Finite Group Databasebeta

← #6498151

Subgroups — #6498151

Order 512.

1508 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35294C211yesmaximal, characteristic, complemented
2#34255C211yesmaximal, characteristic
3#33156C212yesmaximal, complemented
4#32966C212yesmaximal
5#29980C212yesmaximal, complemented
6#29870C212yesmaximal, complemented
7#29578C212yesmaximal, complemented
8#29572C212yesmaximal, complemented
9#29401C214yesmaximal, complemented
10#29310C212yesmaximal, complemented
11#29116C212yesmaximal
12#27615C212yesmaximal, complemented
13#27582C212yesmaximal
14#22126C214yesmaximal, complemented
15#20780C211yesmaximal, characteristic, complemented
16C2×#518C2211yescharacteristic
17C2×#478C2214yes—
18#2868C2211yescharacteristic, complemented
19#2859C2212yescomplemented
20#2859C2211yescharacteristic, complemented
21#2629C2212yes—
22#2628C2212yes—
23#2570—22no—
24#2550—22no—
25#2441—22no—
26#2439—22no—
27#2357—22no—
28#2354—22no—
29#2342—22no—
30#2342—22no—
31#2331C2212yescomplemented
32#2330C2212yescomplemented
33#2319C2212yes—
34#2319C2212yes—
35#2317C2211yescharacteristic
36#2309C2211yescharacteristic
37#2296C2211yescharacteristic
38#2293C2212yes—
39#2293C2212yes—
40C2×#390C2211yescharacteristic
41#2285C2212yes—
42#2274C2211yescharacteristic
43C4×#117C2211yescharacteristic
44#2099C2212yescomplemented
45#2099C2212yescomplemented
46#2093C2212yes—
47#2093C2212yes—
48#2085C2214yescomplemented
49#2084C2212yescomplemented
50#2083C2212yescomplemented
51#2073C2214yes—
52#2071C2212yes—
53#2071C2212yes—
54#2057C2214yescomplemented
55#2023C2214yes—
56#1989C2214yescomplemented
57#1947C2214yescomplemented
58#1944C2214yescomplemented
59#1942C2214yescomplemented
60#1936C2214yes—
61#1934C2214yes—
62#1925C2214yes—
63#1925C2214yes—
64#1921C2214yescomplemented
65#1921C2214yescomplemented
66#1919C2214yescomplemented
67#1918C2214yescomplemented
68#1864C2214yes—
69#1862C2214yes—
70#1854C2214yes—
71#1852C2214yes—
72#1840C2212yescomplemented
73#1831C2212yescomplemented
74#1825C2214yes—
75#1817C2212yes—
76#1802C2214yes—
77#1798C2212yes—
78#1786C2214yes—
79#1768C2212yescomplemented
80#1750C2212yes—
81#1745C2211yescharacteristic, complemented
82#1744C2211yescharacteristic, complemented
83C23×D4C2311yescharacteristic
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8C2312yes—
86C2×#125—22no—
87C2×#125C2312yes—
88#518—28no—
89C2×C4×D4—22no—
90C2×C4×D4C2312yes—
91C2×C4×D4C2312yes—
92C2×#117—22no—
93C2×#117—22no—
94C2×#117—22no—
95C2×#117C2311yescharacteristic
96C2×#117C2311yescharacteristic
97C2×#117C2311yescharacteristic
98C2×#117C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C22⋊C4—22no—
101C22×C22⋊C4—22no—
102C22×C42C2311yescharacteristic
103#478—216no—
104C2×#107C2314yes—
105C2×#107C2314yes—
106C2×#107C2314yes—
107C2×#106C2314yes—
108C2×C4⋊C8C2314yes—
109C2×#102C2314yes—
110C2×#102C2314yes—
111C2×#102C2314yes—
112C2×#102C2314yes—
113C2×C22⋊C8C2314yes—
114C2×#97C2312yes—
115C2×C4×C8C2312yes—
116#400—22no—
117#399—22no—
118#398—24no—
119#396—24no—
120#395—22no—
121#393—22no—
122#391—24no—
123#390—21no—
124#390—21no—
125#390—22no—
126#390C2312yes—
127#390C2312yes—
128#390C2312yes—
129#390C2312yes—
130#389—22no—
131#389—22no—
132#389—22no—
133#389—22no—
134#389—24no—
135#389C2312yes—
136#389C2312yes—
