Finite Group Databasebeta

← #6992878

Subgroups — #6992878

Order 512.

1645 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36481C211yesmaximal, characteristic
2#35281C211yesmaximal, characteristic
3#32801C212yesmaximal, complemented
4#30990C214yesmaximal
5#30449C214yesmaximal
6#30338C212yesmaximal, complemented
7#30042C212yesmaximal, complemented
8#29664C211yesmaximal, characteristic
9#28462C212yesmaximal, complemented
10#28440C212yesmaximal, complemented
11#27749C212yesmaximal, complemented
12#25722C211yesmaximal, characteristic
13#25370C212yesmaximal, complemented
14#24732C212yesmaximal, complemented
15#24212C211yesmaximal, characteristic
16#23571C211yesmaximal, characteristic
17#22639C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#494C2214yes—
21C2×#494C2212yes—
22C2×#462C2214yes—
23C2×#462C2214yes—
24C2×#462C2212yes—
25C2×#462C2212yes—
26C2×#451C2214yes—
27C2×#451C2212yes—
28C2×C4×Q16C2212yes—
29#2868C2212yes—
30#2863C2212yes—
31#2862C2212yes—
32#2857C2211yescharacteristic
33#2660—22no—
34#2606—22no—
35#2567—21no—
36#2567—21no—
37#2552C2211yescharacteristic
38#2546C2212yes—
39#2546C2211yescharacteristic
40#2546C2212yes—
41#2522—22no—
42#2507—21no—
43#2502—22no—
44#2441—21no—
45#2434C2214yes—
46#2429C2212yes—
47#2423C2212yes—
48#2370—21no—
49#2365C2211yescharacteristic
50Q8×C4⋊C4C2211yescharacteristic
51#2347—21no—
52#2317—21no—
53#2317—21no—
54C4×C4⋊Q8C2212yes—
55C4×C22⋊Q8C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2072C2214yes—
58#2068C2214yes—
59#2057C2211yescharacteristic
60#2056C2214yes—
61#2043C2214yes—
62#2043C2211yescharacteristic
63#2043C2214yes—
64#2043C2212yes—
65#2036C2212yes—
66#2024C2211yescharacteristic
67#2024C2212yes—
68#2016C2214yes—
69#2016C2211yescharacteristic
70#2016C2212yes—
71#2016C2212yes—
72#2016C2212yes—
73#2016C2214yes—
74#2016C2212yes—
75#2016C2214yes—
76#2001C2214yes—
77#2001C2212yes—
78#1981C2212yes—
79#1969C2214yes—
80#1956C2214yes—
81#1956C2212yes—
82#1939C2212yes—
83#1933C2214yes—
84#1923C2214yes—
85#1878C2214yes—
86#1878C2214yes—
87#1839C2212yes—
88#1803C2212yes—
89#1797C2211yescharacteristic
90#1787C2212yes—
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2212yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2312yes—
96C22×Q16C2312yes—
97C22×Q16C2314yes—
98C22×Q16C2314yes—
99C22×Q16C2314yes—
100C22×Q16C2314yes—
101#533—44no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—44no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2314yes—
111C2×C4⋊Q8C2314yes—
112C2×C4⋊Q8C2311yescharacteristic
113C2×#125—21no—
114C2×#125—21no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8C2311yescharacteristic
117C2×C22⋊Q8C2312yes—
118C2×C22⋊Q8C2312yes—
119C2×C4×Q8—21no—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2312yes—
124C2×#117—21no—
125C2×#117D411yescharacteristic
126C2×#117D411yescharacteristic
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C42C2311yescharacteristic
130#494—44no—
131#494—48no—
132#462—48no—
133#462—48no—
134#462—44no—
135#462—44no—
136#451—48no—
137#451—44no—
138C4×Q16—44no—
139C2×#107C2312yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2314yes—
142C2×#103C2314yes—
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2312yes—
155C2×C4×C8C2312yes—
156C2×C4×C8C2314yes—
157#399—22no—
158#398—22no—
159#398—22no—
