Finite Group Databasebeta

← #6481751

Subgroups — #6481751

Order 512.

2130 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3101C212yesmaximal
2#35173C211yesmaximal, characteristic, complemented
3C2×#2354C211yesmaximal, characteristic
4#33108C214yesmaximal, complemented
5#29714C214yesmaximal
6#24680C218yesmaximal, complemented
7#24630C218yesmaximal, complemented
8#21951C211yesmaximal, characteristic, complemented
9#21860C212yesmaximal, complemented
10C2×#533C2212yes—
11C22×C4×D4C2211yescharacteristic
12C22×#117—21no—
13#3101—216no—
14C2×#483C2214yes—
15C2×#476C2214yes—
16C2×#461C2214yes—
17C2×#450C2214yes—
18C2×C4×SD16C2218yes—
19C2×#431C2212yes—
20#2857C2214yes—
21#2659—22no—
22#2584—24no—
23#2567C2214yescomplemented
24#2546C2214yescomplemented
25#2354—24no—
26#2354C2218yes—
27#2316—21no—
28#2316—21no—
29C2×#390C2212yes—
30C2×#388—22no—
31C2×#384C2211yescharacteristic
32C2×#380—21no—
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35#2071C2218yes—
36#2055C22116yes—
37#2013C2218yes—
38#1990C2214yescomplemented
39#1981C2218yescomplemented
40#1956C2214yescomplemented
41#1948C2218yescomplemented
42#1939C2218yescomplemented
43#1927C22116yescomplemented
44#1872C2218yes—
45#1803C2218yescomplemented
46#1797C2214yes—
47C23×D4C2311yescharacteristic
48C24×C4—21no—
49C22×SD16C2318yes—
50#533—28no—
51#533—28no—
52C2×C4⋊Q8C2312yes—
53C2×C4⋊Q8C2314yes—
54C2×C4⋊Q8C2312yes—
55C2×#125—22no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8C2314yes—
58C2×C4×Q8—22no—
59C2×C4×Q8C2314yes—
60C2×C4×D4—22no—
61C2×C4×D4—24no—
62C2×C4×D4D418yes—
63C2×C4×D4C2314yes—
64C2×#117—22no—
65C2×#117—28no—
66C2×#117—48no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C42—22no—
77C22×C42C2312yes—
78C22×C42C2311yescharacteristic
79#483—216no—
80#476—216no—
81#461—216no—
82#450—216no—
83C4×SD16—232no—
84#431—28no—
85C2×#106C2318yes—
86C2×C4⋊C8C2314yes—
87C2×C4⋊C8C2318yes—
88C2×#103C2318yes—
89C2×#103C23116yes—
90C2×#102C2318yes—
91C2×C22⋊C8C2314yes—
92C2×C4×C8C2314yes—
93#398—28no—
94#395—28no—
95#391C2318yes—
96#390—24no—
97#390C2318yes—
98#389—24no—
99#389—24no—
100#389—24no—
101#389—24no—
102#389—24no—
103#389—24no—
104#388—216no—
105#384—22no—
106#384C2314yes—
107#384C2314yes—
108#380—24no—
109#380—24no—
110#380—28no—
111C4×C4⋊C4—24no—
112C4×C4⋊C4D414yes—
113C4×C4⋊C4C2318yes—
114C4×C4⋊C4C2314yes—
115C2×#95—22no—
116C2×#95D412yes—
117#340C23116yes—
118#336C2318yes—
119C25—21no—
120C22×Q8—22no—
121C22×Q8C2414yes—
122C22×D4—21no—
123C22×D4—22no—
124C22×D4—42no—
125C22×D4—42no—
126C22×D4C2×D414yes—
127C22×D4C2412yes—
128C23×C4—21no—
129C23×C4—28no—
130C23×C4—24no—
131C23×C4—22no—
132C23×C4—44no—
133C23×C4—44no—
134C23×C4C2×D412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2411yescharacteristic
139C2×SD16—216no—
140C2×SD16—48no—
141C2×SD16—48no—
142C22×C8C2418yes—
143C4⋊Q8—24no—
144C4⋊Q8—24no—
145C4⋊Q8—24no—
146C4⋊Q8—24no—
147C4⋊Q8—48no—
148#125—48no—
149C22⋊Q8—28no—
150C22⋊Q8—28no—
151C22⋊Q8—48no—
152C4×Q8—28no—
153C4×Q8—28no—
154C4×Q8—48no—
155C4×D4—216no—
156C4×D4—28no—
157C4×D4C2×D4116yes—
158#117—48no—
159#117—432no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4SD1618yescomplemented
179C2×C4⋊C4C2×D414yes—
180C2×C4⋊C4C2×D414yes—
181C2×C4⋊C4C2×D414yes—
182C2×C4⋊C4C2414yes—
183C2×C4⋊C4C2414yes—
184C2×C4⋊C4C2412yes—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—48no—
191C2×C22⋊C4—84no—
192C2×C42—28no—
193C2×C42—24no—
194C2×C42—44no—
195C2×C42—48no—
196C2×C42C2×D414yes—
197C2×C42C2×D412yes—
198C2×C42C2×D414yes—
199C2×C42#4114yes—
200C2×C42C2414yes—
201C2×C42C2412yes—
202#106—216no—
203C4⋊C8—28no—
204C4⋊C8—48no—
205#103—216no—
206#103—416no—
207#102—216no—
208C22⋊C8—28no—
209C4×C8—28no—
210#95—28no—
