Finite Group Databasebeta

← #6388990

Subgroups — #6388990

Order 512.

2266 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3038C211yesmaximal, characteristic
2#34364C211yesmaximal, characteristic, complemented
3#34197C211yesmaximal, characteristic, complemented
4#23108C214yesmaximal, complemented
5#23010C214yesmaximal, complemented
6#22584C214yesmaximal, complemented
7#22443C214yesmaximal, complemented
8#22158C214yesmaximal, complemented
9#21965C211yesmaximal, characteristic, complemented
10#21810C211yesmaximal, characteristic, complemented
11#21515C214yesmaximal, complemented
12#20742C211yesmaximal, characteristic, complemented
13#20551C211yesmaximal, characteristic, complemented
14C22×AGL(1,ℤ/8)C2212yes—
15C22×#127C2211yescharacteristic
16C2×C4×#41C2211yescharacteristic
17#3038C4116yescomplemented
18C2×#460C2212yes—
19C2×#460C2212yes—
20C2×#459C2212yes—
21C2×#459C2212yes—
22C2×#440C2214yescomplemented
23C2×C4×SD16C2214yescomplemented
24C2×#432C2211yescharacteristic
25C2×#423C2211yescharacteristic
26C2×#418C2212yes—
27#2380—21no—
28#2380—21no—
29#2366C2214yes—
30#2357—22no—
31#2357C2212yescomplemented
32#2351C2212yescomplemented
33#2346—22no—
34#2346C2212yes—
35#2341C2212yes—
36D4×C22⋊C4—22no—
37D4×C22⋊C4C2214yes—
38#2328—22no—
39#2328C2214yes—
40#2319—22no—
41#2319C2212yescomplemented
42#2308—22no—
43#2283—21no—
44#2283—21no—
45#2283C2211yescharacteristic
46#2283C2211yescharacteristic
47#2283C2211yescharacteristic
48#2276C2212yescomplemented
49#1982C2214yes—
50#1972C2214yescomplemented
51#1970C2214yescomplemented
52#1961C2214yes—
53#1949C2214yescomplemented
54#1949C2214yescomplemented
55#1936C2214yescomplemented
56#1934C2214yescomplemented
57#1926C2214yescomplemented
58#1925C2214yescomplemented
59#1880C2214yescomplemented
60#1880C2214yescomplemented
61#1879C2214yescomplemented
62#1879C2214yescomplemented
63#1864C2214yescomplemented
64#1862C2214yescomplemented
65#1825C2212yes—
66#1809C2214yes—
67#1805C2212yescomplemented
68#1804C2212yescomplemented
69#1788C2212yes—
70#1786C2214yes—
71#1771C2214yescomplemented
72#1768C2214yescomplemented
73#1763C2212yescomplemented
74#1762C2212yescomplemented
75#1762C2214yescomplemented
76C4×#102C2214yescomplemented
77#1753C2214yes—
78C22×#41C2311yescharacteristic
79C23×D4—21no—
80C23×D4—21no—
81C23×D4C2312yes—
82C2×AGL(1,ℤ/8)—44no—
83C2×AGL(1,ℤ/8)—44no—
84C2×AGL(1,ℤ/8)C2×C4116yescomplemented
85C22×SD16C2314yes—
86C22×D8C2314yes—
87C22×M16C2312yes—
88C2×#127—21no—
89C2×#127—21no—
90C2×#127—24no—
91C2×#127C2×C418yes—
92C2×#127C2312yes—
93C2×#127C2312yes—
94C4×#41—24no—
95C4×#41C2×C418yescomplemented
96C2×C4×Q8C2314yes—
97C2×C4×Q8C2311yescharacteristic
98C2×C4×Q8C2311yescharacteristic
99C2×C4×D4—21no—
100C2×C4×D4—21no—
101C2×C4×D4—24no—
102C2×C4×D4C2311yescharacteristic
103C2×C4×D4C2311yescharacteristic
104C2×C4×D4C2314yes—
105C2×C4×D4C2314yes—
106C2×#117—21no—
