Finite Group Databasebeta

← #6477652

Subgroups — #6477652

Order 512.

2066 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#484C211yesmaximal, characteristic, complemented
2C22×#480C211yesmaximal, characteristic
3#37451C211yesmaximal, characteristic
4#34178C211yesmaximal, characteristic, complemented
5#33120C214yesmaximal, complemented
6#29767C214yesmaximal, complemented
7#24582C214yesmaximal, complemented
8#24576C214yesmaximal, complemented
9#24481C214yesmaximal, complemented
10#24479C214yesmaximal, complemented
11#21931C211yesmaximal, characteristic, complemented
12#21762C211yesmaximal, characteristic, complemented
13#21761C211yesmaximal, characteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×#121C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C2×#484—28no—
18C2×#484C2214yes—
19C2×#484C2212yescomplemented
20C2×#484C2212yescomplemented
21C2×#480—28no—
22C2×#480C2214yes—
23C2×#480C2212yes—
24C2×#480C2212yes—
25C22×#107C2212yes—
26C22×#107C2212yescomplemented
27C22×#103C2212yescomplemented
28C22×#102C2212yes—
29C22×C22⋊C8C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31#2816—22no—
32#2626—22no—
33#2620C2214yes—
34#2618—22no—
35#2502C2212yes—
36#2501C2212yes—
37#2436C2212yes—
38#2435C2212yes—
39#2405—22no—
40#2345—22no—
41#2345C2214yes—
42#2344—22no—
43#2344C2214yescomplemented
44#2316C2212yescomplemented
45#2313—22no—
46#2280—22no—
47#2280C2211yescharacteristic
48#2280C2211yescharacteristic
49#2274C2212yescomplemented
50#2083C2218yescomplemented
51#2068C2218yescomplemented
52#2067C2218yescomplemented
53#2042C2218yescomplemented
54#1983C2218yescomplemented
55#1947C2218yescomplemented
56#1928C2218yescomplemented
57#1925C2218yescomplemented
58#1893C2214yes—
59#1892C2214yes—
60#1875C2214yes—
61#1873C2214yescomplemented
62#1850C2214yes—
63#1850C2214yescomplemented
64#1824C2212yes—
65#1823C2212yescomplemented
66#1798C2214yescomplemented
67#1787C2212yescomplemented
68#1786C2212yes—
69C23×Q8C2311yescharacteristic
70C23×D4—21no—
71C23×D4C2311yescharacteristic
72C24×C4—21no—
73C24×C4C2311yescharacteristic
74C23×C8C2312yes—
75C23×C8C2312yes—
76C2×#124C2314yes—
77C2×#123—24no—
78C2×C22⋊Q8—21no—
79C2×C22⋊Q8—21no—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8C2312yes—
83C2×C22⋊Q8C2312yes—
84C2×#121—21no—
85C2×#121—21no—
86C2×#121—24no—
87C2×#121—24no—
88C2×#121C2312yes—
89C2×#121C2312yes—
90C2×#117—24no—
91C2×#117C2312yes—
92C2×#117C2312yes—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4D418yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C22⋊C4—21no—
106C22×C22⋊C4—21no—
107#484—216no—
108#484—28no—
109#484—28no—
110#480—216no—
111#480—28no—
112#480—28no—
113C2×#107—28no—
114C2×#107—28no—
115C2×#107C2314yes—
116C2×#107C2314yes—
117C2×#107C2314yes—
118C2×#107C2314yes—
119C2×C4⋊C8C2318yes—
120C2×#103—28no—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2318yes—
124C2×#102—28no—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2318yes—
128C2×C22⋊C8—24no—
129C2×C22⋊C8—24no—
130C2×C22⋊C8C2312yes—
131C2×C22⋊C8C2312yes—
132C2×C22⋊C8C2312yes—
133C2×C22⋊C8C2312yes—
134#399—24no—
135#398—24no—
136#396—28no—
137#394—28no—
138#394—24no—
139#393—24no—
140#391C2314yes—
141#390C2314yes—
142#389—24no—
143#389C2318yes—
144#387—28no—
145#387—28no—
146#386—24no—
147#385—24no—
148#384C2314yes—
149#381—21no—
150#381—24no—
151#381D412yes—
152#380—24no—
153#380—24no—
154#380D412yes—
155#380D412yes—
156#380C2312yes—
157#380C2312yes—
158#379—21no—
159#379—24no—
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—
173C25C2×D411yescharacteristic
174C22×Q8—21no—
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8—44no—
178C22×Q8C2412yes—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4—21no—
183C22×D4—22no—
184C22×D4—44no—
185C22×D4—44no—
186C22×D4—84no—
187C22×D4C2412yes—
188C23×C4—22no—
189C23×C4—28no—
190C23×C4—21no—
191C23×C4—22no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—44no—
197C23×C4—84no—
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D414yes—
202C23×C4#4114yes—
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C22×C8—24no—
207C22×C8—24no—
208C22×C8—44no—
209C22×C8—44no—
210C22×C8C2414yes—
211C22×C8C2414yes—
212#124—48no—
213#123—88no—
214C22⋊Q8—24no—
215C22⋊Q8—24no—
216C22⋊Q8—24no—
217C22⋊Q8—24no—
218C22⋊Q8—44no—
219C22⋊Q8—44no—
220#121—24no—
221#121—24no—
222#121—24no—
223#121—24no—
224#121—44no—
225#121—44no—
226#117—44no—
227#117—44no—
228#117—88no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
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—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4C2×D418yes—
252C2×C4⋊C4C2×D418yes—
253C2×C4⋊C4C2×D414yes—
254C2×C4⋊C4C2×D414yes—
255C2×C4⋊C4C2×D414yes—
256C2×C4⋊C4C2412yes—
257C2×C4⋊C4C2412yes—
258C2×C4⋊C4C2414yes—
259C2×C4⋊C4C2412yes—
