Finite Group Databasebeta

← #6466070

Subgroups — #6466070

Order 512.

2016 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3080C211yesmaximal, characteristic
2C22×#479C211yesmaximal, characteristic, complemented
3#34217C211yesmaximal, characteristic, complemented
4C4×#536C211yesmaximal, characteristic
5#28294C214yesmaximal, complemented
6#28293C214yesmaximal, complemented
7#28287C214yesmaximal, complemented
8#28285C214yesmaximal, complemented
9#23615C211yesmaximal, characteristic, complemented
10#23615C211yesmaximal, characteristic, complemented
11#23611C211yesmaximal, characteristic
12#23610C211yesmaximal, characteristic
13#21262C211yesmaximal, characteristic
14#21262C211yesmaximal, characteristic
15#21174C211yesmaximal, characteristic, complemented
16#21173C211yesmaximal, characteristic, complemented
17#20793C211yesmaximal, characteristic, complemented
18#20647C211yesmaximal, characteristic, complemented
19#20645C211yesmaximal, characteristic
20C2×#536C2211yescharacteristic
21C22×#125C2211yescharacteristic
22C2×C4×#41C2211yescharacteristic
23C22×C4×Q8C2211yescharacteristic
24#3080—28no—
25C2×#479—28no—
26C2×#479C2214yes—
27C2×#479C2214yes—
28C2×#479C2212yes—
29C2×#479C2212yes—
30C2×#478C2212yes—
31C2×#478C2212yes—
32C2×#428C2211yescharacteristic
33C2×#428C2211yescharacteristic
34C22×#107C2211yescharacteristic
35C22×#106C2211yescharacteristic
36C2×#424C2211yescharacteristic
37C2×#424C2211yescharacteristic
38C2×#424C2211yescharacteristic
39C2×#424C2211yescharacteristic
40C2×#423C2211yescharacteristic
41C22×C4⋊C8C2211yescharacteristic
42C2×#417C2211yescharacteristic
43C2×#417C2211yescharacteristic
44C2×#419C2211yescharacteristic
45C2×#419C2211yescharacteristic
46C2×#416C2211yescharacteristic
47C2×#416C2211yescharacteristic
48C22×#103C2211yescharacteristic
49C22×#103C2211yescharacteristic
50#2370—22no—
51#2370—22no—
52#2345—22no—
53#2345—22no—
54#2330C2214yes—
55#2327C2212yes—
56#2326C2212yes—
57#2323—22no—
58#2322—22no—
59#2317C2211yescharacteristic
60#2316C2211yescharacteristic
61#2316C2211yescharacteristic
62#2311C2211yescharacteristic
63#2310C2211yescharacteristic
64C4×C4⋊Q8C2212yes—
65C4×#125C2214yes—
66C4×#124C2212yes—
67C4×C22⋊Q8—22no—
68C4×#123—22no—
69#2285C2211yescharacteristic
70#2274C2211yescharacteristic
71#2274C2211yescharacteristic
72C4×#117C2211yescharacteristic
73#2068C2214yes—
74#2068C2214yes—
75#2068C2214yes—
76#2067C2214yes—
77#2066C2214yes—
78#2066C2214yes—
79#2066C2214yes—
80#2065C2214yes—
81#1983C2214yes—
82#1983C2214yes—
83#1981C2214yes—
84#1981C2214yes—
85#1949C2214yes—
86#1949C2214yes—
87#1948C2214yes—
88#1947C2214yes—
89#1843C2214yes—
90#1842C2214yes—
91#1805C2214yes—
92#1805C2212yes—
93#1804C2212yes—
94#1803C2214yes—
95#1803C2212yes—
96#1802C2212yes—
97#1736C2211yescharacteristic
98#1736C2211yescharacteristic
99#1735C2211yescharacteristic
100#1735C2211yescharacteristic
101C22×#41C2311yescharacteristic
102C23×Q8C2311yescharacteristic
103C22×M16C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C23×C8C2311yescharacteristic
106C23×C8C2311yescharacteristic
107#536—24no—
108#536Q818yes—
109C2×C4⋊Q8C2312yes—
110C2×#125—21no—
111C2×#125—21no—
112C2×#125—24no—
113C2×#125D418yes—
114C2×#125C2312yes—
115C2×#125C2312yes—
116C2×#124C2312yes—
117C2×#123—22no—
118C2×C22⋊Q8—22no—
119C4×#41—24no—
