Finite Group Databasebeta

← #613010

Subgroups — #613010

Order 512.

4988 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×Q8C211yesmaximal, characteristic, complemented
2C2×#2487C211yesmaximal, characteristic
3C2×#2409C211yesmaximal, characteristic
4C2×C4×#123C211yesmaximal, characteristic
5#15092C212yesmaximal, complemented
6#15091C212yesmaximal
7#14876C214yesmaximal, complemented
8#14875C214yesmaximal, complemented
9#14706C212yesmaximal
10#14705C212yesmaximal, complemented
11#14685C211yesmaximal, characteristic, complemented
12#14017C214yesmaximal, complemented
13#13969C214yesmaximal, complemented
14#13924C211yesmaximal, characteristic, complemented
15#13922C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×#123C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8—28no—
20C22×C4×Q8—28no—
21C22×C4×Q8—28no—
22C22×C4×Q8C2214yes—
23C22×C4×Q8C2214yes—
24C22×C4×D4C2211yescharacteristic
25C23×C4⋊C4C2212yes—
26C23×C4⋊C4C2211yescharacteristic
27C23×C42C2212yes—
28C23×C42C2211yescharacteristic
29#2487—28no—
30#2409—28no—
31C2×#400C2212yes—
32C2×#399C2212yes—
33C2×#395C2212yes—
34C2×#393C2212yes—
35C4×#123—28no—
36C2×#391C2214yes—
37C2×#391C2212yes—
38C2×#389C2212yes—
39C2×#388C2212yes—
40C2×#387C2212yes—
41C2×#386C2212yes—
42C2×#386C2212yes—
43C2×#386C2211yescharacteristic
44C2×#385C2212yes—
45C2×#385C2212yes—
46C2×#385C2211yescharacteristic
47C2×#381C2212yes—
48C2×#381C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×#380C2212yes—
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C22⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4C2212yes—
55#1446C2214yes—
56#1445C2214yes—
57#1444C2218yes—
58#1442C2218yes—
59#1441C2218yes—
60#1439C2218yes—
61#1437C2218yes—
62#1436C2218yes—
63#1436C2218yes—
64#1436C2218yes—
65C4⋊#95C2214yes—
66C4⋊#95C2214yes—
67#1434C2214yes—
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C23×Q8—216no—
74C23×Q8—24no—
75C23×Q8—24no—
76C23×Q8D418yes—
77C23×Q8C2312yes—
78C23×Q8C2312yes—
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2312yes—
86C2×#123—24no—
87C2×#123—24no—
88C2×#123Q818yescomplemented
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—216no—
91C2×C4×Q8—2128no—
92C2×C4×Q8—216no—
93C2×C4×Q8—216no—
94C2×C4×Q8—216no—
95C2×C4×Q8—216no—
96C2×C4×Q8—216no—
97C2×C4×Q8—216no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C22×C4⋊C4—28no—
102C22×C4⋊C4—28no—
103C22×C4⋊C4—28no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4D418yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2318yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C42—28no—
128C22×C42—28no—
129C22×C42—28no—
130C22×C42—24no—
131C22×C42—24no—
132C22×C42D418yes—
133C22×C42C2318yes—
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136C22×C42C2312yes—
137C22×C42C2312yes—
138#400—28no—
139#399—28no—
140#395—28no—
141#393—28no—
142#391—28no—
143#391—216no—
144#389—28no—
145#388—28no—
146#387—28no—
147#386—28no—
148#386—28no—
149#386—24no—
150#385—28no—
151#385—28no—
152#385—24no—
153#381—24no—
154#381—28no—
155#380—24no—
156#380—24no—
157#380—28no—
158C4×C4⋊C4—28no—
159C4×C22⋊C4—24no—
160C4×C22⋊C4—28no—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2318yes—
168C2×#95C2318yes—
169C2×#95C2318yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2318yes—
176C2×#95C2318yes—
177C2×#95C2318yes—
178C25C2411yescharacteristic
179C25C2411yescharacteristic
180C22×Q8—264no—
