Finite Group Databasebeta

← #7186460

Subgroups — #7186460

Order 512.

1958 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3359C212yesmaximal, complemented
2#59828C211yesmaximal, characteristic, complemented
3#59817C211yesmaximal, characteristic
4#30878C214yesmaximal, complemented
5#30808C214yesmaximal, complemented
6#28264C212yesmaximal, complemented
7#28205C211yesmaximal, characteristic, complemented
8#28157C211yesmaximal, characteristic
9#26586C214yesmaximal
10#26373C214yesmaximal, complemented
11#24330C212yesmaximal, complemented
12#24187C212yesmaximal, complemented
13#23799C211yesmaximal, characteristic
14#22238C211yesmaximal, characteristic, complemented
15#22218C211yesmaximal, characteristic, complemented
16C2×#557C2212yes—
17C2×D4×Q8—21no—
18C2×D4×Q8C2212yes—
19#3359—216no—
20C2×#487C2214yes—
21C2×#487C2211yescharacteristic
22C2×#475C2214yes—
23C2×#475C2212yes—
24C2×#461C2212yes—
25C2×#461C2214yes—
26C2×C8×Q8C2212yes—
27C2×C4×SD16C2214yes—
28C2×C4×SD16C2212yes—
29C2×C4×SD16C2212yes—
30#2726—22no—
31#2683—22no—
32#2582C2212yes—
33#2552—21no—
34#2551C2212yes—
35#2546C2212yes—
36#2538—21no—
37#2538C2212yes—
38#2506—22no—
39#2453—22no—
40#2452—22no—
41#2452C2214yes—
42#2446C2214yes—
43#2409—22no—
44#2366C2212yes—
45#2365—21no—
46Q8×C4⋊C4C2212yes—
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49C4×C4⋊Q8C2211yescharacteristic
50C4×C4⋊Q8C2211yescharacteristic
51C4×#127C2211yescharacteristic
52#2096C2212yes—
53#2094C2212yes—
54#2084C2212yes—
55#2072C2212yes—
56#2066C2214yes—
57#2065C2214yes—
58#2058C2214yes—
59#2043C2212yes—
60#2043C2214yes—
61#2024C2214yes—
62#2014C2212yes—
63#2014C2212yes—
64#2013C2214yes—
65#2013C2212yes—
66#1990C2211yescharacteristic
67#1984C2212yes—
68#1968C2214yes—
69#1956C2211yescharacteristic
70#1941C2214yes—
71#1927C2214yes—
72#1927C2214yes—
73#1916C2214yes—
74#1916C2214yes—
75#1880C2214yes—
76#1880C2214yes—
77#1877C2214yes—
78#1865C2214yes—
79#1865C2214yes—
80#1864C2214yes—
81#1844C2211yescharacteristic
82#1831C2211yescharacteristic
83#1803C2212yes—
84#1802C2212yes—
85#1744C2211yescharacteristic
86C23×Q8—22no—
87C22×SD16C2314yes—
88C22×SD16C2312yes—
89C22×SD16C2312yes—
90#557—28no—
91#557—44no—
92D4×Q8—28no—
93D4×Q8—44no—
94D4×Q8—48no—
95C2×C4⋊Q8—22no—
96C2×C4⋊Q8D411yescharacteristic
97C2×C4⋊Q8C2314yes—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2314yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×#127C2311yescharacteristic
103C2×#125—22no—
104C2×#125—24no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8—24no—
109C2×C4×Q8—22no—
110C2×C4×Q8—21no—
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2314yes—
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2312yes—
115C2×C4×D4—22no—
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4C2312yes—
119C2×C4×D4C2314yes—
120#487—216no—
121#487D418yescomplemented
122#475—28no—
123#475—216no—
124#461—28no—
125#461—216no—
126C8×Q8—28no—
127C4×SD16—216no—
128C4×SD16—28no—
129C4×SD16—44no—
130C2×#106C2314yes—
131C2×#106C2312yes—
132C2×#106C2312yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#102C2312yes—
144C2×#102C2314yes—
145C2×#102C2312yes—
146C2×C4×C8C2314yes—
147C2×C4×C8C2311yescharacteristic