120C4×#41—24no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8—24no—
124C2×C4×Q8—24no—
125C2×C4×Q8C2311yescharacteristic
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2312yes—
128C2×C4×Q8C2311yescharacteristic
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4C2311yescharacteristic
132C2×C4×D4C2311yescharacteristic
133C2×#117—21no—
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C22×C4⋊C4—21no—
140C22×C4⋊C4—21no—
141C22×C4⋊C4—21no—
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2311yescharacteristic
144C22×C4⋊C4C2311yescharacteristic
145C22×C42—21no—
146C22×C42—21no—
147C22×C42C2311yescharacteristic
148#479—216no—
149#479—216no—
150#479—28no—
151#479—28no—
152#478—28no—
153#478—28no—
154#428—24no—
155#428—24no—
156C2×#107—24no—
157C2×#107C2312yes—
158C2×#107C2312yes—
159C2×#107C2314yes—
160C2×#106—24no—
161C2×#106C2312yes—
162C2×#106C2312yes—
163C2×#106C2314yes—
164#424—24no—
165#424—24no—
166#424—42no—
167#424—42no—
168#423—24no—
169C2×C4⋊C8—24no—
170C2×C4⋊C8C2312yes—
171C2×C4⋊C8C2312yes—
172C2×C4⋊C8C2312yes—
173C2×C4⋊C8C2312yes—
174C2×C4⋊C8C2314yes—
175C2×C4⋊C8C2314yes—
176C2×C4⋊C8C2314yes—
177C2×C4⋊C8C2314yes—
178#419—42no—
179#419—42no—
180#417—24no—
181#417—24no—
182#416—42no—
183#416—42no—
184C2×#103—24no—
185C2×#103—24no—
186C2×#103C2312yes—
187C2×#103C2314yes—
188C2×#103C2314yes—
189C2×#103C2312yes—
190C2×#103C2312yes—
191C2×#103C2314yes—
192C2×#103C2314yes—
193C2×#103C2312yes—
194C2×#102C2312yes—
195C2×#102C2312yes—
196C2×#102C2312yes—
197C2×#102C2312yes—
198#391—22no—
199#391C2312yes—
200#391C2312yes—
201#391C2312yes—
202#390C2312yes—
203#389—22no—
204#389—22no—
205#389—22no—
206#389—24no—
207#389—24no—
208#389—24no—
209#389C2314yes—
210#389C2314yes—
211#388—24no—
212#388—24no—
213#387—24no—
214#387—24no—
215#387—24no—
216#387—24no—
217#387—24no—
218#387—24no—
219#386—22no—
220#385—22no—
221#385—24no—
222#384C2312yes—
223#384C2312yes—
224#382C2312yes—
225#382C2312yes—
226#381—22no—
227#380—21no—
228#380—21no—
229#380—22no—
230#380—22no—
231#380—22no—
232C4×C4⋊C4—22no—
233C4×C4⋊C4—22no—
234C4×C4⋊C4—24no—
235C4×C4⋊C4—24no—
236C4×C4⋊C4—24no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4D412yes—
240C4×C4⋊C4D412yes—
241C4×C4⋊C4C2314yes—
242C4×C4⋊C4C2314yes—
243C4×C22⋊C4—22no—
244C4×C22⋊C4—22no—
245C4×C22⋊C4—24no—
246C4×C22⋊C4D412yes—
247C4×C22⋊C4D412yes—
248#376C2312yes—
249C43C2312yes—
250#340C2312yes—
251#340C2312yes—
252#340C2312yes—
253#340C2312yes—
254#340C2314yes—
255#340C2314yes—
256C2×#41—22no—
257C2×#41—22no—
258C2×#41—22no—
259C2×#41—22no—
260C2×#41—22no—
261C2×#41C2×Q814yes—
262C22×Q8—21no—
263C22×Q8—22no—
264C22×Q8—22no—
265C22×Q8—22no—
266C22×Q8—22no—
267C22×Q8—22no—
268C22×Q8C2×Q814yes—
269C22×Q8C2412yes—
270C22×Q8C2411yescharacteristic
271C22×D4—21no—
272C22×D4C2411yescharacteristic
273C23×C4—21no—
274C23×C4—21no—
275C23×C4—21no—
276C23×C4—21no—
277C23×C4—21no—
278C23×C4—21no—
279C23×C4—21no—
280C23×C4C2×D411yescharacteristic
281C23×C4C2×D411yescharacteristic
282C23×C4C2411yescharacteristic
283C2×M16—22no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—22no—
288C2×M16—22no—
289C22×C8—22no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8—22no—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C22×C8C2412yes—
299C4⋊Q8—24no—
300C4⋊Q8C2×Q818yes—
301#125—28no—
302#125—28no—
303#125—24no—
304#125—24no—
305#125C2×D418yes—
