Finite Group Databasebeta

← #545108

Subgroups — #545108

Order 512.

2374 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic
2#14639C212yesmaximal, complemented
3#14596C212yesmaximal
4#14584C214yesmaximal
5#14575C212yesmaximal
6#14327C214yesmaximal, complemented
7#14254C212yesmaximal, complemented
8#14130C212yesmaximal
9#14053C212yesmaximal, complemented
10#14051C212yesmaximal, complemented
11#14045C212yesmaximal, complemented
12#14042C212yesmaximal, complemented
13#13852C211yesmaximal, characteristic
14#13801C212yesmaximal
15#13795C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2628C4116yescomplemented
21C2×#400C2212yes—
22C2×#399C2212yes—
23C2×#395C2212yes—
24C2×#393C2212yes—
25C2×#391C2214yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#387C2214yes—
30C2×#385C2214yes—
31C2×#382C2211yescharacteristic
32C2×#380C2212yes—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C22⋊C4C2212yes—
38#1446C2214yes—
39#1444C2214yes—
40#1444C2214yes—
41#1443C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1441C2211yescharacteristic
46#1441C2211yescharacteristic
47#1441C2212yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1439C2211yescharacteristic
52#1439C2211yescharacteristic
53#1439C2212yes—
54#1437C2214yes—
55#1437C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58C4⋊#95C2212yes—
59C4⋊#95C2212yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67#1434C2214yes—
68#1434C2214yes—
69#1434C2214yes—
70#1434C2214yes—
71C22⋊#95C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C4×#95C2212yes—
75C4×#95C2214yes—
76C23×Q8C2312yes—
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C2×C4⋊Q8—42no—
80C2×C4⋊Q8—42no—
81C2×C4⋊Q8C2×C418yescomplemented
82C2×#117—42no—
83C2×#117—42no—
84C2×#117—42no—
85C2×#117—42no—
86C2×#117C2×C418yes—
87C2×#117Q818yescomplemented
88C22×C4⋊C4—24no—
89C22×C4⋊C4—42no—
90C22×C4⋊C4—42no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2312yes—
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2314yes—
116C22×C42C2314yes—
117C22×C42C2312yes—
118#400C2×C4116yescomplemented
119#399C2×C4116yes—
120#395C2×C4116yescomplemented
121#393C2×C4116yes—
122#391—48no—
123#389—44no—
124#389—44no—
125#389C2×C4116yescomplemented
126#387—48no—
127#385—48no—
128#382C2×C418yescomplemented
129#380C2×C4116yes—
130C4×C4⋊C4—42no—
131C4×C4⋊C4—42no—
132C4×C4⋊C4—44no—
133C4×C4⋊C4—44no—
134C4×C22⋊C4—28no—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2311yescharacteristic
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2311yescharacteristic
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C25C2411yescharacteristic
165C22×Q8—42no—
166C22×Q8—42no—
167C22×Q8—44no—
168C22×Q8—44no—
169C22×Q8C22×C418yes—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4C22×C414yes—
180C23×C4C2×Q814yes—
181C23×C4#4114yes—
182C23×C4C2412yes—
183C23×C4C2412yes—
184C23×C4C2414yes—
185C23×C4C2411yescharacteristic
186C23×C4C2414yes—
187C23×C4C2414yes—
188C23×C4C2411yescharacteristic
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191C23×C4C2414yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C4⋊Q8—416no—
197#117—416no—
198#117—416no—
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—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—416no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4C22×C418yes—
234C2×C4⋊C4C22×C418yes—
235C2×C4⋊C4C22×C414yes—
236C2×C4⋊C4C22×C414yes—
237C2×C4⋊C4C22×C418yes—
238C2×C4⋊C4C22×C4116yes—
239C2×C4⋊C4C2×Q818yes—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—41no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4C22×C414yes—
255C2×C22⋊C4C22×C414yes—
256C2×C22⋊C4C22×C4116yes—
257C2×C22⋊C4C2×Q818yes—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—28no—
269C2×C42—28no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42—44no—
273C2×C42—44no—
274C2×C42—44no—
275C2×C42—44no—
276C2×C42C22×C414yes—
277C2×C42C22×C414yes—
278C2×C42C2×Q818yes—
279#95—24no—
280#95—24no—
281#95—28no—
282#95—28no—
283#95—24no—
284#95—24no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—42no—
289#95—44no—
290#95—44no—
291#95—42no—
292#95—42no—
293#95—41no—
294#95—41no—
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—44no—
304#95C22×C4116yes—
305#95C22×C418yes—
306#95C22×C418yes—
307#95C22×C4116yes—
