Finite Group Databasebeta

← #6854299

Subgroups — #6854299

Order 512.

1602 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36560C211yesmaximal, characteristic, complemented
2#36559C211yesmaximal, characteristic
3#33316C211yesmaximal, characteristic, complemented
4#32371C214yesmaximal, complemented
5#32368C214yesmaximal, complemented
6#30307C211yesmaximal, characteristic, complemented
7#30305C211yesmaximal, characteristic, complemented
8#29220C211yesmaximal, characteristic, complemented
9#29207C211yesmaximal, characteristic, complemented
10#29206C211yesmaximal, characteristic, complemented
11#29192C211yesmaximal, characteristic
12#29190C211yesmaximal, characteristic
13#28509C211yesmaximal, characteristic
14#25851C214yesmaximal, complemented
15#25461C214yesmaximal, complemented
16#22220C211yesmaximal, characteristic
17#22219C211yesmaximal, characteristic
18#21646C211yesmaximal, characteristic
19#21340C211yesmaximal, characteristic, complemented
20C2×#557—21no—
21C2×#554C2212yes—
22C2×#501C2214yes—
23C2×#486C2211yescharacteristic
24C2×#476C2214yes—
25#2868C2211yescharacteristic
26#2868C2211yescharacteristic
27#2859C2211yescharacteristic
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2859C2211yescharacteristic
31#2568—21no—
32#2566C2212yes—
33#2566C2212yes—
34#2565—21no—
35#2565—21no—
36#2563C2212yes—
37#2546—21no—
38#2546C2212yes—
39#2538—21no—
40#2538—21no—
41#2506—21no—
42#2506—21no—
43#2506C2212yes—
44#2506C2212yes—
45#2452—21no—
46#2452—21no—
47#2452—21no—
48#2452—21no—
49#2365—21no—
50#2365C2212yes—
51D4×C4⋊C4—21no—
52D4×C4⋊C4—21no—
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C4×#127C2211yescharacteristic
57C4×#127C2211yescharacteristic
58C4×#127C2211yescharacteristic
59C42×Q8C2211yescharacteristic
60#2101C2214yes—
61#2094C2214yes—
62#2085C2214yes—
63#2084C2214yes—
64#2067C2214yes—
65#2065C2214yes—
66#2059C2214yes—
67#2059C2214yes—
68#2058C2214yes—
69#2058C2214yes—
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1989C2214yes—
74#1985C2212yes—
75#1957C2211yescharacteristic
76#1957C2211yescharacteristic
77#1957C2211yescharacteristic
78#1949C2214yes—
79#1948C2214yes—
80#1940C2212yes—
81#1936C2214yes—
82#1926C2214yes—
83#1925C2214yes—
84#1917C2212yes—
85#1917C2212yes—
86#1864C2214yes—
87#1864C2214yes—
88#1862C2214yes—
89#1862C2214yes—
90#1847C2214yes—
91#1845C2211yescharacteristic
92#1845C2211yescharacteristic
93#1844C2211yescharacteristic
94#1844C2211yescharacteristic
95#1844C2211yescharacteristic
96#1844C2211yescharacteristic
97#1834C2211yescharacteristic
98#1834C2211yescharacteristic
99#1834C2211yescharacteristic
100C4×C4⋊C8C2211yescharacteristic
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103#1762C2214yes—
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C4×#97C2211yescharacteristic
107#557—48no—
108#554—48no—
109C2×C4⋊Q8—22no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8C2312yes—
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2312yes—
117C2×#127C2311yescharacteristic
118C2×#125—22no—
119C2×#125—22no—
120C2×#125—21no—
121C2×#125—21no—
122C2×#125C2312yes—
123C2×#125C2312yes—
124C2×C22⋊Q8—22no—
125C2×C22⋊Q8—22no—
126C2×C22⋊Q8—22no—
127C2×C4×Q8—22no—
128C2×C4×Q8—21no—
129C2×C4×Q8—21no—
130C2×C4×Q8—21no—
131C2×C4×Q8D411yescharacteristic
132C2×C4×Q8D411yescharacteristic
133C2×C4×Q8C2311yescharacteristic
134C2×C4×D4—21no—
135C2×C4×D4—21no—
136C2×C4×D4—21no—
137C2×C4×D4—21no—
138C2×C4×D4—21no—
139C2×C4×D4—21no—
140C2×C4×D4C2312yes—
141C2×C4×D4C2312yes—
142C2×C4×D4C2312yes—
143C22×C4⋊C4—22no—
144#501—48no—
145#486Q818yescomplemented
