Finite Group Databasebeta

← #6992606

Subgroups — #6992606

Order 512.

1601 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36456C211yesmaximal, characteristic
2#35280C211yesmaximal, characteristic
3#32771C212yesmaximal, complemented
4#31334C214yesmaximal, complemented
5#30577C214yesmaximal, complemented
6#30296C212yesmaximal, complemented
7#30176C212yesmaximal, complemented
8#29691C211yesmaximal, characteristic, complemented
9#28581C212yesmaximal, complemented
10#28232C212yesmaximal, complemented
11#27948C211yesmaximal, characteristic, complemented
12#27647C211yesmaximal, characteristic, complemented
13#25349C212yesmaximal, complemented
14#25060C212yesmaximal, complemented
15#24333C212yesmaximal, complemented
16#23571C211yesmaximal, characteristic, complemented
17#22473C211yesmaximal, characteristic
18C2×Q82C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#483C2212yes—
22C2×#477C2212yes—
23C2×#470C2214yes—
24C2×C4×Q16C2212yes—
25C2×#436C2212yes—
26#2857C2212yes—
27#2857C2211yescharacteristic
28#2857C2212yes—
29#2857C2212yes—
30#2629—22no—
31#2582—21no—
32#2582C2212yes—
33#2563C2211yescharacteristic
34#2563C2212yes—
35#2538—21no—
36#2538C2212yes—
37#2538C2211yescharacteristic
38#2538C2212yes—
39#2452—22no—
40#2452—21no—
41#2439—21no—
42#2439—22no—
43#2417—22no—
44#2417C2214yes—
45#2351—21no—
46Q8×C4⋊C4C2211yescharacteristic
47#2341—21no—
48#2316—21no—
49#2311—21no—
50C4×C22⋊Q8C2211yescharacteristic
51C4×C22⋊Q8C2212yes—
52C2×#384C2211yescharacteristic
53#2096C2214yes—
54#2084C2212yes—
55#2084C2212yes—
56#2084C2212yes—
57#2084C2214yes—
58#2075C2214yes—
59#2072C2212yes—
60#2057C2214yes—
61#2057C2211yescharacteristic
62#2057C2212yes—
63#2043C2211yescharacteristic
64#2043C2212yes—
65#2036C2212yes—
66#2036C2214yes—
67#2035C2214yes—
68#2035C2214yes—
69#2035C2214yes—
70#2024C2214yes—
71#2024C2211yescharacteristic
72#2016C2211yescharacteristic
73#2001C2212yes—
74#1990C2212yes—
75#1984C2212yes—
76#1948C2212yes—
77#1927C2214yes—
78#1920C2214yes—
79#1894C2214yes—
80#1878C2214yes—
81#1870C2214yes—
82#1863C2214yes—
83#1853C2214yes—
84#1844C2212yes—
85#1823C2212yes—
86#1817C2212yes—
87#1809C2214yes—
88#1809C2214yes—
89#1803C2212yes—
90#1803C2212yes—
91#1797C2211yescharacteristic
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94#1728C2212yes—
95C22×#41D411yescharacteristic
96C22×Q16C2312yes—
97C22×M16C2314yes—
98Q82—44no—
99#533—44no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×#125—21no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8C2311yescharacteristic
112C2×C22⋊Q8C2312yes—
113C2×C22⋊Q8C2314yes—
114C2×C22⋊Q8C2312yes—
115C4×#41—44no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8D411yescharacteristic
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2311yescharacteristic
124C2×C4×D4D412yes—
125C2×C4×D4D411yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2312yes—
128C2×#117—21no—
129C2×#117D411yescharacteristic
130C2×#117D412yes—
131C22×C4⋊C4—21no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4C2312yes—
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136#483—44no—
137#477—44no—
138#470—48no—
139C4×Q16—44no—
140#436—44no—
141C2×#107C2314yes—
142C2×#107C2312yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8C2312yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×#103C2312yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2314yes—
157C2×C4×C8C2312yes—
158#398—24no—
159#398—24no—
160#396—24no—
161#396—24no—
162#396—22no—
163#396—22no—
164#396—22no—
165#393—22no—
166#393—22no—
167#393—22no—
168#391—22no—
169#391—22no—
170#391—21no—
171#391—22no—
172#391C2314yes—
173#391C2311yescharacteristic
174#390—21no—
175#390—22no—
176#390—22no—
177#390—21no—
178#390—22no—
179#389—22no—
180#389—22no—
181#389—21no—
182#389—22no—
183#389—21no—
184#389—22no—
185#389—22no—
186#389—22no—
187#389—22no—
188#389—21no—
189#389—22no—
190#389C2314yes—
191#389C2311yescharacteristic
192#389C2312yes—
193#387—21no—
194#387—22no—
195#386—24no—
196#385—21no—
197#385—22no—
198#384—21no—
199#384—21no—
200#384C2311yescharacteristic
201#384C2312yes—
202#384C2312yes—
203#384C2312yes—
204#382—21no—
205#382—22no—
206#380—22no—
207#380—22no—
208#380—22no—
209#380—21no—
210#380—22no—
211#380—24no—
212#380C2312yes—
213#380C2311yescharacteristic
214#380C2312yes—
215#380C2312yes—
216#380C2312yes—
217#380C2314yes—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C22⋊C4—22no—
223C4×C22⋊C4—21no—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2314yes—
234#336C2311yescharacteristic
235#336C2314yes—
236C2×#41—24no—
237C2×#41—24no—
238C2×#41—42no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—21no—
242C22×Q8C2×D411yescharacteristic
243C22×Q8C2412yes—
244C22×Q8C2411yescharacteristic
245C22×D4C2×D411yescharacteristic
246C22×D4C2×D412yes—
247C23×C4—22no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D412yes—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D412yes—
252C23×C4C2411yescharacteristic
253C23×C4C2412yes—
