Finite Group Databasebeta

← #6449313

Subgroups — #6449313

Order 512.

1816 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35275C211yesmaximal, characteristic, complemented
2#34186C211yesmaximal, characteristic
3#26037C214yesmaximal, complemented
4#26036C214yesmaximal, complemented
5#26025C212yesmaximal, complemented
6#26024C212yesmaximal, complemented
7#26021C212yesmaximal, complemented
8#26020C212yesmaximal, complemented
9#23612C212yesmaximal
10#23456C212yesmaximal
11#21315C212yesmaximal, complemented
12#21169C211yesmaximal, characteristic, complemented
13#21145C211yesmaximal, characteristic, complemented
14#21039C212yesmaximal, complemented
15#20764C211yesmaximal, characteristic, complemented
16C2×#1802C211yesmaximal, characteristic
17C4×#418C211yesmaximal, characteristic
18C2×#544C2212yes—
19C22×C4×D4C2211yescharacteristic
20C2×#427C2212yes—
21C2×#424C2212yes—
22C22×C4⋊C8C2211yescharacteristic
23C2×#418C2211yescharacteristic
24C2×#418C2211yescharacteristic
25C22×#102C2212yes—
26C2×C4×M16C2211yescharacteristic
27#2868—21no—
28#2859—21no—
29#2660—22no—
30#2552C2212yescomplemented
31#2552C2212yescomplemented
32#2551C2212yescomplemented
33#2551C2212yescomplemented
34#2440—22no—
35#2440—22no—
36#2357—22no—
37#2357C2214yes—
38#2354—21no—
39#2354—21no—
40#2354C2212yes—
41#2354C2212yes—
42#2317C2212yes—
43#2312—22no—
44#2309C2212yes—
45#2285C2211yescharacteristic
46#2285C2211yescharacteristic
47C2×#384C2211yescharacteristic
48#2280—22no—
49#2274C2212yes—
50C4×#117C2211yescharacteristic
51#2067C2214yes—
52#2067C2214yes—
53#2065C2214yes—
54#2065C2214yes—
55#2025C2214yes—
56#2025C2214yes—
57#2023C2214yes—
58#2023C2214yes—
59#1989C2214yescomplemented
60#1989C2214yescomplemented
61#1988C2214yescomplemented
62#1987C2214yescomplemented
63#1949C2212yescomplemented
64#1949C2212yescomplemented
65#1948C2211yescharacteristic, complemented
66#1948C2211yescharacteristic, complemented
67#1947C2211yescharacteristic, complemented
68#1947C2211yescharacteristic, complemented
69#1940C2214yescomplemented
70#1939C2212yescomplemented
71#1939C2212yescomplemented
72#1921C2214yescomplemented
73#1920C2214yescomplemented
74#1919C2214yescomplemented
75#1918C2214yescomplemented
76#1836C2214yescomplemented
77#1834C2212yescomplemented
78#1834C2212yescomplemented
79#1802C2212yes—
80#1802C2212yes—
81#1802C2214yes—
82#1762C2212yes—
83#1762C2212yes—
84C4×#102C2212yes—
85C4×#102C2212yes—
86#1736C2212yes—
87#1736C2212yes—
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1735C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93C23×D4C2311yescharacteristic
94C24×C4—21no—
95C22×M16C2311yescharacteristic
96C22×M16C2311yescharacteristic
97C23×C8C2312yes—
98#544—28no—
99#544—44no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8C2314yes—
104C2×#125—22no—
105C2×#125C2314yes—
106C2×C22⋊Q8—24no—
107C2×C4×D4—21no—
108C2×C4×D4—21no—
109C2×C4×D4—24no—
110C2×C4×D4Q818yescomplemented
111C2×C4×D4C2312yes—
112C2×C4×D4C2312yes—
113C2×#117—22no—
114C2×#117—22no—
115C2×#117C2312yes—
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C2×#117C2312yes—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C22⋊C4—21no—
124C22×C42C2311yescharacteristic
125#427—28no—
126C2×#107C2312yes—
127C2×#107C2312yes—
128C2×#106C2312yes—
129C2×#106C2312yes—
130#424—44no—
131C2×C4⋊C8—24no—
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2314yes—
136C2×C4⋊C8C2314yes—
137#418—24no—
138#418—24no—
139C2×#102—44no—
140C2×#102C2312yes—
141C2×#102C2312yes—
142C2×#102C2312yes—
143C2×#102C2312yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C4×M16—24no—
147C2×#97C2312yes—
148C2×C4×C8C2312yes—
149#398—24no—
150#398—24no—
151#395—24no—
152#395—24no—
153#390—22no—
154#390—22no—
155#390C2314yes—
156#390C2314yes—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389C2314yes—
