C4×C22⋊C4—28no—
140C4×C22⋊C4—48no—
141C4×C22⋊C4—44no—
142C4×C22⋊C4—42no—
143C4×C22⋊C4—44no—
144C2×#95C2×C418yescomplemented
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2311yescharacteristic
167C2×#95C2311yescharacteristic
168C25—24no—
169C25—24no—
170C25—24no—
171C25—22no—
172C25—22no—
173C25—44no—
174C25—42no—
175C25C22×C414yes—
176C25C2411yescharacteristic
177C25C2411yescharacteristic
178C25C2411yescharacteristic
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—28no—
190C23×C4—24no—
191C23×C4—28no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—42no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4—42no—
199C23×C4C22×C414yes—
200C23×C4C22×C418yes—
201C23×C4C22×C414yes—
202C23×C4C22×C418yes—
203C23×C4C2414yes—
204C23×C4C2414yes—
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2414yes—
214C23×C4C2412yes—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—432no—
220C2×C4⋊C4—416no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4C22×C418yes—
233C2×C4⋊C4C22×C418yes—
234C2×C4⋊C4C22×C418yes—
235C2×C4⋊C4C22×C414yes—
236C2×C4⋊C4C22×C418yes—
237C2×C4⋊C4C22×C414yes—
238C2×C4⋊C4C22×C418yes—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
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—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—432no—
257C2×C22⋊C4—416no—
258C2×C22⋊C4—416no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—41no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4—41no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4C22×C418yes—
278C2×C22⋊C4C22×C418yes—
279C2×C22⋊C4C22×C418yes—
280C2×C22⋊C4C22×C414yes—
281C2×C22⋊C4C22×C414yes—
282C2×C22⋊C4C22×C418yes—
283C2×C22⋊C4C22×C414yes—
284C2×C22⋊C4C22×C414yes—
285C2×C22⋊C4#4118yes—
286C2×C22⋊C4#4118yes—
287C2×C42—28no—
288C2×C42—28no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42—42no—
292C2×C42—44no—
293C2×C42—44no—
294C2×C42—44no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42—44no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—42no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—42no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95C22×C414yes—
319#95C22×C418yes—
320#95C22×C414yes—
321C24—24no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—24no—
327C24—21no—
328C24—21no—
329C24—22no—
330C24—28no—
331C24—22no—
332C24—22no—
333C24—24no—
334C24—21no—
335C24—21no—
336C24—22no—
337C24—22no—
338C24—216no—
339C24—28no—
340C24—48no—
341C24—44no—
342C24—416no—
343C24—416no—
344C24—48no—
345C24—48no—
346C24—44no—
347C24—48no—
348C24—44no—
349C24—88no—
350C24—88no—
351C24—84no—
352C24—84no—
353C24C23×C412yes—
354C24C23×C412yes—
355C24C23×C412yes—
356C24C2×#4114yes—
357C24C2×#4114yes—
358C24H(5,𝔽2)12yes—
359C24H(5,𝔽2)12yes—
360C242−1+414yes—
361C24C2511yescharacteristic
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—216no—
366C22×C4—216no—
367C22×C4—28no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—28no—
407C22×C4—216no—
408C22×C4—216no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—416no—
429C22×C4—416no—
430C22×C4—416no—
431C22×C4—44no—
432C22×C4—48no—
433C22×C4—44no—
434C22×C4—88no—
435C22×C4—84no—
436C22×C4—84no—
437C22×C4—88no—
438C22×C4—88no—
439C22×C4—84no—
440C22×C4—84no—
441C22×C4C23×C412yes—
442C22×C4C23×C412yes—
443C22×C4C23×C414yes—
444C22×C4C23×C414yes—
445C22×C4C23×C414yes—
446C22×C4C23×C412yes—
447C22×C4C23×C412yes—
448C22×C4C23×C414yes—
449C22×C4C2×#4118yes—
450C22×C4C2×#4118yes—
451C22×C4H(5,𝔽2)18yes—
452C4⋊C4—216no—
453C4⋊C4—216no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—48no—
