Finite Group Databasebeta

← #6479527

Subgroups — #6479527

Order 512.

1992 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2854C211yesmaximal, characteristic
2C2×#2283C211yesmaximal, characteristic, complemented
3#33096C214yesmaximal, complemented
4#29912C214yesmaximal, complemented
5#27859C218yesmaximal, complemented
6#27616C214yesmaximal, complemented
7#27588C214yesmaximal, complemented
8#21940C211yesmaximal, characteristic, complemented
9#21938C211yesmaximal, characteristic, complemented
10#21758C211yesmaximal, characteristic
11#21758C211yesmaximal, characteristic
12C2×#1817C211yesmaximal, characteristic, complemented
13C22×#123—21no—
14C22×#121C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18#2854—21no—
19#2854—21no—
20#2854—22no—
21#2854C2212yes—
22#2854C2214yes—
23#2854C2212yes—
24C2×#390C2212yes—
25C2×#389C2212yes—
26C2×#388—22no—
27C2×#386—21no—
28#2283—22no—
29#2283—22no—
30#2283C2214yes—
31#2283C2214yescomplemented
32C2×#383—21no—
33C2×#382C2212yescomplemented
34C2×#381—21no—
35C2×#380C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×#380C2211yescharacteristic
38C2×C4×C4⋊C4C2212yescomplemented
39C2×C4×C22⋊C4—21no—
40C2×#376—21no—
41#2101C2218yescomplemented
42#2092C2218yescomplemented
43#2085C2218yescomplemented
44#2073C2218yescomplemented
45#1926C2218yescomplemented
46#1925C2218yescomplemented
47#1921C2218yescomplemented
48#1920C2218yescomplemented
49#1893C2214yes—
50#1893C2214yes—
51#1874C2214yes—
52#1874C2214yes—
53#1853C2214yes—
54#1853C2214yescomplemented
55#1852C2214yes—
56#1852C2214yescomplemented
57#1823C2212yes—
58#1823C2212yes—
59#1817C2214yes—
60#1817C2212yescomplemented
61#1817C2212yescomplemented
62#1808C2214yescomplemented
63#1807C2214yescomplemented
64#1786C2212yes—
65#1786C2212yes—
66C2×#336C2214yescomplemented
67C23×D4—21no—
68C23×D4C2311yescharacteristic
69C24×C4D411yescharacteristic
70C24×C4D411yescharacteristic
71C24×C4C2311yescharacteristic
72C23×C8C2312yes—
73C23×C8C2312yes—
74C2×#123—22no—
75C2×#123—22no—
76C2×#123—24no—
77C2×#123—48no—
78C2×#121—21no—
79C2×#121—21no—
80C2×#121—24no—
81C2×#121—24no—
82C2×#121C2312yes—
83C2×#121C2312yes—
84C22×C4⋊C4—22no—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C22⋊C4—21no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C22⋊C4—22no—
94C22×C22⋊C4—22no—
95C22×C42—24no—
96C22×C42—24no—
97C22×C42—22no—
98C22×C42D411yescharacteristic
99C22×C42D411yescharacteristic
100C22×C42D412yes—
101C22×C42D412yes—
102C22×C42C2312yes—
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105C22×C42C2312yes—
106C2×#107C2318yes—
107C2×#106C2318yes—
108C2×#102C2318yes—
109C2×#102C2318yes—
110C2×C22⋊C8—24no—
111C2×C22⋊C8—24no—
112C2×C22⋊C8C2312yes—
113C2×C22⋊C8C2312yes—
114C2×C22⋊C8C2312yes—
115C2×C22⋊C8C2312yes—
116#390—22no—
117#390—22no—
118#390C2314yes—
119#390C2314yes—
120#389—22no—
121#389—22no—
122#389C2314yes—
123#389C2314yes—
124#388—24no—
125#388—24no—
126#388—24no—
127#388—24no—
128#386—22no—
129#386—24no—
130#386—22no—
131#383—24no—
132#383—24no—
133#382—22no—
134#382—22no—
135#382C2314yes—
136#382C2314yes—
137#381—24no—
138#381—24no—
139#380—21no—
140#380—21no—
141#380—21no—
142#380—21no—
143#380—21no—
144#380—21no—
145#380C2312yes—
146#380C2312yes—
147#380C2312yes—
148#380C2312yes—
149#380C2312yes—
150#380C2312yes—
151C4×C4⋊C4—22no—
152C4×C4⋊C4—22no—
153C4×C4⋊C4C2314yes—
154C4×C4⋊C4C2314yes—
155C4×C22⋊C4—24no—
156C4×C22⋊C4—24no—
157#376—24no—
158#376—24no—
159C2×#95—21no—
160C2×#95—21no—
161C2×#95—22no—
162C2×#95—22no—
163C2×#95—22no—
164C2×#95—21no—
165C2×#95—22no—
166#340C2318yes—
167#340C2318yes—
168#340C2318yes—
169#340C2318yes—
170#336C2318yes—
171#336C2318yes—
172C25—21no—
173C25C2×D411yescharacteristic
174C22×D4—22no—
175C22×D4—22no—
176C22×D4—21no—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—44no—
180C22×D4—44no—
181C22×D4—44no—
182C22×D4—44no—
183C22×D4C2412yes—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—21no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4C2×D412yes—
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D412yes—
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D412yes—
200C23×C4C2×D412yes—
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4#4114yes—
206C23×C4#4114yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C22×C8—24no—
211C22×C8—24no—
212C22×C8—44no—
213C22×C8—44no—
214C22×C8C2414yes—
215C22×C8C2414yes—
216#123—416no—
217#123—48no—
218#123—48no—
219#121—24no—
220#121—24no—
221#121—24no—
222#121—24no—
223#121—44no—
224#121—44no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—48no—
237C2×C4⋊C4C2412yes—
238C2×C4⋊C4C2414yes—
239C2×C4⋊C4C2414yes—
240C2×C4⋊C4C2414yes—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—48no—
