Finite Group Databasebeta

← #6381758

Subgroups — #6381758

Order 512.

2180 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#478C211yesmaximal, characteristic
2#34217C211yesmaximal, characteristic, complemented
3#34181C211yesmaximal, characteristic
4#22978C214yesmaximal, complemented
5#22974C214yesmaximal, complemented
6#22972C214yesmaximal, complemented
7#22966C214yesmaximal, complemented
8#21325C211yesmaximal, characteristic, complemented
9#21169C211yesmaximal, characteristic, complemented
10#21158C211yesmaximal, characteristic, complemented
11#21157C211yesmaximal, characteristic, complemented
12#21038C211yesmaximal, characteristic, complemented
13#21037C211yesmaximal, characteristic, complemented
14#20879C211yesmaximal, characteristic
15#20879C211yesmaximal, characteristic
16#20441C211yesmaximal, characteristic, complemented
17#20416C211yesmaximal, characteristic
18#20416C211yesmaximal, characteristic
19C4×#418C211yesmaximal, characteristic
20C22×#125C2211yescharacteristic
21C2×#518C2211yescharacteristic
22C2×#518C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24C2×#478C4116yescomplemented
25C2×#478C2212yes—
26C2×#478C2212yes—
27C2×#478C2212yes—
28C2×#478C2212yes—
29C2×#427C2211yescharacteristic
30C2×#427C2211yescharacteristic
31C22×#107C2211yescharacteristic
32C22×#106C2211yescharacteristic
33C22×C4⋊C8C2211yescharacteristic
34C2×#418C2211yescharacteristic
35C2×#418C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C2×#405C2211yescharacteristic
39C2×#405C2211yescharacteristic
40C2×C4×M16C2211yescharacteristic
41#2370C2214yescomplemented
42#2370C2214yescomplemented
43#2349—22no—
44#2349C2214yes—
45D4×C4⋊C4—22no—
46D4×C4⋊C4C2214yes—
47#2345—22no—
48#2345—22no—
49#2330—22no—
50#2316C2211yescharacteristic
51#2316C2211yescharacteristic
52#2316C2211yescharacteristic
53#2316C2211yescharacteristic
54#2315—22no—
55C4×#125—22no—
56#2285C2211yescharacteristic
57#2285C2211yescharacteristic
58C2×#380—22no—
59#2274C2211yescharacteristic
60#2274C2211yescharacteristic
61C4×#117C2211yescharacteristic
62#1992C2214yescomplemented
63#1991C2214yescomplemented
64#1949C2212yescomplemented
65#1949C2212yescomplemented
66#1948C2212yescomplemented
67#1947C2212yescomplemented
68#1944C2214yescomplemented
69#1943C2214yescomplemented
70#1942C2214yescomplemented
71#1941C2214yescomplemented
72#1921C2214yescomplemented
73#1920C2214yescomplemented
74#1919C2214yescomplemented
75#1918C2214yescomplemented
76#1864C2214yes—
77#1864C2214yes—
78#1862C2214yes—
79#1862C2214yes—
80#1771C2212yescomplemented
81#1771C2212yescomplemented
82#1770C2212yescomplemented
83C8×C4⋊C4C2212yescomplemented
84#1762C2214yes—
85#1762C2214yes—
86#1762C2214yes—
87C4×#102C2214yes—
88C4×#102C2214yes—
89C4×#102C2214yes—
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1736C2211yescharacteristic
96#1735C2211yescharacteristic
97#1735C2211yescharacteristic
98#1729C2211yescharacteristic
99#1729C2211yescharacteristic
100C23×D4C2311yescharacteristic
101C24×C4—21no—
102C22×M16C2311yescharacteristic
103C22×M16C2311yescharacteristic
104C23×C8C2311yescharacteristic
105C23×C8C2311yescharacteristic
106C2×#125—21no—
107C2×#125—21no—
108C2×#125—24no—
109C2×#125C2×C418yescomplemented
110C2×#125C2312yes—
111C2×#125C2312yes—
112#518—24no—
113#518—24no—
114#518—42no—
115#518—42no—
116C2×C4×D4—22no—
