Finite Group Databasebeta

← #6615091

Subgroups — #6615091

Order 512.

1637 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic
2C4×#556C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic, complemented
4#33154C212yesmaximal, complemented
5#29219C211yesmaximal, characteristic, complemented
6#28283C211yesmaximal, characteristic, complemented
7#27169C214yesmaximal, complemented
8#26667C214yesmaximal, complemented
9#24972C212yesmaximal
10#24703C212yesmaximal, complemented
11#24157C212yesmaximal, complemented
12#24123C212yesmaximal
13#22964C212yesmaximal, complemented
14#22795C211yesmaximal, characteristic
15#22771C212yesmaximal, complemented
16C4×#462C211yesmaximal, characteristic, complemented
17#21389C212yesmaximal, complemented
18C2×#556C2211yescharacteristic
19C2×#533C2212yes—
20C2×#475C2212yes—
21C2×#475C2212yes—
22C2×#462C2211yescharacteristic
23C2×C4×Q16C2212yes—
24C2×#431C2212yes—
25#2868C2211yescharacteristic
26#2857C2212yes—
27#2552—21no—
28#2542—21no—
29#2538C2212yes—
30#2441—22no—
31#2429—22no—
32#2423C2214yes—
33#2373—22no—
34#2367—21no—
35#2365—21no—
36#2365C2211yescharacteristic
37#2362C2212yes—
38#2356—21no—
39#2355C2211yescharacteristic
40#2354—21no—
41#2354C2211yescharacteristic
42Q8×C4⋊C4C2212yes—
43#2347C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×#126—22no—
47C4×#124—21no—
48C4×C22⋊Q8C2212yes—
49C42×Q8—21no—
50C42×Q8C2211yescharacteristic
51C42×D4C2211yescharacteristic
52C4×#117C2212yes—
53#2099C2214yes—
54#2077C2214yes—
55#2066C2214yes—
56#2035C2214yes—
57#2035C2214yes—
58#2001C2214yes—
59#1990C2212yes—
60#1988C2212yes—
61#1988C2212yes—
62#1988C2212yes—
63#1981C2212yes—
64#1981C2211yescharacteristic
65#1954C2212yes—
66#1948C2212yes—
67#1948C2212yes—
68#1948C2212yes—
69#1947C2211yescharacteristic
70#1941C2214yes—
71#1933C2214yes—
72#1928C2212yes—
73#1923C2214yes—
74#1921C2214yes—
75#1920C2214yes—
76#1878C2212yes—
77#1878C2214yes—
78#1873C2214yes—
79#1873C2214yes—
80#1865C2214yes—
81#1854C2214yes—
82#1842C2211yescharacteristic
83#1834C2212yes—
84#1825C2214yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1803C2211yescharacteristic
89#1797C2212yes—
90C8×C4⋊C4C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×#103C2214yes—
93C4×#103C2214yes—
94C4×#103C2212yes—
95C22×Q16C2312yes—
96#556—24no—
97#556—24no—
98#533—28no—
99#533—44no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×#126—22no—
106C2×#125—22no—
107C2×#124—21no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2314yes—
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8—22no—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2314yes—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×#117—22no—
125C2×#117C2312yes—
126C2×#117C2312yes—
127C22×C42C2312yes—
128#475—28no—
129#475—44no—
130#462—24no—
131C4×Q16—28no—
132#431—44no—
133C2×#107C2312yes—
134C2×#106C2314yes—
135C2×#106C2314yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2311yescharacteristic
140C2×C4⋊C8C2312yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×#103C2312yes—
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×C22⋊C8C2314yes—
149C2×C22⋊C8C2314yes—
150C2×C4×C8C2312yes—
151C2×C4×C8C2314yes—
152#399—24no—
153#398—24no—
154#396—24no—
155#391—21no—
156#391—21no—
157#391—22no—
158#391C2312yes—
159#391C2312yes—
160#390—22no—
161#390—21no—
162#389—22no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—22no—
167#389—24no—
168#389C2312yes—
169#389C2314yes—
170#389C2312yes—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—22no—
175#388—24no—
176#388—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—24no—
182#385—22no—
183#384—22no—
184#384—21no—
185#384—21no—
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#384C2312yes—
189#384C2312yes—
190#383—22no—
191#382—21no—
192#380—22no—
193#380C2312yes—
194#380C2314yes—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—22no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4D412yes—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4C2312yes—
222#376—22no—
223C43—22no—
224C43—21no—
225C43C2311yescharacteristic
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#336C2314yes—
236#336C2312yes—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8C2412yes—
240C22×Q8C2411yescharacteristic
241C22×D4C2×D411yescharacteristic
242C23×C4C2×D412yes—
243C23×C4C2×D412yes—
244C23×C4C2412yes—
245C2×Q16—24no—
246C2×Q16—24no—
247C2×Q16—42no—
248C22×C8C2412yes—
249C22×C8C2412yes—
250C22×C8C2414yes—
251C4⋊Q8—22no—
252C4⋊Q8—24no—
253C4⋊Q8—22no—
254C4⋊Q8—24no—
255C4⋊Q8—44no—
256C4⋊Q8—82no—
257C4⋊Q8#4114yes—
258C4⋊Q8#4114yes—
259#126—216no—
260#125—48no—
261#124—28no—
262C22⋊Q8—24no—
263C22⋊Q8—24no—
264C22⋊Q8—28no—
