Finite Group Databasebeta

← #6684311

Subgroups — #6684311

Order 512.

1766 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3383C211yesmaximal, characteristic, complemented
2C2×#3328C211yesmaximal, characteristic
3#39744C211yesmaximal, characteristic
4#36446C211yesmaximal, characteristic
5#33196C211yesmaximal, characteristic, complemented
6#30784C214yesmaximal, complemented
7#30686C214yesmaximal, complemented
8#30279C211yesmaximal, characteristic, complemented
9#29831C211yesmaximal, characteristic, complemented
10#28712C211yesmaximal, characteristic
11#28639C211yesmaximal, characteristic, complemented
12#25981C214yesmaximal, complemented
13#25931C211yesmaximal, characteristic, complemented
14#25884C211yesmaximal, characteristic
15#25840C211yesmaximal, characteristic, complemented
16#25705C211yesmaximal, characteristic
17#24906C214yesmaximal, complemented
18#22210C211yesmaximal, characteristic
19#21116C211yesmaximal, characteristic, complemented
20C2×#557C2211yescharacteristic
21C2×#557C2211yescharacteristic
22C2×#556C2211yescharacteristic
23C2×D4×Q8C2211yescharacteristic
24#3383—28no—
25#3328—28no—
26C2×#500C2211yescharacteristic
27C2×#500C2211yescharacteristic
28C2×#499C2212yes—
29C2×#487C2211yescharacteristic
30C2×#487C2211yescharacteristic
31C2×#485C2212yes—
32C2×#479C2212yes—
33C2×#479C2212yes—
34C2×#475C2211yescharacteristic
35C2×#475C2211yescharacteristic
36C2×#475C2211yescharacteristic
37C2×#470C2212yes—
38C2×#462C2211yescharacteristic
39C2×#451C2212yes—
40C2×C8×Q8C2211yescharacteristic
41C2×C4×Q16C2212yes—
42C2×C4×Q16C2211yescharacteristic
43C2×#436C2211yescharacteristic
44C2×#419C2212yes—
45#2868C2211yescharacteristic
46#2862—21no—
47#2862C2211yescharacteristic
48#2721—24no—
49#2658—21no—
50#2568—21no—
51#2568C2211yescharacteristic
52#2547—21no—
53#2547C2211yescharacteristic
54#2547C2211yescharacteristic
55#2546—21no—
56#2546C2211yescharacteristic
57#2545—21no—
58#2489—24no—
59#2489C2214yes—
60#2441C2214yes—
61#2423C2214yes—
62Q8×C22⋊C4C2214yes—
63#2327—21no—
64C4×#124—21no—
65C4×#124C2211yescharacteristic
66#2295C2211yescharacteristic
67C42×Q8C2211yescharacteristic
68#2095C2214yes—
69#2095C2214yes—
70#2076C2214yes—
71#2066C2214yes—
72#2066C2214yes—
73#2059C2214yes—
74#2056C2214yes—
75#2036C2214yes—
76#2035C2214yes—
77#2026C2214yes—
78#1988C2211yescharacteristic
79#1988C2211yescharacteristic
80#1988C2211yescharacteristic
81#1984C2211yescharacteristic
82#1981C2211yescharacteristic
83#1954C2211yescharacteristic
84#1948C2214yes—
85#1933C2214yes—
86#1927C2212yes—
87#1923C2214yes—
88#1918C2212yes—
89#1878C2212yes—
90#1878C2214yes—
91#1865C2212yes—
92#1863C2214yes—
93#1854C2214yes—
94#1851C2214yes—
95#1842C2211yescharacteristic
96#1809C2214yes—
97#1803C2214yes—
98#1787C2214yes—
99C4×#106C2211yescharacteristic
100C4×#103C2211yescharacteristic
101C4×#103C2211yescharacteristic
102#1744C2211yescharacteristic
103C23×Q8C2312yes—
104C22×Q16C2311yescharacteristic
105C22×Q16C2312yes—
106C22×M16C2312yes—
107#557—24no—
108#557—24no—
109#557—42no—
110#557—42no—
111#556—24no—
112#556—42no—
113D4×Q8—24no—
114D4×Q8—42no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8—21no—
119C2×C4⋊Q8—21no—
120C2×C4⋊Q8—21no—
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2311yescharacteristic
123C2×C4⋊Q8C2311yescharacteristic
124C2×C4⋊Q8C2311yescharacteristic
125C2×#126—22no—
126C2×#125—22no—
127C2×#125—22no—
128C2×#125C2312yes—
129C2×#125C2312yes—
130C2×#124—21no—
131C2×#124—21no—
132C2×#124—21no—
133C2×#124D411yescharacteristic
134C2×C22⋊Q8—22no—
135C2×C22⋊Q8C2312yes—
136C2×C22⋊Q8C2312yes—
137C2×C4×Q8—21no—
138C2×C4×Q8—21no—
139C2×C4×Q8—21no—
140C2×C4×Q8C2312yes—
141C2×C4×Q8C2312yes—
142C2×C4×Q8C2312yes—
143C2×C4×Q8C2311yescharacteristic
144C2×C4×Q8C2311yescharacteristic