211#95—28no—
212#95—28no—
213#95C2×D418yes—
214C24—24no—
215C24—22no—
216C24—42no—
217C24—42no—
218C24—42no—
219C24—42no—
220C24—42no—
221C24—42no—
222C24C22×D411yescharacteristic
223C2×Q8—24no—
224C2×Q8—24no—
225C2×Q8—28no—
226C2×Q8—28no—
227C2×Q8—44no—
228C2×Q8—82no—
229C2×Q8—82no—
230C2×D4—24no—
231C2×D4—22no—
232C2×D4—24no—
233C2×D4—24no—
234C2×D4—48no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—48no—
238C2×D4C22×D414yes—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—216no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—48no—
257C22×C4—48no—
258C22×C4—432no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4#12114yes—
263C22×C4#12114yes—
264C22×C4C2×SD1618yes—
265C22×C4C2×SD1614yes—
266C22×C4#13714yes—
267C22×C4C22×D414yes—
268C22×C4C22×D412yes—
269C22×C4C22×D411yescharacteristic
270C22×C4C22×D412yes—
271C22×C4C22×D414yes—
272C22×C4C22×D412yes—
273C22×C4C2×#4112yes—
274C22×C4C2511yescharacteristic
275SD16—432no—
276C2×C8—28no—
277C2×C8—28no—
278C2×C8—28no—
279C4⋊C4—24no—
280C4⋊C4—28no—
281C4⋊C4—22no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—28no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—48no—
289C4⋊C4—48no—
290C4⋊C4—44no—
291C4⋊C4—48no—
292C4⋊C4—84no—
293C4⋊C4—84no—
294C4⋊C4—84no—
295C4⋊C4—84no—
296C4⋊C4—84no—
297C4⋊C4C2×SD16116yes—
298C4⋊C4C22×D414yes—
299C22⋊C4—216no—
300C22⋊C4—48no—
301C22⋊C4—416no—
302C22⋊C4—88no—
303C42—28no—
304C42—44no—
305C42—48no—
306C42—416no—
307C42#12118yes—
308C42C22×D414yes—
309C42C2×#4114yes—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—24no—
315C23—48no—
316C23—48no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—48no—
322C23—42no—
323C23—42no—
324C23—84no—
325C23—84no—
326C23—84no—
327C23—84no—
328C23C2×#12112yes—
329C23C22×SD1612yes—
330C23C2×#13712yes—
331C23C23×D411yescharacteristic
332Q8—48no—
333Q8—48no—
334Q8—88no—
335D4—28no—
336D4—48no—
337D4—44no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—44no—
351C2×C4—48no—
352C2×C4—48no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—432no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—84no—
366C2×C4—88no—
367C2×C4#461116yes—
368C2×C4C2×#12112yes—
369C2×C4C2×#12114yes—
370C2×C4C2×#12114yes—
371C2×C4C22×SD1618yes—
372C2×C4C22×SD1614yes—
373C2×C4C2×#13714yes—
374C2×C4C23×D411yescharacteristic
375C2×C4C22×#4111yescharacteristic
376C2×C4#58512yes—
377C2×C4#58512yes—
378C8—48no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—44no—
386C22—44no—
387C22—44no—
388C22—816no—
389C22—82no—
390C22—82no—
391C22—82no—
392C22—82no—
393C22C2×#46114yes—
394C22C22×#12111yescharacteristic
395C22#348511yescharacteristic
396C22#348711yescharacteristic
397C22C23×SD1611yescharacteristic
398C22C22×#13711yescharacteristic
399C22#358511yescharacteristic
400C22#358611yescharacteristic
401C4—24no—
402C4—22no—
403C4—28no—
404C4—44no—
405C4—44no—
406C4—84no—
407C4—84no—
408C4—84no—
409C4C2×#46118yes—
410C4#344912yes—
411C4#349312yes—
412C2—24no—
413C2—84no—
414C2#3515011yescharacteristic
415C2C22×#46111yescharacteristic
416C2#6042511yescharacteristic
417C2#6169511yescharacteristic
418C2#6172111yescharacteristic
419C2#6188111yescharacteristic
420C2#6188911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35173C2semidirect
4. #33108C2semidirect
6. #24680C2semidirect
7. #24630C2semidirect
8. #21951C2semidirect
9. #21860C2semidirect
23. #2567C22semidirect
24. #2546C22semidirect
38. #1990C22semidirect
39. #1981C22semidirect
40. #1956C22semidirect
41. #1948C22semidirect
42. #1939C22semidirect
43. #1927C22semidirect
45. #1803C22semidirect
178. C2×C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#31012
#351732
C2×#23542
#331082
#297142
#246802
#246302
#219512
#218602