Finite Group Databasebeta

← #6611547

Subgroups — #6611547

Order 512.

1877 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36164C211yesmaximal, characteristic
2D4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#33293C211yesmaximal, characteristic, complemented
4#33051C212yesmaximal, complemented
5#29224C211yesmaximal, characteristic, complemented
6#28726C211yesmaximal, characteristic, complemented
7#27969C212yesmaximal, complemented
8#27823C212yesmaximal, complemented
9#27821C212yesmaximal, complemented
10#27713C214yesmaximal, complemented
11#27711C214yesmaximal, complemented
12#25908C212yesmaximal, complemented
13#22487C212yesmaximal, complemented
14#22485C212yesmaximal
15C4×#489C211yesmaximal, characteristic, complemented
16C2×#1993C212yesmaximal, complemented
17D4×#97C211yesmaximal, characteristic, complemented
18C2×#554C2211yescharacteristic
19C2×D4×Q8—21no—
20C2×#547C2211yescharacteristic
21C22×C4⋊Q8C2212yes—
22C2×#489C2211yescharacteristic
23C2×#489C2212yes—
24C2×C8×D4C2212yes—
25C22×#97C2212yes—
26#2868C2211yescharacteristic
27#2862C2211yescharacteristic
28#2857C2212yes—
29#2850C2212yes—
30#2568C2211yescharacteristic
31#2563C2211yescharacteristic
32#2551—21no—
33#2551C2212yes—
34#2546—21no—
35#2504C2212yes—
36#2503C2212yes—
37#2489—21no—
38#2489—21no—
39#2488—22no—
40#2488—22no—
41#2440—22no—
42#2440C2214yes—
43#2434—22no—
44D4×C4⋊C4—21no—
45D4×C4⋊C4C2212yes—
46#2339—21no—
47#2339—21no—
48C4×C4⋊Q8C2211yescharacteristic
49C4×#124C2211yescharacteristic
50C2×#382C2212yes—
51C42×D4C2211yescharacteristic
52#2101C2214yes—
53#2101C2214yes—
54#2101C2214yes—
55#2072C2212yes—
56#2071C2212yes—
57#2024C2214yes—
58#2023C2214yes—
59#1993C2212yes—
60#1993C2212yes—
61#1993C2212yes—
62#1993C2212yes—
63#1993C2214yes—
64#1993C2214yes—
65#1993C2211yescharacteristic
66#1990C2212yes—
67#1990C2212yes—
68#1988C2212yes—
69#1987C2212yes—
70#1959C2211yescharacteristic
71#1936C2214yes—
72#1936C2212yes—
73#1935C2214yes—
74#1935C2212yes—
75#1893C2214yes—
76#1892C2214yes—
77#1874C2214yes—
78#1872C2214yes—
79#1836C2214yes—
80#1836C2212yes—
81#1836C2211yescharacteristic
82#1824C2214yes—
83#1823C2214yes—
84#1803C2212yes—
85#1802C2212yes—
86#1770C2212yes—
87#1767C2211yescharacteristic
88C4×#103C2212yes—
89C4×#102C2212yes—
90#1752C2214yes—
91#1749C2212yes—
92C2×#340C2214yes—
93C2×#340C2214yes—
94C4×#97C2211yescharacteristic
95C22×#41—21no—
96C23×Q8—22no—
97C23×C8C2314yes—
98#554—28no—
99D4×Q8—48no—
100#547—28no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—28no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—44no—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2314yes—
112C2×C4⋊Q8C2311yescharacteristic
113C2×#125—21no—
114C2×#124C2311yescharacteristic
115C2×#123—22no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—24no—
119C2×C22⋊Q8—22no—
120C2×C22⋊Q8C2312yes—
121C2×#121C2312yes—
122C2×C4×Q8—21no—
123C2×C4×Q8C2311yescharacteristic
124C2×C4×D4—21no—
125C2×C4×D4—22no—
126C2×C4×D4D411yescharacteristic
127C2×C4×D4D411yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2311yescharacteristic
130C22×C4⋊C4—22no—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2314yes—
134C22×C42C2312yes—
135#489—24no—
136#489—44no—
137C8×D4—28no—
138C2×C4⋊C8C2312yes—
139C2×#103C2312yes—
140C2×#103C2312yes—
141C2×#103C2314yes—
142C2×#102C2312yes—
143C2×#102C2312yes—
144C2×#102C2314yes—
