Finite Group Databasebeta

← #6473512

Subgroups — #6473512

Order 512.

1920 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35005C212yesmaximal, complemented
2#33042C212yesmaximal, complemented
3#32936C212yesmaximal, complemented
4#30293C211yesmaximal, characteristic, complemented
5#30239C211yesmaximal, characteristic, complemented
6#29732C212yesmaximal, complemented
7#28921C214yesmaximal, complemented
8#28869C214yesmaximal, complemented
9#28092C214yesmaximal, complemented
10#26984C214yesmaximal, complemented
11#22184C212yesmaximal, complemented
12#21526C212yesmaximal, complemented
13C2×#1753C211yesmaximal, characteristic, complemented
14C23×M16C2211yescharacteristic
15C2×#538—22no—
16C2×#486C2212yescomplemented
17C2×#469C2212yes—
18C2×#469C2212yes—
19C2×#466C2212yes—
20C2×#466C2212yes—
21C2×#442C2212yescomplemented
22C22×C22⋊C8C2211yescharacteristic
23C22×#97C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25#2857C2212yes—
26#2856C2212yes—
27#2847C2214yes—
28#2639—22no—
29#2617—22no—
30#2559—24no—
31#2557—24no—
32#2496C2214yes—
33#2491C2214yes—
34#2428C2214yes—
35#2422C2214yes—
36#2325—22no—
37C4×#121C2214yes—
38C2×C4×C22⋊C4C2211yescharacteristic
39#2093C2214yescomplemented
40#2084C2214yescomplemented
41#2076C2214yes—
42#2058C2214yescomplemented
43#2034C2214yes—
44#2034C2214yes—
45#2011C2214yescomplemented
46#1999C2214yes—
47#1990C2212yescomplemented
48#1972C2212yescomplemented
49#1957C2212yescomplemented
50#1943C2212yescomplemented
51#1936C2214yescomplemented
52#1934C2214yescomplemented
53#1934C2214yescomplemented
54#1926C2214yescomplemented
55#1921C2214yescomplemented
56#1895C2214yes—
57#1895C2214yes—
58#1893C2214yes—
59#1893C2214yes—
60#1879C2214yescomplemented
61#1864C2214yescomplemented
62#1834C2212yescomplemented
63#1809C2212yes—
64#1809C2212yes—
65#1807C2212yes—
66#1807C2212yes—
67#1776C2212yescomplemented
68#1762C2214yescomplemented
69#1753C2214yes—
70#1753C2214yes—
71C2×#336C2211yescharacteristic
72C2×#336C2211yescharacteristic
73C24×C4C2311yescharacteristic
74C22×D8C2312yes—
75C22×D8C2312yes—
76C22×M16—24no—
77C22×M16—42no—
78C22×M16—42no—
79C22×M16C2312yes—
80C22×M16C2312yes—
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85#538—416no—
86C2×C4⋊Q8C2312yes—
87C2×#127C2312yes—
88C2×#126—24no—
89C2×#124—24no—
90C2×#123—24no—
91C2×C22⋊Q8—22no—
92C2×#121—22no—
93C2×#121—22no—
94C2×#121—24no—
95C2×#121C2314yes—
96C2×#121C2314yes—
97C2×#121C2314yes—
98C2×#120—24no—
99C2×C4×D4—24no—
100C2×C4×D4C2314yes—
101C22×C22⋊C4D412yes—
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104#486—44no—
105#469—44no—
106#469—44no—
107#466—28no—
108#466—28no—
109#442—44no—
110C2×#107C2312yes—
111C2×#107C2312yes—
112C2×#106C2312yes—
113C2×#106C2312yes—
114C2×#106C2312yes—
115C2×#106C2312yes—
116C2×#102C2314yes—
117C2×#102C2314yes—
118C2×#102C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×C22⋊C8—24no—
123C2×C22⋊C8C2314yes—
124C2×#97—42no—
125C2×#97C2312yes—
126C2×#97C2312yes—
127C2×C4×C8—42no—
128C2×C4×C8C2312yes—
129C2×C4×C8C2312yes—
130#400—24no—
131#397—24no—
132#397—24no—
133#394—24no—
134#394—24no—
135#394—24no—
136#393—24no—
137#392—24no—
138#391—22no—
139#390—22no—
140#390—22no—
141#390C2314yes—
142#390C2314yes—
143#389—22no—
144#389C2314yes—
145#389C2314yes—
146#388—24no—
147#388—24no—
148#387—24no—
149#386—24no—
150#386—24no—
