Finite Group Databasebeta

← #723278

Subgroups — #723278

Order 512.

2662 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2349C211yesmaximal, characteristic
2C2×#2326C211yesmaximal, characteristic
3#16400C214yesmaximal, complemented
4#16020C214yesmaximal, complemented
5#15357C214yesmaximal, complemented
6#15193C212yesmaximal, complemented
7#15146C212yesmaximal, complemented
8#14920C214yesmaximal, complemented
9#14276C211yesmaximal, characteristic, complemented
10#14027C212yesmaximal
11#14018C212yesmaximal
12#13942C211yesmaximal, characteristic, complemented
13#13926C211yesmaximal, characteristic, complemented
14#13906C211yesmaximal, characteristic
15#13848C211yesmaximal, characteristic, complemented
16C22×#124C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2212yes—
20C2×#400C2214yes—
21#2349—28no—
22#2326—28no—
23C2×#391C2211yescharacteristic
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#390C2211yescharacteristic
27C2×#390C2212yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#388C2214yes—
31C2×#385C2214yes—
32C2×#385C2214yes—
33C2×#385C2214yes—
34C2×#384C2211yescharacteristic
35C2×#384C2211yescharacteristic
36C2×#382C2211yescharacteristic
37C2×#380C2212yes—
38C2×#380C2214yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2212yes—
42C2×#376C2211yescharacteristic
43C22×#95C2212yes—
44#1445C2214yes—
45#1445C2214yes—
46#1445C2214yes—
47#1444C2214yes—
48#1444C2212yes—
49#1442C2212yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2212yes—
53#1441C2212yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2212yes—
58#1441C2212yes—
59#1441C2214yes—
60#1440C2214yes—
61#1440C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2212yes—
65#1439C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2214yes—
71#1434C2212yes—
72#1434C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C4×#95C2212yes—
77C23×Q8C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C24×C4C2314yes—
80C24×C4C2312yes—
81C2×#124—24no—
82C2×#124—24no—
83C2×#124Q818yes—
84C2×C4×Q8—24no—
85C2×C4×Q8—24no—
86C2×C4×Q8—24no—
87C2×C4×D4—28no—
88C2×C4×D4—24no—
89C2×C4×D4—24no—
90C2×C4×D4—44no—
91C2×C4×D4—44no—
92C2×C4×D4D418yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119#400—48no—
120#391—24no—
121#391—44no—
122#391—44no—
123#390—24no—
124#390—28no—
125#389—28no—
126#389—44no—
127#388—216no—
128#385—216no—
129#385—216no—
130#385—48no—
131#384—24no—
132#384—24no—
133#382—24no—
134#380—28no—
135#380—48no—
136C4×C4⋊C4—24no—
137C4×C4⋊C4—24no—
138C4×C4⋊C4—44no—
139#376—24no—
140C2×#95—28no—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2311yescharacteristic
146C2×#95C2311yescharacteristic
147C2×#95C2312yes—
148C2×#95C2311yescharacteristic
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2311yescharacteristic
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C25C2412yes—
171C22×Q8—22no—
172C22×Q8—22no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8C2×Q814yes—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4C2×D414yes—
183C22×D4C2×Q814yes—
184C22×D4#4114yes—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—28no—
188C23×C4—28no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4C2×D418yes—
197C23×C4C2414yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211#124—232no—
212C4×Q8—416no—
213C4×D4—232no—
214C4×D4—432no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4C2×D414yes—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4C2×D418yes—
272C2×C22⋊C4C2×Q8116yes—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—22no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42C2×D414yes—
288C2×C42C2×Q814yes—
289C2×C42#4114yes—
290C2×C42#4114yes—
291#95—22no—
292#95—24no—
293#95—28no—
294#95—24no—
295#95—24no—
296#95—22no—
297#95—24no—
298#95—28no—
299#95—22no—
300#95—22no—
301#95—28no—
302#95—28no—
303#95—24no—
304#95—24no—
305#95—42no—
306#95—44no—
307#95—42no—
308#95—44no—
309#95—42no—
310#95—42no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320C24—22no—
