Finite Group Databasebeta

← #6508726

Subgroups — #6508726

Order 512.

2120 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3193C211yesmaximal, characteristic
2C22×#481C211yesmaximal, characteristic
3C2×#2503C211yesmaximal, characteristic, complemented
4#34246C211yesmaximal, characteristic
5#28115C214yesmaximal, complemented
6#27651C214yesmaximal, complemented
7#25225C218yesmaximal, complemented
8#24579C214yesmaximal, complemented
9#24527C214yesmaximal, complemented
10#21932C211yesmaximal, characteristic, complemented
11#21762C211yesmaximal, characteristic, complemented
12#20700C211yesmaximal, characteristic, complemented
13C22×C22⋊Q8C2211yescharacteristic
14C22×#121C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16#3193—28no—
17C2×#484C2212yes—
18C2×#481—28no—
19C2×#481C2214yes—
20C2×#481C2214yes—
21C2×#480C2212yes—
22C2×#468C2214yes—
23C2×#448C2212yes—
24C2×#450C2212yes—
25C22×#106C2212yes—
26C22×#102C2212yes—
27C2×#407C2211yescharacteristic
28C22×C22⋊C8C2211yescharacteristic
29#2503—21no—
30#2503—22no—
31#2503—21no—
32#2503C2214yes—
33#2503C2214yescomplemented
34C2×#400—22no—
35#2364—22no—
36#2363C2214yes—
37#2345C2212yes—
38D4×C22⋊C4—22no—
39D4×C22⋊C4C2212yes—
40#2333—22no—
41#2312—21no—
42C4×#121C2214yes—
43C2×#391C2212yescomplemented
44C2×#382C2212yescomplemented
45#2280C2211yescharacteristic
46C2×#381—22no—
47#2273—21no—
48C2×C4×C22⋊C4C2211yescharacteristic
49#2085C2218yescomplemented
50#2073C2218yescomplemented
51#2066C2218yescomplemented
52#2055C2218yescomplemented
53#1949C2218yescomplemented
54#1927C2218yescomplemented
55#1892C2214yes—
56#1877C2218yescomplemented
57#1873C2214yescomplemented
58#1851C2214yescomplemented
59#1850C2214yes—
60#1824C2214yescomplemented
61#1823C2214yes—
62#1823C2212yescomplemented
63#1804C2218yescomplemented
64#1797C2214yescomplemented
65#1787C2212yescomplemented
66#1736C2214yescomplemented
67C23×Q8C2311yescharacteristic
68C23×D4—21no—
69C23×D4C2311yescharacteristic
70C24×C4—21no—
71C24×C4C2311yescharacteristic
72C22×SD16C2314yes—
73C22×M16C2312yes—
74C23×C8C2312yes—
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8—21no—
77C2×C22⋊Q8—24no—
78C2×C22⋊Q8—24no—
79C2×C22⋊Q8C2312yes—
80C2×C22⋊Q8C2312yes—
81C2×#121—21no—
82C2×#121—21no—
83C2×#121—24no—
84C2×#121D418yes—
85C2×#121C2312yes—
86C2×#121C2312yes—
87C2×C4×D4—24no—
88C2×C4×D4C2314yes—
89C2×#117C2312yes—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C22⋊C4—21no—
102C22×C22⋊C4—21no—
103C22×C22⋊C4—22no—
104C22×C22⋊C4D412yes—
105C22×C42C2312yes—
106#484—28no—
107#481—216no—
108#481—216no—
109#480—28no—
110#468—216no—
111#450—28no—
112#448—28no—
113C2×#107C2314yes—
114C2×#106—28no—
115C2×#106C2318yes—
116C2×C4⋊C8C2318yes—
117C2×#103C2314yes—
118C2×#103C2318yes—
119C2×#102—28no—
120C2×#102C2318yes—
121C2×#102C2314yes—
122#407—24no—
123C2×C22⋊C8—24no—
124C2×C22⋊C8C2312yes—
125C2×C22⋊C8C2312yes—
126C2×C22⋊C8C2312yes—
127C2×C22⋊C8C2312yes—
128#400—28no—
129#400—24no—
130#400—24no—
131#391—22no—
132#391—22no—
133#391C2318yes—
134#390—24no—
135#390C2314yes—
136#389C2314yes—
137#388—24no—
138#388—24no—
139#387—24no—
140#386—24no—
141#386—24no—
142#386—24no—
143#382—22no—
144#382—22no—
