Finite Group Databasebeta

← #6481784

Subgroups — #6481784

Order 512.

1894 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3110C211yesmaximal, characteristic
2C2×#3101C211yesmaximal, characteristic
3#37641C211yesmaximal, characteristic, complemented
4#34186C211yesmaximal, characteristic
5#33119C214yesmaximal, complemented
6#29723C214yesmaximal
7#24697C214yesmaximal, complemented
8#24681C214yesmaximal, complemented
9#24646C214yesmaximal, complemented
10#24630C214yesmaximal, complemented
11#21954C211yesmaximal, characteristic, complemented
12#21870C211yesmaximal, characteristic, complemented
13#21864C211yesmaximal, characteristic, complemented
14C2×#533C2211yescharacteristic
15C2×#533C2211yescharacteristic
16C22×C4×D4C2211yescharacteristic
17#3110—28no—
18#3101—28no—
19C2×#484C2212yes—
20C2×#483C2212yes—
21C2×#476C2212yes—
22C2×#474C2212yes—
23C2×#461C2212yes—
24C2×#461C2212yes—
25C2×#450C2212yes—
26C2×#450C2212yes—
27C2×#440C2214yes—
28C2×C4×SD16C2214yes—
29C2×#432C2211yescharacteristic
30C2×#431C2211yescharacteristic
31#2822—22no—
32#2625—22no—
33#2625—22no—
34#2621C2214yes—
35#2582—22no—
36#2568C2212yescomplemented
37#2568C2212yescomplemented
38#2546C2212yescomplemented
39#2546C2212yescomplemented
40#2357—22no—
41#2357C2214yes—
42#2354—22no—
43#2354C2214yes—
44#2312—22no—
45#2309C2212yes—
46#2285C2211yescharacteristic
47#2285C2211yescharacteristic
48#2280—22no—
49#2274C2212yes—
50#2073C2218yes—
51#2055C2218yes—
52#2055C2218yes—
53#2013C2218yes—
54#1993C2212yescomplemented
55#1990C2212yescomplemented
56#1981C2218yescomplemented
57#1960C2212yescomplemented
58#1956C2212yescomplemented
59#1949C2218yescomplemented
60#1940C2214yescomplemented
61#1939C2214yescomplemented
62#1928C2218yescomplemented
63#1927C2218yescomplemented
64#1873C2214yes—
65#1873C2214yes—
66#1805C2214yescomplemented
67#1803C2214yescomplemented
68#1798C2214yes—
69C23×D4C2311yescharacteristic
70C24×C4—21no—
71C22×SD16C2314yes—
72C22×SD16C2314yes—
73#533—24no—
74#533—24no—
75#533—24no—
76#533—24no—
77C2×C4⋊Q8C2311yescharacteristic
78C2×C4⋊Q8C2311yescharacteristic
79C2×C4⋊Q8C2314yes—
80C2×C4⋊Q8C2311yescharacteristic
81C2×C4⋊Q8C2311yescharacteristic
82C2×#125—21no—
83C2×#125—21no—
84C2×C22⋊Q8—21no—
85C2×C22⋊Q8—21no—
86C2×C22⋊Q8C2312yes—
87C2×C22⋊Q8C2312yes—
88C2×C4×Q8—21no—
89C2×C4×Q8—21no—
90C2×C4×Q8C2312yes—
91C2×C4×Q8C2312yes—
92C2×C4×D4—21no—
93C2×C4×D4—21no—
94C2×C4×D4—24no—
95C2×C4×D4D418yes—
96C2×C4×D4C2312yes—
97C2×C4×D4C2312yes—
98C2×#117—21no—
99C2×#117—21no—
100C2×#117—24no—
101C2×#117C2312yes—
102C2×#117C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C22⋊C4—21no—
105C22×C42C2311yescharacteristic
106#484—28no—
107#483—28no—
108#476—28no—
109#474—28no—
110#461—28no—
111#461—28no—
112#450—28no—
113#450—28no—
114#440—216no—
115C4×SD16—216no—
116#432—24no—
117#431—24no—
118C2×#107C2314yes—
119C2×#106C2314yes—
120C2×C4⋊C8C2312yes—
121C2×C4⋊C8C2312yes—
122C2×C4⋊C8C2318yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2318yes—
126C2×#103C2318yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×C22⋊C8C2312yes—
130C2×C22⋊C8C2312yes—
131C2×#97C2312yes—
