Finite Group Databasebeta

← #853194

Subgroups — #853194

Order 512.

2738 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16781C214yesmaximal, complemented
2#15496C214yesmaximal, complemented
3#15299C214yesmaximal, complemented
4#15223C211yesmaximal, characteristic, complemented
5#15140C212yesmaximal, complemented
6#15138C212yesmaximal, complemented
7#15124C211yesmaximal, characteristic
8#14698C212yesmaximal, complemented
9#14617C214yesmaximal, complemented
10#14132C212yesmaximal, complemented
11#14002C212yesmaximal, complemented
12#13998C212yesmaximal, complemented
13C4×#381C211yesmaximal, characteristic, complemented
14C22×#127C2211yescharacteristic
15C22×#121C2212yes—
16C23×C42C2211yescharacteristic
17C2×#397C2214yes—
18C2×#394C2214yes—
19C2×#394C2214yes—
20C2×#392C2214yes—
21C2×#390C2212yes—
22C2×#390C2212yes—
23C2×#390C2212yes—
24C2×#390C2212yes—
25C2×#390C2212yes—
26C2×#390C2214yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#386C2214yes—
30C2×#386C2214yes—
31C2×#382C2211yescharacteristic
32C2×#381C2212yes—
33C2×#380C2212yes—
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C22⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4C2211yescharacteristic
37#1445C2214yes—
38#1445C2214yes—
39#1441C2214yes—
40#1440C2214yes—
41#1440C2214yes—
42#1440C2214yes—
43#1440C2214yes—
44#1439C2214yes—
45#1439C2214yes—
46#1439C2214yes—
47#1438C2214yes—
48#1438C2214yes—
49#1438C2214yes—
50#1438C2214yes—
51#1437C2214yes—
52#1437C2214yes—
53#1437C2214yes—
54#1437C2214yes—
55#1437C2214yes—
56#1436C2214yes—
57C4⋊#95C2211yescharacteristic
58C4⋊#95C2211yescharacteristic
59C4⋊#95C2212yes—
60C4⋊#95C2212yes—
61#1434C2212yes—
62#1434C2214yes—
63#1434C2212yes—
64C22⋊#95C2212yes—
65C4×#95C2212yes—
66C4×#95C2212yes—
67C23×D4C2312yes—
68C23×D4C2314yes—
69C23×D4C2314yes—
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C2×#127—42no—
74C2×#127—44no—
75C2×#121—28no—
76C2×#121—28no—
77C2×#121—44no—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2314yes—
83C22×C22⋊C4C2312yes—
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2312yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C42—24no—
98C22×C42—42no—
99C22×C42—42no—
100C22×C42C2312yes—
101C22×C42C2311yescharacteristic
102C22×C42C2311yescharacteristic
103C22×C42C2312yes—
104C22×C42C2312yes—
105C22×C42C2314yes—
106#397—48no—
107#394—216no—
108#394—48no—
109#392—48no—
110#390—28no—
111#390—44no—
112#390—44no—
113#390—44no—
114#390—44no—
115#390—48no—
116#389—28no—
117#389—44no—
118#386—48no—
119#386—48no—
120#382—42no—
121#381—28no—
122#380—44no—
123C4×C4⋊C4—44no—
124C4×C22⋊C4—42no—
125C4×C22⋊C4—42no—
126C2×#95C2312yes—
127C2×#95C2312yes—
128C2×#95C2314yes—
129C2×#95C2312yes—
130C2×#95C2312yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2311yescharacteristic
138C2×#95C2312yes—
139C2×#95C2311yescharacteristic
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C25C2414yes—
149C25C2411yescharacteristic
150C22×D4—28no—
151C22×D4—28no—
152C22×D4—28no—
153C22×D4—28no—
154C22×D4—24no—
155C22×D4—24no—
156C22×D4—24no—
157C22×D4—28no—
158C22×D4—24no—
159C22×D4—24no—
160C22×D4—24no—
161C22×D4—24no—
162C22×D4—44no—
163C22×D4—44no—
164C22×D4—44no—
165C22×D4—44no—
166C22×D4—44no—
167C22×D4—44no—
168C22×D4—44no—
169C22×D4—44no—
170C22×D4—44no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—42no—
181C23×C4—41no—
182C23×C4—41no—
183C23×C4—41no—
184C23×C4—41no—
185C23×C4#4114yes—
186C23×C4C2414yes—
187C23×C4C2412yes—
188C23×C4C2414yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2411yescharacteristic
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2411yescharacteristic
197#127—416no—
198#121—432no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4#4118yes—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—24no—
262C2×C42—22no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42—416no—
270C2×C42—44no—
271C2×C42—44no—
272C2×C42—44no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—28no—
277#95—28no—
278#95—24no—
279#95—24no—
280#95—42no—
281#95—42no—
282#95—41no—
283#95—42no—
284#95—41no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95#4118yes—
295C24—22no—
