Finite Group Databasebeta

← #6435883

Subgroups — #6435883

Order 512.

1673 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35312C211yesmaximal, characteristic
2#35281C211yesmaximal, characteristic, complemented
3#30216C212yesmaximal, complemented
4#30215C212yesmaximal, complemented
5#30211C214yesmaximal, complemented
6#30210C214yesmaximal, complemented
7#30208C212yesmaximal, complemented
8#30205C212yesmaximal, complemented
9#23634C212yesmaximal
10#23633C212yesmaximal, complemented
11#23455C211yesmaximal, characteristic, complemented
12#23452C211yesmaximal, characteristic
13#23451C211yesmaximal, characteristic
14C2×#2024C211yesmaximal, characteristic, complemented
15#21326C212yesmaximal, complemented
16#21205C212yesmaximal
17#20368C211yesmaximal, characteristic, complemented
18C2×#544C2211yescharacteristic
19C2×#536C2211yescharacteristic
20C2×#533C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C2×#417C2212yes—
23C2×#419C2212yes—
24C2×#416C2212yes—
25C22×#103C2212yes—
26C2×#405C2212yes—
27#2868C2212yes—
28#2862C2212yes—
29#2862C2212yes—
30#2861C2212yes—
31#2660C2214yes—
32#2642C2212yes—
33#2640C2212yes—
34#2567—21no—
35#2567—21no—
36#2546—21no—
37#2546—21no—
38#2502—22no—
39#2492—21no—
40#2492—21no—
41#2489—21no—
42#2489—21no—
43#2488—22no—
44#2488—22no—
45#2434—22no—
46#2317C2212yes—
47#2317C2212yes—
48#2311C2211yescharacteristic
49#2310C2211yescharacteristic
50#2297C2211yescharacteristic
51C2×#391C2211yescharacteristic
52C4×#117C2211yescharacteristic
53#2074C2214yes—
54#2074C2214yes—
55#2074C2214yes—
56#2073C2214yes—
57#2072C2214yes—
58#2072C2214yes—
59#2072C2214yes—
60#2071C2214yes—
61#2026C2212yes—
62#2026C2214yes—
63#2025C2212yes—
64#2024C2212yes—
65#2024C2212yes—
66#2024C2212yes—
67#2024C2214yes—
68#2024C2211yescharacteristic
69#2024C2212yes—
70#2024C2212yes—
71#2024C2212yes—
72#2024C2211yescharacteristic
73#2023C2211yescharacteristic
74#2023C2212yes—
75#2023C2212yes—
76#2023C2211yescharacteristic
77#1993C2214yes—
78#1992C2214yes—
79#1991C2214yes—
80#1990C2214yes—
81#1960C2214yes—
82#1959C2214yes—
83#1958C2214yes—
84#1956C2214yes—
85#1746C2214yes—
86#1745C2212yes—
87#1744C2212yes—
88#1736C2212yes—
89#1736C2212yes—
90#1735C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C23×Q8—21no—
97C23×Q8C2311yescharacteristic
98C22×M16C2312yes—
99C23×C8C2312yes—
100#544—28no—
101#536—28no—
102#533—44no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—22no—
107C2×C4⋊Q8—22no—
108C2×C4⋊Q8—44no—
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2312yes—
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2312yes—
115C2×#125—21no—
116C2×#125C2312yes—
117C2×#125C2312yes—
118C2×#124C2312yes—
119C2×#123—22no—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8—21no—
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8—22no—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×#117C2311yescharacteristic
128C2×#117C2312yes—
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C22×C4⋊C4—21no—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2312yes—
134C22×C42C2311yescharacteristic
135#419—44no—
136#417—44no—
137#416—44no—
138C2×#103—44no—
139C2×#103C2312yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#103C2314yes—
146C2×#102C2312yes—
147C2×#102C2314yes—
148C2×#102C2312yes—
149#405—28no—
150C2×#97C2314yes—
151C2×C4×C8C2314yes—
152#400—22no—
153#400—22no—
154#400—22no—
155#400—22no—
156#398—24no—
157#398—24no—
158#397—22no—
159#397—22no—
160#395—24no—
161#395—24no—
162#393—22no—
163#393—22no—
164#391—21no—
165#391—21no—
166#391—22no—
167#391—22no—
168#391—22no—
169#391—22no—
170#391—22no—
171#391—22no—
172#391—21no—
173#391—21no—
174#391C2312yes—
175#391C2312yes—
176#391C2312yes—
177#391C2312yes—
178#391C2311yescharacteristic
179#391C2312yes—
180#391C2311yescharacteristic
181#390C2312yes—
182#390C2311yescharacteristic
183#390C2311yescharacteristic
184#389—22no—
185#389—22no—
186#389—22no—
187#389—22no—
188#388—22no—
189#388—22no—
190#388—22no—
191#388—22no—
192#386—21no—
193#386—21no—
194#386—22no—
195#385—24no—
196#385—24no—
197#384—22no—
198#384—22no—
199#384C2314yes—
200#384C2314yes—
201#382—22no—
202#382—22no—
203#382C2314yes—
204#382C2314yes—
205#381—22no—
206#380—21no—
207#380—21no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C22⋊C4—21no—
213C4×C22⋊C4—21no—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4D412yes—
216#376C2312yes—
217C2×#95—21no—
218C2×#95—21no—
219C43C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#336C2314yes—
231#336C2314yes—
232#336C2311yescharacteristic
233#336C2311yescharacteristic
234#336C2312yes—
235#336C2312yes—
236#336C2312yes—
237C2×#41—24no—
238C2×#41—24no—
239C2×#41—22no—
240C2×#41—22no—
241C22×Q8—21no—
242C22×Q8—21no—
243C22×Q8—21no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—24no—
