Finite Group Databasebeta

← #847421

Subgroups — #847421

Order 512.

3034 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2434C211yesmaximal, characteristic
2C22×#391C211yesmaximal, characteristic
3#16610C214yesmaximal, complemented
4#16110C212yesmaximal, complemented
5#16093C212yesmaximal, complemented
6#15852C214yesmaximal, complemented
7#15092C212yesmaximal
8#14920C212yesmaximal
9#14914C212yesmaximal
10#14695C212yesmaximal
11#14467C212yesmaximal, complemented
12#14450C212yesmaximal, complemented
13#14044C212yesmaximal
14#14040C212yesmaximal
15#13922C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2434—28no—
23C2×#395C2212yes—
24C2×#395C2212yes—
25C2×#391—28no—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#387C2214yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#380C2211yescharacteristic
41C2×C4×C22⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2211yescharacteristic
43C22×#95C2212yes—
44C22×#95C2212yes—
45#1446C2214yes—
46#1446C2212yes—
47#1446C2212yes—
48#1445C2214yes—
49#1445C2214yes—
50#1444C2214yes—
51#1444C2214yes—
52#1444C2214yes—
53#1443C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67C4⋊#95C2214yes—
68#1434C2214yes—
69#1434C2214yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×Q8—28no—
78C23×Q8—42no—
79C23×Q8—42no—
80C23×Q8—44no—
81C23×Q8D418yes—
82C23×Q8C2311yescharacteristic
83C23×Q8C2311yescharacteristic
84C23×Q8C2311yescharacteristic
85C23×Q8C2311yescharacteristic
86C23×Q8C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C24×C4C2312yes—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—28no—
95C2×C4⋊Q8—44no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—42no—
98C2×C22⋊Q8—42no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—42no—
101C22×C4⋊C4—42no—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C42—24no—
119C22×C42—44no—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125#395—28no—
126#395—28no—
127#391—24no—
128#391—24no—
129#391—28no—
130#391—28no—
131#391—28no—
132#391—28no—
133#391—28no—
134#391—44no—
135#387—48no—
136#385—28no—
137#385—28no—
138#385—44no—
139#385—44no—
140#385Q8116yescomplemented
141#380—42no—
142C4×C22⋊C4—24no—
143C4×C22⋊C4—42no—
144C2×#95—28no—
145C2×#95—28no—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C25C2411yescharacteristic
175C22×Q8—232no—
176C22×Q8—22no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—416no—
184C22×Q8—42no—
185C22×Q8—41no—
186C22×Q8—41no—
187C22×Q8—42no—
188C22×Q8—41no—
189C22×Q8—41no—
190C22×Q8—41no—
191C22×Q8—41no—
192C22×Q8—41no—
193C22×Q8—41no—
194C22×Q8—41no—
195C22×Q8—41no—
196C22×Q8—42no—
197C22×Q8—416no—
198C22×Q8—432no—
199C22×Q8—432no—
200C22×Q8C2×D414yes—
201C22×Q8C2×D414yes—
202C22×Q8C2×D414yes—
203C22×Q8C2×D414yes—
204C22×Q8#4118yes—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—44no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4C2×D414yes—
229C23×C4C2×Q818yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2414yes—
233C23×C4C2414yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2414yes—
239C23×C4C2414yes—
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C4⋊Q8—416no—
245C4⋊Q8—416no—
246C22⋊Q8—416no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4—41no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—416no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4C2×Q8116yescomplemented
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4C2×Q818yes—
291C2×C22⋊C4C2×Q818yes—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—28no—
298C2×C42—24no—
299C2×C42—42no—
300C2×C42—42no—
301C2×C42—42no—
302C2×C42—416no—
303C2×C42—42no—
304C2×C42—42no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—42no—
