Finite Group Databasebeta

← #6481764

Subgroups — #6481764

Order 512.

2054 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3110C211yesmaximal, characteristic, complemented
2C2×#3109C211yesmaximal, characteristic
3#35180C211yesmaximal, characteristic, complemented
4C2×#2357C211yesmaximal, characteristic, complemented
5#33110C214yesmaximal, complemented
6#29718C214yesmaximal, complemented
7#24697C214yesmaximal, complemented
8#24693C214yesmaximal, complemented
9#24647C214yesmaximal, complemented
10#24643C214yesmaximal, complemented
11#21955C211yesmaximal, characteristic, complemented
12#21866C211yesmaximal, characteristic, complemented
13#21865C211yesmaximal, characteristic, complemented
14C2×#533C2211yescharacteristic
15C2×#532C2211yescharacteristic
16C22×C4×D4C2211yescharacteristic
17#3110—28no—
18#3109—28no—
19C2×#484C2212yes—
20C2×#481C2212yes—
21C2×#476C2212yes—
22C2×#474C2212yes—
23C2×#461C2212yes—
24C2×#459C2212yes—
25C2×#450C2212yes—
26C2×#447C2212yes—
27C2×#442C2214yes—
28C2×#440C2214yes—
29C2×#432C2211yescharacteristic
30C2×#432C2211yescharacteristic
31#2850C2214yes—
32#2639—21no—
33#2639—21no—
34#2592—22no—
35#2568C2212yescomplemented
36#2563C2212yescomplemented
37#2562—22no—
38#2548C2212yescomplemented
39#2547C2212yescomplemented
40#2357—22no—
41#2357—22no—
42#2357C2214yes—
43#2357C2214yes—
44#2311—22no—
45C2×#390C2212yes—
46C2×#388—22no—
47C2×#382C2211yescharacteristic
48C2×#381—21no—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4—21no—
52#2073C2218yes—
53#2055C2218yes—
54#2054C2218yes—
55#2012C2218yes—
56#1993C2212yescomplemented
57#1993C2212yescomplemented
58#1983C2218yescomplemented
59#1961C2212yescomplemented
60#1960C2212yescomplemented
61#1949C2218yescomplemented
62#1940C2214yescomplemented
63#1940C2214yescomplemented
64#1927C2218yescomplemented
65#1925C2218yescomplemented
66#1874C2214yes—
67#1872C2214yes—
68#1805C2214yescomplemented
69#1804C2214yescomplemented
70#1797C2214yes—
71C22×#41—21no—
72C23×D4C2311yescharacteristic
73C24×C4—21no—
74C22×SD16C2314yes—
75C22×D8C2314yes—
76#533—24no—
77#533—24no—
78#532—24no—
79#532—24no—
80C2×C4⋊Q8C2311yescharacteristic
81C2×C4⋊Q8C2311yescharacteristic
82C2×C4⋊Q8C2311yescharacteristic
83C2×#127C2311yescharacteristic
84C2×#125—21no—
85C2×#124—21no—
86C2×#124C2314yes—
87C2×C22⋊Q8—21no—
88C2×C22⋊Q8C2312yes—
89C2×#121—21no—
90C2×#121C2312yes—
91C2×C4×Q8—21no—
92C2×C4×Q8C2312yes—
93C2×C4×D4—21no—
94C2×C4×D4—21no—
95C2×C4×D4—21no—
96C2×C4×D4—24no—
97C2×C4×D4D418yes—
98C2×C4×D4C2312yes—
99C2×C4×D4C2312yes—
100C2×C4×D4C2312yes—
101C2×#117—21no—
102C22×C4⋊C4D412yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C22⋊C4—21no—
106C22×C22⋊C4—22no—
107C22×C22⋊C4—22no—
108C22×C42—22no—
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111#484—28no—
112#481—28no—
113#476—28no—
114#474—28no—
115#461—28no—
116#459—28no—
117#450—28no—
118#447—28no—
119#442—216no—
120#440—216no—
121#432—24no—
122#432—24no—
123C2×#107C2314yes—
124C2×#106C2314yes—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2318yes—
128C2×#103C2314yes—
129C2×#103C2318yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2318yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2312yes—
136C2×#97C2312yes—
137C2×#97C2312yes—
138#400—22no—
139#400—22no—
140#397—22no—
141#397—22no—
142#394—22no—
143#394—22no—
144#393—22no—
145#393—22no—