248C22×Q8—24no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—44no—
252C22×Q8—82no—
253C22×Q8—82no—
254C22×Q8C2412yes—
255C22×Q8C2411yescharacteristic
256C22×D4—21no—
257C22×D4C2411yescharacteristic
258C23×C4—21no—
259C23×C4—22no—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2411yescharacteristic
265C2×M16—24no—
266C2×M16—24no—
267C2×M16—24no—
268C22×C8—24no—
269C22×C8—24no—
270C22×C8—24no—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C22×C8C2414yes—
274C4⋊Q8—28no—
275C4⋊Q8—28no—
276C4⋊Q8—44no—
277C4⋊Q8—44no—
278C4⋊Q8—42no—
279C4⋊Q8—42no—
280C4⋊Q8—44no—
281C4⋊Q8—44no—
282C4⋊Q8—44no—
283C4⋊Q8—84no—
284C4⋊Q8—84no—
285#125—28no—
286#125—42no—
287#125—42no—
288#125—44no—
289#124—28no—
290#123—48no—
291C22⋊Q8—44no—
292C22⋊Q8—44no—
293C22⋊Q8—44no—
294C22⋊Q8—44no—
295C22⋊Q8—48no—
296C4×Q8—44no—
297C4×Q8—44no—
298#117—22no—
299#117—22no—
300#117—24no—
301#117—24no—
302#117—24no—
303#117#4118yes—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2414yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C42—24no—
348C2×C42—22no—
349C2×C42—21no—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2412yes—
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359C2×C42C2412yes—
360#103—44no—
361#103—44no—
362#103—42no—
363#103—42no—
364#103—44no—
365#103—44no—
366#103—44no—
367#102—42no—
368#102—42no—
369#102—44no—
370#97—28no—
371C4×C8—28no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394C24—21no—
395C24C22×D411yescharacteristic
396#41—48no—
397#41—48no—
398C2×Q8—24no—
399C2×Q8—22no—
400C2×Q8—24no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—42no—
411C2×Q8—44no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×Q8—44no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×Q8—44no—
419C2×Q8—42no—
420C2×Q8—48no—
421C2×Q8—48no—
422C2×Q8—82no—
423C2×Q8—82no—
424C2×Q8—82no—
425C2×Q8—82no—
426C2×Q8—816no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—42no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4—81no—
459C22×C4—81no—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C2×#4114yes—
470C22×C4H(5,𝔽2)14yes—
471C22×C42−1+412yes—
472C22×C42−1+412yes—
473C22×C4C2511yescharacteristic
474M16—48no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—24no—
484C2×C8—48no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—84no—
510C4⋊C4—84no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C4⋊C4#12414yes—
514C4⋊C4#12414yes—
515C22⋊C4—24no—
516C22⋊C4—42no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—82no—
523C22⋊C4—81no—
524C22⋊C4—82no—
525C22⋊C4—81no—
526C22⋊C4#12418yes—
527C42—22no—
528C42—22no—
529C42—24no—
530C42—24no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—42no—
536C42—42no—
537C42—42no—
538C42—44no—
539C42C2×#4118yes—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23C2×#12412yes—
552C23#53512yes—
553C23#53612yes—
554C23C23×D411yescharacteristic
555Q8—44no—
556Q8—44no—
557Q8—42no—
558Q8—42no—
559Q8—84no—
560Q8—84no—
561Q8—88no—
562Q8—88no—
563Q8—84no—
564Q8—164no—
565D4—42no—
566D4—42no—
567D4—84no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—88no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4—162no—
618C2×C4C2×#12414yes—
619C2×C4C2×#12414yes—
620C2×C4C2×#12412yes—
621C2×C4#53512yes—
622C2×C4#53612yes—
623C2×C4#54218yes—
624C2×C4#54218yes—
625C2×C4#57712yes—
626C2×C4#57812yes—
627C2×C4C22×#4112yes—
628C2×C4C2×H(5,𝔽2)12yes—
629C2×C4C2×2−1+411yescharacteristic
630C2×C4C2×2−1+411yescharacteristic
631C8—42no—
632C8—42no—
633C8—44no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22#264014yes—
648C22C22×#12411yescharacteristic
649C22C2×#53511yescharacteristic
650C22C2×#53611yescharacteristic
651C22C2×#57711yescharacteristic
652C22C2×#57811yescharacteristic
653C22#358511yescharacteristic
654C22#358611yescharacteristic
655C4—22no—
656C4—22no—
657C4—24no—
658C4—24no—
659C4—41no—
660C4—41no—
661C4—42no—
662C4—42no—
663C4—81no—
664C4—82no—
665C4—84no—
666C4—82no—
667C4—82no—
668C4—81no—
669C4—82no—
670C4—81no—
671C4#264014yes—
672C4C2×#54212yes—
673C4C2×#54212yes—
674C2—24no—
675C2—161no—
676C2C2×#264011yescharacteristic
677C2#6062311yescharacteristic
678C2#6062511yescharacteristic
679C2#6081211yescharacteristic
680C2#6081411yescharacteristic
681C2#6088911yescharacteristic
682C2#6089311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35281C2semidirect
3. #30216C2semidirect
4. #30215C2semidirect
5. #30211C2semidirect
6. #30210C2semidirect
7. #30208C2semidirect
8. #30205C2semidirect
10. #23633C2semidirect
11. #23455C2semidirect
14. C2×#2024C2semidirect
15. #21326C2semidirect
17. #20368C2semidirect

Maximal subgroups

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

subgroupindex
#353122
#352812
#302162
#302152
#302112
#302102
#302082
#302052
#236342
#236332
#234552
#234522
#234512
C2×#20242
#213262
#212052
#203682