Finite Group Databasebeta

← #6508729

Subgroups — #6508729

Order 512.

2020 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3195C211yesmaximal, characteristic
2C22×#483C211yesmaximal, characteristic
3C2×#2502C211yesmaximal, characteristic
4#34245C211yesmaximal, characteristic
5#28124C214yesmaximal, complemented
6#27647C214yesmaximal, complemented
7#25219C218yesmaximal, complemented
8#24580C214yesmaximal, complemented
9#24532C214yesmaximal, complemented
10#21932C211yesmaximal, characteristic
11#21762C211yesmaximal, characteristic
12#20700C211yesmaximal, characteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×C22⋊Q8C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16#3195—28no—
17C2×#483—28no—
18C2×#483C2212yes—
19C2×#483C2212yes—
20C2×#483C2214yes—
21C2×#483C2214yes—
22C2×#470C2214yes—
23C2×#451C2212yes—
24C2×#451C2212yes—
25C22×#106C2212yes—
26C22×#103C2212yes—
27C2×#407C2211yescharacteristic
28C22×C22⋊C8C2211yescharacteristic
29#2502—21no—
30#2502—22no—
31#2502—21no—
32#2502C2214yes—
33#2502C2214yes—
34C2×#398—22no—
35#2368—22no—
36#2362C2214yes—
37#2344C2212yes—
38#2342—22no—
39Q8×C22⋊C4C2212yes—
40#2332—22no—
41#2313—21no—
42C4×C22⋊Q8C2214yes—
43C2×#391C2212yes—
44C2×#385—22no—
45C2×#384C2212yes—
46#2280C2211yescharacteristic
47#2274—21no—
48C2×C4×C22⋊C4C2211yescharacteristic
49#2084C2218yescomplemented
50#2072C2218yescomplemented
51#2066C2218yescomplemented
52#2056C2218yescomplemented
53#1948C2218yescomplemented
54#1927C2218yescomplemented
55#1892C2214yes—
56#1878C2218yescomplemented
57#1873C2214yes—
58#1851C2214yes—
59#1850C2214yes—
60#1824C2214yes—
61#1823C2212yes—
62#1823C2214yes—
63#1803C2218yescomplemented
64#1797C2214yes—
65#1787C2212yes—
66#1736C2214yes—
67C23×Q8C2311yescharacteristic
68C23×Q8C2311yescharacteristic
69C24×C4—21no—
70C24×C4C2311yescharacteristic
71C22×Q16C2314yes—
72C22×M16C2312yes—
73C23×C8C2312yes—
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8—24no—
77C2×C22⋊Q8—21no—
78C2×C22⋊Q8—21no—
79C2×C22⋊Q8—24no—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8D418yes—
82C2×C22⋊Q8C2312yes—
83C2×C22⋊Q8C2312yes—
84C2×C22⋊Q8C2312yes—
85C2×C22⋊Q8C2312yes—
86C2×C4×Q8C2314yes—
87C2×#117—22no—
88C2×#117C2312yes—
89C22×C4⋊C4—21no—
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⋊C4—24no—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C22⋊C4—21no—
103C22×C22⋊C4—21no—
104C22×C22⋊C4D412yes—
105C22×C42C2312yes—
106#483—216no—
107#483—216no—
108#483—28no—
109#483—28no—
110#470—216no—
111#451—28no—
112#451—28no—
113C2×#106—28no—
114C2×#106C2318yes—
115C2×#106C2314yes—
116C2×C4⋊C8C2318yes—
117C2×#103—28no—
118C2×#103C2318yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2318yes—
122#407—24no—
123C2×C22⋊C8—24no—
124C2×C22⋊C8C2312yes—
125C2×C22⋊C8C2312yes—
126C2×C22⋊C8C2312yes—
127C2×C22⋊C8C2312yes—
128#398—24no—
129#398—24no—
130#398—28no—
131#391—22no—
132#391—22no—
133#391—24no—
134#391C2314yes—
135#391C2318yes—
136#389C2314yes—
137#388—24no—
138#388—24no—
139#387—24no—
140#387—24no—
141#387—24no—
142#385—24no—
143#385—24no—
144#385—28no—
145#385—24no—
146#384—22no—
147#384—22no—
148#384C2318yes—
149#381—21no—
150#381—22no—
151#380—22no—
152#380C2312yes—
153#380C2312yes—
154#379—21no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4—24no—
157C4×C4⋊C4C2314yes—
158C4×C22⋊C4—21no—
159C4×C22⋊C4—24no—
160C4×C22⋊C4—21no—
161C4×C22⋊C4—22no—
162C4×C22⋊C4D411yescharacteristic
163C4×C22⋊C4D412yes—
164C4×C22⋊C4D411yescharacteristic
165C4×C22⋊C4C2312yes—
166C4×C22⋊C4C2312yes—
167C2×#95—24no—
168C2×#95D412yes—
169#340C2318yes—
170#340C2318yes—
171#340C2318yes—
172#336C2318yes—
173C25C2×D411yescharacteristic
174C22×Q8—21no—
175C22×Q8—21no—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8—22no—
179C22×Q8—44no—
180C22×Q8—44no—
181C22×Q8C2×D414yes—
182C22×Q8C2412yes—
183C22×Q8C2412yes—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—84no—
194C23×C4C2×D414yes—
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D414yes—
199C23×C4#4114yes—
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C2×Q16—28no—
204C2×Q16—48no—
205C2×M16—24no—
206C2×M16—44no—
207C22×C8—24no—
208C22×C8—44no—
209C22×C8C2414yes—
210C22×C8C2414yes—
211C22⋊Q8—28no—
212C22⋊Q8—28no—
213C22⋊Q8—24no—
214C22⋊Q8—24no—
215C22⋊Q8—24no—
216C22⋊Q8—24no—
217C22⋊Q8—24no—
218C22⋊Q8—24no—
219C22⋊Q8—44no—
220C22⋊Q8—44no—
