Finite Group Databasebeta

← #6375975

Subgroups — #6375975

Order 512.

2034 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3100C211yesmaximal, characteristic
2#34086C211yesmaximal, characteristic, complemented
3#34058C211yesmaximal, characteristic, complemented
4#22158C214yesmaximal, complemented
5#22127C214yesmaximal, complemented
6#22093C214yesmaximal, complemented
7#22056C214yesmaximal, complemented
8#21390C211yesmaximal, characteristic, complemented
9#21360C211yesmaximal, characteristic, complemented
10#21277C212yesmaximal, complemented
11#21275C212yesmaximal, complemented
12#21118C212yesmaximal, complemented
13#21113C212yesmaximal, complemented
14#20733C211yesmaximal, characteristic, complemented
15#20462C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18#3100C4116yescomplemented
19C2×#480C2212yes—
20C2×#474C2212yes—
21C2×#459C2212yes—
22C2×#447C2212yes—
23C2×C4×D8C2212yes—
24C2×C4×D8C2212yes—
25C2×#431C2211yescharacteristic
26C2×#431C2211yescharacteristic
27#2337—22no—
28#2333—22no—
29#2331C2212yescomplemented
30#2328C2212yescomplemented
31#2324C2214yes—
32#2319—22no—
33#2319C2214yes—
34#2316C2212yes—
35#2312—22no—
36#2309C2212yes—
37#2297—22no—
38#2293—22no—
39#2293C2212yes—
40#2293C2212yes—
41#2293C2212yescomplemented
42#2290—22no—
43#2283C2211yescharacteristic
44#2280—22no—
45C42×D4C2212yescomplemented
46C4×#117C2211yescharacteristic
47C4×#117C2211yescharacteristic
48#1987C2212yescomplemented
49#1982C2212yescomplemented
50#1972C2214yes—
51#1970C2214yes—
52#1949C2214yescomplemented
53#1943C2214yescomplemented
54#1942C2214yescomplemented
55#1936C2214yes—
56#1936C2214yescomplemented
57#1934C2214yes—
58#1934C2214yescomplemented
59#1926C2214yes—
60#1925C2214yes—
61#1921C2214yescomplemented
62#1918C2214yescomplemented
63#1879C2214yes—
64#1879C2214yes—
65#1875C2214yes—
66#1864C2214yescomplemented
67#1862C2214yescomplemented
68#1854C2214yes—
69#1824C2212yes—
70#1804C2214yes—
71#1786C2212yes—
72C4×#107C2212yescomplemented
73C4×#106C2212yescomplemented
74#1768C2214yescomplemented
75C4×#102C2212yes—
76C4×#102C2212yes—
77C4×#102C2212yescomplemented
78C4×#102C2212yescomplemented
79#1750C2214yes—
80#1744C2211yescharacteristic, complemented
81#1744C2211yescharacteristic, complemented
82C4×#97C2211yescharacteristic, complemented
83C42×C8C2211yescharacteristic, complemented
84C22×#41—21no—
85C23×D4C2311yescharacteristic
86C24×C4—21no—
87C22×D8C2312yes—
88C22×D8C2312yes—
89#532—24no—
90#532C2×C418yes—
91C2×C4⋊Q8C2311yescharacteristic
92C2×#127C2311yescharacteristic
93C2×#124—21no—
94C2×#121—21no—
95C2×#121C2312yes—
96C2×C4×D4—21no—
97C2×C4×D4—21no—
98C2×C4×D4—21no—
99C2×C4×D4—24no—
100C2×C4×D4C2×C418yescomplemented
101C2×C4×D4C2312yes—
102C2×C4×D4C2312yes—
103C2×C4×D4C2312yes—
104C2×C4×D4C2314yes—
105C2×#117—22no—
106C2×#117—22no—
107C2×#117—22no—
108C2×#117C2312yes—
109C2×#117C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C22⋊C4—21no—
112C22×C42—22no—
113C22×C42C2311yescharacteristic
114#480C2×C4116yescomplemented
115#474C2×C4116yescomplemented
116#459C2×C4116yes—
117#447C2×C4116yescomplemented
118C4×D8C2×C4116yescomplemented
119C4×D8C2×C4116yescomplemented
120#431—24no—
121#431C2×C418yescomplemented
122C2×#107C2314yes—
123C2×#107C2312yes—
124C2×#107C2312yes—
125C2×#106C2314yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2312yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2312yes—
133C2×#102C2312yes—
134C2×#102C2312yes—
135C2×#102C2312yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2312yes—
138C2×#97C2314yes—
139C2×C4×C8C2311yescharacteristic
140C2×C4×C8C2311yescharacteristic
141C2×C4×C8C2314yes—
142C2×C4×C8C2311yescharacteristic
143C2×C4×C8C2311yescharacteristic
144#391—24no—