296C24—22no—
297C24—21no—
298C24—21no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—24no—
303C24—24no—
304C24—24no—
305C24—24no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—24no—
311C24—24no—
312C24—24no—
313C24—24no—
314C24—24no—
315C24—24no—
316C24—24no—
317C24C2×#4112yes—
318C24H(5,𝔽2)14yes—
319C24C2511yescharacteristic
320C2×D4—416no—
321C2×D4—416no—
322C2×D4—416no—
323C2×D4—416no—
324C2×D4—416no—
325C2×D4—48no—
326C2×D4—48no—
327C2×D4—48no—
328C2×D4—48no—
329C2×D4—48no—
330C2×D4—48no—
331C2×D4—48no—
332C2×D4—88no—
333C2×D4—88no—
334C2×D4—88no—
335C2×D4—88no—
336C2×D4—88no—
337C2×D4—88no—
338C2×D4—88no—
339C2×D4—88no—
340C2×D4—88no—
341C22×C4—28no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—28no—
377C22×C4—216no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
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—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—44no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—48no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4C2×#4114yes—
419C22×C4C2×#4114yes—
420C22×C4C2×#4114yes—
421C22×C4C2×#4118yes—
422C22×C4C2×#4112yes—
423C22×C4C2×#4112yes—
424C22×C4C2×#4114yes—
425C22×C4H(5,𝔽2)14yes—
426C22×C4H(5,𝔽2)12yes—
427C22×C4H(5,𝔽2)12yes—
428C22×C4H(5,𝔽2)14yes—
429C4⋊C4—216no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C4⋊C4—88no—
433C4⋊C4—88no—
434C22⋊C4—416no—
435C22⋊C4—48no—
436C22⋊C4—48no—
437C22⋊C4—88no—
438C22⋊C4—88no—
439C22⋊C4—88no—
440C22⋊C4—88no—
441C22⋊C4—88no—
442C22⋊C4—88no—
443C22⋊C4—88no—
444C22⋊C4—88no—
445C22⋊C4—88no—
446C22⋊C4—88no—
447C22⋊C4—88no—
448C42—48no—
449C42—48no—
450C42—48no—
451C42—44no—
452C42—44no—
453C42—88no—
454C23—22no—
455C23—22no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—42no—
465C23—41no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—42no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23#54014yes—
510C23C22×#4111yescharacteristic
511C23C22×#4112yes—
512C23C22×#4112yes—
513C23C2×H(5,𝔽2)11yescharacteristic
514C23C2×H(5,𝔽2)11yescharacteristic
515C23C2×H(5,𝔽2)12yes—
516C23C2×H(5,𝔽2)12yes—
517C23C2×H(5,𝔽2)12yes—
518C23#58512yes—
519D4—832no—
520D4—832no—
521D4—816no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—416no—
540C2×C4—416no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—416no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4#53814yes—
594C2×C4#54018yes—
595C2×C4#55118yes—
596C2×C4#55914yes—
597C2×C4#56014yes—
598C22—24no—
599C22—28no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—42no—
604C22—44no—
605C22—42no—
606C22—44no—
607C22—84no—
608C22—84no—
609C22—84no—
610C22—84no—
611C22—84no—
612C22—84no—
613C22—84no—
614C22—84no—
615C22—84no—
616C22—84no—
617C22—84no—
618C22—84no—
619C22—84no—
620C22—84no—
621C22—84no—
622C22#344411yescharacteristic
623C22C2×#53811yescharacteristic
624C22C2×#54011yescharacteristic
625C22C2×#54012yes—
626C22C2×#55112yes—
627C22C2×#55911yescharacteristic
628C22#348012yes—
629C22#348312yes—
630C22#349412yes—
631C22#350812yes—
632C22#351312yes—
633C22#351412yes—
634C22#351912yes—
635C22#351912yes—
636C22#351912yes—
637C22C2×#56011yescharacteristic
638C22#353711yescharacteristic
639C22#353911yescharacteristic
640C22#354212yes—
641C22#354312yes—
642C22#355812yes—
643C4—44no—
644C4—44no—
645C4—44no—
646C4—44no—
647C4—48no—
648C4—42no—
649C4—84no—
650C4—82no—
651C4—84no—
652C4—82no—
653C4#275218yes—
654C2—44no—
655C2—164no—
656C2#3508111yescharacteristic
657C2#3639912yes—
658C2#3728311yescharacteristic
659C2#3805612yes—
660C2#3921212yes—
661C2#4012612yes—
662C2#4564712yes—
663C2C2×#275211yescharacteristic
664C2#5963312yes—
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. #16781C2semidirect
2. #15496C2semidirect
3. #15299C2semidirect
4. #15223C2semidirect
5. #15140C2semidirect
6. #15138C2semidirect
8. #14698C2semidirect
9. #14617C2semidirect
10. #14132C2semidirect
11. #14002C2semidirect
12. #13998C2semidirect
13. C4×#381C2semidirect

Maximal subgroups

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

subgroupindex
#167812
#154962
#152992
#152232
#151402
#151382
#151242
#146982
#146172
#141322
#140022
#139982
C4×#3812