151#386—24no—
152#386—24no—
153#386—24no—
154#385—24no—
155#384C2312yes—
156#384C2312yes—
157#380—22no—
158#380C2314yes—
159#380C2314yes—
160#380C2314yes—
161C4×C4⋊C4C2314yes—
162C4×C22⋊C4—24no—
163C4×C22⋊C4D412yes—
164C4×C22⋊C4D412yes—
165C4×C22⋊C4C2314yes—
166C2×#95D412yes—
167#340C2314yes—
168#340C2314yes—
169#340C2314yes—
170#340C2314yes—
171#340C2314yes—
172#340C2314yes—
173#336C2314yes—
174#336C2314yes—
175C25C2×D411yescharacteristic
176C22×Q8—22no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4C2414yes—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—41no—
187C23×C4—41no—
188C23×C4—41no—
189C23×C4—41no—
190C23×C4C2×D412yes—
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D412yes—
193C23×C4C2×D412yes—
194C23×C4C2412yes—
195C23×C4C2411yescharacteristic
196C2×D8—24no—
197C2×D8—24no—
198C2×D8—42no—
199C2×D8—42no—
200C2×D8—42no—
201C2×D8—42no—
202C2×M16—24no—
203C2×M16—24no—
204C2×M16—24no—
205C2×M16—24no—
206C2×M16—42no—
207C2×M16—42no—
208C2×M16—416no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—41no—
214C22×C8—41no—
215C22×C8—41no—
216C22×C8—41no—
217C22×C8—41no—
218C22×C8—41no—
219C22×C8#4114yes—
220C22×C8#4114yes—
221C22×C8C2412yes—
222C22×C8C2412yes—
223C22×C8C2412yes—
224C22×C8C2411yescharacteristic
225C22×C8C2411yescharacteristic
226C4⋊Q8—42no—
227C4⋊Q8—42no—
228#127—42no—
229#127—42no—
230#126—416no—
231#124—416no—
232#123—416no—
233C22⋊Q8—48no—
234#121—28no—
235#121—28no—
236#121—416no—
237#121—48no—
238#121—48no—
239#121—44no—
240#121—44no—
241#121—44no—
242#121—44no—
243#120—416no—
244C4×D4—44no—
245C4×D4—44no—
246C4×D4—416no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4C2414yes—
255C2×C4⋊C4C2414yes—
256C2×C4⋊C4C2414yes—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4C2×D414yes—
270C2×C22⋊C4C2×D414yes—
271C2×C42—24no—
272C2×C42—41no—
273C2×C42—41no—
274C2×C42—41no—
275C2×C42—41no—
276C2×C42C2×D412yes—
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2412yes—
280C2×C42C2412yes—
281#107—24no—
282#107—24no—
283#106—42no—
284#106—42no—
285#106—42no—
286#106—42no—
287#102—28no—
288#102—28no—
289#102—28no—
290#102—28no—
291#102—44no—
292#102—44no—
293C22⋊C8—28no—
294#97—24no—
295#97—42no—
296C4×C8—24no—
297C4×C8—42no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95C2×D414yes—
305#95C2×D414yes—
306C24—22no—
307C24—24no—
308C24—24no—
309C24—41no—
310C24—41no—
311C24—42no—
312C24C22×D412yes—
313C24C22×D411yescharacteristic
314C2×Q8—44no—
315C2×Q8—82no—
316C2×Q8—82no—
317C2×D4—24no—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—24no—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—48no—
324C2×D4—48no—
325C2×D4—48no—
326C2×D4—48no—
327C2×D4—48no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4—82no—
331C2×D4—82no—
332C2×D4—82no—
333C2×D4—82no—
334C2×D4—84no—
335C2×D4—84no—
336C2×D4—84no—
337C2×D4—84no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—48no—
367C22×C4—84no—
368C22×C4—84no—
369C22×C4AGL(1,ℤ/8)18yes—
370C22×C4C22×D412yes—
371C22×C4C22×D414yes—
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D412yes—
374C22×C4C22×D412yes—
375C22×C4C2×#4112yes—
376C22×C4C2×#4112yes—
377C22×C4C2511yescharacteristic
378D8—48no—
379D8—84no—
380M16—48no—
381M16—48no—
382C2×C8—28no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—22no—
387C2×C8—21no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—44no—
