Finite Group Databasebeta

← #7511157

Subgroups — #7511157

Order 512.

1772 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35662C211yesmaximal, characteristic, complemented
2#35515C211yesmaximal, characteristic
3#32657C214yesmaximal, complemented
4#32654C214yesmaximal, complemented
5#32651C214yesmaximal, complemented
6#32649C214yesmaximal, complemented
7#23736C211yesmaximal, characteristic, complemented
8#23735C211yesmaximal, characteristic
9#23661C211yesmaximal, characteristic, complemented
10#23661C211yesmaximal, characteristic, complemented
11#23619C211yesmaximal, characteristic, complemented
12#23618C211yesmaximal, characteristic
13#21300C211yesmaximal, characteristic, complemented
14C2×#1968C211yesmaximal, characteristic
15#21151C211yesmaximal, characteristic, complemented
16#20936C211yesmaximal, characteristic
17#20933C211yesmaximal, characteristic
18#20880C211yesmaximal, characteristic, complemented
19#20879C211yesmaximal, characteristic
20C23×SD16C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C2×#519C2211yescharacteristic
23C2×#518C2211yescharacteristic
24C2×#428C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C2×#427C2211yescharacteristic
27C2×#427C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C2×#423C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#419C2211yescharacteristic
32C2×#418C2211yescharacteristic
33C2×#418C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×C4×C8C2211yescharacteristic
37#2582C2214yescomplemented
38#2581C2214yescomplemented
39#2580C2214yes—
40#2538C2214yes—
41#2454—22no—
42#2453—22no—
43#2452—22no—
44#2451—22no—
45#2438—24no—
46#2437—24no—
47#2317C2211yescharacteristic
48#2317C2211yescharacteristic
49#2316C2211yescharacteristic
50#2316C2211yescharacteristic
51#2310C2211yescharacteristic
52#2309C2211yescharacteristic
53#2296C2211yescharacteristic
54#2296C2211yescharacteristic
55C2×#391C2211yescharacteristic
56C2×#390C2211yescharacteristic
57#2097C2212yescomplemented
58#2097C2212yescomplemented
59#2096C2212yescomplemented
60#2095C2212yescomplemented
61#2094C2212yes—
62#2093C2212yes—
63#2092C2212yes—
64#2092C2212yes—
65#2085C2214yescomplemented
66#2085C2214yescomplemented
67#2084C2214yescomplemented
68#2084C2214yescomplemented
69#2068C2214yescomplemented
70#2067C2214yes—
71#2066C2214yescomplemented
72#2065C2214yes—
73#1984C2214yescomplemented
74#1984C2214yescomplemented
75#1968C2214yes—
76#1968C2214yes—
77#1942C2212yescomplemented
78#1942C2212yescomplemented
79#1941C2212yescomplemented
80#1941C2212yescomplemented
81#1880C2214yes—
82#1880C2214yes—
83#1877C2214yes—
84#1877C2214yes—
85#1865C2214yescomplemented
86#1864C2214yes—
87#1863C2214yescomplemented
88#1862C2214yes—
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96#1729C2211yescharacteristic
97C23×Q8C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C22×SD16—42no—
100C22×SD16—42no—
101C22×SD16Q818yescomplemented
102C22×SD16C2314yes—
103C22×M16C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C23×C8C2311yescharacteristic
106C23×C8C2311yescharacteristic
107#533—42no—
108#533—42no—
109C2×C4⋊Q8D412yes—
110C2×#125D412yes—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8D414yes—
113#519—42no—
114#519—42no—
115#518—42no—
116#518—42no—
117C2×C4×Q8—22no—
118C2×C4×Q8C2314yes—
119C2×C4×Q8C2314yes—
120C2×C4×D4—22no—
121C2×C4×D4C2314yes—
122C2×#117—22no—
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C2×#117C2311yescharacteristic
126C2×#117C2311yescharacteristic
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C22⋊C4—22no—
132C22×C22⋊C4—22no—
133C22×C42C2311yescharacteristic
134#428—42no—
135#428—42no—
136#427—24no—
137#427—42no—
138C2×#107C2311yescharacteristic
139C2×#107C2311yescharacteristic
140C2×#107C2312yes—
141C2×#107C2312yes—
142C2×#107C2312yes—
143C2×#106—42no—
144C2×#106C2311yescharacteristic
145C2×#106C2311yescharacteristic
146C2×#106C2312yes—
