Finite Group Databasebeta

← #6382373

Subgroups — #6382373

Order 512.

2180 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
1C2×#3100C211yesmaximal, characteristic
2#34232C211yesmaximal, characteristic, complemented
3#34194C211yesmaximal, characteristic, complemented
4#23093C212yesmaximal, complemented
5#23066C212yesmaximal, complemented
6#23001C212yesmaximal, complemented
7#22962C212yesmaximal, complemented
8#22931C214yesmaximal, complemented
9#22868C214yesmaximal, complemented
10#21410C211yesmaximal, characteristic, complemented
11#21362C211yesmaximal, characteristic, complemented
12#21282C212yesmaximal, complemented
13#21134C212yesmaximal, complemented
14#21129C212yesmaximal, complemented
15#20503C212yesmaximal, complemented
16#20482C211yesmaximal, characteristic, complemented
17C4×#418C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20#3100C4116yescomplemented
21C2×#480C2212yes—
22C2×#474C2211yescharacteristic
23C2×#474C2211yescharacteristic
24C2×#459C2211yescharacteristic
25C2×#459C2211yescharacteristic
26C2×#447C2212yes—
27C2×#442C2214yes—
28C2×C4×D8C2212yes—
29C2×C4×D8C2212yes—
30C2×#431C2211yescharacteristic
31C2×#418C2212yes—
32C2×C4×M16C2211yescharacteristic
33#2366C2212yescomplemented
34#2365C2212yescomplemented
35#2363C2214yes—
36#2354C2212yes—
37#2353—22no—
38#2352—22no—
39D4×C4⋊C4C2212yes—
40#2345C2212yes—
41#2342—22no—
42#2341—21no—
43#2339—21no—
44D4×C22⋊C4C2212yes—
45#2333—22no—
46#2319C2212yes—
47#2319C2212yescomplemented
48#2318—22no—
49C4×#124—22no—
50#2285C2211yescharacteristic
51#2283C2211yescharacteristic
52#2280—21no—
53C2×#380—21no—
54#2275C2212yes—
55#2275C2212yescomplemented
56C4×#117C2211yescharacteristic
57#1990C2212yescomplemented
58#1970C2214yes—
59#1957C2212yescomplemented
60#1948C2212yescomplemented
61#1947C2212yescomplemented
62#1942C2214yescomplemented
63#1936C2214yes—
64#1936C2214yescomplemented
65#1926C2214yes—
66#1925C2214yescomplemented
67#1921C2214yescomplemented
68#1918C2214yescomplemented
69#1895C2214yes—
70#1879C2214yes—
71#1879C2214yes—
72#1864C2214yescomplemented
73#1862C2214yes—
74#1854C2214yes—
75#1776C2214yescomplemented
76C4×#107C2212yescomplemented
77C4×#106C2212yescomplemented
78#1770C2212yescomplemented
79C8×C4⋊C4C2212yescomplemented
80#1762C2211yescharacteristic
81#1762C2211yescharacteristic
82#1762C2211yescharacteristic
83#1762C2211yescharacteristic
84#1762C2212yescomplemented
85#1762C2212yescomplemented
86C4×#102C2214yes—
87C4×#102C2212yes—
88C4×#102C2212yes—
89C4×#102C2214yescomplemented
90#1753C2214yes—
91#1746C2212yescomplemented
92#1736C2212yes—
93#1726C2212yescomplemented
94C22×#41—21no—
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C22×D8C2312yes—
98C22×D8C2312yes—
99C22×M16C2312yes—
100#532—24no—
101#532C2×C418yescomplemented
102C2×C4⋊Q8C2311yescharacteristic
103C2×#127C2311yescharacteristic
104C2×#124—21no—
105C2×#121—21no—
106C2×#121C2312yes—
107C2×C4×Q8—22no—
108C2×C4×D4—22no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4—24no—
113C2×C4×D4C2×C418yescomplemented
114C2×C4×D4C2314yes—
115C2×C4×D4C2314yes—
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2312yes—
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×#117—21no—
122C2×#117—22no—
