Finite Group Databasebeta

← #7187417

Subgroups — #7187417

Order 512.

1672 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59823C211yesmaximal, characteristic, complemented
2#59821C211yesmaximal, characteristic, complemented
3#32559C214yesmaximal, complemented
4#31656C214yesmaximal, complemented
5#29254C211yesmaximal, characteristic, complemented
6#28160C211yesmaximal, characteristic, complemented
7#27758C212yesmaximal, complemented
8#26188C214yesmaximal, complemented
9#26112C212yesmaximal, complemented
10#25371C212yesmaximal
11#24205C214yesmaximal, complemented
12#23816C211yesmaximal, characteristic, complemented
13#23003C212yesmaximal, complemented
14#22311C211yesmaximal, characteristic, complemented
15#22269C211yesmaximal, characteristic
16C2×#556C2212yes—
17C2×#554—21no—
18C2×#493C2211yescharacteristic
19C2×#460C2212yes—
20C2×#463C2214yes—
21C2×#440C2214yes—
22#2732—22no—
23#2727—22no—
24#2552C2212yes—
25#2548C2212yes—
26#2542—21no—
27#2542C2212yes—
28#2538—21no—
29#2538C2212yes—
30#2538C2214yes—
31#2512—22no—
32#2454—22no—
33#2448C2214yes—
34#2417—22no—
35#2415—22no—
36#2366C2212yes—
37#2354—21no—
38Q8×C4⋊C4C2212yes—
39D4×C4⋊C4C2212yes—
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×#127C2211yescharacteristic
44C4×#127C2211yescharacteristic
45#2100C2214yes—
46#2100C2212yes—
47#2093C2212yes—
48#2087C2214yes—
49#2072C2212yes—
50#2071C2212yes—
51#2071C2214yes—
52#2054C2214yes—
53#2043C2212yes—
54#2042C2214yes—
55#2024C2212yes—
56#2023C2212yes—
57#2013C2212yes—
58#2012C2214yes—
59#1985C2214yes—
60#1982C2212yes—
61#1959C2214yes—
62#1957C2211yescharacteristic
63#1957C2211yescharacteristic
64#1947C2212yes—
65#1939C2212yes—
66#1936C2214yes—
67#1936C2214yes—
68#1917C2214yes—
69#1880C2214yes—
70#1879C2214yes—
71#1879C2214yes—
72#1879C2214yes—
73#1877C2214yes—
74#1863C2214yes—
75#1862C2214yes—
76#1862C2214yes—
77#1834C2211yescharacteristic
78#1831C2211yescharacteristic
79#1803C2212yes—
80#1803C2214yes—
81#1770C2212yes—
82C4×#103C2212yes—
83C4×#103C2212yes—
84#1744C2211yescharacteristic
85C22×SD16C2314yes—
86C22×D8C2314yes—
87#556—48no—
88#554—48no—
89C2×C4⋊Q8D411yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×#127C2311yescharacteristic
93C2×#127C2311yescharacteristic
94C2×#126—24no—
95C2×#125—22no—
96C2×#124C2314yes—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8—22no—
101C2×C22⋊Q8—22no—
102C2×C4×Q8—21no—
103C2×C4×Q8C2312yes—
104C2×C4×Q8C2314yes—
105C2×C4×Q8C2312yes—
106C2×C4×D4—22no—
107C2×C4×D4—21no—
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4C2312yes—
112C2×C4×D4C2312yes—
113C2×C4×D4C2314yes—
114C2×#117—22no—
115C22×C4⋊C4—22no—
116#493—24no—
117#463—48no—
118#460—28no—
119#440—48no—
120C2×#107C2314yes—
121C2×C4⋊C8C2312yes—
122C2×C4⋊C8C2314yes—
123C2×#103C2312yes—
124C2×#103C2312yes—
125C2×#103C2314yes—
126C2×#103C2312yes—
127C2×#103C2312yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#97C2314yes—
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136#399—24no—
137#399—24no—
138#398—24no—
139#397—24no—
140#396—24no—
141#396—24no—
142#396—24no—
143#394—24no—
144#391—21no—
145#391D412yes—
146#391C2312yes—
147#390—22no—
148#390—22no—
149#390—22no—
150#390—21no—
151#390—21no—
152#390—21no—
153#390—21no—
154#390C2312yes—
155#390C2312yes—
156#390C2314yes—
157#390C2314yes—
158#389—22no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—24no—
163#389D412yes—
164#389D412yes—
165#389C2312yes—
166#389C2314yes—
167#389C2314yes—
168#388—24no—
169#388—24no—
170#386—24no—
171#385—24no—
172#384—22no—
173#384D411yescharacteristic
174#384C2311yescharacteristic
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#384C2311yescharacteristic
178#384C2314yes—
179#380—22no—
180#380—22no—
181#380—24no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189C43C2311yescharacteristic
190#340C2312yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2312yes—
195#340C2312yes—
196#340C2312yes—
197#336C2312yes—
198#336C2314yes—
199#336C2312yes—
200#336C2312yes—
201#336C2314yes—
202#336C2312yes—
203C22×Q8—21no—
204C22×Q8C2×D412yes—
205C22×Q8C2412yes—
206C22×D4—21no—
207C22×D4—21no—
208C22×D4—21no—
209C22×D4—21no—
210C22×D4C2412yes—
211C22×D4C2412yes—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—24no—
215C2×SD16—28no—
216C2×SD16—28no—
217C2×SD16—44no—
218C2×D8—28no—
219C2×D8—44no—
220C2×D8—44no—
221C22×C8C2412yes—
222C22×C8C2414yes—
223C22×C8C2412yes—
224C4⋊Q8—24no—
225C4⋊Q8—24no—
226C4⋊Q8—42no—
227#127—22no—
228#127—22no—
229#127—22no—
230#127#4114yes—
231#126—416no—
232#125—48no—
233#124—28no—
234#124—44no—
235C22⋊Q8—48no—
236C22⋊Q8—48no—
237C22⋊Q8—416no—
238C22⋊Q8—84no—
239C22⋊Q8—84no—
240C4×Q8—28no—
