Finite Group Databasebeta

← #522156

Subgroups — #522156

Order 512.

2690 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×#2857C211yesmaximal, characteristic
2C2×#2332C212yesmaximal
3#14432C214yesmaximal, complemented
4#14423C212yesmaximal
5#14422C212yesmaximal
6#14292C212yesmaximal
7#14133C212yesmaximal, complemented
8#14053C212yesmaximal, complemented
9#14033C214yesmaximal, complemented
10#14029C212yesmaximal, complemented
11#13994C212yesmaximal
12#13974C212yesmaximal
13#13821C212yesmaximal, complemented
14C4×#380C211yesmaximal, characteristic
15C4×#381C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C23×C42C2211yescharacteristic
19#2857C4116yescomplemented
20#2332—216no—
21C2×#391C2214yes—
22C2×#391C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#388C2214yes—
28C2×#388C2214yes—
29C2×#387C2214yes—
30C2×#387C2214yes—
31C2×#384C2211yescharacteristic
32C2×#381C2211yescharacteristic
33C2×#381C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×C4×C4⋊C4C2214yes—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2212yes—
43C2×C4×C22⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45#1446C2214yes—
46#1445C2214yes—
47#1443C2214yes—
48#1443C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1440C2214yes—
54#1440C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66#1434C2214yes—
67#1434C2212yes—
68#1434C2212yes—
69#1434C2214yes—
70#1434C2212yes—
71C22⋊#95C2211yescharacteristic
72C22⋊#95C2212yes—
73C22⋊#95C2211yescharacteristic
74C4×#95C2212yes—
75C4×#95C2212yes—
76C4×#95C2212yes—
77C23×Q8C2312yes—
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C2×C4⋊Q8—42no—
82C2×C4⋊Q8—42no—
83C2×C4⋊Q8C2×C418yes—
84C2×C22⋊Q8—28no—
85C2×C22⋊Q8—28no—
86C2×C22⋊Q8C2×C4116yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2311yescharacteristic
100C22×C22⋊C4C2311yescharacteristic
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2311yescharacteristic
103C22×C42—42no—
104C22×C42—42no—
105C22×C42C2×C418yes—
106C22×C42C2312yes—
107C22×C42C2312yes—
108C22×C42C2314yes—
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2314yes—
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116#391—216no—
117#391C2×C4116yes—
118#389—28no—
119#389—44no—
120#389C2×C4116yes—
121#389C2×C4116yes—
122#388—216no—
123#388—216no—
124#387—216no—
125#387—216no—
126#384C2×C418yes—
127#381—24no—
128#381D418yes—
129#380—28no—
130#380C2×C4116yes—
131#380C2×C4116yes—
132C4×C4⋊C4—28no—
133C4×C4⋊C4—216no—
134C4×C4⋊C4—42no—
135C4×C4⋊C4—42no—
136C4×C4⋊C4—44no—
137C4×C22⋊C4—28no—
138C4×C22⋊C4—42no—
139C4×C22⋊C4—42no—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C25C2411yescharacteristic
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—42no—
173C22×Q8—42no—
174C22×Q8—42no—
175C22×Q8—42no—
176C22×Q8C22×C418yes—
177C23×C4—24no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—41no—
183C23×C4—41no—
184C23×C4—42no—
185C23×C4—41no—
186C23×C4—41no—
187C23×C4—41no—
188C23×C4—41no—
189C23×C4C22×C414yes—
190C23×C4C22×C414yes—
191C23×C4C22×C414yes—
192C23×C4C2×D414yes—
193C23×C4#4114yes—
194C23×C4C2414yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2414yes—
198C23×C4C2414yes—
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2414yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C4⋊Q8—416no—
209C22⋊Q8—264no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4C22×C418yes—
233C2×C4⋊C4C22×C4116yes—
234C2×C4⋊C4C22×C418yes—
235C2×C4⋊C4C22×C418yes—
236C2×C4⋊C4C22×C418yes—
237C2×C4⋊C4C22×C418yes—
238C2×C4⋊C4#4118yes—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—41no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4C22×C4116yes—
255C2×C22⋊C4C2×D414yes—
256C2×C22⋊C4C2×D414yes—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—28no—
262C2×C42—28no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—28no—
272C2×C42—416no—
273C2×C42—41no—
274C2×C42—41no—
275C2×C42—41no—
276C2×C42—41no—
277C2×C42—44no—
278C2×C42—42no—
279C2×C42—42no—
280C2×C42—44no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42—44no—
284C2×C42C22×C414yes—
285C2×C42C22×C414yes—
286C2×C42C22×C414yes—
287C2×C42C22×C414yes—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—24no—
