Finite Group Databasebeta

← #6447321

Subgroups — #6447321

Order 512.

1782 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#35555C211yesmaximal, characteristic, complemented
2#34268C211yesmaximal, characteristic
3#26544C214yesmaximal, complemented
4#26539C214yesmaximal, complemented
5#26537C214yesmaximal, complemented
6#26528C214yesmaximal, complemented
7#23738C211yesmaximal, characteristic
8C2×#2096C211yesmaximal, characteristic, complemented
9#23619C211yesmaximal, characteristic, complemented
10#23610C211yesmaximal, characteristic
11#21262C211yesmaximal, characteristic
12#21205C211yesmaximal, characteristic
13#21174C211yesmaximal, characteristic, complemented
14#21148C211yesmaximal, characteristic, complemented
15#20880C211yesmaximal, characteristic, complemented
16#20872C211yesmaximal, characteristic
17#20651C211yesmaximal, characteristic, complemented
18#20648C211yesmaximal, characteristic
19#20454C211yesmaximal, characteristic, complemented
20C2×#536C2211yescharacteristic
21C2×#519C2211yescharacteristic
22C2×C4×#41C2211yescharacteristic
23C2×#428C2211yescharacteristic
24C2×#427C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C2×#424C2211yescharacteristic
28C2×#424C2211yescharacteristic
29C2×#423C2211yescharacteristic
30C2×#417C2211yescharacteristic
31C2×#417C2211yescharacteristic
32C2×#419C2211yescharacteristic
33C2×#419C2211yescharacteristic
34C2×#416C2211yescharacteristic
35C2×#416C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×#103C2211yescharacteristic
38C2×#405C2211yescharacteristic
39#2570C2214yes—
40#2552C2214yes—
41#2507—22no—
42#2506—22no—
43#2473—24no—
44#2367C2214yes—
45#2365C2214yes—
46#2359—24no—
47#2323—22no—
48#2317C2211yescharacteristic
49#2317C2211yescharacteristic
50#2316C2211yescharacteristic
51#2311C2211yescharacteristic
52C4×C22⋊Q8—22no—
53#2296C2211yescharacteristic
54#2296C2211yescharacteristic
55C2×#391C2211yescharacteristic
56C2×#384C2211yescharacteristic
57#2274C2211yescharacteristic
58#2274C2211yescharacteristic
59#2097C2212yes—
60#2096C2212yes—
61#2096C2214yes—
62#2096C2214yes—
63#2094C2212yes—
64#2092C2212yes—
65#2068C2214yes—
66#2066C2214yes—
67#2066C2214yes—
68#2065C2214yes—
69#1983C2214yes—
70#1981C2214yes—
71#1960C2212yes—
72#1959C2212yes—
73#1958C2212yes—
74#1956C2212yes—
75#1949C2214yes—
76#1948C2214yes—
77#1940C2214yes—
78#1939C2214yes—
79#1865C2212yes—
80#1865C2214yes—
81#1864C2212yes—
82#1863C2212yes—
83#1863C2214yes—
84#1862C2212yes—
85#1805C2214yes—
86#1803C2214yes—
87#1803C2214yes—
88#1802C2214yes—
89#1776C2214yes—
90C4×#106C2214yes—
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1736C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C22×M16C2311yescharacteristic
100C22×M16C2311yescharacteristic
101C23×C8C2311yescharacteristic
102C23×C8C2311yescharacteristic
103#536—24no—
104#536Q818yes—
105C2×C4⋊Q8C2312yes—
106C2×#125—24no—
107C2×#124C2312yes—
108C2×#123—22no—
109C2×C22⋊Q8—22no—
110#519—24no—
111#519—42no—
112C4×#41—24no—
113C4×#41—42no—
114C2×C4×Q8—22no—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2314yes—
117C2×C4×D4—22no—
118C2×C4×D4C2312yes—
119C2×#117—22no—
120C2×#117—21no—
121C2×#117C2311yescharacteristic
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C2×#117C2311yescharacteristic
126C22×C4⋊C4—22no—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C42—21no—
131C22×C42C2311yescharacteristic
132#428—42no—
133#427—24no—
134C2×#107C2312yes—
135C2×#107C2312yes—
136C2×#106—24no—
137C2×#106—42no—
138C2×#106C2312yes—
139C2×#106C2312yes—
140C2×#106C2312yes—
141C2×#106C2314yes—
142C2×#106C2312yes—
143#424—24no—
144#424—42no—
145#423—42no—
146C2×C4⋊C8C2314yes—
147C2×C4⋊C8C2314yes—
148C2×C4⋊C8C2314yes—
149#419—42no—
150#419—42no—
151#417—42no—
152#417—42no—
153#416—24no—
