Finite Group Databasebeta

← #6594206

Subgroups — #6594206

Order 512.

1709 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36180C211yesmaximal, characteristic, complemented
2#36164C211yesmaximal, characteristic
3#33293C211yesmaximal, characteristic, complemented
4#33051C212yesmaximal, complemented
5#29224C211yesmaximal, characteristic, complemented
6#29148C211yesmaximal, characteristic, complemented
7#27978C212yesmaximal, complemented
8#27969C212yesmaximal, complemented
9#27737C212yesmaximal, complemented
10#27735C212yesmaximal, complemented
11#27713C214yesmaximal, complemented
12#27711C214yesmaximal, complemented
13#22487C212yesmaximal, complemented
14#22485C212yesmaximal
15C4×#489C211yesmaximal, characteristic, complemented
16C2×#1836C212yesmaximal, complemented
17D4×#97C211yesmaximal, characteristic, complemented
18C2×#554—21no—
19C2×#554C2211yescharacteristic
20C2×#547C2211yescharacteristic
21C2×#489C2211yescharacteristic
22C2×C8×D4C2212yes—
23C22×#97C2212yes—
24#2868C2211yescharacteristic
25#2862C2211yescharacteristic
26#2857C2212yes—
27#2850C2212yes—
28#2568C2211yescharacteristic
29#2567—21no—
30#2563C2211yescharacteristic
31#2552—21no—
32#2552C2212yes—
33#2504C2212yes—
34#2503C2212yes—
35#2502—22no—
36#2489—21no—
37#2489—21no—
38#2488—22no—
39#2488—22no—
40#2440—22no—
41#2440C2214yes—
42D4×C4⋊C4—21no—
43D4×C4⋊C4C2212yes—
44#2339—21no—
45#2339—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#124C2211yescharacteristic
48C2×#384C2212yes—
49C2×#382C2212yes—
50C42×D4C2211yescharacteristic
51#2101C2214yes—
52#2101C2214yes—
53#2101C2214yes—
54#2072C2214yes—
55#2071C2214yes—
56#2024C2212yes—
57#2023C2212yes—
58#1993C2214yes—
59#1993C2212yes—
60#1993C2211yescharacteristic
61#1990C2212yes—
62#1990C2212yes—
63#1988C2212yes—
64#1987C2212yes—
65#1959C2212yes—
66#1959C2211yescharacteristic
67#1936C2214yes—
68#1936C2212yes—
69#1935C2214yes—
70#1935C2212yes—
71#1893C2214yes—
72#1892C2214yes—
73#1874C2214yes—
74#1872C2214yes—
75#1836C2212yes—
76#1836C2212yes—
77#1836C2211yescharacteristic
78#1836C2212yes—
79#1836C2214yes—
80#1836C2214yes—
81#1836C2212yes—
82#1824C2214yes—
83#1823C2214yes—
84#1803C2212yes—
85#1802C2212yes—
86#1770C2212yes—
87#1767C2211yescharacteristic
88C4×#103C2212yes—
89C4×#102C2212yes—
90#1752C2214yes—
91#1749C2212yes—
92C2×#340C2214yes—
93C2×#340C2214yes—
94C4×#97C2211yescharacteristic
95C22×#41—21no—
96C23×C8C2314yes—
97#554—28no—
98#554—48no—
99#547—28no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×#125—22no—
107C2×#125—21no—
108C2×#124C2311yescharacteristic
109C2×#123—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8C2312yes—
115C2×#121C2312yes—
116C2×C4×Q8—21no—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×D4—21no—
119C2×C4×D4—22no—
120C2×C4×D4D411yescharacteristic
121C2×C4×D4D411yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C22×C4⋊C4—22no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2314yes—
129C22×C42C2312yes—
130#489—24no—
131C8×D4—28no—
132C2×C4⋊C8C2312yes—
133C2×#103C2312yes—
134C2×#103C2314yes—
135C2×#103C2312yes—
136C2×#102C2312yes—
137C2×#102C2314yes—
138C2×#102C2312yes—
139C2×C22⋊C8C2314yes—
140C2×#97—28no—
141C2×#97C2314yes—
142C2×#97C2312yes—
143C2×#97C2311yescharacteristic
144C2×#97C2312yes—
145C2×#97C2311yescharacteristic
146C2×C4×C8C2312yes—
147#400—22no—