308C24—22no—
309C24—21no—
310C24—21no—
311C24—21no—
312C24—22no—
313C24—21no—
314C24—22no—
315C24—22no—
316C24C23×C412yes—
317C24C22×Q812yes—
318C24C2×#4112yes—
319C24C2511yescharacteristic
320C2×Q8—48no—
321C2×Q8—416no—
322C2×Q8—88no—
323C2×Q8—84no—
324C2×Q8—84no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—28no—
330C22×C4—28no—
331C22×C4—28no—
332C22×C4—28no—
333C22×C4—28no—
334C22×C4—28no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—48no—
403C22×C4—88no—
404C22×C4—84no—
405C22×C4—88no—
406C22×C4—84no—
407C22×C4C4×Q8116yes—
408C22×C4C23×C412yes—
409C22×C4C23×C418yes—
410C22×C4C23×C412yes—
411C22×C4C23×C412yes—
412C22×C4C23×C418yes—
413C22×C4C23×C412yes—
414C22×C4C23×C412yes—
415C22×C4C23×C412yes—
416C22×C4C22×Q818yes—
417C22×C4C22×Q812yes—
418C22×C4C22×Q812yes—
419C22×C4C2×#4112yes—
420C22×C4C2×#4112yes—
421C22×C4H(5,𝔽2)14yes—
422C22×C4H(5,𝔽2)12yes—
423C22×C4H(5,𝔽2)12yes—
424C22×C4H(5,𝔽2)12yes—
425C22×C42−1+414yes—
426C22×C42−1+412yes—
427C4⋊C4—216no—
428C4⋊C4—216no—
429C4⋊C4—216no—
430C4⋊C4—416no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—48no—
434C4⋊C4—44no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—44no—
438C4⋊C4—48no—
439C4⋊C4—88no—
440C4⋊C4—88no—
441C22⋊C4—416no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—48no—
445C22⋊C4—48no—
446C22⋊C4—48no—
447C42—28no—
448C42—28no—
449C42—48no—
450C42—48no—
451C42—48no—
452C42—88no—
453C42—88no—
454C23—22no—
455C23—22no—
456C23—24no—
457C23—24no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—41no—
467C23—41no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—41no—
473C23—41no—
474C23—42no—
475C23—42no—
476C23C2×C4×Q814yes—
477C23C24×C411yescharacteristic
478C23C23×Q811yescharacteristic
479C23C22×#4111yescharacteristic
480C23C2×H(5,𝔽2)11yescharacteristic
481C23C2×H(5,𝔽2)11yescharacteristic
482C23C2×H(5,𝔽2)11yescharacteristic
483C23C2×H(5,𝔽2)12yes—
484C23C2×2−1+411yescharacteristic
485C23C2×2−1+412yes—
486C23#58512yes—
487C23#58512yes—
488Q8—816no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—232no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—28no—
501C2×C4—28no—
502C2×C4—28no—
503C2×C4—28no—
504C2×C4—24no—
505C2×C4—28no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—28no—
514C2×C4—28no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—41no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—816no—
573C2×C4C2×C4×Q814yes—
574C2×C4C2×C4×Q814yes—
575C2×C4#51814yes—
576C2×C4#52014yes—
577C2×C4#53814yes—
578C2×C4#53914yes—
579C2×C4#54314yes—
580C2×C4#54414yes—
581C22—24no—
582C22—28no—
583C22—42no—
584C22—42no—
585C22—44no—
586C22—44no—
587C22—42no—
588C22—44no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22C22×C4×Q811yescharacteristic
593C22C2×#51811yescharacteristic
594C22C2×#52011yescharacteristic
595C22C4×H(5,𝔽2)12yes—
596C22C4×2−1+412yes—
597C22C2×#53811yescharacteristic
598C22C2×#53911yescharacteristic
599C22C2×#54311yescharacteristic
600C22C2×#54411yescharacteristic
601C22#349412yes—
602C22#349512yes—
603C22#350312yes—
604C22#350412yes—
605C22#353811yescharacteristic
606C22#354011yescharacteristic
607C22#354212yes—
608C22#354412yes—
609C22#354612yes—
610C22#354712yes—
611C22#354911yescharacteristic
612C22#355111yescharacteristic
613C22#356112yes—
614C22#356512yes—
615C4—24no—
616C4—42no—
617C4—42no—
618C4—42no—
619C4—48no—
620C4—42no—
621C4—48no—
622C4—48no—
623C4—44no—
624C4—44no—
625C4—82no—
626C4—82no—
627C4—84no—
628C4#232918yes—
629C2—44no—
630C2C2×#232911yescharacteristic
631C2#3427212yes—
632C2#3441211yescharacteristic
633C2#3441511yescharacteristic
634C2#3442912yes—
635C2#4295412yes—
636C2#4295511yescharacteristic
637C2#4295712yes—
638C2#4295811yescharacteristic
639C2#4297311yescharacteristic
640C2#4297511yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14639C2semidirect
6. #14327C2semidirect
7. #14254C2semidirect
9. #14053C2semidirect
10. #14051C2semidirect
11. #14045C2semidirect
12. #14042C2semidirect
20. #2628C4semidirect
81. C2×C4⋊Q8C2×C4semidirect
87. C2×#117Q8semidirect
118. #400C2×C4semidirect
120. #395C2×C4semidirect
125. #389C2×C4semidirect
128. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
#146392
#145962
#145842
#145752
#143272
#142542
#141302
#140532
#140512
#140452
#140422
#138522
#138012
#137952