459C4⋊C4—416no—
460C4⋊C4—84no—
461C22⋊C4—216no—
462C22⋊C4—216no—
463C22⋊C4—216no—
464C22⋊C4—216no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—48no—
472C22⋊C4—416no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C22⋊C4—88no—
476C22⋊C4—84no—
477C22⋊C4—84no—
478C42—416no—
479C42—84no—
480C42—88no—
481C42—88no—
482C23—24no—
483C23—22no—
484C23—22no—
485C23—28no—
486C23—28no—
487C23—28no—
488C23—28no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—22no—
496C23—22no—
497C23—28no—
498C23—28no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—416no—
507C23—432no—
508C23—41no—
509C23—42no—
510C23—41no—
511C23—44no—
512C23—44no—
513C23—42no—
514C23—42no—
515C23—416no—
516C23—41no—
517C23—42no—
518C23—41no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—88no—
528C23—816no—
529C23—832no—
530C23—816no—
531C23—816no—
532C23C4×#4118yes—
533C23#51814yes—
534C23#51914yes—
535C23#55114yes—
536C23#55318yes—
537C23#55914yes—
538C23C24×C411yescharacteristic
539C23C22×#4112yes—
540C23C22×#4112yes—
541C23C2×H(5,𝔽2)11yescharacteristic
542C23C2×H(5,𝔽2)11yescharacteristic
543C23C2×H(5,𝔽2)14yes—
544C23C2×2−1+412yes—
545C23#58512yes—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—432no—
574C2×C4—416no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—88no—
614C2×C4—816no—
615C2×C4—816no—
616C2×C4#51818yes—
617C22—216no—
618C22—28no—
619C22—28no—
620C22—24no—
621C22—24no—
622C22—28no—
623C22—28no—
624C22—24no—
625C22—28no—
626C22—48no—
627C22—44no—
628C22—44no—
629C22—42no—
630C22—42no—
631C22—44no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—44no—
636C22—44no—
637C22—832no—
638C22—84no—
639C22—82no—
640C22—82no—
641C22#237418yes—
642C22C2×C4×#4112yes—
643C22C2×#51811yescharacteristic
644C22C2×#51812yes—
645C22C2×#51911yescharacteristic
646C22#342811yescharacteristic
647C22C2×#55111yescharacteristic
648C22C2×#55312yes—
649C22C2×#55911yescharacteristic
650C22#351314yes—
651C22#351314yes—
652C22#352014yes—
653C22#352014yes—
654C22#353512yes—
655C22#353512yes—
656C22#354312yes—
657C22#354312yes—
658C4—28no—
659C4—48no—
660C4—48no—
661C4—44no—
662C4—44no—
663C4—42no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—44no—
668C4—84no—
669C4—84no—
670C2—28no—
671C2—44no—
672C2—82no—
673C2C2×#237411yescharacteristic
674C2#3423812yes—
675C2#3424512yes—
676C2#3435911yescharacteristic
677C2#3436411yescharacteristic
678C2#4328714yes—
679C2#4328812yes—
680C2#4330612yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC26C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14618C2semidirect
3. #14592C2semidirect
4. #14578C2semidirect
5. #14240C2semidirect
6. #14215C2semidirect
7. #14194C2semidirect
9. #13882C2semidirect
10. #13874C2semidirect
13. C22⋊C4×C4⋊C4C2semidirect
15. #13776C2semidirect
21. C2×#399C4semidirect
29. C2×#387C22semidirect
36. C2×C4×C4⋊C4C22semidirect
43. #1445C22semidirect
44. #1443C22semidirect
45. #1442C22semidirect
46. #1441C22semidirect
58. #1436C22semidirect
59. #1436C22semidirect
60. #1436C22semidirect
61. #1436C22semidirect
62. #1436C22semidirect
63. #1436C22semidirect
64. #1436C22semidirect
65. #1436C22semidirect
66. C4⋊#95C22semidirect
67. #1434C22semidirect
85. C22×C4⋊C4C2×C4semidirect
86. C22×C4⋊C4C2×C4semidirect
127. #399C2×C4semidirect
128. #399C2×C4semidirect
129. #399C2×C4semidirect
144. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#3992
#146182
#145922
#145782
#142402
#142152
#141942
#139032
#138822
#138742
#138732
#138692
C22⋊C4×C4⋊C42
C22⋊C422
#137762