395C2×C8—44no—
396C2×C8—44no—
397C2×C8C2×#4114yes—
398C2×C8C2×#4114yes—
399C2×C8H(5,𝔽2)12yes—
400C2×C8H(5,𝔽2)12yes—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—48no—
408C4⋊C4—44no—
409C4⋊C4—48no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—82no—
413C4⋊C4—82no—
414C22⋊C4—28no—
415C22⋊C4—28no—
416C22⋊C4—48no—
417C22⋊C4—48no—
418C22⋊C4—48no—
419C22⋊C4—48no—
420C22⋊C4—48no—
421C22⋊C4—48no—
422C22⋊C4—48no—
423C22⋊C4—84no—
424C22⋊C4—84no—
425C22⋊C4—84no—
426C22⋊C4—84no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—41no—
432C42—41no—
433C42—42no—
434C42—48no—
435C23—22no—
436C23—21no—
437C23—21no—
438C23—24no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—24no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—84no—
454C23—84no—
455C23#53212yes—
456C23C2×AGL(1,ℤ/8)14yes—
457C23C23×D411yescharacteristic
458Q8—164no—
459D4—44no—
460D4—44no—
461D4—44no—
462D4—44no—
463D4—816no—
464D4—164no—
465D4—164no—
466D4—168no—
467D4—168no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—48no—
485C2×C4—48no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—84no—
501C2×C4—168no—
502C2×C4#53212yes—
503C2×C4C2×AGL(1,ℤ/8)14yes—
504C2×C4C2×AGL(1,ℤ/8)14yes—
505C2×C4C22×#4111yescharacteristic
506C2×C4C22×#4111yescharacteristic
507C2×C4C2×H(5,𝔽2)11yescharacteristic
508C2×C4C2×H(5,𝔽2)11yescharacteristic
509C2×C4#58511yescharacteristic
510C2×C4#58511yescharacteristic
511C8—24no—
512C8—22no—
513C8—22no—
514C8—24no—
515C8—42no—
516C22—22no—
517C22—24no—
518C22—24no—
519C22—22no—
520C22—42no—
521C22—42no—
522C22—44no—
523C22—44no—
524C22—42no—
525C22—88no—
526C22—84no—
527C22—84no—
528C22—84no—
529C22—84no—
530C22—84no—
531C22—84no—
532C22—84no—
533C22#318414yes—
534C22C2×#53211yescharacteristic
535C22#349611yescharacteristic
536C22#349611yescharacteristic
537C22C22×AGL(1,ℤ/8)12yes—
538C22#358412yes—
539C4—24no—
540C4—24no—
541C4—22no—
542C4—44no—
543C4—42no—
544C4—84no—
545C4—84no—
546C4—84no—
547C4#345911yescharacteristic
548C4#346011yescharacteristic
549C4#349011yescharacteristic
550C4#349311yescharacteristic
551C2—24no—
552C2—82no—
553C2—164no—
554C2#3500311yescharacteristic
555C2#6068511yescharacteristic
556C2C2×#318411yescharacteristic
557C2#6225312yes—
558C2#6225912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
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
1. #35005C2semidirect
2. #33042C2semidirect
3. #32936C2semidirect
4. #30293C2semidirect
5. #30239C2semidirect
6. #29732C2semidirect
7. #28921C2semidirect
8. #28869C2semidirect
9. #28092C2semidirect
10. #26984C2semidirect
11. #22184C2semidirect
12. #21526C2semidirect
13. C2×#1753C2semidirect
16. C2×#486C22semidirect
21. C2×#442C22semidirect
39. #2093C22semidirect
40. #2084C22semidirect
42. #2058C22semidirect
45. #2011C22semidirect
47. #1990C22semidirect
48. #1972C22semidirect
49. #1957C22semidirect
50. #1943C22semidirect
51. #1936C22semidirect
52. #1934C22semidirect
53. #1934C22semidirect
54. #1926C22semidirect
55. #1921C22semidirect
60. #1879C22semidirect
61. #1864C22semidirect
62. #1834C22semidirect
67. #1776C22semidirect
68. #1762C22semidirect

Maximal subgroups

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

subgroupindex
#350052
#330422
#329362
#302932
#302392
#297322
#289212
#288692
#280922
#269842
#221842
#215262
C2×#17532