145#390—24no—
146#390—24no—
147#390C2314yes—
148#390C2314yes—
149#390C2314yes—
150#389—24no—
151#389C2314yes—
152#389C2314yes—
153#389C2314yes—
154#388—24no—
155#388—24no—
156#388—24no—
157#388—24no—
158#388—24no—
159#388—24no—
160#387—24no—
161#386—24no—
162#386—24no—
163#386—24no—
164#385—24no—
165#383—21no—
166#382C2312yes—
167#381—21no—
168#381—24no—
169#380—24no—
170#380—24no—
171#380C2312yes—
172#379—24no—
173C4×C4⋊C4D412yes—
174C4×C4⋊C4D412yes—
175C4×C4⋊C4C2312yes—
176C4×C4⋊C4C2314yes—
177C4×C4⋊C4C2312yes—
178C4×C4⋊C4C2312yes—
179C4×C22⋊C4—21no—
180C4×C22⋊C4—21no—
181C4×C22⋊C4—21no—
182C4×C22⋊C4—24no—
183C4×C22⋊C4—22no—
184C4×C22⋊C4—22no—
185C4×C22⋊C4—22no—
186C4×C22⋊C4—22no—
187C4×C22⋊C4D412yes—
188C4×C22⋊C4D412yes—
189#376—21no—
190#376—21no—
191#376C2312yes—
192C43—21no—
193C43C2311yescharacteristic
194C43C2311yescharacteristic
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#336C2314yes—
202#336C2314yes—
203C25—21no—
204C2×#41—28no—
205C2×#41—84no—
206C22×Q8—21no—
207C22×D4—22no—
208C22×D4—21no—
209C22×D4—21no—
210C22×D4—21no—
211C22×D4—22no—
212C22×D4—44no—
213C22×D4C22×C414yes—
214C22×D4C2412yes—
215C22×D4C2412yes—
216C23×C4—21no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—28no—
220C23×C4—22no—
221C23×C4—24no—
222C23×C4—84no—
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2411yescharacteristic
226C2×D8—44no—
227C2×D8—44no—
228C2×D8C22×C418yes—
229C2×D8C22×C418yes—
230C22×C8C2412yes—
231C22×C8C2412yes—
232C22×C8C2412yes—
233C22×C8C2412yes—
234C4⋊Q8—22no—
235C4⋊Q8C22×C414yes—
236#127—22no—
237#127C22×C414yes—
238#124—28no—
239#121—28no—
240#121—24no—
241#121C22×C418yes—
242C4×D4—28no—
243C4×D4—28no—
244C4×D4—28no—
245C4×D4—24no—
246C4×D4—24no—
247C4×D4—24no—
248C4×D4—44no—
249C4×D4—44no—
250C4×D4C22×C418yes—
251C4×D4C22×C418yes—
252C4×D4C22×C418yes—
253#117—44no—
254#117—44no—
255#117—84no—
256#117—84no—
257#117—84no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4C22×C414yes—
265C2×C4⋊C4C2×D414yes—
266C2×C4⋊C4C2414yes—
267C2×C4⋊C4C2414yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2412yes—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4C2×D414yes—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—24no—
290C2×C42—84no—
291C2×C42C22×C414yes—
292C2×C42C2×D412yes—
293C2×C42C2×D412yes—
294C2×C42C2×D412yes—
295C2×C42C2×D412yes—
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42#4114yes—
299C2×C42C2412yes—
300C2×C42C2412yes—
301C2×C42C2411yescharacteristic
302C2×C42C2411yescharacteristic
303#107—44no—
304#107C22×C418yes—
305#107C22×C418yes—
306#106—44no—
307C4⋊C8—24no—
308C4⋊C8C22×C418yes—
309#102—44no—
310#102—44no—
311#102—44no—
312#102—44no—
313#102C22×C418yes—
314#102C22×C418yes—
315#102C22×C418yes—
316#102C22×C418yes—
317C22⋊C8—24no—
318C22⋊C8C22×C418yes—
319#97—44no—
320C4×C8—22no—
321C4×C8—22no—
322C4×C8—44no—
323C4×C8C22×C414yes—
324C4×C8C22×C414yes—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—22no—
334#95—24no—
335#95C2×D414yes—
336#95C2×D414yes—
337C24—22no—
338C24—22no—
339C24—24no—
340C24—44no—
341C24—82no—
342C24—82no—
343C24C22×D411yescharacteristic
344#41—816no—
345C2×Q8—24no—
346C2×Q8—82no—
347C2×D4—28no—
348C2×D4—24no—
349C2×D4—24no—
350C2×D4—24no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—24no—
357C2×D4—48no—
358C2×D4—44no—
359C2×D4—48no—
360C2×D4—82no—
361C2×D4—82no—
362C2×D4—82no—
363C2×D4—82no—
364C2×D4C23×C414yes—
365C2×D4C23×C414yes—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—42no—
377C22×C4—48no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—816no—
383C22×C4—84no—
384C22×C4—84no—
385C22×C4—82no—
386C22×C4—82no—
387C22×C4—82no—
388C22×C4C4×D414yes—
389C22×C4C4×D414yes—
390C22×C4C23×C412yes—
