Finite Group Databasebeta

← #6555796

Subgroups — #6555796

Order 512.

1809 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic, complemented
2#34313C211yesmaximal, characteristic
3#30039C212yesmaximal, complemented
4#29798C211yesmaximal, characteristic, complemented
5#29560C212yesmaximal, complemented
6#29369C212yesmaximal, complemented
7#29254C211yesmaximal, characteristic, complemented
8#28438C211yesmaximal, characteristic, complemented
9#26603C214yesmaximal, complemented
10#26322C212yesmaximal, complemented
11#26182C214yesmaximal, complemented
12#25627C212yesmaximal
13#23803C212yesmaximal, complemented
14#23774C211yesmaximal, characteristic, complemented
15#23190C212yesmaximal, complemented
16#23002C212yesmaximal, complemented
17#22832C211yesmaximal, characteristic, complemented
18C2×#550C2211yescharacteristic
19C2×#520—21no—
20C2×#492C2212yes—
21C2×#461C2212yes—
22C2×#461C2214yes—
23C2×#440C2214yes—
24C2×#440C2214yes—
25#2868C2211yescharacteristic
26#2862C2212yes—
27#2582—21no—
28#2571—21no—
29#2570—22no—
30#2568C2212yes—
31#2511—22no—
32#2446C2214yes—
33#2366C2211yescharacteristic
34#2366C2212yes—
35#2365C2212yes—
36#2361—22no—
37#2361—21no—
38#2355—21no—
39#2355—21no—
40Q8×C4⋊C4C2211yescharacteristic
41D4×C4⋊C4C2212yes—
42D4×C4⋊C4C2211yescharacteristic
43#2341—21no—
44#2324—21no—
45#2324—22no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2212yes—
48C4×#127C2211yescharacteristic
49C4×#127C2212yes—
50#2293C2211yescharacteristic
51#2293C2212yes—
52C42×Q8C2211yescharacteristic
53#2099C2211yescharacteristic
54#2095C2214yes—
55#2093C2212yes—
56#2093C2211yescharacteristic
57#2086C2212yes—
58#2072C2214yes—
59#2067C2212yes—
60#2057C2211yescharacteristic
61#2043C2212yes—
62#2043C2212yes—
63#2023C2212yes—
64#2023C2211yescharacteristic
65#2014C2214yes—
66#2014C2214yes—
67#2013C2212yes—
68#2013C2212yes—
69#2013C2212yes—
70#2013C2212yes—
71#1984C2212yes—
72#1982C2212yes—
73#1981C2212yes—
74#1973C2214yes—
75#1956C2212yes—
76#1947C2212yes—
77#1935C2214yes—
78#1928C2214yes—
79#1928C2214yes—
80#1917C2214yes—
81#1880C2214yes—
82#1877C2214yes—
83#1877C2214yes—
84#1877C2214yes—
85#1877C2214yes—
86#1863C2214yes—
87#1862C2214yes—
88#1839C2212yes—
89#1831C2211yescharacteristic
90C4×#107C2212yes—
91#1770C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#102C2212yes—
94#1745C2212yes—
95C22×#41—21no—
96C22×SD16C2314yes—
97C22×SD16C2314yes—
98C22×SD16C2314yes—
99#550—24no—
100#550—24no—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×#127C2312yes—
106C2×#127C2311yescharacteristic
107C2×#121—21no—
108C2×#121—22no—
109#520—48no—
110C2×C4×Q8—21no—
111C2×C4×Q8—21no—
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8C2314yes—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×D4—21no—
117C2×C4×D4—22no—
118C2×C4×D4—22no—
119C2×C4×D4—21no—
120C2×C4×D4—22no—
121C2×C4×D4—21no—
122C2×C4×D4C2312yes—
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2312yes—
125C2×C4×D4C2312yes—
126C2×C4×D4C2312yes—
127C2×#117—22no—
128C2×#117—21no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—22no—
131#492—28no—
132#461—28no—
133#461—48no—
134#440—48no—
135#440—48no—
136C2×#107C2312yes—
137C2×#107C2312yes—
138C2×#107C2314yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2312yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#102C2312yes—
147C2×#102C2312yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#97C2314yes—
151C2×C4×C8C2312yes—
152#400—24no—
153#400—24no—
154#398—24no—
155#391—22no—
156#391—21no—
157#391—22no—