145C2×C4×Q8C2311yescharacteristic
146C2×C4×D4—21no—
147C2×C4×D4C2311yescharacteristic
148C2×#117C2312yes—
149#500—24no—
150#500—42no—
151#499—28no—
152#487—24no—
153#487—42no—
154#485—28no—
155#479—28no—
156#479—28no—
157#475—24no—
158#475—24no—
159#475—24no—
160#470—28no—
161#462—24no—
162#451—28no—
163C8×Q8—24no—
164C4×Q16—28no—
165C4×Q16D418yes—
166#436—24no—
167C2×#107C2312yes—
168C2×#107C2312yes—
169C2×#107C2314yes—
170C2×#107C2311yescharacteristic
171C2×#106C2312yes—
172C2×#106C2312yes—
173C2×#106C2312yes—
174C2×#106C2312yes—
175C2×C4⋊C8C2311yescharacteristic
176C2×C4⋊C8C2311yescharacteristic
177C2×C4⋊C8C2312yes—
178C2×C4⋊C8C2314yes—
179#419—28no—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2312yes—
185C2×#103C2312yes—
186C2×#103C2312yes—
187C2×#103C2314yes—
188C2×#103C2314yes—
189C2×#103C2312yes—
190C2×#103C2311yescharacteristic
191C2×#103C2311yescharacteristic
192C2×C22⋊C8C2312yes—
193C2×C22⋊C8C2314yes—
194C2×C4×C8C2311yescharacteristic
195C2×C4×C8C2312yes—
196C2×C4×C8C2311yescharacteristic
197#400—22no—
198#400—24no—
199#399—24no—
200#399—24no—
201#398—22no—
202#398—24no—
203#397—22no—
204#397—22no—
205#397—24no—
206#396—24no—
207#391—21no—
208#391—21no—
209#391—21no—
210#391—21no—
211#391—21no—
212#391—21no—
213#391—24no—
214#391—24no—
215#391D411yescharacteristic
216#391C2312yes—
217#391C2314yes—
218#391C2314yes—
219#390D411yescharacteristic
220#389—24no—
221#389C2314yes—
222#389C2312yes—
223#388—24no—
224#388—24no—
225#388D414yes—
226#387—22no—
227#387—24no—
228#387—24no—
229#384—22no—
230#384C2314yes—
231#382—21no—
232#382—22no—
233#382—24no—
234#382D411yescharacteristic
235#380—24no—
236#380C2314yes—
237#380C2314yes—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—22no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4C2311yescharacteristic
242C4×C4⋊C4C2312yes—
243C4×C4⋊C4C2311yescharacteristic
244C4×C22⋊C4—24no—
245C4×C22⋊C4C2314yes—
246#376—21no—
247C43—21no—
248C43C2311yescharacteristic
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2312yes—
253#340C2312yes—
254#340C2312yes—
255#336C2314yes—
256#336C2312yes—
257#336C2312yes—
258C22×Q8—21no—
259C22×Q8—21no—
260C22×Q8—21no—
261C22×Q8—22no—
262C22×Q8—24no—
263C22×Q8—28no—
264C22×Q8—42no—
265C22×Q8—44no—
266C22×Q8C2×D411yescharacteristic
267C22×Q8C2412yes—
268C22×Q8C2411yescharacteristic
269C22×Q8C2411yescharacteristic
270C22×D4C2×D411yescharacteristic
271C23×C4—22no—
272C23×C4C2412yes—
273C2×Q16—24no—
274C2×Q16—24no—
275C2×Q16—44no—
276C2×Q16C2×D414yes—
277C2×M16—24no—
278C2×M16—44no—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2411yescharacteristic
283C22×C8C2411yescharacteristic
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—44no—
293C4⋊Q8—44no—
294C4⋊Q8—44no—
295C4⋊Q8—44no—
296C4⋊Q8—82no—
297C4⋊Q8—82no—
298#126—48no—
299#125—24no—
300#125—24no—
301#125—42no—
302#125—42no—
303#125—48no—
304#125—48no—
305#124—42no—
306#124—44no—
307#124—44no—
308#124—82no—
309C22⋊Q8—24no—
310C22⋊Q8—24no—
311C22⋊Q8—48no—
312C22⋊Q8—42no—
313C22⋊Q8—42no—
314C4×Q8—22no—
315C4×Q8—22no—
316C4×Q8—22no—
317C4×Q8—22no—
318C4×Q8—24no—
319C4×Q8—24no—
320C4×Q8—24no—
321C4×Q8—42no—
322C4×Q8—42no—
323C4×Q8—42no—
324C4×Q8—44no—
325C4×Q8—44no—
326C4×Q8—44no—
327C4×Q8C2×D414yes—
328C4×Q8C2×D414yes—
329C4×D4—22no—
330C4×D4—22no—
331C4×D4—44no—
332#117—24no—
333#117—42no—
334C2×C4⋊C4—21no—
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—21no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2×D412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2411yescharacteristic
363C2×C4⋊C4C2411yescharacteristic
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4C2×D412yes—
370C2×C22⋊C4C2×D414yes—
371C2×C22⋊C4C2×D414yes—
372C2×C42—21no—
373C2×C42—21no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—21no—