146#476—48no—
147C2×#107C2314yes—
148C2×#107C2314yes—
149C2×#106C2314yes—
150C2×#106C2314yes—
151C2×#106C2314yes—
152C2×#106C2314yes—
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2312yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#97C2312yes—
161C2×#97C2312yes—
162C2×C4×C8C2311yescharacteristic
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2311yescharacteristic
166#400—22no—
167#400—22no—
168#398—22no—
169#398—22no—
170#398—22no—
171#398—22no—
172#396—22no—
173#396—22no—
174#396—22no—
175#396—22no—
176#393—22no—
177#393—22no—
178#391—22no—
179#391—22no—
180#391—21no—
181#391—21no—
182#391—21no—
183#390—21no—
184#390—21no—
185#390—21no—
186#390—21no—
187#390—21no—
188#390—21no—
189#390C2312yes—
190#390C2312yes—
191#390C2312yes—
192#389—21no—
193#389—21no—
194#389—21no—
195#389—21no—
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#389C2312yes—
208#389C2312yes—
209#389C2312yes—
210#389C2312yes—
211#389C2312yes—
212#389C2312yes—
213#389C2312yes—
214#387—22no—
215#387—22no—
216#387—22no—
217#387—22no—
218#385—22no—
219#385—22no—
220#385—22no—
221#385—22no—
222#384—21no—
223#384—21no—
224#384—21no—
225#384—21no—
226#384C2311yescharacteristic
227#384C2311yescharacteristic
228#384C2311yescharacteristic
229#384C2311yescharacteristic
230#384C2311yescharacteristic
231#384C2311yescharacteristic
232#384C2312yes—
233#384C2312yes—
234#382—21no—
235#382—21no—
236#382C2312yes—
237#382C2312yes—
238#380—22no—
239#380—22no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4D411yescharacteristic
245C4×C4⋊C4D411yescharacteristic
246C4×C4⋊C4D411yescharacteristic
247C4×C4⋊C4D411yescharacteristic
248C4×C4⋊C4D411yescharacteristic
249C4×C4⋊C4C2312yes—
250C4×C4⋊C4C2312yes—
251C4×C4⋊C4C2312yes—
252C4×C4⋊C4C2311yescharacteristic
253C4×C4⋊C4C2311yescharacteristic
254C4×C4⋊C4C2311yescharacteristic
255C43C2311yescharacteristic
256C43C2311yescharacteristic
257C43C2311yescharacteristic
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#336C2312yes—
265#336C2312yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271C22×Q8—21no—
272C22×Q8—22no—
273C22×Q8—22no—
274C22×Q8C2×D411yescharacteristic
275C22×D4—21no—
276C22×D4—21no—
277C22×D4C2412yes—
278C23×C4—22no—
279C23×C4—22no—
280C23×C4—22no—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C4⋊Q8—28no—
286C4⋊Q8—22no—
287C4⋊Q8—44no—
288C4⋊Q8—44no—
289C4⋊Q8—44no—
290C4⋊Q8—48no—
291C4⋊Q8—82no—
292C4⋊Q8—82no—
293C4⋊Q8C2×Q814yes—
294#127—22no—
295#127C2×Q814yes—
296#125—28no—
297#125—44no—
298#125—44no—
299#125—48no—
300#125—48no—
301#125—44no—
302C22⋊Q8—48no—
303C22⋊Q8—48no—
304C22⋊Q8—48no—
305C4×Q8—24no—
306C4×Q8—24no—
307C4×Q8—42no—
308C4×Q8—44no—
309C4×Q8—44no—
310C4×Q8—48no—
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—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
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—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—48no—
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2×D411yescharacteristic
357C2×C4⋊C4C2×D411yescharacteristic
358C2×C4⋊C4C2×D411yescharacteristic
359C2×C4⋊C4C2×D411yescharacteristic
360C2×C4⋊C4C2×D411yescharacteristic
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399#107—44no—
400#107—44no—
401#106—44no—
402#106—44no—
403#106—44no—
404#106—44no—
405C4⋊C8—24no—
406C4⋊C8—24no—
407C4⋊C8—24no—
408#102—28no—
409#102—44no—
410#102—44no—
411#102C2×Q8116yes—
412#97—24no—
413#97—24no—
414C4×C8—22no—
415C4×C8—22no—
416C4×C8—22no—
417C4×C8C2×Q814yes—
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#95—22no—
431#95—22no—
