Finite Group Databasebeta

← #6650288

Subgroups — #6650288

Order 512.

2092 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35010C211yesmaximal, characteristic
2C2×#2542C211yesmaximal, characteristic
3#33099C212yesmaximal, complemented
4#32029C214yesmaximal, complemented
5#30707C218yesmaximal, complemented
6#30507C214yesmaximal, complemented
7#29627C212yesmaximal, complemented
8#24380C214yesmaximal, complemented
9#24006C212yesmaximal, complemented
10#23921C212yesmaximal, complemented
11#21738C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C22×C22⋊Q8C2212yes—
15C22×C22⋊Q8C2211yescharacteristic
16C22×C4×D4C2211yescharacteristic
17C2×#484C2214yescomplemented
18C2×#467C2218yescomplemented
19C2×#451C2214yes—
20C2×#451C2214yescomplemented
21C22×C22⋊C8C2212yes—
22#2855C2212yes—
23#2848C2212yes—
24#2627—21no—
25#2626—21no—
26#2579—22no—
27#2564—22no—
28#2542—22no—
29#2542—22no—
30#2542C2214yes—
31#2542C2214yes—
32#2487C2212yes—
33#2417C2214yes—
34#2409C2212yes—
35C2×#399—21no—
36C2×#399—21no—
37#2315—21no—
38#2308—21no—
39C2×#390—21no—
40C2×#384—21no—
41C2×#380C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43#2043C2218yescomplemented
44#2036C2218yescomplemented
45#2036C2218yescomplemented
46#2016C2218yescomplemented
47#2001C2218yescomplemented
48#2001C2218yescomplemented
49#1928C2218yescomplemented
50#1921C2218yescomplemented
51#1892C2218yescomplemented
52#1872C2214yes—
53#1872C2214yescomplemented
54#1872C2214yescomplemented
55#1869C2214yes—
56#1869C2214yes—
57#1852C2214yes—
58#1817C2212yes—
59#1787C2214yes—
60#1779C2212yes—
61C23×Q8C2312yes—
62C23×Q8C2311yescharacteristic
63C23×D4D411yescharacteristic
64C24×C4C2311yescharacteristic
65C24×C4C2311yescharacteristic
66C22×Q16C2318yes—
67C23×C8C2314yes—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—44no—
70C2×C4⋊Q8C2314yes—
71C2×C22⋊Q8—21no—
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8—22no—
74C2×C22⋊Q8—22no—
75C2×C22⋊Q8—24no—
76C2×C22⋊Q8—48no—
77C2×C22⋊Q8—48no—
78C2×C22⋊Q8—44no—
79C2×C22⋊Q8C2312yes—
80C2×C22⋊Q8C2314yes—
81C2×C22⋊Q8C2314yes—
82C2×C22⋊Q8C2318yes—
83C2×C22⋊Q8C2314yes—
84C2×C22⋊Q8C2312yes—
85C2×C4×D4—44no—
86C2×C4×D4D412yes—
87C2×C4×D4D412yes—
88C2×C4×D4C2312yes—
89C2×C4×D4C2312yes—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—22no—
93C22×C4⋊C4—22no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4D411yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C22⋊C4—21no—
100C22×C22⋊C4—21no—
101C22×C22⋊C4—22no—
102C22×C22⋊C4—22no—
103C22×C22⋊C4D411yescharacteristic
104C22×C22⋊C4D411yescharacteristic
105C22×C42—21no—
106C22×C42C2311yescharacteristic
107#484—48no—
108#467—416no—
109#451—48no—
110#451—48no—
111C2×#107C2318yes—
112C2×#103C2318yes—
113C2×#103C2318yes—
114C2×#103C2318yes—
115C2×#103C2318yes—
116C2×C22⋊C8—44no—
117C2×C22⋊C8C2314yes—
118C2×C22⋊C8C2314yes—
119C2×C22⋊C8C2314yes—
120C2×C22⋊C8C2314yes—
121#400—22no—
122#399—22no—
123#399—24no—
124#399—24no—
125#399—24no—
126#399—24no—
127#398—24no—
128#396—24no—
129#395—22no—
130#393—22no—
131#391C2314yes—
132#390—24no—
133#390—24no—
134#389C2314yes—
135#387—24no—
136#386—24no—
137#386—22no—
138#386—22no—
139#385—24no—
140#385—22no—
141#385—22no—
