Finite Group Databasebeta

← #6433102

Subgroups — #6433102

Order 512.

3426 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic, complemented
2C22×#390C211yesmaximal, characteristic
3#29956C218yesmaximal, complemented
4#29952C214yesmaximal, complemented
5#29950C214yesmaximal, complemented
6#23731C212yesmaximal, complemented
7#23618C214yesmaximal, complemented
8#23454C212yesmaximal, complemented
9#20844C212yesmaximal, complemented
10#20626C211yesmaximal, characteristic, complemented
11#20598C212yesmaximal
12C24×D4C2211yescharacteristic
13C2×#533C2211yescharacteristic, complemented
14C23×C4⋊C4C2211yescharacteristic
15C23×C22⋊C4—22no—
16C23×C42C2211yescharacteristic
17C2×#428C2214yescomplemented
18C2×#423C2212yescomplemented
19C2×#418C2214yes—
20C2×#418C2214yes—
21C2×#407C2212yes—
22#2857C2212yescomplemented
23#2857C2212yescomplemented
24#2855—22no—
25#2851C2214yescomplemented
26#2848—22no—
27C4×C22⋊Q8—22no—
28C4×C22⋊Q8—22no—
29#2296C2212yescomplemented
30C2×#390—22no—
31C2×#390—22no—
32C2×#390—22no—
33C2×#390C2214yes—
34C2×#390C2214yes—
35C2×#390C2212yes—
36C2×#390C2212yes—
37#2285C2211yescharacteristic, complemented
38#2280C2212yes—
39#2094C2214yescomplemented
40#2093C2214yescomplemented
41#2092C2218yescomplemented
42#2067C22116yescomplemented
43#2065C22116yescomplemented
44#2025C2218yescomplemented
45#2023C2218yescomplemented
46#1853C2218yescomplemented
47#1852C2218yescomplemented
48#1797C2214yescomplemented
49#1797C2214yescomplemented
50#1786C2218yes—
51#1786C2218yes—
52#1736C2214yescomplemented
53#1728C2212yescomplemented
54C26—21no—
55C23×D4—22no—
56C23×D4—21no—
57C23×D4—24no—
58C23×D4—44no—
59C23×D4—44no—
60C23×D4Q818yescomplemented
61C23×D4C2312yes—
62C23×D4C2314yes—
63C24×C4—21no—
64C24×C4D412yes—
65C24×C4C2311yescharacteristic
66C22×M16C2314yes—
67#533—44no—
68C2×C4⋊Q8C2312yescomplemented
69C2×#125C2312yescomplemented
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—22no—
72C2×C22⋊Q8—22no—
73C2×C4×Q8—22no—
74C2×#117C2312yes—
75C2×#117C2312yes—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—44no—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2314yes—
81C22×C22⋊C4—24no—
82C22×C22⋊C4—24no—
83C22×C22⋊C4—24no—
84C22×C22⋊C4—24no—
85C22×C22⋊C4—24no—
86C22×C22⋊C4—24no—
87C22×C22⋊C4—416no—
88C22×C42—44no—
89C22×C42D412yes—
90C22×C42D412yes—
91C22×C42D412yes—
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94C22×C42C2312yes—
95C22×C42C2312yes—
96#428—48no—
97C2×#107C2318yescomplemented
98C2×#106C2318yescomplemented
99#423—44no—
100C2×C4⋊C8C2318yescomplemented
101#418—48no—
102#418—48no—
103C2×#102C2318yes—
104C2×#102C23116yes—
105C2×#102C2318yes—
106#407—28no—
107C2×C22⋊C8C2318yes—
108#391—22no—
109#391—22no—
110#390—24no—
111#390—22no—
112#390—22no—
113#390—24no—
114#390—24no—
115#390—24no—
116#390—24no—
117#390C2318yes—
118#390C2314yes—
119#390C2314yes—
120#389—22no—
121#389—22no—
122#389C2314yescomplemented
123#389C2314yescomplemented
124#387—24no—
125#387—24no—
126#385—24no—
127#385—24no—
128#385—24no—
129#384C2311yescharacteristic, complemented
130#384C2311yescharacteristic, complemented
131#382C2312yescomplemented
132#381—22no—
133#380—22no—
134#380—22no—
135#380—22no—
136#380C2314yes—
137#380C2314yes—
138#379—22no—
139C4×C4⋊C4—22no—
140C4×C4⋊C4—22no—
141C4×C4⋊C4—22no—
142C4×C22⋊C4—24no—
143#340C23116yescomplemented
144#336C2314yescomplemented
145#336C2314yescomplemented
146C25—28no—
147C25—22no—
148C25—24no—
149C25—48no—
150C25—44no—
151C25—44no—
152C25—48no—
