Finite Group Databasebeta

← #7533926

Subgroups — #7533926

Order 512.

1655 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59823C211yesmaximal, characteristic, complemented
2#36134C211yesmaximal, characteristic, complemented
3#31538C214yesmaximal, complemented
4#29206C211yesmaximal, characteristic, complemented
5#29139C211yesmaximal, characteristic, complemented
6#28800C212yesmaximal, complemented
7#28273C212yesmaximal, complemented
8#27014C214yesmaximal, complemented
9#26530C212yesmaximal, complemented
10#26021C212yesmaximal, complemented
11#25058C212yesmaximal, complemented
12#24327C212yesmaximal, complemented
13#23153C211yesmaximal, characteristic, complemented
14#23099C212yesmaximal
15#22522C212yesmaximal, complemented
16C4×#476C211yesmaximal, characteristic, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#554C2212yes—
19C2×#533—21no—
20C2×#476C2211yescharacteristic
21C2×#459C2211yescharacteristic
22C2×#446C2212yes—
23C2×C4×D8C2212yes—
24#2868C2211yescharacteristic
25#2857—21no—
26#2732—22no—
27#2552C2212yes—
28#2552C2212yes—
29#2542—21no—
30#2542C2212yes—
31#2538—22no—
32#2538C2212yes—
33#2538C2212yes—
34#2454C2214yes—
35#2441—22no—
36#2429—22no—
37#2423—22no—
38#2417—22no—
39#2366C2212yes—
40#2365C2212yes—
41#2354C2212yes—
42Q8×C4⋊C4C2212yes—
43D4×C4⋊C4—21no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C42×D4C2211yescharacteristic
48#2094C2214yes—
49#2084C2214yes—
50#2065C2214yes—
51#2065C2214yes—
52#2059C2214yes—
53#2058C2214yes—
54#2025C2214yes—
55#2023C2214yes—
56#1992C2214yes—
57#1990C2212yes—
58#1989C2214yes—
59#1987C2211yescharacteristic
60#1984C2212yes—
61#1982C2211yescharacteristic
62#1972C2214yes—
63#1957C2212yes—
64#1948C2212yes—
65#1948C2212yes—
66#1948C2211yescharacteristic
67#1947C2211yescharacteristic
68#1944C2214yes—
69#1939C2212yes—
70#1926C2214yes—
71#1925C2212yes—
72#1920C2212yes—
73#1919C2214yes—
74#1917C2214yes—
75#1879C2214yes—
76#1879C2212yes—
77#1864C2214yes—
78#1864C2212yes—
79#1844C2212yes—
80#1834C2212yes—
81#1802C2212yes—
82#1802C2214yes—
83#1802C2211yescharacteristic
84#1802C2212yes—
85#1802C2211yescharacteristic
86C4×#106C2212yes—
87#1770C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90#1762C2214yes—
91C4×#102C2212yes—
92C4×#102C2212yes—
93C22×D8C2312yes—
94#554—28no—
95#554—44no—
96#533—48no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8D411yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#127C2311yescharacteristic
103C2×#125—22no—
104C2×#125C2314yes—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—24no—
110C2×C22⋊Q8—22no—
111C2×C4×Q8—22no—
112C2×C4×Q8—22no—
113C2×C4×Q8—21no—
114C2×C4×Q8C2312yes—
115C2×C4×D4—22no—
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4C2312yes—
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2314yes—
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×#117—22no—
124C22×C4⋊C4—22no—
125C22×C42—21no—
126#476—24no—
127#459Q818yescomplemented
128#446—44no—
129C4×D8—44no—
130C2×#107C2312yes—
131C2×#106C2312yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×#106C2312yes—
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2311yescharacteristic
138C2×C4⋊C8C2314yes—
139C2×C4⋊C8C2311yescharacteristic
140C2×#102C2312yes—
141C2×#102C2314yes—
142C2×#102C2312yes—
143C2×#102C2312yes—
144C2×#102C2312yes—
145C2×#102C2314yes—
146C2×#102C2312yes—
147C2×#102C2312yes—
148C2×#97C2314yes—
149C2×C4×C8C2312yes—
150C2×C4×C8C2312yes—
151#399—24no—
152#399—24no—
153#398—24no—
154#398—24no—
155#396—24no—
156#396—24no—
157#396—24no—
158#391—22no—
159#391—21no—
160#391D412yes—
161#390—22no—
162#390—21no—