137#389C2312yes—
138#389C2314yes—
139#389C2314yes—
140#389C2312yes—
141#388—22no—
142#388—22no—
143#388—22no—
144#388—22no—
145#388—22no—
146#388—22no—
147#388—22no—
148#388—22no—
149#387—24no—
150#387—24no—
151#385—22no—
152#385—21no—
153#385—21no—
154#384—21no—
155#384—22no—
156#384C2312yes—
157#384C2312yes—
158#384C2312yes—
159#382—21no—
160#382—22no—
161#382—22no—
162#382C2312yes—
163#381—22no—
164#381—22no—
165#380—22no—
166#380—22no—
167#380—22no—
168#380C2314yes—
169#379—21no—
170#379—21no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4D412yes—
175C4×C4⋊C4D412yes—
176C4×C4⋊C4C2312yes—
177C4×C4⋊C4C2312yes—
178C4×C4⋊C4C2314yes—
179C4×C22⋊C4—22no—
180C4×C22⋊C4—22no—
181C4×C22⋊C4—22no—
182C4×C22⋊C4D412yes—
183C4×C22⋊C4D412yes—
184#376C2311yescharacteristic
185#376C2311yescharacteristic
186C43C2311yescharacteristic
187C43C2311yescharacteristic
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2312yes—
197#336C2312yes—
198#336C2314yes—
199#336C2312yes—
200#336C2312yes—
201#336C2314yes—
202C25—21no—
203C22×Q8—22no—
204C22×D4—24no—
205C22×D4—21no—
206C22×D4—24no—
207C22×D4—42no—
208C22×D4C2412yes—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—21no—
212C23×C4C2×D412yes—
213C23×C4C2411yescharacteristic
214C22×C8C2414yes—
215C22×C8C2414yes—
216C4⋊Q8—42no—
217C4⋊Q8—42no—
218C4⋊Q8—84no—
219#125—28no—
220#125—84no—
221C4×D4—216no—
222C4×D4—24no—
223C4×D4—24no—
224C4×D4—24no—
225C4×D4#4118yes—
226#117—24no—
227#117—24no—
228#117—24no—
229#117—42no—
230#117—84no—
231#117—84no—
232#117—84no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4C2×D412yes—
249C2×C4⋊C4C2×D411yescharacteristic
250C2×C4⋊C4C2×D411yescharacteristic
251C2×C4⋊C4C2412yes—
252C2×C4⋊C4C2412yes—
253C2×C4⋊C4C2414yes—
254C2×C4⋊C4C2414yes—
255C2×C4⋊C4C2412yes—
256C2×C22⋊C4—216no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—84no—
271C2×C22⋊C4C2×D412yes—
272C2×C22⋊C4C2×D411yescharacteristic
273C2×C22⋊C4C2×D411yescharacteristic
274C2×C42—21no—
275C2×C42—24no—
276C2×C42—21no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—24no—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2411yescharacteristic
286C2×C42C2411yescharacteristic
287C2×C42C2411yescharacteristic
288C2×C42C2411yescharacteristic
289C2×C42C2411yescharacteristic
290C2×C42C2411yescharacteristic
291#107—28no—
292#107—44no—
293#107—44no—
294#106—28no—
295C4⋊C8—28no—
296#102—28no—
297#102—28no—
298#102—44no—
299#102—44no—
300C22⋊C8—28no—
301#97—42no—
302C4×C8—42no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—24no—
309#95—22no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—22no—
314#95—22no—
315#95C2×D412yes—
316#95C2×D412yes—
317#95C2×D412yes—
318#95C2×D412yes—
319C24—24no—
320C24—22no—
321C24—42no—
322C24—42no—
323C24—82no—
324C24—82no—
325C24C22×D411yescharacteristic
326C2×Q8—44no—
327C2×Q8—84no—
328C2×D4—24no—
329C2×D4—22no—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—416no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4C2×#4114yes—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—28no—
341C22×C4—24no—
342C22×C4—28no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—82no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4—82no—
353C22×C4—82no—
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D412yes—
