Finite Group Databasebeta

← #6461280

Subgroups — #6461280

Order 512.

1892 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35334C211yesmaximal, characteristic
2#34268C211yesmaximal, characteristic
3#29171C214yesmaximal, complemented
4#29170C214yesmaximal, complemented
5#29164C214yesmaximal, complemented
6#29158C214yesmaximal, complemented
7#23746C211yesmaximal, characteristic, complemented
8C2×#2097C211yesmaximal, characteristic, complemented
9#23551C211yesmaximal, characteristic, complemented
10#23455C211yesmaximal, characteristic, complemented
11#21038C211yesmaximal, characteristic, complemented
12#21037C211yesmaximal, characteristic, complemented
13#20974C211yesmaximal, characteristic
14#20974C211yesmaximal, characteristic
15#20880C211yesmaximal, characteristic, complemented
16#20880C211yesmaximal, characteristic, complemented
17#20663C211yesmaximal, characteristic
18#20662C211yesmaximal, characteristic
19#20336C211yesmaximal, characteristic
20C22×#131C2211yescharacteristic
21C22×#131C2211yescharacteristic
22C2×#536C2211yescharacteristic
23C2×#533C2211yescharacteristic
24C2×#519C2211yescharacteristic
25C2×C4×#41C2211yescharacteristic
26C2×#427C2211yescharacteristic
27C2×#427C2211yescharacteristic
28C22×#107C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#419C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34#2851C2214yes—
35#2628C2214yes—
36#2581—22no—
37#2580—22no—
38#2564—24no—
39#2367—22no—
40#2365—22no—
41#2359—24no—
42#2323C2214yes—
43#2316C2211yescharacteristic
44#2316C2211yescharacteristic
45#2311C2211yescharacteristic
46#2311C2211yescharacteristic
47#2311C2211yescharacteristic
48C4×C22⋊Q8C2214yes—
49#2296C2211yescharacteristic
50C2×#391C2211yescharacteristic
51C2×#382C2211yescharacteristic
52#2274C2211yescharacteristic
53#2101C2214yes—
54#2101C2214yes—
55#2097C2212yes—
56#2097C2212yes—
57#2097C2212yes—
58#2097C2212yes—
59#2059C2212yes—
60#2059C2212yes—
61#2058C2212yes—
62#2057C2212yes—
63#2026C2214yes—
64#2024C2214yes—
65#1921C2214yes—
66#1920C2214yes—
67#1919C2214yes—
68#1918C2214yes—
69#1894C2214yes—
70#1894C2214yes—
71#1892C2214yes—
72#1892C2214yes—
73#1865C2214yes—
74#1865C2214yes—
75#1863C2214yes—
76#1863C2214yes—
77#1809C2214yes—
78#1809C2214yes—
79#1808C2214yes—
80#1807C2214yes—
81#1736C2211yescharacteristic
82#1736C2211yescharacteristic
83#1736C2211yescharacteristic
84#1736C2211yescharacteristic
85#1736C2211yescharacteristic
86#1736C2211yescharacteristic
87#1735C2211yescharacteristic
88#1735C2211yescharacteristic
89#1729C2211yescharacteristic
90#1729C2211yescharacteristic
91#1729C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94#1728C2212yes—
95C2×#336C2211yescharacteristic
96C2×#336C2212yes—
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C2×#131—24no—
100C2×#131—24no—
101C2×#131—24no—
102C2×#131—24no—
103C2×#131—42no—
104C2×#131—42no—
105C22×M16C2311yescharacteristic
106C22×M16C2311yescharacteristic
107C22×M16C2312yes—
108C22×M16C2312yes—
109C23×C8C2312yes—
110C23×C8C2312yes—
111C23×C8C2311yescharacteristic
112C23×C8C2311yescharacteristic
113#536—24no—
114#536Q818yescomplemented
115#533—42no—
116#533—42no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8C2312yes—
119C2×#125C2312yes—
120C2×#124—21no—
121C2×#123—22no—
122C2×C22⋊Q8—22no—
123C2×C22⋊Q8C2314yes—
124#519—42no—
125#519—42no—
126C4×#41—42no—
127C4×#41—42no—
128C2×C4×Q8—22no—
129C2×C4×Q8—22no—
130C2×C4×Q8D412yes—
131C2×C4×Q8C2314yes—
132C2×C4×D4D414yes—
133C2×C4×D4D412yes—
134C2×#117—22no—
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C2×#117C2312yes—
141C22×C4⋊C4C2311yescharacteristic
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2311yescharacteristic
144C22×C42C2311yescharacteristic
145C22×C42C2312yes—
146#427—42no—
147#427—42no—
148C2×#107—24no—
