Finite Group Databasebeta

← #6434341

Subgroups — #6434341

Order 512.

2354 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36640C211yesmaximal, characteristic, complemented
2C2×C4×#127C211yesmaximal, characteristic
3#30077C218yesmaximal, complemented
4#30076C214yesmaximal, complemented
5#30075C214yesmaximal, complemented
6#23731C212yesmaximal, complemented
7#23658C212yesmaximal, complemented
8#23628C212yesmaximal, complemented
9#23505C212yesmaximal
10#20873C214yesmaximal, complemented
11#20772C211yesmaximal, characteristic, complemented
12C22×#127C2211yescharacteristic
13C22×C4×D4—22no—
14C22×C4×D4C2212yes—
15C2×#428C2214yes—
16C2×#418C2214yes—
17C2×#418C2214yes—
18#2864C2214yescomplemented
19#2859C2212yescomplemented
20#2859C2212yescomplemented
21#2851—22no—
22#2851—22no—
23C4×#127—22no—
24C4×#127—22no—
25C4×#127C2214yes—
26C4×#127C2214yes—
27C4×#125—22no—
28C4×#125—22no—
29#2296C2212yes—
30#2296C2212yes—
31C2×#390—22no—
32C2×#390C2212yes—
33#2285C2211yescharacteristic
34#2285C2211yescharacteristic
35C2×#384C2211yescharacteristic
36C2×C43C2211yescharacteristic
37#2094C2214yescomplemented
38#2093C2214yescomplemented
39#2092C2218yescomplemented
40#2084C2218yescomplemented
41#2083C2218yescomplemented
42#2073C2218yescomplemented
43#2071C2218yescomplemented
44#2041C2218yes—
45#2041C2218yes—
46#1864C22116yescomplemented
47#1862C22116yescomplemented
48#1839C2214yescomplemented
49#1839C2214yescomplemented
50#1736C2214yes—
51#1728C2212yes—
52#1728C2212yes—
53#1728C2212yes—
54C23×D4—22no—
55C23×D4C2312yes—
56C24×C4—22no—
57C22×M16C2314yes—
58C2×#127—28no—
59C2×#127—21no—
60C2×#127—21no—
61C2×#127C2314yes—
62C2×#125—22no—
63C2×#125—22no—
64C2×C4×D4—24no—
65C2×C4×D4—24no—
66C2×C4×D4—24no—
67C2×C4×D4—24no—
68C2×C4×D4—24no—
69C2×C4×D4—48no—
70C2×C4×D4—416no—
71C2×C4×D4C2318yes—
72C2×#117C2312yes—
73C2×#117C2312yes—
74C22×C4⋊C4—22no—
75C22×C4⋊C4C2312yes—
76C22×C22⋊C4—24no—
77C22×C22⋊C4—22no—
78C22×C42D412yes—
79C22×C42D412yes—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83#428—48no—
84C2×#107C2318yes—
85C2×#106C2318yes—
86#418—216no—
87#418—48no—
88C2×#102C23116yes—
89C2×#102C2318yes—
90C2×#102C2318yes—
91#390—24no—
92#390—24no—
93#390—24no—
94#390—24no—
95#390—24no—
96#390C2314yes—
97#390C2314yes—
98#389—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#389C2314yes—
103#389C2314yes—
104#389C2314yes—
105#389C2314yes—
106#387—24no—
107#387—24no—
108#387—24no—
109#387—24no—
110#385—24no—
111#385—22no—
112#385—22no—
113#384—21no—
114#384—21no—
115#384C2314yes—
116#384C2312yes—
117#384C2311yescharacteristic
118#384C2311yescharacteristic
119#382—22no—
120#382—22no—
121#382C2312yes—
122#382C2312yes—
123#380—22no—
124#380—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—24no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C43D412yes—
131C43D412yes—
132C43C2312yes—
133C43C2312yes—
134#340C23116yes—
135#336C2314yes—
136#336C2314yes—
137#336C2318yes—
138#336C2318yes—
139C25—22no—
140C22×D4—216no—
141C22×D4—22no—
142C22×D4—24no—
143C22×D4—24no—
144C22×D4—24no—
145C22×D4—28no—
146C22×D4—28no—
147C22×D4—48no—
148C22×D4—48no—
149C22×D4#4118yes—
150C22×D4C2414yes—
151C23×C4—22no—
152C23×C4—28no—
153C23×C4—48no—
154C23×C4—416no—
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2411yescharacteristic
160C2×M16—28no—
161C2×M16—28no—
162C2×M16—28no—
163C22×C8C2418yes—
164#127—28no—
165#127—28no—
166#127—216no—
167#125—48no—
168#125—48no—
169C4×D4—416no—
170C4×D4—416no—
171C4×D4—416no—
172C4×D4—416no—
173C4×D4—416no—
174C4×D4—416no—
175#117—28no—
176#117—24no—
177#117—24no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—48no—
191C2×C4⋊C4C2414yes—
192C2×C4⋊C4C2414yes—
193C2×C4⋊C4C2412yes—
194C2×C4⋊C4C2412yes—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—48no—
201C2×C22⋊C4—416no—
202C2×C42—24no—
203C2×C42—24no—
204C2×C42—24no—
205C2×C42—28no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—44no—
