Finite Group Databasebeta

← #7538109

Subgroups — #7538109

Order 512.

1691 conjugacy classes of subgroups.

The subgroup lattice has 641 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#32648C214yesmaximal, complemented
4#28827C212yesmaximal, complemented
5#28689C211yesmaximal, characteristic, complemented
6#28527C212yesmaximal, complemented
7#28508C211yesmaximal, characteristic, complemented
8#26526C212yesmaximal, complemented
9#26086C212yesmaximal, complemented
10#25449C212yesmaximal, complemented
11#25056C214yesmaximal, complemented
12#24811C212yesmaximal, complemented
13#23247C211yesmaximal, characteristic, complemented
14#23093C212yesmaximal
15#22962C212yesmaximal, complemented
16C4×#474C211yesmaximal, characteristic, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#554—21no—
19C2×#533—21no—
20C2×#500C2212yes—
21C2×#474C2212yes—
22C2×#474C2211yescharacteristic
23C2×#474C2212yes—
24C2×#459C2211yescharacteristic
25C2×C4×D8C2212yes—
26#2868C2211yescharacteristic
27#2857—21no—
28#2732—22no—
29#2552C2212yes—
30#2552C2212yes—
31#2542C2212yes—
32#2542C2212yes—
33#2538—21no—
34#2538C2212yes—
35#2538C2214yes—
36#2454—22no—
37#2441—22no—
38#2429—22no—
39#2423—22no—
40#2417—22no—
41#2366C2212yes—
42#2365C2212yes—
43#2354C2212yes—
44Q8×C4⋊C4C2212yes—
45D4×C4⋊C4C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×#127C2211yescharacteristic
49C42×D4C2211yescharacteristic
50#2093C2214yes—
51#2093C2214yes—
52#2093C2214yes—
53#2083C2214yes—
54#2083C2214yes—
55#2083C2214yes—
56#2067C2214yes—
57#2067C2214yes—
58#1990C2212yes—
59#1988C2211yescharacteristic
60#1984C2214yes—
61#1982C2211yescharacteristic
62#1970C2214yes—
63#1957C2212yes—
64#1947C2212yes—
65#1947C2212yes—
66#1947C2211yescharacteristic
67#1947C2214yes—
68#1947C2211yescharacteristic
69#1942C2214yes—
70#1939C2212yes—
71#1939C2212yes—
72#1925C2214yes—
73#1925C2212yes—
74#1918C2212yes—
75#1918C2214yes—
76#1879C2214yes—
77#1879C2212yes—
78#1862C2212yes—
79#1862C2214yes—
80#1845C2212yes—
81#1845C2214yes—
82#1802C2211yescharacteristic
83#1802C2214yes—
84#1802C2211yescharacteristic
85C4×#107C2212yes—
86C8×C4⋊C4C2212yes—
87C8×C4⋊C4C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90C4×#102C2212yes—
91C4×#102C2214yes—
92C4×#102C2212yes—
93C22×D8C2312yes—
94#554—48no—
95#533—48no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8D411yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×#127C2311yescharacteristic
102C2×#125—22no—
103C2×#125—22no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—24no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C4×Q8—22no—
111C2×C4×Q8—22no—
112C2×C4×Q8—21no—
113C2×C4×Q8C2312yes—
114C2×C4×D4—22no—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4C2312yes—
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2314yes—
121C2×C4×D4C2311yescharacteristic
122C2×#117—22no—
123C22×C4⋊C4—22no—
124C22×C42—21no—
125#500—44no—
126#474—28no—
127#474—24no—
128#474—44no—
129#459Q818yescomplemented
130C4×D8—44no—
131C2×#107C2314yes—
132C2×#107C2312yes—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#107C2312yes—
136C2×#107C2314yes—
137C2×#107C2312yes—
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2311yescharacteristic
140C2×C4⋊C8C2314yes—
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2311yescharacteristic
143C2×#102C2314yes—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×#102C2312yes—
147C2×#102C2312yes—
148C2×#102C2314yes—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C2×C4×C8C2312yes—
152C2×C4×C8C2312yes—
153C2×C4×C8C2314yes—
154#399—24no—
155#399—24no—
156#398—24no—
157#398—24no—
158#396—24no—
159#396—24no—
160#396—24no—
161#391—22no—
162#391—21no—
163#391D412yes—
164#390—22no—
165#390—21no—
166#390—21no—
167#390C2312yes—
168#390C2314yes—
169#390C2314yes—
