Finite Group Databasebeta

← #6805625

Subgroups — #6805625

Order 512.

1532 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36608C211yesmaximal, characteristic, complemented
2#36559C211yesmaximal, characteristic
3#33290C211yesmaximal, characteristic, complemented
4#32438C214yesmaximal, complemented
5#32435C214yesmaximal, complemented
6#30206C211yesmaximal, characteristic, complemented
7#30205C211yesmaximal, characteristic, complemented
8#29207C211yesmaximal, characteristic, complemented
9#29206C211yesmaximal, characteristic, complemented
10#28718C211yesmaximal, characteristic
11#28613C211yesmaximal, characteristic
12#28613C211yesmaximal, characteristic
13#28515C211yesmaximal, characteristic, complemented
14#24685C214yesmaximal, complemented
15#24337C214yesmaximal, complemented
16#22220C211yesmaximal, characteristic
17#22219C211yesmaximal, characteristic
18C4×#492C211yesmaximal, characteristic
19#21339C211yesmaximal, characteristic, complemented
20C2×#554C2212yes—
21C2×#492C2211yescharacteristic
22C2×#486C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#479C2214yes—
25C2×C4×SD16C2214yes—
26#2868C2211yescharacteristic
27#2862C2211yescharacteristic
28#2862C2211yescharacteristic
29#2862C2211yescharacteristic
30#2859C2211yescharacteristic
31#2859C2211yescharacteristic
32#2581—21no—
33#2581—21no—
34#2581C2212yes—
35#2566C2212yes—
36#2566C2212yes—
37#2563C2212yes—
38#2538—21no—
39#2538—21no—
40#2514—21no—
41#2514—21no—
42#2514—21no—
43#2514—21no—
44#2514C2212yes—
45#2514C2212yes—
46#2452—21no—
47#2452—21no—
48#2452—21no—
49#2452—21no—
50#2356—21no—
51#2356—21no—
52#2356C2212yes—
53D4×C4⋊C4—21no—
54D4×C4⋊C4—21no—
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×#127C2211yescharacteristic
59C4×#127C2211yescharacteristic
60C4×#127C2211yescharacteristic
61C42×Q8C2211yescharacteristic
62#2097C2214yes—
63#2074C2214yes—
64#2073C2214yes—
65#2072C2214yes—
66#2059C2214yes—
67#2058C2214yes—
68#2055C2214yes—
69#2055C2214yes—
70#2026C2214yes—
71#2024C2214yes—
72#1991C2214yes—
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1990C2211yescharacteristic
79#1984C2212yes—
80#1968C2214yes—
81#1957C2211yescharacteristic
82#1956C2211yescharacteristic
83#1956C2211yescharacteristic
84#1956C2211yescharacteristic
85#1949C2214yes—
86#1948C2214yes—
87#1939C2212yes—
88#1926C2214yes—
89#1925C2214yes—
90#1916C2212yes—
91#1916C2212yes—
92#1877C2214yes—
93#1877C2214yes—
94#1864C2214yes—
95#1862C2214yes—
96#1845C2211yescharacteristic
97#1844C2211yescharacteristic
98#1844C2211yescharacteristic
99#1835C2214yes—
100#1804C2214yes—
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×C4⋊C8C2211yescharacteristic
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C22×SD16C2314yes—
108#554—48no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2312yes—
115C2×#127C2311yescharacteristic
116C2×#125C2312yes—
117C2×#125C2312yes—
118C2×C22⋊Q8—22no—
119C2×C22⋊Q8—22no—
120C2×C22⋊Q8—22no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8D411yescharacteristic
124C2×C4×Q8D411yescharacteristic
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2311yescharacteristic
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4C2312yes—
134C2×C4×D4C2312yes—
135C2×C4×D4C2312yes—
