Finite Group Databasebeta

← #7174861

Subgroups — #7174861

Order 512.

1842 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59827C211yesmaximal, characteristic
2#59817C211yesmaximal, characteristic
3#31998C214yesmaximal, complemented
4#31262C214yesmaximal, complemented
5#29255C211yesmaximal, characteristic, complemented
6#28205C211yesmaximal, characteristic, complemented
7#25884C212yesmaximal, complemented
8#25341C212yesmaximal, complemented
9#25014C212yesmaximal, complemented
10#24863C214yesmaximal, complemented
11#24041C214yesmaximal, complemented
12#23816C211yesmaximal, characteristic, complemented
13D4×#103C212yesmaximal, complemented
14#22260C211yesmaximal, characteristic
15#22218C211yesmaximal, characteristic, complemented
16C2×#559—21no—
17C2×D4×Q8C2212yes—
18C2×D4×Q8C2212yes—
19C2×#487C2211yescharacteristic
20C2×#483C2214yes—
21C2×#475C2212yes—
22C2×C8×D4C2212yes—
23C22×#103C2214yes—
24#2724—22no—
25#2683—22no—
26#2582—21no—
27#2551—21no—
28#2547C2212yes—
29#2547C2212yes—
30#2538C2212yes—
31#2538C2212yes—
32#2486C2214yes—
33#2452—22no—
34#2452—22no—
35#2447—22no—
36#2447—22no—
37#2409C2214yes—
38#2366—21no—
39D4×C4⋊C4C2212yes—
40D4×C4⋊C4C2212yes—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×#127C2211yescharacteristic
46#2100C2212yes—
47#2096C2212yes—
48#2084C2212yes—
49#2077C2214yes—
50#2077C2214yes—
51#2072C2212yes—
52#2043C2212yes—
53#2043C2212yes—
54#2035C2214yes—
55#2035C2214yes—
56#2035C2214yes—
57#2035C2214yes—
58#2024C2212yes—
59#2024C2212yes—
60#1990C2211yescharacteristic
61#1981C2212yes—
62#1956C2211yescharacteristic
63#1954C2212yes—
64#1948C2212yes—
65#1927C2214yes—
66#1927C2214yes—
67#1892C2214yes—
68#1892C2214yes—
69#1872C2214yes—
70#1869C2214yes—
71#1869C2214yes—
72#1865C2214yes—
73#1865C2214yes—
74#1853C2214yes—
75#1844C2211yescharacteristic
76#1831C2211yescharacteristic
77#1807C2214yes—
78#1803C2212yes—
79#1787C2214yes—
80#1787C2214yes—
81#1787C2214yes—
82C4×#103C2212yes—
83C4×#103C2212yes—
84#1744C2211yescharacteristic
85C23×Q8C2314yes—
86C23×C8C2314yes—
87#559—48no—
88D4×Q8—28no—
89D4×Q8—44no—
90D4×Q8—48no—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8—22no—
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2311yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2311yescharacteristic
97C2×#127D411yescharacteristic
98C2×#124—22no—
99C2×#124—22no—
100C2×#123—24no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8C2314yes—
104C2×C22⋊Q8C2314yes—
105C2×C4×Q8—21no—
106C2×C4×Q8—21no—
107C2×C4×Q8C2312yes—
108C2×C4×Q8C2312yes—
109C2×C4×D4—22no—
110C2×C4×D4—22no—
111C2×C4×D4—22no—
112C2×C4×D4—22no—
113C2×C4×D4C2312yes—
114C2×C4×D4C2312yes—
115C2×C4×D4C2312yes—
116C22×C4⋊C4C2314yes—
117#487—24no—
118#483—48no—
119#475—44no—
120C8×D4—44no—
121C2×#106C2314yes—
122C2×#106C2314yes—
123C2×#106C2314yes—
124C2×C4⋊C8C2312yes—
125C2×#103—48no—
126C2×#103C2314yes—
127C2×#103C2312yes—
128C2×#103C2312yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2312yes—
135C2×#103C2312yes—