170#389—22no—
171#389—22no—
172#389—24no—
173#389—22no—
174#389—21no—
175#389D412yes—
176#389D412yes—
177#389C2312yes—
178#389C2314yes—
179#389C2314yes—
180#388—24no—
181#387—24no—
182#386—24no—
183#385—24no—
184#384—21no—
185#384—21no—
186#384D411yescharacteristic
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2314yes—
190#384C2312yes—
191#384C2314yes—
192#384C2312yes—
193#380—22no—
194#380—22no—
195#380—22no—
196#380—24no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2311yescharacteristic
200C4×C4⋊C4C2314yes—
201C4×C4⋊C4C2311yescharacteristic
202C4×C4⋊C4C2311yescharacteristic
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C22⋊C4—21no—
206C4×C22⋊C4—21no—
207C4×C22⋊C4—21no—
208C43C2311yescharacteristic
209#340C2312yes—
210#340C2314yes—
211#340C2312yes—
212#340C2312yes—
213#340C2312yes—
214#340C2314yes—
215#340C2312yes—
216#340C2312yes—
217#336C2314yes—
218#336C2312yes—
219#336C2314yes—
220#336C2312yes—
221C22×Q8—21no—
222C22×Q8—22no—
223C22×Q8C2×D412yes—
224C22×D4—21no—
225C22×D4—21no—
226C22×D4C2412yes—
227C22×D4C2411yescharacteristic
228C23×C4—21no—
229C23×C4—22no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—24no—
233C2×D8—42no—
234C2×D8—42no—
235C2×D8C2×Q818yes—
236C22×C8C2412yes—
237C22×C8C2412yes—
238C22×C8C2412yes—
239C22×C8C2412yes—
240C4⋊Q8—22no—
241C4⋊Q8—24no—
242C4⋊Q8—24no—
243C4⋊Q8—22no—
244C4⋊Q8—44no—
245C4⋊Q8—82no—
246C4⋊Q8D818yescomplemented
247#127—22no—
248#127C2×Q814yes—
249#125—48no—
250#125—48no—
251C22⋊Q8—48no—
252C22⋊Q8—416no—
253C22⋊Q8—48no—
254C22⋊Q8—48no—
255C22⋊Q8—84no—
256C22⋊Q8—84no—
257C4×Q8—42no—
258C4×Q8—42no—
259C4×Q8—48no—
260C4×Q8—48no—
261C4×Q8—44no—
262C4×D4—24no—
263C4×D4—22no—
264C4×D4—22no—
265C4×D4—24no—
266C4×D4—22no—
267C4×D4—41no—
268C4×D4—41no—
269C4×D4—44no—
270C4×D4—44no—
271C4×D4—48no—
272C4×D4—44no—
273C4×D4—82no—
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—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—48no—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D414yes—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2411yescharacteristic
302C2×C4⋊C4C2411yescharacteristic
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2411yescharacteristic
307C2×C22⋊C4—21no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—22no—
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×C42C2411yescharacteristic
329C2×C42C2414yes—
330C2×C42C2411yescharacteristic
331C2×C42C2411yescharacteristic
332#107—28no—
333#107—24no—
334#107—42no—
335#107—42no—
336#107—44no—
337#107—44no—
338#107—44no—
339C4⋊C8—22no—
340C4⋊C8—22no—
341C4⋊C8—44no—
342C4⋊C8C2×Q814yes—
343C4⋊C8#4114yes—
344#102—28no—
345#102—24no—
346#102—42no—
347#102—42no—
348#102—42no—
349#102—42no—
350#102—44no—
351#102C2×Q818yes—
352C4×C8—24no—
353C4×C8—24no—
354C4×C8—44no—
355#95—21no—
356#95—21no—
357#95—21no—
358#95—22no—
359#95—24no—
360#95—22no—
361#95—24no—
362#95—24no—
363#95C2×D414yes—
364#95C2×D414yes—
365C24—21no—
366C24—22no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8—44no—
370C2×Q8—42no—
371C2×Q8—42no—
372C2×Q8—44no—
373C2×Q8—42no—
374C2×Q8—44no—
375C2×Q8C2×D818yes—
376C2×D4—24no—
377C2×D4—24no—
378C2×D4—22no—
379C2×D4—21no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—22no—
384C2×D4—21no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—21no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—81no—
391C2×D4—81no—
392C2×D4C22×Q814yes—
393C2×D4C22×Q812yes—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—44no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—42no—
413C22×C4—48no—
414C22×C4—84no—
415C22×C4—84no—
416C22×C4C22×D414yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D414yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C2511yescharacteristic
425D8—48no—