163#390—21no—
164#390C2312yes—
165#390C2314yes—
166#390C2314yes—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—24no—
171#389—21no—
172#389D412yes—
173#389D412yes—
174#389C2312yes—
175#389C2314yes—
176#389C2314yes—
177#388—24no—
178#387—24no—
179#386—24no—
180#385—24no—
181#384—22no—
182#384—21no—
183#384—21no—
184#384D411yescharacteristic
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#384C2312yes—
188#384C2314yes—
189#384C2312yes—
190#380—22no—
191#380—22no—
192#380—22no—
193#380—24no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4C2312yes—
196C4×C4⋊C4C2311yescharacteristic
197C4×C4⋊C4C2314yes—
198C4×C4⋊C4C2311yescharacteristic
199C4×C4⋊C4C2311yescharacteristic
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C22⋊C4—21no—
203C4×C22⋊C4—21no—
204C4×C22⋊C4—21no—
205C43C2311yescharacteristic
206#340C2312yes—
207#340C2314yes—
208#340C2314yes—
209#340C2312yes—
210#340C2314yes—
211#340C2312yes—
212#340C2314yes—
213#340C2312yes—
214#340C2312yes—
215#340C2312yes—
216#336C2312yes—
217#336C2314yes—
218#336C2314yes—
219#336C2312yes—
220C22×Q8—21no—
221C22×Q8—22no—
222C22×Q8C2×D412yes—
223C22×D4—21no—
224C22×D4—21no—
225C22×D4C2412yes—
226C22×D4C2411yescharacteristic
227C23×C4—21no—
228C23×C4—22no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—24no—
232C2×D8—42no—
233C2×D8—42no—
234C2×D8C2×Q818yes—
235C22×C8C2412yes—
236C22×C8C2412yes—
237C22×C8C2412yes—
238C22×C8C2412yes—
239C4⋊Q8—24no—
240C4⋊Q8—22no—
241C4⋊Q8—42no—
242C4⋊Q8—42no—
243C4⋊Q8—44no—
244C4⋊Q8—82no—
245C4⋊Q8#4114yes—
246#127—22no—
247#127C2×Q814yes—
248#125—28no—
249#125—44no—
250#125—48no—
251C22⋊Q8—48no—
252C22⋊Q8—416no—
253C22⋊Q8—48no—
254C22⋊Q8—48no—
255C22⋊Q8—84no—
256C22⋊Q8—84no—
257C4×Q8—24no—
258C4×Q8—24no—
259C4×Q8—48no—
260C4×Q8—48no—
261C4×Q8—82no—
262C4×D4—28no—
263C4×D4—24no—
264C4×D4—22no—
265C4×D4—24no—
266C4×D4—22no—
267C4×D4—22no—
268C4×D4—41no—
269C4×D4—41no—
270C4×D4—44no—
271C4×D4—44no—
272C4×D4—82no—
273C4×D4—84no—
274C4×D4C2×Q814yes—
275#117—48no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—48no—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D414yes—
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2411yescharacteristic
302C2×C4⋊C4C2411yescharacteristic
303C2×C4⋊C4C2414yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2412yes—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—44no—
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2414yes—
329C2×C42C2411yescharacteristic
330C2×C42C2411yescharacteristic
331C2×C42C2411yescharacteristic
332#107—42no—
333#106—24no—
334#106—44no—
335#106—44no—
336#106—42no—
337C4⋊C8—22no—
338C4⋊C8—22no—
339C4⋊C8—22no—
340C4⋊C8—44no—
341C4⋊C8C2×Q814yes—
342#102—24no—
343#102—42no—
344#102—42no—
345#102—44no—
346#102—44no—
347#102—42no—
348#102—42no—
349#102C2×Q818yes—
350#97—44no—
351C4×C8—24no—
352C4×C8—24no—
353#95—21no—
354#95—21no—
355#95—21no—
356#95—22no—
357#95—24no—
358#95—22no—
359#95—24no—
360#95—24no—
361#95C2×D414yes—
362#95C2×D414yes—
363C24—21no—
364C24—22no—
365C2×Q8—24no—
366C2×Q8—24no—
367C2×Q8—24no—
368C2×Q8—24no—
369C2×Q8—44no—
370C2×Q8—44no—
371C2×Q8—44no—
372C2×Q8—81no—
373C2×Q8—81no—
374C2×D4—24no—
375C2×D4—24no—
376C2×D4—21no—
377C2×D4—22no—
378C2×D4—21no—
379C2×D4—22no—
380C2×D4—21no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—21no—
384C2×D4—22no—
385C2×D4—21no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—81no—
389C2×D4—81no—
390C2×D4C22×Q814yes—
391C2×D4C22×Q812yes—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—48no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—42no—
412C22×C4—84no—