153C25C2×D411yescharacteristic
154C22×Q8—22no—
155C22×D4—28no—
156C22×D4—216no—
157C22×D4—24no—
158C22×D4—22no—
159C22×D4—28no—
160C22×D4—24no—
161C22×D4—28no—
162C22×D4—28no—
163C22×D4—432no—
164C22×D4—48no—
165C22×D4—48no—
166C22×D4—44no—
167C22×D4—44no—
168C22×D4—48no—
169C22×D4—48no—
170C22×D4—432no—
171C22×D4C2×Q818yes—
172C22×D4C2×Q8116yes—
173C22×D4C2414yes—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—28no—
177C23×C4—22no—
178C23×C4—24no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4—48no—
182C23×C4—44no—
183C23×C4C2×D412yes—
184C23×C4C2×D412yes—
185C23×C4C2×D412yes—
186C23×C4C2×D412yes—
187C23×C4C2×D412yes—
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D412yes—
190C23×C4C2×D412yes—
191C23×C4C2×Q814yes—
192C23×C4#4114yes—
193C23×C4C2412yes—
194C23×C4C2411yescharacteristic
195C2×M16—28no—
196C2×M16—28no—
197C2×M16—28no—
198C22×C8C2418yes—
199C4⋊Q8—44no—
200#125—44no—
201C22⋊Q8—48no—
202C22⋊Q8—48no—
203C22⋊Q8—48no—
204C4×Q8—48no—
205#117—28no—
206#117—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—48no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—48no—
219C2×C4⋊C4C2414yes—
220C2×C4⋊C4C2414yes—
221C2×C4⋊C4C2412yes—
222C2×C4⋊C4C2412yes—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—416no—
232C2×C22⋊C4—416no—
233C2×C22⋊C4—416no—
234C2×C22⋊C4—416no—
235C2×C22⋊C4—416no—
236C2×C22⋊C4—416no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—44no—
242C2×C42—44no—
243C2×C42—44no—
244C2×C42—44no—
245C2×C42—44no—
246C2×C42C2×D412yes—
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2411yescharacteristic
254C2×C42C2411yescharacteristic
255C2×C42C2412yes—
256#107—48no—
257#106—48no—
258C4⋊C8—48no—
259#102—48no—
260#102—48no—
261#102—416no—
262C22⋊C8—216no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268C24—22no—
269C24—22no—
270C24—24no—
271C24—24no—
272C24—216no—
273C24—28no—
274C24—24no—
275C24—48no—
276C24—48no—
277C24—432no—
278C24—48no—
279C24—48no—
280C24—464no—
281C24—48no—
282C24—48no—
283C24—44no—
284C24—44no—
285C24—48no—
286C24—48no—
287C24—832no—
288C24—816no—
289C24—816no—
290C24C22⋊Q814yes—
291C24C22×D411yescharacteristic
292C24C22×D412yes—
293C2×Q8—44no—
294C2×Q8—44no—
295C2×Q8—44no—
296C2×D4—216no—
297C2×D4—28no—
298C2×D4—232no—
299C2×D4—24no—
300C2×D4—28no—
301C2×D4—28no—
302C2×D4—432no—
303C2×D4—432no—
304C2×D4—48no—
305C2×D4—48no—
306C2×D4—464no—
307C2×D4—432no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—416no—
311C2×D4C22×Q818yes—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—216no—
317C22×C4—216no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—432no—
342C22×C4—48no—
343C22×C4—48no—
344C22×C4—48no—
345C22×C4—44no—
346C22×C4—48no—
347C22×C4—816no—
348C22×C4C22⋊Q814yes—
349C22×C4C22⋊Q818yes—
350C22×C4AGL(1,ℤ/8)18yes—
351C22×C4C22×D412yes—
352C22×C4C22×D412yes—
353C22×C4C22×D412yes—
354C22×C4C22×D412yes—
355C22×C4C22×D412yes—
356C22×C4C22×D411yescharacteristic
357C22×C4C22×Q814yes—
358C22×C4C22×Q812yes—
359C22×C4C2×#4114yes—
360C22×C4C2×#4112yes—
361C22×C4H(5,𝔽2)14yes—
362C22×C4H(5,𝔽2)14yes—
363C22×C4C2511yescharacteristic
364M16—416no—
365C2×C8—28no—
366C2×C8—28no—
367C2×C8—28no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—28no—
371C4⋊C4—28no—
372C4⋊C4—48no—
373C4⋊C4—48no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—48no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—48no—
380C22⋊C4—416no—
381C22⋊C4—416no—
382C22⋊C4—44no—
383C22⋊C4—44no—
384C22⋊C4—48no—
385C22⋊C4—48no—
386C22⋊C4—416no—
387C22⋊C4—416no—