296#95—24no—
297#95—28no—
298#95—24no—
299#95—24no—
300#95—28no—
301#95—42no—
302#95—42no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—42no—
308#95—42no—
309#95—42no—
310#95—42no—
311#95—44no—
312#95—44no—
313#95C22×C4116yes—
314#95C22×C4116yes—
315#95C2×D418yes—
316#95C2×D418yes—
317C24—22no—
318C24—21no—
319C24—21no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—21no—
324C24—21no—
325C24C23×C412yes—
326C24C22×D412yes—
327C24C2×#4112yes—
328C24C2511yescharacteristic
329C2×Q8—216no—
330C2×Q8—48no—
331C2×Q8—48no—
332C2×Q8—84no—
333C2×Q8—84no—
334C2×Q8—84no—
335C2×Q8—84no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—24no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4C4×D4116yes—
423C22×C4C23×C412yes—
424C22×C4C23×C412yes—
425C22×C4C23×C418yes—
426C22×C4C23×C418yes—
427C22×C4C23×C412yes—
428C22×C4C23×C412yes—
429C22×C4C23×C412yes—
430C22×C4C23×C412yes—
431C22×C4C22×D414yes—
432C22×C4C22×D414yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C2×#4118yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4114yes—
439C22×C4H(5,𝔽2)12yes—
440C22×C4H(5,𝔽2)12yes—
441C22×C42−1+414yes—
442C22×C42−1+414yes—
443C4⋊C4—216no—
444C4⋊C4—232no—
445C4⋊C4—216no—
446C4⋊C4—216no—
447C4⋊C4—216no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—416no—
451C4⋊C4—48no—
452C4⋊C4—88no—
453C22⋊C4—232no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—416no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C42—28no—
460C42—28no—
461C42—44no—
462C42—44no—
463C42—416no—
464C42—48no—
465C42—48no—
466C42—84no—
467C42—84no—
468C42—88no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—24no—
476C23—24no—
477C23—22no—
478C23—22no—
479C23—24no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23C2×C4×D414yes—
492C23C24×C411yescharacteristic
493C23C23×D411yescharacteristic
494C23C22×#4111yescharacteristic
495C23C22×#4112yes—
496C23C2×H(5,𝔽2)11yescharacteristic
497C23C2×H(5,𝔽2)11yescharacteristic
498C23C2×2−1+412yes—
499C23C2×2−1+412yes—
500C23#58512yes—
501C23#58511yescharacteristic
502C23#58511yescharacteristic
503Q8—816no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—28no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—28no—
530C2×C4—22no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—416no—
542C2×C4—416no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—416no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4C2×C4×D414yes—
590C2×C4C2×C4×D414yes—
591C2×C4C4×#4114yes—
592C2×C4#51814yes—
593C2×C4#51914yes—
594C2×C4#51914yes—
595C2×C4#53918yes—
596C2×C4#54218yes—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—44no—
604C22—44no—
605C22—42no—
606C22—44no—
607C22—44no—
608C22C22×C4×D411yescharacteristic
609C22C2×C4×#4111yescharacteristic
610C22C2×#51811yescharacteristic
611C22C2×#51911yescharacteristic
612C22C2×#51911yescharacteristic
613C22#342811yescharacteristic
614C22#342811yescharacteristic
615C22C2×#53912yes—
616C22C2×#54212yes—
617C22D4×#4112yes—
618C22#348012yes—
619C22#349611yescharacteristic
620C22#349611yescharacteristic
621C22#349912yes—
622C22#350511yescharacteristic
623C22#350511yescharacteristic
624C22#350912yes—
625C22#351512yes—
626C22#351812yes—
627C22#354112yes—
628C22#354112yes—
629C22#356512yes—
630C22#356512yes—
631C4—24no—
632C4—24no—
633C4—24no—
634C4—42no—
635C4—42no—
636C4—48no—
637C4—48no—
638C4—48no—
639C4—44no—
640C4—44no—
641C4—82no—
642C4—82no—
643C4—84no—
644C4#233218yescomplemented
645C2—44no—
646C2C2×#233211yescharacteristic
647C2#3414111yescharacteristic
648C2#3420511yescharacteristic
649C2#3420911yescharacteristic
650C2#3423111yescharacteristic
651C2#3437011yescharacteristic
652C2#3440311yescharacteristic
653C2#3940012yes—
654C2#3940312yes—
655C2#3978412yes—
656C2#3978712yes—
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
3. #14432C2semidirect
7. #14133C2semidirect
8. #14053C2semidirect
9. #14033C2semidirect
10. #14029C2semidirect
13. #13821C2semidirect
19. #2857C4semidirect
644. C4#2332semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#23322
#144322
#144232
#144222
#142922
#141332
#140532
#140332
#140292
#139942
#139742
#138212
C4×#3802
C4×#3812