306#125C2×D418yes—
307#124—24no—
308#124C2×Q818yes—
309#123—216no—
310C22⋊Q8—216no—
311C4×Q8—24no—
312C4×Q8—24no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—24no—
316C4×Q8—24no—
317C4×Q8—22no—
318C4×Q8—22no—
319C4×Q8—44no—
320C4×Q8—44no—
321C4×D4—22no—
322C4×D4—22no—
323C4×D4—22no—
324C4×D4—22no—
325C4×D4—44no—
326C4×D4—44no—
327#117—22no—
328#117—22no—
329#117—22no—
330#117—22no—
331#117—22no—
332#117—22no—
333#117—22no—
334#117—22no—
335#117—44no—
336#117C2×Q814yes—
337#117#4114yes—
338C2×C4⋊C4—28no—
339C2×C4⋊C4—28no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—24no—
359C2×C4⋊C4—24no—
360C2×C4⋊C4—24no—
361C2×C4⋊C4—44no—
362C2×C4⋊C4C2×D412yes—
363C2×C4⋊C4C2×D414yes—
364C2×C4⋊C4C2×D414yes—
365C2×C4⋊C4C2×D412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2411yescharacteristic
368C2×C4⋊C4C2411yescharacteristic
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2412yes—
372C2×C4⋊C4C2412yes—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—24no—
380C2×C22⋊C4C2×D412yes—
381C2×C22⋊C4C2×D412yes—
382C2×C42—21no—
383C2×C42—21no—
384C2×C42—21no—
385C2×C42—21no—
386C2×C42—21no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—24no—
390C2×C42—24no—
391C2×C42—44no—
392C2×C42—44no—
393C2×C42C2×D414yes—
394C2×C42C2×D414yes—
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D414yes—
398C2×C42#4114yes—
399C2×C42C2412yes—
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402C2×C42C2412yes—
403#107—28no—
404#107—24no—
405#107—24no—
406#106—28no—
407#106—24no—
408#106—24no—
409C4⋊C8—24no—
410C4⋊C8—24no—
411C4⋊C8—24no—
412C4⋊C8—24no—
413C4⋊C8—28no—
414C4⋊C8—28no—
415C4⋊C8—44no—
416C4⋊C8—44no—
417#103—28no—
418#103—28no—
419#103—24no—
420#103—24no—
421#103—42no—
422#103—42no—
423#103—44no—
424#103—44no—
425#102—24no—
426#102—24no—
427#102—42no—
428#102—42no—
429#95—24no—
430#95—24no—
431#95—24no—
432#95—24no—
433#95—22no—
434#95—22no—
435#95—24no—
436#95—24no—
437#95—24no—
438#95—24no—
439#95C2×D414yes—
440#95C2×D414yes—
441C24—21no—
442C24C22×D411yescharacteristic
443#41—28no—
444#41—28no—
445#41—44no—
446#41—44no—
447C2×Q8—24no—
448C2×Q8—22no—
449C2×Q8—21no—
450C2×Q8—28no—
451C2×Q8—28no—
452C2×Q8—22no—
453C2×Q8—21no—
454C2×Q8—22no—
455C2×Q8—21no—
456C2×Q8—22no—
457C2×Q8—21no—
458C2×Q8—22no—
459C2×Q8—21no—
460C2×Q8—42no—
461C2×Q8—42no—
462C2×Q8—44no—
463C2×Q8—44no—
464C2×Q8C22×Q814yes—
465C2×Q8C22×Q812yes—
466C2×D4—24no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4C22×Q812yes—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—21no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—21no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—44no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—82no—
499C22×C4—82no—
500C22×C4—82no—
501C22×C4—82no—
502C22×C4#12114yes—
503C22×C4#12114yes—
504C22×C4C22×D414yes—
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D414yes—
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D412yes—
512C22×C4C22×Q812yes—
513C22×C4C2×#4112yes—
514C22×C4C2×#4112yes—
515C22×C4H(5,𝔽2)12yes—
516C22×C42−1+412yes—
517C22×C4C2511yescharacteristic
518M16—44no—
519M16—44no—
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×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—44no—
533C2×C8—44no—
534C4⋊C4—28no—
535C4⋊C4—28no—
536C4⋊C4—28no—
537C4⋊C4—28no—
538C4⋊C4—24no—
539C4⋊C4—24no—
