Finite Group Databasebeta

← #1394224

Subgroups — #1394224

Order 512.

2530 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2447C211yesmaximal, characteristic
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3#16699C212yesmaximal, complemented
4#16435C214yesmaximal, complemented
5#16429C214yesmaximal, complemented
6#16394C212yesmaximal, complemented
7#15240C211yesmaximal, characteristic, complemented
8#15225C212yesmaximal, complemented
9#15193C211yesmaximal, characteristic, complemented
10#15186C212yesmaximal, complemented
11#15152C211yesmaximal, characteristic, complemented
12#14843C212yesmaximal, complemented
13#14342C212yesmaximal, complemented
14#14027C212yesmaximal
15#13979C212yesmaximal
16#13916C211yesmaximal, characteristic
17C4×#400C211yesmaximal, characteristic, complemented
18C22×#124C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2212yes—
21C23×C4⋊C4C2212yes—
22#2447—28no—
23C2×#400C2211yescharacteristic
24C2×#400C2212yes—
25C2×#397C2212yes—
26C2×#397C2212yes—
27D4×C4⋊C4—28no—
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#390C2212yes—
33C2×#390C2211yescharacteristic
34C2×#389C2212yes—
35C2×#389C2211yescharacteristic
36C2×#389C2212yes—
37C2×#385C2212yes—
38C2×#385C2214yes—
39C2×#384C2211yescharacteristic
40C2×#382C2212yes—
41C2×#381C2212yes—
42C2×#381C2212yes—
43C2×#380C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46#1445C2211yescharacteristic
47#1445C2212yes—
48#1445C2214yes—
49#1445C2212yes—
50#1445C2214yes—
51#1445C2212yes—
52#1445C2214yes—
53#1445C2214yes—
54#1445C2212yes—
55#1444C2212yes—
56#1443C2214yes—
57#1443C2212yes—
58#1443C2214yes—
59#1442C2212yes—
60#1442C2211yescharacteristic
61#1442C2214yes—
62#1442C2214yes—
63#1442C2212yes—
64#1441C2211yescharacteristic
65#1441C2214yes—
66#1441C2212yes—
67#1441C2214yes—
68#1440C2214yes—
69#1440C2214yes—
70#1440C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73C4⋊#95C2212yes—
74C4⋊#95C2211yescharacteristic
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C22⋊#95C2214yes—
78C22⋊#95C2214yes—
79C22⋊#95C2214yes—
80C22⋊#95C2214yes—
81C22⋊#95C2212yes—
82C4×#95C2212yes—
83C4×#95C2211yescharacteristic
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×D4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2314yes—
89C2×#124—24no—
90C2×#124—28no—
91C2×C4×D4—28no—
92C2×C4×D4—28no—
93C2×C4×D4—44no—
94C2×C4×D4—44no—
95C2×C4×D4D418yes—
96C22×C4⋊C4—28no—
97C22×C4⋊C4—28no—
98C22×C4⋊C4—44no—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#400—24no—
126#400—44no—
127#397—28no—
128#397—28no—
129#391—24no—
130#391—24no—
131#391—44no—
132#391—44no—
133#390—24no—
134#390—28no—
135#389—28no—
136#389—24no—
137#389—44no—
138#385—28no—
139#385—216no—
140#384—24no—
141#382—44no—
142#381—28no—
143#381—44no—
144#380—28no—
145C4×C4⋊C4—24no—
146C4×C4⋊C4—24no—
147C2×#95C2311yescharacteristic
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2311yescharacteristic
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2311yescharacteristic
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2311yescharacteristic
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C25C2412yes—
183C22×Q8—22no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×D4—24no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4C2×D414yes—
192C22×D4#4118yes—
193C23×C4—28no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—28no—
197C23×C4—28no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4C2×D418yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2414yes—
209C23×C4C2411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2414yes—
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221#124—416no—
222C4×D4—232no—
223C4×D4—432no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—432no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4#4114yes—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4C2×D418yes—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42C2×D414yes—
292#95—22no—
293#95—24no—
294#95—22no—
295#95—28no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—22no—
303#95—24no—
304#95—22no—
305#95—24no—
306#95—24no—
307#95—42no—
308#95—42no—
