Finite Group Databasebeta

← #6519032

Subgroups — #6519032

Order 512.

1874 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2#34186C211yesmaximal, characteristic
3#27604C214yesmaximal, complemented
4#25017C214yesmaximal, complemented
5#25015C214yesmaximal, complemented
6#24972C214yesmaximal, complemented
7#24970C214yesmaximal, complemented
8#24010C214yesmaximal, complemented
9#21982C211yesmaximal, characteristic, complemented
10#21981C211yesmaximal, characteristic, complemented
11#21786C211yesmaximal, characteristic
12#20734C211yesmaximal, characteristic, complemented
13#20734C211yesmaximal, characteristic, complemented
14C22×C22⋊Q8C2212yes—
15C22×C4×Q8C2211yescharacteristic
16C22×C4×D4C2211yescharacteristic
17C2×#485C2218yescomplemented
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20#2538—22no—
21#2538—22no—
22#2538C2214yes—
23#2538C2214yes—
24C2×#396—22no—
25#2357—22no—
26#2357—22no—
27#2354C2212yes—
28#2354C2212yes—
29#2354C2212yes—
30#2354C2212yes—
31#2312—22no—
32#2309—21no—
33C2×#390—21no—
34C2×#389—21no—
35#2285C2211yescharacteristic
36#2285C2211yescharacteristic
37C2×#384C2211yescharacteristic
38#2280C2214yes—
39C2×#380C2212yes—
40#2274—21no—
41#2077C2218yescomplemented
42#2075C2218yescomplemented
43#2036C2218yescomplemented
44#2035C2218yescomplemented
45#1954C2214yes—
46#1954C2214yes—
47#1948C2214yescomplemented
48#1947C2214yescomplemented
49#1873C2218yescomplemented
50#1854C2218yescomplemented
51#1853C2214yes—
52#1852C2214yes—
53#1845C2212yescomplemented
54#1844C2212yescomplemented
55#1834C2212yescomplemented
56#1834C2212yescomplemented
57#1825C2218yescomplemented
58#1824C2212yes—
59#1824C2212yes—
60#1803C2212yescomplemented
61#1803C2212yescomplemented
62#1803C2212yescomplemented
63#1803C2212yescomplemented
64#1787C2212yes—
65#1787C2212yes—
66#1779C2214yes—
67C4×#103C2214yescomplemented
68C4×#103C2214yescomplemented
69C23×Q8C2311yescharacteristic
70C23×D4D411yescharacteristic
71C24×C4C2312yes—
72C2×C22⋊Q8—24no—
73C2×C22⋊Q8—48no—
74C2×C22⋊Q8C2318yes—
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×C4×Q8—24no—
78C2×C4×Q8—24no—
79C2×C4×Q8C2312yes—
80C2×C4×Q8C2312yes—
81C2×C4×D4—24no—
82C2×C4×D4—24no—
83C2×C4×D4D412yes—
84C2×C4×D4D412yes—
85C2×C4×D4C2312yes—
86C2×C4×D4C2312yes—
87C2×#117—21no—
88C2×#117—21no—
89C2×#117C2314yes—
90C2×#117C2314yes—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—22no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4D411yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C22⋊C4—22no—
99C22×C22⋊C4—22no—
100C22×C22⋊C4D412yes—
101C22×C42—21no—
102C22×C42C2311yescharacteristic
103#485—416no—
104#431—24no—
105#431—24no—
106C2×#107C2318yes—
107C2×#106C2318yes—
108C2×C4⋊C8C2312yes—
109C2×C4⋊C8C2312yes—
110C2×#103C2314yes—
111C2×#103C2314yes—
112C2×#103C2314yes—
113C2×#103C2314yes—
114C2×C22⋊C8C2312yes—
115C2×C22⋊C8C2312yes—
116C2×C22⋊C8C2318yes—
117C2×C22⋊C8C2318yes—
118C2×C4×C8C2312yes—
119C2×C4×C8C2312yes—
120#396—28no—
121#396—28no—
122#390—22no—
123#390—22no—
124#390—22no—
125#390—22no—
126#390—24no—
127#390—24no—
128#389—24no—
129#389—24no—
130#388—24no—
131#388—24no—
132#388—24no—
133#388—24no—
134#384—21no—
135#384—21no—
136#384C2311yescharacteristic
137#384C2311yescharacteristic
138#384C2311yescharacteristic
139#384C2311yescharacteristic
