Finite Group Databasebeta

← #6629632

Subgroups — #6629632

Order 512.

2173 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36626C211yesmaximal, characteristic
2C2×C4×#117C211yesmaximal, characteristic, complemented
3#33101C218yesmaximal, complemented
4#33097C218yesmaximal, complemented
5#21059C214yesmaximal, complemented
6#21033C214yesmaximal, complemented
7#20910C212yesmaximal
8#20845C212yesmaximal, complemented
9#20682C211yesmaximal, characteristic, complemented
10C2×#525C2211yescharacteristic
11C22×#117C2212yes—
12C23×C42C2211yescharacteristic
13C2×#428C2214yes—
14C2×#417C2214yes—
15C2×#407C2212yes—
16#2855C2214yes—
17#2854C2214yes—
18#2853—21no—
19#2853—21no—
20#2849—22no—
21C4×#123—24no—
22#2297C2212yes—
23#2296C2212yes—
24#2280C2211yescharacteristic
25C4×#117—24no—
26C4×#117C2218yes—
27C2×C4×C4⋊C4C2214yes—
28C2×C4×C22⋊C4—22no—
29C2×#376—21no—
30C2×C43—21no—
31#1928C2218yescomplemented
32#1927C2218yescomplemented
33#1926C2218yescomplemented
34#1925C2218yescomplemented
35#1921C2218yescomplemented
36#1920C2218yescomplemented
37#1919C2218yescomplemented
38#1918C2218yescomplemented
39#1875C2218yes—
40#1873C2218yes—
41#1854C22116yes—
42#1817C2218yes—
43#1736C2214yes—
44#1736C2214yes—
45#1728C2214yes—
46C22×#41C2311yescharacteristic
47C24×C4D412yes—
48C24×C4C2311yescharacteristic
49C22×M16C2314yes—
50#525—28no—
51C2×#123—24no—
52C2×#123—21no—
53C2×C22⋊Q8C2312yes—
54C2×#121C2312yes—
55C2×#120—21no—
56C2×C4×D4—22no—
57C2×#117—24no—
58C2×#117—216no—
59C2×#117C2318yes—
60C2×#117C2312yes—
61C2×#117C2311yescharacteristic
62C22×C4⋊C4C2314yes—
63C22×C22⋊C4—22no—
64C22×C42—22no—
65C22×C42—28no—
66C22×C42—24no—
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D414yes—
70C22×C42C2314yes—
71C22×C42C2312yes—
72#428—216no—
73C2×#107C2318yes—
74C2×#106C2318yes—
75#417—216no—
76C2×#103C2318yes—
77C2×#102C2318yes—
78#407—28no—
79C2×C22⋊C8C2318yes—
80#391C2314yes—
81#390C2314yes—
82#389C2314yes—
83#389C2314yes—
84#388—24no—
85#388—24no—
86#387—24no—
87#387—24no—
88#386—22no—
89#386—22no—
90#386—24no—
91#385—22no—
92#385—22no—
93#385—24no—
94#381—24no—
95#381—21no—
96#380C2314yes—
97#379—21no—
98C4×C4⋊C4—24no—
99C4×C4⋊C4—24no—
100C4×C4⋊C4—24no—
101C4×C4⋊C4C2318yes—
102C4×C4⋊C4C2318yes—
103C4×C22⋊C4—28no—
104C4×C22⋊C4—28no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—24no—
107#376—28no—
108C2×#95—24no—
109C43—28no—
110#340C2318yes—
111#340C2318yes—
112#340C23116yes—
113#336C2318yes—
114#336C2318yes—
115C25C2×D411yescharacteristic
116C2×#41—24no—
117C2×#41—28no—
118C22×Q8C2411yescharacteristic
119C22×D4—22no—
120C22×D4—21no—
121C22×D4C2411yescharacteristic
122C23×C4—28no—
123C23×C4—28no—
124C23×C4—24no—
125C23×C4—21no—
126C23×C4—48no—
127C23×C4C2×D412yes—
128C23×C4C2×D414yes—
129C23×C4C2×D414yes—
130C23×C4C2×D412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D412yes—
133C23×C4#4118yes—
134C23×C4C2412yes—
135C23×C4C2411yescharacteristic
136C2×M16—28no—
137C2×M16—28no—
138C2×M16—28no—
139C22×C8C2418yes—
140#123—416no—
141#123—44no—
142C22⋊Q8—28no—
143#121—28no—
144#120—44no—
145C4×D4—48no—
146#117—232no—
147#117—232no—
148#117—24no—
149#117—28no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4#41116yes—
156C2×C4⋊C4C2412yes—
157C2×C4⋊C4C2414yes—
158C2×C4⋊C4C2418yes—
159C2×C22⋊C4—216no—
160C2×C22⋊C4—28no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C42—216no—
169C2×C42—24no—
170C2×C42—216no—
171C2×C42—24no—
172C2×C42—28no—
173C2×C42—28no—
174C2×C42—216no—
175C2×C42—21no—
