Finite Group Databasebeta

← #7598814

Subgroups — #7598814

Order 512.

1664 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59823C211yesmaximal, characteristic, complemented
2#59821C211yesmaximal, characteristic, complemented
3#31814C214yesmaximal, complemented
4#29287C211yesmaximal, characteristic, complemented
5#29080C214yesmaximal, complemented
6#28251C211yesmaximal, characteristic
7#27822C212yesmaximal, complemented
8#26021C212yesmaximal, complemented
9#25930C212yesmaximal, complemented
10#25318C212yesmaximal, complemented
11#25300C212yesmaximal, complemented
12#24992C212yesmaximal, complemented
13#23099C212yesmaximal, complemented
14#22522C212yesmaximal, complemented
15#21712C211yesmaximal, characteristic, complemented
16C4×#486C211yesmaximal, characteristic, complemented
17C4×#486C211yesmaximal, characteristic, complemented
18C2×#556—21no—
19C2×#554C2212yes—
20C2×#486C2211yescharacteristic
21C2×#486C2211yescharacteristic
22C2×#459C2212yes—
23C2×#459C2212yes—
24C2×#446C2212yes—
25#2732—22no—
26#2727—22no—
27#2552C2212yes—
28#2548C2212yes—
29#2542—21no—
30#2542C2212yes—
31#2538—22no—
32#2538—21no—
33#2538C2212yes—
34#2512C2214yes—
35#2454C2214yes—
36#2448—22no—
37#2417—22no—
38#2415—22no—
39#2366C2212yes—
40#2354C2212yes—
41Q8×C4⋊C4C2212yes—
42D4×C4⋊C4—21no—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C4×#127C2211yescharacteristic
48#2100C2214yes—
49#2088C2214yes—
50#2071C2214yes—
51#2065C2214yes—
52#2054C2214yes—
53#2054C2214yes—
54#2025C2214yes—
55#2025C2214yes—
56#1992C2214yes—
57#1990C2211yescharacteristic
58#1990C2211yescharacteristic
59#1989C2214yes—
60#1988C2212yes—
61#1987C2212yes—
62#1984C2212yes—
63#1972C2214yes—
64#1957C2211yescharacteristic
65#1957C2211yescharacteristic
66#1948C2212yes—
67#1948C2212yes—
68#1939C2212yes—
69#1939C2212yes—
70#1936C2214yes—
71#1934C2212yes—
72#1934C2212yes—
73#1934C2212yes—
74#1934C2212yes—
75#1926C2214yes—
76#1919C2214yes—
77#1879C2214yes—
78#1879C2214yes—
79#1842C2211yescharacteristic
80#1834C2211yescharacteristic
81#1834C2211yescharacteristic
82#1802C2214yes—
83#1802C2212yes—
84#1802C2212yes—
85#1770C2212yes—
86#1770C2212yes—
87#1762C2214yes—
88C4×#102C2212yes—
89C4×#102C2212yes—
90C4×#102C2212yes—
91C4×#102C2212yes—
92C42×C8C2211yescharacteristic
93C22×D8C2314yes—
94#556—48no—
95#554—28no—
96#554—44no—
97C2×C4⋊Q8D411yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×#127C2311yescharacteristic
101C2×#127C2311yescharacteristic
102C2×#126—24no—
103C2×#125C2314yes—
104C2×#124—22no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—24no—
109C2×C22⋊Q8—22no—
110C2×C4×Q8—22no—
111C2×C4×Q8—21no—
112C2×C4×Q8—21no—
113C2×C4×Q8C2312yes—
114C2×C4×D4—22no—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4C2312yes—
120C2×C4×D4C2312yes—
121C2×C4×D4C2314yes—
122C2×#117—22no—
123C22×C4⋊C4—22no—
124#486—24no—
125#486—24no—
126#459—28no—
127#459—44no—
128#446—44no—
129C2×#106C2314yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2314yes—
134C2×#102C2312yes—
135C2×#102C2314yes—
136C2×#102C2312yes—
137C2×#102C2312yes—
138C2×#102C2312yes—
139C2×#102C2312yes—
140C2×#102C2312yes—
141C2×#102C2312yes—
142C2×#102C2314yes—
143C2×#102C2312yes—
144C2×#97C2314yes—
145C2×C4×C8C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147C2×C4×C8C2311yescharacteristic
148C2×C4×C8C2311yescharacteristic
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151#399—24no—