181C22×Q8—264no—
182C22×Q8—264no—
183C22×Q8—232no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—216no—
189C22×Q8—28no—
190C22×Q8C2×D418yes—
191C22×Q8C2×D418yes—
192C22×Q8#41116yes—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4C2×Q814yes—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—216no—
207C23×C4—28no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—28no—
216C23×C4—44no—
217C23×C4—48no—
218C23×C4C2×D418yes—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4C2×D418yes—
225C23×C4C2×Q814yes—
226C23×C4#4114yes—
227C23×C4C2411yescharacteristic
228C23×C4C2414yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2414yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2412yes—
238C23×C4C2414yes—
239C23×C4C2411yescharacteristic
240#123—232no—
241C22⋊Q8—432no—
242C4×Q8—2128no—
243C4×Q8—2128no—
244C4×D4—416no—
245C2×C4⋊C4—264no—
246C2×C4⋊C4—232no—
247C2×C4⋊C4—216no—
248C2×C4⋊C4—216no—
249C2×C4⋊C4—216no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—216no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—48no—
269C2×C4⋊C4C2×D418yes—
270C2×C4⋊C4C2×D418yes—
271C2×C4⋊C4C2×Q818yescomplemented
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—216no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—48no—
297C2×C22⋊C4C2×Q814yes—
298C2×C22⋊C4C2×Q818yescomplemented
299C2×C42—264no—
300C2×C42—232no—
301C2×C42—216no—
302C2×C42—216no—
303C2×C42—216no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42—28no—
313C2×C42C2×D418yes—
314C2×C42C2×D418yes—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—28no—
321#95—24no—
322#95—24no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—216no—
327#95—216no—
328#95—216no—
329#95—48no—
330#95—48no—
331#95—48no—
332C24—24no—
333C24—24no—
334C24—21no—
335C24—21no—
336C24—21no—
337C24—24no—
338C24—21no—
339C24—21no—
340C24—21no—
341C24—24no—
342C24C22×D412yes—
343C24C22×D412yes—
344C24C22×D412yes—
345C24C22×Q812yes—
346C24C22×Q812yes—
347C24C2×#4112yes—
348C242−1+412yes—
349C242−1+412yes—
350C24C2511yescharacteristic
351C2×Q8—2256no—
352C2×Q8—232no—
353C2×Q8—232no—
354C2×Q8—216no—
355C2×Q8—216no—
356C2×Q8—232no—
357C2×Q8—216no—
358C2×Q8—232no—
359C2×Q8—232no—
360C2×Q8#123132yescomplemented
361C2×D4—28no—
362C2×D4—28no—
363C2×D4—28no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—44no—
368C22×C4—232no—
369C22×C4—232no—
370C22×C4—28no—
371C22×C4—216no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—28no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—232no—
386C22×C4—216no—
387C22×C4—216no—
388C22×C4—216no—
389C22×C4—216no—
390C22×C4—28no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—28no—
429C22×C4—24no—
430C22×C4—28no—
431C22×C4—24no—
432C22×C4—48no—
433C22×C4—416no—
434C22×C4—416no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—416no—
438C22×C4—44no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4#120116yes—
443C22×C4C22⋊Q8116yescomplemented
444C22×C4C22×D418yes—
445C22×C4C22×D414yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D414yes—
450C22×C4C22×D418yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×Q814yes—
455C22×C4C22×Q812yes—
456C22×C4C22×Q814yes—
457C22×C4C2×#4114yes—
458C22×C4C2×#41116yes—
459C22×C4C2×#4118yes—
460C4⋊C4—264no—
461C4⋊C4—216no—
462C4⋊C4—216no—
463C4⋊C4—216no—
464C4⋊C4—48no—
465C4⋊C4—44no—
466C4⋊C4—48no—
467C4⋊C4—416no—
468C4⋊C4—48no—
469C4⋊C4—88no—
