Finite Group Databasebeta

← #6986812

Subgroups — #6986812

Order 512.

1673 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36456C211yesmaximal, characteristic
2#35316C211yesmaximal, characteristic
3#32788C212yesmaximal, complemented
4#32259C214yesmaximal, complemented
5#31998C214yesmaximal, complemented
6#30293C212yesmaximal, complemented
7#30168C212yesmaximal, complemented
8#29938C211yesmaximal, characteristic, complemented
9#29159C212yesmaximal, complemented
10#28706C212yesmaximal, complemented
11#26278C212yesmaximal, complemented
12#25837C212yesmaximal, complemented
13#25811C211yesmaximal, characteristic, complemented
14#25341C212yesmaximal, complemented
15#24301C211yesmaximal, characteristic, complemented
16#23756C211yesmaximal, characteristic, complemented
17#22469C211yesmaximal, characteristic
18C2×Q82C2211yescharacteristic
19C2×#536—21no—
20C22×C4⋊Q8C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22C2×#498C2212yes—
23C2×#475C2212yes—
24C2×C8×D4C2212yes—
25#2857C2212yes—
26#2857C2212yes—
27#2857C2212yes—
28#2857C2211yescharacteristic
29#2621—22no—
30#2563C2212yes—
31#2563C2211yescharacteristic
32#2547—21no—
33#2547C2212yes—
34#2546—21no—
35#2546C2212yes—
36#2538C2211yescharacteristic
37#2538C2212yes—
38#2486—22no—
39#2486C2214yes—
40#2452—22no—
41#2452—21no—
42#2439—21no—
43#2439—22no—
44#2351—21no—
45Q8×C4⋊C4C2211yescharacteristic
46#2341—21no—
47#2310—21no—
48C4×C22⋊Q8C2211yescharacteristic
49C4×C22⋊Q8C2212yes—
50#2285C2211yescharacteristic
51#2100C2214yes—
52#2099C2211yescharacteristic
53#2099C2212yes—
54#2096C2211yescharacteristic
55#2084C2211yescharacteristic
56#2084C2214yes—
57#2084C2212yes—
58#2077C2212yes—
59#2077C2214yes—
60#2077C2214yes—
61#2077C2214yes—
62#2077C2214yes—
63#2072C2211yescharacteristic
64#2058C2214yes—
65#2058C2214yes—
66#2058C2212yes—
67#2058C2212yes—
68#2058C2212yes—
69#2035C2212yes—
70#2035C2214yes—
71#2024C2212yes—
72#1990C2212yes—
73#1988C2212yes—
74#1987C2212yes—
75#1984C2212yes—
76#1948C2212yes—
77#1919C2214yes—
78#1919C2214yes—
79#1892C2214yes—
80#1872C2212yes—
81#1869C2214yes—
82#1865C2214yes—
83#1865C2214yes—
84#1853C2212yes—
85#1853C2214yes—
86#1817C2212yes—
87#1807C2214yes—
88#1807C2214yes—
89#1807C2214yes—
90#1797C2211yescharacteristic
91C4×#106C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C2×#336C2212yes—
95C22×#41D411yescharacteristic
96C23×Q8—21no—
97C23×C8C2314yes—
98Q82—44no—
99#536—48no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—44no—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2312yes—
110C2×#124—22no—
111C2×#123—24no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8C2311yescharacteristic
117C2×C22⋊Q8C2312yes—
118C4×#41—44no—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8D411yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2311yescharacteristic
125C2×C4×D4D412yes—
126C2×C4×D4D411yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2312yes—
129C2×#117—21no—
130C2×#117D411yescharacteristic
131C2×#117D412yes—
132C22×C4⋊C4—21no—
133C22×C4⋊C4C2312yes—
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136#498—44no—
137#475—44no—
138C8×D4—44no—
139C2×#106C2312yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×#106C2312yes—
144C2×#106C2314yes—
145C2×C4⋊C8C2312yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2314yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2314yes—
152C2×C4×C8C2312yes—
153#398—24no—
154#398—24no—
155#397—24no—
156#397—24no—
157#396—22no—
158#396—22no—
159#396—22no—
160#393—22no—
161#393—22no—
162#393—22no—
163#391—22no—
164#391—22no—
165#391—22no—
166#391—22no—
167#391—21no—
168#391—22no—
169#391C2311yescharacteristic
170#390—21no—
171#390—22no—
172#390—21no—
173#390—22no—
174#389—21no—
175#389—22no—
176#389—21no—
177#389—22no—
178#389—21no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389C2311yescharacteristic
183#389C2312yes—
184#389C2314yes—
185#389C2314yes—
186#387—21no—
187#387—22no—
188#385—21no—
189#385—22no—
190#384C2311yescharacteristic
191#384C2312yes—
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#382—21no—
195#382—22no—
196#382—21no—
197#381—24no—
198#380—21no—
199#380—22no—
200#380—22no—
201#380—22no—
202#380—21no—
203#380C2312yes—
204#380C2311yescharacteristic
205#380C2312yes—
206#380C2312yes—
207#380C2312yes—
208#380C2314yes—
209#380C2314yes—
210#380C2312yes—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—22no—
219#340C2312yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2311yescharacteristic
233#336C2314yes—
234#336C2314yes—
235C2×#41—42no—
236C2×#41—42no—
237C2×#41SD1618yes—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—21no—
242C22×Q8—44no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8C2×D411yescharacteristic
246C22×Q8C2411yescharacteristic
247C22×D4C2×D411yescharacteristic
248C22×D4C2×D412yes—