154#416—42no—
155C2×#103—24no—
156C2×#103—42no—
157C2×#103C2312yes—
158C2×#103C2312yes—
159C2×#103C2311yescharacteristic
160C2×#103C2311yescharacteristic
161C2×#103C2312yes—
162C2×#103C2312yes—
163C2×#103C2312yes—
164C2×#103C2314yes—
165C2×#103C2312yes—
166C2×#103C2312yes—
167C2×#103C2314yes—
168C2×#102C2311yescharacteristic
169C2×#102C2311yescharacteristic
170C2×#102C2312yes—
171C2×#102C2312yes—
172C2×#102C2312yes—
173#405—24no—
174C2×#97C2312yes—
175C2×C4×C8C2312yes—
176#400—24no—
177#398—24no—
178#398—24no—
179#398—24no—
180#391—21no—
181#391—21no—
182#391C2312yes—
183#391C2314yes—
184#390C2312yes—
185#389—22no—
186#389—22no—
187#389—22no—
188#389—24no—
189#389—24no—
190#389D412yes—
191#389D412yes—
192#389C2312yes—
193#389C2314yes—
194#389C2312yes—
195#388—24no—
196#388—24no—
197#387—24no—
198#387—24no—
199#387—24no—
200#387—24no—
201#386—22no—
202#385—22no—
203#385—24no—
204#385—22no—
205#385—22no—
206#385D414yes—
207#384—21no—
208#384—21no—
209#384C2314yes—
210#384C2312yes—
211#384C2312yes—
212#382C2312yes—
213#382C2312yes—
214#380—22no—
215#380—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—24no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4C2314yes—
222C4×C4⋊C4C2314yes—
223C4×C4⋊C4C2314yes—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—24no—
227C2×#95—22no—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#336C2312yes—
234#336C2312yes—
235#336C2312yes—
236#336C2312yes—
237C2×#41—22no—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41C2×Q814yes—
243C22×Q8—21no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8C2×Q814yes—
250C22×Q8C2412yes—
251C22×Q8C2411yescharacteristic
252C22×D4—21no—
253C22×D4C2411yescharacteristic
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—22no—
260C23×C4C2×D412yes—
261C23×C4C2411yescharacteristic
262C2×M16—22no—
263C2×M16—22no—
264C2×M16—22no—
265C2×M16—22no—
266C2×M16—22no—
267C2×M16—22no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8#4114yes—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C22×C8C2411yescharacteristic
278C22×C8C2411yescharacteristic
279C4⋊Q8—24no—
280C4⋊Q8C2×Q818yes—
281#125—88no—
282#124—24no—
283#124C2×Q818yes—
284#123—216no—
285C22⋊Q8—216no—
286C4×Q8—24no—
287C4×Q8—28no—
288C4×Q8—42no—
289C4×Q8—44no—
290C4×Q8—48no—
291C4×D4—24no—
292C4×D4—42no—
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—44no—
302#117—48no—
303#117C2×Q814yes—
304#117#4114yes—
305#117#4114yes—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—84no—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D414yes—
328C2×C4⋊C4C2×D414yes—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2412yes—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C42—21no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—24no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—22no—
352C2×C42—24no—
353C2×C42—24no—
354C2×C42—44no—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2412yes—
358C2×C42C2412yes—
359C2×C42C2412yes—
360#107—24no—
361#107—42no—
362#106—24no—
363#106—24no—
364#106—24no—
365#106—44no—
366#106—42no—
367C4⋊C8—28no—
368C4⋊C8—44no—
369C4⋊C8—44no—
370#103—24no—
371#103—24no—
372#103—22no—
373#103—22no—
374#103—44no—
375#103—42no—
376#103—42no—
377#103—42no—
378#103—44no—
379#103—42no—
380#103—42no—
381#102—22no—
382#102—22no—
383#102—42no—
384#102—42no—
385#102—42no—
386#97—24no—
387C4×C8—24no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95—24no—
392#95—24no—
393#95—22no—
394#95—22no—
395#95—24no—
396#95—24no—
397#95—24no—
398#95C2×D414yes—
399#95C2×D414yes—
400C24—21no—
401C24C22×D411yescharacteristic
402#41—28no—
403#41—28no—
404#41—44no—
405#41—44no—
406C2×Q8—24no—
407C2×Q8—22no—
408C2×Q8—21no—
409C2×Q8—28no—
410C2×Q8—28no—
411C2×Q8—22no—
412C2×Q8—21no—
