Finite Group Databasebeta

← #4416620

Subgroups — #4416620

Order 512.

2234 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2830C212yesmaximal
2#17004C212yesmaximal, complemented
3#16788C214yesmaximal, complemented
4#16786C214yesmaximal, complemented
5#16596C212yesmaximal
6#15151C211yesmaximal, characteristic, complemented
7#15132C212yesmaximal, complemented
8#15132C212yesmaximal, complemented
9#15131C212yesmaximal, complemented
10#15129C212yesmaximal
11#15128C211yesmaximal, characteristic, complemented
12#15125C212yesmaximal, complemented
13#14699C212yesmaximal, complemented
14#14634C212yesmaximal, complemented
15C4×#390C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18#2830—216no—
19C2×#398C2212yes—
20C2×#398C2212yes—
21C2×#397C2214yes—
22C2×#397C2214yes—
23C2×#396C2214yes—
24C2×#395C2212yes—
25C2×#394C2214yes—
26C2×#393C2214yes—
27C2×#390C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#387C2214yes—
33C2×#384C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C2212yes—
40C2×C4×C22⋊C4C2212yes—
41C2×C43C2211yescharacteristic
42#1446C2214yes—
43#1446C2212yes—
44#1445C2212yes—
45#1445C2212yes—
46#1444C2212yes—
47#1443C2214yes—
48#1443C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2212yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1440C2214yes—
57#1440C2214yes—
58#1440C2214yes—
59#1440C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1437C2212yes—
63#1437C2212yes—
64#1436C2214yes—
65#1436C2214yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2211yescharacteristic
71C4⋊#95C2211yescharacteristic
72C4⋊#95C2212yes—
73C4⋊#95C2211yescharacteristic
74C4⋊#95C2211yescharacteristic
75#1434C2214yes—
76C23×Q8C2312yes—
77C23×D4C2311yescharacteristic
78C24×C4C2312yes—
79C2×C4⋊Q8—44no—
80C2×C4⋊Q8—42no—
81C2×C4×D4—24no—
82C2×C4×D4—42no—
83C2×C4×D4—42no—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C42C2311yescharacteristic
107C22×C42C2312yes—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2312yes—
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113#398—28no—
114#398—28no—
115#397—216no—
116#397—216no—
117#396—216no—
118#395—28no—
119#394—216no—
120#393—216no—
121#390—24no—
122#390D418yes—
123#389—28no—
124#389—44no—
125#389—44no—
126#387—216no—
127#384—42no—
128#380—44no—
129#380—44no—
130#380—44no—
131#380—44no—
132C4×C4⋊C4—42no—
133C4×C22⋊C4—44no—
134C4×C22⋊C4—44no—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2311yescharacteristic
156C2×#95C2311yescharacteristic
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C43—42no—
163C25C2412yes—
164C22×Q8—24no—
165C22×Q8—44no—
166C22×Q8—42no—
167C22×Q8—44no—
168C22×Q8—42no—
169C22×D4—22no—
170C22×D4—22no—
171C22×D4—41no—
172C22×D4—41no—
173C22×D4—41no—
174C22×D4—41no—
175C22×D4C2×D414yes—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4C2414yes—
184C23×C4C2414yes—
185C23×C4C2414yes—
186C23×C4C2414yes—
187C23×C4C2412yes—
188C23×C4C2411yescharacteristic
189C23×C4C2412yes—
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192C23×C4C2411yescharacteristic
193C23×C4C2411yescharacteristic
194C23×C4C2412yes—
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C4⋊Q8—416no—
198C4×D4—416no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4C2×D414yes—
236C2×C4⋊C4#4114yes—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4C2×D418yes—
263C2×C22⋊C4C2×D418yes—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42—22no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—41no—
276C2×C42—41no—
277C2×C42—42no—
278C2×C42—41no—
279C2×C42—41no—
280C2×C42—42no—
281C2×C42C2×D414yes—
282C2×C42#4114yes—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—24no—
