Finite Group Databasebeta

← #6460548

Subgroups — #6460548

Order 512.

1892 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
1#35555C211yesmaximal, characteristic, complemented
2#35554C211yesmaximal, characteristic
3#29036C214yesmaximal, complemented
4#29032C214yesmaximal, complemented
5#29011C214yesmaximal, complemented
6#29007C214yesmaximal, complemented
7#23745C211yesmaximal, characteristic, complemented
8#23612C211yesmaximal, characteristic, complemented
9#23612C211yesmaximal, characteristic, complemented
10#23505C211yesmaximal, characteristic
11C2×#1957C211yesmaximal, characteristic
12#21151C211yesmaximal, characteristic, complemented
13#21091C211yesmaximal, characteristic, complemented
14#21091C211yesmaximal, characteristic, complemented
15#21089C211yesmaximal, characteristic, complemented
16#21065C211yesmaximal, characteristic, complemented
17#20879C211yesmaximal, characteristic, complemented
18#20435C211yesmaximal, characteristic, complemented
19#20416C211yesmaximal, characteristic, complemented
20C22×#127C2211yescharacteristic
21C2×#518C2211yescharacteristic
22C2×#427C2211yescharacteristic
23C2×#427C2211yescharacteristic
24C2×#424C2211yescharacteristic
25C2×#418C2211yescharacteristic
26C2×#418C2211yescharacteristic
27C2×#418C2211yescharacteristic
28C2×#418C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C2×#405C2211yescharacteristic
34C2×#405C2211yescharacteristic
35C22×C4×C8C2211yescharacteristic
36#2570—22no—
37#2552—22no—
38#2507C2214yescomplemented
39#2506C2214yescomplemented
40#2473—24no—
41#2445—22no—
42#2445—22no—
43#2445C2214yes—
44#2445C2214yes—
45#2405—24no—
46#2317C2211yescharacteristic
47#2317C2211yescharacteristic
48#2316C2211yescharacteristic
49#2309C2211yescharacteristic
50#2309C2211yescharacteristic
51#2309C2211yescharacteristic
52#2296C2211yescharacteristic
53#2296C2211yescharacteristic
54#2296C2211yescharacteristic
55C2×#384C2211yescharacteristic
56#2274C2211yescharacteristic
57#2100C2214yescomplemented
58#2099C2214yescomplemented
59#2067C2214yescomplemented
60#2067C2214yescomplemented
61#2065C2214yescomplemented
62#2065C2214yescomplemented
63#2041C2214yes—
64#2041C2214yes—
65#1957C2212yes—
66#1957C2212yes—
67#1957C2212yes—
68#1957C2212yes—
69#1942C2212yescomplemented
70#1942C2212yescomplemented
71#1941C2212yescomplemented
72#1941C2212yescomplemented
73#1936C2214yescomplemented
74#1936C2214yescomplemented
75#1936C2214yescomplemented
76#1934C2214yescomplemented
77#1934C2214yescomplemented
78#1934C2214yescomplemented
79#1926C2214yescomplemented
80#1925C2214yescomplemented
81#1864C2214yescomplemented
82#1862C2214yescomplemented
83#1771C2212yescomplemented
84#1771C2212yescomplemented
85C8×C4⋊C4C2212yescomplemented
86C8×C4⋊C4C2212yescomplemented
87#1762C2214yescomplemented
88C4×#102C2214yescomplemented
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96#1728C2211yescharacteristic
97C23×D4—22no—
98C23×D4C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C22×M16C2311yescharacteristic
101C22×M16C2311yescharacteristic
102C23×C8C2311yescharacteristic
103C23×C8C2311yescharacteristic
104C2×#127—24no—
105C2×#127—22no—
106C2×#127—42no—
107C2×#127C2312yes—
108C2×#127C2312yes—
109C2×#125—24no—
110C2×#123—24no—
111C2×#123—24no—
112#518—24no—
113#518—42no—
114C2×C4×D4—22no—
115C2×C4×D4C2314yes—
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C2×#117C2311yescharacteristic
120C2×#117C2311yescharacteristic
121C2×#117C2311yescharacteristic
122C22×C4⋊C4—22no—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C22⋊C4—22no—
126C22×C42C2311yescharacteristic
127#427—24no—
128#427—42no—
129C2×#107C2312yes—
130C2×#107C2312yes—
131C2×#106C2312yes—
132C2×#106C2312yes—
133#424—24no—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2312yes—
136#418—24no—
137#418—24no—
138#418—24no—
139#418—42no—
140C2×#102—24no—
141C2×#102—24no—
142C2×#102—24no—
143C2×#102—24no—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×#102C2314yes—
