Finite Group Databasebeta

← #530935

Subgroups — #530935

Order 512.

2874 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2#14683C211yesmaximal, characteristic, complemented
3#14453C212yesmaximal, complemented
4#14417C212yesmaximal
5#14375C214yesmaximal, complemented
6#14280C212yesmaximal, complemented
7#14038C214yesmaximal, complemented
8#13960C212yesmaximal, complemented
9#13950C212yesmaximal, complemented
10#13919C212yesmaximal, complemented
11#13901C212yesmaximal, complemented
12#13890C212yesmaximal, complemented
13#13813C212yesmaximal, complemented
14#13791C212yesmaximal, complemented
15C4×#381C211yesmaximal, characteristic, complemented
16C22×#121C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×C4×D4C2212yes—
19C23×C42C2211yescharacteristic
20C4×#121C4116yescomplemented
21C2×#390C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#388C2212yes—
24C2×#387C2214yes—
25C2×#386C2214yes—
26C2×#386C2214yes—
27C2×#386C2212yes—
28C2×#383C2211yescharacteristic
29C2×#383C2211yescharacteristic
30C2×#381C2212yes—
31C2×#380C2212yes—
32C2×#380C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4C2214yes—
37C2×C4×C22⋊C4C2212yes—
38C22×#95C2212yes—
39#1445C2214yes—
40#1443C2214yes—
41#1443C2214yes—
42#1443C2214yes—
43#1442C2214yes—
44#1441C2214yes—
45#1440C2214yes—
46#1439C2214yes—
47#1439C2214yes—
48#1438C2214yes—
49#1437C2214yes—
50#1437C2214yes—
51#1437C2214yes—
52#1436C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55#1434C2214yes—
56#1434C2212yes—
57#1434C2212yes—
58#1434C2212yes—
59#1434C2212yes—
60#1434C2212yes—
61#1434C2212yes—
62#1434C2212yes—
63#1434C2212yes—
64C22⋊#95C2212yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C22⋊#95C2214yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C4×#95C2212yes—
74C4×#95C2212yes—
75C4×#95C2212yes—
76C23×D4C2312yes—
77C23×D4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2314yes—
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C2×#121—28no—
84C2×#121C2×C418yes—
85C2×C4×D4—42no—
86C2×C4×D4—42no—
87C2×C4×D4—48no—
88C2×C4×D4C2×C418yes—
89C2×C4×D4C2×C4116yescomplemented
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C42—42no—
107C22×C42—42no—
108C22×C42C2×C418yescomplemented
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2314yes—
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2312yes—
117#390C2×C418yes—
118#389C2×C418yescomplemented
119#388C2×C4116yescomplemented
120#387—48no—
121#386—216no—
122#386—216no—
123#386C2×C4116yescomplemented
124#383—42no—
125#383—42no—
126#381—28no—
127#380—44no—
128#380C2×C418yescomplemented
129C4×C4⋊C4C2×C418yescomplemented
130C4×C22⋊C4—42no—
131C4×C22⋊C4—42no—
132C4×C22⋊C4—48no—
133C4×C22⋊C4C2×C4116yescomplemented
134C2×#95—44no—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C25C2412yes—
164C25C2411yescharacteristic
165C22×D4—24no—
166C22×D4—24no—
167C22×D4—28no—
168C22×D4—41no—
169C22×D4—41no—
170C22×D4—44no—
171C22×D4—42no—
172C22×D4C22×C418yes—
173C22×D4C22×C414yes—
174C22×D4#4118yes—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—28no—
183C23×C4—28no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—41no—
188C23×C4—41no—
189C23×C4—41no—
190C23×C4—41no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4C22×C414yes—
199C23×C4C22×C418yes—
200C23×C4C22×C414yes—
201C23×C4C22×C414yes—
202C23×C4C2414yes—
203C23×C4C2414yes—
204C23×C4C2414yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215#121—232no—
216C4×D4—416no—
217C4×D4—432no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4C22×C418yes—
227C2×C4⋊C4C22×C414yes—
228C2×C4⋊C4C22×C414yes—
229C2×C4⋊C4C22×C414yes—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—42no—
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⋊C4C22×C418yes—
255C2×C22⋊C4C22×C418yes—
256C2×C22⋊C4C22×C418yes—
257C2×C22⋊C4C22×C418yes—
258C2×C22⋊C4C22×C418yes—
259C2×C22⋊C4#4118yes—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—416no—
267C2×C42—44no—
268C2×C42—41no—
269C2×C42—41no—
270C2×C42—41no—
271C2×C42—41no—
272C2×C42—42no—
273C2×C42—42no—
274C2×C42—44no—
275C2×C42—42no—
276C2×C42—42no—
277C2×C42—44no—
278C2×C42—44no—
279C2×C42C22×C414yes—
280C2×C42C22×C414yes—
281C2×C42C22×C418yes—
282C2×C42C22×C414yes—
283C2×C42C22×C414yes—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—28no—
294#95—24no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—42no—
303#95—42no—
304#95—42no—
305#95—42no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95C22×C418yes—
310#95C22×C418yes—
311#95C22×C418yes—
312C24—24no—
313C24—22no—
314C24—24no—
315C24—22no—