140#384C2312yes—
141#384C2312yes—
142#382—21no—
143#382—21no—
144#381—24no—
145#380—21no—
146#380—21no—
147#380—24no—
148#380C2314yes—
149#380C2318yes—
150#380C2314yes—
151#380C2312yes—
152#380C2312yes—
153#379—24no—
154C4×C4⋊C4—22no—
155C4×C4⋊C4—22no—
156C4×C4⋊C4—22no—
157C4×C4⋊C4—22no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4C2312yes—
161C4×C4⋊C4C2312yes—
162C4×C22⋊C4—24no—
163C4×C22⋊C4—24no—
164C2×#95—22no—
165#340C2314yes—
166#340C2314yes—
167#340C2314yes—
168#340C2314yes—
169#340C2318yes—
170#340C2318yes—
171C25C2×D412yes—
172C22×Q8—24no—
173C22×Q8—21no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—42no—
177C22×Q8C2412yes—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4C2×D412yes—
182C22×D4C2×D412yes—
183C23×C4—21no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—22no—
187C23×C4—21no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4C2×D412yes—
191C23×C4C2×D414yes—
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2414yes—
195C23×C4C2411yescharacteristic
196C22×C8C2414yes—
197C22×C8C2414yes—
198C22⋊Q8—216no—
199C22⋊Q8—48no—
200C22⋊Q8—416no—
201C4×Q8—24no—
202C4×Q8—24no—
203C4×Q8—24no—
204C4×Q8—24no—
205C4×Q8—44no—
206C4×Q8—44no—
207C4×D4—28no—
208C4×D4—28no—
209C4×D4—24no—
210C4×D4—24no—
211C4×D4—24no—
212C4×D4—24no—
213#117—44no—
214#117—44no—
215#117—48no—
216#117—48no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—48no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—82no—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2412yes—
245C2×C4⋊C4C2414yes—
246C2×C4⋊C4C2412yes—
247C2×C4⋊C4C2412yes—
248C2×C4⋊C4C2412yes—
249C2×C4⋊C4C2412yes—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—48no—
260C2×C22⋊C4—84no—
261C2×C22⋊C4C2×D414yes—
262C2×C22⋊C4C2×D414yes—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—44no—
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42#4114yes—
273C2×C42#4114yes—
274C2×C42C2411yescharacteristic
275C2×C42C2411yescharacteristic
276#107—48no—
277#106—48no—
278C4⋊C8—24no—
279C4⋊C8—24no—
280#103—28no—
281#103—28no—
282#103—44no—
283#103—44no—
284C22⋊C8—24no—
285C22⋊C8—24no—
286C22⋊C8—48no—
287C22⋊C8—48no—
288C4×C8—24no—
289C4×C8—24no—
290#95—28no—
291#95—22no—
292#95—22no—
293#95—24no—
294#95—24no—
295#95—22no—
296#95—22no—
297C24—24no—
298C24—24no—
299C24—24no—
300C24—42no—
301C24—42no—
302C24—42no—
303C24—42no—
304C24C22×D414yes—
305C24C22×D411yescharacteristic
306C2×Q8—24no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—48no—
313C2×Q8—48no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8H(5,𝔽2)14yes—
317C2×D4—24no—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—24no—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—416no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—28no—
332C22×C4—28no—
333C22×C4—28no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—44no—
349C22×C4—42no—
350C22×C4—44no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—44no—
354C22×C4—88no—
355C22×C4—88no—
356C22×C4#12314yes—
357C22×C4#12314yes—
358C22×C4#13514yes—
359C22×C4#13714yes—
360C22×C4C22×D414yes—
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C2×#4112yes—
367C22×C4C2×#4112yes—
368C22×C4C2511yescharacteristic
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—28no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—416no—