158#391—22no—
159#391C2314yes—
160#391C2314yes—
161#391C2314yes—
162#390—21no—
163#390—22no—
164#390—22no—
165#390—21no—
166#390—22no—
167#390—21no—
168#390C2312yes—
169#390C2311yescharacteristic
170#390C2311yescharacteristic
171#390C2312yes—
172#389—22no—
173#389—21no—
174#389—22no—
175#389—21no—
176#389—22no—
177#389—22no—
178#389C2312yes—
179#389C2311yescharacteristic
180#389C2311yescharacteristic
181#389C2312yes—
182#388—22no—
183#388—21no—
184#388—22no—
185#388—21no—
186#387—24no—
187#387—24no—
188#387—24no—
189#386—22no—
190#386—22no—
191#386—22no—
192#385—22no—
193#385—22no—
194#385—22no—
195#384—21no—
196#384—21no—
197#384—21no—
198#384—21no—
199#384C2312yes—
200#384C2311yescharacteristic
201#384C2311yescharacteristic
202#382—22no—
203#382—22no—
204#382—22no—
205#380—21no—
206#380—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4D412yes—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D412yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—21no—
222C43C2311yescharacteristic
223C43C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#336C2311yescharacteristic
230#336C2314yes—
231#336C2312yes—
232#336C2312yes—
233#336C2311yescharacteristic
234#336C2314yes—
235#336C2314yes—
236C2×#41—28no—
237C2×#41—44no—
238C2×#41—44no—
239C22×Q8—21no—
240C22×Q8C2×D411yescharacteristic
241C22×Q8C2414yes—
242C22×D4—22no—
243C22×D4—21no—
244C22×D4—22no—
245C22×D4—21no—
246C22×D4C2411yescharacteristic
247C22×D4C2412yes—
248C23×C4—22no—
249C23×C4—21no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—22no—
253C2×SD16—28no—
254C2×SD16—28no—
255C2×SD16—28no—
256C2×SD16—28no—
257C2×SD16—44no—
258C2×SD16—44no—
259C2×SD16—44no—
260C2×SD16—44no—
261C2×SD16—44no—
262C22×C8C2412yes—
263C22×C8C2414yes—
264C22×C8C2412yes—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8—42no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270#127—24no—
271#127—24no—
272#127—22no—
273#127#4114yes—
274#121—28no—
275#121—216no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×Q8—44no—
281C4×Q8—44no—
282C4×Q8—44no—
283C4×Q8—44no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—22no—
287C4×D4—48no—
288C4×D4—44no—
289C4×D4—44no—
290C4×D4—44no—
291C4×D4—48no—
292C4×D4—44no—
293C4×D4—44no—
294C4×D4—82no—
295C4×D4—84no—
296C4×D4#4114yes—
297#117—48no—
298#117—44no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—48no—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—22no—
347C2×C42—24no—
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D412yes—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2412yes—
357C2×C42C2411yescharacteristic
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#107—24no—
362#107—42no—
363#107—44no—
364C4⋊C8—24no—
365C4⋊C8—24no—
366#103—28no—
367#103—28no—
368#103—44no—
369#103—44no—
370#103—44no—
371#102—24no—
372#102—44no—
373#102—42no—
374#102—44no—
375#97—28no—
376C4×C8—24no—
377#95—21no—
378#95—22no—
379#95—21no—
380#95—24no—
381#95—24no—
382#95—22no—
383#95—24no—
384#95—22no—
385#95—21no—
386#95—22no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393C24—22no—
394C24—21no—
395#41—432no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—24no—
399C2×Q8—24no—
400C2×Q8—24no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8#13714yes—
408C2×D4—28no—
409C2×D4—28no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4—24no—
416C2×D4—22no—
417C2×D4—24no—
418C2×D4—44no—
419C2×D4—42no—
420C2×D4—44no—
421C2×D4—42no—
422C2×D4—81no—
423C2×D4—82no—
424C2×D4—81no—
425C2×D4—82no—
426C2×D4C2×#4114yes—
