Finite Group Databasebeta

← #6804558

Subgroups — #6804558

Order 512.

1524 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36594C211yesmaximal, characteristic
2#36560C211yesmaximal, characteristic, complemented
3#33286C211yesmaximal, characteristic
4#32589C214yesmaximal, complemented
5#32490C214yesmaximal, complemented
6#30230C211yesmaximal, characteristic, complemented
7#30203C211yesmaximal, characteristic
8#29208C211yesmaximal, characteristic, complemented
9#29192C211yesmaximal, characteristic
10#29179C211yesmaximal, characteristic, complemented
11#28687C211yesmaximal, characteristic, complemented
12#28654C211yesmaximal, characteristic
13#28654C211yesmaximal, characteristic
14#27379C214yesmaximal, complemented
15#27143C214yesmaximal, complemented
16#22246C211yesmaximal, characteristic, complemented
17#22220C211yesmaximal, characteristic
18#21727C211yesmaximal, characteristic, complemented
19#21340C211yesmaximal, characteristic, complemented
20C2×#557—21no—
21C2×#556—21no—
22C2×#498C2211yescharacteristic
23C2×#498C2211yescharacteristic
24C2×#486C2211yescharacteristic
25#2868C2211yescharacteristic
26#2868C2211yescharacteristic
27#2859C2211yescharacteristic
28#2859C2211yescharacteristic
29#2858C2211yescharacteristic
30#2858C2211yescharacteristic
31#2568—21no—
32#2565—21no—
33#2565—21no—
34#2547—21no—
35#2546—21no—
36#2546C2212yes—
37#2545—21no—
38#2545—21no—
39#2538—21no—
40#2538C2212yes—
41#2512—21no—
42#2512—21no—
43#2512C2212yes—
44#2512C2212yes—
45#2506—21no—
46#2506—21no—
47#2506C2212yes—
48#2506C2212yes—
49#2365—21no—
50#2365C2212yes—
51#2354—21no—
52#2354C2212yes—
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×#127C2211yescharacteristic
59C42×Q8C2211yescharacteristic
60#2100C2214yes—
61#2100C2214yes—
62#2087C2214yes—
63#2084C2214yes—
64#2071C2214yes—
65#2067C2214yes—
66#2065C2214yes—
67#2065C2214yes—
68#2057C2214yes—
69#2023C2214yes—
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1985C2212yes—
78#1957C2211yescharacteristic
79#1957C2211yescharacteristic
80#1957C2211yescharacteristic
81#1948C2214yes—
82#1947C2214yes—
83#1944C2214yes—
84#1940C2212yes—
85#1936C2214yes—
86#1936C2214yes—
87#1926C2214yes—
88#1926C2214yes—
89#1925C2214yes—
90#1921C2214yes—
91#1920C2214yes—
92#1919C2214yes—
93#1917C2212yes—
94#1917C2212yes—
95#1862C2214yes—
96#1862C2214yes—
97#1845C2211yescharacteristic
98#1845C2211yescharacteristic
99#1834C2211yescharacteristic
100#1776C2214yes—
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×C4⋊C8C2211yescharacteristic
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C4×#97C2211yescharacteristic
107#557—48no—
108#556—48no—
109C2×C4⋊Q8—22no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2312yes—
118C2×#127C2311yescharacteristic
119C2×#126—22no—
120C2×#126—22no—
121C2×#125—22no—
122C2×#125—22no—
123C2×#125—21no—
124C2×#125—21no—
125C2×#124—21no—
126C2×#124—21no—
127C2×#124—21no—
128C2×#124—21no—
129C2×C22⋊Q8—22no—
130C2×C4×Q8—22no—
131C2×C4×Q8—21no—
132C2×C4×Q8—21no—
133C2×C4×Q8—21no—
134C2×C4×Q8—21no—
135C2×C4×Q8—21no—
136C2×C4×Q8D411yescharacteristic
137C2×C4×Q8D411yescharacteristic
138C2×C4×Q8C2311yescharacteristic
139C2×C4×D4—21no—
140C2×C4×D4—21no—
141C2×C4×D4C2312yes—
142C2×#117—22no—
143#498—24no—
144#498—24no—
145#486—24no—
146C2×#107C2314yes—
147C2×#107C2314yes—
148C2×#106C2314yes—
149C2×#106C2314yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2312yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#97C2312yes—