152#399—24no—
153#398—24no—
154#397—24no—
155#396—24no—
156#396—24no—
157#396—24no—
158#394—24no—
159#391—21no—
160#391—21no—
161#391D412yes—
162#390—22no—
163#390—22no—
164#390—21no—
165#390—21no—
166#390—21no—
167#390—21no—
168#390C2312yes—
169#390C2312yes—
170#390C2314yes—
171#390C2314yes—
172#390C2314yes—
173#389—22no—
174#389—24no—
175#389—22no—
176#389—21no—
177#389—21no—
178#389D412yes—
179#389D412yes—
180#389C2312yes—
181#389C2312yes—
182#389C2314yes—
183#388—24no—
184#388—24no—
185#386—24no—
186#385—24no—
187#384—22no—
188#384D411yescharacteristic
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#384C2314yes—
194#380—22no—
195#380—22no—
196#380—24no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2314yes—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C43C2311yescharacteristic
205#340C2314yes—
206#340C2314yes—
207#340C2312yes—
208#340C2314yes—
209#340C2312yes—
210#340C2312yes—
211#340C2314yes—
212#340C2312yes—
213#340C2312yes—
214#340C2312yes—
215#340C2312yes—
216#340C2312yes—
217#336C2314yes—
218#336C2314yes—
219C22×Q8—21no—
220C22×Q8—21no—
221C22×Q8C2×D412yes—
222C22×D4—21no—
223C22×D4—21no—
224C22×D4—21no—
225C22×D4—21no—
226C22×D4C2412yes—
227C22×D4C2412yes—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—24no—
231C2×D8—28no—
232C2×D8—44no—
233C2×D8—44no—
234C22×C8C2414yes—
235C22×C8C2411yescharacteristic
236C22×C8C2411yescharacteristic
237C22×C8C2411yescharacteristic
238C22×C8C2411yescharacteristic
239C4⋊Q8—24no—
240C4⋊Q8—22no—
241C4⋊Q8—22no—
242C4⋊Q8—22no—
243C4⋊Q8#4114yes—
244#127—22no—
245#127—22no—
246#127—22no—
247#127#4114yes—
248#126—416no—
249#125—28no—
250#125—44no—
251#124—48no—
252C22⋊Q8—48no—
253C22⋊Q8—416no—
254C22⋊Q8—48no—
255C22⋊Q8—84no—
256C22⋊Q8—84no—
257C4×Q8—24no—
258C4×Q8—24no—
259C4×Q8—48no—
260C4×Q8—44no—
261C4×Q8—82no—
262C4×D4—28no—
263C4×D4—24no—
264C4×D4—24no—
265C4×D4—24no—
266C4×D4—24no—
267C4×D4—44no—
268C4×D4—44no—
269C4×D4—44no—
270C4×D4—82no—
271C4×D4—82no—
272C4×D4—84no—
273#117—48no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—48no—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D414yes—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C42—24no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2414yes—
325C2×C42C2411yescharacteristic
326C2×C42C2411yescharacteristic
327C2×C42C2411yescharacteristic
328#106—44no—
329C4⋊C8—24no—
330C4⋊C8—24no—
331C4⋊C8—42no—
332C4⋊C8—44no—
333#102—28no—
334#102—24no—
335#102—24no—
336#102—24no—
337#102—24no—
338#102—24no—
339#102—24no—
340#102—42no—
341#102—42no—
342#102—44no—
343#97—44no—
344C4×C8—22no—
345C4×C8—22no—
346C4×C8—22no—
347C4×C8—22no—
348C4×C8—22no—
349C4×C8—41no—
350#95—22no—
351#95—22no—
352#95—24no—
353#95—22no—
354#95—22no—
355#95—24no—
356#95C2×D414yes—
357#95C2×D414yes—
358C24—22no—
359C24—22no—
360C2×Q8—24no—
361C2×Q8—24no—
362C2×Q8—24no—
363C2×Q8—24no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—81no—
368C2×Q8—81no—
369C2×D4—24no—
370C2×D4—24no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—81no—
389C2×D4—81no—
390C2×D4—81no—
391C2×D4—81no—
392C2×D4C2×#4114yes—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—48no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—84no—
407C22×C4—84no—
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D414yes—
410C22×C4C22×D414yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C2511yescharacteristic