176C2×C42—22no—
177C2×C42—28no—
178C2×C42—28no—
179C2×C42—416no—
180C2×C42C2×D414yes—
181C2×C42C2×D414yes—
182C2×C42C2×D414yes—
183C2×C42C2414yes—
184#107—216no—
185#106—216no—
186#103—216no—
187#102—216no—
188C22⋊C8—216no—
189#95—28no—
190#95—28no—
191#95—24no—
192#95—24no—
193#95—24no—
194#95—24no—
195#95—24no—
196C24—24no—
197C24—21no—
198C24—44no—
199C24#12314yes—
200C24C22×D412yes—
201C24C22×D411yescharacteristic
202#41—416no—
203C2×Q8—24no—
204C2×Q8—22no—
205C2×D4—24no—
206C2×D4—22no—
207C2×D4—48no—
208C2×D4—44no—
209C2×D4—42no—
210C2×D4—44no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—28no—
214C22×C4—232no—
215C22×C4—28no—
216C22×C4—28no—
217C22×C4—28no—
218C22×C4—28no—
219C22×C4—24no—
220C22×C4—216no—
221C22×C4—28no—
222C22×C4—28no—
223C22×C4—28no—
224C22×C4—21no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—432no—
231C22×C4—416no—
232C22×C4—42no—
233C22×C4—44no—
234C22×C4#12318yes—
235C22×C4#12314yes—
236C22×C4C22×D414yes—
237C22×C4C22×D412yes—
238C22×C4C22×D414yes—
239C22×C4C22×D411yescharacteristic
240C22×C4C2×#4118yes—
241C22×C4C2×#4114yes—
242C22×C4C2×#4118yes—
243C22×C4C2511yescharacteristic
244M16—416no—
245C2×C8—28no—
246C2×C8—28no—
247C2×C8—28no—
248C4⋊C4—216no—
249C4⋊C4—28no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—48no—
254C4⋊C4—44no—
255C4⋊C4—48no—
256C4⋊C4C2×#41116yes—
257C22⋊C4—232no—
258C22⋊C4—48no—
259C22⋊C4—48no—
260C22⋊C4—42no—
261C22⋊C4—44no—
262C22⋊C4—44no—
263C22⋊C4—44no—
264C22⋊C4—48no—
265C42—232no—
266C42—28no—
267C42—28no—
268C42—28no—
269C42—28no—
270C42—416no—
271C42—42no—
272C42—44no—
273C42—416no—
274C23—24no—
275C23—28no—
276C23—24no—
277C23—24no—
278C23—24no—
279C23—48no—
280C23—416no—
281C23—41no—
282C23—42no—
283C23—44no—
284C23C2×#12314yes—
285C23C2×#12312yes—
286C23#52514yes—
287C23C23×D411yescharacteristic
288Q8—44no—
289D4—44no—
290D4—88no—
291D4—84no—
292C2×C4—216no—
293C2×C4—28no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—216no—
298C2×C4—216no—
299C2×C4—24no—
300C2×C4—216no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—416no—
304C2×C4—416no—
305C2×C4—432no—
306C2×C4—48no—
307C2×C4—416no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—44no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—84no—
316C2×C4C2×#12314yes—
317C2×C4C2×#12312yes—
318C2×C4#52514yes—
319C2×C4#529116yes—
320C2×C4#57518yes—
321C2×C4C22×#4112yes—
322C2×C4C22×#4114yes—
323C8—48no—
324C22—24no—
325C22—28no—
326C22—24no—
327C22—24no—
328C22—24no—
329C22—416no—
330C22—48no—
331C22—81no—
332C22—82no—
333C22—84no—
334C22#285314yes—
335C22C22×#12311yescharacteristic
336C22C2×#52512yes—
337C22C2×#57514yes—
338C4—24no—
339C4—28no—
340C4—44no—
341C4—416no—
342C4—81no—
343C4—82no—
344C4—84no—
345C4#285314yes—
346C4C2×#52914yes—
347C2—24no—
348C2—44no—
349C2—161no—
350C2C2×#285311yescharacteristic
351C2#6051112yes—
352C2#6055914yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
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
2. C2×C4×#117C2semidirect
3. #33101C2semidirect
4. #33097C2semidirect
5. #21059C2semidirect
6. #21033C2semidirect
8. #20845C2semidirect
9. #20682C2semidirect
31. #1928C22semidirect
32. #1927C22semidirect
33. #1926C22semidirect
34. #1925C22semidirect
35. #1921C22semidirect
36. #1920C22semidirect
37. #1919C22semidirect
38. #1918C22semidirect

Maximal subgroups

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

subgroupindex
#366262
C2×C4×#1172
#331012
#330972
#210592
#210332
#209102
#208452
#206822