Finite Group Databasebeta

← #6647987

Subgroups — #6647987

Order 512.

2062 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35178C211yesmaximal, characteristic, complemented
2C2×#2548C211yesmaximal, characteristic, complemented
3#33095C212yesmaximal, complemented
4#32681C214yesmaximal, complemented
5#32646C214yesmaximal, complemented
6#32623C218yesmaximal, complemented
7#29622C212yesmaximal, complemented
8#27585C212yesmaximal, complemented
9#24506C214yesmaximal, complemented
10#23930C212yesmaximal, complemented
11#21822C211yesmaximal, characteristic, complemented
12C2×#536—21no—
13C2×#532C2211yescharacteristic
14C22×#127C2211yescharacteristic
15C22×#124—21no—
16C22×C4×Q8C2211yescharacteristic
17C2×#447C2214yes—
18C2×#432C2212yes—
19#2854C2212yes—
20#2854C2212yes—
21#2618—21no—
22#2618—21no—
23#2548—22no—
24#2548—22no—
25#2548C2214yes—
26#2548C2214yes—
27#2548C2212yescomplemented
28#2546C2212yescomplemented
29#2545—22no—
30#2538C2214yescomplemented
31#2536—22no—
32C2×#394—21no—
33C2×#394—21no—
34#2311—21no—
35C2×#390—21no—
36C2×#390C2212yes—
37C2×#390C2212yes—
38#2285C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40#2094C2214yescomplemented
41#2071C2214yescomplemented
42#2065C2218yescomplemented
43#2054C2218yescomplemented
44#2041C2218yes—
45#2023C2214yescomplemented
46#2011C2218yes—
47#2011C2214yescomplemented
48#1985C2214yescomplemented
49#1985C2214yescomplemented
50#1972C2218yescomplemented
51#1936C2218yescomplemented
52#1926C2218yescomplemented
53#1926C2218yescomplemented
54#1919C2218yescomplemented
55#1879C2218yescomplemented
56#1874C2214yes—
57#1853C2214yes—
58#1817C2212yes—
59#1817C2212yes—
60#1786C2214yes—
61C22×#41—21no—
62C23×Q8D411yescharacteristic
63C23×D4—21no—
64C23×D4—21no—
65C23×D4C2312yes—
66C22×D8C2318yes—
67#536—48no—
68#532—44no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8C2311yescharacteristic
71C2×#127—22no—
72C2×#127—44no—
73C2×#127C2314yes—
74C2×#127C2311yescharacteristic
75C2×#124—24no—
76C2×#124—24no—
77C2×#124—22no—
78C2×#124—21no—
79C2×#124—48no—
80C2×#123—22no—
81C2×#123—22no—
82C2×C22⋊Q8—24no—
83C2×#121—21no—
84C2×#121C2312yes—
85C2×C4×Q8—44no—
86C2×C4×Q8D412yes—
87C2×C4×Q8D412yes—
88C2×C4×Q8C2312yes—
89C2×C4×Q8C2312yes—
90C2×C4×D4—21no—
91C2×C4×D4C2312yes—
92C2×#117—21no—
93C22×C4⋊C4D411yescharacteristic
94C22×C4⋊C4D411yescharacteristic
95C22×C4⋊C4D411yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C22⋊C4—22no—
98C22×C22⋊C4—22no—
99C22×C22⋊C4—22no—
100C22×C22⋊C4—22no—
101C22×C42—21no—
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105#447—48no—
106#432—44no—
107C2×#106C2318yes—
108C2×C4⋊C8C2314yes—
109C2×#102C2318yes—
110C2×#102C2318yes—
111C2×#102C2318yes—
112C2×#102C2318yes—
113C2×C22⋊C8C2314yes—
114C2×#97C2314yes—
115#398—22no—
116#398—22no—
117#396—24no—
118#396—24no—
119#394—22no—
120#394—22no—
121#394—24no—
122#394—24no—
123#394—24no—
124#394—24no—
125#391—22no—
126#390—22no—
127#390—24no—
128#390—24no—
129#390—22no—
130#390—22no—
131#390—24no—
132#390C2314yes—
133#390C2318yes—