142#385—22no—
143#385—22no—
144#384—24no—
145#384—24no—
146#381—22no—
147#380—21no—
148#380—21no—
149#380—24no—
150#380C2312yes—
151#380C2314yes—
152#380C2312yes—
153#379—22no—
154C4×C4⋊C4—22no—
155C4×C4⋊C4C2314yes—
156C2×#95—21no—
157C2×#95—21no—
158#340C2318yes—
159#340C2318yes—
160#336C2318yes—
161C25C2×D411yescharacteristic
162C25C2×D411yescharacteristic
163C22×Q8—22no—
164C22×Q8—21no—
165C22×Q8—28no—
166C22×Q8—44no—
167C22×Q8—44no—
168C22×Q8—42no—
169C22×Q8—44no—
170C22×Q8C2414yes—
171C22×Q8C2412yes—
172C22×D4—42no—
173C22×D4—42no—
174C22×D4Q1618yescomplemented
175C22×D4C2×D412yes—
176C22×D4C2×D412yes—
177C23×C4—22no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—22no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—22no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D412yes—
193C23×C4C2411yescharacteristic
194C23×C4C2411yescharacteristic
195C23×C4C2412yes—
196C23×C4C2411yescharacteristic
197C2×Q16—48no—
198C2×Q16—48no—
199C2×Q16—48no—
200C22×C8—44no—
201C22×C8—48no—
202C22×C8C2418yes—
203C4⋊Q8—44no—
204C4⋊Q8—44no—
205C4⋊Q8—48no—
206C22⋊Q8—28no—
207C22⋊Q8—28no—
208C22⋊Q8—216no—
209C22⋊Q8—28no—
210C22⋊Q8—44no—
211C22⋊Q8—44no—
212C22⋊Q8—48no—
213C22⋊Q8—44no—
214C22⋊Q8—416no—
215C22⋊Q8—48no—
216C22⋊Q8—48no—
217C22⋊Q8—44no—
218C22⋊Q8—44no—
219C22⋊Q8—48no—
220C4×D4—44no—
221C4×D4—44no—
222C4×D4—42no—
223C4×D4—42no—
224C4×D4—42no—
225C4×D4—42no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—48no—
243C2×C4⋊C4—48no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4C2×D412yes—
248C2×C4⋊C4C2×D412yes—
249C2×C4⋊C4C2412yes—
250C2×C4⋊C4C2414yes—
251C2×C4⋊C4C2414yes—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—48no—
264C2×C22⋊C4—48no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4C2×D412yes—
268C2×C22⋊C4C2×D412yes—
269C2×C22⋊C4C2×D412yes—
270C2×C22⋊C4C2×D412yes—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—44no—
274C2×C42C2×D412yes—
275C2×C42C2412yes—
276#107—216no—
277#103—216no—
278#103—48no—
279#103—48no—
280#103—48no—
281C22⋊C8—28no—
282C22⋊C8—44no—
283C22⋊C8—44no—
284C22⋊C8—44no—
285#95—24no—
286#95—22no—
287#95—22no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—42no—
297C24—42no—
298C24—42no—
299C24—42no—
300C24C2×Q1614yes—
301C24C2×Q1614yes—
302C24C22×D412yes—
303C24C22×D411yescharacteristic
304C24C22×D411yescharacteristic
305C24C22×D411yescharacteristic
306C2×Q8—28no—
307C2×Q8—24no—
308C2×Q8—28no—
309C2×Q8—28no—
310C2×Q8—24no—
311C2×Q8—24no—
312C2×Q8—44no—
313C2×Q8—44no—
314C2×Q8—42no—
315C2×Q8—44no—
316C2×Q8—82no—
317C2×Q8—816no—
318C2×Q8—88no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—416no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4C2×Q1618yes—
325C2×D4C2×Q1618yes—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—28no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—28no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—44no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—88no—
363C22×C4—88no—
364C22×C4—82no—
365C22×C4—82no—
366C22×C4—82no—
367C22×C4—82no—
368C22×C4C2×Q1614yes—
369C22×C4C22×D412yes—
370C22×C4C22×D412yes—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D412yes—
375C22×C4H(5,𝔽2)14yes—
