Finite Group Databasebeta

← #536263

Subgroups — #536263

Order 512.

3040 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#397C211yesmaximal, characteristic
2#14619C214yesmaximal, complemented
3#14610C212yesmaximal, complemented
4#14581C212yesmaximal, complemented
5#14253C212yesmaximal, complemented
6#14248C212yesmaximal, complemented
7#14184C214yesmaximal, complemented
8#13923C212yesmaximal, complemented
9#13874C212yesmaximal, complemented
10#13873C212yesmaximal
11#13869C211yesmaximal, characteristic
12#13868C211yesmaximal, characteristic
13#13799C212yesmaximal, complemented
14C22⋊C42C212yesmaximal
15#13796C212yesmaximal, complemented
16C22×#117C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2212yes—
21C2×#397C4116yescomplemented
22C2×#397C2212yes—
23C2×#397C2211yescharacteristic
24C2×#397C2211yescharacteristic
25C2×#397C2211yescharacteristic
26C2×#397C2212yes—
27C2×#397C2211yescharacteristic
28C2×#388C2214yes—
29C2×#381C2212yes—
30C2×#381C2211yescharacteristic
31C2×#380C2212yes—
32C2×#379C2212yes—
33C2×#379C2211yescharacteristic
34C2×C4×C22⋊C4C2212yes—
35C2×C4×C22⋊C4C2212yes—
36C2×C4×C22⋊C4C2212yes—
37C2×C4×C22⋊C4C2214yes—
38C22×#95C2212yes—
39C22×#95C2211yescharacteristic
40#1443C2214yescomplemented
41#1442C2214yescomplemented
42#1441C2214yescomplemented
43#1441C2214yescomplemented
44#1440C2212yes—
45#1440C2212yes—
46#1440C2214yes—
47#1439C2214yes—
48#1439C2212yes—
49#1439C2212yes—
50#1437C2214yes—
51#1437C2212yes—
52#1437C2214yes—
53#1437C2214yes—
54#1437C2212yes—
55#1436C2214yescomplemented
56#1436C2214yescomplemented
57#1436C2214yescomplemented
58#1436C2214yescomplemented
59#1436C2214yescomplemented
60#1434C2214yescomplemented
61#1434C2214yescomplemented
62#1434C2214yescomplemented
63#1434C2214yescomplemented
64#1434C2214yescomplemented
65#1434C2214yescomplemented
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2212yes—
75C4×#95C2214yescomplemented
76C26C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2312yes—
82C2×#117—28no—
83C2×#117—42no—
84C22×C4⋊C4—42no—
85C22×C4⋊C4—42no—
86C22×C4⋊C4C2×C418yescomplemented
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C22⋊C4—28no—
93C22×C22⋊C4—42no—
94C22×C22⋊C4—48no—
95C22×C22⋊C4—42no—
96C22×C22⋊C4—42no—
97C22×C22⋊C4C2×C4116yes—
98C22×C22⋊C4C2×C418yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C42C2312yes—
117C22×C42C2314yes—
118C22×C42C2314yes—
119C22×C42C2314yes—
120#397C2×C4116yes—
121#397C2×C418yes—
122#397C2×C418yes—
123#397C2×C418yescomplemented
124#397C2×C4116yescomplemented
125#397C2×C418yescomplemented
126#388—48no—
127#381—28no—
128#381—42no—
129#380—44no—
130#379—28no—
131#379—42no—
132C4×C22⋊C4—28no—
133C4×C22⋊C4—28no—
134C4×C22⋊C4—44no—
135C4×C22⋊C4—48no—
136C2×#95C2×C4116yescomplemented
137C2×#95C2×C418yescomplemented
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2311yescharacteristic
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C25—24no—
166C25—24no—
167C25—24no—
168C25—22no—
169C25—22no—
170C25—44no—
171C25—42no—
172C25C22×C414yes—
173C25C2411yescharacteristic
174C25C2411yescharacteristic
175C25C2411yescharacteristic
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—28no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4—42no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—42no—
196C23×C4C22×C414yes—
197C23×C4C22×C414yes—
198C23×C4C22×C418yes—
199C23×C4C22×C418yes—
200C23×C4C2414yes—
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2414yes—
212#117—416no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—416no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4C22×C414yes—
226C2×C4⋊C4C22×C418yes—
227C2×C4⋊C4C22×C414yes—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—432no—
246C2×C22⋊C4—416no—
247C2×C22⋊C4—416no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4C22×C418yes—
267C2×C22⋊C4C22×C418yes—
268C2×C22⋊C4C22×C414yes—
269C2×C22⋊C4C22×C414yes—
270C2×C22⋊C4C22×C418yes—
271C2×C22⋊C4C22×C418yes—
272C2×C22⋊C4C22×C414yes—
273C2×C22⋊C4C22×C414yes—
274C2×C22⋊C4#4118yes—
275C2×C22⋊C4#4118yes—
276C2×C42—28no—
277C2×C42—28no—
278C2×C42—28no—
279C2×C42—42no—
280C2×C42—44no—
281C2×C42—44no—
282C2×C42—44no—
283C2×C42—44no—
284C2×C42—44no—
285C2×C42—44no—
286C2×C42—44no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—44no—
292#95—44no—
293#95—44no—
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—42no—
304#95—42no—
305#95—44no—
306#95—44no—
307#95C22×C418yes—
308#95C22×C418yes—
309#95C22×C414yes—
310#95C22×C414yes—
311#95C22×C418yes—
312#95C22×C418yes—
313#95C22×C418yes—
314C24—22no—
315C24—24no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—21no—
320C24—21no—
321C24—24no—
322C24—22no—