146#391—24no—
147#391C2314yes—
148#390—24no—
149#390—24no—
150#390C2314yes—
151#390C2318yes—
152#388—28no—
153#388—28no—
154#388—24no—
155#388—24no—
156#386—24no—
157#386—24no—
158#382—21no—
159#382—21no—
160#382C2314yes—
161#382C2312yes—
162#382C2312yes—
163#381—24no—
164#381—24no—
165#380—24no—
166#380—24no—
167C4×C4⋊C4—24no—
168C4×C4⋊C4D412yes—
169C4×C4⋊C4D412yes—
170C4×C4⋊C4C2318yes—
171C4×C4⋊C4C2312yes—
172C4×C4⋊C4C2312yes—
173C4×C22⋊C4—24no—
174C4×C22⋊C4—24no—
175C2×#95—21no—
176C2×#95—22no—
177C2×#95—21no—
178C2×#95D411yescharacteristic
179C2×#95D411yescharacteristic
180#340C2318yes—
181#340C2318yes—
182#336C2318yes—
183C25—21no—
184C2×#41—48no—
185C2×#41—44no—
186C22×Q8—21no—
187C22×Q8—21no—
188C22×Q8C2412yes—
189C22×D4—22no—
190C22×D4—21no—
191C22×D4—21no—
192C22×D4—21no—
193C22×D4—22no—
194C22×D4—42no—
195C22×D4—42no—
196C22×D4C2×D414yes—
197C22×D4C2412yes—
198C22×D4C2412yes—
199C23×C4—21no—
200C23×C4—28no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—84no—
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D412yes—
208C23×C4C2×D412yes—
209C23×C4C2×D411yescharacteristic
210C23×C4C2411yescharacteristic
211C2×SD16—28no—
212C2×SD16—44no—
213C2×SD16—44no—
214C2×D8—28no—
215C2×D8—44no—
216C2×D8—44no—
217C22×C8C2414yes—
218C22×C8C2414yes—
219C4⋊Q8—22no—
220C4⋊Q8—22no—
221C4⋊Q8—22no—
222C4⋊Q8—22no—
223C4⋊Q8—22no—
224C4⋊Q8—22no—
225#127—22no—
226#127—22no—
227#125—44no—
228#124—48no—
229#124—44no—
230C22⋊Q8—24no—
231C22⋊Q8—24no—
232C22⋊Q8—44no—
233#121—24no—
234#121—24no—
235#121—44no—
236C4×Q8—24no—
237C4×Q8—24no—
238C4×Q8—44no—
239C4×D4—28no—
240C4×D4—28no—
241C4×D4—24no—
242C4×D4—24no—
243C4×D4—24no—
244C4×D4—24no—
245C4×D4—44no—
246C4×D4C2×D418yes—
247C4×D4C2×D418yes—
248#117—44no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×D414yes—
262C2×C4⋊C4C2×D414yes—
263C2×C4⋊C4C2414yes—
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2412yes—
266C2×C4⋊C4C2412yes—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—48no—
280C2×C22⋊C4—84no—
281C2×C42—22no—
282C2×C42—28no—
283C2×C42—44no—
284C2×C42—84no—
285C2×C42C2×D414yes—
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D414yes—
289C2×C42#4114yes—
290C2×C42C2414yes—
291C2×C42C2411yescharacteristic
292C2×C42C2411yescharacteristic
293#107—28no—
294#106—28no—
295C4⋊C8—24no—
296C4⋊C8—24no—
297C4⋊C8—48no—
298#103—28no—
299#103—48no—
300#102—28no—
301#102—28no—
302#102—28no—
303#102—48no—
304C22⋊C8—24no—
305C22⋊C8—24no—
306#97—24no—
307#97—24no—
308#95—22no—
309#95—22no—
310#95—28no—
311#95—22no—
312#95—22no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95C2×D412yes—
318#95C2×D412yes—
319#95C2×D412yes—
320#95C2×D412yes—
321C24—24no—
322C24—22no—
323C24—22no—
324C24—44no—
325C24—42no—
326C24—42no—
327C24—44no—
328C24C22×D411yescharacteristic
329#41—816no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—44no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—82no—
338C2×D4—24no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—48no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—48no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—48no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—82no—
358C2×D4C22×D414yes—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—28no—
367C22×C4—28no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—48no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—84no—
