Finite Group Databasebeta

← #847424

Subgroups — #847424

Order 512.

2542 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×#2734C212yesmaximal
2C2×#2546C211yesmaximal, characteristic
3C2×C4×C4⋊Q8C211yesmaximal, characteristic
4#16629C212yesmaximal
5#16563C212yesmaximal
6#15860C214yesmaximal
7#15676C214yesmaximal
8#15158C212yesmaximal
9#15133C212yesmaximal
10#15128C212yesmaximal
11#14699C212yesmaximal
12#14450C212yesmaximal
13#14046C212yesmaximal
14#14042C212yesmaximal
15#13926C211yesmaximal, characteristic
16C22×C4⋊Q8C2212yes—
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21#2734—216no—
22#2546—28no—
23C2×#398C2214yes—
24C2×#398C2212yes—
25C2×#398C2212yes—
26C2×#395C2214yes—
27C2×#395C2214yes—
28C4×C4⋊Q8—28no—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2211yescharacteristic
32C2×#391C2211yescharacteristic
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2214yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#387C2214yes—
42C2×#387C2214yes—
43C2×#384C2211yescharacteristic
44C2×#384C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C43C2211yescharacteristic
50#1446C2214yes—
51#1444C2214yes—
52#1443C2214yes—
53#1442C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2212yes—
63#1441C2212yes—
64#1441C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76#1434C2214yes—
77#1434C2214yes—
78C23×Q8C2312yes—
79C23×Q8C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8—28no—
84C2×C4⋊Q8—28no—
85C2×C4⋊Q8—28no—
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8—42no—
88C2×C4⋊Q8—42no—
89C2×C4⋊Q8D418yes—
90C2×C4×Q8—24no—
91C2×C4×Q8—42no—
92C2×C4×Q8—42no—
93C2×C4×Q8—42no—
94C2×C4×Q8—42no—
95C2×C4×Q8D418yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127#398—216no—
128#398—28no—
129#398—28no—
130#395—216no—
131#395—216no—
132#391—24no—
133#391—28no—
134#391—24no—
135#391—28no—
136#389—28no—
137#389—28no—
138#389—216no—
139#389—28no—
140#389—28no—
141#389—44no—
142#389—44no—
143#389Q8116yescomplemented
144#387—216no—
145#387—216no—
146#384—24no—
147#384—42no—
148C4×C4⋊C4—28no—
149C4×C4⋊C4—28no—
150C4×C4⋊C4—24no—
151C4×C4⋊C4—42no—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C43—24no—
178C22×Q8—22no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—22no—
182C22×Q8—24no—
183C22×Q8—41no—
184C22×Q8—41no—
185C22×Q8—41no—
186C22×Q8—41no—
187C22×Q8—41no—
188C22×Q8—41no—
189C22×Q8—41no—
190C22×Q8—41no—
191C22×Q8—42no—
192C22×Q8—42no—
193C22×Q8—42no—
194C22×Q8—42no—
195C22×Q8C2×D414yes—
196C22×Q8C2×D414yes—
197C22×Q8C2×D414yes—
198C22×Q8#4114yes—
199C23×C4C2414yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2414yes—
207C23×C4C2414yes—
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C4⋊Q8—232no—
215C4⋊Q8—416no—
216C4⋊Q8—432no—
217C4×Q8—416no—
218C4×Q8—416no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4C2×D418yes—
274C2×C4⋊C4C2×D418yes—
275C2×C4⋊C4C2×D414yes—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4C2×D414yes—
278C2×C4⋊C4C2×Q818yes—
279C2×C4⋊C4C2×Q8116yes—
280C2×C4⋊C4C2×Q818yes—
281C2×C42—22no—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—42no—
302C2×C42—41no—
303C2×C42—41no—
304C2×C42—42no—
305C2×C42—42no—
306C2×C42C2×D414yes—
307C2×C42C2×D414yes—
308C2×C42C2×D414yes—
309C2×C42C2×D414yes—
310C2×C42C2×Q818yes—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—42no—
333#95—44no—
334#95—44no—
335#95C2×Q8116yes—
336C24C2511yescharacteristic
