Finite Group Databasebeta

← #6989172

Subgroups — #6989172

Order 512.

1707 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36274C211yesmaximal, characteristic
2#35316C211yesmaximal, characteristic
3#32740C212yesmaximal, complemented
4#31772C214yesmaximal, complemented
5#31668C214yesmaximal, complemented
6#30359C212yesmaximal, complemented
7#29951C212yesmaximal, complemented
8#29922C211yesmaximal, characteristic, complemented
9#29105C212yesmaximal, complemented
10#28639C212yesmaximal, complemented
11#26298C212yesmaximal, complemented
12#25902C211yesmaximal, characteristic, complemented
13#25884C212yesmaximal, complemented
14#25472C212yesmaximal, complemented
15#24301C211yesmaximal, characteristic, complemented
16#23756C211yesmaximal, characteristic, complemented
17#22530C211yesmaximal, characteristic
18C2×D4×Q8C2211yescharacteristic
19C2×#536—21no—
20C22×C4⋊Q8C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22C2×#487C2212yes—
23C2×#475C2212yes—
24C2×C8×Q8C2212yes—
25#2857C2212yes—
26#2857C2212yes—
27#2857C2211yescharacteristic
28#2855C2212yes—
29#2621—22no—
30#2582C2211yescharacteristic
31#2568C2211yescharacteristic
32#2547—21no—
33#2547C2212yes—
34#2546—21no—
35#2546C2212yes—
36#2486—22no—
37#2486C2214yes—
38#2451—21no—
39#2450—22no—
40#2439—21no—
41#2438—22no—
42#2417C2212yes—
43#2409C2212yes—
44#2349—21no—
45D4×C4⋊C4C2211yescharacteristic
46#2339—21no—
47#2310—21no—
48C4×C22⋊Q8C2211yescharacteristic
49C2×#391C2212yes—
50#2285C2211yescharacteristic
51#2099C2211yescharacteristic
52#2096C2212yes—
53#2096C2211yescharacteristic
54#2088C2214yes—
55#2084C2211yescharacteristic
56#2077C2212yes—
57#2077C2214yes—
58#2077C2214yes—
59#2072C2211yescharacteristic
60#2072C2212yes—
61#2066C2214yes—
62#2066C2214yes—
63#2066C2214yes—
64#2058C2212yes—
65#2035C2214yes—
66#2035C2214yes—
67#2035C2214yes—
68#2035C2212yes—
69#2024C2212yes—
70#2024C2212yes—
71#2024C2212yes—
72#1990C2212yes—
73#1988C2212yes—
74#1981C2212yes—
75#1954C2212yes—
76#1954C2214yes—
77#1948C2212yes—
78#1941C2214yes—
79#1933C2214yes—
80#1933C2214yes—
81#1933C2214yes—
82#1872C2214yes—
83#1872C2212yes—
84#1853C2212yes—
85#1853C2214yes—
86#1824C2214yes—
87#1797C2211yescharacteristic
88#1797C2212yes—
89#1787C2214yes—
90#1787C2214yes—
91C4×#106C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C4×C22⋊C8C2212yes—
95C22×#41D411yescharacteristic
96C23×Q8—21no—
97C23×Q8C2312yes—
98D4×Q8—44no—
99#536—48no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—44no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×#124—22no—
109C2×#123—24no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8C2311yescharacteristic
114C2×C22⋊Q8C2312yes—
115C2×C22⋊Q8C2312yes—
116C4×#41—44no—
117C2×C4×Q8D411yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4D411yescharacteristic
123C2×C4×D4D412yes—
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2312yes—
126C2×#117—21no—
127C2×#117D411yescharacteristic
128C2×#117D412yes—
129C22×C4⋊C4—21no—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134#487—44no—
135#475—44no—
136C8×Q8—44no—
137C2×#106C2314yes—
138C2×#106C2312yes—
139C2×#106C2312yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2314yes—
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2314yes—
153C2×C22⋊C8C2312yes—
154C2×C4×C8C2312yes—
155C2×C4×C8C2314yes—
156#400—22no—
157#398—22no—
158#398—24no—
159#398—24no—
160#398—22no—
161#397—24no—
162#397—24no—
163#396—22no—
164#395—22no—
165#393—22no—
166#391—21no—
167#391—21no—
168#391—22no—
169#391—22no—
170#391—22no—
171#391—22no—
172#391—22no—
173#391—22no—
174#391C2312yes—
175#391C2312yes—
176#391C2312yes—
177#391C2312yes—
178#391C2311yescharacteristic
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—22no—
184#389—22no—
185#389C2311yescharacteristic
186#389C2314yes—
187#389C2314yes—
188#389C2312yes—
189#387—22no—
190#387—22no—
191#387—21no—
192#386—22no—
193#386—22no—
194#385—22no—
195#385—21no—
196#385—22no—
197#385—22no—
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#384C2311yescharacteristic
201#382—21no—
202#382—21no—
203#381—22no—
204#381—24no—
205#380—21no—
206#380—21no—
207#380—22no—
208#380—22no—
209#380—21no—
210#380—22no—
211#380C2312yes—
212#380C2311yescharacteristic
213#380C2314yes—
214#380C2314yes—
215#380C2312yes—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C22⋊C4—21no—
221C2×#95—22no—
222C2×#95—22no—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2311yescharacteristic
234#336C2311yescharacteristic
235C2×#41—42no—
236C2×#41—42no—
237C2×#41SD1618yes—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—24no—
242C22×Q8—44no—
243C22×Q8—42no—
244C22×Q8—44no—
245C22×Q8—44no—
246C22×Q8—44no—
247C22×Q8—44no—
248C22×Q8C2×D411yescharacteristic
249C22×Q8C2412yes—
250C22×Q8C2411yescharacteristic