427C2×D4C2×#4112yes—
428C22×C4—28no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—42no—
443C22×C4—44no—
444C22×C4—82no—
445C22×C4—82no—
446C22×C4—82no—
447C22×C4—82no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C2511yescharacteristic
461SD16—416no—
462SD16—416no—
463SD16—88no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8H(5,𝔽2)14yes—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—81no—
501C4⋊C4—82no—
502C4⋊C4—84no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4C2×#4112yes—
506C22⋊C4—28no—
507C22⋊C4—24no—
508C22⋊C4—44no—
509C22⋊C4—42no—
510C22⋊C4—44no—
511C22⋊C4—42no—
512C22⋊C4—44no—
513C22⋊C4—42no—
514C22⋊C4—82no—
515C22⋊C4—81no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C22⋊C4—82no—
519C42—24no—
520C42—24no—
521C42—22no—
522C42—22no—
523C42—24no—
524C42—22no—
525C42—22no—
526C42—24no—
527C42—22no—
528C42—22no—
529C42—41no—
530C42—42no—
531C42—44no—
532C42—42no—
533C42—44no—
534C42—84no—
535C42AGL(1,ℤ/8)14yes—
536C42C2×#4112yes—
537C42C2×#4112yes—
538C23—24no—
539C23—22no—
540C23—42no—
541C23—41no—
542C23—41no—
543C23—42no—
544C23—41no—
545C23—42no—
546C23—42no—
547C23—41no—
548C23—42no—
549C23—41no—
550C23—42no—
551C23—41no—
552C23C23×D411yescharacteristic
553Q8—28no—
554Q8—44no—
555Q8—44no—
556Q8—44no—
557Q8—44no—
558Q8—84no—
559D4—24no—
560D4—28no—
561D4—416no—
562D4—48no—
563D4—44no—
564D4—42no—
565D4—84no—
566D4—88no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—416no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—48no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—164no—
613C2×C4—164no—
614C2×C4—164no—
615C2×C4#52512yes—
616C2×C4#52512yes—
617C2×C4C2×AGL(1,ℤ/8)12yes—
618C2×C4C2×AGL(1,ℤ/8)12yes—
619C2×C4C2×AGL(1,ℤ/8)12yes—
620C2×C4C2×#13714yes—
621C2×C4C2×#13712yes—
622C2×C4#57712yes—
623C2×C4C22×#4111yescharacteristic
624C2×C4C22×#4111yescharacteristic
625C2×C4C2×H(5,𝔽2)12yes—
626C2×C4#58512yes—
627C8—24no—
628C8—44no—
629C8—42no—
630C22—42no—
631C22—44no—
632C22—44no—
633C22—42no—
634C22—44no—
635C22—42no—
636C22—82no—
637C22—81no—
638C22—81no—
639C22—82no—
640C22—82no—
641C22—81no—
642C22—81no—
643C22—82no—
644C22C2×#52511yescharacteristic
645C22#348711yescharacteristic
646C22#349811yescharacteristic
647C22C22×AGL(1,ℤ/8)11yescharacteristic
648C22C22×#13711yescharacteristic
649C22C2×#57711yescharacteristic
650C22#358611yescharacteristic
651C4—24no—
652C4—22no—
653C4—22no—
654C4—22no—
655C4—22no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—41no—
661C4—84no—
662C4—81no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4—81no—
667C4#311014yes—
668C4#350612yes—
669C4#351312yes—
670C2—82no—
671C2—84no—
672C2#3584411yescharacteristic
673C2C2×#311011yescharacteristic
674C2#6059111yescharacteristic
675C2#6188811yescharacteristic
676C2#6190611yescharacteristic
677C2#6242111yescharacteristic
678C2#6245411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#550C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36598C2semidirect
3. #30039C2semidirect
4. #29798C2semidirect
5. #29560C2semidirect
6. #29369C2semidirect
7. #29254C2semidirect
8. #28438C2semidirect
9. #26603C2semidirect
10. #26322C2semidirect
11. #26182C2semidirect
13. #23803C2semidirect
14. #23774C2semidirect
15. #23190C2semidirect
16. #23002C2semidirect
17. #22832C2semidirect

Maximal subgroups

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

subgroupindex
#365982
#343132
#300392
#297982
#295602
#293692
#292542
#284382
#266032
#263222
#261822
#256272
#238032
#237742
#231902
#230022
#228322