Finite Group Databasebeta

← #6477650

Subgroups — #6477650

Order 512.

2204 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#481C211yesmaximal, characteristic, complemented
2C22×#480C211yesmaximal, characteristic, complemented
3#37450C211yesmaximal, characteristic
4#34178C211yesmaximal, characteristic, complemented
5#33118C214yesmaximal, complemented
6#29763C214yesmaximal, complemented
7#24583C214yesmaximal, complemented
8#24581C214yesmaximal, complemented
9#24478C214yesmaximal, complemented
10#24476C214yesmaximal, complemented
11#21931C211yesmaximal, characteristic, complemented
12#21761C211yesmaximal, characteristic, complemented
13#21761C211yesmaximal, characteristic, complemented
14C22×#121C2211yescharacteristic
15C22×#121C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C2×#481—28no—
18C2×#481C2214yes—
19C2×#481C2212yescomplemented
20C2×#481C2212yescomplemented
21C2×#480—28no—
22C2×#480C2214yes—
23C2×#480C2212yescomplemented
24C2×#480C2212yescomplemented
25C22×#107C2212yescomplemented
26C22×#106C2212yescomplemented
27C22×#102C2212yescomplemented
28C22×#102C2212yescomplemented
29C22×C22⋊C8C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31#2815—22no—
32#2619C2214yes—
33#2617—22no—
34#2617—22no—
35#2504C2212yes—
36#2504C2212yes—
37#2433C2212yes—
38#2433C2212yes—
39#2403—22no—
40#2345—22no—
41#2345C2214yescomplemented
42#2344—22no—
43#2344C2214yescomplemented
44#2316C2212yescomplemented
45#2313—22no—
46#2280—22no—
47#2280C2211yescharacteristic
48#2280C2211yescharacteristic
49#2274C2212yescomplemented
50#2085C2218yescomplemented
51#2067C2218yescomplemented
52#2065C2218yescomplemented
53#2041C2218yescomplemented
54#1982C2218yescomplemented
55#1949C2218yescomplemented
56#1926C2218yescomplemented
57#1925C2218yescomplemented
58#1893C2214yes—
59#1893C2214yes—
60#1875C2214yescomplemented
61#1875C2214yescomplemented
62#1850C2214yescomplemented
63#1850C2214yescomplemented
64#1824C2212yescomplemented
65#1824C2212yescomplemented
66#1798C2214yescomplemented
67#1786C2212yescomplemented
68#1786C2212yescomplemented
69C23×D4—21no—
70C23×D4—21no—
71C23×D4C2311yescharacteristic
72C23×D4C2311yescharacteristic
73C24×C4—21no—
74C24×C4C2311yescharacteristic
75C23×C8C2312yes—
76C23×C8C2312yes—
77C2×#127C2314yes—
78C2×#121—21no—
79C2×#121—21no—
80C2×#121—21no—
81C2×#121—21no—
82C2×#121—24no—
83C2×#121—24no—
84C2×#121—24no—
85C2×#121—24no—
86C2×#121C2312yes—
87C2×#121C2312yes—
88C2×#121C2312yes—
89C2×#121C2312yes—
90C2×#120—24no—
91C2×#120—24no—
92C2×#117—24no—
93C2×#117C2312yes—
94C2×#117C2312yes—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4D418yescomplemented
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C22⋊C4—21no—
107C22×C22⋊C4—21no—
108#481—216no—
109#481—28no—
110#481—28no—
111#480—216no—
112#480—28no—
113#480—28no—
114C2×#107—28no—
115C2×#107C2314yes—
116C2×#107C2314yes—
117C2×#106—28no—
118C2×#106C2314yes—
119C2×#106C2314yes—
120C2×C4⋊C8C2318yes—
121C2×#102—28no—
122C2×#102—28no—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2318yes—
128C2×#102C2318yes—
129C2×C22⋊C8—24no—
130C2×C22⋊C8—24no—
131C2×C22⋊C8C2312yes—
132C2×C22⋊C8C2312yes—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135#397—24no—
136#397—24no—
137#394—28no—
138#392—28no—
139#392—24no—
140#392—24no—
141#390C2314yes—
142#390C2314yes—
143#389—24no—
144#389C2318yes—
145#387—28no—
146#387—28no—
147#386—24no—
148#386—24no—
149#382C2314yes—
150#381—21no—
151#381—24no—
152#381D412yes—
