Finite Group Databasebeta

← #6461500

Subgroups — #6461500

Order 512.

2118 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34255C211yesmaximal, characteristic, complemented
2#34207C211yesmaximal, characteristic, complemented
3#29381C218yesmaximal, complemented
4#29369C214yesmaximal, complemented
5#29364C214yesmaximal, complemented
6#23745C211yesmaximal, characteristic, complemented
7C2×#2093C211yesmaximal, characteristic, complemented
8#23504C211yesmaximal, characteristic, complemented
9#23428C211yesmaximal, characteristic
10#21067C212yesmaximal, complemented
11#20932C212yesmaximal, complemented
12#20929C212yesmaximal, complemented
13#20881C212yesmaximal, complemented
14#20774C211yesmaximal, characteristic, complemented
15C22×AGL(1,ℤ/8)C2212yes—
16C2×#532C2211yescharacteristic
17C2×#520C2212yes—
18C2×#518C2211yescharacteristic
19C22×#107C2212yes—
20C2×#417C2212yes—
21C2×#416C2212yes—
22C2×#416C2212yes—
23C22×#102C2212yes—
24#2361—24no—
25#2357—22no—
26#2354—22no—
27#2354—22no—
28D4×C4⋊C4—22no—
29#2342—24no—
30#2326C2214yescomplemented
31#2319C2214yes—
32#2317C2211yescharacteristic
33#2311C2212yes—
34#2309C2211yescharacteristic
35C4×C4⋊Q8C2212yescomplemented
36C4×#127C2212yescomplemented
37#2296C2211yescharacteristic
38#2293C2214yes—
39C2×#390C2211yescharacteristic
40C2×#384C2211yescharacteristic
41#2274C2211yescharacteristic
42C4×#117C2211yescharacteristic
43#2100C2214yescomplemented
44#2099C2214yescomplemented
45#2093C2214yes—
46#2093C2214yes—
47#2043C2212yescomplemented
48#2042C2214yescomplemented
49#2041C2212yescomplemented
50#2013C2212yes—
51#2012C2214yes—
52#2011C2212yes—
53#1928C2218yescomplemented
54#1925C2218yescomplemented
55#1880C2218yes—
56#1879C2218yes—
57#1879C2218yes—
58#1877C2218yes—
59#1863C2218yescomplemented
60#1862C2218yescomplemented
61#1840C2214yescomplemented
62#1839C2214yescomplemented
63#1736C2212yes—
64#1736C2212yes—
65#1729C2212yes—
66#1729C2212yes—
67#1728C2211yescharacteristic
68C2×#336C2211yescharacteristic
69C22×#41C2312yes—
70C23×D4C2311yescharacteristic
71C2×AGL(1,ℤ/8)—28no—
72C2×AGL(1,ℤ/8)—28no—
73C2×AGL(1,ℤ/8)—44no—
74C22×SD16C2314yes—
75C22×D8C2314yes—
76C22×M16C2312yes—
77C23×C8C2312yes—
78#532—28no—
79C2×C4⋊Q8C2311yescharacteristic
80C2×#127C2311yescharacteristic
81C2×#124C2312yes—
82C2×#121—22no—
83#520—44no—
84#520—44no—
85#518—24no—
86#518—24no—
87C2×C4×Q8C2314yes—
88C2×C4×D4—22no—
89C2×C4×D4—24no—
90C2×C4×D4—22no—
91C2×C4×D4—24no—
92C2×C4×D4C2314yes—
93C2×C4×D4C2314yes—
94C2×#117—22no—
95C2×#117—24no—
96C2×#117C2311yescharacteristic
97C2×#117C2311yescharacteristic
98C2×#117C2311yescharacteristic
99C2×#117C2311yescharacteristic
100C22×C4⋊C4—22no—
101C22×C4⋊C4—22no—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C22⋊C4—22no—
105C22×C22⋊C4—22no—
106C22×C42C2311yescharacteristic
107C2×#107—44no—
108C2×#107C2318yes—
109#417—44no—
110#416—28no—
111#416—28no—
112C2×#103C2314yes—
113C2×#103C2314yes—
114C2×#103C2314yes—
115C2×#102—44no—
116C2×#102C2314yes—
117C2×#102C2314yes—
118C2×#102C2318yes—
119C2×#102C2314yes—
120#391C2314yes—
121#390—21no—
122#390—21no—
123#390—24no—
124#390—22no—
125#390C2314yes—
126#390C2314yes—
127#390C2312yes—
128#390C2312yes—
129#389—24no—
130#389C2314yes—
131#389C2312yes—
132#389C2312yes—
133#388—24no—
134#388—24no—
135#388—24no—
136#388—24no—
137#388—28no—
138#386—24no—
139#385—22no—
140#385—22no—
141#385—28no—
142#384—21no—
