Finite Group Databasebeta

← #6487712

Subgroups — #6487712

Order 512.

2137 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3105C211yesmaximal, characteristic
2C2×#3101C211yesmaximal, characteristic, complemented
3#37633C211yesmaximal, characteristic, complemented
4#34181C211yesmaximal, characteristic, complemented
5#32917C212yesmaximal
6#32914C212yesmaximal, complemented
7#29768C212yesmaximal, complemented
8#29764C212yesmaximal, complemented
9#24822C214yesmaximal, complemented
10#24815C212yesmaximal, complemented
11#24812C212yesmaximal, complemented
12#24737C214yesmaximal, complemented
13#24730C212yesmaximal, complemented
14#24727C212yesmaximal, complemented
15#21952C211yesmaximal, characteristic, complemented
16#21864C211yesmaximal, characteristic, complemented
17#21862C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#529C2211yescharacteristic
20C2×#518C2212yes—
21C22×C4×D4C2211yescharacteristic
22#3105—28no—
23#3101—28no—
24C2×#484C2211yescharacteristic
25C2×#481C2211yescharacteristic
26C2×#483C2212yes—
27C2×#478C2212yes—
28C2×#476C2212yes—
29C2×#463C2212yes—
30C2×#463C2214yes—
31C2×#461C2212yes—
32C2×#461C2214yes—
33C2×#450C2212yes—
34C2×#450C2211yescharacteristic
35C2×#447C2211yescharacteristic
36C2×C4×SD16C2214yes—
37C2×C4×SD16C2212yes—
38C2×C4×D8C2212yes—
39C2×#431C2211yescharacteristic
40C2×#431C2211yescharacteristic
41#2814—22no—
42#2637—22no—
43#2629—22no—
44#2621C2212yes—
45#2620C2212yes—
46#2567C2211yescharacteristic, complemented
47#2565C2212yescomplemented
48#2563C2211yescharacteristic, complemented
49#2547C2211yescharacteristic, complemented
50#2546C2211yescharacteristic, complemented
51#2545C2212yescomplemented
52#2485—22no—
53#2349—22no—
54#2349C2214yes—
55D4×C4⋊C4—22no—
56D4×C4⋊C4C2212yes—
57D4×C4⋊C4C2212yes—
58#2316C2212yes—
59#2315—22no—
60#2285C2211yescharacteristic
61#2285C2211yescharacteristic
62C2×#380—22no—
63#2085C2214yescomplemented
64#2084C2214yescomplemented
65#2068C2214yescomplemented
66#2066C2214yescomplemented
67#2066C2214yescomplemented
68#2065C2214yescomplemented
69#2043C2214yescomplemented
70#2042C2214yescomplemented
71#1991C2212yescomplemented
72#1990C2212yescomplemented
73#1958C2212yescomplemented
74#1956C2212yescomplemented
75#1939C2212yescomplemented
76#1939C2212yescomplemented
77#1939C2214yescomplemented
78#1879C2214yes—
79#1877C2214yes—
80#1877C2214yes—
81#1877C2214yes—
82#1875C2212yes—
83#1873C2212yes—
84#1873C2212yes—
85#1873C2212yes—
86#1805C2214yescomplemented
87#1804C2214yes—
88#1803C2212yescomplemented
89#1802C2214yes—
90#1802C2212yescomplemented
91#1798C2212yes—
92#1798C2212yes—
93C23×D4C2311yescharacteristic
94C24×C4—21no—
95C22×SD16C2314yes—
96C22×SD16C2312yes—
97C22×D8C2312yes—
98#533—24no—
99#533—24no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103#529—24no—
104#529—24no—
105C2×#126—21no—
106C2×#125—21no—
107C2×#125C2311yescharacteristic
108C2×#124C2311yescharacteristic
109C2×#124C2312yes—
110C2×#123—21no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8C2312yes—
113C2×C22⋊Q8C2311yescharacteristic
114C2×#121C2311yescharacteristic
115#518—48no—
116C2×C4×Q8—21no—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×D4—21no—
120C2×C4×D4—24no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4—24no—
124C2×C4×D4D418yes—
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2314yes—
127C2×C4×D4C2314yes—
128C2×C4×D4C2312yes—
129C2×C4×D4C2312yes—
130C2×#117—21no—
131C2×#117—21no—
132C2×#117C2312yes—
133C2×#117C2312yes—
134C22×C4⋊C4—22no—
135C22×C4⋊C4—22no—
136C22×C4⋊C4C2311yescharacteristic
137C22×C22⋊C4—21no—
138C22×C22⋊C4—24no—
139C22×C42C2311yescharacteristic
140#484—24no—
141#483—28no—
142#481—24no—
143#478—28no—
144#476—28no—
145#463—28no—
146#463—48no—
147#461—28no—
148#461—48no—
149#450—28no—
150#450—24no—
151#447—24no—
152C4×SD16—216no—
153C4×SD16—28no—
154C4×D8—28no—
155#431—24no—
156#431—24no—
157C2×#107C2312yes—
158C2×#106C2312yes—
159C2×#106C2314yes—
160C2×C4⋊C8C2312yes—
161C2×C4⋊C8C2312yes—