107C2×#117C2311yescharacteristic
108C22×C22⋊C4—24no—
109C22×C42—21no—
110C22×C42C2311yescharacteristic
111#460C2×C4116yescomplemented
112#460C2×C4116yescomplemented
113#459C2×C4116yescomplemented
114#459C2×C4116yescomplemented
115#440—48no—
116C4×SD16—48no—
117#432—24no—
118C2×#107C2314yes—
119C2×#106C2314yes—
120#423C2×C418yescomplemented
121C2×C4⋊C8C2312yes—
122C2×C4⋊C8C2312yes—
123C2×C4⋊C8C2312yes—
124#418C2×C4116yescomplemented
125C2×#103C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×C22⋊C8C2312yes—
134C2×#97C2312yes—
135C2×#97C2314yes—
136C2×C4×C8C2314yes—
137#390—24no—
138#390—24no—
139#390C2314yes—
140#390C2314yes—
141#390C2314yes—
142#389—24no—
143#389C2314yes—
144#388—24no—
145#388—24no—
146#388—24no—
147#386—24no—
148#386—24no—
149#386—24no—
150#386—24no—
151#386—24no—
152#386—24no—
153#386—24no—
154#386—24no—
155#383—21no—
156#383—21no—
157#383D411yescharacteristic
158#383D411yescharacteristic
159#382—22no—
160#382—22no—
161#382D412yes—
162#382C2312yes—
163#382C2312yes—
164#381—21no—
165#381—21no—
166#381—22no—
167#381—22no—
168#381D412yes—
169#380—22no—
170#380—22no—
171#380D412yes—
172#380C2312yes—
173#380C2312yes—
174#379—24no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—24no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4C2312yes—
179C4×C4⋊C4C2314yes—
180C4×C4⋊C4C2312yes—
181C4×C4⋊C4C2312yes—
182C4×C22⋊C4—21no—
183C4×C22⋊C4—21no—
184C4×C22⋊C4—22no—
185C4×C22⋊C4—22no—
186C4×C22⋊C4C2312yes—
187#376—21no—
188#376—21no—
189#376—22no—
190#376—22no—
191#376C2312yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2314yes—
197#336C2314yes—
198C25—22no—
199C2×#41—22no—
200C2×#41—24no—
201C2×#41—42no—
202C2×#41C22×C414yes—
203C22×Q8C2411yescharacteristic
204C22×D4—21no—
205C22×D4—28no—
206C22×D4—28no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—24no—
213C22×D4—42no—
214C22×D4—42no—
215C22×D4—42no—
216C22×D4—42no—
217C22×D4—84no—
218C22×D4—84no—
219C22×D4C22×C418yes—
220C22×D4C2414yes—
221C22×D4C2411yescharacteristic
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2411yescharacteristic
230AGL(1,ℤ/8)—432no—
231C2×SD16—28no—
232C2×SD16—28no—
233C2×SD16C22×C4116yes—
234C2×D8—44no—
235C2×D8—44no—
236C2×D8C22×C4116yes—
237C2×M16—42no—
238C2×M16—42no—
239C2×M16C22×C418yes—
240C22×C8C2414yes—
241C22×C8C2414yes—
242#127—28no—
243#127—28no—
244#127—24no—
245#127—24no—
246#127C22×C418yes—
247#127C22×C418yes—
248C4×Q8—22no—
249C4×Q8—22no—
250C4×Q8—28no—
251C4×Q8—44no—
252C4×Q8C22×C414yes—
253C4×Q8C22×C414yes—
254C4×D4—28no—
255C4×D4—28no—
256C4×D4—22no—
257C4×D4—22no—
258C4×D4—44no—
259C4×D4—44no—
260C4×D4—44no—
261C4×D4—44no—
262C4×D4—88no—
263C4×D4C22×C414yes—
264C4×D4C22×C414yes—
265#117—22no—
266#117—28no—
267#117C22×C414yes—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4C2×D414yes—
273C2×C4⋊C4C2414yes—
274C2×C4⋊C4C2414yes—
275C2×C4⋊C4C2411yescharacteristic