470C22⋊C4—28no—
471C22⋊C4—216no—
472C22⋊C4—48no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—44no—
478C22⋊C4—416no—
479C22⋊C4—88no—
480C42—264no—
481C42—216no—
482C42—216no—
483C42—44no—
484C42—48no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—28no—
490C23—28no—
491C23—24no—
492C23—24no—
493C23—28no—
494C23—28no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—28no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—41no—
504C23—42no—
505C23—41no—
506C23—41no—
507C23—44no—
508C23—42no—
509C23—42no—
510C23—44no—
511C23—44no—
512C23C2×#12014yes—
513C23C2×C22⋊Q814yes—
514C23#53614yes—
515C23#54814yes—
516C23#54814yes—
517C23D4×Q814yes—
518C23D4×Q814yes—
519C23C23×D411yescharacteristic
520C23C23×D411yescharacteristic
521C23C23×D411yescharacteristic
522C23C23×Q811yescharacteristic
523C23C22×#4111yescharacteristic
524C23C22×#4114yes—
525C23C2×2−1+411yescharacteristic
526C23C2×2−1+411yescharacteristic
527C23#58514yes—
528Q8—264no—
529Q8—264no—
530D4—416no—
531C2×C4—264no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—232no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—216no—
540C2×C4—216no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—232no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—416no—
559C2×C4—432no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—48no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—48no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—48no—
585C2×C4—48no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—48no—
593C2×C4—48no—
594C2×C4—88no—
595C2×C4C2×#12014yes—
596C2×C4C2×#12014yes—
597C2×C4C2×C22⋊Q818yes—
598C2×C4C2×#123116yes—
599C2×C4C2×#12318yes—
600C2×C4#52518yes—
601C2×C4#52518yes—
602C22—24no—
603C22—24no—
604C22—28no—
605C22—28no—
606C22—28no—
607C22—28no—
608C22—28no—
609C22—28no—
610C22—42no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—81no—
615C22—81no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22#240918yes—
620C22C22×#12011yescharacteristic
621C22C22×C22⋊Q811yescharacteristic
622C22C22×#12312yes—
623C22C2×#52512yes—
624C22C2×#52512yes—
625C22C2×#53611yescharacteristic
626C22#344912yes—
627C22C2×#54811yescharacteristic
628C22C2×#54811yescharacteristic
629C22C2×D4×Q811yescharacteristic
630C22C2×D4×Q811yescharacteristic
631C22Q8×#4114yes—
632C22#348014yes—
633C22#348512yes—
634C22#348712yes—
635C22#351614yes—
636C22#351614yes—
637C4—24no—
638C4—216no—
639C4—28no—
640C4—44no—
641C4—48no—
642C4—42no—
643C4—44no—
644C4—81no—
645C4—81no—
646C4—82no—
647C4—84no—
648C4—84no—
649C4#284918yes—
650C2—28no—
651C2—82no—
652C2C2×#240911yescharacteristic
653C2#3495212yes—
654C2#3514911yescharacteristic
655C2#3524512yes—
656C2Q8×#12312yes—
657C2#3654712yes—
658C2#3657812yes—
659C2#3658212yes—
660C2C2×#284911yescharacteristic
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
1. C23×C4×Q8C2semidirect
5. #15092C2semidirect
7. #14876C2semidirect
8. #14875C2semidirect
10. #14705C2semidirect
11. #14685C2semidirect
12. #14017C2semidirect
13. #13969C2semidirect
14. #13924C2semidirect
15. #13922C2semidirect
88. C2×#123Q8semidirect
271. C2×C4⋊C4C2×Q8semidirect
298. C2×C22⋊C4C2×Q8semidirect
360. C2×Q8#123semidirect
443. C22×C4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C23×C4×Q82
C2×#24872
C2×#24092
C2×C4×#1232
#150922
#150912
#148762
#148752
#147062
#147052
#146852
#140172
#139692
#139242
#139222