149C2×#107C2312yes—
150C2×#107C2312yes—
151C2×#107C2312yes—
152C2×#107C2312yes—
153C2×#106—24no—
154C2×#106C2312yes—
155C2×#106C2312yes—
156C2×#106C2312yes—
157C2×#106C2312yes—
158#419—42no—
159#419—42no—
160C2×#103—24no—
161C2×#103—24no—
162C2×#103C2312yes—
163C2×#103C2312yes—
164C2×#103C2312yes—
165C2×#103C2312yes—
166C2×#103C2314yes—
167C2×#103C2314yes—
168#400—24no—
169#399—24no—
170#395—24no—
171#393—24no—
172#391—21no—
173#391—21no—
174#391—21no—
175#391—22no—
176#391C2312yes—
177#391C2312yes—
178#390—21no—
179#390—21no—
180#390—21no—
181#389—22no—
182#389—22no—
183#389—24no—
184#389C2312yes—
185#389C2312yes—
186#389C2314yes—
187#389C2314yes—
188#388—24no—
189#388—24no—
190#387—24no—
191#387—24no—
192#387—24no—
193#387—24no—
194#386—22no—
195#386—22no—
196#386—22no—
197#385—24no—
198#385—24no—
199#385—22no—
200#385—24no—
201#382—22no—
202#382C2312yes—
203#382C2312yes—
204#380—22no—
205#380—22no—
206#380C2314yes—
207#380C2314yes—
208#380C2314yes—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—24no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4C2314yes—
213C4×C22⋊C4—24no—
214C4×C22⋊C4—22no—
215C2×#95—22no—
216C2×#95—22no—
217#340C2312yes—
218#340C2312yes—
219#340C2312yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#336C2311yescharacteristic
228#336C2311yescharacteristic
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2312yes—
233#336C2312yes—
234#336C2312yes—
235#336C2312yes—
236#336C2312yes—
237C2×#41—22no—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41C2×Q814yes—
242C2×#41#4114yes—
243C22×Q8—21no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8C2×D411yescharacteristic
251C22×Q8C2×Q814yes—
252C22×Q8C2412yes—
253C22×D4C2×D412yes—
254C22×D4C2×D411yescharacteristic
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4C2×D412yes—
260C23×C4C2×D414yes—
261C23×C4C2412yes—
262C23×C4C2411yescharacteristic
263#131—416no—
264#131—416no—
265C2×M16—24no—
266C2×M16—24no—
267C2×M16—22no—
268C2×M16—22no—
269C2×M16—22no—
270C2×M16—22no—
271C2×M16—22no—
272C2×M16—22no—
273C2×M16—24no—
274C2×M16—24no—
275C2×M16—42no—
276C2×M16—42no—
277C22×C8—22no—
278C22×C8—22no—
279C22×C8—22no—
280C22×C8—22no—
281C22×C8—24no—
282C22×C8—24no—
283C22×C8—24no—
284C22×C8—24no—
285C22×C8—42no—
286C22×C8—42no—
287C22×C8#4114yes—
288C22×C8#4114yes—
289C22×C8C2412yes—
290C22×C8C2412yes—
291C22×C8C2411yescharacteristic
292C22×C8C2411yescharacteristic
293C22×C8C2411yescharacteristic
294C22×C8C2411yescharacteristic
295C4⋊Q8—28no—
296C4⋊Q8—42no—
297C4⋊Q8—42no—
298#125—42no—
299#125—42no—
300#124—28no—
301#123—216no—
302C22⋊Q8—28no—
303C22⋊Q8—48no—
304C22⋊Q8C2×Q8116yes—
305C4×Q8—44no—
306C4×Q8—44no—
307C4×Q8—44no—
308C4×Q8—48no—
309C4×Q8—48no—
310C4×D4—44no—
311C4×D4—48no—
312#117—22no—
313#117—22no—
314#117—22no—
315#117—22no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—22no—
321#117—42no—
322#117—42no—
323#117—48no—
324#117C2×Q814yes—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4C2×D412yes—
344C2×C4⋊C4C2×D414yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4C2×D412yes—
361C2×C22⋊C4C2×D414yes—
362C2×C42—21no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—24no—
366C2×C42—21no—
367C2×C42—21no—
368C2×C42—21no—
369C2×C42—24no—
370C2×C42—42no—
371C2×C42—42no—
372C2×C42C2×D412yes—
373C2×C42C2×D414yes—
374C2×C42C2412yes—
375C2×C42C2411yescharacteristic
376C2×C42C2411yescharacteristic
377#107—24no—
378#107—24no—
379#107—42no—
380#107—42no—
381#106—24no—
382#106—24no—
383#106—42no—
384#106—42no—
385#103—24no—
386#103—24no—
387#103—24no—
388#103—24no—
389#103—44no—
390#103—44no—
391#95—24no—
392#95—24no—
393#95—24no—
394#95—24no—
395#95—22no—
396#95—24no—
397#95—24no—
398#95—22no—
399#95—24no—
400#95—24no—
401#95—24no—
402#95—24no—
403C24C22×D412yes—