143#384—22no—
144#384C2312yes—
145#384C2312yes—
146#382—21no—
147#381—24no—
148#380—24no—
149#380—24no—
150#379—22no—
151C4×C4⋊C4—22no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—22no—
154C4×C4⋊C4D412yes—
155C4×C4⋊C4D412yes—
156C4×C4⋊C4C2314yes—
157C4×C22⋊C4—22no—
158C4×C22⋊C4—24no—
159C4×C22⋊C4D414yes—
160#376C2312yes—
161C43C2312yes—
162#340C2318yes—
163#340C2318yes—
164#336C2312yes—
165#336C2312yes—
166#336C2312yes—
167#336C2318yes—
168#336C2318yes—
169#336C2312yes—
170C25—21no—
171C2×#41—24no—
172C2×#41—24no—
173C2×#41—24no—
174C2×#41—24no—
175C2×#41—24no—
176C2×#41#4118yes—
177C22×Q8C2412yes—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—21no—
181C22×D4—24no—
182C22×D4—22no—
183C22×D4—24no—
184C22×D4#4114yes—
185C22×D4C2412yes—
186C22×D4C2412yes—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—21no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4C2×D412yes—
193C23×C4C2411yescharacteristic
194AGL(1,ℤ/8)—432no—
195C2×SD16—28no—
196C2×SD16—28no—
197C2×SD16—44no—
198C2×D8—28no—
199C2×D8—28no—
200C2×D8—44no—
201C2×M16—24no—
202C2×M16—24no—
203C2×M16—24no—
204C22×C8—24no—
205C22×C8—24no—
206C22×C8—24no—
207C22×C8C2414yes—
208C22×C8C2414yes—
209C4⋊Q8—24no—
210#127—24no—
211#124—24no—
212#124—24no—
213#121—216no—
214C4×Q8—44no—
215C4×Q8—44no—
216C4×D4—216no—
217C4×D4—216no—
218C4×D4—28no—
219C4×D4—28no—
220C4×D4—44no—
221C4×D4—44no—
222C4×D4—416no—
223C4×D4—88no—
224#117—22no—
225#117—22no—
226#117—22no—
227#117—24no—
228#117—24no—
229#117—48no—
230#117—88no—
231#117#4114yes—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—48no—
245C2×C4⋊C4—84no—
246C2×C4⋊C4C2×D411yescharacteristic
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C4⋊C4C2×D412yes—
249C2×C4⋊C4C2414yes—
250C2×C4⋊C4C2412yes—
251C2×C4⋊C4C2412yes—
252C2×C4⋊C4C2412yes—
253C2×C22⋊C4—216no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—84no—
265C2×C22⋊C4C2×D412yes—
266C2×C22⋊C4C2×D412yes—
267C2×C42—21no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—21no—
271C2×C42—24no—
272C2×C42C2×D414yes—
273C2×C42C2×D412yes—
274C2×C42C2×D414yes—
275C2×C42C2412yes—
276C2×C42C2412yes—
277C2×C42C2411yescharacteristic
278C2×C42C2411yescharacteristic
279#107—48no—
280#103—28no—
281#103—28no—
282#103—44no—
283#102—28no—
284#102—28no—
285#102—48no—
286#102—44no—
287#95—22no—
288#95—24no—
289#95—24no—
290#95—28no—
291#95—22no—
292#95C2×D414yes—
293#95C2×D414yes—
294C24—22no—
295C24—22no—
296C24—24no—
297C24—42no—
298C24—42no—
299C24—82no—
300C24—82no—
301C24C22×D411yescharacteristic
302#41—216no—
303#41—216no—
304#41—48no—
305#41—48no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8C2×#4114yes—
312C2×D4—28no—
313C2×D4—28no—
314C2×D4—24no—
315C2×D4—216no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—24no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—24no—
323C2×D4—22no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—48no—
330C2×D4—44no—
331C2×D4C2×#4114yes—
332C2×D4C2×#4114yes—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—216no—
337C22×C4—28no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—42no—
343C22×C4—48no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—42no—
348C22×C4—84no—
349C22×C4—84no—
350C22×C4—88no—
351C22×C4—82no—
352C22×C4—84no—
353C22×C4C22×D414yes—
354C22×C4C22×D412yes—
355C22×C4C22×D412yes—
356C22×C4C22×D412yes—
357C22×C4C22×D412yes—
358C22×C4C22×D411yescharacteristic