117C2×C4×D4—22no—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—24no—
121C2×C4×D4C2×C418yes—
122C2×C4×D4C2314yes—
123C2×C4×D4C2314yes—
124C2×C4×D4C2312yes—
125C2×C4×D4C2312yes—
126C2×#117C2311yescharacteristic
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C22×C4⋊C4—21no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4—24no—
135C22×C4⋊C4C2311yescharacteristic
136C22×C4⋊C4C2311yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C22⋊C4—21no—
139C22×C22⋊C4—22no—
140C22×C22⋊C4—22no—
141C22×C42C2311yescharacteristic
142#478C2×C4116yescomplemented
143#478C2×C4116yescomplemented
144#478C2×C4116yescomplemented
145#478C2×C4116yescomplemented
146#427—42no—
147#427—42no—
148C2×#107C2×C418yescomplemented
149C2×#107C2312yes—
150C2×#107C2312yes—
151C2×#107C2312yes—
152C2×#107C2312yes—
153C2×#106C2×C418yescomplemented
154C2×#106C2312yes—
155C2×#106C2312yes—
156C2×#106C2312yes—
157C2×#106C2312yes—
158C2×C4⋊C8C2×C418yescomplemented
159C2×C4⋊C8C2311yescharacteristic
160C2×C4⋊C8C2311yescharacteristic
161C2×C4⋊C8C2311yescharacteristic
162C2×C4⋊C8C2311yescharacteristic
163#418—24no—
164#418Q818yes—
165C2×#102C2×C418yes—
166C2×#102C2×C418yes—
167C2×#102C2314yes—
168C2×#102C2312yes—
169C2×#102C2312yes—
170C2×#102C2312yes—
171C2×#102C2312yes—
172C2×#102C2314yes—
173#405—42no—
174#405—42no—
175C4×M16—24no—
176C2×#97C2312yes—
177C2×#97C2312yes—
178C2×#97C2312yes—
179C2×C4×C8C2312yes—
180C2×C4×C8C2312yes—
181C2×C4×C8C2312yes—
182#389—22no—
183#389—22no—
184#389—22no—
185#389—22no—
186#389—24no—
187#389—24no—
188#389C2314yes—
189#389C2314yes—
190#389C2314yes—
191#389C2314yes—
192#387—24no—
193#387—24no—
194#387—24no—
195#387—24no—
196#385—24no—
197#385—24no—
198#385—24no—
199#385—24no—
200#384D412yes—
201#384C2312yes—
202#382D412yes—
203#382C2312yes—
204#380—21no—
205#380—21no—
206#380—24no—
207#380—22no—
208#380—22no—
209#380—22no—
210#380—22no—
211#380—24no—
212#380D412yes—
213#380D412yes—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—24no—
219C4×C4⋊C4—24no—
220C4×C4⋊C4C2314yes—
221C4×C4⋊C4C2314yes—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2312yes—
226C4×C22⋊C4—21no—
227C4×C22⋊C4—21no—
228C4×C22⋊C4—22no—
229C4×C22⋊C4—22no—
230#376—21no—
231C2×#95—24no—
232C43—21no—
233#340C2314yes—
234#340C2314yes—
235#340C2312yes—
236#340C2314yes—
237#340C2312yes—
238#340C2312yes—
239#340C2312yes—
240#340C2314yes—
241#340C2314yes—
242#340C2314yes—
243#336C2312yes—
244#336C2312yes—
245#336C2312yes—
246#336C2312yes—
247C25—21no—
248C22×D4—21no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—22no—
252C22×D4C22×C414yes—
253C22×D4C2×Q814yes—
254C22×D4#4114yes—
255C22×D4C2412yes—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—22no—
260C23×C4—28no—
261C23×C4—22no—
262C23×C4—24no—
263C23×C4—24no—
264C23×C4—84no—
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2411yescharacteristic
268C2×M16—22no—
269C2×M16—22no—
270C2×M16—22no—
271C2×M16—22no—
272C2×M16—22no—
273C2×M16C2×Q814yes—
274C22×C8—22no—
275C22×C8—22no—
276C22×C8—22no—
277C22×C8—22no—
278C22×C8C22×C414yes—
279C22×C8C22×C414yes—
280C22×C8C2412yes—
281C22×C8C2411yescharacteristic
282C22×C8C2411yescharacteristic
283C22×C8C2411yescharacteristic
284C22×C8C2411yescharacteristic
285C22×C8C2412yes—
286#125—28no—
287#125—28no—
288#125—24no—
289#125—24no—
290#125C22×C418yes—
291#125C22×C418yes—
