Finite Group Databasebeta

← #6496421

Subgroups — #6496421

Order 512.

1984 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
1C2×#3109C211yesmaximal, characteristic
2C2×#3077C211yesmaximal, characteristic, complemented
3#37634C211yesmaximal, characteristic, complemented
4#34190C211yesmaximal, characteristic, complemented
5#33205C212yesmaximal, complemented
6#33038C212yesmaximal, complemented
7#29874C212yesmaximal, complemented
8#29834C212yesmaximal, complemented
9#24830C212yesmaximal, complemented
10#24829C212yesmaximal, complemented
11#24693C212yesmaximal, complemented
12#24692C212yesmaximal, complemented
13#24589C214yesmaximal, complemented
14#24539C214yesmaximal, complemented
15#21979C211yesmaximal, characteristic, complemented
16#21891C211yesmaximal, characteristic, complemented
17#21842C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#532C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#3109—28no—
22C2×#481C2212yes—
23#3077—28no—
24C2×#478C2212yes—
25C2×#478C2212yes—
26C2×#476C2211yescharacteristic
27C2×#476C2211yescharacteristic
28C2×#476C2211yescharacteristic
29C2×#476C2211yescharacteristic
30C2×#474C2211yescharacteristic
31C2×#474C2211yescharacteristic
32C2×#459C2211yescharacteristic
33C2×#459C2211yescharacteristic
34C2×#447C2212yes—
35C2×#442C2212yes—
36C2×#442C2212yes—
37C2×#432C2211yescharacteristic
38C2×#428C2212yes—
39C2×#423C2211yescharacteristic
40C2×#418C2212yes—
41#2816—21no—
42#2816—21no—
43#2660C2212yescomplemented
44#2641C2212yescomplemented
45#2632—22no—
46#2616—22no—
47#2563C2212yescomplemented
48#2542C2212yescomplemented
49#2536—22no—
50#2500C2214yes—
51#2357C2212yes—
52#2353—22no—
53#2349C2212yes—
54#2344C2212yes—
55#2342—22no—
56D4×C22⋊C4C2212yes—
57#2319C2212yes—
58#2318—22no—
59#2283C2211yescharacteristic
60#2283C2211yescharacteristic
61#2280—21no—
62C2×#381—21no—
63#2275C2212yes—
64#2085C2214yescomplemented
65#2084C2214yescomplemented
66#2073C2214yes—
67#2071C2214yes—
68#2067C2214yescomplemented
69#2065C2214yescomplemented
70#2054C2214yes—
71#2054C2214yes—
72#1993C2212yescomplemented
73#1993C2212yescomplemented
74#1989C2212yescomplemented
75#1989C2212yescomplemented
76#1988C2212yescomplemented
77#1982C2212yescomplemented
78#1949C2214yescomplemented
79#1948C2214yescomplemented
80#1940C2212yescomplemented
81#1940C2212yescomplemented
82#1926C2214yescomplemented
83#1925C2214yescomplemented
84#1895C2214yes—
85#1879C2214yes—
86#1879C2214yes—
87#1851C2214yes—
88#1843C2212yescomplemented
89#1832C2212yescomplemented
90#1825C2214yes—
91#1804C2214yes—
92#1804C2212yescomplemented
93#1804C2212yescomplemented
94#1802C2214yes—
95C22×#41—21no—
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×D8C2312yes—
99C22×D8C2312yes—
100C22×M16C2312yes—
101#533—24no—
102#533—24no—
103#532—24no—
104#532—24no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×#127C2311yescharacteristic
109C2×#125C2312yes—
110C2×#124—21no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—24no—
114C2×#121—21no—
115C2×#121C2312yes—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4—24no—
122C2×C4×D4D418yes—
123C2×C4×D4C2312yes—
124C2×C4×D4C2314yes—
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×#117—22no—
130C2×#117C2312yes—
131C22×C4⋊C4C2311yescharacteristic
132C22×C22⋊C4—21no—
133C22×C22⋊C4—22no—
134C22×C42C2311yescharacteristic
135#481—28no—
136#478—28no—
137#478—28no—
138#476—24no—
139#476—24no—
140#476—24no—
141#476—24no—
142#474—24no—
143#474—24no—
144#459—24no—
145#459—24no—
146#447—28no—
147#442—28no—
148#442—28no—
149#432—24no—
150#428—28no—
151C2×#107C2312yes—
152C2×#107C2312yes—
153C2×#106C2312yes—
154C2×#106C2312yes—
155C2×#106C2312yes—
156C2×#106C2312yes—
157#423—24no—
158C2×C4⋊C8C2312yes—
159C2×C4⋊C8C2311yescharacteristic
160C2×C4⋊C8C2311yescharacteristic
161C2×C4⋊C8C2311yescharacteristic
162C2×C4⋊C8C2311yescharacteristic
163C2×C4⋊C8C2314yes—
164C2×C4⋊C8C2314yes—
165#418—28no—
166C2×#102C2312yes—
167C2×#102C2312yes—
