Finite Group Databasebeta

← #4443904

Subgroups — #4443904

Order 512.

3052 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2619C211yesmaximal, characteristic
2C2×#2366C211yesmaximal, characteristic, complemented
3#16573C214yesmaximal, complemented
4#16530C214yesmaximal, complemented
5#16074C212yesmaximal, complemented
6#15984C212yesmaximal, complemented
7#15342C212yesmaximal, complemented
8#15260C212yesmaximal, complemented
9#14731C214yesmaximal, complemented
10#14721C214yesmaximal, complemented
11#14661C211yesmaximal, characteristic, complemented
12#14298C212yesmaximal, complemented
13#13938C212yesmaximal, complemented
14C22×#127C2211yescharacteristic
15C22×#123C2212yes—
16C22×#121C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×C4×D4C2214yes—
19C22×#117C2211yescharacteristic
20#2619—28no—
21C2×#397C2214yes—
22C2×#394C2214yes—
23C2×#394C2214yes—
24C2×#392C2214yes—
25#2366—28no—
26C2×#390C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#390C2214yes—
29C2×#386C2212yes—
30C2×#386C2212yes—
31C2×#386C2212yes—
32C2×#385C2214yes—
33C2×#385C2212yes—
34C2×#383C2212yes—
35C2×#382C2211yescharacteristic
36C2×#381C2212yes—
37C2×C4×C4⋊C4C2212yes—
38#1444C2214yes—
39#1444C2214yes—
40#1442C2214yes—
41#1442C2214yes—
42#1441C2212yes—
43#1441C2212yes—
44#1440C2214yes—
45#1440C2212yes—
46#1440C2214yes—
47#1440C2214yes—
48#1440C2214yes—
49#1440C2214yes—
50#1440C2212yes—
51#1439C2212yes—
52#1439C2212yes—
53#1439C2214yes—
54#1439C2214yes—
55#1438C2212yes—
56#1438C2214yes—
57#1438C2214yes—
58#1438C2212yes—
59#1438C2214yes—
60#1438C2212yes—
61#1438C2212yes—
62#1437C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1434C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C23×D4C2312yes—
70C23×D4C2312yes—
71C23×D4C2314yes—
72C23×D4C2311yescharacteristic
73C23×D4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2312yes—
76C24×C4C2312yes—
77C2×#127—28no—
78C2×#127D418yes—
79C2×#123—44no—
80C2×#123—48no—
81C2×#121—24no—
82C2×#121—24no—
83C2×#121—42no—
84C2×#121—42no—
85C2×#121—42no—
86C2×#121—42no—
87C2×C4×D4—216no—
88C2×C4×D4—48no—
89C2×C4×D4—48no—
90C2×#117—24no—
91C2×#117—28no—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2314yes—
118C22×C42C2311yescharacteristic
119#397—48no—
120#394—216no—
121#394—48no—
122#392—216no—
123#390—24no—
124#390—24no—
125#390—216no—
126#386—28no—
127#386—28no—
128#386—44no—
129#385—44no—
130#385—48no—
131#383—28no—
132#382—24no—
133#381—44no—
134C4×C4⋊C4—28no—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C25C2412yes—
154C25C2412yes—
155C25C2411yescharacteristic
156C22×D4—24no—
157C22×D4—24no—
158C22×D4—24no—
159C22×D4—24no—
160C22×D4—28no—
161C22×D4—28no—
162C22×D4—24no—
163C22×D4—24no—
164C22×D4—44no—
165C22×D4—44no—
166C22×D4—44no—
167C22×D4—44no—
168C22×D4—41no—
169C22×D4—41no—
170C22×D4—41no—
171C22×D4—41no—
172C22×D4—42no—
173C22×D4—42no—
174C22×D4—44no—
175C22×D4—42no—
176C22×D4—42no—
177C22×D4—44no—
178C22×D4—44no—
179C22×D4—44no—
180C22×D4C2×D4116yes—
181C22×D4C2×D414yes—
182C22×D4C2×D414yes—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—28no—
190C23×C4—28no—
191C23×C4—42no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4C2414yes—
199C23×C4C2414yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208#127—232no—
209#123—432no—
210#121—416no—
211#121—416no—
212C4×D4—464no—
213#117—416no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C42—22no—
279C2×C42—28no—
280C2×C42—24no—
281C2×C42—44no—
282C2×C42—44no—
283C2×C42C2×D414yes—
284C2×C42#4118yes—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—24no—
289#95—24no—
290#95—28no—
291#95—28no—
292#95—28no—
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—
303C24—21no—
304C24—22no—
305C24—24no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—22no—
311C24—22no—
312C24—24no—
313C24—24no—
314C24—22no—
315C24—22no—
316C24—21no—
317C24—22no—
318C24—24no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24C22×D414yes—
326C24C22×D414yes—
327C24H(5,𝔽2)12yes—
328C24C2511yescharacteristic
329C2×D4—232no—
330C2×D4—28no—
331C2×D4—28no—
332C2×D4—416no—
