Finite Group Databasebeta

← #6985997

Subgroups — #6985997

Order 512.

1374 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36594C211yesmaximal, characteristic
2#36447C211yesmaximal, characteristic, complemented
3#32819C212yesmaximal
4#32401C214yesmaximal, complemented
5#32253C214yesmaximal, complemented
6#30305C211yesmaximal, characteristic, complemented
7#30094C212yesmaximal, complemented
8#30066C212yesmaximal
9#29278C212yesmaximal, complemented
10#29213C211yesmaximal, characteristic, complemented
11#29130C211yesmaximal, characteristic
12#26091C212yesmaximal, complemented
13#26021C212yesmaximal, complemented
14#25058C212yesmaximal, complemented
15#24990C212yesmaximal, complemented
16#23819C211yesmaximal, characteristic
17#22022C211yesmaximal, characteristic, complemented
18C2×#556—21no—
19#2868C2211yescharacteristic
20#2859C2212yes—
21#2859C2211yescharacteristic
22#2858C2212yes—
23#2571C2212yes—
24#2571C2211yescharacteristic
25#2552C2211yescharacteristic
26#2552C2212yes—
27#2547—21no—
28#2545—22no—
29#2538—21no—
30#2538C2212yes—
31#2512—22no—
32#2512C2214yes—
33#2508—22no—
34#2508—21no—
35#2506—21no—
36#2506—22no—
37#2354—21no—
38#2354C2212yes—
39#2351—21no—
40Q8×C4⋊C4—21no—
41Q8×C4⋊C4C2211yescharacteristic
42Q8×C4⋊C4C2211yescharacteristic
43C4×C4⋊Q8C2212yes—
44C4×#127C2211yescharacteristic
45C4×#125C2211yescharacteristic
46C4×#125C2212yes—
47C42×Q8C2211yescharacteristic
48#2100C2211yescharacteristic
49#2100C2211yescharacteristic
50#2100C2212yes—
51#2099C2214yes—
52#2094C2214yes—
53#2092C2214yes—
54#2084C2214yes—
55#2084C2214yes—
56#2073C2212yes—
57#2071C2214yes—
58#2065C2214yes—
59#2065C2214yes—
60#2058C2212yes—
61#2058C2212yes—
62#2058C2212yes—
63#2057C2212yes—
64#2041C2211yescharacteristic
65#2041C2211yescharacteristic
66#2041C2212yes—
67#2025C2212yes—
68#2025C2214yes—
69#1992C2212yes—
70#1989C2214yes—
71#1984C2212yes—
72#1949C2214yes—
73#1948C2212yes—
74#1947C2212yes—
75#1939C2212yes—
76#1936C2214yes—
77#1926C2214yes—
78#1925C2214yes—
79#1919C2214yes—
80#1919C2214yes—
81#1919C2214yes—
82#1918C2214yes—
83#1917C2214yes—
84#1847C2212yes—
85#1844C2212yes—
86#1839C2212yes—
87#1834C2212yes—
88#1831C2212yes—
89#1802C2212yes—
90#1802C2212yes—
91#1767C2211yescharacteristic
92C4×#102C2212yes—
93C4×#102C2212yes—
94#1744C2211yescharacteristic
95#556—48no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2312yes—
99C2×#127C2311yescharacteristic
100C2×#126—24no—
101C2×#125—22no—
102C2×#125—21no—
103C2×#125C2312yes—
104C2×#125C2311yescharacteristic
105C2×#124—22no—
106C2×#124—22no—
107C2×C22⋊Q8—22no—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8—22no—
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4C2312yes—
117C2×#117—22no—
118C2×#107C2314yes—
119C2×#106C2314yes—
120C2×#106C2314yes—
121C2×#106C2314yes—
122C2×C4⋊C8C2312yes—
123C2×C4⋊C8C2314yes—
124C2×#102C2312yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2312yes—
128C2×#102C2312yes—
129C2×#102C2312yes—
130C2×#97C2312yes—
131C2×C4×C8C2312yes—
132#400—22no—
133#400—22no—
134#400—22no—
135#398—22no—
136#398—22no—
137#398—22no—
138#397—24no—
139#396—24no—
140#396—24no—
141#391—22no—
142#390—21no—
143#390—21no—
144#390—22no—