265C22⋊Q8—44no—
266C22⋊Q8—48no—
267C22⋊Q8—84no—
268C4×Q8—28no—
269C4×Q8—22no—
270C4×Q8—28no—
271C4×Q8—22no—
272C4×Q8—24no—
273C4×Q8—22no—
274C4×Q8—24no—
275C4×Q8—22no—
276C4×Q8—41no—
277C4×Q8—44no—
278C4×Q8—48no—
279C4×Q8—44no—
280C4×Q8—44no—
281C4×Q8#4114yes—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—22no—
285C4×D4—22no—
286#117—24no—
287#117—24no—
288#117—24no—
289#117—42no—
290#117—48no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C42—22no—
333C2×C42—21no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—22no—
344C2×C42—21no—
345C2×C42—22no—
346C2×C42—21no—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357#107—24no—
358#106—28no—
359#106—44no—
360C4⋊C8—22no—
361C4⋊C8—24no—
362C4⋊C8—22no—
363C4⋊C8—24no—
364C4⋊C8—42no—
365#103—24no—
366#103—24no—
367#103—24no—
368#103—28no—
369#103—24no—
370#103—44no—
371#103—44no—
372C22⋊C8—44no—
373C22⋊C8—44no—
374C4×C8—24no—
375C4×C8—44no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—21no—
385#95—21no—
386#95—24no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—24no—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394C24C22×D412yes—
395C2×Q8—24no—
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—22no—
403C2×Q8—21no—
404C2×Q8—22no—
405C2×Q8—21no—
406C2×Q8—44no—
407C2×Q8—42no—
408C2×Q8—44no—
409C2×Q8—42no—
410C2×Q8—44no—
411C2×Q8C2×#4112yes—
412C2×Q8C2×#4114yes—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—42no—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C2511yescharacteristic
447Q16—48no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C4⋊C4—28no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—42no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—81no—
489C4⋊C4—84no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4#13714yes—
493C4⋊C4C2×#4114yes—
494C4⋊C4C2×#4112yes—
495C4⋊C42−1+412yes—
496C22⋊C4—28no—
497C22⋊C4—28no—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—44no—
502C22⋊C4—48no—
503C22⋊C4—82no—
504C22⋊C4—84no—
505C42—28no—
506C42—24no—
507C42—24no—
508C42—24no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—21no—
514C42—21no—
515C42—41no—
516C42—44no—
517C42—48no—
518C42—42no—
519C42—44no—
520C42—42no—
521C42—44no—
522C42—44no—
523C42—42no—
524C42—42no—
525C42—81no—
526C42—82no—
527C42#13514yes—
528C42C2×#4112yes—
529C42C2×#4112yes—
530C42C2×#4112yes—
531C42H(5,𝔽2)12yes—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23C23×D411yescharacteristic
540Q8—22no—
541Q8—24no—
542Q8—24no—
543Q8—22no—
544Q8—48no—
545Q8—41no—
546Q8—42no—
547Q8—41no—
548Q8—42no—
549Q8—88no—
550D4—44no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—48no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—48no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4#52512yes—
602C2×C4#52512yes—
603C2×C4#54612yes—
604C2×C4#54612yes—
605C2×C4C2×#13512yes—
606C2×C4C2×#13512yes—
607C2×C4C2×#13512yes—
608C2×C4C2×#13714yes—
609C2×C4C2×#13712yes—
610C2×C4#57712yes—
611C2×C4C22×#4111yescharacteristic
612C2×C4C22×#4111yescharacteristic
613C2×C4C22×#4111yescharacteristic
614C2×C4C2×H(5,𝔽2)11yescharacteristic
615C2×C4C2×2−1+411yescharacteristic
616C2×C4#58511yescharacteristic
617C8—42no—
618C8—42no—
619C8—44no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22C2×#52511yescharacteristic
628C22C2×#54611yescharacteristic
629C22#348911yescharacteristic
630C22C22×#13511yescharacteristic
631C22C22×#13711yescharacteristic
632C22C2×#57711yescharacteristic
633C22#358611yescharacteristic
634C4—22no—
635C4—22no—
636C4—22no—
637C4—24no—
638C4—42no—
639C4—42no—
640C4—44no—
641C4—44no—
642C4—42no—
643C4—42no—
644C4—41no—
645C4—42no—
646C4—82no—
647C4—82no—
648C4—84no—
649C4—81no—
650C4#310714yes—
651C4#330414yes—
652C4#347511yescharacteristic
653C4#348211yescharacteristic
654C4#351111yescharacteristic
655C4#351711yescharacteristic
656C2—82no—
657C2#3640411yescharacteristic
658C2C2×#310711yescharacteristic
659C2#6059911yescharacteristic
660C2C2×#330411yescharacteristic
661C2#6115711yescharacteristic
662C2#6210111yescharacteristic
663C2#6214711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
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
3. #33196C2semidirect
4. #33154C2semidirect
5. #29219C2semidirect
6. #28283C2semidirect
7. #27169C2semidirect
8. #26667C2semidirect
10. #24703C2semidirect
11. #24157C2semidirect
13. #22964C2semidirect
15. #22771C2semidirect
16. C4×#462C2semidirect
17. #21389C2semidirect

Maximal subgroups

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

subgroupindex
#361342
C4×#5562
#331962
#331542
#292192
#282832
#271692
#266672
#249722
#247032
#241572
#241232
#229642
#227952
#227712
C4×#4622
#213892