148C2×C4×C8C2311yescharacteristic
149#400—24no—
150#398—24no—
151#398—24no—
152#398—24no—
153#396—24no—
154#396—24no—
155#395—24no—
156#393—24no—
157#391—22no—
158#391—21no—
159#391D412yes—
160#391C2312yes—
161#391C2312yes—
162#391C2314yes—
163#390—22no—
164#390—21no—
165#390—21no—
166#390C2312yes—
167#390C2314yes—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—24no—
178#389D412yes—
179#389D412yes—
180#389C2312yes—
181#389C2314yes—
182#389C2314yes—
183#387—24no—
184#387—24no—
185#386—24no—
186#385—24no—
187#384—22no—
188#384D411yescharacteristic
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#380—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C43C2311yescharacteristic
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#336C2314yes—
208#336C2312yes—
209#336C2312yes—
210#336C2312yes—
211#336C2312yes—
212#336C2314yes—
213C22×Q8—21no—
214C22×Q8—24no—
215C22×Q8—48no—
216C22×Q8—48no—
217C22×Q8—84no—
218C22×Q8C2×D412yes—
219C22×Q8C2412yes—
220C22×Q8C2412yes—
221C22×D4—21no—
222C22×D4—21no—
223C22×D4C2412yes—
224C23×C4—22no—
225C23×C4—24no—
226C2×SD16—28no—
227C2×SD16—24no—
228C2×SD16—28no—
229C2×SD16—24no—
230C2×SD16—24no—
231C2×SD16—42no—
232C2×SD16—44no—
233C2×SD16—42no—
234C2×SD16—42no—
235C22×C8C2412yes—
236C22×C8C2414yes—
237C22×C8C2412yes—
238C4⋊Q8—28no—
239C4⋊Q8—28no—
240C4⋊Q8—22no—
241C4⋊Q8—22no—
242C4⋊Q8—44no—
243C4⋊Q8—42no—
244C4⋊Q8—44no—
245C4⋊Q8—48no—
246C4⋊Q8SD1618yescomplemented
247C4⋊Q8C2×D414yes—
248C4⋊Q8C2×D414yes—
249#127—24no—
250#125—416no—
251#125—48no—
252C22⋊Q8—416no—
253C22⋊Q8—48no—
254C22⋊Q8—48no—
255C22⋊Q8—416no—
256C4×Q8—24no—
257C4×Q8—24no—
258C4×Q8—28no—
259C4×Q8—24no—
260C4×Q8—24no—
261C4×Q8—44no—
262C4×Q8—44no—
263C4×Q8—48no—
264C4×Q8—44no—
265C4×D4—28no—
266C4×D4—24no—
267C4×D4—24no—
268C4×D4—44no—
269C4×D4—48no—
270C4×D4—44no—
271C4×D4—82no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D414yes—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—24no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2414yes—
320C2×C42C2411yescharacteristic
321C2×C42C2411yescharacteristic
322C2×C42C2411yescharacteristic
323#106—24no—
324#106—28no—
325#106—44no—
326#106—42no—
327C4⋊C8—28no—
328C4⋊C8—24no—
329#103—28no—
330#103—24no—
331#103—28no—
332#103—44no—
333#103—44no—
334#103—42no—
335#103C2×D4116yes—
336#102—24no—
337#102—44no—
338#102—42no—
339C4×C8—28no—
340C4×C8—22no—
341C4×C8C2×D414yes—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—24no—
348#95—22no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95C2×D414yes—
353#95C2×D414yes—
354C24—22no—
355C2×Q8—24no—
356C2×Q8—24no—
357C2×Q8—24no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—24no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—24no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—48no—
368C2×Q8—42no—
369C2×Q8—48no—
370C2×Q8—42no—
371C2×Q8—84no—
372C2×Q8—832no—
373C2×Q8C2×SD1618yes—
374C2×Q8C22×D414yes—
375C2×D4—24no—
376C2×D4—24no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—81no—
385C2×D4—81no—
386C2×D42−1+414yes—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—48no—
395C22×C4—44no—
396C22×C4—48no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—84no—
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D414yes—