158C2×#97C2312yes—
159C2×C4×C8C2311yescharacteristic
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2311yescharacteristic
163#400—22no—
164#400—22no—
165#398—22no—
166#398—22no—
167#398—22no—
168#398—22no—
169#397—22no—
170#397—22no—
171#396—22no—
172#396—22no—
173#396—22no—
174#396—22no—
175#391—22no—
176#391—22no—
177#391—21no—
178#391—21no—
179#391—21no—
180#391—21no—
181#391—21no—
182#390—21no—
183#390—21no—
184#390—21no—
185#390—21no—
186#390—21no—
187#390—21no—
188#390C2312yes—
189#390C2312yes—
190#390C2312yes—
191#390C2312yes—
192#390C2312yes—
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#389C2312yes—
203#389C2312yes—
204#389C2312yes—
205#389C2312yes—
206#389C2312yes—
207#389C2312yes—
208#389C2312yes—
209#389C2312yes—
210#389C2312yes—
211#388—22no—
212#388—22no—
213#388—22no—
214#388—22no—
215#387—22no—
216#387—22no—
217#387—22no—
218#387—22no—
219#384—21no—
220#384—21no—
221#384—21no—
222#384—21no—
223#384C2311yescharacteristic
224#384C2311yescharacteristic
225#384C2311yescharacteristic
226#384C2311yescharacteristic
227#384C2312yes—
228#384C2312yes—
229#384C2312yes—
230#382—21no—
231#382—21no—
232#380—22no—
233#380—22no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4—21no—
236C4×C4⋊C4—21no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4D411yescharacteristic
240C4×C4⋊C4D411yescharacteristic
241C4×C4⋊C4D411yescharacteristic
242C4×C4⋊C4D411yescharacteristic
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4D411yescharacteristic
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2312yes—
248C4×C4⋊C4C2312yes—
249C4×C4⋊C4C2312yes—
250C4×C4⋊C4C2311yescharacteristic
251C4×C4⋊C4C2311yescharacteristic
252C4×C4⋊C4C2311yescharacteristic
253C43C2311yescharacteristic
254C43C2311yescharacteristic
255C43C2311yescharacteristic
256#340C2314yes—
257#340C2314yes—
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—21no—
273C22×Q8—21no—
274C22×Q8—22no—
275C22×Q8C2×D411yescharacteristic
276C22×D4—21no—
277C22×D4—21no—
278C22×D4C2412yes—
279C23×C4—22no—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C22×C8C2412yes—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—44no—
290C4⋊Q8—44no—
291C4⋊Q8—44no—
292C4⋊Q8—48no—
293C4⋊Q8—44no—
294C4⋊Q8—82no—
295C4⋊Q8#4114yes—
296#127—22no—
297#127—22no—
298#126—48no—
299#126—48no—
300#125—44no—
301#125—48no—
302#125—48no—
303#125—44no—
304#124—44no—
305#124—44no—
306#124—82no—
307#124—82no—
308C22⋊Q8—48no—
309C4×Q8—24no—
310C4×Q8—24no—
311C4×Q8—42no—
312C4×Q8—44no—
313C4×Q8—44no—
314C4×Q8—44no—
315C4×Q8—48no—
316C4×Q8—44no—
317C4×Q8—44no—
318C4×D4—44no—
319C4×D4—44no—
320C4×D4—44no—
321#117—48no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D411yescharacteristic
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⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
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×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
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×C42C2×D411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399C2×C42C2411yescharacteristic
400#107—44no—
401#107—44no—
402#106—28no—
403#106—28no—
404C4⋊C8—24no—
405C4⋊C8—24no—
406C4⋊C8—42no—
407#102—28no—
408#102—28no—
409#102—44no—
410#102—44no—
411#97—24no—
412#97—42no—
413C4×C8—22no—
414C4×C8—22no—
415C4×C8—22no—
416C4×C8—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#95—22no—
431#95C2×D411yescharacteristic