241C4×Q8—24no—
242C4×Q8—24no—
243C4×Q8—44no—
244C4×Q8—44no—
245C4×Q8—44no—
246C4×D4—44no—
247C4×D4—44no—
248C4×D4—44no—
249C4×D4—44no—
250C4×D4—48no—
251C4×D4—44no—
252C4×D4—44no—
253C4×D4—82no—
254C4×D4—82no—
255#117—48no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—48no—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D414yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2411yescharacteristic
307C2×C42C2411yescharacteristic
308C2×C42C2414yes—
309C2×C42C2411yescharacteristic
310#107—44no—
311C4⋊C8—24no—
312C4⋊C8—44no—
313#103—24no—
314#103—24no—
315#103—44no—
316#103—42no—
317#103—42no—
318#102—28no—
319#102—44no—
320#102—44no—
321#102—44no—
322#102—44no—
323#97—44no—
324C4×C8—22no—
325C4×C8—22no—
326#95—22no—
327#95—24no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—24no—
332#95C2×D414yes—
333#95C2×D414yes—
334C24—22no—
335C24—22no—
336C2×Q8—24no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×Q8H(5,𝔽2)14yes—
348C2×D4—24no—
349C2×D4—24no—
350C2×D4—22no—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—24no—
356C2×D4—24no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—81no—
364C2×D4—81no—
365C2×D4—81no—
366C2×D4—81no—
367C2×D4C2×#4114yes—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—48no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—84no—
382C22×C4—84no—
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D414yes—
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D414yes—
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C2511yescharacteristic
392SD16—416no—
393D8—88no—
394C2×C8—22no—
395C2×C8—24no—
396C2×C8—22no—
397C2×C8—24no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—24no—
402C2×C8—22no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—42no—
417C4⋊C4—44no—
418C4⋊C4—42no—
419C4⋊C4—48no—
420C4⋊C4—48no—
421C4⋊C4—48no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—44no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—44no—
429C4⋊C4—81no—
430C4⋊C4—81no—
431C4⋊C4—81no—
432C4⋊C4—81no—
433C4⋊C4—84no—
434C4⋊C4—84no—
435C22⋊C4—48no—
436C22⋊C4—48no—
437C22⋊C4—44no—
438C22⋊C4—48no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—84no—
443C22⋊C4—84no—
444C22⋊C4—82no—
445C22⋊C4—82no—
446C22⋊C4—84no—
447C22⋊C4—84no—
448C42—22no—
449C42—24no—
450C42—24no—
451C42—22no—
452C42—22no—
453C42—41no—
454C42—41no—
455C42—48no—
456C42—44no—
457C42—44no—
458C42AGL(1,ℤ/8)14yes—
459C42C2×#4112yes—
460C422−1+412yes—
461C23—24no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23C23×D411yescharacteristic
476Q8—24no—
477Q8—24no—
478Q8—42no—
479Q8—42no—
480Q8—84no—
481Q8—84no—
482D4—28no—
483D4—48no—
484D4—44no—
485D4—44no—
486D4—44no—
487D4—84no—
488D4—84no—
489D4—84no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—21no—
505C2×C4—24no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—24no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
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—88no—
527C2×C4—88no—
528C2×C4—168no—
529C2×C4#54812yes—
530C2×C4#54812yes—
531C2×C4C2×AGL(1,ℤ/8)12yes—
532C2×C4C2×AGL(1,ℤ/8)12yes—
533C2×C4C2×AGL(1,ℤ/8)12yes—
534C2×C4#57812yes—
535C2×C4C22×#4111yescharacteristic
536C2×C4C2×H(5,𝔽2)12yes—
537C2×C4C2×2−1+411yescharacteristic
538C2×C4#58512yes—
539C8—42no—
540C8—44no—
541C8—42no—
542C22—44no—
543C22—44no—
544C22—44no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22C2×#54811yescharacteristic
557C22#348411yescharacteristic
558C22#348711yescharacteristic
559C22C22×AGL(1,ℤ/8)11yescharacteristic
560C22C2×#57811yescharacteristic
561C22#358511yescharacteristic
562C22#358511yescharacteristic
563C4—22no—
564C4—22no—
565C4—22no—
566C4—44no—
567C4—44no—
568C4—41no—
569C4—41no—
570C4—44no—
571C4—82no—
572C4—82no—
573C4—82no—
574C4—82no—
575C4—82no—
576C4#336714yes—
577C4#351312yes—
578C4#355912yes—
579C2—84no—
580C2—162no—
581C2#4527111yescharacteristic
582C2C2×#336711yescharacteristic
583C2#6133211yescharacteristic
584C2#6158711yescharacteristic
585C2#6168711yescharacteristic
586C2#6189011yescharacteristic
587C2#6191711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#493C22
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. #59823C2semidirect
2. #59821C2semidirect
3. #32559C2semidirect
4. #31656C2semidirect
5. #29254C2semidirect
6. #28160C2semidirect
7. #27758C2semidirect
8. #26188C2semidirect
9. #26112C2semidirect
11. #24205C2semidirect
12. #23816C2semidirect
13. #23003C2semidirect
14. #22311C2semidirect

Maximal subgroups

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

subgroupindex
#598232
#598212
#325592
#316562
#292542
#281602
#277582
#261882
#261122
#253712
#242052
#238162
#230032
#223112
#222692