Finite Group Databasebeta

← #6458259

Subgroups — #6458259

Order 512.

1896 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34268C211yesmaximal, characteristic
2#34184C211yesmaximal, characteristic
3#27690C214yesmaximal, complemented
4#27689C214yesmaximal, complemented
5#27686C214yesmaximal, complemented
6#27682C214yesmaximal, complemented
7#23746C211yesmaximal, characteristic, complemented
8#23744C211yesmaximal, characteristic, complemented
9#23732C211yesmaximal, characteristic, complemented
10C2×#2097C211yesmaximal, characteristic, complemented
11#21174C211yesmaximal, characteristic, complemented
12#21173C211yesmaximal, characteristic, complemented
13#20913C211yesmaximal, characteristic
14#20913C211yesmaximal, characteristic
15#20848C211yesmaximal, characteristic
16#20847C211yesmaximal, characteristic
17#20651C211yesmaximal, characteristic, complemented
18#20651C211yesmaximal, characteristic, complemented
19#20630C211yesmaximal, characteristic
20C2×#536C2211yescharacteristic
21C2×#520C2211yescharacteristic
22C2×#520C2211yescharacteristic
23C2×#519C2211yescharacteristic
24C2×C4×#41C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C2×#428C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C2×#424C2211yescharacteristic
30C2×#424C2211yescharacteristic
31C2×#419C2211yescharacteristic
32C2×#419C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C2×#408C2211yescharacteristic
36C2×#408C2211yescharacteristic
37#2367—22no—
38#2365—22no—
39#2359—24no—
40#2351—22no—
41#2349—24no—
42#2349—22no—
43#2323C2214yes—
44#2317C2212yes—
45#2317C2212yes—
46#2317C2211yescharacteristic
47#2317C2211yescharacteristic
48#2311C2211yescharacteristic
49C4×C22⋊Q8C2214yes—
50#2296C2211yescharacteristic
51C2×#391C2211yescharacteristic
52#2285C2212yes—
53#2285C2211yescharacteristic
54C2×#382C2212yes—
55C2×#382C2211yescharacteristic
56#2274C2211yescharacteristic
57#2101C2212yes—
58#2101C2214yes—
59#2101C2212yes—
60#2101C2214yes—
61#2100C2212yes—
62#2099C2212yes—
63#2097C2212yes—
64#2097C2214yes—
65#2097C2212yes—
66#2097C2214yes—
67#2096C2212yes—
68#2095C2212yes—
69#1949C2214yes—
70#1949C2214yes—
71#1948C2214yes—
72#1947C2214yes—
73#1873C2214yes—
74#1873C2214yes—
75#1872C2214yes—
76#1872C2214yes—
77#1854C2214yes—
78#1854C2214yes—
79#1853C2214yes—
80#1852C2214yes—
81#1805C2214yes—
82#1805C2214yes—
83#1803C2214yes—
84#1803C2214yes—
85#1798C2214yes—
86#1797C2214yes—
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C22×M16C2311yescharacteristic
100C22×M16C2311yescharacteristic
101C23×C8C2311yescharacteristic
102C23×C8C2311yescharacteristic
103#536—24no—
104#536Q818yes—
105C2×C4⋊Q8—21no—
106C2×#124—21no—
107C2×#123—22no—
108C2×C22⋊Q8C2314yes—
109#520—42no—
110#520—42no—
111#520—42no—
112#520—42no—
113#519—24no—
114#519—24no—
115C4×#41—24no—
116C4×#41—24no—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8—22no—
120C2×C4×Q8D412yes—
121C2×C4×Q8C2314yes—
122C2×C4×D4—22no—
123C2×C4×D4—22no—
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4D414yes—
127C2×C4×D4D412yes—
128C2×#117—22no—
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2312yes—
134C2×#117C2312yes—
135C2×#117C2312yes—
136C2×#117C2311yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2312yes—
141C22×C4⋊C4C2312yes—
142C22×C42C2312yes—
143C22×C42C2311yescharacteristic
144#428—42no—
145#428—42no—
146C2×#107—42no—
147C2×#107C2312yes—
148C2×#107C2312yes—
149C2×#107C2314yes—
150C2×#106—42no—
151C2×#106C2312yes—
152C2×#106C2312yes—
153C2×#106C2314yes—
154#424—24no—
155#424—24no—
156C2×C4⋊C8C2314yes—