148#400—22no—
149#400—22no—
150#398—24no—
151#398—24no—
152#397—22no—
153#397—22no—
154#395—24no—
155#393—22no—
156#391—21no—
157#391—21no—
158#391—21no—
159#391—22no—
160#391—21no—
161#391—21no—
162#391—21no—
163#391—21no—
164#391C2312yes—
165#391C2311yescharacteristic
166#390C2312yes—
167#390C2311yescharacteristic
168#389—22no—
169#389—22no—
170#389—21no—
171#389—22no—
172#389C2312yes—
173#389C2314yes—
174#388—22no—
175#388—22no—
176#388—21no—
177#388—21no—
178#386—22no—
179#386—22no—
180#386—22no—
181#386—22no—
182#385—24no—
183#385—24no—
184#384—22no—
185#384—21no—
186#384—21no—
187#384—22no—
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2314yes—
191#384C2312yes—
192#382—22no—
193#382—22no—
194#382C2312yes—
195#382C2312yes—
196#382C2312yes—
197#382C2311yescharacteristic
198#382C2312yes—
199#381—22no—
200#381—22no—
201#380—22no—
202#380—21no—
203#380—21no—
204#380C2314yes—
205C4×C4⋊C4—21no—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C22⋊C4—21no—
211C4×C22⋊C4—21no—
212C4×C22⋊C4D412yes—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4C2312yes—
215C2×#95—22no—
216C2×#95—22no—
217C43C2311yescharacteristic
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#336C2314yes—
233#336C2312yes—
234C2×#41—44no—
235C2×#41—44no—
236C2×#41—44no—
237C22×Q8—21no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8—21no—
241C22×Q8C2411yescharacteristic
242C22×D4—22no—
243C22×D4C2×D411yescharacteristic
244C22×D4C2411yescharacteristic
245C23×C4—22no—
246C23×C4—24no—
247C23×C4—22no—
248C23×C4—21no—
249C23×C4C2×D412yes—
250C23×C4C2×D412yes—
251C23×C4C2412yes—
252C22×C8—28no—
253C22×C8—28no—
254C22×C8—44no—
255C22×C8C2414yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—22no—
261C4⋊Q8—22no—
262C4⋊Q8—44no—
263C4⋊Q8—44no—
264C4⋊Q8—82no—
265#125—48no—
266#125—44no—
267#124—22no—
268#124—22no—
269#123—48no—
270C22⋊Q8—28no—
271C22⋊Q8—416no—
272C22⋊Q8—48no—
273C22⋊Q8—44no—
274C22⋊Q8—48no—
275#121—28no—
276C4×Q8—24no—
277C4×Q8—44no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—44no—
282C4×D4—48no—
283C4×D4#4118yes—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—48no—
308C2×C4⋊C4—84no—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2412yes—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—22no—
335C2×C42—21no—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2411yescharacteristic
343C2×C42C2412yes—
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346C4⋊C8—24no—
347#103—24no—
348#103—42no—
349#103—44no—
350#102—24no—
351#102—42no—
352#102—44no—
353C22⋊C8—28no—
354#97—28no—
355#97—24no—
356#97—22no—
357#97—42no—
358#97#4114yescomplemented
359C4×C8—24no—
360#95—21no—
361#95—21no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—21no—
369#95—21no—
370#95—22no—
371#95—22no—
372#95—24no—
373#95—24no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95C2×D412yes—
379#95C2×D412yes—
380#95C2×D412yes—
381C24—21no—
382C24C22×D412yes—
383#41—816no—
384C2×Q8—21no—
385C2×Q8—22no—
386C2×Q8—21no—
387C2×Q8—22no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—44no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—82no—
397C2×Q8—82no—
398C2×Q8—81no—
399C2×Q8—81no—
400C2×D4—21no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—22no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4#12414yes—
408C2×D4AGL(1,ℤ/8)14yes—