413C22×C4—84no—
414C22×C4C22×D414yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D414yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C2511yescharacteristic
423D8—48no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8C22×Q814yes—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—28no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—42no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—81no—
465C4⋊C4—81no—
466C4⋊C4—82no—
467C4⋊C4—84no—
468C4⋊C4—84no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C4⋊C4C22×Q812yes—
472C4⋊C4C2×#4114yes—
473C22⋊C4—24no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—82no—
481C22⋊C4—84no—
482C22⋊C4—84no—
483C22⋊C4—84no—
484C42—22no—
485C42—22no—
486C42—24no—
487C42—24no—
488C42—21no—
489C42—21no—
490C42—41no—
491C42—48no—
492C42—42no—
493C42—42no—
494C42—42no—
495C42—44no—
496C42AGL(1,ℤ/8)14yes—
497C42C22×Q812yes—
498C42C2×#4112yes—
499C42H(5,𝔽2)12yes—
500C422−1+412yes—
501C23—22no—
502C23—24no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23—41no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—41no—
514C23—41no—
515C23C23×D411yescharacteristic
516Q8—48no—
517Q8—84no—
518Q8—88no—
519D4—22no—
520D4—24no—
521D4—22no—
522D4—24no—
523D4—48no—
524D4—41no—
525D4—42no—
526D4—41no—
527D4—42no—
528D4—84no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—48no—
551C2×C4—44no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—84no—
567C2×C4—88no—
568C2×C4—84no—
569C2×C4—168no—
570C2×C4#54612yes—
571C2×C4#54612yes—
572C2×C4D4×Q812yes—
573C2×C4D4×Q812yes—
574C2×C4C2×AGL(1,ℤ/8)12yes—
575C2×C4C2×AGL(1,ℤ/8)12yes—
576C2×C4C2×AGL(1,ℤ/8)12yes—
577C2×C4#57712yes—
578C2×C4#57712yes—
579C2×C4C23×Q811yescharacteristic
580C2×C4C22×#4111yescharacteristic
581C2×C4C2×H(5,𝔽2)11yescharacteristic
582C2×C4C2×2−1+411yescharacteristic
583C2×C4#58511yescharacteristic
584C2×C4#58511yescharacteristic
585C8—24no—
586C8—42no—
587C8—42no—
588C8—42no—
589C22—42no—
590C22—44no—
591C22—42no—
592C22—44no—
593C22—44no—
594C22—42no—
595C22—82no—
596C22—81no—
597C22—81no—
598C22—82no—
599C22—82no—
600C22—81no—
601C22—81no—
602C22—82no—
603C22C2×#54611yescharacteristic
604C22C2×D4×Q811yescharacteristic
605C22#348711yescharacteristic
606C22C22×AGL(1,ℤ/8)11yescharacteristic
607C22C2×#57711yescharacteristic
608C22C2×#57711yescharacteristic
609C22#358511yescharacteristic
610C4—22no—
611C4—22no—
612C4—22no—
613C4—44no—
614C4—44no—
615C4—42no—
616C4—44no—
617C4—41no—
618C4—81no—
619C4—84no—
620C4—82no—
621C4—81no—
622C4—82no—
623C4#333214yes—
624C4#338414yes—
625C4Q8×#4111yescharacteristic
626C4#350111yescharacteristic
627C4#351811yescharacteristic
628C4#354711yescharacteristic
629C2—82no—
630C2—84no—
631C2#5903611yescharacteristic
632C2C2×#333211yescharacteristic
633C2#6120011yescharacteristic
634C2C2×#338411yescharacteristic
635C2#6138211yescharacteristic
636C2#6189011yescharacteristic
637C2#6193511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#459C22
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. #36134C2semidirect
3. #31538C2semidirect
4. #29206C2semidirect
5. #29139C2semidirect
6. #28800C2semidirect
7. #28273C2semidirect
8. #27014C2semidirect
9. #26530C2semidirect
10. #26021C2semidirect
11. #25058C2semidirect
12. #24327C2semidirect
13. #23153C2semidirect
15. #22522C2semidirect
16. C4×#476C2semidirect
127. #459Q8semidirect

Maximal subgroups

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

subgroupindex
#598232
#361342
#315382
#292062
#291392
#288002
#282732
#270142
#265302
#260212
#250582
#243272
#231532
#230992
#225222
C4×#4762
C4×#4592