289#95—24no—
290#95—28no—
291#95—24no—
292#95—28no—
293#95—28no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—42no—
298#95—42no—
299#95—42no—
300#95—42no—
301#95—42no—
302#95—41no—
303#95—41no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24C22×D414yes—
325C24C2511yescharacteristic
326C2×Q8—48no—
327C2×Q8—88no—
328C2×Q8—84no—
329C2×Q8—84no—
330C2×Q8—88no—
331C2×D4—28no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—44no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
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—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4C22×D414yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C2×#4112yes—
421C22×C4C2×#4112yes—
422C22×C4C2×#4112yes—
423C22×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4H(5,𝔽2)12yes—
427C22×C4H(5,𝔽2)12yes—
428C22×C4H(5,𝔽2)14yes—
429C22×C42−1+414yes—
430C22×C42−1+414yes—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—48no—
435C4⋊C4—416no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—88no—
441C4⋊C4—88no—
442C4⋊C4—84no—
443C4⋊C4—84no—
444C22⋊C4—48no—
445C22⋊C4—48no—
446C22⋊C4—48no—
447C22⋊C4—48no—
448C22⋊C4—48no—
449C22⋊C4—88no—
450C22⋊C4—88no—
451C22⋊C4—88no—
452C22⋊C4—88no—
453C42—28no—
454C42—44no—
455C42—44no—
456C42—48no—
457C42—44no—
458C42—44no—
459C42—48no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23C23×D411yescharacteristic
496C23C22×#4111yescharacteristic
497C23C22×#4111yescharacteristic
498C23C2×H(5,𝔽2)11yescharacteristic
499C23C2×H(5,𝔽2)11yescharacteristic
500C23C2×H(5,𝔽2)12yes—
501C23C2×2−1+412yes—
502C23C2×2−1+412yes—
503C23#58511yescharacteristic
504C23#58511yescharacteristic
505C23#58512yes—
506Q8—816no—
507D4—416no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
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—42no—
578C2×C4—44no—
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—816no—
589C2×C4#53214yes—
590C2×C4#53814yes—
591C2×C4#54218yes—
592C2×C4#56014yes—
593C2×C4#56218yes—
594C2×C4#56314yes—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22C2×#53211yescharacteristic
611C22C2×#53811yescharacteristic
612C22C2×#54212yes—
613C22#345511yescharacteristic
614C22#346112yes—
615C22#349611yescharacteristic
616C22#349612yes—
617C22#349611yescharacteristic
618C22#349912yes—
619C22#350511yescharacteristic
620C22#350611yescharacteristic
621C22#351812yes—
622C22#352012yes—
623C22C2×#56011yescharacteristic
624C22C2×#56212yes—
625C22C2×#56311yescharacteristic
626C22#354112yes—
627C22#354712yes—
628C22#355011yescharacteristic
629C22#355111yescharacteristic
630C22#355112yes—
631C22#356312yes—
632C22#356512yes—
633C4—24no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—44no—
638C4—44no—
639C4—42no—
640C4—42no—
641C4—82no—
642C4—84no—
643C4—84no—
644C4—84no—
645C4—84no—
646C4#283018yescomplemented
647C2—84no—
648C2#3500511yescharacteristic
649C2#3517012yes—
650C2C2×#283011yescharacteristic
651C2#3751411yescharacteristic
652C2#3758212yes—
653C2#4548212yes—
654C2#4571712yes—
655C2#4614711yescharacteristic
656C2#4619611yescharacteristic
657C2#4632811yescharacteristic
658C2#4634211yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17004C2semidirect
3. #16788C2semidirect
4. #16786C2semidirect
6. #15151C2semidirect
7. #15132C2semidirect
8. #15132C2semidirect
9. #15131C2semidirect
11. #15128C2semidirect
12. #15125C2semidirect
13. #14699C2semidirect
14. #14634C2semidirect
646. C4#2830semidirect

Maximal subgroups

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

subgroupindex
C2×#28302
#170042
#167882
#167862
#165962
#151512
#151322
#151312
#151292
#151282
#151252
#146992
#146342
C4×#3902