134#390C2314yes—
135#390C2314yes—
136#390C2314yes—
137#389C2314yes—
138#389C2314yes—
139#388—24no—
140#388—24no—
141#386—22no—
142#386—22no—
143#384C2312yes—
144#382—21no—
145#380—21no—
146#380—24no—
147#380C2312yes—
148C4×C4⋊C4—22no—
149C4×C4⋊C4—21no—
150C4×C4⋊C4C2314yes—
151C4×C4⋊C4C2312yes—
152C2×#95—21no—
153#340C2318yes—
154#340C2318yes—
155#336C2314yes—
156#336C2318yes—
157#336C2314yes—
158C25—22no—
159C2×#41—44no—
160C2×#41—44no—
161C2×#41—44no—
162C22×Q8—24no—
163C22×Q8—24no—
164C22×Q8—21no—
165C22×Q8—42no—
166C22×Q8C2×D412yes—
167C22×Q8C2×D412yes—
168C22×D4—22no—
169C22×D4—21no—
170C22×D4—24no—
171C22×D4—22no—
172C22×D4—21no—
173C22×D4—24no—
174C22×D4—28no—
175C22×D4—44no—
176C22×D4—44no—
177C22×D4—44no—
178C22×D4—44no—
179C22×D4—44no—
180C22×D4—44no—
181C22×D4—84no—
182C22×D4C2412yes—
183C22×D4C2414yes—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2411yescharacteristic
193C2×D8—48no—
194C2×D8—48no—
195C2×D8—48no—
196C22×C8C2418yes—
197C4⋊Q8—42no—
198C4⋊Q8—48no—
199#127—44no—
200#127—44no—
201#127—42no—
202#127—48no—
203#124—44no—
204#124—416no—
205#124—416no—
206#124—48no—
207#123—48no—
208#123—48no—
209C22⋊Q8—416no—
210#121—28no—
211#121—44no—
212C4×Q8—44no—
213C4×Q8—44no—
214C4×Q8—42no—
215C4×Q8—42no—
216C4×Q8—42no—
217C4×Q8—42no—
218C4×D4—44no—
219C4×D4—44no—
220#117—44no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2414yes—
238C2×C4⋊C4C2412yes—
239C2×C4⋊C4C2414yes—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
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—84no—
256C2×C22⋊C4—84no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—24no—
262C2×C42—44no—
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2411yescharacteristic
268C2×C42C2412yes—
269C2×C42C2411yescharacteristic
270C2×C42C2412yes—
271#106—48no—
272C4⋊C8—44no—
273#102—48no—
274#102—48no—
275#102—48no—
276#102—48no—
277C22⋊C8—28no—
278#97—44no—
279#95—24no—
280#95—22no—
281#95—22no—
282#95—24no—
283#95—24no—
284C24—22no—
285C24—24no—
286C24—44no—
287C24—44no—
288C24—44no—
289C24—44no—
290C24—44no—
291C24—44no—
292C24—44no—
293C24C22×D411yescharacteristic
294#41—816no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—416no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×D4—28no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—28no—
309C2×D4—28no—
310C2×D4—24no—
311C2×D4—42no—
312C2×D4—48no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—48no—
316C2×D4—48no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—48no—
324C2×D4—42no—
325C2×D4—816no—
326C2×D4—82no—
327C2×D4—88no—
328C2×D4—816no—
329C2×D4—816no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4AGL(1,ℤ/8)14yes—
354C22×C4AGL(1,ℤ/8)14yes—
355C22×C4C22×D412yes—
356C22×C4C22×D412yes—
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D412yes—
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D412yes—
363C22×C4C22×D411yescharacteristic
364C22×C4H(5,𝔽2)14yes—
365C22×C4C2511yescharacteristic
366D8—816no—
367C2×C8—28no—
368C2×C8—28no—
369C2×C8—28no—
370C4⋊C4—24no—