153#380—24no—
154#380—24no—
155#380D412yes—
156#380D412yes—
157#380C2312yes—
158#380C2312yes—
159#379—21no—
160#379D412yes—
161C4×C4⋊C4—24no—
162C4×C4⋊C4C2318yes—
163C4×C22⋊C4—21no—
164C4×C22⋊C4—21no—
165C4×C22⋊C4—24no—
166C4×C22⋊C4—24no—
167C4×C22⋊C4C2312yes—
168C4×C22⋊C4C2312yes—
169#340C2318yes—
170#340C2318yes—
171#336C2318yes—
172C25—21no—
173C25—21no—
174C25C2×D411yescharacteristic
175C22×D4—22no—
176C22×D4—22no—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4—21no—
182C22×D4—21no—
183C22×D4—28no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—44no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4—84no—
191C22×D4—84no—
192C22×D4C2412yes—
193C22×D4C2412yes—
194C23×C4—22no—
195C23×C4—28no—
196C23×C4—21no—
197C23×C4—22no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—44no—
202C23×C4—84no—
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D414yes—
207C23×C4#4114yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C22×C8—24no—
212C22×C8—24no—
213C22×C8—44no—
214C22×C8—44no—
215C22×C8C2414yes—
216C22×C8C2414yes—
217#127—48no—
218#121—24no—
219#121—24no—
220#121—24no—
221#121—24no—
222#121—24no—
223#121—24no—
224#121—24no—
225#121—24no—
226#121—44no—
227#121—44no—
228#121—44no—
229#121—44no—
230#120—88no—
231#120—88no—
232#117—44no—
233#117—44no—
234#117—88no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4C2×D418yes—
254C2×C4⋊C4C2×D418yes—
255C2×C4⋊C4C2×D414yes—
256C2×C4⋊C4C2×D414yes—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4C2412yes—
259C2×C4⋊C4C2412yes—
260C2×C4⋊C4C2414yes—
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2412yes—
263C2×C22⋊C4—22no—
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—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4C2×D414yes—
275C2×C22⋊C4C2×D414yes—
276C2×C22⋊C4C2×D414yes—
277C2×C42—22no—
278C2×C42—28no—
279C2×C42C2414yes—
280#107—28no—
281#107—28no—
282#106—28no—
283#106—28no—
284C4⋊C8—48no—
285#102—28no—
286#102—28no—
287#102—28no—
288#102—28no—
289#102—48no—
290#102—48no—
291C22⋊C8—24no—
292C22⋊C8—24no—
293C22⋊C8—24no—
294C22⋊C8—24no—
295#95—28no—
296#95—28no—
297#95—22no—
298#95—22no—
299#95—28no—
300#95C2×D414yes—
301#95C2×D414yes—
302C24—24no—
303C24—24no—
304C24—22no—
305C24—22no—
306C24—42no—
307C24—42no—
308C24—42no—
309C24—44no—
310C24—44no—
311C24—82no—
312C24—82no—
313C24—82no—
314C24—82no—
315C24#12114yes—
316C24C22×D411yescharacteristic
317C24C22×D411yescharacteristic
318C24C22×D411yescharacteristic
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—28no—
324C2×D4—28no—
325C2×D4—48no—
326C2×D4—48no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—48no—
332C2×D4—48no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—84no—
336C2×D4—84no—
337C2×D4—82no—
338C2×D4—82no—
339C2×D4—88no—
340C2×D4—88no—
341C2×D4—88no—
342C2×D4—84no—
343C2×D4—84no—
344C2×D4—816no—
345C2×D4—816no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—28no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—28no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—48no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—816no—
367C22×C4—82no—
368C22×C4—84no—
369C22×C4—82no—
370C22×C4—82no—
371C22×C4—82no—
372C22×C4#12114yes—
373C22×C4#12114yes—
374C22×C4#12114yes—
375C22×C4C22×D418yes—
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D412yes—
380C22×C4C22×D412yes—
381C22×C4C22×D412yes—
382C22×C4C2×#4112yes—
383C22×C4C2×#4112yes—
384C22×C4C2×#4112yes—
385C22×C4C2511yescharacteristic