432#95C2×D411yescharacteristic
433#95C2×D411yescharacteristic
434#95C2×D411yescharacteristic
435#95C2×D411yescharacteristic
436#95C2×D411yescharacteristic
437C24—22no—
438C2×Q8—24no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—41no—
442C2×Q8—44no—
443C2×Q8—44no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—44no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×D4—22no—
454C2×D4—24no—
455C2×D4—22no—
456C2×D4—24no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—42no—
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×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
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×C4C2511yescharacteristic
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C82−1+414yes—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—41no—
517C4⋊C4—41no—
518C4⋊C4—41no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—44no—
533C4⋊C4—42no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—82no—
544C4⋊C4—82no—
545C4⋊C4C2×#4114yes—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—82no—
554C22⋊C4—82no—
555C22⋊C4—82no—
556C22⋊C4—82no—
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—22no—
572C42—22no—
573C42—44no—
574C42—44no—
575C42—44no—
576C42—44no—
577C42—44no—
578C42AGL(1,ℤ/8)14yes—
579C42AGL(1,ℤ/8)14yes—
580C42C2×#4112yes—
581C422−1+412yes—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23C23×D411yescharacteristic
590Q8—44no—
591Q8—48no—
592Q8—88no—
593Q8—84no—
594Q8—84no—
595D4—44no—
596D4—44no—
597D4—84no—
598D4—84no—
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—24no—
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#54812yes—
654C2×C4#54812yes—
655C2×C4C2×AGL(1,ℤ/8)12yes—
656C2×C4C2×AGL(1,ℤ/8)12yes—
657C2×C4C2×AGL(1,ℤ/8)12yes—
658C2×C4C2×AGL(1,ℤ/8)12yes—
659C2×C4C2×AGL(1,ℤ/8)12yes—
660C2×C4C2×AGL(1,ℤ/8)12yes—
661C2×C4#57712yes—
662C2×C4#57812yes—
663C2×C4C22×#4111yescharacteristic
664C2×C4C2×2−1+412yes—
665C2×C4C2×2−1+411yescharacteristic
666C2×C4#58512yes—
667C8—24no—
668C8—42no—
669C8—42no—
670C8—42no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22#344911yescharacteristic
679C22C2×#54811yescharacteristic
680C22#349911yescharacteristic
681C22C22×AGL(1,ℤ/8)11yescharacteristic
682C22C22×AGL(1,ℤ/8)11yescharacteristic
683C22C2×#57711yescharacteristic
684C22C2×#57811yescharacteristic
685C4—22no—
686C4—22no—
687C4—22no—
688C4—22no—
689C4—41no—
690C4—41no—
691C4—41no—
692C4—41no—
693C4—41no—
694C4—41no—
695C4—41no—
696C4—41no—
697C4—44no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4—82no—
703C4—82no—
704C4#336714yes—
705C4#336814yes—
706C4#351612yes—
707C4#355512yes—
708C2—162no—
709C2#3916211yescharacteristic
710C2#6096811yescharacteristic
711C2#6100811yescharacteristic
712C2C2×#336711yescharacteristic
713C2C2×#336811yescharacteristic
714C2#6247711yescharacteristic
715C2#6247911yescharacteristic
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
2. #36560C2semidirect
4. #32589C2semidirect
5. #32490C2semidirect
6. #30230C2semidirect
8. #29208C2semidirect
10. #29179C2semidirect
11. #28687C2semidirect
14. #27379C2semidirect
15. #27143C2semidirect
16. #22246C2semidirect
18. #21727C2semidirect
19. #21340C2semidirect

Maximal subgroups

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

subgroupindex
#365942
#365602
#332862
#325892
#324902
#302302
#302032
#292082
#291922
#291792
#286872
#286542
#273792
#271432
#222462
#222202
#217272
#213402