Finite Group Databasebeta

← #6803031

Subgroups — #6803031

Order 512.

1655 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36577C211yesmaximal, characteristic
2C2×#2568C211yesmaximal, characteristic
3#33108C211yesmaximal, characteristic
4#32679C214yesmaximal, complemented
5#32625C214yesmaximal, complemented
6#29715C212yesmaximal
7#28744C212yesmaximal, complemented
8#28313C212yesmaximal
9#28124C212yesmaximal
10#26320C212yesmaximal, complemented
11#25722C211yesmaximal, characteristic
12#25634C212yesmaximal, complemented
13#25302C212yesmaximal, complemented
14#24212C211yesmaximal, characteristic
15#22510C211yesmaximal, characteristic
16#22050C212yesmaximal
17C4×#483C211yesmaximal, characteristic
18C2×#557C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C2×#494C2212yes—
22C2×#483C2211yescharacteristic
23C2×#483C2212yes—
24C2×#462C2212yes—
25C2×#451C2212yes—
26C2×#451C2212yes—
27C2×#451C2212yes—
28C2×#446C2212yes—
29#2857C2212yes—
30#2857C2211yescharacteristic
31#2582C2212yes—
32#2582C2211yescharacteristic
33#2568—21no—
34#2568—21no—
35#2568—22no—
36#2568C2212yes—
37#2568C2211yescharacteristic
38#2568C2212yes—
39#2568C2214yes—
40#2568C2212yes—
41#2453—21no—
42#2453—22no—
43#2442—22no—
44#2442—21no—
45C2×#400—22no—
46#2351—21no—
47Q8×C4⋊C4C2211yescharacteristic
48Q8×C22⋊C4—21no—
49C4×C22⋊Q8C2212yes—
50C4×C22⋊Q8C2211yescharacteristic
51C2×#391—21no—
52C2×#391C2212yes—
53C2×#391C2212yes—
54C2×#382—21no—
55C2×C4×C4⋊C4C2211yescharacteristic
56#2096C2214yes—
57#2072C2214yes—
58#2066C2214yes—
59#2056C2214yes—
60#2056C2214yes—
61#2056C2214yes—
62#2043C2212yes—
63#2016C2214yes—
64#2016C2214yes—
65#2016C2214yes—
66#2016C2212yes—
67#1985C2214yes—
68#1984C2212yes—
69#1981C2212yes—
70#1981C2212yes—
71#1971C2214yes—
72#1971C2214yes—
73#1971C2214yes—
74#1956C2212yes—
75#1948C2212yes—
76#1935C2214yes—
77#1927C2214yes—
78#1927C2212yes—
79#1927C2214yes—
80#1927C2214yes—
81#1920C2212yes—
82#1878C2214yes—
83#1878C2214yes—
84#1872C2211yescharacteristic
85#1872C2212yes—
86#1853C2211yescharacteristic
87#1823C2211yescharacteristic
88#1797C2212yes—
89#1797C2211yescharacteristic
90#1787C2212yes—
91C4×#106C2212yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8D411yescharacteristic
96C23×Q8C2312yes—
97C22×Q16C2314yes—
98C22×Q16C2314yes—
99C22×Q16C2314yes—
100#557—44no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—44no—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×#125—21no—
109C2×#125—22no—
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2311yescharacteristic
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8—44no—
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8D412yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2312yes—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4D412yes—
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132#494—44no—
133#483—24no—
134#483—44no—
135#462—44no—
136#451—28no—
137#451—44no—
138#451—44no—
139#446—44no—
140C2×#106C2312yes—
141C2×#106C2314yes—
142C2×#106C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2314yes—
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#103C2312yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2311yescharacteristic
156C2×C22⋊C8C2311yescharacteristic
157C2×#97C2314yes—
158C2×C4×C8C2312yes—
159#400—24no—
160#400—24no—
161#400—24no—
162#400—24no—
163#400—22no—
164#400—22no—
165#400—22no—
166#398—22no—
167#398—22no—
168#398—22no—
169#391—22no—
170#391—24no—
171#391—22no—
172#391—21no—
173#391—22no—
174#391—21no—
175#391—22no—
176#391—22no—
177#391—22no—
178#391—22no—
179#391—22no—
180#391C2314yes—
181#391C2312yes—
182#391C2312yes—
183#391C2312yes—
184#391C2314yes—
185#391C2314yes—
186#391C2314yes—
187#391C2311yescharacteristic
188#391C2312yes—
189#389—21no—
190#389—22no—
191#389—21no—
192#389—22no—
193#389—21no—
194#389—22no—
195#389—21no—
196#389—22no—
197#389C2312yes—
198#389C2311yescharacteristic
199#387—21no—
200#387—22no—
201#385—21no—
202#385—22no—
203#384C2312yes—
204#384C2311yescharacteristic
205#382—22no—
206#382—22no—
207#382—24no—
208#382—21no—
209#382—22no—
210#380—22no—
211#380—21no—
212#380—21no—
213#380—22no—
214#380C2311yescharacteristic
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2311yescharacteristic
223C4×C22⋊C4—22no—
224C4×C22⋊C4—21no—
225C2×#95—21no—
226C2×#95—22no—
227C2×#95—22no—
228C2×#95—22no—
229C2×#95—22no—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#340C2312yes—
234#340C2312yes—
235#336C2314yes—
236#336C2312yes—
237#336C2314yes—
238#336C2312yes—
239C22×Q8—24no—
240C22×Q8—24no—
241C22×Q8—22no—
242C22×Q8—24no—
243C22×Q8—24no—
244C22×Q8—42no—
245C22×Q8—44no—
246C22×Q8—44no—
247C22×Q8C2×D411yescharacteristic
248C22×Q8C2×D412yes—
249C22×Q8C2×D411yescharacteristic
250C22×Q8C2411yescharacteristic
251C22×Q8C2412yes—
252C22×Q8C2414yes—