413C2×Q8—22no—
414C2×Q8—21no—
415C2×Q8—22no—
416C2×Q8—21no—
417C2×Q8—22no—
418C2×Q8—21no—
419C2×Q8—44no—
420C2×Q8—44no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8C22×Q814yes—
424C2×Q8C22×Q812yes—
425C2×D4—24no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4C22×Q812yes—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4C22×D414yes—
462C22×C4C22×D412yes—
463C22×C4C22×D414yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×Q812yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4H(5,𝔽2)12yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C4C2511yescharacteristic
473M16—44no—
474M16—44no—
475C2×C8—28no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—44no—
490C2×C8C2×#4112yes—
491C2×C8C2×#4112yes—
492C4⋊C4—216no—
493C4⋊C4—24no—
494C4⋊C4—28no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—48no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—48no—
516C4⋊C4—44no—
517C4⋊C4—84no—
518C4⋊C4—84no—
519C4⋊C4—84no—
520C4⋊C4—84no—
521C4⋊C4C22⋊Q818yes—
522C4⋊C4C2×#4114yes—
523C22⋊C4—24no—
524C22⋊C4—24no—
525C22⋊C4—216no—
526C22⋊C4—44no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—84no—
530C22⋊C4C22⋊Q818yes—
531C42—22no—
532C42—22no—
533C42—24no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—44no—
539C42—48no—
540C42—42no—
541C42—42no—
542C42—84no—
543C42—84no—
544C42C22×Q814yes—
545C42C2×#4114yes—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—42no—
550C23—41no—
551C23—41no—
552C23—42no—
553C23C2×C22⋊Q812yes—
554C23#54612yes—
555C23#55012yes—
556C23C23×D411yescharacteristic
557Q8—24no—
558Q8—24no—
559Q8—22no—
560Q8—22no—
561Q8—42no—
562Q8—41no—
563Q8—42no—
564Q8—41no—
565Q8—48no—
566D4—22no—
567D4—22no—
568D4—48no—
569D4—41no—
570D4—41no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—48no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—48no—
591C2×C4—42no—
592C2×C4—48no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—88no—
607C2×C4—162no—
608C2×C4—162no—
609C2×C4—162no—
610C2×C4C2×C22⋊Q818yes—
611C2×C4C2×C22⋊Q812yes—
612C2×C4#52914yes—
613C2×C4#54014yes—
614C2×C4#54612yes—
615C2×C4#55012yes—
616C2×C4#55414yes—
617C2×C4#55414yes—
618C2×C4#57714yes—
619C2×C4C23×Q811yescharacteristic
620C2×C4C22×#4111yescharacteristic
621C2×C4C22×#4111yescharacteristic
622C2×C4C22×#4111yescharacteristic
623C2×C4C2×H(5,𝔽2)11yescharacteristic
624C2×C4C2×H(5,𝔽2)11yescharacteristic
625C8—22no—
626C8—22no—
627C8—42no—
628C8—42no—
629C8—42no—
630C22—24no—
631C22—24no—
632C22—22no—
633C22—22no—
634C22—44no—
635C22—42no—
636C22—82no—
637C22—81no—
638C22—81no—
639C22#245414yes—
640C22C22×C22⋊Q811yescharacteristic
641C22C2×#54611yescharacteristic
642C22C2×#55011yescharacteristic
643C22C2×#57712yes—
644C22#358612yes—
645C4—24no—
646C4—28no—
647C4—44no—
648C4—42no—
649C4—44no—
650C4—42no—
651C4—44no—
652C4—82no—
653C4—81no—
654C4—82no—
655C4—82no—
656C4—81no—
657C4—82no—
658C4#245414yes—
659C4C2×#52911yescharacteristic
660C4C2×#54011yescharacteristic
661C4C2×#55411yescharacteristic
662C4C2×#55411yescharacteristic
663C2—24no—
664C2—82no—
665C2C2×#245411yescharacteristic
666C2#6049712yes—
667C2#6119312yes—
668C2#6141011yescharacteristic
669C2#6141811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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. #26544C2semidirect
4. #26539C2semidirect
5. #26537C2semidirect
6. #26528C2semidirect
8. C2×#2096C2semidirect
9. #23619C2semidirect
13. #21174C2semidirect
14. #21148C2semidirect
15. #20880C2semidirect
17. #20651C2semidirect
19. #20454C2semidirect

Maximal subgroups

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

subgroupindex
#355552
#342682
#265442
#265392
#265372
#265282
#237382
C2×#20962
#236192
#236102
#212622
#212052
#211742
#211482
#208802
#208722
#206512
#206482
#204542