Finite Group Databasebeta

← #6683836

Subgroups — #6683836

Order 512.

1784 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3380C211yesmaximal, characteristic
2C2×#3360C211yesmaximal, characteristic
3#39490C211yesmaximal, characteristic, complemented
4#36180C211yesmaximal, characteristic, complemented
5#33026C211yesmaximal, characteristic, complemented
6#30952C214yesmaximal, complemented
7#30811C214yesmaximal, complemented
8#30279C211yesmaximal, characteristic, complemented
9#29868C211yesmaximal, characteristic, complemented
10#28676C211yesmaximal, characteristic, complemented
11#28638C211yesmaximal, characteristic, complemented
12#26374C214yesmaximal, complemented
13#26164C214yesmaximal
14#25934C211yesmaximal, characteristic, complemented
15#25839C211yesmaximal, characteristic, complemented
16#25839C211yesmaximal, characteristic, complemented
17#25646C211yesmaximal, characteristic, complemented
18#22238C211yesmaximal, characteristic, complemented
19#21113C211yesmaximal, characteristic
20C2×#556C2211yescharacteristic
21C2×#554C2211yescharacteristic
22C2×#554C2211yescharacteristic
23C2×#554C2211yescharacteristic
24#3380—28no—
25#3360—28no—
26C2×#500C2211yescharacteristic
27C2×#500C2211yescharacteristic
28C2×#499C2212yes—
29C2×#488C2212yes—
30C2×#487C2211yescharacteristic
31C2×#487C2211yescharacteristic
32C2×#479C2212yes—
33C2×#478C2212yes—
34C2×#478C2212yes—
35C2×#476C2211yescharacteristic
36C2×#476C2211yescharacteristic
37C2×#476C2211yescharacteristic
38C2×#463C2212yes—
39C2×#461C2211yescharacteristic
40C2×C8×Q8C2211yescharacteristic
41C2×C8×Q8C2211yescharacteristic
42C2×C4×SD16C2212yes—
43C2×C4×D8C2212yes—
44C2×C4×D8C2211yescharacteristic
45#2868C2211yescharacteristic
46#2862C2211yescharacteristic
47#2857—21no—
48#2689—24no—
49#2621—21no—
50#2568C2211yescharacteristic
51#2567C2211yescharacteristic
52#2567C2211yescharacteristic
53#2567C2211yescharacteristic
54#2552C2214yes—
55#2503—21no—
56#2502—21no—
57#2502—21no—
58#2499—21no—
59#2453C2214yes—
60#2447C2214yes—
61#2440—24no—
62D4×C4⋊C4C2214yes—
63#2317—21no—
64C4×C4⋊Q8C2211yescharacteristic
65#2293C2211yescharacteristic
66C2×#384—21no—
67C42×D4C2211yescharacteristic
68#2095C2214yes—
69#2095C2214yes—
70#2087C2214yes—
71#2084C2214yes—
72#2071C2214yes—
73#2068C2214yes—
74#2059C2214yes—
75#2059C2214yes—
76#2042C2214yes—
77#2026C2214yes—
78#1987C2211yescharacteristic
79#1987C2211yescharacteristic
80#1987C2211yescharacteristic
81#1984C2211yescharacteristic
82#1984C2211yescharacteristic
83#1981C2211yescharacteristic
84#1942C2214yes—
85#1933C2214yes—
86#1927C2214yes—
87#1925C2212yes—
88#1918C2212yes—
89#1916C2214yes—
90#1916C2214yes—
91#1879C2212yes—
92#1877C2214yes—
93#1877C2214yes—
94#1865C2214yes—
95#1862C2212yes—
96#1862C2214yes—
97#1862C2214yes—
98#1831C2211yescharacteristic
99C4×#107C2211yescharacteristic
100C4×#102C2211yescharacteristic
101C4×#102C2211yescharacteristic
102#1744C2211yescharacteristic
103C22×SD16C2312yes—
104C22×D8C2311yescharacteristic
105#556—24no—
106#556—42no—
107#554—24no—
108#554—24no—
109#554—24no—
110#554—42no—
111#554—42no—
112#554—42no—
113C2×C4⋊Q8—22no—
114C2×C4⋊Q8D411yescharacteristic
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×#126—22no—
120C2×#125C2312yes—
121C2×#125C2312yes—
122C2×#125C2312yes—
123C2×#124C2312yes—
124C2×C22⋊Q8—21no—
125C2×C22⋊Q8—21no—
126C2×C22⋊Q8—21no—
127C2×C22⋊Q8—21no—
128C2×C22⋊Q8—22no—
129C2×C22⋊Q8—22no—
130C2×C22⋊Q8—22no—
131C2×C4×Q8C2312yes—
132C2×C4×Q8C2311yescharacteristic
133C2×C4×Q8C2311yescharacteristic
134C2×C4×D4C2312yes—
135C2×C4×D4C2312yes—
136C2×C4×D4C2312yes—
137C2×C4×D4C2312yes—
138C2×C4×D4C2311yescharacteristic
139C2×C4×D4C2311yescharacteristic
140C2×#117—21no—
141C2×#117—21no—
142C22×C4⋊C4—21no—
143C22×C4⋊C4—21no—
144C22×C4⋊C4—21no—
145C22×C4⋊C4—24no—
146C22×C42—21no—
147#500—24no—
148#500—42no—
149#499—28no—
150#488—28no—
151#487—24no—
152#487—42no—
153#479—28no—
154#478—28no—