254C2×Q16—24no—
255C2×Q16—24no—
256C2×Q16—42no—
257C2×M16—28no—
258C2×M16—44no—
259C2×M16—44no—
260C22×C8C2414yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C4⋊Q8—24no—
264C4⋊Q8—42no—
265C4⋊Q8—44no—
266C4⋊Q8—42no—
267C4⋊Q8—48no—
268C4⋊Q8—44no—
269#125—44no—
270C22⋊Q8—24no—
271C22⋊Q8—22no—
272C22⋊Q8—41no—
273C22⋊Q8—42no—
274C22⋊Q8—42no—
275C22⋊Q8—42no—
276C22⋊Q8—44no—
277C22⋊Q8—44no—
278C22⋊Q8—44no—
279C22⋊Q8—82no—
280C22⋊Q8—84no—
281C22⋊Q8—84no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—42no—
287C4×Q8—44no—
288C4×Q8—44no—
289C4×Q8—44no—
290C4×Q8—44no—
291C4×D4—24no—
292C4×D4—42no—
293C4×D4—44no—
294C4×D4—44no—
295#117—44no—
296#117—42no—
297#117—44no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—42no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—82no—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2414yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2411yescharacteristic
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C42—24no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—21no—
354C2×C42—22no—
355C2×C42—21no—
356C2×C42—44no—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D412yes—
360C2×C42C2×D412yes—
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#107—42no—
365#107—44no—
366#106—44no—
367#106—44no—
368#106—44no—
369C4⋊C8—24no—
370#103—28no—
371#103—24no—
372#103—42no—
373#103—42no—
374#103—42no—
375#103—44no—
376#103—44no—
377C22⋊C8—24no—
378C22⋊C8—44no—
379C22⋊C8—42no—
380C4×C8—24no—
381#95—21no—
382#95—22no—
383#95—21no—
384#95—21no—
385#95—22no—
386#95—21no—
387#95—22no—
388#95—21no—
389#95—24no—
390#95—24no—
391#95—24no—
392#95—24no—
393#95—22no—
394#95—22no—
395#95—22no—
396C24C22×D412yes—
397C24C22×D411yescharacteristic
398#41—416no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—21no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—44no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8—81no—
416C2×Q8—82no—
417C2×Q8—81no—
418C2×Q8—82no—
419C2×Q8#13714yes—
420C2×Q8H(5,𝔽2)12yes—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—24no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—42no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—24no—
451C22×C4—42no—
452C22×C4—41no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4#13714yes—
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C2511yescharacteristic
468Q16—48no—
469M16—88no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C82−1+414yes—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—84no—
515C4⋊C4—81no—
516C4⋊C4—84no—
517C4⋊C4—82no—
518C22⋊C4—24no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—81no—
525C22⋊C4—82no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C22⋊C4—82no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—24no—
535C42—42no—
536C42—42no—
537C42—44no—
538C42—44no—
539C42—42no—
540C42—44no—
541C42—42no—
542C42H(5,𝔽2)14yes—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23C2×#13712yes—
557C23C23×D411yescharacteristic
558Q8—28no—
559Q8—24no—
560Q8—42no—
561Q8—42no—
562Q8—42no—
563Q8—42no—
564Q8—42no—
565Q8—84no—
566Q8—162no—
567Q8—164no—
568D4—48no—
569D4—44no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—28no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—44no—
590C2×C4—48no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—84no—
622C2×C4—164no—
623C2×C4#53412yes—
624C2×C4#53412yes—
625C2×C4C2×#13714yes—
626C2×C4C2×#13712yes—
627C2×C4C2×#13712yes—
628C2×C4C2×#13712yes—
629C2×C4#57712yes—
630C2×C4C2×H(5,𝔽2)11yescharacteristic
631C2×C4C2×H(5,𝔽2)12yes—
632C2×C4C2×2−1+412yes—
633C2×C4#58511yescharacteristic
634C8—24no—
635C8—44no—
636C8—42no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22C2×#53411yescharacteristic
652C22#348811yescharacteristic
653C22#349911yescharacteristic
654C22C22×#13711yescharacteristic
655C22C22×#13711yescharacteristic
656C22C2×#57711yescharacteristic
657C22#358511yescharacteristic
658C4—22no—
659C4—24no—
660C4—22no—
661C4—22no—
662C4—42no—
663C4—41no—
664C4—42no—
665C4—42no—
666C4—84no—
667C4—81no—
668C4—82no—
669C4—81no—
670C4—82no—
671C4—82no—
672C4—81no—
673C4—82no—
674C4#319514yes—
675C4#353612yes—
676C4#354712yes—
677C2—42no—
678C2—82no—
679C2#4298411yescharacteristic
680C2C2×#319511yescharacteristic
681C2#6076511yescharacteristic
682C2#6196511yescharacteristic
683C2#6200211yescharacteristic
684C2#6247811yescharacteristic
685C2#6251011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32771C2semidirect
4. #31334C2semidirect
5. #30577C2semidirect
6. #30296C2semidirect
7. #30176C2semidirect
8. #29691C2semidirect
9. #28581C2semidirect
10. #28232C2semidirect
11. #27948C2semidirect
12. #27647C2semidirect
13. #25349C2semidirect
14. #25060C2semidirect
15. #24333C2semidirect
16. #23571C2semidirect

Maximal subgroups

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

subgroupindex
#364562
#352802
#327712
#313342
#305772
#302962
#301762
#296912
#285812
#282322
#279482
#276472
#253492
#250602
#243332
#235712
#224732