Finite Group Databasebeta

← #1409536

Subgroups — #1409536

Order 512.

2072 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2571C211yesmaximal, characteristic, complemented
2C2×#2513C211yesmaximal, characteristic
3#16863C214yesmaximal, complemented
4#16842C214yesmaximal, complemented
5#16821C212yesmaximal, complemented
6#16701C212yesmaximal, complemented
7#15225C212yesmaximal, complemented
8#15151C211yesmaximal, characteristic, complemented
9#15148C211yesmaximal, characteristic
10#15141C212yesmaximal
11#15136C211yesmaximal, characteristic
12#15020C212yesmaximal, complemented
13#14324C212yesmaximal, complemented
14#14006C212yesmaximal, complemented
15#14002C212yesmaximal
16C4×#400C211yesmaximal, characteristic, complemented
17#13818C211yesmaximal, characteristic, complemented
18#2571—28no—
19#2513—28no—
20C2×#400C2211yescharacteristic
21C2×#400C2214yes—
22C2×#399C2212yes—
23C2×#398C2212yes—
24C2×#398C2212yes—
25C2×#397C2212yes—
26C2×#397C2212yes—
27C2×#397C2212yes—
28C2×#397C2212yes—
29C2×#396C2214yes—
30C2×#390C2211yescharacteristic
31C2×#390C2211yescharacteristic
32C2×#390C2212yes—
33C2×#390C2211yescharacteristic
34C2×#390C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2211yescharacteristic
40C2×#389C2212yes—
41C2×#388C2212yes—
42C2×#388C2212yes—
43C2×#384C2211yescharacteristic
44C2×#382C2212yes—
45C2×#382C2211yescharacteristic
46C2×#382C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51#1445C2211yescharacteristic
52#1445C2212yes—
53#1445C2212yes—
54#1445C2214yes—
55#1443C2212yes—
56#1442C2212yes—
57#1442C2211yescharacteristic
58#1442C2214yes—
59#1442C2214yes—
60#1441C2211yescharacteristic
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1440C2214yes—
65#1440C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2212yes—
71#1439C2214yes—
72#1437C2214yes—
73#1437C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2211yescharacteristic
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2214yes—
81C4×#95C2211yescharacteristic
82C4×#95C2212yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×D4C2311yescharacteristic
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2314yes—
118C22×C42C2312yes—
119#400—216no—
120#400—24no—
121#399—44no—
122#398—44no—
123#398—44no—
124#397—28no—
125#397—28no—
126#397—28no—
127#397—44no—
128#396—48no—
129#390—24no—
130#390—24no—
131#390—28no—
132#390—24no—
133#390—44no—
134#389—28no—
135#389—24no—
136#389—28no—
137#389—28no—
138#389—28no—
139#389—44no—
140#388—28no—
141#388—28no—
142#384—24no—
143#382—28no—
144#382—24no—
145#382—28no—
146C4×C4⋊C4—24no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—44no—
149C4×C4⋊C4D418yes—
150C2×#95C2312yes—
151C2×#95C2311yescharacteristic
152C2×#95C2311yescharacteristic
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2311yescharacteristic
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2311yescharacteristic
181C25C2412yes—
182C22×D4—22no—
183C22×D4—24no—
184C22×D4—22no—
185C22×D4—24no—
186C22×D4—22no—
187C23×C4C2414yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2411yescharacteristic
192C23×C4C2412yes—
193C23×C4C2411yescharacteristic
194C23×C4C2411yescharacteristic
195C23×C4C2412yes—
196C23×C4C2411yescharacteristic
197C23×C4C2412yes—
198C23×C4C2414yes—
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4C2×D414yes—
245C2×C4⋊C4C2×D414yes—
246C2×C4⋊C4#4118yes—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
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⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—22no—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—44no—
286C2×C42—44no—
287C2×C42—44no—
288C2×C42—44no—
289C2×C42C2×D414yes—
290C2×C42C2×D418yes—
291#95—22no—
292#95—22no—
293#95—24no—
294#95—28no—
295#95—24no—
296#95—24no—
297#95—28no—
298#95—28no—
299#95—24no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—22no—
307#95—24no—
308#95—42no—
309#95—42no—