391C22×C4C22×D414yes—
392C22×C4C22×D414yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C2×#4112yes—
399C22×C4C2511yescharacteristic
400D8—48no—
401D8—48no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8C23×C414yes—
409C2×C8C23×C414yes—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—44no—
419C4⋊C4—84no—
420C4⋊C4—84no—
421C4⋊C4—84no—
422C4⋊C4C23×C414yes—
423C4⋊C4C23×C414yes—
424C22⋊C4—28no—
425C22⋊C4—28no—
426C22⋊C4—28no—
427C22⋊C4—28no—
428C22⋊C4—44no—
429C22⋊C4—48no—
430C22⋊C4—44no—
431C22⋊C4—84no—
432C22⋊C4—84no—
433C22⋊C4—84no—
434C22⋊C4—84no—
435C22⋊C4—84no—
436C22⋊C4—84no—
437C22⋊C4—82no—
438C42—24no—
439C42—24no—
440C42—24no—
441C42—24no—
442C42—42no—
443C42—42no—
444C42—44no—
445C42—44no—
446C42—44no—
447C42—82no—
448C42—84no—
449C42C4×D414yes—
450C42C4×D414yes—
451C42C23×C412yes—
452C42C23×C412yes—
453C42C2×#4112yes—
454C42C2×#4112yes—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—24no—
459C23—24no—
460C23—48no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—84no—
470C23—84no—
471C23—816no—
472C23C2×C4×D412yes—
473C23C23×D411yescharacteristic
474Q8—84no—
475D4—24no—
476D4—24no—
477D4—24no—
478D4—24no—
479D4—48no—
480D4—44no—
481D4—44no—
482D4—84no—
483D4—84no—
484D4—88no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—28no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—416no—
506C2×C4—416no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—88no—
514C2×C4—84no—
515C2×C4—816no—
516C2×C4C2×C4×D412yes—
517C2×C4C2×C4×D412yes—
518C2×C4C2×C4×D412yes—
519C2×C4C2×C4×D412yes—
520C2×C4C2×C4×D412yes—
521C2×C4#57614yes—
522C2×C4C24×C411yescharacteristic
523C2×C4C22×#4111yescharacteristic
524C2×C4#58514yes—
525C8—24no—
526C8—24no—
527C8—42no—
528C8—42no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—44no—
533C22—44no—
534C22—44no—
535C22—44no—
536C22—44no—
537C22—44no—
538C22—44no—
539C22—816no—
540C22—84no—
541C22—84no—
542C22—84no—
543C22—84no—
544C22C22×C4×D411yescharacteristic
545C22#344912yes—
546C22C2×#57612yes—
547C22#358412yes—
548C4—22no—
549C4—22no—
550C4—22no—
551C4—22no—
552C4—44no—
553C4—44no—
554C4—44no—
555C4—44no—
556C4—84no—
557C4—84no—
558C4#294814yes—
559C4#342812yes—
560C4#345512yes—
561C2—42no—
562C2—84no—
563C2—84no—
564C2#3406511yescharacteristic
565C2#6016111yescharacteristic
566C2C2×#294811yescharacteristic
567C2#6095514yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC2×#3100C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34086C2semidirect
3. #34058C2semidirect
4. #22158C2semidirect
5. #22127C2semidirect
6. #22093C2semidirect
7. #22056C2semidirect
8. #21390C2semidirect
9. #21360C2semidirect
10. #21277C2semidirect
11. #21275C2semidirect
12. #21118C2semidirect
13. #21113C2semidirect
14. #20733C2semidirect
15. #20462C2semidirect
18. #3100C4semidirect
29. #2331C22semidirect
30. #2328C22semidirect
41. #2293C22semidirect
45. C42×D4C22semidirect
48. #1987C22semidirect
49. #1982C22semidirect
52. #1949C22semidirect
53. #1943C22semidirect
54. #1942C22semidirect
56. #1936C22semidirect
58. #1934C22semidirect
61. #1921C22semidirect
62. #1918C22semidirect
66. #1864C22semidirect
67. #1862C22semidirect
72. C4×#107C22semidirect
73. C4×#106C22semidirect
74. #1768C22semidirect
77. C4×#102C22semidirect
78. C4×#102C22semidirect
80. #1744C22semidirect
81. #1744C22semidirect
82. C4×#97C22semidirect
83. C42×C8C22semidirect
100. C2×C4×D4C2×C4semidirect
114. #480C2×C4semidirect
115. #474C2×C4semidirect
117. #447C2×C4semidirect
118. C4×D8C2×C4semidirect
119. C4×D8C2×C4semidirect
121. #431C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31002
#340862
#340582
#221582
#221272
#220932
#220562
#213902
#213602
#212772
#212752
#211182
#211132
#207332
#204622