Finite Group Databasebeta

← #6482129

Subgroups — #6482129

Order 512.

1844 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3236C211yesmaximal, characteristic, complemented
2C2×#3234C211yesmaximal, characteristic
3#35180C211yesmaximal, characteristic
4C2×#2370C211yesmaximal, characteristic, complemented
5#33111C212yesmaximal, complemented
6#33110C212yesmaximal, complemented
7#29761C212yesmaximal, complemented
8#29760C212yesmaximal, complemented
9#25468C214yesmaximal, complemented
10#25466C214yesmaximal, complemented
11#25291C214yesmaximal, complemented
12#25289C214yesmaximal, complemented
13#21955C211yesmaximal, characteristic, complemented
14#21395C211yesmaximal, characteristic
15#21394C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#532C2211yescharacteristic
18C22×#125C2211yescharacteristic
19#3236—28no—
20#3234—28no—
21C2×#501C2212yes—
22C2×#501C2212yes—
23C2×#491C2212yes—
24C2×#490C2212yes—
25C2×#484C2212yes—
26C2×#481C2212yes—
27C2×#483C2212yes—
28C2×#480C2212yes—
29C2×#479C2214yes—
30C2×#478C2214yes—
31C2×#432C2211yescharacteristic
32C2×#432C2211yescharacteristic
33#2850C2212yes—
34#2850C2212yes—
35#2639—21no—
36#2639—21no—
37#2592—22no—
38#2568C2212yes—
39#2563C2212yes—
40#2562—22no—
41#2548C2212yes—
42#2547C2212yes—
43#2370—22no—
44#2370—22no—
45#2370C2214yes—
46#2370C2214yes—
47#2311—21no—
48#2311—21no—
49C2×#389—21no—
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#387—21no—
53C2×#387—21no—
54C2×C4×C4⋊C4—21no—
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59#2085C2214yes—
60#2085C2214yes—
61#2068C2214yes—
62#2067C2214yes—
63#2066C2214yes—
64#2065C2214yes—
65#2042C2214yes—
66#2042C2214yes—
67#1988C2212yes—
68#1987C2212yes—
69#1983C2214yes—
70#1983C2214yes—
71#1982C2212yes—
72#1981C2212yes—
73#1949C2214yes—
74#1949C2214yes—
75#1940C2214yes—
76#1940C2214yes—
77#1928C2214yes—
78#1927C2214yes—
79#1926C2214yes—
80#1925C2214yes—
81#1874C2212yes—
82#1874C2212yes—
83#1872C2212yes—
84#1872C2212yes—
85#1797C2212yes—
86#1797C2212yes—
87#1763C2214yes—
88#1762C2214yes—
89C22×#41—21no—
90#533—24no—
91#533—24no—
92#532—24no—
93#532—24no—
94C2×C4⋊Q8C2311yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2311yescharacteristic
97C2×#127C2311yescharacteristic
98C2×#125—21no—
99C2×#125—21no—
100C2×#125—21no—
101C2×#125—24no—
102C2×#125D418yes—
103C2×#125C2312yes—
104C2×#125C2312yes—
105C2×#124—21no—
106C2×#124C2312yes—
107C2×#124C2312yes—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8C2312yes—
110C2×#121—21no—
111C2×#121C2312yes—
112C2×C4×Q8—21no—
113C2×C4×Q8C2312yes—
114C2×C4×D4—21no—
115C2×C4×D4C2312yes—
116C2×#117—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C42—21no—
128C22×C42—21no—
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133#501—28no—
134#501—28no—
135#491—28no—
136#490—28no—
137#484—28no—
138#483—28no—
139#481—28no—
140#480—28no—
141#479—216no—
142#478—216no—
143#432—24no—
144#432—24no—
145C2×#107C2314yes—
146C2×#107C2314yes—
147C2×#106C2314yes—
148C2×#106C2314yes—
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2314yes—
152C2×C4⋊C8C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×C22⋊C8C2312yes—
162C2×C22⋊C8C2312yes—
163C2×#97C2312yes—
164C2×#97C2312yes—
165#400—22no—
166#400—22no—
167#397—22no—
168#397—22no—
169#394—22no—
170#394—22no—
171#393—22no—
172#393—22no—
173#391—22no—
174#391—22no—
175#391C2312yes—
176#391C2312yes—
177#390—22no—
178#390—22no—
179#390C2312yes—
180#390C2312yes—
181#389—24no—
182#389—24no—
183#389—22no—
184#389—22no—
185#389C2314yes—
186#389C2314yes—
187#388—22no—
188#388—22no—
189#388—22no—
190#388—22no—
191#387—24no—
192#387—24no—
193#387—24no—
194#387—24no—
195#386—22no—
196#386—22no—
197#386—22no—
198#386—22no—
199#382C2312yes—
200#382C2312yes—
201#380—22no—
202#380—22no—
203#380—22no—
204#380—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—24no—
210C4×C4⋊C4—24no—
211C4×C4⋊C4D412yes—
212C4×C4⋊C4D412yes—
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219C2×#95—21no—
220C2×#95—21no—
221C2×#95—21no—
222C2×#95—21no—
223C2×#95—21no—
224C2×#95D411yescharacteristic
225C2×#95D411yescharacteristic
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#336C2314yes—
231#336C2314yes—
232C2×#41—44no—
233C2×#41—44no—
234C2×#41—44no—
235C22×Q8—21no—
236C22×Q8—21no—
237C22×Q8C2412yes—
238C22×D4—22no—
239C22×D4—21no—
240C22×D4—21no—
241C22×D4C2412yes—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2411yescharacteristic
254C22×C8C2414yes—
255C22×C8C2414yes—
256C4⋊Q8—22no—
257C4⋊Q8—22no—
258C4⋊Q8—22no—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261C4⋊Q8—22no—
262#127—22no—