333C2×D4—416no—
334C2×D4—416no—
335C2×D4—416no—
336C2×D4—416no—
337C2×D4—416no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—84no—
347C2×D4—84no—
348C2×D4—84no—
349C2×D4—84no—
350C2×D4—88no—
351C2×D4—88no—
352C2×D4—84no—
353C2×D4—84no—
354C2×D4—88no—
355C2×D4—88no—
356C22×C4—24no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—28no—
362C22×C4—24no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—21no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
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—24no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—44no—
413C22×C4—416no—
414C22×C4—416no—
415C22×C4—44no—
416C22×C4—88no—
417C22×C4—88no—
418C22×C4—84no—
419C22×C4—84no—
420C22×C4—84no—
421C22×C4—84no—
422C22×C4C22×D412yes—
423C22×C4C22×D414yes—
424C22×C4C2×#4114yes—
425C22×C4C2×#4114yes—
426C22×C4C2×#4114yes—
427C22×C4H(5,𝔽2)12yes—
428C22×C4H(5,𝔽2)18yes—
429C4⋊C4—216no—
430C4⋊C4—48no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—416no—
434C4⋊C4—48no—
435C4⋊C4—88no—
436C22⋊C4—416no—
437C22⋊C4—416no—
438C22⋊C4—48no—
439C22⋊C4—48no—
440C22⋊C4—416no—
441C22⋊C4—416no—
442C22⋊C4—48no—
443C22⋊C4—48no—
444C22⋊C4—48no—
445C22⋊C4—88no—
446C22⋊C4—84no—
447C22⋊C4—84no—
448C22⋊C4—88no—
449C22⋊C4—84no—
450C22⋊C4—84no—
451C22⋊C4—88no—
452C22⋊C4—88no—
453C42—416no—
454C42—44no—
455C42#123116yescomplemented
456C23—24no—
457C23—24no—
458C23—28no—
459C23—28no—
460C23—24no—
461C23—24no—
462C23—22no—
463C23—24no—
464C23—24no—
465C23—28no—
466C23—28no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—41no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—42no—
477C23—41no—
478C23—42no—
479C23—42no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—42no—
490C23—42no—
491C23—44no—
492C23—42no—
493C23—42no—
494C23—44no—
495C23—42no—
496C23—42no—
497C23—44no—
498C23—42no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—42no—
505C23—42no—
506C23—44no—
507C23—42no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23C23×D411yescharacteristic
514C23C22×#4112yes—
515C23C2×H(5,𝔽2)14yes—
516C23C2×H(5,𝔽2)11yescharacteristic
517C23C2×H(5,𝔽2)11yescharacteristic
518C23#58514yes—
519C23#58512yes—
520D4—464no—
521D4—88no—
522D4—88no—
523D4—816no—
524D4—816no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—22no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—416no—
539C2×C4—41no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
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—816no—
583C2×C4—816no—
584C2×C4C2×#12314yes—
585C2×C4C2×#12318yes—
586C22—44no—
587C22—44no—
588C22—42no—
589C22—48no—
590C22—48no—
591C22—48no—
592C22—48no—
593C22—42no—
594C22—44no—
595C22—44no—
596C22—48no—
597C22—42no—
598C22—44no—
599C22—44no—
600C22—82no—
601C22—82no—
602C22—84no—
603C22—84no—
604C22—84no—
605C22—84no—
606C22—84no—
607C22—84no—
608C22—84no—
609C22—84no—
610C22—84no—
611C22C22×#12311yescharacteristic
612C22#346014yes—
613C22#348412yes—
614C22#348411yescharacteristic
615C22#348511yescharacteristic
616C22#348512yes—
617C22#349412yes—
618C22#349412yes—
619C22#350614yes—
620C22#353212yes—
621C22#353312yes—
622C22#353712yes—
623C22#353914yes—
624C22#354012yes—
625C22#356214yes—
626C4—28no—
627C4—24no—
628C4—48no—
629C4—42no—
630C4—82no—
631C4—84no—
632C4—82no—
633C4—82no—
634C4—84no—
635C4—84no—
636C2—84no—
637C2—84no—
638C2—82no—
639C2#3515212yes—
640C2#3554011yescharacteristic
641C2#4112214yes—
642C2#4113714yes—
643C2#5200411yescharacteristic
644C2#5200912yes—
645C2#5205411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2619C2
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×#2366C2semidirect
3. #16573C2semidirect
4. #16530C2semidirect
5. #16074C2semidirect
6. #15984C2semidirect
7. #15342C2semidirect
8. #15260C2semidirect
9. #14731C2semidirect
10. #14721C2semidirect
11. #14661C2semidirect
12. #14298C2semidirect
13. #13938C2semidirect
455. C42#123semidirect

Maximal subgroups

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

subgroupindex
C2×#26192
C2×#23662
#165732
#165302
#160742
#159842
#153422
#152602
#147312
#147212
#146612
#142982
#139382