157C2×C4⋊C8C2314yes—
158#419—42no—
159#419—42no—
160C2×#103—42no—
161C2×#103—42no—
162C2×#103C2312yes—
163C2×#103C2312yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#103C2312yes—
167C2×#103C2312yes—
168#408—24no—
169#408—24no—
170C2×C22⋊C8C2314yes—
171C2×C22⋊C8C2314yes—
172#391—21no—
173#391—22no—
174#391C2312yes—
175#391C2312yes—
176#390—21no—
177#389—24no—
178#389—24no—
179#389—22no—
180#389—22no—
181#389—24no—
182#389C2312yes—
183#389C2312yes—
184#388—24no—
185#388—24no—
186#387—24no—
187#387—24no—
188#386—22no—
189#385—22no—
190#385—24no—
191#385—24no—
192#385—24no—
193#385—22no—
194#385—22no—
195#384C2312yes—
196#384C2312yes—
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#382—24no—
200#382—22no—
201#382C2312yes—
202#382C2312yes—
203#382C2312yes—
204#382C2312yes—
205#382C2311yescharacteristic
206#382C2311yescharacteristic
207#380—22no—
208#380—24no—
209#380C2314yes—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—24no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—24no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4C2314yes—
216C4×C22⋊C4—24no—
217C4×C22⋊C4—22no—
218C2×#95—22no—
219C2×#95—24no—
220C2×#95—22no—
221#340C2312yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
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×Q8C2×D411yescharacteristic
250C22×Q8C2×Q814yes—
251C22×Q8C2412yes—
252C22×D4C2×D412yes—
253C22×D4C2×D411yescharacteristic
254C23×C4—21no—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4C2×D412yes—
261C23×C4C2×D414yes—
262C23×C4C2412yes—
263C23×C4C2411yescharacteristic
264C2×M16—22no—
265C2×M16—22no—
266C2×M16—22no—
267C2×M16—22no—
268C2×M16—22no—
269C2×M16—22no—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8—22no—
274C22×C8—22no—
275C22×C8—22no—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C4⋊Q8—28no—
281#124—28no—
282#123—216no—
283C22⋊Q8—28no—
284C22⋊Q8C2×Q8116yes—
285C4×Q8—28no—
286C4×Q8—28no—
287C4×Q8—28no—
288C4×Q8—44no—
289C4×Q8—44no—
290C4×Q8—48no—
291C4×D4—28no—
292C4×D4—216no—
293C4×D4—48no—
294C4×D4—48no—
295C4×D4—44no—
296C4×D4—44no—
297#117—22no—
298#117—24no—
299#117—24no—
300#117—22no—
301#117—22no—
302#117—22no—
303#117—22no—
304#117—22no—
305#117—22no—
306#117—42no—
307#117—42no—
308#117—42no—
309#117—42no—
310#117—48no—
311#117C2×Q814yes—
312#117#4114yes—
313#117#4114yes—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—42no—
334C2×C4⋊C4—42no—
335C2×C4⋊C4—42no—
336C2×C4⋊C4—42no—
337C2×C4⋊C4C2×D414yes—
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2411yescharacteristic
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4C2×D414yes—
357C2×C22⋊C4C2×D412yes—
358C2×C42—21no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—24no—
362C2×C42—24no—
363C2×C42—24no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—21no—
367C2×C42—21no—
368C2×C42—21no—
369C2×C42—24no—
370C2×C42C2×D414yes—
371C2×C42C2×D412yes—
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374C2×C42C2412yes—
375#107—44no—
376#107—42no—
377#107—42no—
378#106—44no—
379#106—42no—
380#106—42no—
381C4⋊C8—28no—
382C4⋊C8—28no—
383#103—44no—
384#103—44no—
385#103—42no—
386#103—42no—
387#103—42no—
388#103—42no—
389C22⋊C8—28no—
390C22⋊C8—28no—
391#95—24no—
392#95—22no—
393#95—24no—
394#95—24no—
395#95—24no—
396#95—24no—
397#95—24no—
398#95—22no—
399#95—24no—
400#95—22no—
401#95—24no—
402#95—22no—
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—42no—
420C2×Q8—42no—
421C2×Q8—44no—
422C2×Q8—44no—
423C2×Q8C22⋊Q814yes—
424C2×Q8C22×Q814yes—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4C22⋊Q818yes—