136C22×C4⋊C4—22no—
137#492—24no—
138#486—24no—
139#486—24no—
140#479—48no—
141C4×SD16—48no—
142C2×#107C2314yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×C4×C8C2312yes—
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2312yes—
159C2×C4×C8C2311yescharacteristic
160#401—22no—
161#401—22no—
162#400—22no—
163#400—22no—
164#400—22no—
165#396—22no—
166#396—22no—
167#396—22no—
168#396—22no—
169#395—22no—
170#393—22no—
171#393—22no—
172#391—21no—
173#391—21no—
174#391—21no—
175#391—21no—
176#391—21no—
177#391—21no—
178#391—21no—
179#391—21no—
180#391—21no—
181#391—21no—
182#391C2312yes—
183#391C2312yes—
184#391C2312yes—
185#391C2312yes—
186#391C2312yes—
187#390—21no—
188#390—21no—
189#390—21no—
190#390—21no—
191#390—21no—
192#390—21no—
193#390C2312yes—
194#390C2312yes—
195#390C2312yes—
196#389—21no—
197#389—21no—
198#389—21no—
199#389—21no—
200#389—21no—
201#389—21no—
202#389—21no—
203#389—21no—
204#389—21no—
205#389—21no—
206#389—21no—
207#389—21no—
208#389C2312yes—
209#389C2312yes—
210#387—22no—
211#387—22no—
212#387—22no—
213#387—22no—
214#387—22no—
215#387—22no—
216#385—22no—
217#385—22no—
218#385—22no—
219#385—22no—
220#384—21no—
221#384—21no—
222#384C2311yescharacteristic
223#384C2311yescharacteristic
224#384C2311yescharacteristic
225#384C2311yescharacteristic
226#384C2312yes—
227#383—22no—
228#383—22no—
229#383—22no—
230#382—21no—
231#382—21no—
232#382—21no—
233#382—21no—
234#382C2312yes—
235#382C2312yes—
236#382C2312yes—
237#382C2312yes—
238#380—22no—
239#380—22no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4D411yescharacteristic
245C4×C4⋊C4D411yescharacteristic
246C4×C4⋊C4D411yescharacteristic
247C4×C4⋊C4D411yescharacteristic
248C4×C4⋊C4D411yescharacteristic
249C4×C4⋊C4D411yescharacteristic
250C4×C4⋊C4C2312yes—
251C4×C4⋊C4C2312yes—
252C4×C4⋊C4C2311yescharacteristic
253C4×C4⋊C4C2311yescharacteristic
254C4×C4⋊C4C2311yescharacteristic
255#376—22no—
256C43C2311yescharacteristic
257C43C2311yescharacteristic
258C43C2311yescharacteristic
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#336C2312yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272C22×Q8—21no—
273C22×Q8—21no—
274C22×Q8—22no—
275C22×Q8C2×D411yescharacteristic
276C22×Q8C2412yes—
277C22×D4—21no—
278C22×D4—21no—
279C22×D4C2412yes—
280C23×C4—22no—
281C23×C4—22no—
282C23×C4—22no—
283C2×SD16—28no—
284C2×SD16—28no—
285C2×SD16—44no—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—44no—
297C4⋊Q8—82no—
298C4⋊Q8—82no—
299#127—22no—
300#127#4114yes—
301#125—28no—
302#125—44no—
303C22⋊Q8—48no—
304C22⋊Q8—48no—
305C22⋊Q8—48no—
306C4×Q8—24no—
307C4×Q8—24no—
308C4×Q8—28no—
309C4×Q8—42no—
310C4×Q8—44no—
311C4×Q8—44no—
312C4×D4—28no—
313C4×D4—44no—
314C4×D4—44no—
315C4×D4—44no—
316C4×D4—44no—
317C4×D4—82no—
318C4×D4—82no—
319C4×D4—82no—
320C4×D4—82no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—48no—
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390#107—44no—
391#106—44no—
392#106—44no—
393C4⋊C8—24no—
394C4⋊C8—24no—
395C4⋊C8—24no—
396#103—44no—
397#103—44no—
398#103—44no—
399#102—28no—
400#102—28no—
401#102—44no—
402C4×C8—24no—
403C4×C8—22no—
404C4×C8—22no—
405C4×C8—22no—
406C4×C8—24no—
407C4×C8—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95C2×D411yescharacteristic