325#95—42no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95C2×Q8116yescomplemented
333C24—22no—
334C24—22no—
335C24—24no—
336C24C22×D412yes—
337C24C22×Q814yes—
338C24H(5,𝔽2)12yes—
339C24H(5,𝔽2)12yes—
340C242−1+414yes—
341C24C2511yescharacteristic
342C2×Q8—28no—
343C2×Q8—28no—
344C2×Q8—28no—
345C2×Q8—28no—
346C2×Q8—28no—
347C2×Q8—44no—
348C2×Q8—44no—
349C2×Q8—44no—
350C2×Q8—44no—
351C2×Q8—44no—
352C2×Q8—44no—
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—48no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8—48no—
359C2×Q8—4128no—
360C2×Q8—48no—
361C2×Q8—48no—
362C2×Q8—44no—
363C2×Q8—44no—
364C2×Q8—44no—
365C2×Q8—84no—
366C2×Q8—84no—
367C2×Q8#123116yes—
368C22×C4—216no—
369C22×C4—216no—
370C22×C4—28no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—216no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—416no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—416no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—88no—
456C22×C4—88no—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×Q814yes—
464C22×C4C22×Q818yes—
465C22×C4C22×Q818yes—
466C22×C4C22×Q814yes—
467C22×C4C2×#4114yes—
468C22×C4C2×#4114yes—
469C22×C4C2×#4114yes—
470C22×C4H(5,𝔽2)14yes—
471C4⋊C4—416no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—44no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—84no—
481C4⋊C4—88no—
482C4⋊C4—88no—
483C22⋊C4—216no—
484C22⋊C4—216no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C42—48no—
488C42—44no—
489C42—44no—
490C42—48no—
491C42—84no—
492C23—22no—
493C23—22no—
494C23—28no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—28no—
499C23—28no—
500C23—28no—
501C23—24no—
502C23—24no—
503C23#53414yes—
504C23#54418yes—
505C23D4×Q818yes—
506C23Q8214yes—
507C23#56314yes—
508C23C23×D411yescharacteristic
509C23C23×Q812yes—
510C23C22×#4112yes—
511C23C2×H(5,𝔽2)11yescharacteristic
512C23C2×H(5,𝔽2)11yescharacteristic
513C23C2×H(5,𝔽2)12yes—
514C23C2×2−1+412yes—
515C23#58514yes—
516Q8—416no—
517Q8—416no—
518Q8—416no—
519Q8—416no—
520Q8—88no—
521C2×C4—232no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—416no—
555C2×C4—416no—
556C2×C4—432no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
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—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—816no—
600C2×C4C2×#12314yes—
601C2×C4C2×#12314yes—
602C2×C4C2×#12314yes—
603C2×C4#53414yes—
604C22—28no—
605C22—28no—
606C22—216no—
607C22—24no—
608C22—28no—
609C22—24no—
610C22—28no—
611C22#273418yes—
612C22C22×#12311yescharacteristic
613C22C2×#53411yescharacteristic
614C22C2×#53411yescharacteristic
615C22C2×#54412yes—
616C22C2×D4×Q812yes—
617C22C2×Q8211yescharacteristic
618C22Q8×#4114yes—
619C22#348712yes—
620C22#349012yes—
621C22#349012yes—
622C22#350114yes—
623C22#351614yes—
624C22C2×#56311yescharacteristic
625C22#354114yes—
626C22#354512yes—
627C22#354512yes—
628C4—24no—
629C4—24no—
630C4—24no—
631C4—24no—
632C4—48no—
633C4—44no—
634C4—48no—
635C4—42no—
636C4—42no—
637C4—44no—
638C4—84no—
639C4—82no—
640C4—84no—
641C4—82no—
642C2—28no—
643C2#3499211yescharacteristic
644C2Q8×#12312yes—
645C2#3718711yescharacteristic
646C2#3844014yes—
647C2#3959512yes—
648C2#4634812yes—
649C2C2×#273411yescharacteristic
650C2#5992312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #16610C2semidirect
4. #16110C2semidirect
5. #16093C2semidirect
6. #15852C2semidirect
11. #14467C2semidirect
12. #14450C2semidirect
140. #385Q8semidirect
279. C2×C4⋊C4C2×Q8semidirect
332. #95C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24342
C22×#3912
#166102
#161102
#160932
#158522
#150922
#149202
#149142
#146952
#144672
#144502
#140442
#140402
#139222