123C2×#117C2312yes—
124C22×C4⋊C4—22no—
125C22×C4⋊C4C2311yescharacteristic
126C22×C22⋊C4—21no—
127C22×C22⋊C4—22no—
128C22×C42C2311yescharacteristic
129#480C2×C4116yescomplemented
130#474C2×C418yescomplemented
131#474C2×C418yescomplemented
132#459C2×C418yes—
133#459C2×C418yescomplemented
134#447C2×C4116yescomplemented
135#442—216no—
136C4×D8C2×C4116yescomplemented
137C4×D8C2×C4116yescomplemented
138#431C2×C418yescomplemented
139C2×#107C2314yes—
140C2×#107C2312yes—
141C2×#107C2312yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2311yescharacteristic
144C2×C4⋊C8C2311yescharacteristic
145#418—28no—
146C2×#102C2312yes—
147C2×#102C2312yes—
148C2×#102C2312yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×#102C2312yes—
155C2×#102C2312yes—
156C2×C22⋊C8C2312yes—
157C4×M16—24no—
158C2×#97C2311yescharacteristic
159C2×#97C2311yescharacteristic
160C2×#97C2314yes—
161C2×C4×C8C2312yes—
162C2×C4×C8C2314yes—
163C2×C4×C8C2312yes—
164#391—22no—
165#390—22no—
166#390—24no—
167#390C2314yes—
168#390C2314yes—
169#389C2314yes—
170#389C2314yes—
171#388—24no—
172#388—24no—
173#388—24no—
174#388—24no—
175#387—24no—
176#387—24no—
177#386—24no—
178#386—24no—
179#386—24no—
180#385—24no—
181#384C2312yes—
182#383D412yes—
183#382—21no—
184#382D412yes—
185#381—22no—
186#381—22no—
187#381D412yes—
188#380—21no—
189#380—22no—
190#380—22no—
191#380—22no—
192#380—22no—
193#380—22no—
194#380D411yescharacteristic
195#380D411yescharacteristic
196#380C2312yes—
197#379—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2311yescharacteristic
201C4×C4⋊C4C2314yes—
202C4×C4⋊C4C2314yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2312yes—
206C4×C22⋊C4—21no—
207C4×C22⋊C4—21no—
208C4×C22⋊C4—22no—
209C4×C22⋊C4—22no—
210C4×C22⋊C4—24no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—24no—
213C4×C22⋊C4—24no—
214C4×C22⋊C4C2312yes—
215#376—21no—
216#376C2312yes—
217C2×#95—22no—
218C2×#95—22no—
219C43—21no—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#336C2314yes—
229#336C2314yes—
230C25—21no—
231C2×#41—28no—
232C2×#41—84no—
233C22×Q8—21no—
234C22×D4—24no—
235C22×D4—22no—
236C22×D4—21no—
237C22×D4—21no—
238C22×D4—21no—
239C22×D4—22no—
240C22×D4—42no—
241C22×D4C22×C414yes—
242C22×D4C2412yes—
243C22×D4C2411yescharacteristic
244C22×D4C2411yescharacteristic
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—28no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4—24no—
253C23×C4—24no—
254C23×C4—84no—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2411yescharacteristic
258C2×D8—28no—
259C2×D8—44no—
260C2×D8C22×C418yes—
261C2×D8C22×C418yes—
262C2×M16—24no—
263C2×M16—24no—
264C2×M16—42no—
265C22×C8C2412yes—
266C22×C8C2412yes—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C4⋊Q8—22no—
270C4⋊Q8C22×C414yes—
271#127—22no—
272#127C22×C414yes—
273#124—28no—
274#121—28no—
275#121—24no—
276#121C22×C418yes—
277C4×Q8—84no—
278C4×D4—28no—
279C4×D4—28no—
280C4×D4—28no—
281C4×D4—28no—
282C4×D4—28no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—24no—
288C4×D4—44no—
289C4×D4—44no—
290C4×D4—84no—
291C4×D4C22×C414yes—
292C4×D4C22×C414yes—
293C4×D4C22×C414yes—
294C4×D4C22×C414yes—
295C4×D4C22×C418yes—
296#117—24no—
297#117—42no—
298#117—82no—