160#398—22no—
161#398—22no—
162#398—24no—
163#398—24no—
164#396—22no—
165#395—24no—
166#395—24no—
167#391—22no—
168#391—22no—
169#391C2311yescharacteristic
170#391C2312yes—
171#391C2312yes—
172#391C2314yes—
173#391C2314yes—
174#391C2312yes—
175#389—22no—
176#389—22no—
177#389—22no—
178#389—22no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—22no—
184#389—22no—
185#389—21no—
186#389—21no—
187#389—22no—
188#389—22no—
189#389C2311yescharacteristic
190#389C2312yes—
191#389C2312yes—
192#388—22no—
193#388—22no—
194#387—21no—
195#387—22no—
196#387—22no—
197#387—22no—
198#387—22no—
199#387—22no—
200#385—21no—
201#385—24no—
202#385—24no—
203#384—21no—
204#384—22no—
205#384—22no—
206#384—22no—
207#384—22no—
208#384C2312yes—
209#384C2311yescharacteristic
210#383—22no—
211#382—22no—
212#382—22no—
213#380—21no—
214#380—21no—
215#380C2311yescharacteristic
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4D411yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C22⋊C4—21no—
227C4×C22⋊C4D412yes—
228C4×C22⋊C4C2312yes—
229#376D412yes—
230C43C2312yes—
231#340C2312yes—
232#340C2314yes—
233#340C2312yes—
234#336C2312yes—
235#336C2311yescharacteristic
236#336C2314yes—
237#336C2311yescharacteristic
238#336C2314yes—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—44no—
242C22×Q8—42no—
243C22×Q8—44no—
244C22×Q8—42no—
245C22×Q8C2412yes—
246C22×Q8C2412yes—
247C22×Q8C2414yes—
248C22×Q8C2411yescharacteristic
249C23×C4—22no—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2411yescharacteristic
253C2×Q16—24no—
254C2×Q16—28no—
255C2×Q16—24no—
256C2×Q16—28no—
257C2×Q16—42no—
258C2×Q16—44no—
259C2×Q16—44no—
260C2×Q16—44no—
261C2×Q16—44no—
262C2×Q16—44no—
263C2×Q16—44no—
264C2×Q16—44no—
265C2×Q16—44no—
266C2×Q16—44no—
267C2×Q16—44no—
268C22×C8C2412yes—
269C22×C8C2414yes—
270C22×C8C2412yes—
271C4⋊Q8—24no—
272C4⋊Q8—24no—
273C4⋊Q8—24no—
274C4⋊Q8—24no—
275C4⋊Q8—42no—
276C4⋊Q8—44no—
277C4⋊Q8—42no—
278C4⋊Q8—42no—
279C4⋊Q8—44no—
280C4⋊Q8—44no—
281C4⋊Q8—44no—
282C4⋊Q8—44no—
283C4⋊Q8—44no—
284C4⋊Q8—44no—
285#125—44no—
286#125—44no—
287C22⋊Q8—22no—
288C22⋊Q8—41no—
289C22⋊Q8—42no—
290C22⋊Q8—42no—
291C22⋊Q8—44no—
292C22⋊Q8—44no—
293C4×Q8—24no—
294C4×Q8—24no—
295C4×Q8—24no—
296C4×Q8—44no—
297C4×Q8—44no—
298C4×Q8—44no—
299#117—24no—
300#117—44no—
301#117Q1618yescomplemented
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—44no—
332C2×C4⋊C4—44no—
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2411yescharacteristic
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C42—24no—
349C2×C42—22no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D412yes—
358C2×C42C2×D412yes—
359C2×C42C2×D412yes—
360C2×C42C2412yes—
361C2×C42C2411yescharacteristic
362C2×C42C2412yes—
363C2×C42C2411yescharacteristic
364#107—42no—
365C4⋊C8—28no—
366C4⋊C8—24no—
367#103—28no—
368#103—24no—
369#103—42no—
370#103—42no—
371#103—42no—
372#103—44no—
373#103—44no—
374#103—44no—
375#103—44no—
376#103—44no—
377#103—44no—
378C22⋊C8—24no—
379C22⋊C8—42no—
380C4×C8—24no—
381C4×C8—44no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95—21no—
391#95—24no—
392#95—24no—
393#95—24no—
394#95—24no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95C2×D412yes—
400#95C2×D412yes—
401#95C2×D412yes—
402#95C2×D412yes—
403C24C22×D411yescharacteristic