260C2×C4⋊C4C2412yes—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4C2×D414yes—
273C2×C22⋊C4C2×D414yes—
274C2×C22⋊C4C2×D414yes—
275C2×C42—22no—
276C2×C42—28no—
277C2×C42C2414yes—
278#107—28no—
279#107—28no—
280#107—28no—
281#107—28no—
282C4⋊C8—48no—
283#103—28no—
284#103—28no—
285#103—48no—
286#102—28no—
287#102—28no—
288#102—48no—
289C22⋊C8—24no—
290C22⋊C8—24no—
291C22⋊C8—24no—
292C22⋊C8—24no—
293#95—28no—
294#95—28no—
295#95—22no—
296#95—22no—
297#95—28no—
298#95C2×D414yes—
299#95C2×D414yes—
300C24—24no—
301C24—24no—
302C24—22no—
303C24—42no—
304C24—42no—
305C24—44no—
306C24—82no—
307C24—82no—
308C24#12114yes—
309C24C22×D411yescharacteristic
310C24C22×D411yescharacteristic
311C24C22×D411yescharacteristic
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—28no—
315C2×Q8—48no—
316C2×Q8—42no—
317C2×Q8—82no—
318C2×Q8—84no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—28no—
322C2×D4—48no—
323C2×D4—44no—
324C2×D4—44no—
325C2×D4—48no—
326C2×D4—42no—
327C2×D4—84no—
328C2×D4—82no—
329C2×D4—84no—
330C2×D4—816no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—28no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—28no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—48no—
345C22×C4—48no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—48no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—44no—
354C22×C4—816no—
355C22×C4—82no—
356C22×C4—84no—
357C22×C4—82no—
358C22×C4—82no—
359C22×C4—82no—
360C22×C4#12114yes—
361C22×C4#12114yes—
362C22×C4#12114yes—
363C22×C4C22×D418yes—
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D412yes—
368C22×C4C22×D412yes—
369C22×C4C22×D412yes—
370C22×C4C2×#4112yes—
371C22×C4C2×#4112yes—
372C22×C4C2×#4112yes—
373C22×C4C2511yescharacteristic
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—28no—
377C2×C8—28no—
378C2×C8—48no—
379C2×C8—48no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—28no—
383C4⋊C4—28no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—28no—
391C4⋊C4—44no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—88no—
397C4⋊C4—88no—
398C4⋊C4C22×D414yes—
399C4⋊C4C22×D414yes—
400C22⋊C4—28no—
401C22⋊C4—28no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—88no—
411C22⋊C4—88no—
412C42—28no—
413C42—44no—
414C42—88no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—28no—
421C23—28no—
422C23—22no—
423C23—22no—
424C23—44no—
425C23—416no—
426C23—42no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—84no—
431C23—84no—
432C23—84no—
433C23—168no—
434C23C2×#12112yes—
435C23C2×#12112yes—
436C23C2×#12112yes—
437C23#57614yes—
438C23#57814yes—
439C23C23×D411yescharacteristic
440Q8—44no—
441Q8—44no—
442Q8—84no—
443D4—44no—
444D4—44no—
445D4—84no—
446D4—88no—
447D4—88no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—28no—
460C2×C4—28no—
461C2×C4—28no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—416no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—88no—
478C2×C4—84no—
479C2×C4—84no—
480C2×C4—84no—
481C2×C4—84no—
482C2×C4—84no—
483C2×C4C2×#12112yes—
484C2×C4C2×#12112yes—
485C2×C4C2×#12112yes—
486C2×C4C23×D411yescharacteristic
487C2×C4C22×#4111yescharacteristic
488C2×C4#58514yes—
489C8—44no—
490C8—44no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—28no—
496C22—28no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—416no—
501C22—44no—
502C22—82no—
503C22—82no—
504C22—82no—
505C22—88no—
506C22—168no—
507C22#304114yes—
508C22#304414yes—
509C22C22×#12111yescharacteristic
510C22#348512yes—
511C22C2×#57612yes—
512C22C2×#57812yes—
513C4—22no—
514C4—22no—
515C4—22no—
516C4—44no—
517C4—44no—
518C4—48no—
519C4—82no—
520C4—82no—
521C4—82no—
522C4—84no—
523C4#344912yes—
524C4#349112yes—
525C2—24no—
526C2—24no—
527C2—42no—
528C2—162no—
529C2#3508911yescharacteristic
530C2C2×#304111yescharacteristic
531C2C2×#304411yescharacteristic
532C2#6176414yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
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×#484C2semidirect
4. #34178C2semidirect
5. #33120C2semidirect
6. #29767C2semidirect
7. #24582C2semidirect
8. #24576C2semidirect
9. #24481C2semidirect
10. #24479C2semidirect
11. #21931C2semidirect
12. #21762C2semidirect
19. C2×#484C22semidirect
20. C2×#484C22semidirect
26. C22×#107C22semidirect
27. C22×#103C22semidirect
43. #2344C22semidirect
44. #2316C22semidirect
49. #2274C22semidirect
50. #2083C22semidirect
51. #2068C22semidirect
52. #2067C22semidirect
53. #2042C22semidirect
54. #1983C22semidirect
55. #1947C22semidirect
56. #1928C22semidirect
57. #1925C22semidirect
61. #1873C22semidirect
63. #1850C22semidirect
65. #1823C22semidirect
66. #1798C22semidirect
67. #1787C22semidirect

Maximal subgroups

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

subgroupindex
C22×#4842
C22×#4802
#374512
#341782
#331202
#297672
#245822
#245762
#244812
#244792
#219312
#217622
#217612