276C2×C4⋊C4C2411yescharacteristic
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—88no—
291C2×C22⋊C4C2×D414yes—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—28no—
296C2×C42—22no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—22no—
300C2×C42C22×C414yes—
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42#4114yes—
304C2×C42C2414yes—
305C2×C42C2411yescharacteristic
306C2×C42C2411yescharacteristic
307#107—44no—
308#106—44no—
309C4⋊C8—24no—
310C4⋊C8C22×C418yes—
311C4⋊C8C22×C418yes—
312#103—28no—
313#102—44no—
314#102—44no—
315#102—44no—
316#102—44no—
317#102—44no—
318#102C22×C4116yes—
319#102C22×C4116yes—
320C22⋊C8—24no—
321#97—24no—
322#97—44no—
323C4×C8—44no—
324#95—22no—
325#95—22no—
326#95—24no—
327#95—22no—
328#95—24no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95C2×D412yes—
335#95C2×D412yes—
336#95C2×D412yes—
337#95C2×D414yes—
338#95C2×D412yes—
339#95C2×D412yes—
340#95C2×D412yes—
341C24—21no—
342C24—24no—
343C24—28no—
344C24—48no—
345C24—82no—
346C24—82no—
347C24—84no—
348C24C22×D411yescharacteristic
349#41—28no—
350#41—44no—
351#41—48no—
352C2×Q8—21no—
353C2×Q8—22no—
354C2×Q8C23×C412yes—
355C2×Q8H(5,𝔽2)14yes—
356C2×D4—24no—
357C2×D4—28no—
358C2×D4—28no—
359C2×D4—28no—
360C2×D4—28no—
361C2×D4—28no—
362C2×D4—21no—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—24no—
369C2×D4—24no—
370C2×D4—48no—
371C2×D4—48no—
372C2×D4—44no—
373C2×D4—48no—
374C2×D4—84no—
375C2×D4—84no—
376C2×D4—82no—
377C2×D4—82no—
378C2×D4—84no—
379C2×D4—82no—
380C2×D4—82no—
381C2×D4—82no—
382C2×D4—82no—
383C2×D4—816no—
384C2×D4—816no—
385C2×D4C23×C418yes—
386C2×D4C23×C412yes—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—44no—
400C22×C4—82no—
401C22×C4—82no—
402C22×C4—84no—
403C22×C4—84no—
404C22×C4—84no—
405C22×C4—84no—
406C22×C4—84no—
407C22×C4—84no—
408C22×C4C4×D414yes—
409C22×C4C4×D414yes—
410C22×C4C23×C412yes—
411C22×C4C22×D414yes—
412C22×C4C22×D414yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C2×#4112yes—
419C22×C4C2511yescharacteristic
420SD16—232no—
421D8—416no—
422M16—48no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8C23×C418yes—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—44no—
438C4⋊C4—48no—
439C4⋊C4—84no—
440C4⋊C4C23×C412yes—
441C4⋊C4C23×C412yes—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—24no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—84no—
448C22⋊C4—84no—
449C22⋊C4—84no—
450C22⋊C4—84no—
451C22⋊C4—84no—
452C22⋊C4—84no—
453C22⋊C4—84no—
454C22⋊C4—84no—
455C22⋊C4—82no—
456C42—24no—
457C42—24no—
458C42—24no—
459C42—24no—
460C42—24no—
461C42—48no—
462C42—44no—
463C42—82no—
464C42—84no—
465C42C4×D414yes—
466C42C4×D414yes—
467C42C23×C412yes—
468C42C23×C412yes—
469C42C2×#4112yes—
470C42C2×#4112yes—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—28no—
476C23—416no—
477C23—42no—
478C23—42no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—42no—
