Finite Group Databasebeta

← #393459

Subgroups — #393459

Order 512.

1642 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2274C211yesmaximal, characteristic
2#20846C212yesmaximal
3C2×#1821C211yesmaximal, characteristic
4#13904C211yesmaximal, characteristic
5#10899C218yesmaximal, complemented
6#10733C212yesmaximal
7C22×#117C2212yes—
8C23×C4⋊C4C2211yescharacteristic
9C2×#428C2214yes—
10C2×#407C2212yes—
11C2×#381—22no—
12C2×#380C2211yescharacteristic
13#2274—24no—
14#2274—22no—
15#2274C2214yes—
16C2×C4×C4⋊C4—22no—
17C2×C4×C4⋊C4—22no—
18C2×C4×C22⋊C4—22no—
19C2×C4×C22⋊C4C2211yescharacteristic
20#1854—24no—
21#1853—22no—
22#1852—22no—
23#1821C418yescomplemented
24C2×#343C2214yes—
25#1736C2214yes—
26C2×#339C2218yes—
27#1445—22no—
28C4⋊#95C2214yes—
29C24×C4D412yes—
30C24×C4C2311yescharacteristic
31C22×M16C2314yes—
32C2×#117—22no—
33C2×#117—22no—
34C2×#117—216no—
35C2×#117D414yes—
36C2×#117C2314yes—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—24no—
40C22×C4⋊C4C2×C414yes—
41C22×C4⋊C4D412yes—
42C22×C4⋊C4D412yes—
43C22×C4⋊C4D412yes—
44C22×C4⋊C4Q814yes—
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2312yes—
47C22×C22⋊C4—22no—
48C22×C22⋊C4—21no—
49C22×C22⋊C4—21no—
50C22×C42—22no—
51C22×C42—22no—
52C22×C42—22no—
53C22×C42C2312yes—
54#428—216no—
55C2×#107—24no—
56C2×#106—24no—
57#407—24no—
58#407C2×C418yes—
59C2×C22⋊C8—22no—
60C2×C22⋊C8—22no—
61#381—48no—
62#380—22no—
63#380—22no—
64C4×C4⋊C4—28no—
65C4×C4⋊C4—28no—
66C4×C4⋊C4—28no—
67C4×C4⋊C4—28no—
68C4×C22⋊C4—28no—
69C4×C22⋊C4—28no—
70C4×C22⋊C4—24no—
71C2×#95—24no—
72C2×#95—28no—
73C2×#95—21no—
74C2×#95—21no—
75#343C2×C4116yescomplemented
76#340—24no—
77#340—24no—
78#339—216no—
79C25C2×D411yescharacteristic
80C23×C4—28no—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4—22no—
84C23×C4—22no—
85C23×C4—24no—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4—44no—
89C23×C4—44no—
90C23×C4C4⋊C418yes—
91C23×C4C22×C412yes—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D412yes—
94C23×C4C2×D412yes—
95C23×C4C2×D414yes—
96C23×C4C2×Q812yes—
97C23×C4#4114yes—
98C23×C4C2411yescharacteristic
99C2×M16—24no—
100C2×M16—28no—
101C2×M16—28no—
102C2×M16C22×C418yes—
103C22×C8—24no—
104#117—216no—
105#117—216no—
106#117—28no—
107#117—28no—
108#117—28no—
109#117#41116yes—
110C2×C4⋊C4—28no—
111C2×C4⋊C4—28no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—28no—
116C2×C4⋊C4—28no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—28no—
121C2×C4⋊C4—28no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—28no—
125C2×C4⋊C4C4⋊C418yes—
126C2×C4⋊C4C22×C414yes—
127C2×C4⋊C4C2×D414yes—
128C2×C4⋊C4C2×D414yes—
129C2×C4⋊C4C2×Q814yes—
130C2×C22⋊C4—216no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—42no—
135C2×C22⋊C4—42no—
136C2×C22⋊C4—44no—
137C2×C22⋊C4—41no—
138C2×C22⋊C4—41no—
139C2×C42—216no—
140C2×C42—24no—
141C2×C42—24no—
142C2×C42—28no—
143C2×C42—24no—
144C2×C42—28no—
145C2×C42—28no—
146C2×C42—84no—
147C2×C42—84no—
148#107—48no—
149#106—48no—
150C22⋊C8—28no—
151C22⋊C8—28no—
152#95—216no—
153#95—416no—
154#95—42no—
155#95—42no—
156C24—24no—
157C24—22no—
158C24—22no—
159C24—42no—
160C24C2×C4⋊C412yes—
161C24#12011yescharacteristic
162C24#12011yescharacteristic