132C2×C4×C8C2312yes—
133#400—24no—
134#400—24no—
135#398—28no—
136#398—24no—
137#398—24no—
138#397—28no—
139#391—24no—
140#391—24no—
141#391C2314yes—
142#391C2314yes—
143#390—24no—
144#390C2318yes—
145#389—24no—
146#389—24no—
147#388—28no—
148#388—28no—
149#384D412yes—
150#384C2312yes—
151#382D412yes—
152#382C2314yes—
153#382C2312yes—
154#381—24no—
155#380—21no—
156#380—21no—
157#380—24no—
158#380D412yes—
159#380D412yes—
160#379—24no—
161C4×C4⋊C4—24no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4—21no—
164C4×C4⋊C4C2318yes—
165C4×C4⋊C4C2312yes—
166C4×C4⋊C4C2312yes—
167C4×C22⋊C4—24no—
168C4×C22⋊C4—24no—
169#340C2318yes—
170#340C2318yes—
171#336C2318yes—
172C25—21no—
173C22×Q8—21no—
174C22×Q8—21no—
175C22×Q8C2412yes—
176C22×Q8C2412yes—
177C22×D4—21no—
178C22×D4—22no—
179C22×D4—42no—
180C22×D4—42no—
181C22×D4C2×D414yes—
182C22×D4C2412yes—
183C23×C4—21no—
184C23×C4—22no—
185C23×C4—28no—
186C23×C4—22no—
187C23×C4—84no—
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2411yescharacteristic
191C2×SD16—28no—
192C2×SD16—28no—
193C2×SD16—44no—
194C2×SD16—44no—
195C2×SD16—44no—
196C2×SD16—44no—
197C22×C8C2414yes—
198C22×C8C2414yes—
199C4⋊Q8—22no—
200C4⋊Q8—22no—
201C4⋊Q8—22no—
202C4⋊Q8—22no—
203C4⋊Q8—22no—
204C4⋊Q8—22no—
205C4⋊Q8—22no—
206C4⋊Q8—22no—
207C4⋊Q8—48no—
208#125—44no—
209#125—44no—
210C22⋊Q8—24no—
211C22⋊Q8—24no—
212C22⋊Q8—24no—
213C22⋊Q8—24no—
214C22⋊Q8—44no—
215C22⋊Q8—44no—
216C4×Q8—24no—
217C4×Q8—24no—
218C4×Q8—24no—
219C4×Q8—24no—
220C4×Q8—44no—
221C4×Q8—44no—
222C4×D4—28no—
223C4×D4—28no—
224C4×D4—24no—
225C4×D4—24no—
226C4×D4C2×D418yes—
227C4×D4C2×D418yes—
228#117—44no—
229#117—44no—
230#117—44no—
231#117—44no—
232#117—88no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4C2×D414yes—
249C2×C4⋊C4C2×D414yes—
250C2×C4⋊C4C2×D414yes—
251C2×C4⋊C4C2×D414yes—
252C2×C4⋊C4C2414yes—
253C2×C4⋊C4C2412yes—
254C2×C4⋊C4C2412yes—
255C2×C4⋊C4C2412yes—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4C2×D414yes—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—28no—
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D414yes—
272C2×C42#4114yes—
273C2×C42C2414yes—
274C2×C42C2411yescharacteristic
275C2×C42C2411yescharacteristic
276#107—28no—
277#106—28no—
278C4⋊C8—24no—
279C4⋊C8—24no—
280C4⋊C8—48no—
281#103—28no—
282#103—28no—
283#103—48no—
284#103—48no—
285#102—28no—
286#102—28no—
287C22⋊C8—24no—
288C22⋊C8—24no—
289#97—24no—
290C4×C8—24no—
291#95—22no—
292#95—28no—
293#95—22no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95C2×D414yes—
298#95C2×D414yes—
299C24—22no—
300C24—24no—
301C24—42no—
302C24—42no—
303C24—82no—
304C24—82no—
305C24C22×D411yescharacteristic
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—24no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—44no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—44no—
318C2×D4—24no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—48no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—48no—
328C2×D4C22×D414yes—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—28no—
335C22×C4—28no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—42no—
342C22×C4—48no—
