Finite Group Databasebeta

← #6804376

Subgroups — #6804376

Order 512.

1654 conjugacy classes of subgroups.

The subgroup lattice has 759 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36609C211yesmaximal, characteristic, complemented
2#36559C211yesmaximal, characteristic, complemented
3#33278C211yesmaximal, characteristic, complemented
4#32605C214yesmaximal, complemented
5#31155C214yesmaximal, complemented
6#30219C211yesmaximal, characteristic, complemented
7#30203C211yesmaximal, characteristic, complemented
8#29192C211yesmaximal, characteristic, complemented
9#29179C211yesmaximal, characteristic, complemented
10#28615C211yesmaximal, characteristic
11#28613C211yesmaximal, characteristic, complemented
12#28511C211yesmaximal, characteristic, complemented
13#28443C211yesmaximal, characteristic
14#26890C214yesmaximal, complemented
15#26164C214yesmaximal, complemented
16#22268C211yesmaximal, characteristic, complemented
17#22220C211yesmaximal, characteristic, complemented
18C4×#493C211yesmaximal, characteristic
19#21339C211yesmaximal, characteristic, complemented
20C2×#554—21no—
21C2×#493C2211yescharacteristic
22C2×#492C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#477C2214yes—
25C2×#463C2214yes—
26C2×C4×D8C2214yes—
27#2868C2211yescharacteristic
28#2862C2211yescharacteristic
29#2859C2211yescharacteristic
30#2859C2211yescharacteristic
31#2858C2211yescharacteristic
32#2858C2211yescharacteristic
33#2592—21no—
34#2592C2212yes—
35#2580—21no—
36#2566—21no—
37#2566—21no—
38#2563—21no—
39#2538—21no—
40#2538C2212yes—
41#2513—21no—
42#2513—21no—
43#2452—21no—
44#2452—21no—
45#2452C2212yes—
46#2452C2212yes—
47#2448—21no—
48#2448C2212yes—
49#2447—21no—
50#2447C2212yes—
51#2367—21no—
52#2367C2212yes—
53#2357—21no—
54D4×C4⋊C4—21no—
55D4×C4⋊C4C2212yes—
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×#127C2211yescharacteristic
59C4×#127C2211yescharacteristic
60C4×#127C2211yescharacteristic
61C4×#127C2211yescharacteristic
62C42×Q8C2211yescharacteristic
63#2095C2214yes—
64#2087C2214yes—
65#2071C2214yes—
66#2067C2214yes—
67#2055C2214yes—
68#2054C2214yes—
69#2042C2214yes—
70#2042C2214yes—
71#2023C2214yes—
72#2012C2214yes—
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1984C2212yes—
77#1970C2214yes—
78#1957C2211yescharacteristic
79#1957C2211yescharacteristic
80#1957C2211yescharacteristic
81#1957C2211yescharacteristic
82#1956C2211yescharacteristic
83#1956C2211yescharacteristic
84#1956C2211yescharacteristic
85#1939C2212yes—
86#1933C2214yes—
87#1933C2214yes—
88#1925C2214yes—
89#1921C2214yes—
90#1916C2212yes—
91#1916C2212yes—
92#1880C2214yes—
93#1879C2214yes—
94#1877C2214yes—
95#1877C2214yes—
96#1862C2214yes—
97#1862C2214yes—
98#1845C2211yescharacteristic
99#1845C2211yescharacteristic
100#1834C2211yescharacteristic
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×C4⋊C8C2211yescharacteristic
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C22×SD16C2314yes—
108C22×D8C2314yes—
109#554—48no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2312yes—
115C2×#127C2311yescharacteristic
116C2×#127C2311yescharacteristic
117C2×#125—21no—
118C2×#125—21no—
119C2×#124—21no—
120C2×#124—21no—
121C2×#124C2312yes—
122C2×#124C2312yes—
123C2×C22⋊Q8—22no—
124C2×C22⋊Q8—22no—
125C2×C22⋊Q8—22no—
126C2×C4×Q8—21no—
127C2×C4×Q8D411yescharacteristic
128C2×C4×Q8D411yescharacteristic
129C2×C4×Q8C2312yes—
130C2×C4×Q8C2311yescharacteristic
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4—21no—
134C2×C4×D4—21no—
135C2×C4×D4—21no—
136C2×C4×D4—21no—
137C2×C4×D4—21no—
138C2×C4×D4—21no—
139C2×C4×D4C2312yes—
140C2×C4×D4C2312yes—
141C2×C4×D4C2312yes—
142C2×C4×D4C2312yes—
143C22×C4⋊C4—22no—
144#493—24no—
145#492—24no—
146#486—24no—
147#477—48no—
148#463—48no—
149C4×D8—48no—
150C2×#107C2314yes—
151C2×#107C2314yes—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2312yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2312yes—
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2312yes—
169#399—22no—
170#397—22no—
171#397—22no—
172#397—22no—
173#397—22no—
174#396—22no—