162#389C2314yes—
163#388—24no—
164#388—24no—
165#388—24no—
166#388—24no—
167#385—24no—
168#384—22no—
169#384—22no—
170#384—22no—
171#384—21no—
172#384—21no—
173#384D411yescharacteristic
174#384D411yescharacteristic
175#384C2312yes—
176#384C2312yes—
177#384C2311yescharacteristic
178#384C2311yescharacteristic
179#382—22no—
180#382D412yes—
181#382C2312yes—
182#381—24no—
183#380—21no—
184#380—21no—
185#380—22no—
186#380—22no—
187#380D412yes—
188#380D412yes—
189#379—24no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4C2314yes—
195C4×C4⋊C4C2314yes—
196C4×C4⋊C4C2311yescharacteristic
197C4×C4⋊C4C2311yescharacteristic
198C4×C4⋊C4C2311yescharacteristic
199C4×C4⋊C4C2311yescharacteristic
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C22⋊C4—21no—
203C4×C22⋊C4—21no—
204C4×C22⋊C4—24no—
205C4×C22⋊C4—24no—
206#376—21no—
207C43—21no—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2312yes—
215#340C2312yes—
216#340C2312yes—
217#340C2312yes—
218#340C2312yes—
219#340C2312yes—
220#340C2312yes—
221#340C2312yes—
222#336C2314yes—
223#336C2314yes—
224C25—21no—
225C22×Q8—24no—
226C22×D4—21no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4C2×Q814yes—
233C22×D4C2412yes—
234C23×C4—21no—
235C23×C4—22no—
236C23×C4—28no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—82no—
240C23×C4—82no—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2411yescharacteristic
244C2×M16—22no—
245C2×M16—22no—
246C2×M16—22no—
247C2×M16—22no—
248C2×M16—22no—
249C2×M16—22no—
250C22×C8—24no—
251C22×C8—24no—
252C22×C8—24no—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C4⋊Q8—28no—
258C4⋊Q8—44no—
259C4⋊Q8—48no—
260C4⋊Q8—84no—
261C4⋊Q8—84no—
262#125—28no—
263#125—44no—
264#125—48no—
265C22⋊Q8—416no—
266C4×D4—28no—
267C4×D4—28no—
268C4×D4—24no—
269C4×D4—24no—
270C4×D4C2×Q818yes—
271C4×D4C2×Q818yes—
272#117—24no—
273#117—22no—
274#117—24no—
275#117—22no—
276#117—24no—
277#117—24no—
278#117—84no—
279#117—84no—
280#117#4114yes—
281#117#4114yes—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—82no—
301C2×C4⋊C4—82no—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×Q814yes—
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—28no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4C2×D414yes—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×Q814yes—
340C2×C42#4114yes—
341C2×C42C2414yes—
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344#107—24no—
345#107—24no—
346#106—24no—
347#106—24no—
348C4⋊C8—22no—
349C4⋊C8—22no—
350C4⋊C8—24no—
351C4⋊C8—44no—
352C4⋊C8—44no—
353#102—24no—
354#102—24no—
355#102—24no—
356#102—24no—
357#102—44no—
358#102—44no—
359#97—24no—
360C4×C8—24no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—22no—
366#95—24no—
367#95—24no—
368#95—22no—
369#95—24no—
370#95—24no—
371#95C2×D414yes—
372#95C2×D414yes—
373C24—22no—
374C24—24no—
375C24—42no—
376C24—42no—
377C24—81no—
378C24—81no—
379C24—81no—
380C24—81no—
381C24C22×D411yescharacteristic
382C2×Q8—48no—
383C2×Q8—84no—
384C2×Q8—84no—
385C2×D4—24no—
386C2×D4—28no—
387C2×D4—28no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4C22×Q814yes—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—816no—
417C22×C4—82no—
418C22×C4—82no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4—82no—
422C22×C4—82no—
423C22×C4C22⋊Q814yes—
424C22×C4C22⋊Q814yes—
425C22×C4C22×D414yes—
426C22×C4C22×D414yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×Q812yes—
433C22×C4C2×#4112yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4112yes—
436C22×C4H(5,𝔽2)12yes—
437C22×C42−1+412yes—
438C22×C4C2511yescharacteristic
439M16—44no—
440M16—44no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—48no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—81no—
481C4⋊C4—81no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4—84no—