486C23—88no—
487C23—88no—
488C23—168no—
489C23—168no—
490C23C2×C4×D412yes—
491C23C23×D411yescharacteristic
492Q8—22no—
493Q8—24no—
494Q8#51814yes—
495D4—22no—
496D4—28no—
497D4—28no—
498D4—22no—
499D4—48no—
500D4—416no—
501D4—44no—
502D4—44no—
503D4—42no—
504D4—88no—
505D4—88no—
506D4—88no—
507D4—88no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—48no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—84no—
538C2×C4—84no—
539C2×C4—168no—
540C2×C4—168no—
541C2×C4C2×C4×D412yes—
542C2×C4C2×C4×D412yes—
543C2×C4C2×C4×D412yes—
544C2×C4C2×C4×D412yes—
545C2×C4C2×C4×D412yes—
546C2×C4C24×C411yescharacteristic
547C2×C4C22×#4111yescharacteristic
548C2×C4C2×H(5,𝔽2)12yes—
549C2×C4#58512yes—
550C8—28no—
551C8—44no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—44no—
556C22—42no—
557C22—42no—
558C22—48no—
559C22—48no—
560C22—42no—
561C22—48no—
562C22—84no—
563C22—84no—
564C22—84no—
565C22—84no—
566C22—84no—
567C22—1616no—
568C22C22×C4×D411yescharacteristic
569C22#348812yes—
570C22#358512yes—
571C22#358512yes—
572C4—22no—
573C4—22no—
574C4—22no—
575C4—22no—
576C4—22no—
577C4—24no—
578C4—42no—
579C4—42no—
580C4—44no—
581C4—44no—
582C4—84no—
583C4—84no—
584C4C2×#51811yescharacteristic
585C4#342811yescharacteristic
586C4#349412yes—
587C2—42no—
588C2—82no—
589C2—88no—
590C2#3420411yescharacteristic
591C2#6016811yescharacteristic
592C2#6016811yescharacteristic
593C2#6208114yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#3038C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2283 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34364C2semidirect
3. #34197C2semidirect
4. #23108C2semidirect
5. #23010C2semidirect
6. #22584C2semidirect
7. #22443C2semidirect
8. #22158C2semidirect
9. #21965C2semidirect
10. #21810C2semidirect
11. #21515C2semidirect
12. #20742C2semidirect
13. #20551C2semidirect
17. #3038C4semidirect
22. C2×#440C22semidirect
23. C2×C4×SD16C22semidirect
31. #2357C22semidirect
32. #2351C22semidirect
41. #2319C22semidirect
48. #2276C22semidirect
50. #1972C22semidirect
51. #1970C22semidirect
53. #1949C22semidirect
54. #1949C22semidirect
55. #1936C22semidirect
56. #1934C22semidirect
57. #1926C22semidirect
58. #1925C22semidirect
59. #1880C22semidirect
60. #1880C22semidirect
61. #1879C22semidirect
62. #1879C22semidirect
63. #1864C22semidirect
64. #1862C22semidirect
67. #1805C22semidirect
68. #1804C22semidirect
71. #1771C22semidirect
72. #1768C22semidirect
73. #1763C22semidirect
74. #1762C22semidirect
75. #1762C22semidirect
76. C4×#102C22semidirect
84. C2×AGL(1,ℤ/8)C2×C4semidirect
95. C4×#41C2×C4semidirect
111. #460C2×C4semidirect
112. #460C2×C4semidirect
113. #459C2×C4semidirect
114. #459C2×C4semidirect
120. #423C2×C4semidirect
124. #418C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30382
#343642
#341972
#231082
#230102
#225842
#224432
#221582
#219652
#218102
#215152
#207422
#205512