155#478—28no—
156#476—24no—
157#476—24no—
158#476—24no—
159#463—28no—
160#461—24no—
161C8×Q8—24no—
162C8×Q8—24no—
163C4×SD16—28no—
164C4×D8—28no—
165C4×D8D418yes—
166C2×#107C2312yes—
167C2×#107C2312yes—
168C2×#107C2312yes—
169C2×#107C2314yes—
170C2×#107C2311yescharacteristic
171C2×#106C2312yes—
172C2×#106C2312yes—
173C2×#106C2312yes—
174C2×C4⋊C8C2312yes—
175C2×C4⋊C8C2312yes—
176C2×C4⋊C8C2311yescharacteristic
177C2×C4⋊C8C2311yescharacteristic
178C2×#103C2312yes—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2314yes—
182C2×#103C2314yes—
183C2×#102C2312yes—
184C2×#102C2312yes—
185C2×#102C2312yes—
186C2×#102C2312yes—
187C2×#102C2312yes—
188C2×#102C2311yescharacteristic
189C2×#102C2311yescharacteristic
190C2×C4×C8C2311yescharacteristic
191C2×C4×C8C2312yes—
192C2×C4×C8C2312yes—
193C2×C4×C8C2311yescharacteristic
194#400—22no—
195#400—24no—
196#398—24no—
197#398—24no—
198#398—24no—
199#397—24no—
200#397—24no—
201#395—22no—
202#395—22no—
203#395—22no—
204#391—21no—
205#391—21no—
206#391—21no—
207#391—21no—
208#391—22no—
209#391D411yescharacteristic
210#391D411yescharacteristic
211#391C2314yes—
212#391C2314yes—
213#390C2312yes—
214#389—24no—
215#389—24no—
216#389D414yes—
217#389C2314yes—
218#389C2314yes—
219#389C2314yes—
220#389C2312yes—
221#388—22no—
222#387—24no—
223#385—24no—
224#385—24no—
225#385—24no—
226#384—21no—
227#384—21no—
228#384—21no—
229#384—24no—
230#384—21no—
231#384—21no—
232#384D411yescharacteristic
233#384C2314yes—
234#382—21no—
235#382—21no—
236#382—21no—
237#381—24no—
238#380—24no—
239C4×C4⋊C4C2311yescharacteristic
240C4×C4⋊C4C2311yescharacteristic
241C4×C4⋊C4C2312yes—
242C4×C4⋊C4C2314yes—
243C4×C22⋊C4—22no—
244C4×C22⋊C4—21no—
245C4×C22⋊C4—21no—
246C43C2311yescharacteristic
247#340C2312yes—
248#340C2312yes—
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2312yes—
253#336C2314yes—
254#336C2314yes—
255#336C2314yes—
256#336C2312yes—
257#336C2312yes—
258C22×Q8—21no—
259C22×Q8—21no—
260C22×Q8—21no—
261C22×Q8C2×D411yescharacteristic
262C22×Q8C2×D411yescharacteristic
263C22×Q8C2412yes—
264C22×D4C2411yescharacteristic
265C22×D4C2411yescharacteristic
266C23×C4—21no—
267C23×C4—22no—
268C23×C4—21no—
269C23×C4—22no—
270C23×C4—22no—
271C23×C4—22no—
272C2×SD16—24no—
273C2×SD16—28no—
274C2×D8—24no—
275C2×D8C2×D414yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22×C8C2411yescharacteristic
280C22×C8C2411yescharacteristic
281C4⋊Q8—24no—
282C4⋊Q8—22no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—84no—
291#126—48no—
292#125—24no—
293#125—24no—
294#125—24no—
295#125—42no—
296#125—42no—
297#125—42no—
298#124—24no—
299#124—42no—
300C22⋊Q8—48no—
301C22⋊Q8—48no—
302C22⋊Q8—48no—
303C22⋊Q8—44no—
304C22⋊Q8—44no—
305C22⋊Q8—44no—
306C22⋊Q8—44no—
307C4×Q8—22no—
308C4×Q8—22no—
309C4×Q8—22no—
310C4×Q8—22no—
311C4×Q8—24no—
312C4×Q8—42no—
313C4×D4—24no—
314C4×D4—24no—
315C4×D4—24no—
316C4×D4—24no—
317C4×D4—24no—
318C4×D4—22no—
319C4×D4—22no—
320C4×D4—42no—
321C4×D4—42no—
322C4×D4—42no—
323C4×D4C2×D414yes—
324C4×D4C2×D414yes—
325#117—44no—
326#117—44no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—21no—
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—24no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—44no—
346C2×C4⋊C4—44no—
347C2×C4⋊C4—44no—
348C2×C4⋊C4—88no—
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D412yes—
352C2×C4⋊C4C2×D412yes—
353C2×C4⋊C4C2×D414yes—
354C2×C4⋊C4C2×D414yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2411yescharacteristic
360C2×C4⋊C4C2411yescharacteristic
361C2×C4⋊C4C2412yes—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—22no—
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×C42—22no—
373C2×C42—21no—
374C2×C42—22no—
375C2×C42—21no—
376C2×C42—44no—
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D412yes—
379C2×C42C2×D411yescharacteristic
380C2×C42C2412yes—
381C2×C42C2412yes—
382C2×C42C2412yes—
383C2×C42C2411yescharacteristic
