Finite Group Databasebeta

← #6526700

Subgroups — #6526700

Order 512.

1792 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3236C211yesmaximal, characteristic
2C2×#3113C211yesmaximal, characteristic
3C2×#2565C211yesmaximal, characteristic
4#34213C211yesmaximal, characteristic
5#28124C212yesmaximal
6#28121C212yesmaximal
7#27653C212yesmaximal
8#27647C212yesmaximal
9#25464C214yesmaximal, complemented
10#25333C214yesmaximal, complemented
11#24826C214yesmaximal, complemented
12#24691C214yesmaximal, complemented
13#21955C211yesmaximal, characteristic
14#21838C211yesmaximal, characteristic
15#20740C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19#3236—28no—
20C2×#501C2212yes—
21C2×#491C2212yes—
22#3113—28no—
23C2×#484C2212yes—
24C2×#483C2212yes—
25C2×#483C2212yes—
26C2×#479C2214yes—
27C2×#475C2212yes—
28C2×#462C2212yes—
29C2×#451C2212yes—
30C2×#441C2214yes—
31C2×#432C2211yescharacteristic
32C2×#432C2211yescharacteristic
33#2565—22no—
34#2565—22no—
35#2565C2214yes—
36#2565C2214yes—
37C2×#398—21no—
38C2×#398—21no—
39#2370—22no—
40#2369C2212yes—
41#2365C2212yes—
42#2355—22no—
43#2355C2212yes—
44Q8×C4⋊C4C2212yes—
45#2329—21no—
46#2329—21no—
47#2317—21no—
48C4×C22⋊Q8C2212yes—
49C4×C22⋊Q8C2212yes—
50C2×#391C2211yescharacteristic
51C2×#391C2211yescharacteristic
52C2×#387—21no—
53C2×#387—21no—
54#2285C2211yescharacteristic
55C2×#384C2211yescharacteristic
56C2×#382C2211yescharacteristic
57#2274—21no—
58C2×C4×C4⋊C4C2211yescharacteristic
59#2085C2214yes—
60#2084C2214yes—
61#2074C2214yes—
62#2072C2214yes—
63#2068C2214yes—
64#2066C2214yes—
65#2056C2214yes—
66#2056C2214yes—
67#1992C2212yes—
68#1992C2212yes—
69#1989C2214yes—
70#1949C2214yes—
71#1948C2214yes—
72#1940C2214yes—
73#1940C2214yes—
74#1928C2214yes—
75#1927C2214yes—
76#1878C2214yes—
77#1878C2214yes—
78#1824C2212yes—
79#1823C2212yes—
80#1823C2212yes—
81#1805C2214yes—
82#1803C2212yes—
83#1803C2212yes—
84#1803C2214yes—
85#1797C2212yes—
86#1797C2212yes—
87#1787C2212yes—
88#1763C2214yes—
89C23×Q8C2311yescharacteristic
90C22×Q16C2314yes—
91#533—24no—
92#533D418yes—
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2311yescharacteristic
95C2×#125—21no—
96C2×#125—21no—
97C2×#125—21no—
98C2×#125—24no—
99C2×#125—24no—
100C2×#125C2312yes—
101C2×#125C2312yes—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8C2312yes—
104C2×C4×Q8—21no—
105C2×C4×Q8—21no—
106C2×C4×Q8—21no—
107C2×C4×Q8—24no—
108C2×C4×Q8—24no—
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2312yes—
114C2×#117—21no—
115C2×#117—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C42—21no—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130#501—28no—
131#491—28no—
132#484—28no—
133#483—28no—
134#483—28no—
135#479—216no—
136#475—28no—
137#462—28no—
138#451—28no—
139#441—216no—
140#432—24no—
141#432—24no—
142C2×#107C2314yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2314yes—
148C2×C4⋊C8C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2312yes—
157C2×#97C2312yes—
158C2×#97C2312yes—
159#398—24no—
160#398—24no—
161#398—24no—
162#398—24no—
163#391—22no—
164#391—22no—
165#391—22no—
166#391—22no—
167#391C2314yes—
168#391C2314yes—
169#391C2312yes—
170#391C2312yes—
171#389—22no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389C2312yes—
176#389C2312yes—
177#387—22no—
178#387—22no—
179#387—24no—
180#387—24no—
181#387—24no—
182#387—24no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#385—22no—
188#385—22no—
189#384—22no—
190#384—22no—
191#384C2314yes—
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#382—22no—
195#382—22no—
196#382—21no—
197#382C2314yes—
198#380—21no—
199#380C2312yes—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4D412yes—
208C4×C4⋊C4D412yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—22no—
216C2×#95—21no—
217C2×#95—21no—
218C2×#95—21no—
219C2×#95—21no—
220C2×#95—21no—
221C2×#95—21no—
222C2×#95D411yescharacteristic
223C2×#95D411yescharacteristic
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#336C2314yes—
231#336C2314yes—
232C22×Q8—24no—