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—24no—
409C2×Q8—22no—
410C2×Q8—24no—
411C2×Q8—22no—
412C2×Q8—24no—
413C2×Q8—24no—
414C2×Q8—24no—
415C2×Q8—22no—
416C2×Q8—24no—
417C2×Q8—24no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—24no—
423C2×Q8—42no—
424C2×Q8—44no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×Q8—82no—
428C2×Q8—82no—
429C2×Q8—88no—
430C2×Q8—88no—
431C2×Q8H(5,𝔽2)12yes—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—41no—
450C22×C4—44no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4C2×Q1614yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C2511yescharacteristic
464Q16—48no—
465Q16—88no—
466Q16—88no—
467Q16—88no—
468Q16—88no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8H(5,𝔽2)14yes—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—82no—
506C4⋊C4—81no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C4⋊C4—84no—
514C4⋊C4—84no—
515C4⋊C4C2×Q1614yes—
516C4⋊C4C2×Q1614yes—
517C4⋊C4#13714yes—
518C22⋊C4—24no—
519C22⋊C4—42no—
520C22⋊C4—44no—
521C22⋊C4—81no—
522C22⋊C4—82no—
523C22⋊C4C2×Q1618yes—
524C42—22no—
525C42—22no—
526C42—24no—
527C42—24no—
528C42—22no—
529C42—24no—
530C42—22no—
531C42—22no—
532C42—42no—
533C42—44no—
534C42—42no—
535C42—42no—
536C42—82no—
537C42—82no—
538C42C2×Q1618yes—
539C42H(5,𝔽2)14yes—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23C22×Q1612yes—
547C23C23×D411yescharacteristic
548Q8—24no—
549Q8—42no—
550Q8—42no—
551Q8—44no—
552Q8—42no—
553Q8—44no—
554Q8—44no—
555Q8—44no—
556Q8—44no—
557Q8—42no—
558Q8—42no—
559Q8—44no—
560Q8—84no—
561Q8—88no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—28no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—44no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—88no—
612C2×C4#53512yes—
613C2×C4#53512yes—
614C2×C4C22×Q1612yes—
615C2×C4C22×Q1614yes—
616C2×C4C22×Q1614yes—
617C2×C4C22×Q1612yes—
618C2×C4C2×#13714yes—
619C2×C4C2×#13712yes—
620C2×C4#57812yes—
621C2×C4C2×H(5,𝔽2)11yescharacteristic
622C2×C4C2×H(5,𝔽2)12yes—
623C2×C4C2×H(5,𝔽2)12yes—
624C2×C4#58511yescharacteristic
625C8—24no—
626C8—44no—
627C8—42no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22C2×#53511yescharacteristic
636C22#348811yescharacteristic
637C22#349811yescharacteristic
638C22C23×Q1611yescharacteristic
639C22C22×#13711yescharacteristic
640C22C2×#57811yescharacteristic
641C22#358611yescharacteristic
642C4—24no—
643C4—24no—
644C4—22no—
645C4—22no—
646C4—22no—
647C4—24no—
648C4—41no—
649C4—42no—
650C4—42no—
651C4—81no—
652C4—81no—
653C4—82no—
654C4—84no—
655C4—82no—
656C4—82no—
657C4—81no—
658C4—82no—
659C4#320914yes—
660C4#353712yes—
661C4#354612yes—
662C2—42no—
663C2#4298611yescharacteristic
664C2C2×#320911yescharacteristic
665C2#6081411yescharacteristic
666C2#6195211yescharacteristic
667C2#6198411yescharacteristic
668C2#6241611yescharacteristic
669C2#6245411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
3. #32801C2semidirect
6. #30338C2semidirect
7. #30042C2semidirect
9. #28462C2semidirect
10. #28440C2semidirect
11. #27749C2semidirect
13. #25370C2semidirect
14. #24732C2semidirect
301. #117Q16semidirect

Maximal subgroups

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

subgroupindex
#364812
#352812
#328012
#309902
#304492
#303382
#300422
#296642
#284622
#284402
#277492
#257222
#253702
#247322
#242122
#235712
#226392