253C23×C4—22no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D412yes—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D412yes—
260C23×C4C2411yescharacteristic
261C2×Q16—28no—
262C2×Q16—44no—
263C2×Q16—44no—
264C2×Q16—44no—
265C2×Q16—44no—
266C2×Q16—44no—
267C2×Q16—44no—
268C2×Q16—44no—
269C2×Q16—44no—
270C22×C8C2412yes—
271C22×C8C2414yes—
272C22×C8C2412yes—
273C4⋊Q8—24no—
274C4⋊Q8—24no—
275C4⋊Q8—24no—
276C4⋊Q8—44no—
277C4⋊Q8—44no—
278C4⋊Q8—44no—
279C4⋊Q8—44no—
280#125—48no—
281#125—44no—
282C22⋊Q8—24no—
283C22⋊Q8—22no—
284C22⋊Q8—22no—
285C22⋊Q8—24no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—44no—
290C4×Q8—44no—
291C4×Q8—42no—
292C4×Q8—42no—
293C4×Q8—44no—
294C4×Q8—44no—
295C4×Q8—44no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
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—21no—
318C2×C4⋊C4—42no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—42no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2411yescharacteristic
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42—21no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—44no—
347C2×C42—44no—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2412yes—
355C2×C42C2411yescharacteristic
356#106—24no—
357#106—42no—
358#106—44no—
359C4⋊C8—24no—
360C4⋊C8—44no—
361#103—28no—
362#103—24no—
363#103—44no—
364#103—44no—
365#103—42no—
366#103—44no—
367#103—44no—
368#103—44no—
369#103—44no—
370C22⋊C8—22no—
371C22⋊C8—24no—
372C22⋊C8#4114yes—
373#97—44no—
374C4×C8—24no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—22no—
386#95—21no—
387#95—22no—
388#95—21no—
389#95—21no—
390#95—22no—
391#95—21no—
392#95—22no—
393#95—21no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400C24C22×D411yescharacteristic
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—21no—
409C2×Q8—24no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—24no—
413C2×Q8—24no—
414C2×Q8—22no—
415C2×Q8—24no—
416C2×Q8—21no—
417C2×Q8—24no—
418C2×Q8—24no—
419C2×Q8—22no—
420C2×Q8—24no—
421C2×Q8—24no—
422C2×Q8—42no—
423C2×Q8—416no—
424C2×Q8—44no—
425C2×Q8—44no—
426C2×Q8—44no—
427C2×Q8—88no—
428C2×Q8—88no—
429C2×Q8#13714yes—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—42no—
449C22×C4—41no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—44no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4#13714yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C2×#4112yes—
472C22×C4H(5,𝔽2)12yes—
473C22×C4C2511yescharacteristic
474Q16—88no—
475Q16—88no—
476Q16—88no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—24no—
484C2×C8—22no—
485C2×C8C2×#4114yes—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—81no—
520C4⋊C4—81no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C22⋊C4—44no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C42—22no—
527C42—22no—
528C42—24no—
529C42—22no—
530C42—24no—
531C42—42no—
532C42—42no—
533C42—44no—
534C42—42no—
535C42—44no—
536C42—44no—
537C42—44no—
538C42—42no—
539C42H(5,𝔽2)14yes—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23#55012yes—
545C23C2×#13712yes—
546C23#57812yes—
547C23C23×D411yescharacteristic
548Q8—28no—
549Q8—48no—
550Q8—44no—
551Q8—44no—
552Q8—44no—
553Q8—42no—
554Q8—44no—
555Q8—42no—
556Q8—44no—
557Q8—44no—
558Q8—44no—
559Q8—88no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—44no—
584C2×C4—48no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—88no—
615C2×C4—84no—
616C2×C4#53412yes—
617C2×C4#53412yes—
618C2×C4#55012yes—
619C2×C4C2×#13712yes—
620C2×C4C2×#13714yes—
621C2×C4C2×#13712yes—
622C2×C4C2×#13712yes—
623C2×C4C22×#4111yescharacteristic
624C2×C4C2×H(5,𝔽2)12yes—
625C2×C4C2×H(5,𝔽2)11yescharacteristic
626C2×C4#58512yes—
627C8—24no—
628C8—44no—
629C8—42no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22#339814yes—
637C22C2×#53411yescharacteristic
638C22C2×#55011yescharacteristic
639C22#348711yescharacteristic
640C22C22×#13711yescharacteristic
641C22C22×#13711yescharacteristic
642C22C2×#57811yescharacteristic
643C22#358611yescharacteristic
644C4—22no—
645C4—24no—
646C4—22no—
647C4—22no—
648C4—42no—
649C4—41no—
650C4—42no—
651C4—42no—
652C4—81no—
653C4—82no—
654C4—81no—
655C4—82no—
656C4—82no—
657C4—81no—
658C4—84no—
659C4—82no—
660C4—81no—
661C4#319514yes—
662C4#351412yes—
663C4#355812yes—
664C2—24no—
665C2#3913511yescharacteristic
666C2C2×#319511yescharacteristic
667C2#6075911yescharacteristic
668C2C2×#339811yescharacteristic
669C2#6140411yescharacteristic
670C2#6188911yescharacteristic
671C2#6190011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #32679C2semidirect
5. #32625C2semidirect
7. #28744C2semidirect
10. #26320C2semidirect
12. #25634C2semidirect
13. #25302C2semidirect

Maximal subgroups

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

subgroupindex
#365772
C2×#25682
#331082
#326792
#326252
#297152
#287442
#283132
#281242
#263202
#257222
#256342
#253022
#242122
#225102
#220502
C4×#4832