299#117—84no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—84no—
308C2×C4⋊C4C22×C414yes—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2414yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—84no—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—24no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42C22×C414yes—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42#4114yes—
346C2×C42C2414yes—
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349#107—44no—
350#107C22×C418yes—
351#107C22×C418yes—
352#106—28no—
353C4⋊C8C22×C414yes—
354C4⋊C8C22×C414yes—
355#102—24no—
356#102—24no—
357#102—28no—
358#102—24no—
359#102—44no—
360#102C22×C418yes—
361#102C22×C418yes—
362#102C22×C418yes—
363#102C22×C418yes—
364#102#4118yescomplemented
365C22⋊C8C22×C418yes—
366#97—28no—
367#97—22no—
368#97—22no—
369C4×C8—24no—
370C4×C8—44no—
371C4×C8C22×C418yes—
372#95—21no—
373#95—22no—
374#95—24no—
375#95—24no—
376#95—21no—
377#95—22no—
378#95—24no—
379#95—24no—
380#95—22no—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D414yes—
384#95C2×D414yes—
385#95C2×D414yes—
386C24—21no—
387C24—21no—
388C24—22no—
389C24—24no—
390C24—44no—
391C24—42no—
392C24—42no—
393C24—82no—
394C24C22×D411yescharacteristic
395#41—816no—
396C2×Q8—24no—
397C2×Q8—44no—
398C2×D4—28no—
399C2×D4—28no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—21no—
404C2×D4—22no—
405C2×D4—24no—
406C2×D4—21no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—24no—
410C2×D4—22no—
411C2×D4—44no—
412C2×D4—48no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—81no—
416C2×D4—81no—
417C2×D4—84no—
418C2×D4C23×C412yes—
419C2×D4C23×C414yes—
420C2×D4C23×C412yes—
421C2×D4C2×#4114yes—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—28no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—816no—
444C22×C4—82no—
445C22×C4—82no—
446C22×C4—84no—
447C22×C4—84no—
448C22×C4—84no—
449C22×C4—84no—
450C22×C4—84no—
451C22×C4—82no—
452C22×C4C4×D414yes—
453C22×C4C4×D414yes—
454C22×C4C23×C412yes—
455C22×C4C22×D414yes—
456C22×C4C22×D414yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C2×#4112yes—
463C22×C4C2511yescharacteristic
464D8—216no—
465D8—48no—
466M16—48no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8C23×C414yes—
475C2×C8C23×C414yes—
476C2×C8C2×#4114yes—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—44no—
490C4⋊C4—84no—
491C4⋊C4—82no—
492C4⋊C4—84no—
493C4⋊C4C23×C412yes—
494C4⋊C4C23×C412yes—
495C4⋊C4C23×C414yes—
496C4⋊C4C2×#4114yes—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—28no—
502C22⋊C4—28no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—48no—
506C22⋊C4—44no—
507C22⋊C4—48no—
508C22⋊C4—44no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C22⋊C4—84no—
512C22⋊C4—84no—
513C42—24no—
514C42—24no—
515C42—44no—
516C42—44no—
517C42—81no—
518C42—81no—
519C42—82no—
520C42—84no—
521C42—84no—
522C42C4×D414yes—
523C42C4×D414yes—
524C42C23×C412yes—
525C42C23×C412yes—
526C42C2×#4112yes—
527C42C2×#4112yes—
528C23—21no—
529C23—21no—
530C23—22no—
531C23—22no—
532C23—24no—
533C23—22no—
534C23—22no—
535C23—24no—
536C23—48no—
537C23—48no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23—41no—
542C23—42no—
543C23—42no—
544C23—41no—
545C23—44no—
546C23—42no—
547C23—42no—
548C23—44no—