385C4⋊C4—44no—
386C4⋊C4—42no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—81no—
396C4⋊C4—81no—
397C4⋊C4—81no—
398C4⋊C4—81no—
399C4⋊C4—82no—
400C4⋊C4—82no—
401C22⋊C4—28no—
402C22⋊C4—28no—
403C22⋊C4—416no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—84no—
407C22⋊C4—84no—
408C22⋊C4—88no—
409C22⋊C4—84no—
410C22⋊C4—84no—
411C42—44no—
412C42—44no—
413C42—44no—
414C42—44no—
415C42—84no—
416C42—84no—
417C42#12314yes—
418C42#12314yes—
419C42C2×#4112yes—
420C42C2×#4112yes—
421C42C2×#4112yes—
422C42C2×#4112yes—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—24no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—48no—
435C23—48no—
436C23—48no—
437C23—84no—
438C23—84no—
439C23—84no—
440C23—84no—
441C23C2×#12312yes—
442C23C2×#13512yes—
443C23C2×#13712yes—
444C23C23×D411yescharacteristic
445Q8—24no—
446Q8—24no—
447Q8—48no—
448Q8—42no—
449Q8—42no—
450D4—48no—
451D4—48no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—44no—
465C2×C4—48no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—48no—
469C2×C4—48no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—88no—
489C2×C4—84no—
490C2×C4—84no—
491C2×C4—88no—
492C2×C4#485116yescomplemented
493C2×C4C2×#12312yes—
494C2×C4C2×#12312yes—
495C2×C4C2×#12312yes—
496C2×C4C2×#12312yes—
497C2×C4C2×#12312yes—
498C2×C4C2×#13512yes—
499C2×C4C2×#13512yes—
500C2×C4C2×#13712yes—
501C2×C4C2×#13712yes—
502C2×C4C22×#4111yescharacteristic
503C2×C4C22×#4111yescharacteristic
504C2×C4C2×H(5,𝔽2)12yes—
505C2×C4#58512yes—
506C8—44no—
507C8—44no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—44no—
515C22—44no—
516C22—44no—
517C22—44no—
518C22—44no—
519C22—44no—
520C22—44no—
521C22—816no—
522C22C2×#48514yes—
523C22C22×#12311yescharacteristic
524C22#348411yescharacteristic
525C22#348511yescharacteristic
526C22C22×#13511yescharacteristic
527C22C22×#13711yescharacteristic
528C22#358511yescharacteristic
529C22#358611yescharacteristic
530C4—22no—
531C4—22no—
532C4—22no—
533C4—44no—
534C4—44no—
535C4—82no—
536C4—82no—
537C4—82no—
538C4—82no—
539C4—84no—
540C4—82no—
541C4C2×#48514yes—
542C4C2×#48514yes—
543C4#349412yes—
544C4#349412yes—
545C2—24no—
546C2—84no—
547C2#3554011yescharacteristic
548C2C22×#48511yescharacteristic
549C2#6052011yescharacteristic
550C2#6156611yescharacteristic
551C2#6158411yescharacteristic
552C2#6170111yescharacteristic
553C2#6171911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27604C2semidirect
4. #25017C2semidirect
5. #25015C2semidirect
6. #24972C2semidirect
7. #24970C2semidirect
8. #24010C2semidirect
9. #21982C2semidirect
10. #21981C2semidirect
12. #20734C2semidirect
13. #20734C2semidirect
17. C2×#485C22semidirect
41. #2077C22semidirect
42. #2075C22semidirect
43. #2036C22semidirect
44. #2035C22semidirect
47. #1948C22semidirect
48. #1947C22semidirect
49. #1873C22semidirect
50. #1854C22semidirect
53. #1845C22semidirect
54. #1844C22semidirect
55. #1834C22semidirect
56. #1834C22semidirect
57. #1825C22semidirect
60. #1803C22semidirect
61. #1803C22semidirect
62. #1803C22semidirect
63. #1803C22semidirect
67. C4×#103C22semidirect
68. C4×#103C22semidirect
492. C2×C4#485semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
#341862
#276042
#250172
#250152
#249722
#249702
#240102
#219822
#219812
#217862
#207342