254C2×C22⋊C4—48no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C42—24no—
260C2×C42—28no—
261C2×C42—24no—
262C2×C42—28no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—48no—
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42C2412yes—
281C2×C42C2412yes—
282#107—48no—
283#106—48no—
284#102—48no—
285#102—48no—
286C22⋊C8—24no—
287C22⋊C8—24no—
288C22⋊C8—24no—
289C22⋊C8—24no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—22no—
301#95—22no—
302#95—24no—
303#95—24no—
304C24—22no—
305C24—42no—
306C24—42no—
307C24—44no—
308C24—44no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24#12314yes—
313C24#13514yes—
314C24AGL(1,ℤ/8)14yes—
315C24C22×D411yescharacteristic
316C24C22×D411yescharacteristic
317C24C22×D411yescharacteristic
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—28no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—42no—
329C2×D4—84no—
330C2×D4—84no—
331C2×D4—816no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—416no—
367C22×C4—416no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—42no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—42no—
380C22×C4#12314yes—
381C22×C4#12314yes—
382C22×C4#12314yes—
383C22×C4C22×D412yes—
384C22×C4C22×D412yes—
385C22×C4C22×D412yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D412yes—
390C22×C4C2×#4112yes—
391C22×C4C2×#4112yes—
392C22×C4C2×#4112yes—
393C22×C4C2×#4112yes—
394C22×C4C2×#4112yes—
395C22×C4C2×#4112yes—
396C22×C4C2511yescharacteristic
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—28no—
400C2×C8—28no—
401C2×C8—84no—
402C2×C8—84no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—28no—
406C4⋊C4—28no—
407C4⋊C4—28no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—48no—
412C4⋊C4—48no—
413C4⋊C4—42no—
414C22⋊C4—48no—
415C22⋊C4—48no—
416C22⋊C4—48no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—44no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—48no—
426C42—24no—
427C42—24no—
428C42—24no—
429C42—24no—
430C42—24no—
431C42—24no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—48no—
437C42—48no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—44no—
445C23—44no—
446C23—48no—
447C23—48no—
448C23—42no—
449C23—44no—
450C23—42no—
451C23—42no—
452C23—44no—
453C23—84no—
454C23—88no—
455C23—88no—
456C23—84no—
457C23—84no—
458C23#482116yes—
459C23C2×#12312yes—
460C23C2×#12312yes—
461C23C2×#12312yes—
462C23C2×#13512yes—
463C23C2×#13512yes—
464C23C2×#13512yes—
465C23C2×AGL(1,ℤ/8)12yes—
466C23C2×AGL(1,ℤ/8)12yes—
467C23C2×AGL(1,ℤ/8)12yes—
468C23C23×D411yescharacteristic
469D4—44no—
470D4—44no—
471D4—84no—
472D4—88no—
473D4—88no—
474C2×C4—28no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—48no—
499C2×C4—48no—
500C2×C4—44no—
501C2×C4—48no—
502C2×C4—48no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—88no—
518C2×C4—88no—
519C2×C4—88no—
520C2×C4—84no—
521C2×C4C2×#12312yes—
522C2×C4C2×#12312yes—
523C2×C4C2×#12312yes—
524C2×C4#57514yes—
525C2×C4C22×#4111yescharacteristic
526C2×C4C22×#4111yescharacteristic
527C2×C4#58514yes—
528C8—44no—
529C8—44no—
530C22—24no—
531C22—24no—
532C22—24no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—24no—
537C22—24no—
538C22—24no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—416no—
543C22—84no—
544C22—82no—
545C22—84no—
546C22—82no—
547C22—82no—
548C22—168no—
549C22C2×#48214yes—
550C22C2×#48214yes—
551C22C2×#48214yes—
552C22C22×#12311yescharacteristic
553C22#344912yes—
554C22C22×#13511yescharacteristic
555C22C22×AGL(1,ℤ/8)11yescharacteristic
556C22C2×#57512yes—
557C4—22no—
558C4—22no—
559C4—22no—
560C4—24no—
561C4—42no—
562C4—42no—
563C4—42no—
564C4—84no—
565C4—84no—
566C4—84no—
567C4—82no—
568C4#349112yes—
569C4#349112yes—
570C2—24no—
571C2—24no—
572C2—24no—
573C2—162no—
574C2#3509711yescharacteristic
575C2C22×#48211yescharacteristic
576C2#6051111yescharacteristic
577C2#6094512yes—
578C2#6095812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2283C2semidirect
3. #33096C2semidirect
4. #29912C2semidirect
5. #27859C2semidirect
6. #27616C2semidirect
7. #27588C2semidirect
8. #21940C2semidirect
9. #21938C2semidirect
12. C2×#1817C2semidirect
31. #2283C22semidirect
33. C2×#382C22semidirect
38. C2×C4×C4⋊C4C22semidirect
41. #2101C22semidirect
42. #2092C22semidirect
43. #2085C22semidirect
44. #2073C22semidirect
45. #1926C22semidirect
46. #1925C22semidirect
47. #1921C22semidirect
48. #1920C22semidirect
54. #1853C22semidirect
56. #1852C22semidirect
60. #1817C22semidirect
61. #1817C22semidirect
62. #1808C22semidirect
63. #1807C22semidirect
66. C2×#336C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28542
C2×#22832
#330962
#299122
#278592
#276162
#275882
#219402
#219382
#217582
C2×#18172