136C2×C22⋊C8C2314yes—
137C2×C22⋊C8C2314yes—
138C2×C22⋊C8C2314yes—
139C2×C4×C8C2311yescharacteristic
140C2×C4×C8C2311yescharacteristic
141#398—24no—
142#397—24no—
143#397—24no—
144#397—24no—
145#396—24no—
146#396—24no—
147#395—24no—
148#393—24no—
149#391—22no—
150#391—22no—
151#391—21no—
152#391—21no—
153#391C2312yes—
154#391C2312yes—
155#390—22no—
156#390—22no—
157#390D412yes—
158#390D412yes—
159#390D412yes—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389C2312yes—
171#389C2312yes—
172#386—24no—
173#385—24no—
174#385—24no—
175#385—24no—
176#384D411yescharacteristic
177#384C2311yescharacteristic
178#384C2311yescharacteristic
179#384C2311yescharacteristic
180#384C2311yescharacteristic
181#381—24no—
182#380C2314yes—
183#380C2314yes—
184#380C2314yes—
185#380C2314yes—
186C4×C4⋊C4—21no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4C2312yes—
193C4×C4⋊C4C2312yes—
194C43C2311yescharacteristic
195#340C2314yes—
196#340C2312yes—
197#340C2314yes—
198#340C2312yes—
199#340C2312yes—
200#340C2312yes—
201#340C2314yes—
202#336C2312yes—
203#336C2312yes—
204#336C2312yes—
205#336C2312yes—
206C22×Q8—21no—
207C22×Q8—21no—
208C22×Q8—24no—
209C22×Q8—28no—
210C22×Q8—28no—
211C22×Q8—44no—
212C22×Q8—44no—
213C22×Q8—44no—
214C22×Q8—44no—
215C22×Q8C2412yes—
216C22×Q8C2412yes—
217C22×D4C2×D412yes—
218C22×D4C2×D412yes—
219C22×D4C2×D412yes—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4C2414yes—
223C22×C8—44no—
224C22×C8—44no—
225C22×C8—44no—
226C22×C8C2412yes—
227C22×C8C2414yes—
228C22×C8C2412yes—
229C4⋊Q8—24no—
230C4⋊Q8—22no—
231C4⋊Q8—22no—
232C4⋊Q8—22no—
233C4⋊Q8—42no—
234C4⋊Q8—48no—
235C4⋊Q8—48no—
236C4⋊Q8#4114yes—
237#127SD1618yes—
238#124—48no—
239#124—48no—
240#123—416no—
241C22⋊Q8—28no—
242C22⋊Q8—28no—
243C22⋊Q8—416no—
244C22⋊Q8—44no—
245C22⋊Q8—44no—
246C22⋊Q8—416no—
247C4×Q8—24no—
248C4×Q8—24no—
249C4×Q8—44no—
250C4×Q8—44no—
251C4×Q8—44no—
252C4×D4—24no—
253C4×D4—24no—
254C4×D4—44no—
255C4×D4—44no—
256C4×D4—48no—
257C4×D4—48no—
258C4×D4—48no—
259C4×D4—48no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4C2×D414yes—
299C2×C22⋊C4C2×D414yes—
300C2×C22⋊C4C2×D414yes—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2411yescharacteristic
310C2×C42C2411yescharacteristic
311C2×C42C2411yescharacteristic
312#106—44no—
313#106—44no—
314#106—44no—
315C4⋊C8—24no—
316#103—28no—
317#103—24no—
318#103—28no—
319#103—24no—
320#103—44no—
321#103—44no—
322#103—42no—
323#103—44no—
324#103—42no—
325#103—44no—
326C22⋊C8—44no—
327C22⋊C8—44no—
328C22⋊C8—44no—
329C4×C8—22no—
330C4×C8—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343C24C22×D414yes—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—24no—
349C2×Q8—22no—
350C2×Q8—24no—
351C2×Q8—22no—
352C2×Q8—48no—
353C2×Q8—48no—
354C2×Q8—42no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—416no—
361C2×Q8—84no—
362C2×Q8—88no—
363C2×Q8—88no—
364C2×Q8—88no—
365C2×Q8C2×#4114yes—
366C2×Q8H(5,𝔽2)14yes—
367C2×D4—24no—
368C2×D4—24no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4C2×SD1618yes—
