Finite Group Databasebeta

← #7509825

Subgroups — #7509825

Order 512.

1877 conjugacy classes of subgroups.

The subgroup lattice has 762 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34809C211yesmaximal, characteristic
2C2×#2487C211yesmaximal, characteristic
3#32468C212yesmaximal, complemented
4#32467C212yesmaximal, complemented
5#32465C214yesmaximal, complemented
6#32464C214yesmaximal, complemented
7#32460C212yesmaximal, complemented
8#32459C212yesmaximal, complemented
9#23670C212yesmaximal, complemented
10#23642C212yesmaximal, complemented
11#23642C212yesmaximal, complemented
12#21190C211yesmaximal, characteristic
13#21159C212yesmaximal, complemented
14#20846C212yesmaximal, complemented
15#20844C212yesmaximal, complemented
16C22×C4×D4C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C2×#428C2212yes—
20C2×#428C2212yes—
21C2×#428C2212yes—
22C2×#423C2211yescharacteristic
23C2×#407C2212yes—
24C2×#407C2212yes—
25C2×C4×M16C2211yescharacteristic
26#2487—21no—
27#2487—21no—
28#2487—21no—
29#2487—21no—
30#2487C2212yes—
31#2487C2211yescharacteristic
32#2487C2212yes—
33#2487C2212yes—
34#2487C2212yes—
35#2487C2211yescharacteristic
36#2486C2211yescharacteristic
37#2486C2211yescharacteristic
38#2485C2212yes—
39#2485C2212yes—
40#2484—21no—
41#2483—21no—
42#2482—21no—
43#2405—21no—
44#2395—21no—
45#2395—21no—
46C2×#400—21no—
47C2×#399—21no—
48C4×#123—21no—
49C4×#123—21no—
50#2296C2212yes—
51C2×#389C2212yes—
52C2×#386—21no—
53C2×#385—21no—
54#2280C2211yescharacteristic
55#2280C2212yes—
56C2×#380C2211yescharacteristic
57C2×#380C2212yes—
58#2088C2214yescomplemented
59#2088C2214yescomplemented
60#2087C2214yescomplemented
61#2086C2214yescomplemented
62#2077C2214yescomplemented
63#2077C2214yescomplemented
64#2077C2214yescomplemented
65#2077C2214yescomplemented
66#2076C2214yescomplemented
67#2076C2214yescomplemented
68#2075C2214yescomplemented
69#2075C2214yescomplemented
70#1955C2212yes—
71#1955C2212yes—
72#1954C2212yes—
73#1954C2212yes—
74#1944C2214yescomplemented
75#1943C2214yescomplemented
76#1942C2214yescomplemented
77#1941C2214yescomplemented
78#1854C2214yescomplemented
79#1854C2214yescomplemented
80#1853C2214yescomplemented
81#1853C2214yescomplemented
82#1853C2214yescomplemented
83#1852C2214yescomplemented
84#1852C2214yescomplemented
85#1852C2214yescomplemented
86#1736C2212yes—
87#1736C2212yes—
88#1736C2212yes—
89#1736C2212yes—
90#1736C2212yes—
91C23×D4D411yescharacteristic
92C24×C4—21no—
93C24×C4C2311yescharacteristic
94C24×C4C2311yescharacteristic
95C22×M16C2312yes—
96C22×M16C2312yes—
97C2×#123—21no—
98C2×#123—21no—
99C2×C4×D4—42no—
100C2×C4×D4—42no—
101C2×C4×D4D411yescharacteristic
102C2×C4×D4D411yescharacteristic
103C2×C4×D4D411yescharacteristic
104C2×C4×D4D411yescharacteristic
105C2×C4×D4C2311yescharacteristic
106C2×C4×D4C2311yescharacteristic
107C2×C4×D4C2311yescharacteristic
108C2×C4×D4C2311yescharacteristic
109C2×#117—21no—
110C2×#117—21no—
111C2×#117—42no—
112C2×#117—42no—
113C2×#117C2312yes—
114C2×#117C2312yes—
115C2×#117C2312yes—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4—42no—
121C22×C4⋊C4—42no—
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2311yescharacteristic