485C4⋊C4—84no—
486C4⋊C4C22×Q814yes—
487C4⋊C4C2×#4112yes—
488C4⋊C4C2×#4112yes—
489C4⋊C4C2×#4112yes—
490C4⋊C4C2×#4112yes—
491C22⋊C4—24no—
492C22⋊C4—28no—
493C22⋊C4—24no—
494C22⋊C4—28no—
495C22⋊C4—28no—
496C22⋊C4—44no—
497C22⋊C4—84no—
498C22⋊C4—84no—
499C22⋊C4—84no—
500C22⋊C4—84no—
501C42—24no—
502C42—24no—
503C42—24no—
504C42—24no—
505C42—42no—
506C42—42no—
507C42—44no—
508C42—42no—
509C42—42no—
510C42—84no—
511C42—84no—
512C42C22⋊Q814yes—
513C42C22⋊Q814yes—
514C42C22×Q812yes—
515C42C22×Q812yes—
516C42C2×#4112yes—
517C42C2×#4112yes—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—24no—
523C23—42no—
524C23—42no—
525C23—48no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—84no—
531C23—84no—
532C23—84no—
533C23—84no—
534C23—84no—
535C23—84no—
536C23C2×C22⋊Q812yes—
537C23#54612yes—
538C23#54812yes—
539C23C23×D411yescharacteristic
540Q8—168no—
541D4—24no—
542D4—24no—
543D4—48no—
544D4—42no—
545D4—42no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—44no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—416no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—88no—
586C2×C4—84no—
587C2×C4—164no—
588C2×C4C2×C22⋊Q812yes—
589C2×C4C2×C22⋊Q812yes—
590C2×C4C2×C22⋊Q812yes—
591C2×C4C2×C22⋊Q812yes—
592C2×C4C2×C22⋊Q812yes—
593C2×C4#53314yes—
594C2×C4#54414yes—
595C2×C4#54612yes—
596C2×C4#54812yes—
597C2×C4#55514yes—
598C2×C4#55614yes—
599C2×C4#57612yes—
600C2×C4#57712yes—
601C2×C4C23×Q811yescharacteristic
602C2×C4C22×#4111yescharacteristic
603C2×C4C22×#4111yescharacteristic
604C2×C4C22×#4111yescharacteristic
605C2×C4C2×H(5,𝔽2)11yescharacteristic
606C2×C4C2×2−1+411yescharacteristic
607C8—42no—
608C8—42no—
609C8—42no—
610C8—42no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—816no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22#243914yes—
626C22C22×C22⋊Q811yescharacteristic
627C22C2×#54611yescharacteristic
628C22C2×#54811yescharacteristic
629C22C2×#57611yescharacteristic
630C22C2×#57711yescharacteristic
631C22#358511yescharacteristic
632C22#358611yescharacteristic
633C4—22no—
634C4—22no—
635C4—22no—
636C4—44no—
637C4—42no—
638C4—44no—
639C4—42no—
640C4—44no—
641C4—42no—
642C4—42no—
643C4—84no—
644C4—82no—
645C4—82no—
646C4#243914yes—
647C4#308114yes—
648C4C2×#53311yescharacteristic
649C4C2×#54411yescharacteristic
650C4C2×#55511yescharacteristic
651C4C2×#55611yescharacteristic
652C2—24no—
653C2—84no—
654C2C2×#243911yescharacteristic
655C2C2×#308111yescharacteristic
656C2#6049311yescharacteristic
657C2#6120511yescharacteristic
658C2#6120911yescharacteristic
659C2#6134511yescharacteristic
660C2#6134711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×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
1. #35275C2semidirect
3. #26037C2semidirect
4. #26036C2semidirect
5. #26025C2semidirect
6. #26024C2semidirect
7. #26021C2semidirect
8. #26020C2semidirect
11. #21315C2semidirect
12. #21169C2semidirect
13. #21145C2semidirect
14. #21039C2semidirect
15. #20764C2semidirect
30. #2552C22semidirect
31. #2552C22semidirect
32. #2551C22semidirect
33. #2551C22semidirect
59. #1989C22semidirect
60. #1989C22semidirect
61. #1988C22semidirect
62. #1987C22semidirect
63. #1949C22semidirect
64. #1949C22semidirect
65. #1948C22semidirect
66. #1948C22semidirect
67. #1947C22semidirect
68. #1947C22semidirect
69. #1940C22semidirect
70. #1939C22semidirect
71. #1939C22semidirect
72. #1921C22semidirect
73. #1920C22semidirect
74. #1919C22semidirect
75. #1918C22semidirect
76. #1836C22semidirect
77. #1834C22semidirect
78. #1834C22semidirect
110. C2×C4×D4Q8semidirect

Maximal subgroups

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

subgroupindex
#352752
#341862
#260372
#260362
#260252
#260242
#260212
#260202
#236122
#234562
#213152
#211692
#211452
#210392
#207642
C2×#18022
C4×#4182