309#95—44no—
310#95—42no—
311#95—42no—
312#95—44no—
313#95—44no—
314#95—42no—
315#95—42no—
316#95—42no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327C24—22no—
328C24—22no—
329C24—24no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—24no—
335C24—24no—
336C24C22×D414yes—
337C24C2×#4118yes—
338C24C2511yescharacteristic
339C2×Q8—44no—
340C2×Q8—44no—
341C2×Q8—44no—
342C2×Q8—48no—
343C2×Q8—48no—
344C2×D4—28no—
345C2×D4—28no—
346C2×D4—216no—
347C2×D4—216no—
348C2×D4#123116yes—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—216no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—216no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—48no—
434C22×C4—416no—
435C22×C4—416no—
436C22×C4—416no—
437C22×C4—416no—
438C22×C4—88no—
439C22×C4—88no—
440C22×C4—88no—
441C22×C4—88no—
442C22×C4—88no—
443C22×C4C22×D412yes—
444C22×C4C22×D414yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C2×#4114yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4114yes—
450C22×C4H(5,𝔽2)12yes—
451C22×C4H(5,𝔽2)14yes—
452C22×C42−1+412yes—
453C22×C42−1+412yes—
454C22×C42−1+412yes—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—44no—
458C4⋊C4—416no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4—44no—
464C4⋊C4—48no—
465C4⋊C4—84no—
466C4⋊C4—84no—
467C4⋊C4—88no—
468C22⋊C4—216no—
469C22⋊C4—48no—
470C22⋊C4—416no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—84no—
474C22⋊C4—88no—
475C22⋊C4—88no—
476C42—28no—
477C42—44no—
478C42—48no—
479C42—44no—
480C42—48no—
481C23—24no—
482C23—24no—
483C23—28no—
484C23—28no—
485C23—24no—
486C23—28no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—28no—
491C23—28no—
492C23—28no—
493C23—44no—
494C23—42no—
495C23—42no—
496C23—44no—
497C23—42no—
498C23—42no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23C2×#12318yes—
504C23C23×D411yescharacteristic
505C23C22×#4112yes—
506C23C22×#4111yescharacteristic
507C23C2×H(5,𝔽2)12yes—
508C23C2×H(5,𝔽2)11yescharacteristic
509C23C2×2−1+411yescharacteristic
510C23C2×2−1+411yescharacteristic
511C23C2×2−1+411yescharacteristic
512C23#58512yes—
513C23#58511yescharacteristic
514C23#58512yes—
515Q8—88no—
516D4—232no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—432no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—832no—
607C2×C4C2×#12314yes—
608C2×C4#53514yes—
609C2×C4#53614yes—
610C2×C4#53918yes—
611C2×C4#53918yes—
612C2×C4#54614yes—
613C22—28no—
614C22—28no—
615C22—28no—
616C22—44no—
617C22—48no—
618C22—48no—
619C22—48no—
620C22—48no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22C22×#12311yescharacteristic
625C22C2×#53511yescharacteristic
626C22C2×#53611yescharacteristic
627C22C2×#53912yes—
628C22C2×#53912yes—
629C22C2×#54611yescharacteristic
630C22#347512yes—
631C22#348511yescharacteristic
632C22#348911yescharacteristic
633C22#350011yescharacteristic
634C22#351312yes—
635C22#351412yes—
636C22#351512yes—
637C22#351811yescharacteristic
638C22#352112yes—
639C22#352311yescharacteristic
640C22#354612yes—
641C22#355211yescharacteristic
642C22#355411yescharacteristic
643C22#355412yes—
644C22#355712yes—
645C22#356212yes—
646C22#356512yes—
647C4—24no—
648C4—44no—
649C4—44no—
650C4—42no—
651C4—42no—
652C4—82no—
653C4—84no—
654C4—81no—
655C4—84no—
656C4—82no—
657C4—82no—
658C4—81no—
659C4—82no—
660C4—82no—
661C4—81no—
662C4#262318yes—
663C2—48no—
664C2C2×#262311yescharacteristic
665C2#3606711yescharacteristic
666C2#3615511yescharacteristic
667C2#3891111yescharacteristic
668C2#3895711yescharacteristic
669C2#3902512yes—
670C2#3903512yes—
671C2#4256712yes—
672C2#4262212yes—
673C2#4264711yescharacteristic
674C2#4268311yescharacteristic
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
3. #16699C2semidirect
4. #16435C2semidirect
5. #16429C2semidirect
6. #16394C2semidirect
7. #15240C2semidirect
8. #15225C2semidirect
9. #15193C2semidirect
10. #15186C2semidirect
11. #15152C2semidirect
12. #14843C2semidirect
13. #14342C2semidirect
17. C4×#400C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24472
C2×D4×C4⋊C42
#166992
#164352
#164292
#163942
#152402
#152252
#151932
#151862
#151522
#148432
#143422
#140272
#139792
#139162
C4×#4002