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—22no—
132C22×C22⋊C4D411yescharacteristic
133C22×C22⋊C4D411yescharacteristic
134C22×C42—21no—
135C22×C42C2311yescharacteristic
136#428—28no—
137#428—44no—
138#428—44no—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145#423—42no—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148#407—28no—
149#407—28no—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2314yes—
152C2×C22⋊C8C2314yes—
153C2×C22⋊C8C2314yes—
154C4×M16—42no—
155C2×#97C2312yes—
156C2×C4×C8C2312yes—
157#400—22no—
158#400—22no—
159#400—22no—
160#400—22no—
161#400—21no—
162#400—21no—
163#399—22no—
164#399—22no—
165#399—22no—
166#399—22no—
167#399—21no—
168#399—21no—
169#398—21no—
170#398—21no—
171#397—21no—
172#397—21no—
173#397—22no—
174#397—22no—
175#394—22no—
176#394—22no—
177#389—22no—
178#389—22no—
179#389C2312yes—
180#389C2314yes—
181#389C2312yes—
182#389C2314yes—
183#389C2312yes—
184#389C2312yes—
185#388—22no—
186#388—22no—
187#387—22no—
188#387—22no—
189#386—21no—
190#386—21no—
191#386—22no—
192#386—22no—
193#386—22no—
194#386—22no—
195#385—22no—
196#385—22no—
197#385—21no—
198#385—21no—
199#385—22no—
200#385—22no—
201#385—22no—
202#385—22no—
203#381—21no—
204#381—22no—
205#381—21no—
206#381—21no—
207#380—22no—
208#380—22no—
209#380—21no—
210#380—21no—
211#380C2312yes—
212#380C2312yes—
213#380C2312yes—
214#380C2314yes—
215#380C2311yescharacteristic
216#380C2312yes—
217#380C2312yes—
218#380C2314yes—
219#380C2311yescharacteristic
220#380C2312yes—
221#380C2312yes—
222#379—22no—
223#379—21no—
224C4×C4⋊C4—22no—
225C4×C4⋊C4—22no—
226C4×C22⋊C4—21no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—22no—
229C4×C22⋊C4—21no—
230C2×#95—21no—
231C2×#95—22no—
232C2×#95—22no—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#340C2314yes—
240#340C2314yes—
241#340C2314yes—
242#340C2314yes—
243C25C2×D411yescharacteristic
244C25C2×D411yescharacteristic
245C22×D4—24no—
246C22×D4—42no—
247C22×D4—42no—
248C22×D4C2×D411yescharacteristic
249C22×D4C2×D411yescharacteristic
250C22×D4C2×D411yescharacteristic
251C22×D4C2×D411yescharacteristic
252C23×C4—24no—
253C23×C4—24no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—24no—
259C23×C4—24no—
260C23×C4—22no—
261C23×C4—22no—
262C23×C4—21no—
263C23×C4—21no—
264C23×C4—22no—
265C23×C4—22no—
266C23×C4—44no—
267C23×C4—84no—
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2411yescharacteristic
277C23×C4C2411yescharacteristic
278C23×C4C2411yescharacteristic
279C23×C4C2411yescharacteristic
280C23×C4C2411yescharacteristic
281C2×M16—24no—
282C2×M16—24no—
283C2×M16—24no—
284C2×M16—24no—
285C2×M16—24no—
286C2×M16#4118yes—
287C22×C8C2414yes—
288C22×C8C2414yes—
289#123—44no—
290#123—44no—
291C4×D4—42no—
292C4×D4—42no—
293C4×D4—42no—
294C4×D4—42no—
295C4×D4—41no—
296C4×D4—41no—
297C4×D4—41no—
298C4×D4—41no—
299C4×D4—41no—
300C4×D4—41no—
301C4×D4—41no—
302C4×D4—41no—
303#117—24no—
304#117—24no—
305#117—24no—
306#117—24no—
307#117—42no—
308#117—42no—
309#117—44no—
310#117—44no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—42no—
328C2×C4⋊C4—42no—
329C2×C4⋊C4—42no—
330C2×C4⋊C4—42no—
331C2×C4⋊C4—41no—
332C2×C4⋊C4—41no—