376C22×C4C2511yescharacteristic
377Q16—816no—
378C2×C8—28no—
379C2×C8—28no—
380C2×C8—28no—
381C2×C8—88no—
382C4⋊C4—28no—
383C4⋊C4—24no—
384C4⋊C4—28no—
385C4⋊C4—24no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—416no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—44no—
392C4⋊C4—42no—
393C4⋊C4—48no—
394C4⋊C4—48no—
395C4⋊C4—44no—
396C4⋊C4—84no—
397C4⋊C4—82no—
398C4⋊C4—84no—
399C4⋊C4H(5,𝔽2)18yes—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—416no—
405C22⋊C4—48no—
406C22⋊C4—48no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—84no—
412C22⋊C4—84no—
413C42—24no—
414C42—22no—
415C42—22no—
416C42—48no—
417C23—22no—
418C23—22no—
419C23—24no—
420C23—22no—
421C23—22no—
422C23—21no—
423C23—21no—
424C23—22no—
425C23—24no—
426C23—22no—
427C23—22no—
428C23—21no—
429C23—21no—
430C23—22no—
431C23—28no—
432C23—28no—
433C23—48no—
434C23—48no—
435C23—82no—
436C23—82no—
437C23—82no—
438C23—82no—
439C23—84no—
440C23—84no—
441C23#53412yes—
442C23C22×Q1614yes—
443C23C22×Q1612yes—
444C23C22×Q1612yes—
445C23C22×Q1612yes—
446C23#57512yes—
447C23C23×D411yescharacteristic
448Q8—48no—
449Q8—48no—
450Q8—48no—
451Q8—88no—
452Q8—84no—
453D4—48no—
454D4—48no—
455C2×C4—28no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—28no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—28no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—44no—
470C2×C4—416no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—82no—
488C2×C4—84no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—88no—
493C2×C4—84no—
494C2×C4—84no—
495C2×C4—164no—
496C2×C4#53412yes—
497C2×C4C22×Q1614yes—
498C2×C4C2×H(5,𝔽2)12yes—
499C2×C4C2×H(5,𝔽2)14yes—
500C8—48no—
501C22—22no—
502C22—22no—
503C22—24no—
504C22—24no—
505C22—22no—
506C22—22no—
507C22—24no—
508C22—28no—
509C22—24no—
510C22—24no—
511C22—41no—
512C22—42no—
513C22—41no—
514C22—41no—
515C22—42no—
516C22—41no—
517C22—42no—
518C22—44no—
519C22—42no—
520C22—88no—
521C22—88no—
522C22#318914yes—
523C22C2×#53411yescharacteristic
524C22#348412yes—
525C22C23×Q1611yescharacteristic
526C22C2×#57511yescharacteristic
527C22#358612yes—
528C4—24no—
529C4—22no—
530C4—42no—
531C4—41no—
532C4—42no—
533C4—41no—
534C4—48no—
535C4—82no—
536C4—82no—
537C4—84no—
538C4—84no—
539C4#354314yes—
540C2—24no—
541C2—42no—
542C2—42no—
543C2—44no—
544C2#3723211yescharacteristic
545C2C2×#318911yescharacteristic
546C2#6075411yescharacteristic
547C2#6156312yes—
548C2#6157512yes—
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. #33099C2semidirect
4. #32029C2semidirect
5. #30707C2semidirect
6. #30507C2semidirect
7. #29627C2semidirect
8. #24380C2semidirect
9. #24006C2semidirect
10. #23921C2semidirect
17. C2×#484C22semidirect
18. C2×#467C22semidirect
20. C2×#451C22semidirect
43. #2043C22semidirect
44. #2036C22semidirect
45. #2036C22semidirect
46. #2016C22semidirect
47. #2001C22semidirect
48. #2001C22semidirect
49. #1928C22semidirect
50. #1921C22semidirect
51. #1892C22semidirect
53. #1872C22semidirect
54. #1872C22semidirect
174. C22×D4Q16semidirect

Maximal subgroups

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

subgroupindex
#350102
C2×#25422
#330992
#320292
#307072
#305072
#296272
#243802
#240062
#239212
#217382