323C24—28no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—21no—
328C24—21no—
329C24—24no—
330C24—22no—
331C24—216no—
332C24—28no—
333C24—48no—
334C24—44no—
335C24—416no—
336C24—416no—
337C24—48no—
338C24—48no—
339C24—44no—
340C24—44no—
341C24—48no—
342C24—88no—
343C24—88no—
344C24—84no—
345C24—84no—
346C24C23×C412yes—
347C24C23×C412yes—
348C24C23×C412yes—
349C24C2×#4114yes—
350C24C2×#4114yes—
351C24H(5,𝔽2)14yes—
352C24H(5,𝔽2)12yes—
353C24H(5,𝔽2)12yes—
354C24C2511yescharacteristic
355C22×C4—216no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—216no—
401C22×C4—216no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—416no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—48no—
428C22×C4—88no—
429C22×C4—88no—
430C22×C4—88no—
431C22×C4—84no—
432C22×C4—88no—
433C22×C4—88no—
434C22×C4—84no—
435C22×C4C23×C412yes—
436C22×C4C23×C412yes—
437C22×C4C23×C414yes—
438C22×C4C23×C414yes—
439C22×C4C23×C414yes—
440C22×C4C23×C414yes—
441C22×C4C23×C412yes—
442C22×C4C23×C412yes—
443C22×C4C2×#4118yes—
444C22×C4C2×#4118yes—
445C4⋊C4—44no—
446C4⋊C4—48no—
447C4⋊C4—44no—
448C4⋊C4—48no—
449C4⋊C4—88no—
450C22⋊C4—216no—
451C22⋊C4—216no—
452C22⋊C4—216no—
453C22⋊C4—216no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—48no—
461C22⋊C4—416no—
462C22⋊C4—48no—
463C22⋊C4—48no—
464C22⋊C4—88no—
465C22⋊C4—84no—
466C22⋊C4—84no—
467C42—48no—
468C42—88no—
469C42—88no—
470C42—88no—
471C23—22no—
472C23—24no—
473C23—22no—
474C23—28no—
475C23—28no—
476C23—28no—
477C23—28no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—22no—
484C23—22no—
485C23—24no—
486C23—28no—
487C23—28no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—22no—
495C23—416no—
496C23—432no—
497C23—41no—
498C23—41no—
499C23—42no—
500C23—44no—
501C23—44no—
502C23—42no—
503C23—42no—
504C23—416no—
505C23—41no—
506C23—41no—
507C23—42no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—88no—
517C23—816no—
518C23—816no—
519C23—832no—
520C23—816no—
521C23C4×#4118yes—
522C23#51814yes—
523C23#51814yes—
524C23#55114yes—
525C23#55114yes—
526C23#55518yes—
527C23C24×C411yescharacteristic
528C23C22×#4112yes—
529C23C22×#4112yes—
530C23C2×H(5,𝔽2)11yescharacteristic
531C23C2×H(5,𝔽2)11yescharacteristic
532C23C2×H(5,𝔽2)12yes—
533C23#58514yes—
534C23#58512yes—
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—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—416no—
562C2×C4—416no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
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—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—816no—
604C2×C4—816no—
605C2×C4—816no—
606C22—24no—
607C22—24no—
608C22—28no—
609C22—28no—
610C22—28no—
611C22—216no—
612C22—28no—
613C22—24no—
614C22—28no—
615C22—42no—
616C22—42no—
617C22—44no—
618C22—44no—
619C22—48no—
620C22—44no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—44no—
625C22—44no—
626C22—832no—
627C22—82no—
628C22—84no—
629C22—82no—
630C22#237218yes—
631C22C2×C4×#4112yes—
632C22C2×#51811yescharacteristic
633C22C2×#51811yescharacteristic
634C22#342812yes—
635C22#342811yescharacteristic
636C22C2×#55111yescharacteristic
637C22C2×#55111yescharacteristic
638C22C2×#55512yes—
639C22#351314yes—
640C22#351314yes—
641C22#352114yes—
642C22#352114yes—
643C22#353312yes—
644C22#353312yes—
645C22#353412yes—
646C22#353412yes—
647C4—42no—
648C4—42no—
649C4—44no—
650C4—44no—
651C4—42no—
652C4—42no—
653C4—44no—
654C4—44no—
655C4—48no—
656C4—48no—
657C4—84no—
658C4—84no—
659C2—28no—
660C2—44no—
661C2—82no—
662C2C2×#237211yescharacteristic
663C2#3423812yes—
664C2#3424712yes—
665C2#3435711yescharacteristic
666C2#3435811yescharacteristic
667C2#4322012yes—
668C2#4322214yes—
669C2#4324412yes—
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. #14619C2semidirect
3. #14610C2semidirect
4. #14581C2semidirect
5. #14253C2semidirect
6. #14248C2semidirect
7. #14184C2semidirect
8. #13923C2semidirect
9. #13874C2semidirect
13. #13799C2semidirect
15. #13796C2semidirect
21. C2×#397C4semidirect
40. #1443C22semidirect
41. #1442C22semidirect
42. #1441C22semidirect
43. #1441C22semidirect
55. #1436C22semidirect
56. #1436C22semidirect
57. #1436C22semidirect
58. #1436C22semidirect
59. #1436C22semidirect
60. #1434C22semidirect
61. #1434C22semidirect
62. #1434C22semidirect
63. #1434C22semidirect
64. #1434C22semidirect
65. #1434C22semidirect
75. C4×#95C22semidirect
86. C22×C4⋊C4C2×C4semidirect
123. #397C2×C4semidirect
124. #397C2×C4semidirect
125. #397C2×C4semidirect
136. C2×#95C2×C4semidirect
137. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#3972
#146192
#146102
#145812
#142532
#142482
#141842
#139232
#138742
#138732
#138692
#138682
#137992
C22⋊C422
#137962