333C2×C4⋊C4—41no—
334C2×C4⋊C4—41no—
335C2×C4⋊C4—42no—
336C2×C4⋊C4—44no—
337C2×C4⋊C4—44no—
338C2×C4⋊C4—44no—
339C2×C4⋊C4—84no—
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2414yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2414yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—42no—
365C2×C22⋊C4—42no—
366C2×C22⋊C4—44no—
367C2×C22⋊C4—44no—
368C2×C22⋊C4—84no—
369C2×C22⋊C4C2×D411yescharacteristic
370C2×C22⋊C4C2×D411yescharacteristic
371C2×C22⋊C4C2×D411yescharacteristic
372C2×C22⋊C4C2×D411yescharacteristic
373C2×C22⋊C4C2×D411yescharacteristic
374C2×C22⋊C4C2×D411yescharacteristic
375C2×C22⋊C4C2×D411yescharacteristic
376C2×C22⋊C4C2×D411yescharacteristic
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—44no—
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387#107—28no—
388#107—44no—
389#107—44no—
390#106—28no—
391#106—44no—
392#106—44no—
393C4⋊C8—42no—
394C4⋊C8—42no—
395C22⋊C8—28no—
396C22⋊C8—28no—
397C22⋊C8—28no—
398C22⋊C8—28no—
399#97—42no—
400C4×C8—42no—
401#95—24no—
402#95—24no—
403#95—22no—
404#95—22no—
405#95—24no—
406#95—24no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—21no—
413#95—21no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24—24no—
422C24—22no—
423C24—22no—
424C24—24no—
425C24—42no—
426C24—42no—
427C24—41no—
428C24C22×D411yescharacteristic
429C24C22×D411yescharacteristic
430C24C22×D411yescharacteristic
431C24C22×D411yescharacteristic
432C24C22×D411yescharacteristic
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—41no—
439C2×D4—41no—
440C2×D4—41no—
441C2×D4—41no—
442C2×D4—41no—
443C2×D4—41no—
444C2×D4—416no—
445C2×D4#13714yes—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—24no—
483C22×C4—41no—
484C22×C4—42no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—42no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—42no—
503C22×C4—82no—
504C22×C4—84no—
505C22×C4—84no—
506C22×C4—816no—
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C2×#4114yes—
517C22×C4H(5,𝔽2)12yes—
518C22×C4H(5,𝔽2)12yes—
519C22×C42−1+414yes—
520C22×C4C2511yescharacteristic
521M16—216no—
522M16—48no—
523C2×C8—24no—
524C2×C8—24no—
525C2×C8—24no—
526C2×C8—24no—
527C2×C8—24no—
528C2×C8C2×#4118yes—
529C4⋊C4—22no—
530C4⋊C4—24no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—24no—
535C4⋊C4—24no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—24no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—41no—
542C4⋊C4—41no—
543C4⋊C4—41no—
544C4⋊C4—41no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—44no—
550C4⋊C4—44no—
551C4⋊C4—44no—
552C4⋊C4—44no—
553C4⋊C4—82no—
554C4⋊C4—82no—
555C4⋊C4—84no—
556C4⋊C4—84no—
557C22⋊C4—41no—
558C22⋊C4—41no—
559C22⋊C4—41no—
560C22⋊C4—41no—
561C22⋊C4—41no—
562C22⋊C4—41no—
563C22⋊C4—41no—
564C22⋊C4—41no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—42no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—42no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—82no—
575C22⋊C4—82no—
576C22⋊C4—82no—
577C22⋊C4—84no—
578C22⋊C4—84no—
579C42—22no—
580C42—22no—
581C42—21no—
582C42—21no—
583C42—21no—
584C42—21no—
585C42—44no—
586C42—44no—
587C42—44no—
588C23—22no—
589C23—21no—
590C23—21no—
591C23—28no—