162C2×C4⋊C8C2314yes—
163C2×C4⋊C8C2314yes—
164C2×#103C2314yes—
165C2×#103C2312yes—
166C2×#103C2314yes—
167C2×#103C2314yes—
168C2×#103C2314yes—
169C2×#102C2312yes—
170C2×#102C2312yes—
171C2×#102C2314yes—
172C2×#102C2312yes—
173C2×#102C2314yes—
174C2×C22⋊C8C2312yes—
175C2×C22⋊C8C2311yescharacteristic
176C2×C22⋊C8C2311yescharacteristic
177C2×C4×C8C2312yes—
178C2×C4×C8C2312yes—
179#398—24no—
180#398—24no—
181#398—22no—
182#397—24no—
183#397—22no—
184#396—24no—
185#396—24no—
186#395—22no—
187#394—24no—
188#393—22no—
189#391C2312yes—
190#391C2312yes—
191#391C2312yes—
192#390C2312yes—
193#389—22no—
194#389—24no—
195#389—22no—
196#389C2314yes—
197#389C2314yes—
198#388—24no—
199#387—24no—
200#385—24no—
201#385—24no—
202#385—24no—
203#385—24no—
204#384D412yes—
205#384C2312yes—
206#384C2312yes—
207#382D412yes—
208#382C2312yes—
209#382C2312yes—
210#380—21no—
211#380—21no—
212#380—22no—
213#380—22no—
214#380—22no—
215#380—22no—
216#380—24no—
217#380—24no—
218#380D412yes—
219#380D412yes—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2311yescharacteristic
225C2×#95—24no—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#336C2314yes—
231#336C2314yes—
232C25—21no—
233C22×Q8—21no—
234C22×Q8C2411yescharacteristic
235C22×Q8C2412yes—
236C22×D4—24no—
237C22×D4—24no—
238C22×D4—21no—
239C22×D4—21no—
240C22×D4—22no—
241C22×D4C2×D414yes—
242C22×D4C2411yescharacteristic
243C22×D4C2412yes—
244C23×C4—21no—
245C23×C4—22no—
246C23×C4—28no—
247C23×C4—22no—
248C23×C4—21no—
249C23×C4—24no—
250C23×C4—24no—
251C23×C4—84no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2411yescharacteristic
255C2×SD16—28no—
256C2×SD16—24no—
257C2×SD16—24no—
258C2×SD16—28no—
259C2×SD16—28no—
260C2×SD16—24no—
261C2×D8—24no—
262C2×D8—24no—
263C2×D8—24no—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C22×C8C2414yes—
267C4⋊Q8—22no—
268C4⋊Q8—22no—
269C4⋊Q8—22no—
270C4⋊Q8—22no—
271C4⋊Q8—44no—
272#126—44no—
273#125—22no—
274#125—22no—
275#125—44no—
276#124—22no—
277#124—22no—
278#124—44no—
279#123—44no—
280C22⋊Q8—22no—
281C22⋊Q8—24no—
282C22⋊Q8—22no—
283C22⋊Q8—24no—
284C22⋊Q8—44no—
285#121—22no—
286#121—22no—
287C4×Q8—22no—
288C4×Q8—24no—
289C4×Q8—22no—
290C4×Q8—24no—
291C4×Q8—44no—
292C4×D4—28no—
293C4×D4—28no—
294C4×D4—28no—
295C4×D4—28no—
296C4×D4—22no—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—22no—
300C4×D4—44no—
301C4×D4—44no—
302C4×D4—416no—
303C4×D4—44no—
304C4×D4C2×D418yes—
305C4×D4C2×D418yes—
306#117—44no—
307#117—44no—
308#117—44no—
309#117—44no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—84no—
328C2×C4⋊C4—84no—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D414yes—
332C2×C4⋊C4C2×D414yes—
333C2×C4⋊C4C2×D414yes—
334C2×C4⋊C4C2414yes—
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C22⋊C4—28no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—416no—
354C2×C22⋊C4C2×D414yes—
355C2×C42—21no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D414yes—
361C2×C42#4114yes—
362C2×C42C2412yes—
363C2×C42C2412yes—
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366#107—24no—
367#106—28no—
368#106—24no—
369C4⋊C8—24no—
370C4⋊C8—24no—
371C4⋊C8—44no—
372C4⋊C8—44no—
373#103—28no—
374#103—24no—
375#103—44no—
376#103—44no—
377#103—44no—
378#102—24no—
379#102—24no—
380#102—28no—
381#102—24no—
382#102—44no—
383C22⋊C8—22no—
384C22⋊C8—22no—
385C22⋊C8—24no—
386C4×C8—24no—
387C4×C8—24no—
388#95—22no—
389#95—24no—
390#95—24no—
391#95—22no—
392#95—24no—
393#95—24no—
394#95—24no—
395#95—24no—
396#95C2×D414yes—
397#95C2×D412yes—
398#95C2×D412yes—
399C24—24no—
400C24—24no—
401C24—22no—
402C24—24no—
403C24—21no—
404C24—82no—
405C24—82no—
406C24C22×D411yescharacteristic
407C2×Q8—21no—
408C2×Q8—22no—
409C2×Q8—21no—
410C2×Q8—22no—
411C2×Q8—24no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—24no—
415C2×Q8—42no—
416C2×Q8—82no—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—216no—
420C2×D4—216no—
421C2×D4—24no—
422C2×D4—21no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—42no—
430C2×D4—82no—
431C2×D4C22×D414yes—