417D8—88no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—24no—
425C2×C8—21no—
426C2×C8—24no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—24no—
432C2×C82−1+412yes—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—28no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—42no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—81no—
461C4⋊C4—81no—
462C4⋊C4—81no—
463C4⋊C4—81no—
464C4⋊C4—82no—
465C4⋊C4—84no—
466C4⋊C4—84no—
467C4⋊C4C2×#4114yes—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—84no—
476C22⋊C4—82no—
477C22⋊C4—84no—
478C22⋊C4—82no—
479C22⋊C4—84no—
480C22⋊C4—84no—
481C42—22no—
482C42—22no—
483C42—24no—
484C42—24no—
485C42—21no—
486C42—21no—
487C42—41no—
488C42—48no—
489C42—44no—
490C42—44no—
491C42AGL(1,ℤ/8)14yes—
492C42C2×#4112yes—
493C42C2×#4112yes—
494C42H(5,𝔽2)12yes—
495C422−1+412yes—
496C23—24no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23C23×D411yescharacteristic
511Q8—48no—
512Q8—84no—
513Q8—84no—
514D4—24no—
515D4—24no—
516D4—48no—
517D4—42no—
518D4—42no—
519D4—42no—
520D4—42no—
521D4—42no—
522D4—42no—
523D4—84no—
524D4—84no—
525D4—84no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—21no—
537C2×C4—24no—
538C2×C4—21no—
539C2×C4—24no—
540C2×C4—21no—
541C2×C4—24no—
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—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—88no—
560C2×C4—88no—
561C2×C4—168no—
562C2×C4#54612yes—
563C2×C4#54612yes—
564C2×C4#54812yes—
565C2×C4#54812yes—
566C2×C4C2×AGL(1,ℤ/8)12yes—
567C2×C4C2×AGL(1,ℤ/8)12yes—
568C2×C4C2×AGL(1,ℤ/8)12yes—
569C2×C4#57612yes—
570C2×C4#57712yes—
571C2×C4C22×#4111yescharacteristic
572C2×C4C22×#4111yescharacteristic
573C2×C4C2×H(5,𝔽2)11yescharacteristic
574C2×C4C2×2−1+411yescharacteristic
575C2×C4C2×2−1+411yescharacteristic
576C2×C4#58511yescharacteristic
577C8—22no—
578C8—41no—
579C8—41no—
580C8—44no—
581C8—41no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22C2×#54611yescharacteristic
597C22C2×#54811yescharacteristic
598C22#349711yescharacteristic
599C22C22×AGL(1,ℤ/8)11yescharacteristic
600C22C2×#57611yescharacteristic
601C22C2×#57711yescharacteristic
602C22#358411yescharacteristic
603C4—22no—
604C4—22no—
605C4—22no—
606C4—44no—
607C4—44no—
608C4—41no—
609C4—44no—
610C4—82no—
611C4—82no—
612C4—82no—
613C4—82no—
614C4—82no—
615C4#333214yes—
616C4#336614yes—
617C4#348211yescharacteristic
618C4#351411yescharacteristic
619C4#351811yescharacteristic
620C4#355211yescharacteristic
621C2—84no—
622C2—162no—
623C2#5942111yescharacteristic
624C2C2×#333211yescharacteristic
625C2#6121111yescharacteristic
626C2C2×#336611yescharacteristic
627C2#6132611yescharacteristic
628C2#6236511yescharacteristic
629C2#6238111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#28251C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59823C2semidirect
2. #59821C2semidirect
3. #31814C2semidirect
4. #29287C2semidirect
5. #29080C2semidirect
7. #27822C2semidirect
8. #26021C2semidirect
9. #25930C2semidirect
10. #25318C2semidirect
11. #25300C2semidirect
12. #24992C2semidirect
13. #23099C2semidirect
14. #22522C2semidirect
15. #21712C2semidirect
16. C4×#486C2semidirect
17. C4×#486C2semidirect

Maximal subgroups

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

subgroupindex
#598232
#598212
#318142
#292872
#290802
#282512
#278222
#260212
#259302
#253182
#253002
#249922
#230992
#225222
#217122
C4×#4862