221C22⋊Q8C2×D418yes—
222C22⋊Q8C2×D418yes—
223C4×Q8—48no—
224#117—48no—
225#117—44no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—88no—
254C2×C4⋊C4C2×D414yes—
255C2×C4⋊C4C2414yes—
256C2×C4⋊C4C2412yes—
257C2×C4⋊C4C2414yes—
258C2×C4⋊C4C2412yes—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4C2×D412yes—
270C2×C22⋊C4C2×D414yes—
271C2×C22⋊C4C2×D412yes—
272C2×C42—24no—
273C2×C42—22no—
274C2×C42—24no—
275C2×C42—44no—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42C2414yes—
279#106—216no—
280#106—28no—
281C4⋊C8—48no—
282#103—216no—
283#103—28no—
284#103—28no—
285#103—48no—
286C22⋊C8—24no—
287C22⋊C8—24no—
288C22⋊C8—24no—
289C22⋊C8—24no—
290#95—28no—
291#95—28no—
292#95—22no—
293#95—24no—
294#95—24no—
295#95—22no—
296#95—24no—
297#95—24no—
298#95C2×D412yes—
299#95C2×D412yes—
300#95C2×D414yes—
301C24—24no—
302C24—24no—
303C24—42no—
304C24#12114yes—
305C24C22×D411yescharacteristic
306C24C22×D411yescharacteristic
307C24C22×D411yescharacteristic
308C2×Q8—24no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—28no—
313C2×Q8—28no—
314C2×Q8—44no—
315C2×Q8—48no—
316C2×Q8—42no—
317C2×Q8—82no—
318C2×Q8—84no—
319C2×Q8—84no—
320C2×Q8—84no—
321C2×Q8C22×D414yes—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—28no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—28no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—48no—
336C22×C4—48no—
337C22×C4—44no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—48no—
350C22×C4—84no—
351C22×C4—84no—
352C22×C4—82no—
353C22×C4—816no—
354C22×C4#12114yes—
355C22×C4#12114yes—
356C22×C4#12114yes—
357C22×C4C22×D414yes—
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D412yes—
362C22×C4C22×D412yes—
363C22×C4C22×D412yes—
364C22×C4C22×D414yes—
365C22×C4C2×#4112yes—
366C22×C4C2×#4112yes—
367C22×C4C2×#4112yes—
368C22×C4C2511yescharacteristic
369Q16—88no—
370M16—84no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—28no—
374C2×C8—28no—
375C2×C8—48no—
376C4⋊C4—28no—
377C4⋊C4—28no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—24no—
383C4⋊C4—22no—
384C4⋊C4—28no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—42no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—48no—
392C4⋊C4—88no—
393C4⋊C4—88no—
394C4⋊C4—84no—
395C4⋊C4C22×D414yes—
396C22⋊C4—28no—
397C22⋊C4—28no—
398C22⋊C4—24no—
399C22⋊C4—28no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—82no—
405C22⋊C4#13718yes—
406C42—28no—
407C42—42no—
408C42—42no—
409C42—48no—
410C42—82no—
411C42—84no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—28no—
418C23—28no—
419C23—22no—
420C23—22no—
421C23—44no—
422C23—416no—
423C23C2×#12112yes—
424C23C2×#12112yes—
425C23C2×#12112yes—
426C23C2×#13714yes—
427C23#57714yes—
428C23C23×D411yescharacteristic
429Q8—28no—
430Q8—44no—
431Q8—44no—
432Q8—44no—
433Q8—84no—
434Q8—84no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—416no—
455C2×C4—44no—
456C2×C4—48no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—48no—
468C2×C4—84no—
469C2×C4—84no—
470C2×C4—88no—
471C2×C4—88no—
472C2×C4—84no—
473C2×C4—88no—
474C2×C4C2×#12112yes—
475C2×C4C2×#12112yes—
476C2×C4C2×#12112yes—
477C2×C4C2×#13718yes—
478C2×C4C23×D411yescharacteristic
479C2×C4C22×#4111yescharacteristic
480C2×C4#58514yes—
481C8—44no—
482C8—44no—
483C22—22no—
484C22—28no—
485C22—28no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—416no—
493C22—44no—
494C22#303914yes—
495C22#304314yes—
496C22C22×#12111yescharacteristic
497C22#348512yes—
498C22C22×#13712yes—
499C22C2×#57712yes—
500C4—28no—
501C4—22no—
502C4—22no—
503C4—22no—
504C4—44no—
505C4—44no—
506C4—44no—
507C4—82no—
508C4—84no—
509C4—84no—
510C4—82no—
511C4#348512yes—
512C4#349112yes—
513C2—24no—
514C2—24no—
515C2—42no—
516C2#3544711yescharacteristic
517C2C2×#303911yescharacteristic
518C2C2×#304311yescharacteristic
519C2#6173514yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
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
5. #28124C2semidirect
6. #27647C2semidirect
7. #25219C2semidirect
8. #24580C2semidirect
9. #24532C2semidirect
49. #2084C22semidirect
50. #2072C22semidirect
51. #2066C22semidirect
52. #2056C22semidirect
53. #1948C22semidirect
54. #1927C22semidirect
56. #1878C22semidirect
63. #1803C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31952
C22×#4832
C2×#25022
#342452
#281242
#276472
#252192
#245802
#245322
#219322
#217622
#207002