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2×D412yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42C2412yes—
222C2×C42C2412yes—
223C2×C42C2411yescharacteristic
224C2×C42C2411yescharacteristic
225#107—48no—
226#106—48no—
227#102—232no—
228#102—48no—
229#102—48no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236C24—28no—
237C24—24no—
238C24—48no—
239C24—48no—
240C24—44no—
241C24—44no—
242C24C22×D411yescharacteristic
243C2×D4—216no—
244C2×D4—216no—
245C2×D4—28no—
246C2×D4—232no—
247C2×D4—24no—
248C2×D4—28no—
249C2×D4—28no—
250C2×D4—48no—
251C2×D4—464no—
252C2×D4—48no—
253C2×D4—44no—
254C2×D4—44no—
255C2×D4—416no—
256C2×D4—48no—
257C2×D4—48no—
258C2×D4C2×#4118yes—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—416no—
274C22×C4—416no—
275C22×C4—48no—
276C22×C4—44no—
277C22×C4—416no—
278C22×C4—832no—
279C22×C4C22×D412yes—
280C22×C4C22×D412yes—
281C22×C4C22×D412yes—
282C22×C4C22×D412yes—
283C22×C4C22×D412yes—
284C22×C4C22×D412yes—
285C22×C4C22×D411yescharacteristic
286C22×C4C2×#4114yes—
287C22×C4H(5,𝔽2)14yes—
288C22×C42−1+414yes—
289C22×C4C2511yescharacteristic
290M16—416no—
291C2×C8—28no—
292C2×C8—28no—
293C2×C8—28no—
294C4⋊C4—28no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—28no—
299C4⋊C4—22no—
300C4⋊C4—48no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—48no—
304C4⋊C4—44no—
305C4⋊C4—48no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—48no—
309C4⋊C4—48no—
310C4⋊C4—48no—
311C22⋊C4—416no—
312C22⋊C4—416no—
313C22⋊C4—44no—
314C22⋊C4—44no—
315C22⋊C4—416no—
316C42—22no—
317C42—22no—
318C42—24no—
319C42—24no—
320C42—24no—
321C42—24no—
322C42—24no—
323C42—24no—
324C42—24no—
325C42—24no—
326C42—24no—
327C42—22no—
328C42—22no—
329C42—42no—
330C42—42no—
331C42—44no—
332C42—44no—
333C42AGL(1,ℤ/8)18yes—
334C23—22no—
335C23—22no—
336C23—28no—
337C23—432no—
338C23—416no—
339C23—44no—
340C23—44no—
341C23—48no—
342C23—48no—
343C23—88no—
344C23—88no—
345C23—816no—
346C23#52512yes—
347C23#53512yes—
348C23#53612yes—
349C23C23×D411yescharacteristic
350D4—216no—
351D4—432no—
352D4—432no—
353D4—416no—
354D4—48no—
355C2×C4—28no—
356C2×C4—216no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
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—48no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—48no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—832no—
383C2×C4—88no—
384C2×C4#52512yes—
385C2×C4#53512yes—
386C2×C4#53612yes—
387C2×C4#539116yes—
388C2×C4C2×AGL(1,ℤ/8)14yes—
389C2×C4C2×AGL(1,ℤ/8)14yes—
390C2×C4C2×AGL(1,ℤ/8)14yes—
391C2×C4#57514yes—
392C2×C4C22×#4112yes—
393C2×C4C2×H(5,𝔽2)12yes—
394C2×C4C2×2−1+412yes—
395C8—48no—
396C22—22no—
397C22—22no—
398C22—24no—
399C22—24no—
400C22—48no—
401C22—416no—
402C22—832no—
403C22—84no—
404C22—84no—
405C22—88no—
406C22#263314yes—
407C22C2×#52511yescharacteristic
408C22C2×#53511yescharacteristic
409C22C2×#53611yescharacteristic
410C22C22×AGL(1,ℤ/8)12yes—
411C22C2×#57512yes—
412C4—24no—
413C4—24no—
414C4—24no—
415C4—24no—
416C4—42no—
417C4—42no—
418C4—82no—
419C4—84no—
420C4—82no—
421C4—84no—
422C4#263314yes—
423C4C2×#53914yes—
424C2—24no—
425C2—88no—
426C2C2×#263311yescharacteristic
427C2#6055812yes—
428C2#6079811yescharacteristic
429C2#6080211yescharacteristic
430C2#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
1. #36640C2semidirect
3. #30077C2semidirect
4. #30076C2semidirect
5. #30075C2semidirect
6. #23731C2semidirect
7. #23658C2semidirect
8. #23628C2semidirect
10. #20873C2semidirect
11. #20772C2semidirect
18. #2864C22semidirect
19. #2859C22semidirect
20. #2859C22semidirect
37. #2094C22semidirect
38. #2093C22semidirect
39. #2092C22semidirect
40. #2084C22semidirect
41. #2083C22semidirect
42. #2073C22semidirect
43. #2071C22semidirect
46. #1864C22semidirect
47. #1862C22semidirect
48. #1839C22semidirect
49. #1839C22semidirect

Maximal subgroups

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

subgroupindex
#366402
C2×C4×#1272
#300772
#300762
#300752
#237312
#236582
#236282
#235052
#208732
#207722