316C24—21no—
317C24—21no—
318C24—22no—
319C24—22no—
320C24—24no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—24no—
326C24—22no—
327C24C23×C414yes—
328C24C23×C412yes—
329C24C2×#4114yes—
330C24C2×#4114yes—
331C24C2×#4114yes—
332C24C2511yescharacteristic
333C2×D4—216no—
334C2×D4—216no—
335C2×D4—44no—
336C2×D4—48no—
337C2×D4—44no—
338C2×D4—48no—
339C2×D4—48no—
340C2×D4—416no—
341C2×D4—416no—
342C2×D4#117116yes—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—28no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—216no—
386C22×C4—24no—
387C22×C4—24no—
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—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—416no—
419C22×C4—416no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—88no—
423C22×C4—88no—
424C22×C4—88no—
425C22×C4—88no—
426C22×C4—88no—
427C22×C4—88no—
428C22×C4—88no—
429C22×C4C23×C412yes—
430C22×C4C23×C412yes—
431C22×C4C23×C412yes—
432C22×C4C23×C414yes—
433C22×C4C23×C414yes—
434C22×C4C23×C414yes—
435C22×C4C23×C412yes—
436C22×C4C23×C412yes—
437C22×C4C23×C412yes—
438C22×C4C2×#4118yes—
439C22×C4C2×#4114yes—
440C22×C4H(5,𝔽2)12yes—
441C22×C4H(5,𝔽2)12yes—
442C22×C4H(5,𝔽2)14yes—
443C4⋊C4—28no—
444C4⋊C4—216no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—88no—
448C22⋊C4—216no—
449C22⋊C4—216no—
450C22⋊C4—48no—
451C22⋊C4—48no—
452C22⋊C4—48no—
453C22⋊C4—48no—
454C22⋊C4—416no—
455C22⋊C4—48no—
456C22⋊C4—88no—
457C22⋊C4—88no—
458C22⋊C4—88no—
459C42—28no—
460C42—28no—
461C42—44no—
462C42—44no—
463C42—48no—
464C42—84no—
465C42—84no—
466C42—88no—
467C23—22no—
468C23—24no—
469C23—24no—
470C23—22no—
471C23—24no—
472C23—22no—
473C23—24no—
474C23—22no—
475C23—24no—
476C23—22no—
477C23—22no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—28no—
482C23—28no—
483C23—28no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—28no—
489C23—28no—
490C23—28no—
491C23—44no—
492C23—44no—
493C23—42no—
494C23—42no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23C2×#11718yes—
506C23C4×#4114yes—
507C23C24×C411yescharacteristic
508C23C22×#4112yes—
509C23C22×#4112yes—
510C23C2×H(5,𝔽2)11yescharacteristic
511C23C2×H(5,𝔽2)11yescharacteristic
512C23C2×H(5,𝔽2)12yes—
513C23#58512yes—
514C23#58512yes—
515C23#58512yes—
516D4—416no—
517D4—432no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—416no—
552C2×C4—416no—
553C2×C4—416no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
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—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
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—816no—
592C2×C4—832no—
593C2×C4C2×#11714yes—
594C2×C4#51814yes—
595C2×C4#51814yes—
596C22—24no—
597C22—24no—
598C22—28no—
599C22—28no—
600C22—28no—
601C22—24no—
602C22—44no—
603C22—42no—
604C22—44no—
605C22—44no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—44no—
610C22—48no—
611C22—48no—
612C22—48no—
613C22—48no—
614C22—48no—
615C22—48no—
616C22C22×#11711yescharacteristic
617C22C2×C4×#4111yescharacteristic
618C22C2×#51811yescharacteristic
619C22C2×#51811yescharacteristic
620C22#342811yescharacteristic
621C22#342811yescharacteristic
622C22#342811yescharacteristic
623C22#345912yes—
624C22#346012yes—
625C22#348114yes—
626C22#350512yes—
627C22#350512yes—
628C22#352114yes—
629C22#352114yes—
630C22#353412yes—
631C22#353412yes—
632C22#356214yes—
633C4—24no—
634C4—24no—
635C4—24no—
636C4—42no—
637C4—42no—
638C4—44no—
639C4—44no—
640C4—44no—
641C4—42no—
642C4—48no—
643C4—84no—
644C4—84no—
645C4—84no—
646C2—44no—
647C2—48no—
648C2#3408211yescharacteristic
649C2#3412611yescharacteristic
650C2#3418511yescharacteristic
651C2#3419511yescharacteristic
652C2#3423411yescharacteristic
653C2#3435811yescharacteristic
654C2#3440211yescharacteristic
655C2#4042414yes—
656C2#4042614yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC22×#121C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14683C2semidirect
3. #14453C2semidirect
5. #14375C2semidirect
6. #14280C2semidirect
7. #14038C2semidirect
8. #13960C2semidirect
9. #13950C2semidirect
10. #13919C2semidirect
11. #13901C2semidirect
12. #13890C2semidirect
13. #13813C2semidirect
14. #13791C2semidirect
15. C4×#381C2semidirect
20. C4×#121C4semidirect
89. C2×C4×D4C2×C4semidirect
108. C22×C42C2×C4semidirect
118. #389C2×C4semidirect
119. #388C2×C4semidirect
123. #386C2×C4semidirect
128. #380C2×C4semidirect
129. C4×C4⋊C4C2×C4semidirect
133. C4×C22⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
#146832
#144532
#144172
#143752
#142802
#140382
#139602
#139502
#139192
#139012
#138902
#138132
#137912
C4×#3812