540C4⋊C4—216no—
541C4⋊C4—24no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—22no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—42no—
555C4⋊C4—44no—
556C4⋊C4—44no—
557C4⋊C4—42no—
558C4⋊C4—42no—
559C4⋊C4—84no—
560C4⋊C4—82no—
561C4⋊C4—82no—
562C4⋊C4—84no—
563C4⋊C4—84no—
564C4⋊C4C22⋊Q818yes—
565C4⋊C4C22×D414yes—
566C4⋊C4C22×D414yes—
567C22⋊C4—24no—
568C22⋊C4—216no—
569C22⋊C4—42no—
570C22⋊C4—42no—
571C22⋊C4—42no—
572C22⋊C4—42no—
573C22⋊C4—42no—
574C22⋊C4—84no—
575C22⋊C4C22⋊Q818yes—
576C42—22no—
577C42—22no—
578C42—44no—
579C42—44no—
580C42—42no—
581C42—42no—
582C42—42no—
583C42—44no—
584C42—44no—
585C42—42no—
586C42—42no—
587C42—84no—
588C42—84no—
589C42#12114yes—
590C42#12114yes—
591C42C22×D412yes—
592C42C22×D412yes—
593C42C22×Q814yes—
594C42C2×#4112yes—
595C42C2×#4112yes—
596C42C2×#4114yes—
597C23—22no—
598C23—22no—
599C23—22no—
600C23—42no—
601C23—42no—
602C23—41no—
603C23—41no—
604C23C2×#12112yes—
605C23C2×C22⋊Q812yes—
606C23#53712yes—
607C23C23×D411yescharacteristic
608Q8—24no—
609Q8—22no—
610Q8—24no—
611Q8—22no—
612Q8—42no—
613Q8—41no—
614Q8—42no—
615Q8—41no—
616Q8—48no—
617D4—22no—
618D4—22no—
619D4—48no—
620D4—41no—
621D4—41no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—24no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—22no—
634C2×C4—24no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—48no—
644C2×C4—48no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—84no—
658C2×C4—84no—
659C2×C4—84no—
660C2×C4—84no—
661C2×C4—84no—
662C2×C4C2×#12112yes—
663C2×C4C2×#12112yes—
664C2×C4C2×#12112yes—
665C2×C4C2×#12112yes—
666C2×C4C2×#12112yes—
667C2×C4C2×C22⋊Q818yes—
668C2×C4C2×C22⋊Q812yes—
669C2×C4#53712yes—
670C2×C4D4×Q814yes—
671C2×C4#55014yes—
672C2×C4#55414yes—
673C2×C4#55614yes—
674C2×C4C23×D411yescharacteristic
675C2×C4C23×Q811yescharacteristic
676C2×C4C22×#4111yescharacteristic
677C2×C4C22×#4111yescharacteristic
678C2×C4C2×H(5,𝔽2)11yescharacteristic
679C2×C4C2×2−1+411yescharacteristic
680C8—42no—
681C8—42no—
682C8—42no—
683C8—42no—
684C22—24no—
685C22—22no—
686C22—22no—
687C22—24no—
688C22—44no—
689C22—42no—
690C22—82no—
691C22—81no—
692C22—81no—
693C22#253814yes—
694C22C22×#12111yescharacteristic
695C22C22×C22⋊Q811yescharacteristic
696C22C2×#53711yescharacteristic
697C22#358412yes—
698C22#358612yes—
699C4—22no—
700C4—22no—
701C4—22no—
702C4—22no—
703C4—28no—
704C4—44no—
705C4—44no—
706C4—44no—
707C4—42no—
708C4—44no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4—81no—
713C4—81no—
714C4#253814yes—
715C4C2×D4×Q811yescharacteristic
716C4C2×#55011yescharacteristic
717C4C2×#55411yescharacteristic
718C4C2×#55611yescharacteristic
719C2—24no—
720C2—82no—
721C2C2×#253811yescharacteristic
722C2#6042211yescharacteristic
723C2#6042711yescharacteristic
724C2#6049912yes—
725C2#6093212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#479C2semidirect
3. #34217C2semidirect
5. #28294C2semidirect
6. #28293C2semidirect
7. #28287C2semidirect
8. #28285C2semidirect
9. #23615C2semidirect
10. #23615C2semidirect
15. #21174C2semidirect
16. #21173C2semidirect
17. #20793C2semidirect
18. #20647C2semidirect

Maximal subgroups

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

subgroupindex
C2×#30802
C22×#4792
#342172
C4×#5362
#282942
#282932
#282872
#282852
#236152
#236112
#236102
#212622
#211742
#211732
#207932
#206472
#206452