388C42—22no—
389C42—22no—
390C42—24no—
391C42—24no—
392C42—22no—
393C42—22no—
394C42—42no—
395C42—42no—
396C42—42no—
397C42—42no—
398C42—44no—
399C42—44no—
400C23—216no—
401C23—28no—
402C23—28no—
403C23—216no—
404C23—28no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—28no—
412C23—432no—
413C23—432no—
414C23—432no—
415C23—432no—
416C23—416no—
417C23—44no—
418C23—44no—
419C23—48no—
420C23—48no—
421C23—8128no—
422C23—816no—
423C23—816no—
424C23—816no—
425C23—88no—
426C23—816no—
427C23—88no—
428C23—816no—
429C23C2×C22⋊Q814yes—
430C23C2×C22⋊Q812yes—
431C23#53412yes—
432C23#53512yes—
433C23C2×AGL(1,ℤ/8)14yes—
434C23#57514yes—
435C23C23×D411yescharacteristic
436Q8—88no—
437D4—216no—
438D4—432no—
439D4—416no—
440D4—48no—
441C2×C4—28no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—216no—
449C2×C4—28no—
450C2×C4—28no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—48no—
458C2×C4—48no—
459C2×C4—48no—
460C2×C4—48no—
461C2×C4—48no—
462C2×C4—48no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—48no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—88no—
473C2×C4—816no—
474C2×C4C2×C22⋊Q814yes—
475C2×C4C2×C22⋊Q812yes—
476C2×C4#53412yes—
477C2×C4#53512yes—
478C2×C4#53818yes—
479C2×C4#54318yes—
480C2×C4C2×AGL(1,ℤ/8)14yes—
481C2×C4C2×AGL(1,ℤ/8)14yes—
482C2×C4C23×Q811yescharacteristic
483C2×C4C22×#4111yescharacteristic
484C2×C4C2×H(5,𝔽2)12yes—
485C2×C4C2×H(5,𝔽2)12yes—
486C8—48no—
487C22—232no—
488C22—24no—
489C22—24no—
490C22—24no—
491C22—24no—
492C22—24no—
493C22—22no—
494C22—24no—
495C22—22no—
496C22—24no—
497C22—24no—
498C22—416no—
499C22—48no—
500C22—864no—
501C22—832no—
502C22—88no—
503C22—84no—
504C22—84no—
505C22#262614yes—
506C22#307718yes—
507C22C22×C22⋊Q811yescharacteristic
508C22C2×#53411yescharacteristic
509C22C2×#53511yescharacteristic
510C22C22×AGL(1,ℤ/8)12yes—
511C22C2×#57512yes—
512C4—24no—
513C4—24no—
514C4—24no—
515C4—42no—
516C4—42no—
517C4—84no—
518C4—82no—
519C4—84no—
520C4—82no—
521C4#262614yes—
522C4C2×#53812yes—
523C4C2×#54312yes—
524C2—28no—
525C2—24no—
526C2—88no—
527C2C2×#262611yescharacteristic
528C2C2×#307712yes—
529C2#6075012yes—
530C2#6079811yescharacteristic
531C2#6079911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24×D4C22
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
1. #36625C2semidirect
3. #29956C2semidirect
4. #29952C2semidirect
5. #29950C2semidirect
6. #23731C2semidirect
7. #23618C2semidirect
8. #23454C2semidirect
9. #20844C2semidirect
10. #20626C2semidirect
13. C2×#533C22semidirect
17. C2×#428C22semidirect
18. C2×#423C22semidirect
22. #2857C22semidirect
23. #2857C22semidirect
25. #2851C22semidirect
29. #2296C22semidirect
37. #2285C22semidirect
39. #2094C22semidirect
40. #2093C22semidirect
41. #2092C22semidirect
42. #2067C22semidirect
43. #2065C22semidirect
44. #2025C22semidirect
45. #2023C22semidirect
46. #1853C22semidirect
47. #1852C22semidirect
48. #1797C22semidirect
49. #1797C22semidirect
52. #1736C22semidirect
53. #1728C22semidirect
60. C23×D4Q8semidirect
68. C2×C4⋊Q8C23semidirect
69. C2×#125C23semidirect
97. C2×#107C23semidirect
98. C2×#106C23semidirect
100. C2×C4⋊C8C23semidirect
122. #389C23semidirect
123. #389C23semidirect
129. #384C23semidirect
130. #384C23semidirect
131. #382C23semidirect
143. #340C23semidirect
144. #336C23semidirect
145. #336C23semidirect

Maximal subgroups

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

subgroupindex
#366252
C22×#3902
#299562
#299522
#299502
#237312
#236182
#234542
#208442
#206262
#205982