145#390—22no—
146#390C2312yes—
147#390C2314yes—
148#390C2314yes—
149#389—21no—
150#389—22no—
151#389—21no—
152#389—22no—
153#389—21no—
154#389—22no—
155#389—21no—
156#389—22no—
157#389—22no—
158#389—21no—
159#389—22no—
160#389—21no—
161#389—22no—
162#389C2311yescharacteristic
163#389C2311yescharacteristic
164#389C2312yes—
165#389C2314yes—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#388—24no—
170#388—24no—
171#387—22no—
172#387—21no—
173#387—22no—
174#387—21no—
175#384—21no—
176#384—22no—
177#384—21no—
178#384—21no—
179#384C2311yescharacteristic
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#384C2312yes—
183#384C2312yes—
184#384C2312yes—
185#384C2312yes—
186#382—21no—
187#382—22no—
188#382—22no—
189#382—21no—
190#382C2311yescharacteristic
191#382C2312yes—
192#380—22no—
193#380—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4D411yescharacteristic
200C4×C4⋊C4D412yes—
201C4×C4⋊C4D412yes—
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2312yes—
210C43C2311yescharacteristic
211C43C2312yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2312yes—
219#340C2312yes—
220#340C2312yes—
221#340C2312yes—
222#336C2312yes—
223#336C2311yescharacteristic
224#336C2311yescharacteristic
225#336C2314yes—
226#336C2312yes—
227#336C2314yes—
228#336C2314yes—
229C22×Q8—22no—
230C22×Q8C2×D411yescharacteristic
231C22×D4—21no—
232C22×D4—21no—
233C22×D4C2412yes—
234C23×C4—22no—
235C22×C8C2412yes—
236C22×C8C2412yes—
237C22×C8C2414yes—
238C4⋊Q8—44no—
239C4⋊Q8—44no—
240C4⋊Q8—82no—
241#127—42no—
242#126—416no—
243#125—24no—
244#125—22no—
245#125—41no—
246#125—42no—
247#125—82no—
248#125—84no—
249#124—48no—
250#124—84no—
251C22⋊Q8—48no—
252C4×Q8—24no—
253C4×Q8—24no—
254C4×Q8—42no—
255C4×Q8—48no—
256C4×Q8—44no—
257C4×Q8—44no—
258C4×D4—24no—
259C4×D4—24no—
260C4×D4—44no—
261C4×D4—44no—
262#117—48no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4C2×D411yescharacteristic
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D411yescharacteristic
292C2×C4⋊C4C2×D411yescharacteristic
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2411yescharacteristic
298C2×C4⋊C4C2414yes—
299C2×C4⋊C4C2411yescharacteristic
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2411yescharacteristic
302C2×C4⋊C4C2411yescharacteristic
303C2×C22⋊C4—24no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—21no—
313C2×C42—22no—
314C2×C42—21no—
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D412yes—
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D412yes—
323C2×C42C2412yes—
324C2×C42C2412yes—
325C2×C42C2411yescharacteristic
326C2×C42C2411yescharacteristic
327C2×C42C2411yescharacteristic
328#107—44no—
329#106—44no—
330#106—44no—
331#106—44no—
332C4⋊C8—24no—
333C4⋊C8—44no—
334#102—24no—
335#102—44no—
336#102—42no—
337#102—42no—
338#102—44no—
339#102—42no—
340#97—42no—
341C4×C8—24no—
342#95—21no—
343#95—24no—
344#95—22no—
345#95—21no—
346#95—21no—
347#95—22no—
348#95—22no—
349#95—21no—
350#95—24no—
351#95—22no—
352#95—21no—
353#95—22no—
354#95—21no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95C2×D412yes—
359#95C2×D412yes—
360#95C2×D412yes—
361C24—22no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—44no—