147C2×#106C2312yes—
148C2×#106C2314yes—
149C2×#106C2312yes—
150#423—42no—
151C2×C4⋊C8C2314yes—
152#419—42no—
153#419—42no—
154#418—42no—
155#418—42no—
156C2×#103—24no—
157C2×#103C2312yes—
158C2×#103C2312yes—
159C2×#103C2312yes—
160C2×#103C2314yes—
161C2×#103C2312yes—
162C2×#102—24no—
163C2×#102C2312yes—
164C2×#102C2312yes—
165C2×#102C2312yes—
166C2×#102C2314yes—
167C2×#102C2312yes—
168C2×C4×C8—42no—
169C2×C4×C8C2312yes—
170C2×C4×C8C2312yes—
171#400—24no—
172#399—24no—
173#398—24no—
174#398—24no—
175#396—24no—
176#396—24no—
177#395—24no—
178#393—24no—
179#391—24no—
180#391—22no—
181#391—21no—
182#391—21no—
183#391C2312yes—
184#391C2312yes—
185#391C2314yes—
186#390—22no—
187#390—21no—
188#390—21no—
189#390C2312yes—
190#390C2312yes—
191#390C2314yes—
192#389—22no—
193#389—22no—
194#389—22no—
195#389—22no—
196#389—24no—
197#389C2314yes—
198#389C2314yes—
199#389C2312yes—
200#389C2312yes—
201#389C2312yes—
202#389C2312yes—
203#387—24no—
204#387—24no—
205#385—22no—
206#385—22no—
207#385—24no—
208#385—24no—
209#385—22no—
210#385—22no—
211#384C2312yes—
212#384C2312yes—
213#382C2312yes—
214#382C2312yes—
215#381—22no—
216#380—24no—
217#380—22no—
218#380—22no—
219#379—22no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C2×#95—22no—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#336C2314yes—
232#336C2314yes—
233#336C2312yes—
234#336C2312yes—
235#336C2312yes—
236#336C2312yes—
237C25—21no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8C2×D414yes—
245C22×Q8C2×Q814yes—
246C22×Q8C2412yes—
247C22×D4—21no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4C2×Q814yes—
254C22×D4C2412yes—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—24no—
259C23×C4C2×D412yes—
260C23×C4C2411yescharacteristic
261C2×SD16—416no—
262C2×SD16—44no—
263C2×SD16—44no—
264C2×SD16C2×Q8116yes—
265C2×M16—22no—
266C2×M16—22no—
267C2×M16—22no—
268C2×M16—22no—
269C2×M16—22no—
270C2×M16—22no—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8—22no—
274C22×C8—22no—
275C22×C8C2×Q814yes—
276C22×C8#4114yes—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C22×C8C2411yescharacteristic
281C22×C8C2411yescharacteristic
282C4⋊Q8—44no—
283#125—44no—
284C22⋊Q8—48no—
285C22⋊Q8—88no—
286C4×Q8—44no—
287C4×Q8—44no—
288C4×Q8—44no—
289C4×Q8—44no—
290C4×Q8—48no—
291C4×D4—44no—
292C4×D4—44no—
293C4×D4—48no—
294#117—22no—
295#117—22no—
296#117—22no—
297#117—22no—
298#117—22no—
299#117—22no—
300#117—22no—
301#117—22no—
302#117—22no—
303#117—22no—
304#117—22no—
305#117—22no—
306#117—48no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D414yes—
325C2×C4⋊C4C2×D414yes—
326C2×C4⋊C4C2×D414yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—48no—
345C2×C22⋊C4—84no—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—24no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2412yes—
358C2×C42C2412yes—
359C2×C42C2412yes—
360#107—22no—
361#107—22no—
362#107—42no—
363#107—42no—
364#107—42no—
365#106—22no—
366#106—22no—
367#106—44no—
368#106—42no—
369#106—42no—
370#106—42no—
371C4⋊C8—44no—
372#103—24no—
373#103—24no—
374#103—42no—
375#103—42no—
376#103—44no—
377#102—24no—
378#102—24no—
379#102—42no—
380#102—42no—
381#102—44no—
382C4×C8—42no—
383C4×C8—42no—
384#95—24no—
385#95—24no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—24no—
391#95—24no—
392#95—24no—
393#95—24no—
394#95—24no—
395#95—24no—
396C24—22no—
397C24—24no—
398C24—42no—
399C24—42no—
400C24—82no—
401C24—82no—
402C24C22×D411yescharacteristic
403C2×Q8—24no—
404C2×Q8—28no—
405C2×Q8—22no—
406C2×Q8—28no—
407C2×Q8—28no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—42no—
415C2×Q8—44no—
416C2×Q8—42no—
417C2×Q8—44no—
418C2×Q8C22×Q814yes—
419C2×D4—28no—
420C2×D4—24no—
421C2×D4—28no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—42no—
428C2×D4—44no—
429C2×D4—42no—
430C2×D4—44no—
431C2×D4C22×Q814yes—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—42no—