343C22×C4—816no—
344C22×C4—84no—
345C22×C4—82no—
346C22×C4#12114yes—
347C22×C4#12114yes—
348C22×C4C22×D418yes—
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D412yes—
354C22×C4C22×D411yescharacteristic
355C22×C4C2×#4112yes—
356C22×C4C2511yescharacteristic
357SD16—416no—
358SD16—416no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C4⋊C4—24no—
366C4⋊C4—28no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—28no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—44no—
386C4⋊C4—82no—
387C4⋊C4—88no—
388C4⋊C4C22×D414yes—
389C22⋊C4—28no—
390C22⋊C4—28no—
391C22⋊C4—44no—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—88no—
396C22⋊C4—88no—
397C42—28no—
398C42—44no—
399C42—44no—
400C42—44no—
401C42—88no—
402C42#12114yes—
403C42#12114yes—
404C42C22×D412yes—
405C42C22×D412yes—
406C42C2×#4112yes—
407C42C2×#4112yes—
408C23—22no—
409C23—21no—
410C23—21no—
411C23—22no—
412C23—24no—
413C23—48no—
414C23—44no—
415C23—42no—
416C23—42no—
417C23—44no—
418C23—84no—
419C23—84no—
420C23—88no—
421C23—88no—
422C23C2×#12112yes—
423C23C23×D411yescharacteristic
424Q8—44no—
425Q8—44no—
426Q8—44no—
427Q8—44no—
428Q8—84no—
429Q8—84no—
430D4—24no—
431D4—24no—
432D4—48no—
433D4—42no—
434D4—42no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—28no—
443C2×C4—28no—
444C2×C4—24no—
445C2×C4—28no—
446C2×C4—28no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—416no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—88no—
464C2×C4—84no—
465C2×C4C2×#12112yes—
466C2×C4C2×#12112yes—
467C2×C4C2×#12112yes—
468C2×C4C2×#12112yes—
469C2×C4C2×#12112yes—
470C2×C4#57714yes—
471C2×C4C23×D411yescharacteristic
472C2×C4C22×#4111yescharacteristic
473C2×C4#58512yes—
474C2×C4#58512yes—
475C8—44no—
476C8—44no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—42no—
481C22—42no—
482C22—44no—
483C22—44no—
484C22—44no—
485C22—816no—
486C22—84no—
487C22—84no—
488C22C22×#12111yescharacteristic
489C22#348511yescharacteristic
490C22#348711yescharacteristic
491C22C2×#57712yes—
492C22#358612yes—
493C4—22no—
494C4—22no—
495C4—22no—
496C4—22no—
497C4—44no—
498C4—48no—
499C4—82no—
500C4—82no—
501C4—82no—
502C4—82no—
503C4—84no—
504C4#304214yes—
505C4#344912yes—
506C4#349312yes—
507C2—42no—
508C2—84no—
509C2#3515011yescharacteristic
510C2C2×#304211yescharacteristic
511C2#6042711yescharacteristic
512C2#6180312yes—
513C2#6191112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #37641C2semidirect
5. #33119C2semidirect
7. #24697C2semidirect
8. #24681C2semidirect
9. #24646C2semidirect
10. #24630C2semidirect
11. #21954C2semidirect
12. #21870C2semidirect
13. #21864C2semidirect
36. #2568C22semidirect
37. #2568C22semidirect
38. #2546C22semidirect
39. #2546C22semidirect
54. #1993C22semidirect
55. #1990C22semidirect
56. #1981C22semidirect
57. #1960C22semidirect
58. #1956C22semidirect
59. #1949C22semidirect
60. #1940C22semidirect
61. #1939C22semidirect
62. #1928C22semidirect
63. #1927C22semidirect
66. #1805C22semidirect
67. #1803C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31102
C2×#31012
#376412
#341862
#331192
#297232
#246972
#246812
#246462
#246302
#219542
#218702
#218642