337C2×Q8—28no—
338C2×Q8—44no—
339C2×Q8—44no—
340C2×Q8—44no—
341C2×Q8—44no—
342C2×Q8—44no—
343C2×Q8—44no—
344C2×Q8—44no—
345C2×Q8—44no—
346C2×Q8—44no—
347C2×Q8—44no—
348C2×Q8—44no—
349C2×Q8—44no—
350C2×Q8—48no—
351C2×Q8—48no—
352C2×Q8—48no—
353C2×Q8—84no—
354C2×Q8—84no—
355C2×Q8—84no—
356C2×Q8—84no—
357C2×Q8#121116yes—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—24no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D414yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D414yes—
442C22×C4C22×D412yes—
443C22×C4C22×Q814yes—
444C22×C4C22×Q818yes—
445C22×C4C22×Q818yes—
446C22×C4C22×Q814yes—
447C22×C4C22×Q814yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4H(5,𝔽2)12yes—
452C22×C4H(5,𝔽2)12yes—
453C22×C42−1+414yes—
454C22×C42−1+414yes—
455C4⋊C4—216no—
456C4⋊C4—216no—
457C4⋊C4—216no—
458C4⋊C4—232no—
459C4⋊C4—216no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—44no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—416no—
469C4⋊C4—416no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—84no—
473C4⋊C4—88no—
474C4⋊C4—88no—
475C4⋊C4—84no—
476C42—28no—
477C42—28no—
478C42—28no—
479C42—28no—
480C42—216no—
481C42—48no—
482C42—44no—
483C42—48no—
484C42—48no—
485C42—84no—
486C23C23×D411yescharacteristic
487C23C23×D411yescharacteristic
488C23C23×Q812yes—
489C23C22×#4111yescharacteristic
490C23C2×H(5,𝔽2)11yescharacteristic
491C23C2×H(5,𝔽2)11yescharacteristic
492C23C2×2−1+412yes—
493C23C2×2−1+412yes—
494C23#58511yescharacteristic
495C23#58511yescharacteristic
496C23#58512yes—
497Q8—416no—
498Q8—416no—
499Q8—816no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—44no—
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—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
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—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4C2×#12114yes—
589C2×C4C2×#12114yes—
590C2×C4C2×#12114yes—
591C2×C4#53414yes—
592C2×C4#54418yes—
593C2×C4D4×Q818yes—
594C2×C4D4×Q818yes—
595C2×C4Q8214yes—
596C2×C4#56314yes—
597C22C22×#12111yescharacteristic
598C22C2×#53411yescharacteristic
599C22#344911yescharacteristic
600C22C2×#54412yes—
601C22C2×D4×Q812yes—
602C22C2×D4×Q812yes—
603C22C2×Q8211yescharacteristic
604C22Q8×#4112yes—
605C22#348912yes—
606C22#349611yescharacteristic
607C22#349611yescharacteristic
608C22#349712yes—
609C22#350212yes—
610C22#350412yes—
611C22#350511yescharacteristic
612C22#350511yescharacteristic
613C22#352312yes—
614C22C2×#56311yescharacteristic
615C22#354712yes—
616C22#355011yescharacteristic
617C22#355111yescharacteristic
618C22#355512yes—
619C22#355712yes—
620C4—24no—
621C4—24no—
622C4—24no—
623C4—28no—
624C4—42no—
625C4—42no—
626C4—42no—
627C4—44no—
628C4—44no—
629C4—48no—
630C4—48no—
631C4—82no—
632C4—84no—
633C4—84no—
634C4#273418yescomplemented
635C2#3500011yescharacteristic
636C2Q8×#12112yes—
637C2#3737611yescharacteristic
638C2#3937012yes—
639C2#3959112yes—
640C2#3997512yes—
641C2#4632911yescharacteristic
642C2#4634611yescharacteristic
643C2C2×#273411yescharacteristic
644C2#5983211yescharacteristic
645C2#5992511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
143. #389Q8semidirect
634. C4#2734semidirect

Maximal subgroups

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

subgroupindex
C2×#27342
C2×#25462
C2×C4×C4⋊Q82
#166292
#165632
#158602
#156762
#151582
#151332
#151282
#146992
#144502
#140462
#140422
#139262