Finite Group Databasebeta

← #6473108

Subgroups — #6473108

Order 512.

2052 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3202C212yesmaximal
2#35316C211yesmaximal, characteristic
3#34268C211yesmaximal, characteristic
4#33117C212yesmaximal, complemented
5#32915C212yesmaximal, complemented
6#29759C212yesmaximal, complemented
7#29715C212yesmaximal, complemented
8#28581C212yesmaximal, complemented
9#28575C212yesmaximal, complemented
10#28132C214yesmaximal, complemented
11#27652C212yesmaximal, complemented
12#24764C214yesmaximal, complemented
13#24212C212yesmaximal, complemented
14#21771C212yesmaximal
15#20630C211yesmaximal, characteristic
16C2×#536C2212yes—
17C2×#536C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C2×#519C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21#3202—216no—
22C2×#485C2214yes—
23C2×#485C2214yes—
24C2×#470C2214yes—
25C2×#467C2214yes—
26C2×#462C2214yes—
27C2×#451C2214yes—
28C2×#451C2214yes—
29C2×#424C2212yes—
30C2×#408C2212yes—
31#2857C2212yes—
32#2857C2212yes—
33#2621C2212yes—
34#2621C2212yes—
35#2547—22no—
36#2546—22no—
37#2486—22no—
38#2486—22no—
39#2367—22no—
40#2365—22no—
41#2359—22no—
42#2359—22no—
43#2323C2212yes—
44#2323C2212yes—
45#2311C2211yescharacteristic
46#2310C2212yes—
47C4×C22⋊Q8C2212yes—
48C4×C22⋊Q8C2212yes—
49#2296C2211yescharacteristic
50C2×#391C2211yescharacteristic
51#2285C2211yescharacteristic
52#2274C2211yescharacteristic
53#2084C2212yes—
54#2083C2212yes—
55#2072C2212yes—
56#2072C2212yes—
57#2068C2214yes—
58#2066C2214yes—
59#2056C2214yes—
60#2056C2214yes—
61#2043C2212yes—
62#2043C2212yes—
63#2016C2212yes—
64#2016C2212yes—
65#1981C2214yes—
66#1949C2214yes—
67#1928C2214yes—
68#1927C2214yes—
69#1894C2214yes—
70#1892C2214yes—
71#1878C2214yes—
72#1878C2214yes—
73#1873C2214yes—
74#1872C2214yes—
75#1825C2214yes—
76#1805C2214yes—
77#1798C2212yes—
78#1798C2212yes—
79#1797C2212yes—
80#1797C2212yes—
81#1787C2214yes—
82#1728C2211yescharacteristic
83C2×#336C2211yescharacteristic
84C22×#41C2311yescharacteristic
85C23×Q8C2312yes—
86C23×Q8C2311yescharacteristic
87C22×Q16C2314yes—
88C22×Q16C2314yes—
89C22×M16C2312yes—
90C23×C8C2312yes—
91#536—24no—
92#536—28no—
93#536—28no—
94#536D418yes—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—44no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2312yes—
101C2×#124—21no—
102C2×#124—22no—
103C2×#123—21no—
104C2×#123—21no—
105C2×#123—22no—
106C2×#123—22no—
107C2×C22⋊Q8C2314yes—
108C2×C22⋊Q8C2314yes—
109C2×C22⋊Q8C2312yes—
110C2×C22⋊Q8C2312yes—
111#519—44no—
112C4×#41—44no—
113C2×C4×Q8—22no—
114C2×C4×Q8D412yes—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2312yes—
117C2×C4×D4D412yes—
118C2×C4×D4D412yes—
119C2×C4×D4D412yes—
120C2×#117—22no—
121C2×#117C2312yes—
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C2×#117C2311yescharacteristic
126C2×#117C2311yescharacteristic
127C22×C4⋊C4—22no—
128C22×C4⋊C4C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#485—216no—
133#485—216no—
134#470—216no—
135#467—216no—
136#462—48no—
137#451—216no—
138#451—216no—
139C2×#107C2314yes—
140C2×#106C2314yes—
141#424—44no—
142C2×C4⋊C8C2314yes—
143C2×C4⋊C8C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152#408—28no—
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2314yes—
155#398—24no—
156#398—24no—
157#397—24no—
158#397—24no—
159#391—21no—
160#391—24no—
161#391—22no—
162#391C2314yes—
163#391C2314yes—
164#391C2312yes—
165#391C2312yes—
166#390—21no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389C2312yes—
174#389C2312yes—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#386—21no—
184#386—21no—
185#385—22no—
186#385—22no—
187#385—22no—
188#384C2312yes—
189#382C2312yes—
190#381—24no—
191#380—22no—
192#380—22no—
193#380—22no—
194#380—22no—
195#380—22no—
196#380C2312yes—
197#380C2312yes—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C22⋊C4—22no—
206C4×C22⋊C4—22no—
207C4×C22⋊C4—22no—
208C2×#95—22no—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#336C2312yes—
214#336C2312yes—
215#336C2312yes—
216#336C2312yes—
217C2×#41—42no—
218C2×#41—42no—
219C2×#41C2×D414yes—
220C2×#41#4114yes—
221C22×Q8—21no—
222C22×Q8—22no—
223C22×Q8—22no—
224C22×Q8—24no—
225C22×Q8—24no—
226C22×Q8—42no—
227C22×Q8—42no—
228C22×Q8—42no—
229C22×Q8—42no—
230C22×Q8—42no—
231C22×Q8—42no—
232C22×Q8C2×D414yes—
233C22×Q8C2×D411yescharacteristic
234C22×Q8C2414yes—
235C22×Q8C2412yes—
236C22×D4C2×D411yescharacteristic
237C22×D4C2×D411yescharacteristic
238C22×D4C2×D411yescharacteristic
239C23×C4—21no—
240C23×C4—22no—
241C23×C4—21no—
