Finite Group Databasebeta

← #6434751

Subgroups — #6434751

Order 512.

2186 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2858C211yesmaximal, characteristic
2C4×#533C211yesmaximal, characteristic, complemented
3#30066C218yesmaximal, complemented
4#30062C214yesmaximal, complemented
5#30059C214yesmaximal, complemented
6#23731C212yesmaximal, complemented
7#23552C212yesmaximal, complemented
8#23505C212yesmaximal
9#23454C212yesmaximal, complemented
10#21060C214yesmaximal, complemented
11#20772C211yesmaximal, characteristic, complemented
12C2×#533C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×#127C2211yescharacteristic
15C22×#124—22no—
16C2×#428C2214yes—
17C2×#418C2214yes—
18C2×#418C2214yes—
19#2858—22no—
20#2858—22no—
21#2858C2214yes—
22#2858C2214yes—
23C4×C4⋊Q8C2212yescomplemented
24C4×C4⋊Q8C2212yescomplemented
25C4×#125C2214yescomplemented
26C4×C22⋊Q8—22no—
27C4×C22⋊Q8—22no—
28#2296C2212yes—
29#2295—22no—
30C2×#390—22no—
31C2×#390C2212yes—
32C2×#390C2212yes—
33#2285C2211yescharacteristic
34C42×Q8—22no—
35C4×#117C2212yes—
36C2×C43C2211yescharacteristic
37#2094C2214yescomplemented
38#2093C2214yescomplemented
39#2092C2218yescomplemented
40#2058C2218yescomplemented
41#2057C2218yescomplemented
42#2041C2218yes—
43#2041C2218yes—
44#2025C2218yescomplemented
45#2023C2218yescomplemented
46#1926C22116yescomplemented
47#1925C22116yescomplemented
48#1839C2214yescomplemented
49#1839C2214yescomplemented
50#1736C2214yes—
51#1728C2212yes—
52#1728C2212yes—
53#1728C2212yes—
54C23×Q8—21no—
55C23×D4—21no—
56C23×D4—21no—
57C23×D4C2312yes—
58C22×M16C2314yes—
59#533—28no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—44no—
62C2×C4⋊Q8C2312yes—
63C2×C4⋊Q8C2314yes—
64C2×#127—22no—
65C2×#127—44no—
66C2×#127C2314yes—
67C2×#125C2312yes—
68C2×#124—24no—
69C2×#124—24no—
70C2×#124—24no—
71C2×#124—24no—
72C2×#124—416no—
73C2×C22⋊Q8—22no—
74C2×C4×Q8—22no—
75C2×C4×Q8—22no—
76C2×C4×Q8—22no—
77C2×#117C2312yes—
78C2×#117C2312yes—
79C22×C4⋊C4—21no—
80C22×C4⋊C4C2312yes—
81C22×C22⋊C4—24no—
82C22×C22⋊C4—24no—
83C22×C42D412yes—
84C22×C42D412yes—
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#428—48no—
89C2×#107C2318yes—
90C2×#106C2318yes—
91#418—216no—
92#418—48no—
93C2×#102C23116yes—
94C2×#102C2318yes—
95C2×#102C2318yes—
96#391—22no—
97#391—22no—
98#390—24no—
99#390—22no—
100#390—22no—
101#390—24no—
102#390—24no—
103#390—24no—
104#390—24no—
105#390C2314yes—
106#390C2314yes—
107#390C2318yes—
108#389—22no—
109#389—22no—
110#389C2314yes—
111#389C2314yes—
112#387—24no—
113#387—24no—
114#385—22no—
115#385—22no—
116#384C2311yescharacteristic
117#384C2311yescharacteristic
118#382C2312yes—
119#380—22no—
120C4×C4⋊C4—24no—
121C4×C4⋊C4—24no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—24no—
126C4×C4⋊C4C2314yes—
127C4×C4⋊C4C2314yes—
128C4×C22⋊C4—24no—
129#376—22no—
130C43—22no—
131C43D412yes—
132C43D412yes—
133C43C2312yes—
134C43C2312yes—
135#340C23116yes—
136#336C2318yes—
137#336C2314yes—
138#336C2314yes—
139#336C2318yes—
140C25—22no—
141C22×Q8—22no—
142C22×Q8—24no—
143C22×Q8—48no—
144C22×Q8—44no—
145C22×D4—24no—
146C22×D4—22no—
147C22×D4—24no—
148C22×D4—28no—
149C22×D4—28no—
150C22×D4—28no—
151C22×D4—48no—
152C22×D4—44no—
153C22×D4—44no—
154C22×D4—48no—
155C22×D4C2414yes—
156C23×C4—22no—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2411yescharacteristic
162C2×M16—28no—
163C2×M16—28no—
164C2×M16—28no—
165C22×C8C2418yes—
166C4⋊Q8—28no—
167C4⋊Q8—48no—
168C4⋊Q8—48no—
169#127—48no—
170#127—48no—
171#125—28no—
172#124—416no—
173#124—416no—
174#124—416no—
175#124—416no—
176C22⋊Q8—216no—
177C4×Q8—216no—
178C4×Q8—48no—
179C4×Q8—48no—
180#117—24no—
181#117—28no—
182#117#4118yes—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4C2414yes—
194C2×C4⋊C4C2412yes—
195C2×C4⋊C4C2412yes—
196C2×C4⋊C4C2414yes—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—416no—
204C2×C22⋊C4—416no—
205C2×C42—22no—
206C2×C42—24no—
207C2×C42—24no—
208C2×C42—24no—