409C2×D4#13714yes—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—48no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—84no—
436C22×C4—84no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C2×#4118yes—
449C22×C4C2511yescharacteristic
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—24no—
458C2×C8—416no—
459C2×C8C2×#4114yes—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—42no—
473C4⋊C4—48no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—48no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—84no—
488C4⋊C4—84no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4#12414yes—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—44no—
495C22⋊C4—48no—
496C22⋊C4—42no—
497C22⋊C4—44no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—81no—
501C22⋊C4—82no—
502C22⋊C4—81no—
503C22⋊C4—82no—
504C22⋊C4#12418yes—
505C42—24no—
506C42—24no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—41no—
513C42—41no—
514C42—42no—
515C42—44no—
516C42C2×#4114yes—
517C42C2×#4112yes—
518C422−1+412yes—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23C2×#12414yes—
531C23C2×AGL(1,ℤ/8)14yes—
532C23C2×#13714yes—
533C23C23×D411yescharacteristic
534Q8—42no—
535Q8—42no—
536Q8—84no—
537Q8—84no—
538Q8—162no—
539Q8—164no—
540D4—42no—
541D4—42no—
542D4—88no—
543D4#489116yescomplemented
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—48no—
567C2×C4—48no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—88no—
593C2×C4—84no—
594C2×C4—168no—
595C2×C4—164no—
596C2×C4C2×#12412yes—
597C2×C4C2×#12414yes—
598C2×C4C2×#12412yes—
599C2×C4#54712yes—
600C2×C4#54712yes—
601C2×C4C2×AGL(1,ℤ/8)12yes—
602C2×C4C2×#13712yes—
603C2×C4C22×#4111yescharacteristic
604C2×C4C22×#4112yes—
605C2×C4C2×2−1+411yescharacteristic
606C2×C4#58512yes—
607C8—24no—
608C8—44no—
609C8—42no—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22C2×#48918yes—
624C22C22×#12411yescharacteristic
625C22C2×#54711yescharacteristic
626C22#350011yescharacteristic
627C22C22×AGL(1,ℤ/8)11yescharacteristic
628C22C22×#13711yescharacteristic
629C22#358511yescharacteristic
630C22#358611yescharacteristic
631C4—24no—
632C4—22no—
633C4—22no—
634C4—22no—
635C4—42no—
636C4—42no—
637C4—41no—
638C4—41no—
639C4—81no—
640C4—82no—
641C4—82no—
642C4—84no—
643C4—81no—
644C4—82no—
645C4—81no—
646C4—82no—
647C4C2×#48914yes—
648C4#347512yes—
649C4#351512yes—
650C2—28no—
651C2—161no—
652C2#3616411yescharacteristic
653C2C22×#48911yescharacteristic
654C2#6062511yescharacteristic
655C2#6125211yescharacteristic
656C2#6125411yescharacteristic
657C2#6257611yescharacteristic
658C2#6257711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
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. #36180C2semidirect
3. #33293C2semidirect
4. #33051C2semidirect
5. #29224C2semidirect
6. #29148C2semidirect
7. #27978C2semidirect
8. #27969C2semidirect
9. #27737C2semidirect
10. #27735C2semidirect
11. #27713C2semidirect
12. #27711C2semidirect
13. #22487C2semidirect
15. C4×#489C2semidirect
16. C2×#1836C2semidirect
17. D4×#97C2semidirect
358. #97#41semidirect
543. D4#489semidirect

Maximal subgroups

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

subgroupindex
#361802
#361642
#332932
#330512
#292242
#291482
#279782
#279692
#277372
#277352
#277132
#277112
#224872
#224852
C4×#4892
C2×#18362
D4×#972