371C4⋊C4—28no—
372C4⋊C4—28no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—42no—
382C4⋊C4—48no—
383C4⋊C4—48no—
384C4⋊C4—82no—
385C22⋊C4—416no—
386C22⋊C4—44no—
387C22⋊C4—48no—
388C22⋊C4—44no—
389C22⋊C4—44no—
390C22⋊C4—48no—
391C22⋊C4—48no—
392C22⋊C4—48no—
393C22⋊C4—84no—
394C22⋊C4—84no—
395C22⋊C4—88no—
396C22⋊C4—84no—
397C22⋊C4—84no—
398C42—22no—
399C42—24no—
400C42—22no—
401C42—24no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—48no—
409C42—44no—
410C23—22no—
411C23—22no—
412C23—44no—
413C23—44no—
414C23—44no—
415C23—42no—
416C23—44no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23—42no—
421C23—44no—
422C23—42no—
423C23—44no—
424C23—88no—
425C23—88no—
426C23—88no—
427C23—88no—
428C23—88no—
429C23—88no—
430C23—88no—
431C23#53412yes—
432C23C2×AGL(1,ℤ/8)12yes—
433C23C2×AGL(1,ℤ/8)12yes—
434C23C23×D411yescharacteristic
435Q8—48no—
436Q8—48no—
437Q8—84no—
438D4—48no—
439D4—48no—
440D4—48no—
441D4—816no—
442D4—88no—
443D4—84no—
444D4—84no—
445D4—816no—
446D4—88no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—28no—
451C2×C4—28no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—28no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—88no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—88no—
483C2×C4#53412yes—
484C2×C4C2×AGL(1,ℤ/8)12yes—
485C2×C4C2×AGL(1,ℤ/8)12yes—
486C2×C4C2×AGL(1,ℤ/8)14yes—
487C2×C4C2×H(5,𝔽2)12yes—
488C2×C4#58514yes—
489C8—48no—
490C22—22no—
491C22—22no—
492C22—24no—
493C22—24no—
494C22—82no—
495C22—84no—
496C22—84no—
497C22—82no—
498C22—84no—
499C22—82no—
500C22—84no—
501C22—84no—
502C22—84no—
503C22—84no—
504C22—84no—
505C22—84no—
506C22—1616no—
507C22#319214yes—
508C22C2×#53411yescharacteristic
509C22#348712yes—
510C22C22×AGL(1,ℤ/8)11yescharacteristic
511C22C22×AGL(1,ℤ/8)11yescharacteristic
512C22#358512yes—
513C4—22no—
514C4—24no—
515C4—22no—
516C4—22no—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—41no—
521C4—41no—
522C4—82no—
523C4—84no—
524C4—84no—
525C4#354514yes—
526C2—24no—
527C2—162no—
528C2—164no—
529C2#3718711yescharacteristic
530C2C2×#319211yescharacteristic
531C2#6075311yescharacteristic
532C2#6189012yes—
533C2#6189112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35178C2semidirect
2. C2×#2548C2semidirect
3. #33095C2semidirect
4. #32681C2semidirect
5. #32646C2semidirect
6. #32623C2semidirect
7. #29622C2semidirect
8. #27585C2semidirect
9. #24506C2semidirect
10. #23930C2semidirect
11. #21822C2semidirect
27. #2548C22semidirect
28. #2546C22semidirect
30. #2538C22semidirect
40. #2094C22semidirect
41. #2071C22semidirect
42. #2065C22semidirect
43. #2054C22semidirect
45. #2023C22semidirect
47. #2011C22semidirect
48. #1985C22semidirect
49. #1985C22semidirect
50. #1972C22semidirect
51. #1936C22semidirect
52. #1926C22semidirect
53. #1926C22semidirect
54. #1919C22semidirect
55. #1879C22semidirect

Maximal subgroups

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

subgroupindex
#351782
C2×#25482
#330952
#326812
#326462
#326232
#296222
#275852
#245062
#239302
#218222