145#382C2318yes—
146#381—21no—
147#381—24no—
148#381—24no—
149#381—28no—
150#380—22no—
151#380C2312yes—
152#380C2312yes—
153#379—21no—
154#379—22no—
155C4×C4⋊C4C2314yes—
156C4×C22⋊C4—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4—22no—
159C4×C22⋊C4—24no—
160C4×C22⋊C4—24no—
161C4×C22⋊C4D411yescharacteristic
162C4×C22⋊C4D412yes—
163C4×C22⋊C4D411yescharacteristic
164C4×C22⋊C4C2312yes—
165C4×C22⋊C4C2312yes—
166C2×#95—24no—
167C2×#95—24no—
168C2×#95D412yes—
169#340C2318yes—
170#340C2318yes—
171#340C2318yes—
172#336C2318yes—
173C25—21no—
174C25C2×D411yescharacteristic
175C22×Q8—21no—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8—44no—
179C22×Q8C2412yes—
180C22×D4—22no—
181C22×D4—28no—
182C22×D4—22no—
183C22×D4—21no—
184C22×D4—22no—
185C22×D4—44no—
186C22×D4—84no—
187C22×D4C2×D414yes—
188C22×D4C2412yes—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—44no—
197C23×C4—48no—
198C23×C4—44no—
199C23×C4C2×D414yes—
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D414yes—
204C23×C4#4114yes—
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C2×SD16—28no—
209C2×SD16—48no—
210C2×M16—24no—
211C2×M16—44no—
212C22×C8—24no—
213C22×C8—44no—
214C22×C8C2414yes—
215C22×C8C2414yes—
216C22⋊Q8—24no—
217C22⋊Q8—24no—
218C22⋊Q8—24no—
219C22⋊Q8—24no—
220C22⋊Q8—44no—
221C22⋊Q8—44no—
222#121—28no—
223#121—28no—
224#121—24no—
225#121—24no—
226#121C2×D418yes—
227#121C2×D418yes—
228C4×D4—48no—
229C4×D4—88no—
230#117—44no—
231C2×C4⋊C4—21no—
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—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4C2×D414yes—
253C2×C4⋊C4C2414yes—
254C2×C4⋊C4C2412yes—
255C2×C4⋊C4C2414yes—
256C2×C4⋊C4C2412yes—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—48no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4C2×D412yes—
272C2×C22⋊C4C2×D412yes—
273C2×C22⋊C4C2×D414yes—
274C2×C42—22no—
275C2×C42—24no—
276C2×C42—44no—
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42C2414yes—
280#107—28no—
281#106—216no—
282C4⋊C8—48no—
283#103—28no—
284#103—48no—
285#102—216no—
286#102—28no—
287C22⋊C8—24no—
288C22⋊C8—24no—
289C22⋊C8—24no—
290C22⋊C8—24no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—22no—
296#95—24no—
297#95—22no—
298#95—24no—
299#95C2×D412yes—
300#95C2×D414yes—
301#95C2×D412yes—
302C24—24no—
303C24—24no—
304C24—22no—
305C24—24no—
306C24—42no—
307C24—44no—
308C24—82no—
309C24—82no—
310C24#12114yes—
311C24C22×D411yescharacteristic
312C24C22×D411yescharacteristic
313C24C22×D411yescharacteristic
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—28no—
317C2×Q8—44no—
318C2×Q8—42no—
319C2×Q8—84no—
320C2×Q8—84no—
321C2×D4—28no—
322C2×D4—28no—
323C2×D4—24no—
324C2×D4—22no—
325C2×D4—28no—
326C2×D4—48no—
327C2×D4—48no—
328C2×D4—84no—
329C2×D4—82no—
330C2×D4—84no—
331C2×D4—816no—
332C2×D4C22×D414yes—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—28no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—48no—
344C22×C4—48no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—48no—
353C22×C4—44no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—48no—
360C22×C4—44no—
361C22×C4—84no—
362C22×C4—84no—
363C22×C4—82no—
364C22×C4—816no—
365C22×C4#12114yes—