145C2×C22⋊C8C2314yes—
146C2×#97—44no—
147C2×#97C2312yes—
148C2×#97C2312yes—
149C2×#97C2311yescharacteristic
150C2×#97C2311yescharacteristic
151C2×#97C2314yes—
152C2×C4×C8C2312yes—
153#400—22no—
154#400—22no—
155#400—22no—
156#398—24no—
157#397—22no—
158#397—22no—
159#395—24no—
160#395—24no—
161#393—22no—
162#391—21no—
163#391—21no—
164#391—21no—
165#391—21no—
166#391—21no—
167#391—22no—
168#391—21no—
169#391—21no—
170#391C2312yes—
171#391C2311yescharacteristic
172#390C2312yes—
173#390C2311yescharacteristic
174#389—22no—
175#389—22no—
176#389—21no—
177#389—22no—
178#389C2312yes—
179#389C2314yes—
180#388—22no—
181#388—22no—
182#388—21no—
183#388—21no—
184#386—22no—
185#386—22no—
186#386—22no—
187#386—22no—
188#385—24no—
189#385—24no—
190#384—21no—
191#384—21no—
192#384C2311yescharacteristic
193#384C2312yes—
194#384C2312yes—
195#382—22no—
196#382—22no—
197#382C2312yes—
198#382C2312yes—
199#382C2312yes—
200#382C2312yes—
201#382C2311yescharacteristic
202#381—22no—
203#381—22no—
204#380—22no—
205#380—21no—
206#380—21no—
207#380C2314yes—
208C4×C4⋊C4—21no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—21no—
214C4×C22⋊C4—21no—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4C2312yes—
218C2×#95—22no—
219C2×#95—22no—
220C43C2311yescharacteristic
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#340C2312yes—
235#336C2312yes—
236#336C2314yes—
237C2×#41—44no—
238C2×#41—44no—
239C2×#41—44no—
240C22×Q8—21no—
241C22×Q8—22no—
242C22×Q8—21no—
243C22×Q8—24no—
244C22×Q8—21no—
245C22×Q8—48no—
246C22×Q8—48no—
247C22×Q8—84no—
248C22×Q8C2411yescharacteristic
249C22×D4—22no—
250C22×D4C2×D411yescharacteristic
251C22×D4C2411yescharacteristic
252C23×C4—22no—
253C23×C4—24no—
254C23×C4—22no—
255C23×C4—21no—
256C23×C4C2×D412yes—
257C23×C4C2×D412yes—
258C23×C4C2412yes—
259C22×C8—28no—
260C22×C8—28no—
261C22×C8—44no—
262C22×C8C2414yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C4⋊Q8—28no—
266C4⋊Q8—24no—
267C4⋊Q8—24no—
268C4⋊Q8—24no—
269C4⋊Q8—22no—
270C4⋊Q8—24no—
271C4⋊Q8—44no—
272C4⋊Q8—48no—
273C4⋊Q8—48no—
274C4⋊Q8—44no—
275C4⋊Q8—48no—
276C4⋊Q8—82no—
277C4⋊Q8#4114yescomplemented
278#125—44no—
279#124—22no—
280#124—22no—
281#123—48no—
282C22⋊Q8—28no—
283C22⋊Q8—416no—
284C22⋊Q8—48no—
285C22⋊Q8—44no—
286C22⋊Q8—48no—
287#121—28no—
288C4×Q8—24no—
289C4×Q8—44no—
290C4×D4—24no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—44no—
294C4×D4—48no—
295C4×D4#4118yes—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—28no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—48no—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2414yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356C4⋊C8—24no—
357#103—24no—
358#103—24no—
359#103—44no—
360#102—24no—
361#102—24no—
362#102—44no—
363C22⋊C8—28no—
364#97—24no—
365#97—22no—
366#97—22no—
367#97—24no—
368#97—44no—
369C4×C8—24no—
370#95—21no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—21no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—24no—
383#95—24no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391C24—21no—
392C24C22×D412yes—
393#41—816no—
394C2×Q8—24no—
395C2×Q8—21no—
396C2×Q8—22no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—48no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—44no—
403C2×Q8—48no—
404C2×Q8—44no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—44no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—84no—