251C22×D4C2×D411yescharacteristic
252C22×D4C2×D412yes—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D412yes—
259C23×C4C2×D412yes—
260C23×C4C2411yescharacteristic
261C23×C4C2412yes—
262C22×C8C2412yes—
263C22×C8C2412yes—
264C22×C8C2414yes—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8—24no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271C4⋊Q8—44no—
272C4⋊Q8—48no—
273#124—48no—
274#123—416no—
275C22⋊Q8—22no—
276C22⋊Q8—48no—
277C22⋊Q8—41no—
278C22⋊Q8—44no—
279C22⋊Q8—48no—
280C22⋊Q8—48no—
281C22⋊Q8—44no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—42no—
285C4×D4—24no—
286C4×D4—42no—
287C4×D4—44no—
288C4×D4—44no—
289C4×D4—44no—
290C4×D4—44no—
291#117—44no—
292#117—42no—
293#117—44no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—44no—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351#106—42no—
352#106—42no—
353#106—44no—
354#106—44no—
355C4⋊C8—24no—
356C4⋊C8—44no—
357#103—24no—
358#103—44no—
359#103—44no—
360#103—42no—
361#103—44no—
362#103—44no—
363#103—44no—
364C22⋊C8—44no—
365C22⋊C8—42no—
366C22⋊C8—42no—
367C22⋊C8—44no—
368C4×C8—24no—
369C4×C8—44no—
370#95—21no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—21no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—21no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391C24C22×D411yescharacteristic
392C24C22×D412yes—
393#41—416no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8—24no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—24no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—82no—
412C2×Q8—816no—
413C2×Q8—88no—
414C2×Q8—88no—
415C2×Q8C2×SD1614yes—
416C2×Q82−1+412yes—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4C2×SD1614yes—
422C2×D4C2×SD1618yes—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—24no—
445C22×C4—42no—
446C22×C4—41no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4C2×SD1614yes—
460C22×C4C2×SD1618yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C2511yescharacteristic
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—24no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—81no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C22⋊C4—24no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—81no—
524C22⋊C4—82no—
525C42—22no—
526C42—22no—
527C42—24no—
528C42—22no—
529C42—42no—
530C42—42no—
531C42—42no—
532C42—44no—
533C42—42no—
534C42H(5,𝔽2)14yes—
535C23—21no—
536C23—22no—
537C23—22no—
538C23—21no—
539C23—21no—
540C23—22no—
541C23—21no—
542C23—22no—
543C23—22no—
544C23—21no—
545C23—21no—
546C23—22no—
547C23C22×SD1614yes—
548C23C22×SD1612yes—
549C23C23×D411yescharacteristic
550Q8—28no—
551Q8—24no—
552Q8—42no—
553Q8—44no—
554Q8—44no—
555Q8—88no—
556Q8—88no—
557Q8—88no—
558D4—28no—
559D4—48no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—28no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—48no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—84no—
612C2×C4—88no—
613C2×C4—88no—
614C2×C4—164no—
615C2×C4#53412yes—
616C2×C4#53412yes—
617C2×C4C22×SD1614yes—
618C2×C4C22×SD1612yes—
619C2×C4C22×SD1612yes—
620C2×C4#57512yes—
621C2×C4#57812yes—
622C2×C4C2×H(5,𝔽2)12yes—
623C2×C4C2×2−1+411yescharacteristic
624C2×C4#58512yes—
625C2×C4#58511yescharacteristic
626C8—42no—
627C8—42no—
628C8—44no—
629C22—22no—
630C22—24no—
631C22—22no—
632C22—24no—
633C22—22no—
634C22—24no—
635C22—41no—
636C22—42no—
637C22—42no—
638C22—41no—
639C22—42no—
640C22—41no—
641C22—41no—
642C22—42no—
643C22C2×#53411yescharacteristic
644C22#348911yescharacteristic
645C22#350011yescharacteristic
646C22C23×SD1611yescharacteristic
647C22C2×#57511yescharacteristic
648C22C2×#57811yescharacteristic
649C22#358611yescharacteristic
650C4—22no—
651C4—24no—
652C4—22no—
653C4—22no—
654C4—42no—
655C4—41no—
656C4—42no—
657C4—42no—
658C4—84no—
659C4—82no—
660C4—81no—
661C4—82no—
662C4—81no—
663C4—82no—
664C4—82no—
665C4—81no—
666C4#318714yes—
667C4#356112yes—
668C4#356212yes—
669C2—42no—
670C2—44no—
671C2#4294311yescharacteristic
672C2C2×#318711yescharacteristic
673C2#6075911yescharacteristic
674C2#6209911yescharacteristic
675C2#6214611yescharacteristic
676C2#6252611yescharacteristic
677C2#6259311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
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. #32740C2semidirect
4. #31772C2semidirect
5. #31668C2semidirect
6. #30359C2semidirect
7. #29951C2semidirect
8. #29922C2semidirect
9. #29105C2semidirect
10. #28639C2semidirect
11. #26298C2semidirect
12. #25902C2semidirect
13. #25884C2semidirect
14. #25472C2semidirect
15. #24301C2semidirect
16. #23756C2semidirect

Maximal subgroups

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

subgroupindex
#362742
#353162
#327402
#317722
#316682
#303592
#299512
#299222
#291052
#286392
#262982
#259022
#258842
#254722
#243012
#237562
#225302