175#396—22no—
176#396—22no—
177#396—22no—
178#393—22no—
179#393—22no—
180#393—22no—
181#391—21no—
182#391—21no—
183#391—21no—
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#390—21no—
194#390—21no—
195#390—21no—
196#390—21no—
197#390—21no—
198#390—21no—
199#390—21no—
200#390—21no—
201#390C2312yes—
202#390C2312yes—
203#390C2312yes—
204#390C2312yes—
205#390C2312yes—
206#390C2312yes—
207#389—21no—
208#389—21no—
209#389—21no—
210#389—21no—
211#389—21no—
212#389—21no—
213#389—21no—
214#389C2312yes—
215#389C2312yes—
216#389C2312yes—
217#389C2312yes—
218#389C2312yes—
219#388—22no—
220#388—22no—
221#388—22no—
222#388—22no—
223#388—22no—
224#388—22no—
225#385—22no—
226#385—22no—
227#385—22no—
228#385—22no—
229#385—22no—
230#385—22no—
231#384—21no—
232#384—21no—
233#384C2311yescharacteristic
234#384C2311yescharacteristic
235#384C2311yescharacteristic
236#384C2311yescharacteristic
237#384C2312yes—
238#382—21no—
239#382—21no—
240#382—21no—
241#382—21no—
242#381—22no—
243#380—22no—
244#380—22no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—21no—
247C4×C4⋊C4—21no—
248C4×C4⋊C4D411yescharacteristic
249C4×C4⋊C4D411yescharacteristic
250C4×C4⋊C4D411yescharacteristic
251C4×C4⋊C4D411yescharacteristic
252C4×C4⋊C4D411yescharacteristic
253C4×C4⋊C4D411yescharacteristic
254C4×C4⋊C4C2312yes—
255C4×C4⋊C4C2311yescharacteristic
256C4×C4⋊C4C2311yescharacteristic
257C4×C4⋊C4C2311yescharacteristic
258C4×C22⋊C4—22no—
259C43C2311yescharacteristic
260C43C2311yescharacteristic
261C43C2311yescharacteristic
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272#336C2312yes—
273C22×Q8—21no—
274C22×Q8—22no—
275C22×Q8C2×D411yescharacteristic
276C22×Q8C2412yes—
277C22×D4—21no—
278C22×D4—21no—
279C22×D4—21no—
280C22×D4—21no—
281C22×D4C2412yes—
282C22×D4C2412yes—
283C23×C4—22no—
284C23×C4—22no—
285C23×C4—22no—
286C23×C4—22no—
287C2×SD16—28no—
288C2×SD16—44no—
289C2×SD16—44no—
290C2×D8—28no—
291C2×D8—28no—
292C2×D8—44no—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—44no—
302C4⋊Q8—44no—
303C4⋊Q8—82no—
304#127—22no—
305#127—22no—
306#127—22no—
307#127#4114yes—
308#125—44no—
309#125—44no—
310#124—44no—
311#124—44no—
312#124—44no—
313#124—82no—
314C22⋊Q8—48no—
315C22⋊Q8—48no—
316C22⋊Q8—48no—
317C4×Q8—24no—
318C4×Q8—24no—
319C4×Q8—42no—
320C4×Q8—44no—
321C4×Q8—82no—
322C4×D4—44no—
323C4×D4—44no—
324C4×D4—44no—
325C4×D4—44no—
326C4×D4—44no—
327C4×D4—44no—
328C4×D4—44no—
329C4×D4—44no—
330C4×D4—44no—
331C4×D4—82no—
332C4×D4—82no—
333C4×D4—82no—
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—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—48no—
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⋊C4C2×D411yescharacteristic
362C2×C4⋊C4C2×D411yescharacteristic
363C2×C4⋊C4C2×D411yescharacteristic
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
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×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—22no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—22no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42C2×D411yescharacteristic
401C2×C42C2×D411yescharacteristic
402C2×C42C2×D411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406C2×C42C2411yescharacteristic
407C2×C42C2411yescharacteristic
408C2×C42C2411yescharacteristic
409C2×C42C2411yescharacteristic
410#107—28no—
411#107—44no—
412C4⋊C8—24no—
413C4⋊C8—24no—
414C4⋊C8—42no—
415#103—28no—
416#103—44no—
417#103—44no—
418#102—28no—
419#102—44no—
420#102—44no—
421#102—44no—
422#102—44no—
423C4×C8—24no—
424C4×C8—22no—
425C4×C8—22no—
426C4×C8—22no—
427C4×C8—22no—
428C4×C8—42no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95C2×D411yescharacteristic
442#95C2×D411yescharacteristic
443#95C2×D411yescharacteristic
444#95C2×D411yescharacteristic
445#95C2×D411yescharacteristic
446#95C2×D411yescharacteristic
447C24—22no—
448C24—22no—
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—24no—
453C2×Q8—22no—
454C2×Q8—24no—
455C2×Q8—41no—
456C2×Q8—44no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×Q8—82no—
461C2×Q8—82no—
462C2×D4—24no—
463C2×D4—24no—