209C2×C42—28no—
210C2×C42—22no—
211C2×C42—24no—
212C2×C42—24no—
213C2×C42—44no—
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42C2411yescharacteristic
227C2×C42C2412yes—
228C2×C42C2411yescharacteristic
229C2×C42C2412yes—
230#107—48no—
231#106—48no—
232#102—232no—
233#102—48no—
234#102—48no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240C24—24no—
241C24—48no—
242C24—48no—
243C24—44no—
244C24—44no—
245C24—44no—
246C24C22×D411yescharacteristic
247C2×Q8—28no—
248C2×Q8—44no—
249C2×Q8—48no—
250C2×Q8—48no—
251C2×Q8—44no—
252C2×Q8—48no—
253C2×Q8—816no—
254C2×D4—28no—
255C2×D4—28no—
256C2×D4—28no—
257C2×D4—48no—
258C2×D4—48no—
259C2×D4—48no—
260C2×D4—48no—
261C2×D4—44no—
262C2×D4—44no—
263C2×D4—44no—
264C2×D4—432no—
265C2×D4—48no—
266C2×D4—48no—
267C2×D4—816no—
268C2×D4—816no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—48no—
284C22×C4—44no—
285C22×C4C22×D412yes—
286C22×C4C22×D412yes—
287C22×C4C22×D412yes—
288C22×C4C22×D412yes—
289C22×C4C22×D412yes—
290C22×C4C22×D412yes—
291C22×C4C22×D411yescharacteristic
292C22×C4C2×#4114yes—
293C22×C4H(5,𝔽2)14yes—
294C22×C42−1+414yes—
295C22×C4C2511yescharacteristic
296M16—416no—
297C2×C8—28no—
298C2×C8—28no—
299C2×C8—28no—
300C4⋊C4—216no—
301C4⋊C4—28no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—28no—
306C4⋊C4—48no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—48no—
310C4⋊C4—44no—
311C4⋊C4—48no—
312C4⋊C4C2×#4118yes—
313C22⋊C4—28no—
314C22⋊C4—416no—
315C22⋊C4—416no—
316C22⋊C4—44no—
317C22⋊C4—416no—
318C22⋊C4—416no—
319C42—24no—
320C42—28no—
321C42—24no—
322C42—22no—
323C42—22no—
324C42—24no—
325C42—24no—
326C42—24no—
327C42—24no—
328C42—24no—
329C42—22no—
330C42—22no—
331C42—24no—
332C42—24no—
333C42—42no—
334C42—42no—
335C42—48no—
336C42—44no—
337C42—48no—
338C42AGL(1,ℤ/8)18yes—
339C23—22no—
340C23—22no—
341C23—44no—
342C23—44no—
343C23—44no—
344C23—48no—
345C23—48no—
346C23—88no—
347C23—88no—
348C23—816no—
349C23—816no—
350C23—88no—
351C23#52512yes—
352C23#53512yes—
353C23#53612yes—
354C23C23×D411yescharacteristic
355Q8—416no—
356Q8—816no—
357D4—416no—
358D4—816no—
359D4—816no—
360D4—88no—
361C2×C4—28no—
362C2×C4—28no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—28no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—48no—
379C2×C4—48no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—48no—
387C2×C4—44no—
388C2×C4—88no—
389C2×C4#52512yes—
390C2×C4#53512yes—
391C2×C4#53612yes—
392C2×C4#542116yes—
393C2×C4C2×AGL(1,ℤ/8)14yes—
394C2×C4C2×AGL(1,ℤ/8)14yes—
395C2×C4C2×AGL(1,ℤ/8)14yes—
396C2×C4#57514yes—
397C2×C4C22×#4112yes—
398C2×C4C2×H(5,𝔽2)12yes—
399C2×C4C2×2−1+412yes—
400C8—48no—
401C22—22no—
402C22—22no—
403C22—24no—
404C22—24no—
405C22—84no—
406C22—84no—
407C22—84no—
408C22—88no—
409C22—88no—
410C22—1616no—
411C22#263214yes—
412C22C2×#52511yescharacteristic
413C22C2×#53511yescharacteristic
414C22C2×#53611yescharacteristic
415C22C22×AGL(1,ℤ/8)12yes—
416C22C2×#57512yes—
417C4—24no—
418C4—24no—
419C4—24no—
420C4—24no—
421C4—42no—
422C4—48no—
423C4—42no—
424C4—82no—
425C4—84no—
426C4—82no—
427C4#263214yes—
428C4C2×#54214yes—
429C2—24no—
430C2—164no—
431C2C2×#263211yescharacteristic
432C2#6056112yes—
433C2#6079811yescharacteristic
434C2#6079911yescharacteristic
435C2#6086412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#533C2semidirect
3. #30066C2semidirect
4. #30062C2semidirect
5. #30059C2semidirect
6. #23731C2semidirect
7. #23552C2semidirect
9. #23454C2semidirect
10. #21060C2semidirect
11. #20772C2semidirect
23. C4×C4⋊Q8C22semidirect
24. C4×C4⋊Q8C22semidirect
25. C4×#125C22semidirect
37. #2094C22semidirect
38. #2093C22semidirect
39. #2092C22semidirect
40. #2058C22semidirect
41. #2057C22semidirect
44. #2025C22semidirect
45. #2023C22semidirect
46. #1926C22semidirect
47. #1925C22semidirect
48. #1839C22semidirect
49. #1839C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28582
C4×#5332
#300662
#300622
#300592
#237312
#235522
#235052
#234542
#210602
#207722