432#95C2×D411yescharacteristic
433#95C2×D411yescharacteristic
434#95C2×D411yescharacteristic
435#95C2×D411yescharacteristic
436#95C2×D411yescharacteristic
437#95C2×D411yescharacteristic
438C24—22no—
439C2×Q8—22no—
440C2×Q8—24no—
441C2×Q8—22no—
442C2×Q8—41no—
443C2×Q8—44no—
444C2×Q8—44no—
445C2×Q8—44no—
446C2×Q8—44no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—82no—
450C2×Q8—82no—
451C2×D4—24no—
452C2×D4—24no—
453C2×D4—24no—
454C2×D4—24no—
455C2×D4—22no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4C22×Q814yes—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—82no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
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×C4C2511yescharacteristic
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8C22×Q814yes—
503C2×C82−1+414yes—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4—41no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—44no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—82no—
549C4⋊C4—82no—
550C4⋊C4C22×Q814yes—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—82no—
557C22⋊C4—82no—
558C22⋊C4—82no—
559C22⋊C4—82no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—44no—
577C42—44no—
578C42—44no—
579C42AGL(1,ℤ/8)14yes—
580C42C22×Q812yes—
581C42H(5,𝔽2)14yes—
582C422−1+412yes—
583C23—24no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23C23×D411yescharacteristic
591Q8—44no—
592Q8—48no—
593Q8—88no—
594Q8—164no—
595D4—28no—
596D4—48no—
597D4—48no—
598D4—44no—
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—24no—
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—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
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—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—88no—
653C2×C4—164no—
654C2×C4—164no—
655C2×C4#53512yes—
656C2×C4#53512yes—
657C2×C4D4×Q812yes—
658C2×C4D4×Q812yes—
659C2×C4C2×AGL(1,ℤ/8)12yes—
660C2×C4C2×AGL(1,ℤ/8)12yes—
661C2×C4C2×AGL(1,ℤ/8)12yes—
662C2×C4#57512yes—
663C2×C4#57712yes—
664C2×C4#57712yes—
665C2×C4C23×Q811yescharacteristic
666C2×C4C2×H(5,𝔽2)12yes—
667C2×C4C2×2−1+412yes—
668C2×C4C2×2−1+411yescharacteristic
669C8—24no—
670C8—24no—
671C8—42no—
672C8—42no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22C2×#53511yescharacteristic
681C22C2×D4×Q811yescharacteristic
682C22#349911yescharacteristic
683C22C22×AGL(1,ℤ/8)11yescharacteristic
684C22C2×#57511yescharacteristic
685C22C2×#57711yescharacteristic
686C22C2×#57711yescharacteristic
687C4—22no—
688C4—22no—
689C4—22no—
690C4—22no—
691C4—41no—
692C4—41no—
693C4—41no—
694C4—41no—
695C4—41no—
696C4—41no—
697C4—41no—
698C4—41no—
699C4—44no—
700C4—82no—
701C4—82no—
702C4—82no—
703C4—82no—
704C4—82no—
705C4—82no—
706C4#338414yes—
707C4#338814yes—
708C4#350312yes—
709C4#355212yes—
710C2—84no—
711C2#3961611yescharacteristic
712C2#6080311yescharacteristic
713C2#6081811yescharacteristic
714C2C2×#338411yescharacteristic
715C2C2×#338811yescharacteristic
716C2#6247711yescharacteristic
717C2#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. #36560C2semidirect
3. #33316C2semidirect
4. #32371C2semidirect
5. #32368C2semidirect
6. #30307C2semidirect
7. #30305C2semidirect
8. #29220C2semidirect
9. #29207C2semidirect
10. #29206C2semidirect
14. #25851C2semidirect
15. #25461C2semidirect
19. #21340C2semidirect
145. #486Q8semidirect

Maximal subgroups

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

subgroupindex
#365602
#365592
#333162
#323712
#323682
#303072
#303052
#292202
#292072
#292062
#291922
#291902
#285092
#258512
#254612
#222202
#222192
#216462
#213402