321C24—22no—
322C24—24no—
323C24—22no—
324C24—22no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24C22×D414yes—
329C24C22×Q814yes—
330C24C2×#4114yes—
331C24C2511yescharacteristic
332C2×Q8—28no—
333C2×Q8—28no—
334C2×Q8—28no—
335C2×Q8—44no—
336C2×Q8—44no—
337C2×Q8—44no—
338C2×Q8—44no—
339C2×D4—28no—
340C2×D4—28no—
341C2×D4—28no—
342C2×D4—28no—
343C2×D4—28no—
344C2×D4—28no—
345C2×D4C22⋊Q8116yes—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—216no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—416no—
420C22×C4—416no—
421C22×C4—416no—
422C22×C4—88no—
423C22×C4—84no—
424C22×C4—84no—
425C22×C4—88no—
426C22×C4—88no—
427C22×C4—88no—
428C22×C4—88no—
429C22×C4—88no—
430C22×C4C22×D414yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×Q814yes—
435C22×C4C22×Q814yes—
436C22×C4C22×Q812yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4112yes—
439C22×C4C2×#4112yes—
440C22×C4C2×#4112yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4H(5,𝔽2)14yes—
445C22×C4H(5,𝔽2)12yes—
446C22×C4H(5,𝔽2)12yes—
447C22×C42−1+414yes—
448C4⋊C4—28no—
449C4⋊C4—48no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—44no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—84no—
458C4⋊C4—84no—
459C4⋊C4—88no—
460C4⋊C4—88no—
461C4⋊C4—88no—
462C22⋊C4—232no—
463C22⋊C4—216no—
464C22⋊C4—416no—
465C22⋊C4—416no—
466C22⋊C4—88no—
467C22⋊C4—88no—
468C42—28no—
469C42—48no—
470C42—44no—
471C42—48no—
472C42—48no—
473C42#123116yes—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—28no—
484C23—28no—
485C23—28no—
486C23—44no—
487C23—44no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23C2×C22⋊Q818yes—
497C23C23×D411yescharacteristic
498C23C23×Q811yescharacteristic
499C23C22×#4111yescharacteristic
500C23C22×#4111yescharacteristic
501C23C22×#4111yescharacteristic
502C23C2×H(5,𝔽2)11yescharacteristic
503C23C2×H(5,𝔽2)11yescharacteristic
504C23C2×H(5,𝔽2)12yes—
505C23C2×2−1+412yes—
506C23#58512yes—
507C23#58512yes—
508Q8—416no—
509D4—232no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
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—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—816no—
592C2×C4—832no—
593C2×C4C2×C22⋊Q814yes—
594C2×C4C2×#12314yes—
595C2×C4C2×#12314yes—
596C2×C4C2×#12314yes—
597C2×C4#53514yes—
598C2×C4#55314yes—
599C2×C4#55314yes—
600C2×C4#55414yes—
601C2×C4#55414yes—
602C22—28no—
603C22—216no—
604C22—44no—
605C22—48no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—48no—
610C22—44no—
611C22—44no—
612C22—48no—
613C22C22×C22⋊Q811yescharacteristic
614C22C22×#12311yescharacteristic
615C22C2×#53511yescharacteristic
616C22#346112yes—
617C22C2×#55311yescharacteristic
618C22C2×#55311yescharacteristic
619C22C2×#55411yescharacteristic
620C22C2×#55411yescharacteristic
621C22#348912yes—
622C22#349412yes—
623C22#349612yes—
624C22#350312yes—
625C22#350412yes—
626C22#350811yescharacteristic
627C22#350911yescharacteristic
628C22#351011yescharacteristic
629C22#351011yescharacteristic
630C22#351312yes—
631C22#351512yes—
632C22#353612yes—
633C22#354012yes—
634C22#354612yes—
635C22#356112yes—
636C4—24no—
637C4—24no—
638C4—44no—
639C4—44no—
640C4—44no—
641C4—44no—
642C4—44no—
643C4—84no—
644C4—82no—
645C4—82no—
646C4—82no—
647C4—84no—
648C4—82no—
649C4#248718yes—
650C2—48no—
651C2C2×#248711yescharacteristic
652C2#3530711yescharacteristic
653C2#3554111yescharacteristic
654C2#3580612yes—
655C2#3925812yes—
656C2#3967512yes—
657C2#4244312yes—
658C2#4507011yescharacteristic
659C2#4522111yescharacteristic
660C2#4542511yescharacteristic
661C2#4543411yescharacteristic
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. #16400C2semidirect
4. #16020C2semidirect
5. #15357C2semidirect
6. #15193C2semidirect
7. #15146C2semidirect
8. #14920C2semidirect
9. #14276C2semidirect
12. #13942C2semidirect
13. #13926C2semidirect
15. #13848C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23492
C2×#23262
#164002
#160202
#153572
#151932
#151462
#149202
#142762
#140272
#140182
#139422
#139262
#139062
#138482