592C23—22no—
593C23—22no—
594C23—22no—
595C23—21no—
596C23—21no—
597C23—21no—
598C23—22no—
599C23—22no—
600C23—22no—
601C23—21no—
602C23—21no—
603C23—21no—
604C23—22no—
605C23—21no—
606C23—22no—
607C23—21no—
608C23—21no—
609C23—48no—
610C23—48no—
611C23—44no—
612C23—48no—
613C23—44no—
614C23—44no—
615C23—44no—
616C23—84no—
617C23—82no—
618C23#53412yes—
619C23#54714yes—
620C23C2×#13712yes—
621C23C2×#13712yes—
622C23#57512yes—
623C23C23×D411yescharacteristic
624D4—44no—
625D4—44no—
626D4—44no—
627D4—44no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—41no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—44no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—44no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—44no—
681C2×C4—42no—
682C2×C4—82no—
683C2×C4—88no—
684C2×C4—84no—
685C2×C4—88no—
686C2×C4—84no—
687C2×C4—88no—
688C2×C4—88no—
689C2×C4#53412yes—
690C2×C4#53918yes—
691C2×C4#54714yes—
692C2×C4#55914yes—
693C2×C4#56314yes—
694C2×C4C2×#13712yes—
695C2×C4C22×#4112yes—
696C2×C4C2×H(5,𝔽2)11yescharacteristic
697C2×C4C2×H(5,𝔽2)11yescharacteristic
698C2×C4C2×2−1+412yes—
699C8—28no—
700C8—44no—
701C22—24no—
702C22—24no—
703C22—22no—
704C22—22no—
705C22—24no—
706C22—24no—
707C22—24no—
708C22—22no—
709C22—22no—
710C22—22no—
711C22—416no—
712C22—42no—
713C22—42no—
714C22—42no—
715C22—41no—
716C22—41no—
717C22—41no—
718C22—41no—
719C22—41no—
720C22—41no—
721C22—42no—
722C22—42no—
723C22—41no—
724C22—41no—
725C22—41no—
726C22—88no—
727C22#272414yes—
728C22C2×#53411yescharacteristic
729C22C2×#54712yes—
730C22C22×#13711yescharacteristic
731C22C2×#57511yescharacteristic
732C22#358512yes—
733C4—22no—
734C4—41no—
735C4—41no—
736C4—41no—
737C4—41no—
738C4—41no—
739C4—41no—
740C4—41no—
741C4—82no—
742C4—82no—
743C4—84no—
744C4—82no—
745C4—82no—
746C4—84no—
747C4#272414yes—
748C4C2×#53912yes—
749C4C2×#55911yescharacteristic
750C4C2×#56311yescharacteristic
751C2—24no—
752C2—42no—
753C2—42no—
754C2—42no—
755C2—81no—
756C2C2×#272411yescharacteristic
757C2#6075111yescharacteristic
758C2#6075311yescharacteristic
759C2#6125412yes—
760C2#6125512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32468C2semidirect
4. #32467C2semidirect
5. #32465C2semidirect
6. #32464C2semidirect
7. #32460C2semidirect
8. #32459C2semidirect
9. #23670C2semidirect
10. #23642C2semidirect
11. #23642C2semidirect
13. #21159C2semidirect
14. #20846C2semidirect
15. #20844C2semidirect
58. #2088C22semidirect
59. #2088C22semidirect
60. #2087C22semidirect
61. #2086C22semidirect
62. #2077C22semidirect
63. #2077C22semidirect
64. #2077C22semidirect
65. #2077C22semidirect
66. #2076C22semidirect
67. #2076C22semidirect
68. #2075C22semidirect
69. #2075C22semidirect
74. #1944C22semidirect
75. #1943C22semidirect
76. #1942C22semidirect
77. #1941C22semidirect
78. #1854C22semidirect
79. #1854C22semidirect
80. #1853C22semidirect
81. #1853C22semidirect
82. #1853C22semidirect
83. #1852C22semidirect
84. #1852C22semidirect
85. #1852C22semidirect

Maximal subgroups

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

subgroupindex
#348092
C2×#24872
#324682
#324672
#324652
#324642
#324602
#324592
#236702
#236422
#211902
#211592
#208462
#208442