249C23×C4—22no—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D412yes—
253C23×C4C2×D412yes—
254C23×C4C2411yescharacteristic
255C23×C4C2412yes—
256C22×C8—28no—
257C22×C8—44no—
258C22×C8—44no—
259C22×C8C2414yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—44no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269C4⋊Q8—48no—
270C4⋊Q8—44no—
271C4⋊Q8—48no—
272#124—48no—
273#123—416no—
274C22⋊Q8—22no—
275C22⋊Q8—24no—
276C22⋊Q8—41no—
277C22⋊Q8—42no—
278C22⋊Q8—48no—
279C22⋊Q8—48no—
280C22⋊Q8—82no—
281C22⋊Q8—84no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—42no—
285C4×Q8—44no—
286C4×Q8—44no—
287C4×Q8—44no—
288C4×D4—24no—
289C4×D4—42no—
290C4×D4—44no—
291C4×D4—44no—
292#117—44no—
293#117—42no—
294#117—44no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—21no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—44no—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2412yes—
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357#106—24no—
358#106—42no—
359#106—44no—
360#106—44no—
361#106—44no—
362#106—44no—
363C4⋊C8—24no—
364#103—24no—
365#103—24no—
366#103—44no—
367C22⋊C8—24no—
368C22⋊C8—42no—
369C22⋊C8—44no—
370C4×C8—24no—
371#95—21no—
372#95—21no—
373#95—22no—
374#95—21no—
375#95—22no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—22no—
386#95—22no—
387#95—22no—
388C24C22×D412yes—
389C24C22×D411yescharacteristic
390#41—416no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—21no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—42no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—81no—
405C2×Q8—82no—
406C2×Q8—81no—
407C2×Q8—82no—
408C2×Q8—816no—
409C2×Q8C2×SD1614yes—
410C2×Q82−1+412yes—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4C2×SD1614yes—
416C2×D4C2×SD1618yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—24no—
439C22×C4—42no—
440C22×C4—41no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4C2×SD1614yes—
448C22×C4C2×SD1618yes—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C2511yescharacteristic
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—24no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—88no—
467C2×C82−1+414yes—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—81no—
505C4⋊C4—82no—
506C22⋊C4—24no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—81no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C22⋊C4—82no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—24no—
523C42—42no—
524C42—42no—
525C42—44no—
526C42—44no—
527C42—42no—
528C42—44no—
529C42—42no—
530C42H(5,𝔽2)14yes—
531C23—21no—
532C23—22no—
533C23—22no—
534C23—21no—
535C23—21no—
536C23—22no—
537C23—21no—
538C23—22no—
539C23—21no—
540C23—22no—
541C23—21no—
542C23—22no—
543C23C22×SD1614yes—
544C23C22×SD1612yes—
545C23C23×D411yescharacteristic
546Q8—28no—
547Q8—24no—
548Q8—42no—
549Q8—88no—
550Q8—88no—
551Q8—162no—
552Q8—164no—
553D4—28no—
554D4—48no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—28no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—48no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—84no—
607C2×C4—88no—
608C2×C4#53412yes—
609C2×C4#53412yes—
610C2×C4C22×SD1614yes—
611C2×C4C22×SD1612yes—
612C2×C4C22×SD1612yes—
613C2×C4#57512yes—
614C2×C4#57712yes—
615C2×C4C2×H(5,𝔽2)12yes—
616C2×C4C2×2−1+411yescharacteristic
617C2×C4C2×2−1+412yes—
618C2×C4#58511yescharacteristic
619C8—24no—
620C8—42no—
621C8—44no—
622C22—24no—
623C22—22no—
624C22—22no—
625C22—24no—
626C22—24no—
627C22—22no—
628C22—41no—
629C22—42no—
630C22—41no—
631C22—42no—
632C22—42no—
633C22—41no—
634C22—41no—
635C22—42no—
636C22C2×#53411yescharacteristic
637C22#348911yescharacteristic
638C22#349911yescharacteristic
639C22C23×SD1611yescharacteristic
640C22C2×#57511yescharacteristic
641C22C2×#57711yescharacteristic
642C22#358611yescharacteristic
643C4—22no—
644C4—24no—
645C4—22no—
646C4—22no—
647C4—42no—
648C4—41no—
649C4—42no—
650C4—42no—
651C4—82no—
652C4—81no—
653C4—84no—
654C4—82no—
655C4—81no—
656C4—82no—
657C4—82no—
658C4—81no—
659C4#318714yes—
660C4#355212yes—
661C4#356112yes—
662C2—42no—
663C2—44no—
664C2#4291211yescharacteristic
665C2C2×#318711yescharacteristic
666C2#6075811yescharacteristic
667C2#6209911yescharacteristic
668C2#6213511yescharacteristic
669C2#6246911yescharacteristic
670C2#6250711yescharacteristic
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. #32788C2semidirect
4. #32259C2semidirect
5. #31998C2semidirect
6. #30293C2semidirect
7. #30168C2semidirect
8. #29938C2semidirect
9. #29159C2semidirect
10. #28706C2semidirect
11. #26278C2semidirect
12. #25837C2semidirect
13. #25811C2semidirect
14. #25341C2semidirect
15. #24301C2semidirect
16. #23756C2semidirect

Maximal subgroups

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

subgroupindex
#364562
#353162
#327882
#322592
#319982
#302932
#301682
#299382
#291592
#287062
#262782
#258372
#258112
#253412
#243012
#237562
#224692