263#127—22no—
264#125—28no—
265#125—28no—
266#125—24no—
267#125—24no—
268#125—44no—
269#125C2×D418yes—
270#125C2×D418yes—
271#124—44no—
272#124—44no—
273#124—44no—
274C22⋊Q8—24no—
275C22⋊Q8—24no—
276C22⋊Q8—44no—
277#121—24no—
278#121—24no—
279#121—44no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—44no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—44no—
286#117—44no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—42no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—82no—
313C2×C4⋊C4C2×D414yes—
314C2×C4⋊C4C2×D414yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C42—22no—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—24no—
341C2×C42—42no—
342C2×C42—42no—
343C2×C42—44no—
344C2×C42—82no—
345C2×C42—82no—
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D414yes—
351C2×C42#4114yes—
352C2×C42C2412yes—
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#107—28no—
357#107—28no—
358#106—28no—
359#106—28no—
360C4⋊C8—24no—
361C4⋊C8—24no—
362C4⋊C8—44no—
363C4⋊C8—44no—
364#103—28no—
365#103—28no—
366#103—28no—
367#103—44no—
368#102—28no—
369#102—28no—
370#102—28no—
371#102—44no—
372C22⋊C8—24no—
373C22⋊C8—24no—
374#97—24no—
375#97—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—22no—
380#95—22no—
381#95—24no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394C24—22no—
395C24C22×D411yescharacteristic
396#41—816no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—24no—
400C2×Q8—24no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—81no—
406C2×Q8—81no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—81no—
419C2×D4—81no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—44no—
460C22×C4#12114yes—
461C22×C4#12114yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472C22×C4C2×#4112yes—
473C22×C4C2511yescharacteristic
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—24no—
480C4⋊C4—24no—
481C4⋊C4—28no—
482C4⋊C4—28no—
483C4⋊C4—28no—
484C4⋊C4—28no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—48no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—84no—
509C4⋊C4C22×D414yes—
510C4⋊C4C22×D414yes—
511C4⋊C42−1+414yes—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C22⋊C4—82no—
519C22⋊C4—82no—
520C22⋊C4—82no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C42—24no—
524C42—24no—
525C42—42no—
526C42—42no—
527C42—48no—
528C42—44no—
529C42—82no—
530C42—82no—
531C42—84no—
532C42#12114yes—
533C42#12114yes—
534C42C22×D412yes—
535C42C22×D412yes—
536C42C2×#4112yes—
537C42C2×#4112yes—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23C2×#12112yes—
550C23#57512yes—
551C23#57512yes—
552C23C23×D411yescharacteristic
553Q8—44no—
554Q8—44no—
555Q8—84no—
556Q8—84no—
557D4—44no—
558D4—44no—
559D4—84no—
560D4—84no—
561D4—88no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—28no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—44no—
591C2×C4—48no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—48no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—48no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—88no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—84no—
618C2×C4C2×#12112yes—
619C2×C4C2×#12112yes—
620C2×C4C2×#12112yes—
621C2×C4C2×#12112yes—
622C2×C4C2×#12112yes—
623C2×C4C23×D411yescharacteristic
624C2×C4C22×#4111yescharacteristic
625C2×C4C2×2−1+412yes—
626C2×C4#58512yes—
627C8—44no—
628C8—44no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22#304014yes—
643C22C22×#12111yescharacteristic
644C22#348511yescharacteristic
645C22#348611yescharacteristic
646C22C2×#57511yescharacteristic
647C22C2×#57511yescharacteristic
648C22#358511yescharacteristic
649C22#358611yescharacteristic
650C4—22no—
651C4—22no—
652C4—22no—
653C4—22no—
654C4—22no—
655C4—44no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—44no—
661C4—44no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4#346112yes—
667C4#349712yes—
668C2—24no—
669C2—162no—
670C2#3515311yescharacteristic
671C2C2×#304011yescharacteristic
672C2#6042511yescharacteristic
673C2#6171111yescharacteristic
674C2#6171311yescharacteristic
675C2#6183111yescharacteristic
676C2#6183211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3236C2semidirect
4. C2×#2370C2semidirect
5. #33111C2semidirect
6. #33110C2semidirect
7. #29761C2semidirect
8. #29760C2semidirect
9. #25468C2semidirect
10. #25466C2semidirect
11. #25291C2semidirect
12. #25289C2semidirect
13. #21955C2semidirect
15. #21394C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32362
C2×#32342
#351802
C2×#23702
#331112
#331102
#297612
#297602
#254682
#254662
#252912
#252892
#219552
#213952
#213942