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—28no—
389C2×C8—28no—
390C2×C8—48no—
391C2×C8—48no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—28no—
395C4⋊C4—28no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—28no—
403C4⋊C4—44no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—88no—
407C4⋊C4C22×D414yes—
408C4⋊C4C22×D414yes—
409C22⋊C4—28no—
410C22⋊C4—28no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—88no—
420C22⋊C4—88no—
421C42—28no—
422C42—44no—
423C42—88no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—28no—
430C23—28no—
431C23—22no—
432C23—22no—
433C23—44no—
434C23—416no—
435C23—42no—
436C23—42no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—84no—
444C23—84no—
445C23—84no—
446C23—84no—
447C23—84no—
448C23—84no—
449C23—168no—
450C23—168no—
451C23C2×#12112yes—
452C23C2×#12112yes—
453C23C2×#12112yes—
454C23#57614yes—
455C23#57714yes—
456C23C23×D411yescharacteristic
457D4—44no—
458D4—44no—
459D4—44no—
460D4—44no—
461D4—84no—
462D4—84no—
463D4—88no—
464D4—88no—
465D4—88no—
466D4—88no—
467D4—1616no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—28no—
480C2×C4—28no—
481C2×C4—28no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—416no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—88no—
496C2×C4—84no—
497C2×C4—84no—
498C2×C4—84no—
499C2×C4—84no—
500C2×C4C2×#12112yes—
501C2×C4C2×#12112yes—
502C2×C4C2×#12112yes—
503C2×C4C23×D411yescharacteristic
504C2×C4C22×#4111yescharacteristic
505C2×C4#58514yes—
506C8—44no—
507C8—44no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—28no—
513C22—28no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—416no—
518C22—44no—
519C22—82no—
520C22—82no—
521C22—82no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22—88no—
526C22—88no—
527C22—168no—
528C22—168no—
529C22#304114yes—
530C22#304314yes—
531C22C22×#12111yescharacteristic
532C22#348512yes—
533C22C2×#57612yes—
534C22C2×#57712yes—
535C4—22no—
536C4—22no—
537C4—22no—
538C4—44no—
539C4—44no—
540C4—48no—
541C4—82no—
542C4—82no—
543C4—84no—
544C4#344912yes—
545C4#349112yes—
546C2—24no—
547C2—24no—
548C2—42no—
549C2—162no—
550C2—162no—
551C2#3508911yescharacteristic
552C2C2×#304111yescharacteristic
553C2C2×#304311yescharacteristic
554C2#6176614yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#37450C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#481C2semidirect
2. C22×#480C2semidirect
4. #34178C2semidirect
5. #33118C2semidirect
6. #29763C2semidirect
7. #24583C2semidirect
8. #24581C2semidirect
9. #24478C2semidirect
10. #24476C2semidirect
11. #21931C2semidirect
12. #21761C2semidirect
13. #21761C2semidirect
19. C2×#481C22semidirect
20. C2×#481C22semidirect
23. C2×#480C22semidirect
24. C2×#480C22semidirect
25. C22×#107C22semidirect
26. C22×#106C22semidirect
27. C22×#102C22semidirect
28. C22×#102C22semidirect
41. #2345C22semidirect
43. #2344C22semidirect
44. #2316C22semidirect
49. #2274C22semidirect
50. #2085C22semidirect
51. #2067C22semidirect
52. #2065C22semidirect
53. #2041C22semidirect
54. #1982C22semidirect
55. #1949C22semidirect
56. #1926C22semidirect
57. #1925C22semidirect
60. #1875C22semidirect
61. #1875C22semidirect
62. #1850C22semidirect
63. #1850C22semidirect
64. #1824C22semidirect
65. #1824C22semidirect
66. #1798C22semidirect
67. #1786C22semidirect
68. #1786C22semidirect
99. C22×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C22×#4812
C22×#4802
#374502
#341782
#331182
#297632
#245832
#245812
#244782
#244762
#219312
#217612