366C22×C4#12114yes—
367C22×C4#12114yes—
368C22×C4C22×D414yes—
369C22×C4C22×D414yes—
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D412yes—
374C22×C4C22×D412yes—
375C22×C4C22×D412yes—
376C22×C4C2×#4112yes—
377C22×C4C2×#4112yes—
378C22×C4C2×#4112yes—
379C22×C4C2511yescharacteristic
380SD16—88no—
381M16—84no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—28no—
385C2×C8—28no—
386C2×C8—48no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—28no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—42no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—84no—
401C4⋊C4C22×D414yes—
402C22⋊C4—24no—
403C22⋊C4—24no—
404C22⋊C4—28no—
405C22⋊C4—28no—
406C22⋊C4—44no—
407C22⋊C4—48no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—84no—
413C22⋊C4—84no—
414C22⋊C4—82no—
415C42—28no—
416C42—42no—
417C42—42no—
418C42—82no—
419C42—84no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—28no—
426C23—28no—
427C23—22no—
428C23—22no—
429C23—24no—
430C23—44no—
431C23—416no—
432C23—44no—
433C23—42no—
434C23—42no—
435C23—44no—
436C23—48no—
437C23—84no—
438C23—84no—
439C23—168no—
440C23C2×#12112yes—
441C23C2×#12112yes—
442C23C2×#12112yes—
443C23#57514yes—
444C23#57714yes—
445C23C23×D411yescharacteristic
446Q8—44no—
447Q8—44no—
448Q8—84no—
449D4—28no—
450D4—416no—
451D4—44no—
452D4—84no—
453D4—88no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—48no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—88no—
484C2×C4—84no—
485C2×C4—88no—
486C2×C4—88no—
487C2×C4—84no—
488C2×C4—88no—
489C2×C4—84no—
490C2×C4—168no—
491C2×C4C2×#12112yes—
492C2×C4C2×#12112yes—
493C2×C4C2×#12112yes—
494C2×C4C23×D411yescharacteristic
495C2×C4C22×#4111yescharacteristic
496C2×C4#58514yes—
497C8—44no—
498C8—44no—
499C22—22no—
500C22—28no—
501C22—28no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—416no—
509C22—44no—
510C22—44no—
511C22—44no—
512C22—44no—
513C22—84no—
514C22—84no—
515C22—168no—
516C22#304014yes—
517C22#304214yes—
518C22C22×#12111yescharacteristic
519C22#348512yes—
520C22C2×#57512yes—
521C22C2×#57712yes—
522C4—22no—
523C4—22no—
524C4—22no—
525C4—44no—
526C4—44no—
527C4—44no—
528C4—82no—
529C4—84no—
530C4—82no—
531C4—84no—
532C4#348512yes—
533C4#349112yes—
534C2—24no—
535C2—24no—
536C2—42no—
537C2—84no—
538C2#3544711yescharacteristic
539C2C2×#304011yescharacteristic
540C2C2×#304211yescharacteristic
541C2#6172914yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2503C2semidirect
5. #28115C2semidirect
6. #27651C2semidirect
7. #25225C2semidirect
8. #24579C2semidirect
9. #24527C2semidirect
10. #21932C2semidirect
11. #21762C2semidirect
12. #20700C2semidirect
33. #2503C22semidirect
43. C2×#391C22semidirect
44. C2×#382C22semidirect
49. #2085C22semidirect
50. #2073C22semidirect
51. #2066C22semidirect
52. #2055C22semidirect
53. #1949C22semidirect
54. #1927C22semidirect
56. #1877C22semidirect
57. #1873C22semidirect
58. #1851C22semidirect
60. #1824C22semidirect
62. #1823C22semidirect
63. #1804C22semidirect
64. #1797C22semidirect
65. #1787C22semidirect
66. #1736C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31932
C22×#4812
C2×#25032
#342462
#281152
#276512
#252252
#245792
#245272
#219322
#217622
#207002