Finite Group Databasebeta

← #6435213

Subgroups — #6435213

Order 512.

1682 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35281C211yesmaximal, characteristic, complemented
2#34253C211yesmaximal, characteristic
3#29993C214yesmaximal, complemented
4#29990C214yesmaximal, complemented
5#29985C212yesmaximal, complemented
6#29983C212yesmaximal, complemented
7#29982C212yesmaximal, complemented
8#29981C212yesmaximal, complemented
9#23749C212yesmaximal, complemented
10#23738C212yesmaximal
11#23554C211yesmaximal, characteristic, complemented
12C2×#2058C211yesmaximal, characteristic, complemented
13#23452C211yesmaximal, characteristic
14#23451C211yesmaximal, characteristic
15#21174C212yesmaximal, complemented
16#20754C211yesmaximal, characteristic, complemented
17#20652C212yesmaximal
18C2×#533C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#520C2211yescharacteristic
21C2×#428C2212yes—
22C22×#106C2212yes—
23C2×#424C2212yes—
24C2×#417C2212yes—
25C2×#416C2212yes—
26#2868C2212yes—
27#2862C2212yes—
28#2660C2214yes—
29#2567—21no—
30#2567—21no—
31#2546—21no—
32#2546—21no—
33#2502—22no—
34#2434—22no—
35#2353—22no—
36#2352—22no—
37#2347—21no—
38#2346—21no—
39#2345—21no—
40#2344—21no—
41#2320C2212yes—
42#2319C2212yes—
43#2317C2212yes—
44#2317C2212yes—
45#2316C2211yescharacteristic
46#2311C2211yescharacteristic
47#2297C2211yescharacteristic
48#2295C2212yes—
49#2293C2212yes—
50C2×#389C2211yescharacteristic
51#2274C2211yescharacteristic
52C4×#117C2211yescharacteristic
53#2101C2214yes—
54#2101C2214yes—
55#2100C2214yes—
56#2099C2214yes—
57#2097C2214yes—
58#2096C2214yes—
59#2094C2214yes—
60#2092C2214yes—
61#2059C2214yes—
62#2058C2212yes—
63#2058C2212yes—
64#2058C2212yes—
65#2058C2214yes—
66#2057C2212yes—
67#2026C2212yes—
68#2025C2212yes—
69#2024C2211yescharacteristic
70#2024C2212yes—
71#2024C2211yescharacteristic
72#2024C2212yes—
73#2023C2211yescharacteristic
74#2023C2212yes—
75#2023C2211yescharacteristic
76#2023C2212yes—
77#1949C2214yes—
78#1949C2214yes—
79#1948C2214yes—
80#1948C2214yes—
81#1832C2214yes—
82#1831C2212yes—
83#1831C2212yes—
84#1805C2214yes—
85#1804C2214yes—
86#1803C2214yes—
87#1802C2214yes—
88#1736C2212yes—
89#1736C2212yes—
90#1735C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C23×Q8—21no—
97C22×M16C2312yes—
98C23×C8C2312yes—
99#533—44no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—44no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2312yes—
109C2×#125C2312yes—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112#520—24no—
113#520Q818yescomplemented
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8C2312yes—
117C2×C4×D4—22no—
118C2×C4×D4C2312yes—
119C2×#117—21no—
120C2×#117—22no—
121C2×#117C2311yescharacteristic
122C2×#117C2312yes—
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#428—44no—
133C2×#107C2314yes—
134C2×#106—44no—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138#424—28no—
139C2×C4⋊C8C2314yes—
140C2×C4⋊C8C2314yes—
141#417—44no—
142#416—44no—
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×#103C2312yes—
146C2×#102C2312yes—
147C2×#102C2314yes—
148C2×#102C2312yes—
149#398—24no—
150#398—24no—
151#395—24no—
152#395—24no—
153#391—22no—
154#391—22no—
155#391—22no—
156#391—22no—
157#391C2312yes—
158#391C2311yescharacteristic
159#391C2311yescharacteristic
160#390C2312yes—
161#390C2311yescharacteristic
162#390C2311yescharacteristic
163#389—22no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#389C2312yes—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#386—21no—
184#386—21no—
185#386—22no—
186#385—24no—
187#385—24no—
188#384—22no—
189#384—22no—
190#384C2314yes—
191#384C2314yes—
192#383—22no—
193#382—22no—
194#382C2314yes—
195#382C2314yes—
196#381—22no—
197#380—21no—
198#380—21no—
199#380—22no—
200#380—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C22⋊C4—22no—
211C4×C22⋊C4—21no—
212C4×C22⋊C4—21no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4D412yes—
216#376—22no—
217#376C2312yes—
218C2×#95—21no—
219C43C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2311yescharacteristic
233#336C2312yes—
234#336C2312yes—
235#336C2314yes—
236#336C2314yes—
237C2×#41—22no—
238C2×#41—24no—
239C2×#41—24no—
240C2×#41C2×Q814yes—
241C22×Q8—21no—
242C22×Q8—21no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—44no—
246C22×Q8—82no—
247C22×Q8—82no—
248C22×Q8C2411yescharacteristic
249C22×D4—21no—
250C22×D4C2411yescharacteristic
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—21no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2411yescharacteristic
259C2×M16—24no—
260C2×M16—24no—
261C2×M16—24no—
262C22×C8—24no—
263C22×C8—24no—
264C22×C8—24no—
265C22×C8C2412yes—
266C22×C8C2412yes—
267C22×C8C2414yes—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270C4⋊Q8—42no—