379C2×C42—21no—
380C2×C42—22no—
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D412yes—
384C2×C42C2412yes—
385C2×C42C2412yes—
386C2×C42C2411yescharacteristic
387#107—24no—
388#107—24no—
389#107—44no—
390#107C2×D414yes—
391#106—24no—
392#106—24no—
393#106—24no—
394#106—42no—
395C4⋊C8—24no—
396C4⋊C8—22no—
397C4⋊C8—22no—
398C4⋊C8—44no—
399#103—24no—
400#103—24no—
401#103—24no—
402#103—24no—
403#103—24no—
404#103—24no—
405#103—24no—
406#103—44no—
407#103—42no—
408#103—44no—
409#103C2×D414yes—
410#103C2×D414yes—
411C22⋊C8—24no—
412C22⋊C8—44no—
413C4×C8—24no—
414C4×C8—22no—
415C4×C8C2×D414yes—
416#95—22no—
417#95—22no—
418#95—24no—
419#95—24no—
420#95—22no—
421#95—21no—
422#95—22no—
423#95—21no—
424#95—24no—
425#95—24no—
426#95—24no—
427#95—24no—
428#95—24no—
429#95C2×D414yes—
430#95C2×D414yes—
431C24C22×D412yes—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—24no—
442C2×Q8—22no—
443C2×Q8—24no—
444C2×Q8—44no—
445C2×Q8—48no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—48no—
456C2×Q8—84no—
457C2×Q8—88no—
458C2×Q8C22×D412yes—
459C2×Q8C22×D412yes—
460C2×Q82−1+414yes—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—22no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—44no—
479C22×C4—48no—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C2511yescharacteristic
488Q16—28no—
489Q16—84no—
490M16—84no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8C22×D412yes—
501C2×C8C22×D412yes—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
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—48no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—44no—
535C4⋊C4—84no—
536C4⋊C4—84no—
537C4⋊C4C22×D412yes—
538C4⋊C4C22×D412yes—
539C22⋊C4—24no—
540C22⋊C4—28no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—48no—
546C22⋊C4—84no—
547C42—24no—
548C42—24no—
549C42—22no—
550C42—22no—
551C42—24no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—44no—
557C42—44no—
558C42—44no—
559C42—44no—
560C42—42no—
561C42—42no—
562C42—44no—
563C42C22×D412yes—
564C42H(5,𝔽2)12yes—
565C23—22no—
566C23—24no—
567C23—22no—
568C23—22no—
569C23—24no—
570C23C23×D411yescharacteristic
571Q8—22no—
572Q8—24no—
573Q8—22no—
574Q8—22no—
575Q8—44no—
576Q8—42no—
577Q8—42no—
578Q8—42no—
579Q8—44no—
580Q8—88no—
581Q8—84no—
582Q8—84no—
583Q8—84no—
584Q8#53614yes—
585D4—44no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—88no—
628C2×C4D4212yes—
629C2×C4D4212yes—
630C2×C4#57512yes—
631C2×C4#57712yes—
632C2×C4#57814yes—
633C2×C4C23×D411yescharacteristic
634C2×C4C2×H(5,𝔽2)11yescharacteristic
635C2×C4C2×2−1+412yes—
636C2×C4#58512yes—
637C8—22no—
638C8—22no—
639C8—42no—
640C8—42no—
641C8—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—44no—
646C22—44no—
647C22C2×D4211yescharacteristic
648C22#348912yes—
649C22C2×#57511yescharacteristic
650C22C2×#57711yescharacteristic
651C22C2×#57812yes—
652C4—24no—
653C4—22no—
654C4—24no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4#327114yes—
669C4C2×#53611yescharacteristic
670C4#350011yescharacteristic
671C4#356112yes—
672C2—82no—
673C2#3767611yescharacteristic
674C2C2×#327111yescharacteristic
675C2#6109111yescharacteristic
676C2#6214612yes—
677C2#6220312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#124 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3383C2semidirect
5. #33196C2semidirect
6. #30784C2semidirect
7. #30686C2semidirect
8. #30279C2semidirect
9. #29831C2semidirect
11. #28639C2semidirect
12. #25981C2semidirect
13. #25931C2semidirect
15. #25840C2semidirect
17. #24906C2semidirect
19. #21116C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33832
C2×#33282
#397442
#364462
#331962
#307842
#306862
#302792
#298312
#287122
#286392
#259812
#259312
#258842
#258402
#257052
#249062
#222102
#211162