147C2×#102C2312yes—
148C2×#102C2312yes—
149C2×#102C2314yes—
150C2×#102C2312yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156#405—24no—
157#405—42no—
158C2×#97C2312yes—
159C2×#97C2312yes—
160C2×C4×C8—42no—
161C2×C4×C8C2312yes—
162C2×C4×C8C2311yescharacteristic
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2312yes—
167#400—24no—
168#398—24no—
169#398—24no—
170#398—24no—
171#394—24no—
172#394—24no—
173#394—24no—
174#394—24no—
175#390—21no—
176#390—21no—
177#390—22no—
178#390—21no—
179#390—21no—
180#390C2314yes—
181#390C2314yes—
182#390C2314yes—
183#389—22no—
184#389—24no—
185#389D412yes—
186#389D412yes—
187#389C2314yes—
188#389C2314yes—
189#389C2314yes—
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#387—24no—
195#387—24no—
196#386—24no—
197#386—24no—
198#385—22no—
199#385—22no—
200#385—22no—
201#385—22no—
202#385D414yes—
203#384—21no—
204#384—21no—
205#384—22no—
206#384C2312yes—
207#384C2312yes—
208#382—21no—
209#382—21no—
210#380—22no—
211#379—22no—
212#379—22no—
213#379—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4C2314yes—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2312yes—
220C4×C22⋊C4—22no—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#336C2312yes—
230#336C2312yes—
231#336C2312yes—
232#336C2312yes—
233C25—21no—
234C25—21no—
235C22×D4—21no—
236C22×D4—21no—
237C22×D4—24no—
238C22×D4—24no—
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—48no—
251C22×D4—84no—
252C22×D4—84no—
253C22×D4#4114yes—
254C22×D4C2412yes—
255C22×D4C2412yes—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—24no—
259C23×C4C2×D412yes—
260C23×C4C2411yescharacteristic
261C2×M16—22no—
262C2×M16—22no—
263C2×M16—22no—
264C2×M16—22no—
265C2×M16—22no—
266C2×M16—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8#4114yes—
272C22×C8#4114yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22×C8C2411yescharacteristic
276C22×C8C2411yescharacteristic
277C22×C8C2411yescharacteristic
278C22×C8C2411yescharacteristic
279#127—24no—
280#127—24no—
281#127—42no—
282#127—42no—
283#127—48no—
284#125—88no—
285#123—88no—
286#123—88no—
287C4×D4—28no—
288C4×D4—44no—
289C4×D4—48no—
290#117—22no—
291#117—22no—
292#117—22no—
293#117—22no—
294#117—22no—
295#117—22no—
296#117—22no—
297#117—22no—
298#117—22no—
299#117—22no—
300#117—22no—
301#117#4114yes—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—84no—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D414yes—
322C2×C4⋊C4C2×D414yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—48no—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C42—21no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—24no—
350C2×C42C2×D412yes—
351C2×C42C2412yes—
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355#107—24no—
356#107—42no—
357#106—24no—
358#106—42no—
359C4⋊C8—24no—
360C4⋊C8—24no—
361#102—28no—
362#102—24no—
363#102—24no—
364#102—24no—
365#102—24no—
366#102—24no—
367#102—24no—
368#102—24no—
369#102—24no—
370#102—28no—
371#102—28no—
372#102—44no—
373#97—24no—
374#97—42no—
375C4×C8—24no—
376C4×C8—41no—
377C4×C8—41no—
378C4×C8—41no—
379C4×C8—41no—
380C4×C8—42no—
381#95—24no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95C2×D414yes—
390#95C2×D414yes—
391C24—22no—
392C24—24no—
393C24—22no—
394C24—42no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24—82no—
400C24—82no—
401C24—82no—
402C24—82no—
403C24C22×D411yescharacteristic
404C2×D4—28no—
405C2×D4—24no—
406C2×D4—28no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—44no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—48no—
432C2×D4—48no—
433C2×D4—84no—
434C2×D4—84no—
435C2×D4—832no—
436C2×D4C2×#4114yes—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—48no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—84no—
450C22×C4—84no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4C22×D412yes—
454C22×C4C22×D414yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C42−1+412yes—