163C24#12312yes—
164C24C22×D411yescharacteristic
165C22×C4—24no—
166C22×C4—28no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—216no—
172C22×C4—28no—
173C22×C4—28no—
174C22×C4—28no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—28no—
179C22×C4—24no—
180C22×C4—28no—
181C22×C4—24no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4—416no—
190C22×C4—44no—
191C22×C4—42no—
192C22×C4—44no—
193C22×C4—42no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—44no—
197C22×C4—42no—
198C22×C4—44no—
199C22×C4—88no—
200C22×C4—82no—
201C22×C4—82no—
202C22×C4C2×C4⋊C412yes—
203C22×C4C2×C4⋊C418yes—
204C22×C4C4×D414yes—
205C22×C4C4×D414yes—
206C22×C4#12012yes—
207C22×C4C22⋊Q814yes—
208C22×C4C22⋊Q814yes—
209C22×C4#12312yes—
210C22×C4C23×C411yescharacteristic
211C22×C4C22×D412yes—
212C22×C4C22×Q811yescharacteristic
213C22×C4C2×#4112yes—
214C22×C4C2×#4118yes—
215M16—216no—
216M16—48no—
217C2×C8—28no—
218C2×C8—44no—
219C2×C8—44no—
220C4⋊C4—28no—
221C4⋊C4—28no—
222C4⋊C4—28no—
223C4⋊C4—28no—
224C4⋊C4—24no—
225C4⋊C4—24no—
226C4⋊C4—216no—
227C4⋊C4—44no—
228C4⋊C4—44no—
229C4⋊C4#12318yes—
230C4⋊C4#12318yes—
231C22⋊C4—216no—
232C22⋊C4—216no—
233C22⋊C4—48no—
234C22⋊C4—82no—
235C22⋊C4—82no—
236C42—216no—
237C42—216no—
238C42—48no—
239C42—88no—
240C42—88no—
241C23—22no—
242C23—21no—
243C23—21no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—24no—
248C23—24no—
249C23—24no—
250C23—21no—
251C23—21no—
252C23—22no—
253C23—24no—
254C23—42no—
255C23—42no—
256C23—48no—
257C23—48no—
258C23#38512yes—
259C23#38512yes—
260C23#45312yes—
261C23#45512yes—
262C23C22×C4⋊C411yescharacteristic
263C23C2×#12011yescharacteristic
264C23C2×#12311yescharacteristic
265C23#52514yes—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—28no—
271C2×C4—28no—
272C2×C4—28no—
273C2×C4—28no—
274C2×C4—28no—
275C2×C4—28no—
276C2×C4—28no—
277C2×C4—416no—
278C2×C4—416no—
279C2×C4—416no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—48no—
287C2×C4—44no—
288C2×C4—48no—
289C2×C4—84no—
290C2×C4—88no—
291C2×C4—82no—
292C2×C4—84no—
293C2×C4—82no—
294C2×C4—88no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4#38514yes—
299C2×C4C2×C4×D412yes—
300C2×C4C4×#4114yes—
301C2×C4C2×C22⋊Q812yes—
302C2×C4C2×#12314yes—
303C2×C4#52514yes—
304C2×C4#52918yes—
305C2×C4#53314yes—
306C8—48no—
307C22—22no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—22no—
313C22—24no—
314C22—24no—
315C22—24no—
316C22—24no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—48no—
321C22—48no—
322C22—42no—
323C22—42no—
324C22#185614yes—
325C22C2×#38511yescharacteristic
326C22#229612yes—
327C22#284912yes—
328C22#285312yes—
329C22C2×#45311yescharacteristic
330C22C2×#45511yescharacteristic
331C22#302012yes—
332C4—24no—
333C4—48no—
334C4—48no—
335C4—48no—
336C4—84no—
337C4—162no—
338C4C4×#12314yes—
339C4#285514yes—
340C2—22no—
341C2—24no—
342C2—22no—
343C2—42no—
344C2#1470511yescharacteristic
345C2C2×#185611yescharacteristic
346C2#2085711yescharacteristic
347C2#3299612yes—
348C2#3299912yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10899C2semidirect
23. #1821C4semidirect
75. #343C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22742
#208462
C2×#18212
#139042
#108992
#107332