Finite Group Databasebeta

← #6487700

Subgroups — #6487700

Order 512.

2220 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3109C211yesmaximal, characteristic
2C2×#3101C211yesmaximal, characteristic, complemented
3#37640C211yesmaximal, characteristic, complemented
4#34181C211yesmaximal, characteristic, complemented
5#32917C214yesmaximal, complemented
6#29768C214yesmaximal, complemented
7#24829C214yesmaximal, complemented
8#24815C214yesmaximal, complemented
9#24745C214yesmaximal, complemented
10#24731C214yesmaximal, complemented
11#21954C211yesmaximal, characteristic, complemented
12#21870C211yesmaximal, characteristic, complemented
13#21863C211yesmaximal, characteristic, complemented
14C2×#533C2211yescharacteristic
15C2×#532C2211yescharacteristic
16C2×#518C2212yes—
17C22×C4×D4C2211yescharacteristic
18#3109—28no—
19#3101—28no—
20C2×#481C2212yes—
21C2×#483C2212yes—
22C2×#476C2212yes—
23C2×#476C2212yes—
24C2×#463C2218yes—
25C2×#461C2212yes—
26C2×#459C2212yes—
27C2×#450C2212yes—
28C2×#447C2212yes—
29C2×#442C2214yes—
30C2×C4×SD16C2214yes—
31C2×#432C2211yescharacteristic
32C2×#431C2211yescharacteristic
33#2820—22no—
34#2624—22no—
35#2620C2214yes—
36#2616—22no—
37#2567C2212yescomplemented
38#2548C2212yescomplemented
39#2547C2212yescomplemented
40#2542C2212yescomplemented
41#2538—22no—
42#2349—22no—
43#2349C2214yes—
44D4×C4⋊C4—22no—
45D4×C4⋊C4C2214yes—
46#2316C2212yes—
47#2315—22no—
48#2285C2211yescharacteristic
49#2285C2211yescharacteristic
50C2×#380—22no—
51#2084C2218yescomplemented
52#2066C2218yescomplemented
53#2065C2218yescomplemented
54#2042C2218yescomplemented
55#1993C2212yescomplemented
56#1990C2212yescomplemented
57#1960C2212yescomplemented
58#1957C2212yescomplemented
59#1940C2214yescomplemented
60#1939C2214yescomplemented
61#1879C2218yes—
62#1877C2218yes—
63#1875C2214yes—
64#1873C2214yes—
65#1804C2214yescomplemented
66#1803C2214yescomplemented
67#1802C2218yes—
68#1798C2214yes—
69C22×#41—21no—
70C23×D4C2311yescharacteristic
71C24×C4—21no—
72C22×SD16C2314yes—
73C22×D8C2314yes—
74#533—24no—
75#533—24no—
76#532—24no—
77#532—24no—
78C2×C4⋊Q8C2311yescharacteristic
79C2×C4⋊Q8C2311yescharacteristic
80C2×C4⋊Q8C2311yescharacteristic
81C2×#127C2311yescharacteristic
82C2×#125—21no—
83C2×#124—21no—
84C2×#124C2314yes—
85C2×C22⋊Q8—21no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8C2312yes—
88C2×#121—21no—
89C2×#121C2312yes—
90#518—48no—
91C2×C4×Q8—21no—
92C2×C4×Q8C2312yes—
93C2×C4×D4—24no—
94C2×C4×D4—21no—
95C2×C4×D4—21no—
96C2×C4×D4—21no—
97C2×C4×D4—24no—
98C2×C4×D4D418yes—
99C2×C4×D4C2312yes—
100C2×C4×D4C2318yes—
101C2×C4×D4C2312yes—
102C2×C4×D4C2312yes—
103C2×#117—21no—
104C2×#117C2312yes—
105C2×#117C2312yes—
106C22×C4⋊C4—24no—
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4—21no—
109C22×C22⋊C4—24no—
110C22×C42C2311yescharacteristic
111#483—28no—
112#481—28no—
113#476—28no—
114#476—28no—
115#463—416no—
116#461—28no—
117#459—28no—
118#450—28no—
119#447—28no—
120#442—216no—
121C4×SD16—216no—
122#432—24no—
123#431—24no—
124C2×#106C2314yes—
125C2×#106C2314yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2318yes—
129C2×#103C2314yes—
130C2×#103C2318yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2318yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×#97C2312yes—
138C2×C4×C8C2312yes—
139#399—24no—
140#397—24no—
141#396—28no—
142#395—24no—