384#107—24no—
385#107—24no—
386#107—24no—
387#107—44no—
388#107C2×D414yes—
389#106—24no—
390#106—24no—
391#106—24no—
392C4⋊C8—24no—
393C4⋊C8—24no—
394C4⋊C8—22no—
395C4⋊C8—22no—
396#103—24no—
397#103—24no—
398#103—24no—
399#103—28no—
400#103—44no—
401#102—24no—
402#102—24no—
403#102—24no—
404#102—24no—
405#102—24no—
406#102C2×D414yes—
407#102C2×D414yes—
408C4×C8—24no—
409C4×C8—24no—
410C4×C8—22no—
411C4×C8C2×D414yes—
412#95—21no—
413#95—22no—
414#95—22no—
415#95—21no—
416#95—24no—
417#95—24no—
418#95—24no—
419#95—24no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95C2×D414yes—
424#95C2×D414yes—
425C24—21no—
426C24—21no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—24no—
436C2×Q8—24no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—82no—
441C2×Q8—82no—
442C2×Q8—82no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4C22×D412yes—
449C2×D4C22×D412yes—
450C2×D42−1+414yes—
451C22×C4—28no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—84no—
472C22×C4—84no—
473C22×C4—84no—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C2511yescharacteristic
483SD16—48no—
484D8—28no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8C22×D412yes—
495C2×C8C22×D412yes—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—28no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—48no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—44no—
527C4⋊C4—84no—
528C4⋊C4—84no—
529C4⋊C4—84no—
530C4⋊C4—84no—
531C4⋊C4C22×D412yes—
532C4⋊C4C22×D412yes—
533C4⋊C42−1+414yes—
534C22⋊C4—28no—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—82no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—42no—
552C42—44no—
553C42—44no—
554C42—42no—
555C42—42no—
556C42#13514yes—
557C42C22×D412yes—
558C42H(5,𝔽2)12yes—
559C23—22no—
560C23—24no—
561C23—22no—
562C23—42no—
563C23—41no—
564C23—41no—
565C23—42no—
566C23—42no—
567C23—41no—
568C23—41no—
569C23C23×D411yescharacteristic
570Q8—44no—
571Q8—44no—
572Q8—42no—
573Q8—42no—
574Q8—44no—
575Q8—162no—
576Q8—162no—
577Q8—162no—
578D4—22no—
579D4—22no—
580D4—22no—
581D4—24no—
582D4—42no—
583D4#53614yes—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—416no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—48no—
611C2×C4—48no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—164no—
624C2×C4—164no—
625C2×C4—164no—
626C2×C4D4212yes—
627C2×C4D4212yes—
628C2×C4C2×#13514yes—
629C2×C4C2×#13512yes—
630C2×C4#57812yes—
631C2×C4C23×D411yescharacteristic
632C2×C4C2×H(5,𝔽2)11yescharacteristic
633C2×C4C2×2−1+412yes—
634C2×C4C2×2−1+412yes—
635C8—22no—
636C8—22no—
637C8—42no—
638C8—42no—
639C8—42no—
640C22—24no—
641C22—42no—
642C22—44no—
643C22—42no—
644C22—44no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—82no—
649C22—82no—
650C22C2×D4211yescharacteristic
651C22#348612yes—
652C22C22×#13511yescharacteristic
653C22C2×#57811yescharacteristic
654C22#358512yes—
655C4—24no—
656C4—22no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—44no—
662C4—42no—
663C4—42no—
664C4—42no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4#328314yes—
671C4C2×#53611yescharacteristic
672C4#349911yescharacteristic
673C4#355212yes—
674C2—44no—
675C2—82no—
676C2#3767511yescharacteristic
677C2C2×#328311yescharacteristic
678C2#6108211yescharacteristic
679C2#6181912yes—
680C2#6185912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D8C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #39490C2semidirect
4. #36180C2semidirect
5. #33026C2semidirect
6. #30952C2semidirect
7. #30811C2semidirect
8. #30279C2semidirect
9. #29868C2semidirect
10. #28676C2semidirect
11. #28638C2semidirect
12. #26374C2semidirect
14. #25934C2semidirect
15. #25839C2semidirect
16. #25839C2semidirect
17. #25646C2semidirect
18. #22238C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33802
C2×#33602
#394902
#361802
#330262
#309522
#308112
#302792
#298682
#286762
#286382
#263742
#261642
#259342
#258392
#256462
#222382
#211132