367C2×Q8AGL(1,ℤ/8)14yes—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—42no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—24no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4C22×D412yes—
392C22×C4C22×D412yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D412yes—
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D412yes—
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C2511yescharacteristic
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—24no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—21no—
421C4⋊C4—22no—
422C4⋊C4—21no—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—44no—
431C4⋊C4—42no—
432C4⋊C4—44no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—42no—
441C4⋊C4—48no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—44no—
445C4⋊C4—42no—
446C4⋊C4—81no—
447C4⋊C4—81no—
448C4⋊C4—82no—
449C4⋊C4—82no—
450C4⋊C4—82no—
451C4⋊C4—82no—
452C4⋊C4—82no—
453C4⋊C4—82no—
454C4⋊C4—82no—
455C4⋊C42−1+412yes—
456C22⋊C4—48no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—48no—
461C22⋊C4—84no—
462C22⋊C4—84no—
463C42—22no—
464C42—24no—
465C42—22no—
466C42—24no—
467C42—22no—
468C42—24no—
469C42—22no—
470C42—24no—
471C42—22no—
472C42—21no—
473C42—21no—
474C42—22no—
475C42—22no—
476C42—42no—
477C42—48no—
478C42—44no—
479C42—44no—
480C42—42no—
481C42—42no—
482C422−1+414yes—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23C23×D411yescharacteristic
491Q8—28no—
492Q8—88no—
493D4—42no—
494D4—42no—
495D4—42no—
496D4—42no—
497D4—84no—
498D4—84no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—88no—
547C2×C4#53612yes—
548C2×C4#53612yes—
549C2×C4C2×AGL(1,ℤ/8)14yes—
550C2×C4C2×AGL(1,ℤ/8)12yes—
551C2×C4#57512yes—
552C2×C4#57712yes—
553C2×C4#57712yes—
554C2×C4C2×2−1+411yescharacteristic
555C2×C4C2×2−1+412yes—
556C2×C4#58512yes—
557C2×C4#58511yescharacteristic
558C8—42no—
559C8—42no—
560C8—44no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22C2×#53611yescharacteristic
569C22#348711yescharacteristic
570C22#348911yescharacteristic
571C22C22×AGL(1,ℤ/8)11yescharacteristic
572C22C2×#57511yescharacteristic
573C22C2×#57711yescharacteristic
574C22C2×#57711yescharacteristic
575C4—24no—
576C4—22no—
577C4—22no—
578C4—22no—
579C4—41no—
580C4—42no—
581C4—42no—
582C4—42no—
583C4—41no—
584C4—42no—
585C4—82no—
586C4—82no—
587C4—81no—
588C4—82no—
589C4—81no—
590C4—81no—
591C4—84no—
592C4#323414yes—
593C4#355512yes—
594C4#356512yes—
595C2—162no—
596C2#4289111yescharacteristic
597C2C2×#323411yescharacteristic
598C2#6088111yescharacteristic
599C2#6189911yescharacteristic
600C2#6190111yescharacteristic
601C2#6213411yescharacteristic
602C2#6215011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36447C2semidirect
4. #32401C2semidirect
5. #32253C2semidirect
6. #30305C2semidirect
7. #30094C2semidirect
9. #29278C2semidirect
10. #29213C2semidirect
12. #26091C2semidirect
13. #26021C2semidirect
14. #25058C2semidirect
15. #24990C2semidirect
17. #22022C2semidirect

Maximal subgroups

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

subgroupindex
#365942
#364472
#328192
#324012
#322532
#303052
#300942
#300662
#292782
#292132
#291302
#260912
#260212
#250582
#249902
#238192
#220222