357C22×C4C22×D412yes—
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D412yes—
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D412yes—
363C22×C42−1+414yes—
364C22×C4C2511yescharacteristic
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—44no—
382C4⋊C4—42no—
383C4⋊C4—44no—
384C4⋊C4—48no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—82no—
390C4⋊C4—82no—
391C4⋊C4—82no—
392C4⋊C4—84no—
393C4⋊C4—84no—
394C4⋊C4—84no—
395C4⋊C4C2×#4114yes—
396C22⋊C4—24no—
397C22⋊C4—28no—
398C22⋊C4—24no—
399C22⋊C4—28no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—42no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—82no—
406C22⋊C4—82no—
407C22⋊C4—82no—
408C22⋊C4—82no—
409C22⋊C4—82no—
410C22⋊C4—82no—
411C22⋊C4—84no—
412C42—24no—
413C42—24no—
414C42—24no—
415C42—24no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—42no—
422C42—42no—
423C42—42no—
424C42—82no—
425C42—82no—
426C42—84no—
427C42C2×#4114yes—
428C23—22no—
429C23—21no—
430C23—21no—
431C23—22no—
432C23—24no—
433C23—48no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—88no—
441C23—164no—
442C23—162no—
443C23—162no—
444C23#53612yes—
445C23C23×D411yescharacteristic
446Q8—164no—
447D4—28no—
448D4—48no—
449D4—44no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—28no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—48no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—88no—
488C2×C4—88no—
489C2×C4—162no—
490C2×C4#52514yes—
491C2×C4#53612yes—
492C2×C4#57614yes—
493C2×C4C22×#4112yes—
494C2×C4C2×2−1+412yes—
495C2×C4#58512yes—
496C8—44no—
497C8—44no—
498C22—24no—
499C22—22no—
500C22—41no—
501C22—41no—
502C22—42no—
503C22—44no—
504C22—44no—
505C22—44no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22—82no—
510C22—168no—
511C22C2×#52511yescharacteristic
512C22C2×#53611yescharacteristic
513C22#344911yescharacteristic
514C22C2×#57612yes—
515C22#358612yes—
516C4—24no—
517C4—24no—
518C4—22no—
519C4—41no—
520C4—41no—
521C4—41no—
522C4—41no—
523C4—42no—
524C4—44no—
525C4—42no—
526C4—82no—
527C4—84no—
528C4—84no—
529C4—82no—
530C4#349514yes—
531C2—42no—
532C2—84no—
533C2#3532511yescharacteristic
534C2#6059612yes—
535C2#6087112yes—
536C2#6101111yescharacteristic
537C2#6101811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35294C2semidirect
3. #33156C2semidirect
5. #29980C2semidirect
6. #29870C2semidirect
7. #29578C2semidirect
8. #29572C2semidirect
9. #29401C2semidirect
10. #29310C2semidirect
12. #27615C2semidirect
14. #22126C2semidirect
15. #20780C2semidirect
18. #2868C22semidirect
19. #2859C22semidirect
20. #2859C22semidirect
31. #2331C22semidirect
32. #2330C22semidirect
44. #2099C22semidirect
45. #2099C22semidirect
48. #2085C22semidirect
49. #2084C22semidirect
50. #2083C22semidirect
54. #2057C22semidirect
56. #1989C22semidirect
57. #1947C22semidirect
58. #1944C22semidirect
59. #1942C22semidirect
64. #1921C22semidirect
65. #1921C22semidirect
66. #1919C22semidirect
67. #1918C22semidirect
72. #1840C22semidirect
73. #1831C22semidirect
79. #1768C22semidirect
81. #1745C22semidirect
82. #1744C22semidirect

Maximal subgroups

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

subgroupindex
#352942
#342552
#331562
#329662
#299802
#298702
#295782
#295722
#294012
#293102
#291162
#276152
#275822
#221262
#207802