143#394—28no—
144#394—24no—
145#391C2314yes—
146#390—24no—
147#390C2314yes—
148#389—24no—
149#389—24no—
150#389C2318yes—
151#385—28no—
152#385—28no—
153#384D412yes—
154#384C2314yes—
155#384C2312yes—
156#382D412yes—
157#382C2312yes—
158#380—21no—
159#380—21no—
160#380—24no—
161#380—24no—
162#380—24no—
163#380—24no—
164#380D412yes—
165#380D412yes—
166C4×C4⋊C4—21no—
167C4×C4⋊C4—21no—
168C4×C4⋊C4C2312yes—
169C4×C4⋊C4C2312yes—
170C2×#95—24no—
171#340C2318yes—
172#340C2318yes—
173#336C2318yes—
174C25—21no—
175C2×#41—44no—
176C2×#41—84no—
177C22×Q8—21no—
178C22×Q8—21no—
179C22×Q8C2412yes—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—22no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—22no—
187C22×D4C2×D414yes—
188C22×D4C2412yes—
189C22×D4C2412yes—
190C23×C4—21no—
191C23×C4—22no—
192C23×C4—28no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—28no—
196C23×C4—84no—
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2411yescharacteristic
200C2×SD16—28no—
201C2×SD16—28no—
202C2×SD16—28no—
203C2×D8—28no—
204C2×D8—28no—
205C2×D8—28no—
206C22×C8C2414yes—
207C22×C8C2414yes—
208C4⋊Q8—22no—
209C4⋊Q8—22no—
210C4⋊Q8—22no—
211C4⋊Q8—22no—
212C4⋊Q8—22no—
213C4⋊Q8—22no—
214#127—22no—
215#127—22no—
216#125—44no—
217#124—48no—
218#124—44no—
219C22⋊Q8—24no—
220C22⋊Q8—24no—
221C22⋊Q8—44no—
222C22⋊Q8—88no—
223#121—24no—
224#121—24no—
225#121—44no—
226C4×Q8—24no—
227C4×Q8—24no—
228C4×Q8—44no—
229C4×D4—216no—
230C4×D4—28no—
231C4×D4—28no—
232C4×D4—24no—
233C4×D4—24no—
234C4×D4—24no—
235C4×D4—24no—
236C4×D4—48no—
237C4×D4—416no—
238C4×D4—44no—
239C4×D4C2×D418yes—
240C4×D4C2×D418yes—
241#117—44no—
242#117—44no—
243#117—44no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—88no—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4C2×D414yes—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2414yes—
262C2×C4⋊C4C2412yes—
263C2×C4⋊C4C2412yes—
264C2×C4⋊C4C2412yes—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—416no—
276C2×C22⋊C4C2×D414yes—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D414yes—
282C2×C42#4114yes—
283C2×C42C2414yes—
284C2×C42C2411yescharacteristic
285C2×C42C2411yescharacteristic
286#106—28no—
287#106—28no—
288C4⋊C8—24no—
289C4⋊C8—24no—
290C4⋊C8—48no—
291#103—28no—
292#103—48no—
293#102—28no—
294#102—28no—
295#102—28no—
296#102—48no—
297C22⋊C8—24no—
298C22⋊C8—24no—
299#97—24no—
300C4×C8—24no—
301#95—22no—
302#95—28no—
303#95—22no—
304#95—28no—
305#95C2×D414yes—
306#95C2×D414yes—
307C24—24no—
308C24—24no—
309C24—22no—
310C24—24no—
311C24—22no—
312C24—82no—
313C24—82no—
314C24C22×D411yescharacteristic
315#41—816no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—24no—
319C2×Q8—24no—
320C2×Q8—44no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—82no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—216no—
327C2×D4—216no—
328C2×D4—24no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—24no—
334C2×D4—22no—
335C2×D4—24no—
336C2×D4—44no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—44no—
340C2×D4—82no—
341C2×D4—84no—
342C2×D4C22×D414yes—
343C2×D4H(5,𝔽2)14yes—
344C2×D42−1+414yes—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—42no—
357C22×C4—48no—
358C22×C4—416no—
359C22×C4—48no—
360C22×C4—44no—
361C22×C4—816no—
362C22×C4—84no—
363C22×C4—88no—
364C22×C4—88no—
365C22×C4—82no—
366C22×C4#12114yes—
367C22×C4#12114yes—
368C22×C4C22×D418yes—