411C2×Q8—81no—
412C2×Q8—81no—
413C2×Q8—832no—
414C2×D4—21no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4#12414yes—
422C2×D4AGL(1,ℤ/8)14yes—
423C2×D4#13714yes—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—48no—
442C22×C4—44no—
443C22×C4—48no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—84no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C2×#4118yes—
463C22×C4C2511yescharacteristic
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—416no—
473C2×C82−1+414yes—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—48no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4#12414yes—
504C4⋊C4C2×#4114yes—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—44no—
508C22⋊C4—48no—
509C22⋊C4—42no—
510C22⋊C4—44no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—81no—
514C22⋊C4—82no—
515C22⋊C4—81no—
516C22⋊C4—82no—
517C22⋊C4#12418yes—
518C42—24no—
519C42—24no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—41no—
526C42—41no—
527C42—42no—
528C42—44no—
529C42C2×#4114yes—
530C42C2×#4112yes—
531C42H(5,𝔽2)12yes—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23C2×#12414yes—
544C23C2×AGL(1,ℤ/8)14yes—
545C23C2×#13714yes—
546C23C23×D411yescharacteristic
547Q8—48no—
548Q8—42no—
549Q8—42no—
550Q8—88no—
551Q8—816no—
552Q8—84no—
553Q8—162no—
554D4—42no—
555D4—42no—
556D4—88no—
557D4#489116yescomplemented
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—48no—
582C2×C4—48no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—816no—
607C2×C4—88no—
608C2×C4—84no—
609C2×C4—164no—
610C2×C4C2×#12412yes—
611C2×C4C2×#12414yes—
612C2×C4C2×#12412yes—
613C2×C4#54612yes—
614C2×C4#54612yes—
615C2×C4C2×AGL(1,ℤ/8)12yes—
616C2×C4C2×#13712yes—
617C2×C4C22×#4111yescharacteristic
618C2×C4C22×#4112yes—
619C2×C4C2×H(5,𝔽2)11yescharacteristic
620C2×C4C2×2−1+412yes—
621C8—24no—
622C8—42no—
623C8—44no—
624C22—24no—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22C2×#48918yes—
638C22C22×#12411yescharacteristic
639C22C2×#54611yescharacteristic
640C22#349911yescharacteristic
641C22C22×AGL(1,ℤ/8)11yescharacteristic
642C22C22×#13711yescharacteristic
643C22#358511yescharacteristic
644C22#358611yescharacteristic
645C4—24no—
646C4—22no—
647C4—22no—
648C4—22no—
649C4—41no—
650C4—41no—
651C4—42no—
652C4—44no—
653C4—42no—
654C4—82no—
655C4—81no—
656C4—82no—
657C4—81no—
658C4—81no—
659C4—82no—
660C4—84no—
661C4C2×#48914yes—
662C4#348212yes—
663C4#352012yes—
664C2—28no—
665C2—161no—
666C2#3630311yescharacteristic
667C2C22×#48911yescharacteristic
668C2#6062511yescharacteristic
669C2#6115511yescharacteristic
670C2#6115711yescharacteristic
671C2#6250311yescharacteristic
672C2#6250411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. D4×C4⋊Q8C2semidirect
3. #33293C2semidirect
4. #33051C2semidirect
5. #29224C2semidirect
6. #28726C2semidirect
7. #27969C2semidirect
8. #27823C2semidirect
9. #27821C2semidirect
10. #27713C2semidirect
11. #27711C2semidirect
12. #25908C2semidirect
13. #22487C2semidirect
15. C4×#489C2semidirect
16. C2×#1993C2semidirect
17. D4×#97C2semidirect
277. C4⋊Q8#41semidirect
557. D4#489semidirect

Maximal subgroups

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

subgroupindex
#361642
D4×C4⋊Q82
#332932
#330512
#292242
#287262
#279692
#278232
#278212
#277132
#277112
#259082
#224872
#224852
C4×#4892
C2×#19932
D4×#972