464C2×D4—22no—
465C2×D4—24no—
466C2×D4—24no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—24no—
470C2×D4—24no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4—42no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4—42no—
480C2×D4—42no—
481C2×D4C2×#4114yes—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—82no—
495C22×C4—82no—
496C22×C4—82no—
497C22×C4—82no—
498C22×C4—82no—
499C22×C4—82no—
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C2511yescharacteristic
515SD16—88no—
516D8—416no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8H(5,𝔽2)14yes—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—22no—
540C4⋊C4—24no—
541C4⋊C4—41no—
542C4⋊C4—41no—
543C4⋊C4—41no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—44no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—44no—
557C4⋊C4—42no—
558C4⋊C4—42no—
559C4⋊C4—42no—
560C4⋊C4—82no—
561C4⋊C4—82no—
562C4⋊C4—82no—
563C4⋊C4—82no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—82no—
575C22⋊C4—82no—
576C22⋊C4—82no—
577C22⋊C4—82no—
578C22⋊C4—82no—
579C22⋊C4—82no—
580C22⋊C4—82no—
581C22⋊C4—82no—
582C22⋊C4—82no—
583C22⋊C4—82no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—22no—
593C42—22no—
594C42—22no—
595C42—22no—
596C42—22no—
597C42—22no—
598C42—22no—
599C42—22no—
600C42—82no—
601C42—82no—
602C42#13514yes—
603C42AGL(1,ℤ/8)14yes—
604C42C2×#4112yes—
605C422−1+412yes—
606C23—24no—
607C23—42no—
608C23—42no—
609C23—42no—
610C23—42no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23—42no—
615C23—42no—
616C23—42no—
617C23—42no—
618C23—42no—
619C23—42no—
620C23C23×D411yescharacteristic
621Q8—44no—
622Q8—44no—
623Q8—44no—
624Q8—84no—
625Q8—164no—
626D4—28no—
627D4—48no—
628D4—48no—
629D4—44no—
630D4—44no—
631D4—44no—
632D4—84no—
633D4—84no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—21no—
662C2×C4—21no—
663C2×C4—21no—
664C2×C4—21no—
665C2×C4—21no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—88no—
685C2×C4—164no—
686C2×C4—164no—
687C2×C4—164no—
688C2×C4#54812yes—
689C2×C4#54812yes—
690C2×C4C2×#13512yes—
691C2×C4C2×#13512yes—
692C2×C4C2×#13512yes—
693C2×C4C2×AGL(1,ℤ/8)12yes—
694C2×C4C2×AGL(1,ℤ/8)12yes—
695C2×C4C2×AGL(1,ℤ/8)12yes—
696C2×C4#57612yes—
697C2×C4#57812yes—
698C2×C4C22×#4111yescharacteristic
699C2×C4C2×H(5,𝔽2)12yes—
700C2×C4C2×2−1+411yescharacteristic
701C2×C4#58512yes—
702C8—24no—
703C8—42no—
704C8—42no—
705C8—42no—
706C22—44no—
707C22—44no—
708C22—44no—
709C22—82no—
710C22—82no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22—82no—
715C22—82no—
716C22—82no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22#344911yescharacteristic
721C22C2×#54811yescharacteristic
722C22#349811yescharacteristic
723C22C22×#13511yescharacteristic
724C22C22×AGL(1,ℤ/8)11yescharacteristic
725C22C2×#57611yescharacteristic
726C22C2×#57811yescharacteristic
727C4—22no—
728C4—22no—
729C4—22no—
730C4—22no—
731C4—41no—
732C4—41no—
733C4—41no—
734C4—41no—
735C4—41no—
736C4—41no—
737C4—41no—
738C4—41no—
739C4—82no—
740C4—82no—
741C4—82no—
742C4—82no—
743C4—82no—
744C4—82no—
745C4#336114yes—
746C4#336714yes—
747C4#351312yes—
748C4#355412yes—
749C2—84no—
750C2—162no—
751C2#3916111yescharacteristic
752C2#6094611yescharacteristic
753C2#6101411yescharacteristic
754C2C2×#336111yescharacteristic
755C2C2×#336711yescharacteristic
756C2#6241811yescharacteristic
757C2#6242311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#493C22
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. #36609C2semidirect
2. #36559C2semidirect
3. #33278C2semidirect
4. #32605C2semidirect
5. #31155C2semidirect
6. #30219C2semidirect
7. #30203C2semidirect
8. #29192C2semidirect
9. #29179C2semidirect
11. #28613C2semidirect
12. #28511C2semidirect
14. #26890C2semidirect
15. #26164C2semidirect
16. #22268C2semidirect
17. #22220C2semidirect
19. #21339C2semidirect

Maximal subgroups

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

subgroupindex
#366092
#365592
#332782
#326052
#311552
#302192
#302032
#291922
#291792
#286152
#286132
#285112
#284432
#268902
#261642
#222682
#222202
C4×#4932
#213392