430C2×D4C22⋊Q814yes—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—28no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4—84no—
462C22×C4—82no—
463C22×C4—84no—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D414yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×Q814yes—
470C22×C4C22×Q812yes—
471C22×C4C2×#4114yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C22×C4C2×#4112yes—
475C22×C4H(5,𝔽2)12yes—
476C22×C4H(5,𝔽2)12yes—
477C22×C4C2511yescharacteristic
478M16—44no—
479M16—44no—
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—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—44no—
493C2×C8—44no—
494C4⋊C4—216no—
495C4⋊C4—28no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—48no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—84no—
521C4⋊C4—84no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4—82no—
525C4⋊C4#12318yes—
526C4⋊C4C22×Q814yes—
527C22⋊C4—24no—
528C22⋊C4—216no—
529C22⋊C4—28no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C22⋊C4—84no—
536C22⋊C4#12318yes—
537C42—24no—
538C42—28no—
539C42—24no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—44no—
545C42—44no—
546C42—42no—
547C42—48no—
548C42—42no—
549C42—42no—
550C42—84no—
551C42C2×#4114yes—
552C42C2×#4114yes—
553C23—22no—
554C23—22no—
555C23—24no—
556C23—24no—
557C23—22no—
558C23—22no—
559C23C2×C22⋊Q814yes—
560C23C2×C22⋊Q812yes—
561C23C2×#12312yes—
562C23#53512yes—
563C23C23×D411yescharacteristic
564Q8—28no—
565Q8—24no—
566Q8—24no—
567Q8—42no—
568Q8—42no—
569Q8—48no—
570D4—216no—
571D4—28no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—48no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—88no—
611C2×C4—84no—
612C2×C4—162no—
613C2×C4—162no—
614C2×C4—162no—
615C2×C4—164no—
616C2×C4—164no—
617C2×C4C2×C22⋊Q814yes—
618C2×C4C2×C22⋊Q812yes—
619C2×C4C2×#12318yes—
620C2×C4C2×#12312yes—
621C2×C4#53512yes—
622C2×C4#55314yes—
623C2×C4#55314yes—
624C2×C4#55414yes—
625C2×C4#55414yes—
626C2×C4C23×Q811yescharacteristic
627C2×C4C22×#4111yescharacteristic
628C2×C4C22×#4111yescharacteristic
629C2×C4C22×#4111yescharacteristic
630C2×C4C2×H(5,𝔽2)11yescharacteristic
631C2×C4C2×H(5,𝔽2)11yescharacteristic
632C8—42no—
633C8—42no—
634C8—42no—
635C8—42no—
636C22—24no—
637C22—24no—
638C22—22no—
639C22—22no—
640C22—28no—
641C22—24no—
642C22—44no—
643C22—42no—
644C22—42no—
645C22#248714yes—
646C22C22×C22⋊Q811yescharacteristic
647C22C22×#12311yescharacteristic
648C22C2×#53511yescharacteristic
649C22#358412yes—
650C22#358612yes—
651C4—28no—
652C4—24no—
653C4—44no—
654C4—42no—
655C4—42no—
656C4—44no—
657C4—82no—
658C4—81no—
659C4—82no—
660C4—82no—
661C4—81no—
662C4—81no—
663C4—82no—
664C4—81no—
665C4#248714yes—
666C4C2×#55311yescharacteristic
667C4C2×#55311yescharacteristic
668C4C2×#55411yescharacteristic
669C4C2×#55411yescharacteristic
670C2—24no—
671C2—44no—
672C2C2×#248711yescharacteristic
673C2#6049912yes—
674C2#6052412yes—
675C2#6081911yescharacteristic
676C2#6082611yescharacteristic
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. #27690C2semidirect
4. #27689C2semidirect
5. #27686C2semidirect
6. #27682C2semidirect
7. #23746C2semidirect
8. #23744C2semidirect
9. #23732C2semidirect
10. C2×#2097C2semidirect
11. #21174C2semidirect
12. #21173C2semidirect
17. #20651C2semidirect
18. #20651C2semidirect

Maximal subgroups

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

subgroupindex
#342682
#341842
#276902
#276892
#276862
#276822
#237462
#237442
#237322
C2×#20972
#211742
#211732
#209132
#208482
#208472
#206512
#206302