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D414yes—
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C2511yescharacteristic
409SD16—48no—
410SD16—416no—
411SD16—48no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—24no—
415C2×C8—22no—
416C2×C8—24no—
417C2×C8—22no—
418C2×C8—24no—
419C2×C8—22no—
420C2×C8C22×D414yes—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—28no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—42no—
440C4⋊C4—44no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—44no—
444C4⋊C4—42no—
445C4⋊C4—44no—
446C4⋊C4—42no—
447C4⋊C4—48no—
448C4⋊C4—42no—
449C4⋊C4—81no—
450C4⋊C4—81no—
451C4⋊C4—81no—
452C4⋊C4—84no—
453C4⋊C4—81no—
454C4⋊C4C2×SD1618yes—
455C4⋊C4C2×SD1618yes—
456C4⋊C4C22×D414yes—
457C22⋊C4—48no—
458C22⋊C4—44no—
459C22⋊C4—48no—
460C22⋊C4—44no—
461C22⋊C4—84no—
462C22⋊C4—82no—
463C42—22no—
464C42—24no—
465C42—24no—
466C42—22no—
467C42—22no—
468C42—41no—
469C42—41no—
470C42—48no—
471C42—44no—
472C42—44no—
473C42—44no—
474C42C2×SD1614yes—
475C42C22×D412yes—
476C42H(5,𝔽2)12yes—
477C23—24no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23C23×D411yescharacteristic
485Q8—216no—
486Q8—28no—
487Q8—48no—
488Q8—44no—
489Q8—42no—
490Q8—42no—
491Q8—42no—
492Q8—42no—
493Q8—816no—
494D4—24no—
495D4—24no—
496D4—48no—
497D4—42no—
498D4—42no—
499D4—84no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—24no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—24no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
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—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—816no—
539C2×C4—88no—
540C2×C4D4212yes—
541C2×C4D4212yes—
542C2×C4C22×SD1612yes—
543C2×C4C22×SD1612yes—
544C2×C4C22×SD1618yes—
545C2×C4C22×SD1612yes—
546C2×C4#57712yes—
547C2×C4C23×D411yescharacteristic
548C2×C4C2×H(5,𝔽2)11yescharacteristic
549C2×C4C2×2−1+412yes—
550C2×C4#58512yes—
551C8—24no—
552C8—44no—
553C8—42no—
554C22—44no—
555C22—44no—
556C22—44no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22C2×D4211yescharacteristic
562C22#348611yescharacteristic
563C22#348711yescharacteristic
564C22C23×SD1611yescharacteristic
565C22C2×#57711yescharacteristic
566C22#358611yescharacteristic
567C22#358611yescharacteristic
568C4—28no—
569C4—22no—
570C4—22no—
571C4—22no—
572C4—44no—
573C4—41no—
574C4—44no—
575C4—44no—
576C4—41no—
577C4—82no—
578C4—82no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4D4×SD1614yes—
583C4#348912yes—
584C4#354112yes—
585C2—84no—
586C2#4526711yescharacteristic
587C2C2×D4×SD1611yescharacteristic
588C2#6108111yescharacteristic
589C2#6181811yescharacteristic
590C2#6184811yescharacteristic
591C2#6188311yescharacteristic
592C2#6191611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3359C2semidirect
2. #59828C2semidirect
4. #30878C2semidirect
5. #30808C2semidirect
6. #28264C2semidirect
7. #28205C2semidirect
10. #26373C2semidirect
11. #24330C2semidirect
12. #24187C2semidirect
14. #22238C2semidirect
15. #22218C2semidirect
121. #487D4semidirect
246. C4⋊Q8SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#33592
#598282
#598172
#308782
#308082
#282642
#282052
#281572
#265862
#263732
#243302
#241872
#237992
#222382
#222182