168C2×#102C2312yes—
169C2×#102C2312yes—
170C2×#102C2312yes—
171C2×#102C2312yes—
172C2×#102C2312yes—
173C2×#102C2312yes—
174C2×#102C2314yes—
175C2×#102C2314yes—
176C2×C22⋊C8C2312yes—
177C2×#97C2312yes—
178#399—22no—
179#399—22no—
180#398—22no—
181#398—22no—
182#395—24no—
183#394—24no—
184#394—24no—
185#394—22no—
186#394—22no—
187#394—24no—
188#393—22no—
189#393—22no—
190#390C2314yes—
191#390C2314yes—
192#390C2314yes—
193#389C2314yes—
194#388—24no—
195#388—24no—
196#388—24no—
197#388—24no—
198#387—24no—
199#386—24no—
200#385—24no—
201#384C2314yes—
202#383—21no—
203#383D412yes—
204#382D412yes—
205#382C2314yes—
206#382C2312yes—
207#381—21no—
208#381—22no—
209#381—22no—
210#381—22no—
211#381D411yescharacteristic
212#381D411yescharacteristic
213#380—24no—
214#380—22no—
215#380—22no—
216#380D412yes—
217#380C2312yes—
218#379—22no—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2314yes—
221C4×C4⋊C4C2312yes—
222C4×C22⋊C4—21no—
223C4×C22⋊C4—22no—
224C4×C22⋊C4—24no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4C2312yes—
227#376—21no—
228#376C2312yes—
229C2×#95—22no—
230C2×#95—22no—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#336C2314yes—
238#336C2314yes—
239C25—21no—
240C2×#41—44no—
241C2×#41—84no—
242C22×Q8—21no—
243C22×Q8—21no—
244C22×Q8—21no—
245C22×D4—24no—
246C22×D4—22no—
247C22×D4—21no—
248C22×D4—21no—
249C22×D4—21no—
250C22×D4—22no—
251C22×D4—42no—
252C22×D4C2×D414yes—
253C22×D4C2412yes—
254C22×D4C2411yescharacteristic
255C22×D4C2411yescharacteristic
256C23×C4—21no—
257C23×C4—28no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4—22no—
263C23×C4—24no—
264C23×C4—84no—
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2411yescharacteristic
268C2×D8—24no—
269C2×D8—24no—
270C2×D8—24no—
271C2×D8—24no—
272C2×D8—42no—
273C2×D8—42no—
274C2×M16—24no—
275C2×M16—44no—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C4⋊Q8—22no—
281C4⋊Q8—22no—
282C4⋊Q8—22no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286#127—22no—
287#127—22no—
288#125—24no—
289#125—24no—
290#124—44no—
291C22⋊Q8—44no—
292C22⋊Q8—44no—
293C22⋊Q8—88no—
294#121—24no—
295#121—24no—
296#121—44no—
297C4×Q8—44no—
298C4×Q8—44no—
299C4×D4—28no—
300C4×D4—28no—
301C4×D4—28no—
302C4×D4—24no—
303C4×D4—22no—
304C4×D4—22no—
305C4×D4—22no—
306C4×D4—22no—
307C4×D4—24no—
308C4×D4—44no—
309C4×D4—44no—
310C4×D4C2×D414yes—
311C4×D4C2×D414yes—
312C4×D4C2×D414yes—
313C4×D4C2×D414yes—
314#117—24no—
315#117—24no—
316#117—84no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4C2×D414yes—
328C2×C4⋊C4C2×D414yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2414yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C22⋊C4—28no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—84no—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C42—22no—
355C2×C42—24no—
356C2×C42—22no—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D414yes—
360C2×C42#4114yes—
361C2×C42C2414yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#107—24no—
365#107—24no—
366#106—24no—
367#106—24no—
368#106—24no—
369#106—24no—
370C4⋊C8—22no—
371C4⋊C8—22no—
372C4⋊C8—24no—
373C4⋊C8—22no—
374C4⋊C8—22no—
375C4⋊C8—44no—
376C4⋊C8—44no—
377#102—24no—
378#102—24no—
379#102—24no—
380#102—24no—
381#102—24no—
382#102—24no—
383#102—24no—
384#102—24no—
385#102—44no—
386#102—44no—
387C22⋊C8—24no—
388#97—24no—
389#95—22no—
390#95—22no—
391#95—24no—
392#95—24no—
393#95—22no—
394#95—22no—
395#95—24no—
396#95—24no—
397#95C2×D412yes—
398#95C2×D412yes—
399#95C2×D412yes—
400#95C2×D414yes—
401#95C2×D412yes—
402#95C2×D414yes—
403C24—21no—
404C24—21no—
405C24—24no—
406C24—22no—
407C24—44no—
408C24—44no—
409C24—82no—
410C24C22×D411yescharacteristic
411#41—816no—
412C2×Q8—44no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8—82no—
417C2×Q8—82no—
418C2×D4—28no—
419C2×D4—24no—
420C2×D4—22no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—44no—
432C2×D4—48no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—82no—