292C4×D4—28no—
293C4×D4—28no—
294C4×D4—28no—
295C4×D4—28no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—44no—
299C4×D4—44no—
300C4×D4—48no—
301C4×D4—48no—
302C4×D4C22×C418yes—
303C4×D4C22×C418yes—
304#117—22no—
305#117—22no—
306#117—22no—
307#117—22no—
308#117—22no—
309#117—22no—
310#117—22no—
311#117—22no—
312#117—22no—
313#117—22no—
314#117—22no—
315#117C2×Q814yes—
316C2×C4⋊C4—28no—
317C2×C4⋊C4—28no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—88no—
335C2×C4⋊C4C22×C414yes—
336C2×C4⋊C4C22×C414yes—
337C2×C4⋊C4C22×C414yes—
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4C2×D412yes—
340C2×C4⋊C4C2×D414yes—
341C2×C4⋊C4C2×D414yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—28no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—48no—
359C2×C22⋊C4—48no—
360C2×C22⋊C4C2×D412yes—
361C2×C22⋊C4C2×D412yes—
362C2×C42—21no—
363C2×C42—21no—
364C2×C42—21no—
365C2×C42—21no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—24no—
369C2×C42—24no—
370C2×C42C22×C414yes—
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42#4114yes—
374C2×C42C2412yes—
375C2×C42C2412yes—
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378#107—42no—
379#107—42no—
380#107C22×C418yes—
381#107C22×C418yes—
382#106—42no—
383#106—42no—
384#106C22×C418yes—
385#106C22×C418yes—
386C4⋊C8C22×C414yes—
387C4⋊C8C22×C414yes—
388C4⋊C8C22×C414yes—
389C4⋊C8C22×C414yes—
390#102—28no—
391#102C22×C418yes—
392#102C22×C418yes—
393#102C22×C418yes—
394#102C22×C418yes—
395#102C2×Q8116yes—
396#97—24no—
397#97—42no—
398#97—42no—
399C4×C8—24no—
400C4×C8—42no—
401C4×C8—42no—
402#95—22no—
403#95—24no—
404#95—24no—
405#95—22no—
406#95—22no—
407#95—24no—
408#95—24no—
409#95—22no—
410#95—24no—
411#95—24no—
412#95C2×D414yes—
413#95C2×D414yes—
414C24—22no—
415C24—24no—
416C24—24no—
417C24—24no—
418C24—82no—
419C24—82no—
420C24C22×D411yescharacteristic
421C2×D4—24no—
422C2×D4—28no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—28no—
426C2×D4—28no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4C4×Q8116yes—
431C2×D4C23×C414yes—
432C2×D4C22×Q814yes—
433C2×D4C2×#4114yes—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—28no—
442C22×C4—28no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—84no—
452C22×C4—84no—
453C22×C4—84no—
454C22×C4—84no—
455C22×C4—84no—
456C22×C4—816no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4C4×D414yes—
461C22×C4C4×D414yes—
462C22×C4C23×C412yes—
463C22×C4C22×D414yes—
464C22×C4C22×D414yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×Q812yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4H(5,𝔽2)12yes—
474C22×C42−1+412yes—
475C22×C4C2511yescharacteristic
476M16—28no—
477M16—44no—
478C2×C8—28no—
479C2×C8—28no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8C23×C412yes—
494C2×C8C23×C412yes—
495C2×C8C23×C412yes—
496C2×C8C23×C412yes—
497C2×C8C22×Q814yes—
498C4⋊C4—28no—
499C4⋊C4—28no—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—22no—
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—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—82no—
521C4⋊C4—84no—
522C4⋊C4—84no—
523C4⋊C4—84no—
524C4⋊C4—84no—
525C4⋊C4C23×C414yes—
526C4⋊C4C23×C414yes—
527C4⋊C4C23×C414yes—
528C4⋊C4C22×Q814yes—
529C22⋊C4—24no—
530C22⋊C4—24no—
531C22⋊C4—24no—
532C22⋊C4—28no—