431#95C2×D411yescharacteristic
432#95C2×D411yescharacteristic
433#95C2×D411yescharacteristic
434#95C2×D411yescharacteristic
435#95C2×D411yescharacteristic
436C24—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—24no—
441C2×Q8—22no—
442C2×Q8—24no—
443C2×Q8—41no—
444C2×Q8—44no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—82no—
452C2×Q8—82no—
453C2×D4—24no—
454C2×D4—24no—
455C2×D4—24no—
456C2×D4—24no—
457C2×D4—22no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4C2×#4114yes—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4—82no—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C2511yescharacteristic
494SD16—416no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C82−1+414yes—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—41no—
522C4⋊C4—41no—
523C4⋊C4—41no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—44no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—82no—
545C4⋊C4—82no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—82no—
552C22⋊C4—82no—
553C22⋊C4—82no—
554C22⋊C4—82no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—44no—
572C42—44no—
573C42—44no—
574C42#13514yes—
575C42C2×#4112yes—
576C42H(5,𝔽2)14yes—
577C422−1+412yes—
578C23—24no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23C23×D411yescharacteristic
586Q8—44no—
587Q8—44no—
588Q8—44no—
589Q8—84no—
590Q8—84no—
591Q8—164no—
592D4—28no—
593D4—48no—
594D4—48no—
595D4—44no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—88no—
650C2×C4—164no—
651C2×C4—164no—
652C2×C4#53512yes—
653C2×C4#53512yes—
654C2×C4#54812yes—
655C2×C4#54812yes—
656C2×C4C2×#13512yes—
657C2×C4C2×#13512yes—
658C2×C4C2×#13512yes—
659C2×C4#57512yes—
660C2×C4#57712yes—
661C2×C4#57712yes—
662C2×C4C22×#4111yescharacteristic
663C2×C4C2×H(5,𝔽2)12yes—
664C2×C4C2×2−1+412yes—
665C2×C4C2×2−1+411yescharacteristic
666C8—24no—
667C8—42no—
668C8—42no—
669C8—42no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22C2×#53511yescharacteristic
678C22C2×#54811yescharacteristic
679C22#349911yescharacteristic
680C22C22×#13511yescharacteristic
681C22C2×#57511yescharacteristic
682C22C2×#57711yescharacteristic
683C22C2×#57711yescharacteristic
684C4—22no—
685C4—22no—
686C4—22no—
687C4—22no—
688C4—41no—
689C4—41no—
690C4—41no—
691C4—41no—
692C4—41no—
693C4—41no—
694C4—41no—
695C4—41no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4—82no—
703C4#336014yes—
704C4#337614yes—
705C4#352012yes—
706C4#355012yes—
707C2—84no—
708C2#3916711yescharacteristic
709C2#6078211yescharacteristic
710C2#6081811yescharacteristic
711C2C2×#336011yescharacteristic
712C2C2×#337611yescharacteristic
713C2#6247011yescharacteristic
714C2#6248311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36608C2semidirect
3. #33290C2semidirect
4. #32438C2semidirect
5. #32435C2semidirect
6. #30206C2semidirect
7. #30205C2semidirect
8. #29207C2semidirect
9. #29206C2semidirect
13. #28515C2semidirect
14. #24685C2semidirect
15. #24337C2semidirect
19. #21339C2semidirect

Maximal subgroups

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

subgroupindex
#366082
#365592
#332902
#324382
#324352
#302062
#302052
#292072
#292062
#287182
#286132
#285152
#246852
#243372
#222202
#222192
C4×#4922
#213392