Finite Group Databasebeta

← #6971693

Subgroups — #6971693

Order 512.

1800 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#36261C211yesmaximal, characteristic, complemented
2#36127C211yesmaximal, characteristic
3#32786C212yesmaximal, complemented
4#31350C214yesmaximal, complemented
5#31078C214yesmaximal, complemented
6#30304C211yesmaximal, characteristic, complemented
7#29966C212yesmaximal, complemented
8#29687C212yesmaximal, complemented
9#29208C211yesmaximal, characteristic, complemented
10#29121C211yesmaximal, characteristic, complemented
11#26106C212yesmaximal, complemented
12#26040C212yesmaximal, complemented
13#25010C212yesmaximal, complemented
14#24965C212yesmaximal, complemented
15#24060C212yesmaximal, complemented
16#23815C211yesmaximal, characteristic, complemented
17#22008C211yesmaximal, characteristic, complemented
18C2×#554—21no—
19C2×#547—21no—
20C2×#532C2212yes—
21C2×#480C2214yes—
22C2×#474C2214yes—
23#2868C2211yescharacteristic
24#2859C2211yescharacteristic
25#2857C2212yes—
26#2857C2212yes—
27#2584—21no—
28#2583—22no—
29#2552C2211yescharacteristic
30#2551C2211yescharacteristic
31#2550—21no—
32#2549—22no—
33#2547—21no—
34#2537—21no—
35#2537C2212yes—
36#2440C2212yes—
37#2440C2212yes—
38#2432—22no—
39#2422—22no—
40#2422C2214yes—
41#2349—21no—
42D4×C4⋊C4—21no—
43D4×C4⋊C4C2212yes—
44D4×C4⋊C4C2211yescharacteristic
45D4×C4⋊C4C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×#127C2211yescharacteristic
48C2×#389C2212yes—
49C42×D4C2211yescharacteristic
50#2099C2211yescharacteristic
51#2099C2211yescharacteristic
52#2099C2212yes—
53#2099C2214yes—
54#2086C2214yes—
55#2086C2214yes—
56#2083C2214yes—
57#2075C2214yes—
58#2075C2214yes—
59#2071C2212yes—
60#2057C2212yes—
61#2057C2212yes—
62#2057C2212yes—
63#2057C2212yes—
64#2057C2214yes—
65#2041C2211yescharacteristic
66#2041C2211yescharacteristic
67#2041C2212yes—
68#2033C2214yes—
69#2033C2214yes—
70#2023C2212yes—
71#1990C2212yes—
72#1954C2212yes—
73#1947C2212yes—
74#1947C2212yes—
75#1939C2212yes—
76#1934C2214yes—
77#1923C2214yes—
78#1874C2214yes—
79#1862C2214yes—
80#1862C2214yes—
81#1852C2214yes—
82#1845C2212yes—
83#1845C2212yes—
84#1834C2212yes—
85#1823C2214yes—
86#1817C2212yes—
87#1802C2214yes—
88#1802C2212yes—
89#1802C2212yes—
90#1797C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×#102C2212yes—
93C4×#102C2212yes—
94#1744C2211yescharacteristic
95C22×#41—22no—
96#554—48no—
97#547—48no—
98#532—48no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106C2×#125—21no—
107C2×#125—21no—
108C2×#124—22no—
109C2×#123—24no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×#121—22no—
115C2×#121—22no—
116C2×#121C2314yes—
117C2×C4×Q8—21no—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—22no—
123C2×C4×D4D411yescharacteristic
124C2×C4×D4D411yescharacteristic
125C2×C4×D4C2312yes—
126C2×C4×D4C2314yes—
127C2×C4×D4C2311yescharacteristic
128C22×C4⋊C4—22no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2312yes—
132C22×C42C2312yes—
133#480—48no—
134#474—48no—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2314yes—
141C2×#102C2314yes—
142C2×#102C2312yes—
143C2×#102C2312yes—
144C2×#102C2312yes—
145C2×#102C2314yes—
146C2×#102C2312yes—
147C2×C22⋊C8C2314yes—
148C2×C22⋊C8C2314yes—
149C2×C4×C8C2312yes—
150C2×C4×C8C2312yes—
151#398—22no—
152#398—22no—
153#398—22no—
154#397—24no—
155#395—22no—
156#395—22no—
157#395—22no—
158#394—24no—
159#394—24no—
160#391—21no—
161#391—22no—
162#391—22no—
163#390—21no—
164#390—21no—
165#390C2312yes—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389—21no—
176#389—22no—
177#389C2311yescharacteristic
178#389C2311yescharacteristic
179#389C2312yes—
180#389C2314yes—
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#385—22no—
185#385—22no—
186#385—22no—
187#385—22no—
188#385—24no—
189#385—24no—
190#384—21no—
191#384—21no—
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#384C2312yes—
195#384C2311yescharacteristic
196#384C2311yescharacteristic
197#384C2312yes—
198#384C2312yes—
199#382—21no—
200#380—22no—
201#380—22no—
202#380—22no—
203#380—22no—
204#380C2314yes—
205#380C2312yes—
206#380C2312yes—
207C4×C4⋊C4—21no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4D412yes—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4C2312yes—
216C2×#95—22no—
217C43C2311yescharacteristic
218#340C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#336C2311yescharacteristic
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2314yes—
232#336C2312yes—
233C2×#41—48no—
234C2×#41—48no—
235C2×#41—84no—
236C22×Q8—22no—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×D4—24no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4C2×D411yescharacteristic