549C23—84no—
550C23—88no—
551C23—88no—
552C23C2×C4×D412yes—
553C23C23×D411yescharacteristic
554Q8—48no—
555D4—24no—
556D4—22no—
557D4—22no—
558D4—22no—
559D4—24no—
560D4—24no—
561D4—48no—
562D4—48no—
563D4—44no—
564D4—42no—
565D4—84no—
566D4—88no—
567D4C4×#4114yes—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—48no—
595C2×C4—48no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4—88no—
605C2×C4—84no—
606C2×C4—168no—
607C2×C4—168no—
608C2×C4C2×C4×D412yes—
609C2×C4C2×C4×D412yes—
610C2×C4C2×C4×D412yes—
611C2×C4C2×C4×D412yes—
612C2×C4C2×C4×D412yes—
613C2×C4#57614yes—
614C2×C4C24×C411yescharacteristic
615C2×C4C22×#4112yes—
616C2×C4C22×#4111yescharacteristic
617C2×C4#58512yes—
618C8—24no—
619C8—24no—
620C8—24no—
621C8—42no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—24no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—44no—
630C22—44no—
631C22—42no—
632C22—42no—
633C22—44no—
634C22—44no—
635C22—42no—
636C22—42no—
637C22—44no—
638C22—816no—
639C22—84no—
640C22—82no—
641C22—84no—
642C22—82no—
643C22C22×C4×D411yescharacteristic
644C22D4×#4112yes—
645C22C2×#57612yes—
646C22#358511yescharacteristic
647C22#358611yescharacteristic
648C4—22no—
649C4—22no—
650C4—22no—
651C4—22no—
652C4—42no—
653C4—42no—
654C4—44no—
655C4—44no—
656C4—44no—
657C4—44no—
658C4—82no—
659C4—84no—
660C4#294814yes—
661C4C2×C4×#4111yescharacteristic
662C4#342811yescharacteristic
663C4#347512yes—
664C2—42no—
665C2—44no—
666C2—84no—
667C2—82no—
668C2#3413611yescharacteristic
669C2C2×#294811yescharacteristic
670C2#6017011yescharacteristic
671C2#6155312yes—
672C2#6155512yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#3100C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2283 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34232C2semidirect
3. #34194C2semidirect
4. #23093C2semidirect
5. #23066C2semidirect
6. #23001C2semidirect
7. #22962C2semidirect
8. #22931C2semidirect
9. #22868C2semidirect
10. #21410C2semidirect
11. #21362C2semidirect
12. #21282C2semidirect
13. #21134C2semidirect
14. #21129C2semidirect
15. #20503C2semidirect
16. #20482C2semidirect
17. C4×#418C2semidirect
20. #3100C4semidirect
33. #2366C22semidirect
34. #2365C22semidirect
47. #2319C22semidirect
55. #2275C22semidirect
57. #1990C22semidirect
59. #1957C22semidirect
60. #1948C22semidirect
61. #1947C22semidirect
62. #1942C22semidirect
64. #1936C22semidirect
66. #1925C22semidirect
67. #1921C22semidirect
68. #1918C22semidirect
72. #1864C22semidirect
75. #1776C22semidirect
76. C4×#107C22semidirect
77. C4×#106C22semidirect
78. #1770C22semidirect
79. C8×C4⋊C4C22semidirect
84. #1762C22semidirect
85. #1762C22semidirect
89. C4×#102C22semidirect
91. #1746C22semidirect
93. #1726C22semidirect
101. #532C2×C4semidirect
113. C2×C4×D4C2×C4semidirect
129. #480C2×C4semidirect
130. #474C2×C4semidirect
131. #474C2×C4semidirect
133. #459C2×C4semidirect
134. #447C2×C4semidirect
136. C4×D8C2×C4semidirect
137. C4×D8C2×C4semidirect
138. #431C2×C4semidirect
364. #102#41semidirect

Maximal subgroups

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

subgroupindex
C2×#31002
#342322
#341942
#230932
#230662
#230012
#229622
#229312
#228682
#214102
#213622
#212822
#211342
#211292
#205032
#204822
C4×#4182