404C24C22×D411yescharacteristic
405#41—28no—
406#41—28no—
407#41—28no—
408#41—28no—
409C2×Q8—24no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—28no—
414C2×Q8—28no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—44no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—44no—
423C2×Q8—44no—
424C2×Q8—82no—
425C2×Q8—82no—
426C2×Q8C22⋊Q814yes—
427C2×Q8C22×Q814yes—
428C2×D4—24no—
429C2×D4—24no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4C22⋊Q818yes—
433C2×D4C22⋊Q814yes—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D414yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×Q814yes—
467C22×C4C22×Q812yes—
468C22×C4C2×#4114yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4H(5,𝔽2)12yes—
473C22×C4H(5,𝔽2)12yes—
474C22×C4C2511yescharacteristic
475M16—48no—
476M16—48no—
477M16—44no—
478M16—44no—
479C2×C8—28no—
480C2×C8—28no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—48no—
496C2×C8—48no—
497C2×C8C2×#4112yes—
498C2×C8C2×#4112yes—
499C2×C8C2×#4112yes—
500C2×C8C2×#4112yes—
501C4⋊C4—216no—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—48no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—84no—
526C4⋊C4—82no—
527C4⋊C4—84no—
528C4⋊C4C22×Q814yes—
529C22⋊C4—24no—
530C22⋊C4—216no—
531C22⋊C4—24no—
532C22⋊C4—44no—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—84no—
537C22⋊C4—84no—
538C22⋊C4—84no—
539C42—24no—
540C42—24no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—22no—
546C42—22no—
547C42—44no—
548C42—48no—
549C42—42no—
550C42—42no—
551C42—42no—
552C42—84no—
553C23—22no—
554C23—22no—
555C23—24no—
556C23—24no—
557C23—22no—
558C23—22no—
559C23C2×C22⋊Q814yes—
560C23C2×C22⋊Q812yes—
561C23#53212yes—
562C23#53412yes—
563C23C23×D411yescharacteristic
564Q8—28no—
565Q8—24no—
566Q8—24no—
567Q8—48no—
568Q8—42no—
569Q8—42no—
570Q8—88no—
571D4—216no—
572D4—28no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—48no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—84no—
611C2×C4—88no—
612C2×C4—162no—
613C2×C4—162no—
614C2×C4—162no—
615C2×C4C2×C22⋊Q814yes—
616C2×C4C2×C22⋊Q812yes—
617C2×C4#53212yes—
618C2×C4#53412yes—
619C2×C4#55114yes—
620C2×C4#55114yes—
621C2×C4#55414yes—
622C2×C4#55414yes—
623C2×C4C23×Q811yescharacteristic
624C2×C4C22×#4111yescharacteristic
625C2×C4C22×#4111yescharacteristic
626C2×C4C22×#4111yescharacteristic
627C2×C4C2×H(5,𝔽2)11yescharacteristic
628C2×C4C2×H(5,𝔽2)11yescharacteristic
629C8—22no—
630C8—22no—
631C8—22no—
632C8—22no—
633C8—42no—
634C8—42no—
635C22—22no—
636C22—22no—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—28no—
641C22—42no—
642C22—42no—
643C22—44no—
644C22#256414yes—
645C22C22×C22⋊Q811yescharacteristic
646C22C2×#53211yescharacteristic
647C22C2×#53411yescharacteristic
648C22#358412yes—
649C22#358612yes—
650C4—24no—
651C4—44no—
652C4—44no—
653C4—44no—
654C4—42no—
655C4—42no—
656C4—82no—
657C4—82no—
658C4—82no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4#256414yes—
663C4C2×#55111yescharacteristic
664C4C2×#55111yescharacteristic
665C4C2×#55411yescharacteristic
666C4C2×#55411yescharacteristic
667C2—24no—
668C2—44no—
669C2C2×#256411yescharacteristic
670C2#6049912yes—
671C2#6071311yescharacteristic
672C2#6072011yescharacteristic
673C2#6076612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29171C2semidirect
4. #29170C2semidirect
5. #29164C2semidirect
6. #29158C2semidirect
7. #23746C2semidirect
8. C2×#2097C2semidirect
9. #23551C2semidirect
10. #23455C2semidirect
11. #21038C2semidirect
12. #21037C2semidirect
15. #20880C2semidirect
16. #20880C2semidirect
114. #536Q8semidirect

Maximal subgroups

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

subgroupindex
#353342
#342682
#291712
#291702
#291642
#291582
#237462
C2×#20972
#235512
#234552
#210382
#210372
#209742
#208802
#206632
#206622
#203362