233C22×Q8—21no—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—42no—
238C22×Q8C2412yes—
239C22×Q8C2412yes—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2411yescharacteristic
251C2×Q16—28no—
252C2×Q16—28no—
253C2×Q16—44no—
254C22×C8C2414yes—
255C22×C8C2414yes—
256C4⋊Q8—22no—
257C4⋊Q8—22no—
258C4⋊Q8C2×D414yes—
259C4⋊Q8C2×D414yes—
260#125—28no—
261#125—24no—
262#125—24no—
263#125—24no—
264#125—24no—
265#125—44no—
266#125—44no—
267C22⋊Q8—28no—
268C22⋊Q8—24no—
269C22⋊Q8C2×D418yes—
270C4×Q8—28no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—44no—
277C4×Q8—44no—
278C4×Q8—44no—
279C4×Q8—44no—
280C4×Q8C2×D418yes—
281#117—28no—
282#117—44no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—42no—
341C2×C42—42no—
342C2×C42—44no—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D414yes—
348C2×C42#4114yes—
349C2×C42C2412yes—
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#107—28no—
354#106—28no—
355#106—28no—
356C4⋊C8—24no—
357C4⋊C8—24no—
358C4⋊C8—44no—
359C4⋊C8—44no—
360#103—28no—
361#103—28no—
362#103—28no—
363#103—28no—
364#103—28no—
365#103—44no—
366C22⋊C8—24no—
367C22⋊C8—24no—
368#97—24no—
369#97—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390C24C22×D411yescharacteristic
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—24no—
398C2×Q8—22no—
399C2×Q8—24no—
400C2×Q8—22no—
401C2×Q8—48no—
402C2×Q8—48no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—81no—
406C2×Q8—81no—
407C2×Q8C22×D414yes—
408C2×Q82−1+414yes—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4#12114yes—
447C22×C4#12114yes—
448C22×C4#13714yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C2×#4112yes—
460C22×C4C2511yescharacteristic
461Q16—416no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—28no—
472C4⋊C4—28no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—48no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—48no—
499C4⋊C4—81no—
500C4⋊C4—81no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4C22×D414yes—
506C22⋊C4—28no—
507C22⋊C4—44no—
508C42—28no—
509C42—24no—
510C42—24no—
511C42—42no—
512C42—42no—
513C42—44no—
514C42—44no—
515C42—44no—
516C42—82no—
517C42—82no—
518C42—82no—
519C42—82no—
520C42#12114yes—
521C42#12114yes—
522C42C22×D412yes—
523C42C22×D412yes—
524C42C2×#4112yes—
525C42C2×#4112yes—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23C2×#12112yes—
531C23C2×#13712yes—
532C23#57512yes—
533C23C23×D411yescharacteristic
534Q8—24no—
535Q8—28no—
536Q8—24no—
537Q8—48no—
538Q8—42no—
539Q8—44no—
540Q8—42no—
541Q8—84no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—28no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—48no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4C2×#12112yes—
600C2×C4C2×#12112yes—
601C2×C4C2×#12112yes—
602C2×C4C2×#12112yes—
603C2×C4C2×#12112yes—
604C2×C4C2×#13714yes—
605C2×C4C23×D411yescharacteristic
606C2×C4C22×#4111yescharacteristic
607C2×C4C2×2−1+412yes—
608C2×C4#58512yes—
609C8—44no—
610C8—44no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22#303714yes—
618C22C22×#12111yescharacteristic
619C22#348511yescharacteristic
620C22#348611yescharacteristic
621C22C22×#13711yescharacteristic
622C22C2×#57511yescharacteristic
623C22#358611yescharacteristic
624C22#358611yescharacteristic
625C4—24no—
626C4—22no—
627C4—22no—
628C4—22no—
629C4—22no—
630C4—22no—
631C4—42no—
632C4—44no—
633C4—42no—
634C4—42no—
635C4—44no—
636C4—44no—
637C4—82no—
638C4—82no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4#348912yes—
643C4#349512yes—
644C2—24no—
645C2#3563311yescharacteristic
646C2C2×#303711yescharacteristic
647C2#6042711yescharacteristic
648C2#6170511yescharacteristic
649C2#6171111yescharacteristic
650C2#6182911yescharacteristic
651C2#6183211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #25464C2semidirect
10. #25333C2semidirect
11. #24826C2semidirect
12. #24691C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32362
C2×#31132
C2×#25652
#342132
#281242
#281212
#276532
#276472
#254642
#253332
#248262
#246912
#219552
#218382
#207402