243C22×D4C2412yes—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4—24no—
248C23×C4C2×D412yes—
249C23×C4C2×D412yes—
250C23×C4C2412yes—
251C22×C8C2412yes—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C4⋊Q8—22no—
255C4⋊Q8—41no—
256C4⋊Q8—44no—
257C4⋊Q8—44no—
258C4⋊Q8—44no—
259C4⋊Q8—44no—
260C4⋊Q8—82no—
261#127—24no—
262#125—44no—
263#125—44no—
264#124—48no—
265#123—416no—
266C22⋊Q8—48no—
267C22⋊Q8—48no—
268C22⋊Q8—48no—
269C22⋊Q8—48no—
270#121—28no—
271#121—44no—
272#121—48no—
273#121—48no—
274C4×Q8—44no—
275C4×D4—28no—
276C4×D4—24no—
277C4×D4—24no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—44no—
281C4×D4—48no—
282C4×D4—44no—
283C4×D4—44no—
284C4×D4—44no—
285C4×D4—44no—
286C4×D4D818yes—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—48no—
317C2×C4⋊C4—48no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—44no—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#107—44no—
354#107—44no—
355#107—44no—
356#107—44no—
357C4⋊C8—24no—
358C4⋊C8—44no—
359#102—24no—
360#102—28no—
361#102—24no—
362#102—24no—
363#102—44no—
364#102—42no—
365C22⋊C8—44no—
366C22⋊C8—44no—
367C4×C8—24no—
368C4×C8—42no—
369#95—21no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—24no—
377#95—24no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384C24—22no—
385C24C22×D412yes—
386#41—832no—
387C2×Q8—42no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—42no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8—81no—
394C2×Q8—82no—
395C2×Q8—82no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—48no—
408C2×D4—48no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—84no—
412C2×D4C2×D814yes—
413C2×D4H(5,𝔽2)14yes—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—44no—
432C22×C4—48no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—48no—
438C22×C4—44no—
439C22×C4—84no—
440C22×C4—82no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4C2×D818yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C2511yescharacteristic
454C2×C8—22no—
455C2×C8—24no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—81no—
496C4⋊C4—81no—
497C4⋊C4—81no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4C2×D814yes—
504C4⋊C42−1+412yes—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—48no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4C2×D818yes—
516C42—24no—
517C42—22no—
518C42—21no—
519C42—21no—
520C42—42no—
521C42—42no—
522C42—41no—
523C42—41no—
524C42—42no—
525C42—42no—
526C42C2×D814yes—
527C42#13714yes—
528C422−1+414yes—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—24no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23C22×D814yes—
543C23C23×D411yescharacteristic
544Q8—84no—
545Q8—88no—
546Q8—164no—
547D4—28no—
548D4—24no—
549D4—24no—
550D4—48no—
551D4—48no—
552D4—42no—
553D4—42no—
554D4—816no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—48no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—84no—
601C2×C4—816no—
602C2×C4—88no—
603C2×C4—88no—
604C2×C4—164no—
605C2×C4#53612yes—
606C2×C4#53612yes—
607C2×C4C22×D812yes—
608C2×C4C22×D812yes—
609C2×C4C22×D814yes—
610C2×C4C22×D812yes—
611C2×C4C2×#13712yes—
612C2×C4C2×#13712yes—
613C2×C4C2×#13712yes—
614C2×C4C2×H(5,𝔽2)12yes—
615C2×C4C2×2−1+411yescharacteristic
616C2×C4C2×2−1+412yes—
617C2×C4#58511yescharacteristic
618C8—42no—
619C8—44no—
620C8—42no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22C2×#53611yescharacteristic
636C22#348411yescharacteristic
637C22#348911yescharacteristic
638C22C23×D811yescharacteristic
639C22C22×#13711yescharacteristic
640C22#358511yescharacteristic
641C22#358611yescharacteristic
642C4—22no—
643C4—24no—
644C4—22no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—84no—
652C4—81no—
653C4—82no—
654C4—81no—
655C4—82no—
656C4—81no—
657C4—82no—
658C4#322614yes—
659C4#354412yes—
660C4#355912yes—
661C2—44no—
662C2—84no—
663C2#4248211yescharacteristic
664C2C2×#322611yescharacteristic
665C2#6088611yescharacteristic
666C2#6155911yescharacteristic
667C2#6158611yescharacteristic
668C2#6209511yescharacteristic
669C2#6211411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36127C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36261C2semidirect
3. #32786C2semidirect
4. #31350C2semidirect
5. #31078C2semidirect
6. #30304C2semidirect
7. #29966C2semidirect
8. #29687C2semidirect
9. #29208C2semidirect
10. #29121C2semidirect
11. #26106C2semidirect
12. #26040C2semidirect
13. #25010C2semidirect
14. #24965C2semidirect
15. #24060C2semidirect
16. #23815C2semidirect
17. #22008C2semidirect

Maximal subgroups

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

subgroupindex
#362612
#361272
#327862
#313502
#310782
#303042
#299662
#296872
#292082
#291212
#261062
#260402
#250102
#249652
#240602
#238152
#220082