384C22×C4—84no—
385C22×C4—816no—
386C22×C4#12114yes—
387C22×C4#12114yes—
388C22×C4C22×D414yes—
389C22×C4C22×D414yes—
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D412yes—
396C22×C4C2×#4112yes—
397C22×C4C2511yescharacteristic
398SD16—416no—
399D8—416no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—28no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—42no—
422C4⋊C4C22×D414yes—
423C22⋊C4—28no—
424C22⋊C4—28no—
425C22⋊C4—416no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—84no—
430C22⋊C4—84no—
431C22⋊C4—88no—
432C22⋊C4—84no—
433C22⋊C4—84no—
434C42—28no—
435C42—44no—
436C42—44no—
437C42—88no—
438C42#12114yes—
439C42#12114yes—
440C42C22×D412yes—
441C42C22×D412yes—
442C42C2×#4112yes—
443C42C2×#4112yes—
444C23—22no—
445C23—21no—
446C23—21no—
447C23—24no—
448C23—48no—
449C23—48no—
450C23—44no—
451C23—42no—
452C23—42no—
453C23—48no—
454C23—44no—
455C23—44no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—44no—
460C23—88no—
461C23—88no—
462C23C2×#12112yes—
463C23#57514yes—
464C23C23×D411yescharacteristic
465Q8—44no—
466Q8—44no—
467Q8—84no—
468Q8—84no—
469D4—24no—
470D4—24no—
471D4—48no—
472D4—44no—
473D4—42no—
474D4—42no—
475D4—44no—
476D4—84no—
477D4—84no—
478D4—88no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—28no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—44no—
496C2×C4—48no—
497C2×C4—48no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—88no—
510C2×C4—88no—
511C2×C4—88no—
512C2×C4—84no—
513C2×C4—88no—
514C2×C4C2×#12112yes—
515C2×C4C2×#12112yes—
516C2×C4C2×#12112yes—
517C2×C4C2×#12112yes—
518C2×C4C2×#12112yes—
519C2×C4C23×D411yescharacteristic
520C2×C4C22×#4111yescharacteristic
521C2×C4#58512yes—
522C2×C4#58512yes—
523C8—44no—
524C8—44no—
525C22—22no—
526C22—24no—
527C22—22no—
528C22—24no—
529C22—44no—
530C22—44no—
531C22—44no—
532C22—816no—
533C22—84no—
534C22—84no—
535C22—82no—
536C22—84no—
537C22—82no—
538C22—84no—
539C22—82no—
540C22#304014yes—
541C22C22×#12111yescharacteristic
542C22#348511yescharacteristic
543C22#348711yescharacteristic
544C22C2×#57512yes—
545C22#358512yes—
546C4—22no—
547C4—22no—
548C4—22no—
549C4—22no—
550C4—22no—
551C4—44no—
552C4—44no—
553C4—44no—
554C4—82no—
555C4—82no—
556C4—84no—
557C4—82no—
558C4#344912yes—
559C4#349312yes—
560C2—24no—
561C2—84no—
562C2—162no—
563C2#3515011yescharacteristic
564C2C2×#304011yescharacteristic
565C2#6042311yescharacteristic
566C2#6171312yes—
567C2#6189512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3109C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3110C2semidirect
3. #35180C2semidirect
4. C2×#2357C2semidirect
5. #33110C2semidirect
6. #29718C2semidirect
7. #24697C2semidirect
8. #24693C2semidirect
9. #24647C2semidirect
10. #24643C2semidirect
11. #21955C2semidirect
12. #21866C2semidirect
13. #21865C2semidirect
35. #2568C22semidirect
36. #2563C22semidirect
38. #2548C22semidirect
39. #2547C22semidirect
56. #1993C22semidirect
57. #1993C22semidirect
58. #1983C22semidirect
59. #1961C22semidirect
60. #1960C22semidirect
61. #1949C22semidirect
62. #1940C22semidirect
63. #1940C22semidirect
64. #1927C22semidirect
65. #1925C22semidirect
68. #1805C22semidirect
69. #1804C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31102
C2×#31092
#351802
C2×#23572
#331102
#297182
#246972
#246932
#246472
#246432
#219552
#218662
#218652