465C22×C4C2511yescharacteristic
466M16—44no—
467M16—44no—
468C2×C8—28no—
469C2×C8—28no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8C2×#4112yes—
485C2×C8C2×#4112yes—
486C2×C8C2×#4112yes—
487C2×C8C2×#4112yes—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—48no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C4⋊C4—84no—
514C4⋊C4—84no—
515C4⋊C4C2×#4114yes—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—48no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C22⋊C4—84no—
529C42—24no—
530C42—24no—
531C42—22no—
532C42—22no—
533C42—21no—
534C42—21no—
535C42—22no—
536C42—21no—
537C42—21no—
538C42—22no—
539C42—42no—
540C42—42no—
541C42—42no—
542C42—84no—
543C23—22no—
544C23—22no—
545C23—24no—
546C23—42no—
547C23—42no—
548C23—44no—
549C23—44no—
550C23—48no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—44no—
555C23—44no—
556C23—88no—
557C23—88no—
558C23—84no—
559C23—162no—
560C23—162no—
561C23—162no—
562C23—162no—
563C23—164no—
564C23—164no—
565C23#53212yes—
566C23#54612yes—
567C23#54812yes—
568C23C23×D411yescharacteristic
569D4—24no—
570D4—24no—
571D4—42no—
572D4—42no—
573D4—42no—
574D4—42no—
575D4—42no—
576D4—42no—
577D4—48no—
578D4—816no—
579D4—816no—
580D4—84no—
581C2×C4—216no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—168no—
610C2×C4—162no—
611C2×C4—162no—
612C2×C4#52914yes—
613C2×C4#53212yes—
614C2×C4#53914yes—
615C2×C4#54612yes—
616C2×C4#54812yes—
617C2×C4#55214yes—
618C2×C4#55314yes—
619C2×C4#57614yes—
620C2×C4#57714yes—
621C2×C4C22×#4111yescharacteristic
622C2×C4C22×#4111yescharacteristic
623C2×C4C22×#4111yescharacteristic
624C2×C4C22×#4111yescharacteristic
625C2×C4C2×H(5,𝔽2)11yescharacteristic
626C2×C4C2×2−1+411yescharacteristic
627C8—22no—
628C8—22no—
629C8—22no—
630C8—22no—
631C8—42no—
632C8—42no—
633C22—22no—
634C22—22no—
635C22—24no—
636C22—24no—
637C22—44no—
638C22—48no—
639C22—82no—
640C22—82no—
641C22—84no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—84no—
646C22—84no—
647C22—168no—
648C22—168no—
649C22#255814yes—
650C22C2×#53211yescharacteristic
651C22C2×#54611yescharacteristic
652C22C2×#54811yescharacteristic
653C22C2×#57612yes—
654C22C2×#57712yes—
655C4—24no—
656C4—24no—
657C4—42no—
658C4—42no—
659C4—44no—
660C4—44no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4#255814yes—
667C4C2×#52911yescharacteristic
668C4C2×#53911yescharacteristic
669C4C2×#55211yescharacteristic
670C4C2×#55311yescharacteristic
671C2—24no—
672C2—84no—
673C2—162no—
674C2C2×#255811yescharacteristic
675C2#6071211yescharacteristic
676C2#6071411yescharacteristic
677C2#6116712yes—
678C2#6132912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35555C2semidirect
3. #29036C2semidirect
4. #29032C2semidirect
5. #29011C2semidirect
6. #29007C2semidirect
7. #23745C2semidirect
8. #23612C2semidirect
9. #23612C2semidirect
12. #21151C2semidirect
13. #21091C2semidirect
14. #21091C2semidirect
15. #21089C2semidirect
16. #21065C2semidirect
17. #20879C2semidirect
18. #20435C2semidirect
19. #20416C2semidirect
38. #2507C22semidirect
39. #2506C22semidirect
57. #2100C22semidirect
58. #2099C22semidirect
59. #2067C22semidirect
60. #2067C22semidirect
61. #2065C22semidirect
62. #2065C22semidirect
69. #1942C22semidirect
70. #1942C22semidirect
71. #1941C22semidirect
72. #1941C22semidirect
73. #1936C22semidirect
74. #1936C22semidirect
75. #1936C22semidirect
76. #1934C22semidirect
77. #1934C22semidirect
78. #1934C22semidirect
79. #1926C22semidirect
80. #1925C22semidirect
81. #1864C22semidirect
82. #1862C22semidirect
83. #1771C22semidirect
84. #1771C22semidirect
85. C8×C4⋊C4C22semidirect
86. C8×C4⋊C4C22semidirect
87. #1762C22semidirect
88. C4×#102C22semidirect

Maximal subgroups

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

subgroupindex
#355552
#355542
#290362
#290322
#290112
#290072
#237452
#236122
#235052
C2×#19572
#211512
#210912
#210892
#210652
#208792
#204352
#204162