374C2×D4C2×SD1618yes—
375C2×D4C2×SD1618yes—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—84no—
394C22×C4—84no—
395C22×C4—84no—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D414yes—
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C2511yescharacteristic
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—24no—
412C2×C8—22no—
413C2×C8—88no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—42no—
434C4⋊C4—44no—
435C4⋊C4—42no—
436C4⋊C4—44no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—42no—
443C4⋊C4—81no—
444C4⋊C4—81no—
445C4⋊C4—81no—
446C4⋊C4—81no—
447C4⋊C4—84no—
448C22⋊C4—28no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—48no—
452C22⋊C4—48no—
453C22⋊C4—48no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—84no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—41no—
461C42—41no—
462C42—44no—
463C42—44no—
464C42—44no—
465C42—44no—
466C42C2×SD1614yes—
467C42C2×#4112yes—
468C42H(5,𝔽2)12yes—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23C22×SD1618yes—
476C23C23×D411yescharacteristic
477Q8—24no—
478Q8—28no—
479Q8—24no—
480Q8—42no—
481Q8—42no—
482Q8—44no—
483Q8—84no—
484Q8—816no—
485Q8—84no—
486D4—216no—
487D4—48no—
488D4—48no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—21no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—21no—
503C2×C4—24no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—816no—
533C2×C4—168no—
534C2×C4#54612yes—
535C2×C4#54612yes—
536C2×C4C22×SD1612yes—
537C2×C4C22×SD1612yes—
538C2×C4C22×SD1612yes—
539C2×C4#57812yes—
540C2×C4C22×#4111yescharacteristic
541C2×C4C2×H(5,𝔽2)12yes—
542C2×C4C2×H(5,𝔽2)11yescharacteristic
543C2×C4#58512yes—
544C8—42no—
545C8—44no—
546C8—42no—
547C22—28no—
548C22—28no—
549C22—28no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—44no—
554C22C2×#54611yescharacteristic
555C22#348411yescharacteristic
556C22#348511yescharacteristic
557C22C23×SD1611yescharacteristic
558C22C2×#57811yescharacteristic
559C22#358611yescharacteristic
560C22#358611yescharacteristic
561C4—22no—
562C4—22no—
563C4—22no—
564C4—44no—
565C4—44no—
566C4—44no—
567C4—41no—
568C4—41no—
569C4—82no—
570C4—82no—
571C4—82no—
572C4—82no—
573C4—82no—
574C4—82no—
575C4#329814yes—
576C4#351312yes—
577C4#355812yes—
578C2—48no—
579C2#4514011yescharacteristic
580C2C2×#329811yescharacteristic
581C2#6122111yescharacteristic
582C2#6156211yescharacteristic
583C2#6168311yescharacteristic
584C2#6169511yescharacteristic
585C2#6180611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #31998C2semidirect
4. #31262C2semidirect
5. #29255C2semidirect
6. #28205C2semidirect
7. #25884C2semidirect
8. #25341C2semidirect
9. #25014C2semidirect
10. #24863C2semidirect
11. #24041C2semidirect
12. #23816C2semidirect
13. D4×#103C2semidirect
15. #22218C2semidirect

Maximal subgroups

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

subgroupindex
#598272
#598172
#319982
#312622
#292552
#282052
#258842
#253412
#250142
#248632
#240412
#238162
D4×#1032
#222602
#222182