359C22×C4C2×#4112yes—
360C22×C4C2×#4114yes—
361C22×C4C2×#4112yes—
362C22×C4H(5,𝔽2)14yes—
363C22×C4C2511yescharacteristic
364SD16—416no—
365D8—416no—
366M16—48no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—48no—
374C4⋊C4—28no—
375C4⋊C4—24no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—48no—
384C4⋊C4—42no—
385C4⋊C4—44no—
386C4⋊C4—48no—
387C4⋊C4—44no—
388C4⋊C4—82no—
389C4⋊C4—84no—
390C4⋊C4—84no—
391C4⋊C4C2×#4114yes—
392C22⋊C4—216no—
393C22⋊C4—216no—
394C22⋊C4—24no—
395C22⋊C4—24no—
396C22⋊C4—24no—
397C22⋊C4—48no—
398C22⋊C4—42no—
399C22⋊C4—48no—
400C22⋊C4—84no—
401C22⋊C4—84no—
402C22⋊C4—84no—
403C22⋊C4—88no—
404C42—24no—
405C42—24no—
406C42—28no—
407C42—22no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—24no—
412C42—44no—
413C42—48no—
414C42—42no—
415C42—84no—
416C23—22no—
417C23—22no—
418C23—24no—
419C23—24no—
420C23—44no—
421C23—44no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—44no—
427C23—44no—
428C23—48no—
429C23—88no—
430C23—162no—
431C23—162no—
432C23—164no—
433C23#52512yes—
434C23#52512yes—
435C23#53512yes—
436C23C23×D411yescharacteristic
437Q8—24no—
438Q8—24no—
439Q8—42no—
440Q8—42no—
441D4—24no—
442D4—28no—
443D4—24no—
444D4—416no—
445D4—416no—
446D4—48no—
447D4—42no—
448D4—44no—
449D4—42no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—48no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—416no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—48no—
477C2×C4—816no—
478C2×C4—168no—
479C2×C4—162no—
480C2×C4—164no—
481C2×C4#52512yes—
482C2×C4#52512yes—
483C2×C4#53512yes—
484C2×C4#55118yes—
485C2×C4#55318yes—
486C2×C4#57614yes—
487C2×C4C22×#4111yescharacteristic
488C2×C4C22×#4112yes—
489C2×C4C22×#4111yescharacteristic
490C2×C4C2×H(5,𝔽2)12yes—
491C8—44no—
492C8—44no—
493C22—22no—
494C22—22no—
495C22—24no—
496C22—24no—
497C22—44no—
498C22—44no—
499C22—48no—
500C22—48no—
501C22—82no—
502C22—82no—
503C22—82no—
504C22—82no—
505C22—84no—
506C22—84no—
507C22—168no—
508C22#259814yes—
509C22C2×#52511yescharacteristic
510C22C2×#52511yescharacteristic
511C22C2×#53511yescharacteristic
512C22C2×#57612yes—
513C22#358612yes—
514C4—24no—
515C4—44no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—44no—
520C4—44no—
521C4—82no—
522C4—84no—
523C4—82no—
524C4#259814yes—
525C4C2×#55112yes—
526C4C2×#55312yes—
527C2—24no—
528C2—84no—
529C2—84no—
530C2C2×#259811yescharacteristic
531C2#6058412yes—
532C2#6059612yes—
533C2#6081711yescharacteristic
534C2#6082811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#23428C2
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. #34255C2semidirect
2. #34207C2semidirect
3. #29381C2semidirect
4. #29369C2semidirect
5. #29364C2semidirect
6. #23745C2semidirect
7. C2×#2093C2semidirect
8. #23504C2semidirect
10. #21067C2semidirect
11. #20932C2semidirect
12. #20929C2semidirect
13. #20881C2semidirect
14. #20774C2semidirect
30. #2326C22semidirect
35. C4×C4⋊Q8C22semidirect
36. C4×#127C22semidirect
43. #2100C22semidirect
44. #2099C22semidirect
47. #2043C22semidirect
48. #2042C22semidirect
49. #2041C22semidirect
53. #1928C22semidirect
54. #1925C22semidirect
59. #1863C22semidirect
60. #1862C22semidirect
61. #1840C22semidirect
62. #1839C22semidirect

Maximal subgroups

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

subgroupindex
#342552
#342072
#293812
#293692
#293642
#237452
C2×#20932
#235042
#234282
#210672
#209322
#209292
#208812
#207742