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×C8—22no—
436C2×C8C22×Q814yes—
437C2×C8C2×#4114yes—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—21no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—42no—
454C4⋊C4—44no—
455C4⋊C4—42no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—48no—
459C4⋊C4—44no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—44no—
465C4⋊C4—48no—
466C4⋊C4—81no—
467C4⋊C4—81no—
468C4⋊C4—84no—
469C4⋊C4—84no—
470C4⋊C4—82no—
471C4⋊C4—82no—
472C4⋊C4C2×D818yes—
473C4⋊C4C2×D818yes—
474C4⋊C4C22×Q812yes—
475C22⋊C4—24no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—44no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—44no—
482C22⋊C4—84no—
483C22⋊C4—82no—
484C22⋊C4—84no—
485C22⋊C4—84no—
486C42—22no—
487C42—22no—
488C42—24no—
489C42—24no—
490C42—21no—
491C42—21no—
492C42—41no—
493C42—42no—
494C42—42no—
495C42—42no—
496C42—48no—
497C42—44no—
498C42C2×D814yes—
499C42C22×Q812yes—
500C42C2×#4112yes—
501C42H(5,𝔽2)12yes—
502C422−1+412yes—
503C23—22no—
504C23—24no—
505C23—41no—
506C23—42no—
507C23—42no—
508C23—41no—
509C23—41no—
510C23—42no—
511C23—41no—
512C23—42no—
513C23—41no—
514C23—42no—
515C23—41no—
516C23—42no—
517C23C23×D411yescharacteristic
518Q8—48no—
519Q8—84no—
520Q8—88no—
521D4—22no—
522D4—24no—
523D4—22no—
524D4—24no—
525D4—48no—
526D4—41no—
527D4—42no—
528D4—42no—
529D4—41no—
530D4—84no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—24no—
535C2×C4—21no—
536C2×C4—24no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—21no—
543C2×C4—24no—
544C2×C4—21no—
545C2×C4—24no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—48no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—41no—
567C2×C4—84no—
568C2×C4—84no—
569C2×C4—88no—
570C2×C4—168no—
571C2×C4D4×Q812yes—
572C2×C4D4×Q812yes—
573C2×C4#55012yes—
574C2×C4#55012yes—
575C2×C4C22×D812yes—
576C2×C4C22×D818yes—
577C2×C4C22×D812yes—
578C2×C4C22×D812yes—
579C2×C4#57612yes—
580C2×C4#57812yes—
581C2×C4C23×Q811yescharacteristic
582C2×C4C22×#4111yescharacteristic
583C2×C4C2×H(5,𝔽2)11yescharacteristic
584C2×C4C2×2−1+411yescharacteristic
585C2×C4#58511yescharacteristic
586C2×C4#58511yescharacteristic
587C8—24no—
588C8—24no—
589C8—42no—
590C8—42no—
591C22—42no—
592C22—44no—
593C22—42no—
594C22—44no—
595C22—44no—
596C22—42no—
597C22—81no—
598C22—82no—
599C22—81no—
600C22—82no—
601C22—81no—
602C22—82no—
603C22—81no—
604C22—82no—
605C22C2×D4×Q811yescharacteristic
606C22C2×#55011yescharacteristic
607C22#348511yescharacteristic
608C22C23×D811yescharacteristic
609C22C2×#57611yescharacteristic
610C22C2×#57811yescharacteristic
611C22#358611yescharacteristic
612C4—22no—
613C4—22no—
614C4—22no—
615C4—28no—
616C4—41no—
617C4—44no—
618C4—42no—
619C4—81no—
620C4—82no—
621C4—82no—
622C4—84no—
623C4—81no—
624C4—82no—
625C4Q8×D814yes—
626C4#339114yes—
627C4Q8×#4111yescharacteristic
628C4#350111yescharacteristic
629C4#351511yescharacteristic
630C4#354711yescharacteristic
631C2—82no—
632C2—84no—
633C2#5905211yescharacteristic
634C2C2×Q8×D811yescharacteristic
635C2#6138111yescharacteristic
636C2C2×#339111yescharacteristic
637C2#6142211yescharacteristic
638C2#6169411yescharacteristic
639C2#6179111yescharacteristic
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. #32648C2semidirect
4. #28827C2semidirect
5. #28689C2semidirect
6. #28527C2semidirect
7. #28508C2semidirect
8. #26526C2semidirect
9. #26086C2semidirect
10. #25449C2semidirect
11. #25056C2semidirect
12. #24811C2semidirect
13. #23247C2semidirect
15. #22962C2semidirect
16. C4×#474C2semidirect
129. #459Q8semidirect
246. C4⋊Q8D8semidirect

Maximal subgroups

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

subgroupindex
#598232
#361342
#326482
#288272
#286892
#285272
#285082
#265262
#260862
#254492
#250562
#248112
#232472
#230932
#229622
C4×#4742
C4×#4592