533C22⋊C4—28no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—42no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C42—24no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—42no—
545C42—42no—
546C42—42no—
547C42—82no—
548C42—82no—
549C42—84no—
550C42—84no—
551C42C4×D414yes—
552C42C4×D414yes—
553C42C23×C412yes—
554C42C23×C412yes—
555C42C2×#4112yes—
556C42C2×#4112yes—
557C23—22no—
558C23—22no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—216no—
563C23—44no—
564C23—44no—
565C23—84no—
566C23—84no—
567C23—88no—
568C23—88no—
569C23C2×C4×D412yes—
570C23#54612yes—
571C23D4×Q812yes—
572C23C23×D411yescharacteristic
573D4—216no—
574D4—24no—
575D4—24no—
576D4—24no—
577D4C2×C4×Q818yes—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—416no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—88no—
610C2×C4—84no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4—168no—
614C2×C4—168no—
615C2×C4C2×C4×D412yes—
616C2×C4C2×C4×D412yes—
617C2×C4C2×C4×D412yes—
618C2×C4C2×C4×D412yes—
619C2×C4C2×C4×D412yes—
620C2×C4C2×C4×Q814yes—
621C2×C4#52014yes—
622C2×C4#54612yes—
623C2×C4D4×Q812yes—
624C2×C4#55214yes—
625C2×C4#55414yes—
626C2×C4C24×C411yescharacteristic
627C2×C4C23×Q811yescharacteristic
628C2×C4C22×#4111yescharacteristic
629C2×C4C22×#4111yescharacteristic
630C2×C4C2×H(5,𝔽2)11yescharacteristic
631C2×C4C2×2−1+411yescharacteristic
632C8—22no—
633C8—22no—
634C8—22no—
635C8—22no—
636C8—24no—
637C8—42no—
638C22—22no—
639C22—22no—
640C22—24no—
641C22—24no—
642C22—28no—
643C22—44no—
644C22—44no—
645C22—48no—
646C22—48no—
647C22—816no—
648C22#236014yes—
649C22C22×C4×D411yescharacteristic
650C22C2×#54611yescharacteristic
651C22C2×D4×Q811yescharacteristic
652C22#358412yes—
653C22#358512yes—
654C4—22no—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—44no—
659C4—44no—
660C4—44no—
661C4—44no—
662C4—44no—
663C4—44no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4#236014yes—
668C4C22×C4×Q811yescharacteristic
669C4C2×#52011yescharacteristic
670C4C2×#55211yescharacteristic
671C4C2×#55411yescharacteristic
672C2—24no—
673C2—48no—
674C2C2×#236011yescharacteristic
675C2#6016711yescharacteristic
676C2#6016811yescharacteristic
677C2#6120112yes—
678C2#6138512yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34217C2semidirect
4. #22978C2semidirect
5. #22974C2semidirect
6. #22972C2semidirect
7. #22966C2semidirect
8. #21325C2semidirect
9. #21169C2semidirect
10. #21158C2semidirect
11. #21157C2semidirect
12. #21038C2semidirect
13. #21037C2semidirect
16. #20441C2semidirect
24. C2×#478C4semidirect
41. #2370C22semidirect
42. #2370C22semidirect
62. #1992C22semidirect
63. #1991C22semidirect
64. #1949C22semidirect
65. #1949C22semidirect
66. #1948C22semidirect
67. #1947C22semidirect
68. #1944C22semidirect
69. #1943C22semidirect
70. #1942C22semidirect
71. #1941C22semidirect
72. #1921C22semidirect
73. #1920C22semidirect
74. #1919C22semidirect
75. #1918C22semidirect
80. #1771C22semidirect
81. #1771C22semidirect
82. #1770C22semidirect
83. C8×C4⋊C4C22semidirect
109. C2×#125C2×C4semidirect
142. #478C2×C4semidirect
143. #478C2×C4semidirect
144. #478C2×C4semidirect
145. #478C2×C4semidirect
148. C2×#107C2×C4semidirect
153. C2×#106C2×C4semidirect
158. C2×C4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4782
#342172
#341812
#229782
#229742
#229722
#229662
#213252
#211692
#211582
#211572
#210382
#210372
#208792
#204412
#204162
C4×#4182