242C23×C4C2×D412yes—
243C23×C4C2×D412yes—
244C23×C4C2×D412yes—
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C2×Q16—28no—
249C2×Q16—28no—
250C2×Q16—44no—
251C2×Q16—44no—
252C2×Q16—44no—
253C2×Q16—44no—
254C2×M16—24no—
255C2×M16—42no—
256C2×M16—42no—
257C22×C8—24no—
258C22×C8—42no—
259C22×C8—42no—
260C22×C8C2414yes—
261C22×C8C2414yes—
262C4⋊Q8—28no—
263C4⋊Q8—28no—
264C4⋊Q8—44no—
265C4⋊Q8—48no—
266C4⋊Q8—48no—
267#124—28no—
268#124—48no—
269#123—28no—
270#123—28no—
271#123—48no—
272#123—48no—
273C22⋊Q8—24no—
274C22⋊Q8—24no—
275C22⋊Q8—28no—
276C22⋊Q8—28no—
277C22⋊Q8—28no—
278C22⋊Q8—28no—
279C22⋊Q8C2×D418yes—
280C22⋊Q8C2×D418yes—
281C4×Q8—28no—
282C4×Q8—44no—
283C4×Q8—44no—
284C4×Q8—48no—
285C4×D4—44no—
286C4×D4—44no—
287C4×D4—44no—
288#117—22no—
289#117—24no—
290#117—24no—
291#117—24no—
292#117—24no—
293#117—42no—
294#117—48no—
295#117—42no—
296#117C2×D414yes—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—48no—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C42—21no—
333C2×C42—24no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—21no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—42no—
341C2×C42—42no—
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349#107—28no—
350#106—28no—
351C4⋊C8—28no—
352C4⋊C8—44no—
353#103—28no—
354#103—28no—
355#103—28no—
356#103—28no—
357#103—44no—
358#103—44no—
359#103—44no—
360#103—44no—
361C22⋊C8—28no—
362C22⋊C8—28no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—24no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378C24C22×D411yescharacteristic
379C24C22×D411yescharacteristic
380C24C22×D411yescharacteristic
381#41—48no—
382#41—48no—
383C2×Q8—24no—
384C2×Q8—22no—
385C2×Q8—24no—
386C2×Q8—24no—
387C2×Q8—24no—
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—41no—
394C2×Q8—41no—
395C2×Q8—48no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—48no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—84no—
403C2×Q8—84no—
404C2×Q8—84no—
405C2×Q8—84no—
406C2×Q8#12114yes—
407C2×Q8C22×D414yes—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4#12114yes—
415C2×D4#12114yes—
416C2×D4#12114yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—42no—
445C22×C4—82no—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4C2511yescharacteristic
460Q16—88no—
461Q16—88no—
462M16—84no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—84no—
470C4⋊C4—28no—
471C4⋊C4—28no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—82no—
492C4⋊C4C22×D414yes—
493C22⋊C4—24no—
494C22⋊C4—28no—
495C22⋊C4—28no—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—44no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—82no—
504C22⋊C4—82no—
505C42—24no—
506C42—24no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—42no—
511C42—42no—
512C42—42no—
513C42—44no—
514C42—44no—
515C42—44no—
516C42—82no—
517C42—82no—
518C42H(5,𝔽2)14yes—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23C2×#12112yes—
532C23C2×#12112yes—
533C23C2×#12112yes—
534C23C23×D411yescharacteristic
535Q8—28no—
536Q8—44no—
537Q8—48no—
538Q8—44no—
539Q8—44no—
540Q8—44no—
541Q8—44no—
542Q8—44no—
543Q8—88no—
544D4—44no—
545D4—44no—
546D4—44no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—48no—
572C2×C4—48no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4—88no—
592C2×C4—162no—
593C2×C4C2×#12112yes—
594C2×C4C2×#12112yes—
595C2×C4C2×#12112yes—
596C2×C4#53414yes—
597C2×C4#57814yes—
598C2×C4C23×D411yescharacteristic
599C2×C4C22×#4111yescharacteristic
600C2×C4C2×H(5,𝔽2)12yes—
601C2×C4#58512yes—
602C8—44no—
603C8—44no—
604C22—24no—
605C22—24no—
606C22—24no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22C22×#12111yescharacteristic
620C22C2×#53411yescharacteristic
621C22#344911yescharacteristic
622C22C2×#57812yes—
623C22#358612yes—
624C4—24no—
625C4—24no—
626C4—22no—
627C4—22no—
628C4—22no—
629C4—44no—
630C4—42no—
631C4—42no—
632C4—44no—
633C4—84no—
634C4—82no—
635C4—82no—
636C4—84no—
637C4#349612yes—
638C4#350512yes—
639C2—42no—
640C2—42no—
641C2—42no—
642C2#3500011yescharacteristic
643C2#6043212yes—
644C2#6075912yes—
645C2#6101311yescharacteristic
646C2#6101811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #33117C2semidirect
5. #32915C2semidirect
6. #29759C2semidirect
7. #29715C2semidirect
8. #28581C2semidirect
9. #28575C2semidirect
10. #28132C2semidirect
11. #27652C2semidirect
12. #24764C2semidirect
13. #24212C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32022
#353162
#342682
#331172
#329152
#297592
#297152
#285812
#285752
#281322
#276522
#247642
#242122
#217712
#206302