438C2×D4—82no—
439C2×D4—84no—
440C2×D4C22×D412yes—
441C2×D4C22×D412yes—
442C2×D42−1+414yes—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—82no—
463C22×C4—82no—
464C22×C4—82no—
465C22×C4—82no—
466C22×C4—84no—
467C22×C4—84no—
468C22×C4—816no—
469C22×C4#12114yes—
470C22×C4#12114yes—
471C22×C4C22×D414yes—
472C22×C4C22×D414yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D412yes—
479C22×C4C2×#4112yes—
480C22×C4C2511yescharacteristic
481D8—48no—
482D8—48no—
483M16—48no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—22no—
493C2×C8—22no—
494C4⋊C4—24no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C4⋊C4—84no—
514C4⋊C4C22×D412yes—
515C4⋊C4C22×D412yes—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—48no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—84no—
528C22⋊C4—84no—
529C22⋊C4—84no—
530C22⋊C4—84no—
531C22⋊C4—84no—
532C22⋊C4—84no—
533C42—24no—
534C42—24no—
535C42—44no—
536C42—82no—
537C42—84no—
538C42#12114yes—
539C42#12114yes—
540C42C22×D412yes—
541C42C22×D412yes—
542C42C2×#4112yes—
543C42C2×#4112yes—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—24no—
548C23—22no—
549C23—22no—
550C23—48no—
551C23—48no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—44no—
558C23—44no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—84no—
563C23—816no—
564C23C2×#12112yes—
565C23C23×D411yescharacteristic
566Q8—84no—
567Q8—84no—
568Q8—84no—
569D4—22no—
570D4—22no—
571D4—22no—
572D4—24no—
573D4—48no—
574D4—42no—
575D4—42no—
576D4—42no—
577D4—42no—
578D4—42no—
579D4—84no—
580D4—84no—
581D4—88no—
582D4#53614yes—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—48no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—88no—
619C2×C4—168no—
620C2×C4C2×#12112yes—
621C2×C4C2×#12112yes—
622C2×C4C2×#12112yes—
623C2×C4C2×#12112yes—
624C2×C4C2×#12112yes—
625C2×C4C23×D411yescharacteristic
626C2×C4C22×#4111yescharacteristic
627C2×C4C2×2−1+412yes—
628C2×C4#58512yes—
629C8—42no—
630C8—42no—
631C8—42no—
632C8—42no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—24no—
637C22—42no—
638C22—42no—
639C22—48no—
640C22—44no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—816no—
645C22—84no—
646C22—82no—
647C22—82no—
648C22—84no—
649C22—84no—
650C22—82no—
651C22C22×#12111yescharacteristic
652C22#348912yes—
653C22#358512yes—
654C22#358512yes—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—22no—
659C4—22no—
660C4—42no—
661C4—42no—
662C4—44no—
663C4—44no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4—84no—
668C4C2×#53611yescharacteristic
669C4#344911yescharacteristic
670C4#349512yes—
671C2—42no—
672C2—44no—
673C2—82no—
674C2—162no—
675C2#3530611yescharacteristic
676C2#6042311yescharacteristic
677C2#6042611yescharacteristic
678C2#6223114yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3109C2
LCSG ▷ C22×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. C2×#3077C2semidirect
3. #37634C2semidirect
4. #34190C2semidirect
5. #33205C2semidirect
6. #33038C2semidirect
7. #29874C2semidirect
8. #29834C2semidirect
9. #24830C2semidirect
10. #24829C2semidirect
11. #24693C2semidirect
12. #24692C2semidirect
13. #24589C2semidirect
14. #24539C2semidirect
15. #21979C2semidirect
16. #21891C2semidirect
17. #21842C2semidirect
43. #2660C22semidirect
44. #2641C22semidirect
47. #2563C22semidirect
48. #2542C22semidirect
64. #2085C22semidirect
65. #2084C22semidirect
68. #2067C22semidirect
69. #2065C22semidirect
72. #1993C22semidirect
73. #1993C22semidirect
74. #1989C22semidirect
75. #1989C22semidirect
76. #1988C22semidirect
77. #1982C22semidirect
78. #1949C22semidirect
79. #1948C22semidirect
80. #1940C22semidirect
81. #1940C22semidirect
82. #1926C22semidirect
83. #1925C22semidirect
88. #1843C22semidirect
89. #1832C22semidirect
92. #1804C22semidirect
93. #1804C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31092
C2×#30772
#376342
#341902
#332052
#330382
#298742
#298342
#248302
#248292
#246932
#246922
#245892
#245392
#219792
#218912
#218422