432C2×D4H(5,𝔽2)14yes—
433C2×D42−1+414yes—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—28no—
440C22×C4—28no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—42no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—42no—
453C22×C4—44no—
454C22×C4—816no—
455C22×C4—82no—
456C22×C4—82no—
457C22×C4—84no—
458C22×C4—84no—
459C22×C4—84no—
460C22×C4—84no—
461C22×C4—82no—
462C22×C4#12114yes—
463C22×C4#12114yes—
464C22×C4C22×D414yes—
465C22×C4C22×D414yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C2×#4112yes—
473C22×C4C2511yescharacteristic
474SD16—416no—
475SD16—48no—
476D8—48no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—24no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—82no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C4⋊C4—84no—
514C4⋊C4—84no—
515C4⋊C4C22×D414yes—
516C22⋊C4—28no—
517C22⋊C4—28no—
518C22⋊C4—44no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—84no—
529C22⋊C4—84no—
530C42—24no—
531C42—24no—
532C42—44no—
533C42—42no—
534C42—44no—
535C42#12114yes—
536C42#12114yes—
537C42C22×D412yes—
538C42C22×D412yes—
539C42C2×#4112yes—
540C42C2×#4112yes—
541C23—216no—
542C23—22no—
543C23—24no—
544C23—21no—
545C23—24no—
546C23—21no—
547C23—24no—
548C23—22no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—42no—
558C23—84no—
559C23—84no—
560C23—88no—
561C23—88no—
562C23C2×#12112yes—
563C23C23×D411yescharacteristic
564Q8—42no—
565Q8—44no—
566Q8—42no—
567Q8—44no—
568Q8—84no—
569D4—216no—
570D4—24no—
571D4—24no—
572D4—24no—
573D4—42no—
574D4—42no—
575D4—84no—
576D4#53718yes—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—416no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—88no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—168no—
615C2×C4—168no—
616C2×C4C2×#12112yes—
617C2×C4C2×#12112yes—
618C2×C4C2×#12112yes—
619C2×C4C2×#12112yes—
620C2×C4C2×#12112yes—
621C2×C4#57714yes—
622C2×C4C23×D411yescharacteristic
623C2×C4C22×#4111yescharacteristic
624C2×C4C2×H(5,𝔽2)12yes—
625C2×C4C2×2−1+412yes—
626C8—42no—
627C8—42no—
628C8—44no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—28no—
633C22—44no—
634C22—44no—
635C22—42no—
636C22—44no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22—42no—
641C22—816no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—82no—
646C22—82no—
647C22C22×#12111yescharacteristic
648C22#348411yescharacteristic
649C22#348611yescharacteristic
650C22C2×#57712yes—
651C22#358412yes—
652C4—22no—
653C4—22no—
654C4—22no—
655C4—22no—
656C4—44no—
657C4—44no—
658C4—44no—
659C4—81no—
660C4—81no—
661C4—82no—
662C4—81no—
663C4—82no—
664C4—84no—
665C4#304214yes—
666C4C2×#53712yes—
667C4#349212yes—
668C2—42no—
669C2—48no—
670C2—161no—
671C2#3522311yescharacteristic
672C2C2×#304211yescharacteristic
673C2#6042211yescharacteristic
674C2#6165812yes—
675C2#6186412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#447C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3101C2semidirect
3. #37633C2semidirect
4. #34181C2semidirect
6. #32914C2semidirect
7. #29768C2semidirect
8. #29764C2semidirect
9. #24822C2semidirect
10. #24815C2semidirect
11. #24812C2semidirect
12. #24737C2semidirect
13. #24730C2semidirect
14. #24727C2semidirect
15. #21952C2semidirect
16. #21864C2semidirect
17. #21862C2semidirect
46. #2567C22semidirect
47. #2565C22semidirect
48. #2563C22semidirect
49. #2547C22semidirect
50. #2546C22semidirect
51. #2545C22semidirect
63. #2085C22semidirect
64. #2084C22semidirect
65. #2068C22semidirect
66. #2066C22semidirect
67. #2066C22semidirect
68. #2065C22semidirect
69. #2043C22semidirect
70. #2042C22semidirect
71. #1991C22semidirect
72. #1990C22semidirect
73. #1958C22semidirect
74. #1956C22semidirect
75. #1939C22semidirect
76. #1939C22semidirect
77. #1939C22semidirect
86. #1805C22semidirect
88. #1803C22semidirect
90. #1802C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31052
C2×#31012
#376332
#341812
#329172
#329142
#297682
#297642
#248222
#248152
#248122
#247372
#247302
#247272
#219522
#218642
#218622