445C22×C4—48no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—84no—
451C22×C4—84no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4C22×D412yes—
455C22×C4C22×D414yes—
456C22×C4C22×D414yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×Q812yes—
460C22×C4C2×#4112yes—
461C22×C4H(5,𝔽2)12yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C42−1+412yes—
465C22×C4C2511yescharacteristic
466SD16—416no—
467M16—44no—
468M16—44no—
469C2×C8—28no—
470C2×C8—28no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8C22×Q814yes—
484C2×C8C2×#4112yes—
485C2×C8C2×#4112yes—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—48no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—84no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4—84no—
516C4⋊C4—84no—
517C4⋊C4—84no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—48no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—48no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C22⋊C4—84no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—48no—
538C42—42no—
539C42—42no—
540C42—42no—
541C42—42no—
542C23—22no—
543C23—22no—
544C23—24no—
545C23—44no—
546C23—42no—
547C23—42no—
548C23—44no—
549C23—48no—
550C23—88no—
551C23—164no—
552C23—162no—
553C23—162no—
554C23#53412yes—
555C23D4×Q812yes—
556C23#55012yes—
557C23C23×D411yescharacteristic
558Q8—24no—
559Q8—24no—
560Q8—42no—
561Q8—48no—
562Q8—42no—
563Q8—88no—
564D4—24no—
565D4—24no—
566D4—42no—
567D4—48no—
568D4—42no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—48no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—48no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—168no—
604C2×C4—162no—
605C2×C4—162no—
606C2×C4#53412yes—
607C2×C4#53914yes—
608C2×C4#54314yes—
609C2×C4D4×Q812yes—
610C2×C4#55012yes—
611C2×C4#55414yes—
612C2×C4#56214yes—
613C2×C4#57714yes—
614C2×C4C23×Q811yescharacteristic
615C2×C4C22×#4111yescharacteristic
616C2×C4C2×H(5,𝔽2)11yescharacteristic
617C2×C4C2×H(5,𝔽2)11yescharacteristic
618C2×C4C2×H(5,𝔽2)11yescharacteristic
619C2×C4C2×2−1+411yescharacteristic
620C8—22no—
621C8—22no—
622C8—24no—
623C8—42no—
624C8—42no—
625C22—24no—
626C22—24no—
627C22—22no—
628C22—22no—
629C22—48no—
630C22—44no—
631C22—84no—
632C22—82no—
633C22—82no—
634C22—168no—
635C22#273214yes—
636C22C2×#53411yescharacteristic
637C22C2×D4×Q811yescharacteristic
638C22C2×#55011yescharacteristic
639C22C2×#57712yes—
640C22#358412yes—
641C4—24no—
642C4—42no—
643C4—44no—
644C4—44no—
645C4—44no—
646C4—82no—
647C4—82no—
648C4—81no—
649C4—81no—
650C4—82no—
651C4—81no—
652C4—81no—
653C4—82no—
654C4#273214yes—
655C4C2×#53911yescharacteristic
656C4C2×#54311yescharacteristic
657C4C2×#55411yescharacteristic
658C4C2×#56211yescharacteristic
659C2—24no—
660C2—84no—
661C2C2×#273211yescharacteristic
662C2#6076912yes—
663C2#6137112yes—
664C2#6141011yescharacteristic
665C2#6141311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35662C2semidirect
3. #32657C2semidirect
4. #32654C2semidirect
5. #32651C2semidirect
6. #32649C2semidirect
7. #23736C2semidirect
9. #23661C2semidirect
10. #23661C2semidirect
11. #23619C2semidirect
13. #21300C2semidirect
15. #21151C2semidirect
18. #20880C2semidirect
37. #2582C22semidirect
38. #2581C22semidirect
57. #2097C22semidirect
58. #2097C22semidirect
59. #2096C22semidirect
60. #2095C22semidirect
65. #2085C22semidirect
66. #2085C22semidirect
67. #2084C22semidirect
68. #2084C22semidirect
69. #2068C22semidirect
71. #2066C22semidirect
73. #1984C22semidirect
74. #1984C22semidirect
77. #1942C22semidirect
78. #1942C22semidirect
79. #1941C22semidirect
80. #1941C22semidirect
85. #1865C22semidirect
87. #1863C22semidirect
101. C22×SD16Q8semidirect

Maximal subgroups

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

subgroupindex
#356622
#355152
#326572
#326542
#326512
#326492
#237362
#237352
#236612
#236192
#236182
#213002
C2×#19682
#211512
#209362
#209332
#208802
#208792