271C4⋊Q8—42no—
272C4⋊Q8—44no—
273C4⋊Q8—44no—
274C4⋊Q8—84no—
275C4⋊Q8—84no—
276#125—42no—
277#125—42no—
278C22⋊Q8—44no—
279C22⋊Q8—44no—
280C4×Q8—24no—
281C4×Q8—44no—
282C4×Q8—44no—
283C4×Q8C2×Q818yescomplemented
284C4×D4—216no—
285C4×D4—24no—
286C4×D4C2×Q818yescomplemented
287#117—24no—
288#117—28no—
289#117—24no—
290#117—24no—
291#117—24no—
292#117—84no—
293#117C2×Q814yes—
294#117#4114yes—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—82no—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×Q814yes—
325C2×C4⋊C4C2414yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C42—22no—
344C2×C42—24no—
345C2×C42—22no—
346C2×C42—21no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42C2×D412yes—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358C2×C42C2412yes—
359#107—44no—
360#106—44no—
361#106—44no—
362#106—44no—
363C4⋊C8—28no—
364C4⋊C8—28no—
365#103—42no—
366#103—42no—
367#103—44no—
368#102—42no—
369#102—42no—
370#102—44no—
371#95—22no—
372#95—22no—
373#95—21no—
374#95—22no—
375#95—22no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—21no—
382#95—22no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391C24—21no—
392C24C22×D411yescharacteristic
393#41—216no—
394#41—48no—
395C2×Q8—21no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—44no—
407C2×Q8—42no—
408C2×Q8—82no—
409C2×Q8—82no—
410C2×Q8—816no—
411C2×Q8C22×Q812yes—
412C2×D4—24no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4C22×Q812yes—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—28no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—42no—
440C22×C4—82no—
441C22×C4—82no—
442C22×C4—81no—
443C22×C4—81no—
444C22×C4—82no—
445C22×C4—82no—
446C22×C4—81no—
447C22×C4—81no—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×Q812yes—
458C22×C4C2×#4112yes—
459C22×C4H(5,𝔽2)14yes—
460C22×C42−1+414yes—
461C22×C4C2511yescharacteristic
462M16—48no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—48no—
473C4⋊C4—24no—
474C4⋊C4—28no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4—84no—
500C4⋊C4—84no—
501C4⋊C4—84no—
502C4⋊C4—84no—
503C4⋊C4C22⋊Q814yes—
504C4⋊C4C22⋊Q814yes—
505C4⋊C4C22×Q814yes—
506C22⋊C4—24no—
507C22⋊C4—28no—
508C22⋊C4—44no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—81no—
513C22⋊C4—82no—
514C22⋊C4—81no—
515C22⋊C4—82no—
516C22⋊C4C22⋊Q818yes—
517C42—28no—
518C42—22no—
519C42—22no—
520C42—24no—
521C42—24no—
522C42—22no—
523C42—22no—
524C42—24no—
525C42—42no—
526C42—42no—
527C42—44no—
528C42—42no—
529C42—82no—
530C42—82no—
531C42C22×Q814yes—
532C42C2×#4114yes—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—22no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23C2×C22⋊Q812yes—
545C23#53512yes—
546C23#53612yes—
547C23C23×D411yescharacteristic
548Q8—24no—
549Q8—42no—
550Q8—84no—
551Q8—84no—
552Q8—88no—
553Q8—88no—
554D4—24no—
555D4—48no—
556D4—42no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—48no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—88no—
605C2×C4—84no—
606C2×C4—88no—
607C2×C4—162no—
608C2×C4C2×C22⋊Q814yes—
609C2×C4C2×C22⋊Q814yes—
610C2×C4C2×C22⋊Q812yes—
611C2×C4#53512yes—
612C2×C4#53612yes—
613C2×C4#54218yes—
614C2×C4#54418yes—
615C2×C4#57712yes—
616C2×C4#57712yes—
617C2×C4C23×Q811yescharacteristic
618C2×C4C22×#4111yescharacteristic
619C2×C4C2×H(5,𝔽2)12yes—
620C2×C4C2×2−1+412yes—
621C8—42no—
622C8—42no—
623C8—44no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22#262814yes—
638C22C22×C22⋊Q811yescharacteristic
639C22C2×#53511yescharacteristic
640C22C2×#53611yescharacteristic
641C22C2×#57711yescharacteristic
642C22C2×#57711yescharacteristic
643C22#358511yescharacteristic
644C22#358611yescharacteristic
645C4—24no—
646C4—24no—
647C4—22no—
648C4—22no—
649C4—42no—
650C4—44no—
651C4—41no—
652C4—41no—
653C4—42no—
654C4—42no—
655C4—84no—
656C4—82no—
657C4—82no—
658C4—82no—
659C4—81no—
660C4—81no—
661C4#262814yes—
662C4C2×#54212yes—
663C4C2×#54412yes—
664C2—24no—
665C2—82no—
666C2C2×#262811yescharacteristic
667C2#6049511yescharacteristic
668C2#6049711yescharacteristic
669C2#6081211yescharacteristic
670C2#6081311yescharacteristic
671C2#6088111yescharacteristic
672C2#6088711yescharacteristic
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
1. #35281C2semidirect
3. #29993C2semidirect
4. #29990C2semidirect
5. #29985C2semidirect
6. #29983C2semidirect
7. #29982C2semidirect
8. #29981C2semidirect
9. #23749C2semidirect
11. #23554C2semidirect
12. C2×#2058C2semidirect
15. #21174C2semidirect
16. #20754C2semidirect
113. #520Q8semidirect
283. C4×Q8C2×Q8semidirect
286. C4×D4C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#352812
#342532
#299932
#299902
#299852
#299832
#299822
#299812
#237492
#237382
#235542
C2×#20582
#234522
#234512
#211742
#207542
#206522