310#95—42no—
311#95—42no—
312#95—42no—
313#95—42no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—42no—
319#95—44no—
320#95C2×D418yes—
321#95#4114yes—
322C24—22no—
323C24—22no—
324C24—24no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—24no—
329C24—22no—
330C24—24no—
331C24—24no—
332C24—22no—
333C24C2511yescharacteristic
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—48no—
338C2×D4—48no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4C22×D414yes—
420C22×C4C22×D414yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C2×#4114yes—
425C22×C4C2×#4118yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4114yes—
428C22×C4H(5,𝔽2)12yes—
429C22×C4H(5,𝔽2)12yes—
430C22×C4H(5,𝔽2)12yes—
431C22×C42−1+414yes—
432C22×C42−1+412yes—
433C4⋊C4—28no—
434C4⋊C4—28no—
435C4⋊C4—216no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—48no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—48no—
443C4⋊C4—88no—
444C4⋊C4—84no—
445C4⋊C4—84no—
446C4⋊C4—88no—
447C4⋊C4—88no—
448C4⋊C4—88no—
449C22⋊C4—48no—
450C22⋊C4—48no—
451C22⋊C4—84no—
452C22⋊C4—84no—
453C22⋊C4—84no—
454C22⋊C4—84no—
455C22⋊C4—88no—
456C22⋊C4—88no—
457C22⋊C4—88no—
458C22⋊C4—88no—
459C42—28no—
460C42—48no—
461C42—48no—
462C42—44no—
463C42—44no—
464C42—48no—
465C42—88no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—44no—
470C23—44no—
471C23—42no—
472C23—44no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—42no—
481C23—44no—
482C23—42no—
483C23—44no—
484C23—42no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23C23×D411yescharacteristic
490C23C22×#4112yes—
491C23C22×#4111yescharacteristic
492C23C2×H(5,𝔽2)11yescharacteristic
493C23C2×H(5,𝔽2)11yescharacteristic
494C23C2×H(5,𝔽2)11yescharacteristic
495C23C2×2−1+412yes—
496C23C2×2−1+411yescharacteristic
497C23#58512yes—
498C23#58512yes—
499C23#58511yescharacteristic
500D4—88no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—28no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—44no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—41no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4#53214yes—
590C2×C4#53614yes—
591C2×C4#53714yes—
592C2×C4#53918yes—
593C2×C4#54118yes—
594C2×C4#55014yes—
595C22—84no—
596C22—82no—
597C22—84no—
598C22—84no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—84no—
605C22—82no—
606C22C2×#53211yescharacteristic
607C22C2×#53611yescharacteristic
608C22C2×#53711yescharacteristic
609C22C2×#53912yes—
610C22C2×#54112yes—
611C22C2×#55011yescharacteristic
612C22#348012yes—
613C22#348511yescharacteristic
614C22#348811yescharacteristic
615C22#350011yescharacteristic
616C22#351412yes—
617C22#351412yes—
618C22#351411yescharacteristic
619C22#351512yes—
620C22#352211yescharacteristic
621C22#352412yes—
622C22#354311yescharacteristic
623C22#354712yes—
624C22#355411yescharacteristic
625C22#355712yes—
626C22#355912yes—
627C22#356212yes—
628C22#356512yes—
629C4—24no—
630C4—44no—
631C4—44no—
632C4—44no—
633C4—42no—
634C4—42no—
635C4—48no—
636C4—81no—
637C4—82no—
638C4—81no—
639C4—81no—
640C4—84no—
641C4—82no—
642C4—82no—
643C4—82no—
644C4#266418yes—
645C2—162no—
646C2C2×#266411yescharacteristic
647C2#3605111yescharacteristic
648C2#3659711yescharacteristic
649C2#3913411yescharacteristic
650C2#3913711yescharacteristic
651C2#3921712yes—
652C2#3922612yes—
653C2#4264511yescharacteristic
654C2#4273912yes—
655C2#4275512yes—
656C2#4296511yescharacteristic
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. C2×#2571C2semidirect
3. #16863C2semidirect
4. #16842C2semidirect
5. #16821C2semidirect
6. #16701C2semidirect
7. #15225C2semidirect
8. #15151C2semidirect
12. #15020C2semidirect
13. #14324C2semidirect
14. #14006C2semidirect
16. C4×#400C2semidirect
17. #13818C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25712
C2×#25132
#168632
#168422
#168212
#167012
#152252
#151512
#151482
#151412
#151362
#150202
#143242
#140062
#140022
C4×#4002
#138182