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D412yes—
374C22×C4C22×D411yescharacteristic
375C22×C4C2×#4112yes—
376C22×C4C2511yescharacteristic
377SD16—416no—
378D8—416no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C4⋊C4—24no—
386C4⋊C4—28no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—28no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—42no—
401C4⋊C4—44no—
402C4⋊C4—82no—
403C4⋊C4—88no—
404C4⋊C4—88no—
405C4⋊C4C22×D414yes—
406C22⋊C4—28no—
407C22⋊C4—28no—
408C22⋊C4—44no—
409C22⋊C4—416no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—88no—
415C22⋊C4—88no—
416C42—28no—
417C42—44no—
418C42—44no—
419C42#12114yes—
420C42#12114yes—
421C42C22×D412yes—
422C42C22×D412yes—
423C42C2×#4112yes—
424C42C2×#4112yes—
425C23—216no—
426C23—22no—
427C23—24no—
428C23—21no—
429C23—24no—
430C23—21no—
431C23—24no—
432C23—22no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—44no—
440C23—84no—
441C23—84no—
442C23—88no—
443C23—88no—
444C23C2×#12112yes—
445C23C23×D411yescharacteristic
446Q8—44no—
447Q8—44no—
448Q8—84no—
449Q8—84no—
450D4—216no—
451D4—24no—
452D4—24no—
453D4—24no—
454D4—44no—
455D4—44no—
456D4—84no—
457D4—88no—
458D4—84no—
459D4#53718yes—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—28no—
468C2×C4—28no—
469C2×C4—24no—
470C2×C4—28no—
471C2×C4—28no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—416no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—88no—
486C2×C4—88no—
487C2×C4—84no—
488C2×C4—1616no—
489C2×C4C2×#12112yes—
490C2×C4C2×#12112yes—
491C2×C4C2×#12112yes—
492C2×C4C2×#12112yes—
493C2×C4C2×#12112yes—
494C2×C4#57714yes—
495C2×C4C23×D411yescharacteristic
496C2×C4C22×#4111yescharacteristic
497C2×C4C2×H(5,𝔽2)12yes—
498C2×C4C2×2−1+412yes—
499C8—44no—
500C8—44no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—28no—
505C22—44no—
506C22—44no—
507C22—42no—
508C22—48no—
509C22—48no—
510C22—42no—
511C22—816no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22—84no—
516C22—84no—
517C22C22×#12111yescharacteristic
518C22#348411yescharacteristic
519C22#348611yescharacteristic
520C22C2×#57712yes—
521C22#358512yes—
522C4—22no—
523C4—22no—
524C4—22no—
525C4—22no—
526C4—44no—
527C4—48no—
528C4—82no—
529C4—82no—
530C4—82no—
531C4—84no—
532C4#304214yes—
533C4C2×#53712yes—
534C4#349212yes—
535C2—42no—
536C2—48no—
537C2—162no—
538C2#3522311yescharacteristic
539C2C2×#304211yescharacteristic
540C2#6042611yescharacteristic
541C2#6168512yes—
542C2#6185012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3109C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3101C2semidirect
3. #37640C2semidirect
4. #34181C2semidirect
5. #32917C2semidirect
6. #29768C2semidirect
7. #24829C2semidirect
8. #24815C2semidirect
9. #24745C2semidirect
10. #24731C2semidirect
11. #21954C2semidirect
12. #21870C2semidirect
13. #21863C2semidirect
37. #2567C22semidirect
38. #2548C22semidirect
39. #2547C22semidirect
40. #2542C22semidirect
51. #2084C22semidirect
52. #2066C22semidirect
53. #2065C22semidirect
54. #2042C22semidirect
55. #1993C22semidirect
56. #1990C22semidirect
57. #1960C22semidirect
58. #1957C22semidirect
59. #1940C22semidirect
60. #1939C22semidirect
65. #1804C22semidirect
66. #1803C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31092
C2×#31012
#376402
#341812
#329172
#297682
#248292
#248152
#247452
#247312
#219542
#218702
#218632