Finite Group Databasebeta

← #393461

Subgroups — #393461

Order 512.

1614 conjugacy classes of subgroups.

The subgroup lattice has 342 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#13924C211yesmaximal, characteristic
5#10899C218yesmaximal, complemented
6#10733C212yesmaximal
7C22×#117C2212yes—
8C23×C4⋊C4C2211yescharacteristic
9C2×#428C2214yes—
10C2×#407C2212yes—
11C2×#380C2211yescharacteristic
12#2274—24no—
13#2274—22no—
14#2274C2214yes—
15C2×C4×C4⋊C4—22no—
16C2×C4×C4⋊C4—22no—
17C2×C4×C22⋊C4—22no—
18C2×C4×C22⋊C4C2211yescharacteristic
19#1854—24no—
20#1853—22no—
21#1852—22no—
22#1821C418yescomplemented
23C2×#343C2214yes—
24#1736C2214yes—
25C2×#339C2218yes—
26#1445—22no—
27C4⋊#95C2214yes—
28C22⋊#95—22no—
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—24no—
52C22×C42C2312yes—
53#428—216no—
54C2×#107—24no—
55C2×#106—24no—
56#407—24no—
57#407C2×C418yes—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—22no—
60#380—22no—
61#380—22no—
62C4×C4⋊C4—28no—
63C4×C4⋊C4—28no—
64C4×C4⋊C4—28no—
65C4×C4⋊C4—28no—
66C4×C22⋊C4—28no—
67C4×C22⋊C4—28no—
68C4×C22⋊C4—24no—
69C2×#95—22no—
70C2×#95—22no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—28no—
74#343C2×C4116yescomplemented
75#340—24no—
76#340—24no—
77#339—216no—
78C25C2×D411yescharacteristic
79C23×C4—28no—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4—22no—
84C23×C4—24no—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4—84no—
88C23×C4C4⋊C418yes—
89C23×C4C22×C412yes—
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D412yes—
92C23×C4C2×D412yes—
93C23×C4C2×D414yes—
94C23×C4C2×Q812yes—
95C23×C4#4114yes—
96C23×C4C2411yescharacteristic
97C2×M16—24no—
98C2×M16—28no—
99C2×M16—28no—
100C2×M16C22×C418yes—
101C22×C8—24no—
102#117—216no—
103#117—216no—
104#117—28no—
105#117—28no—
106#117—28no—
107#117#41116yes—
108C2×C4⋊C4—28no—
109C2×C4⋊C4—28no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—28no—
114C2×C4⋊C4—28no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—28no—
119C2×C4⋊C4—28no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—28no—
123C2×C4⋊C4C4⋊C418yes—
124C2×C4⋊C4C22×C414yes—
125C2×C4⋊C4C2×D414yes—
126C2×C4⋊C4C2×D414yes—
127C2×C4⋊C4C2×Q814yes—
128C2×C22⋊C4—216no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4—42no—
134C2×C22⋊C4—44no—
135C2×C22⋊C4—41no—
136C2×C22⋊C4—41no—
137C2×C42—216no—
138C2×C42—24no—
139C2×C42—24no—
140C2×C42—28no—
141C2×C42—24no—
142C2×C42—216no—
143C2×C42—88no—
144#107—48no—
145#106—48no—
146C22⋊C8—28no—
147C22⋊C8—28no—
148#95—28no—
149#95—28no—
150#95—42no—
151#95—42no—
152#95—88no—
153C24—24no—
154C24—22no—
155C24—22no—
156C24—42no—
157C24C2×C4⋊C412yes—
158C24#12011yescharacteristic
159C24#12011yescharacteristic
160C24#12312yes—
161C24C22×D411yescharacteristic
162C22×C4—24no—
163C22×C4—28no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—216no—
169C22×C4—28no—
170C22×C4—28no—
171C22×C4—28no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—28no—
176C22×C4—24no—
177C22×C4—28no—
178C22×C4—24no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—48no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—48no—
185C22×C4—44no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—42no—
192C22×C4—44no—
193C22×C4—88no—
194C22×C4—84no—
195C22×C4—88no—
196C22×C4C2×C4⋊C412yes—
197C22×C4C2×C4⋊C418yes—
198C22×C4C4×D414yes—
199C22×C4C4×D414yes—
200C22×C4#12012yes—
201C22×C4C22⋊Q814yes—
202C22×C4C22⋊Q814yes—
203C22×C4#12312yes—
204C22×C4C23×C411yescharacteristic
205C22×C4C22×D412yes—
206C22×C4C22×Q811yescharacteristic
207C22×C4C2×#4112yes—
208C22×C4C2×#4118yes—
209M16—216no—
210M16—48no—
211C2×C8—28no—
212C2×C8—44no—
213C2×C8—44no—
214C4⋊C4—28no—
215C4⋊C4—28no—
216C4⋊C4—28no—
217C4⋊C4—28no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—216no—
221C4⋊C4—44no—
222C4⋊C4—44no—
223C4⋊C4#12318yes—
224C4⋊C4#12318yes—
225C22⋊C4—216no—
226C22⋊C4—216no—
227C22⋊C4—48no—
228C22⋊C4—82no—
229C22⋊C4—82no—
230C42—216no—
231C42—216no—
232C42—48no—
233C42—816no—
234C23—22no—
235C23—21no—
236C23—21no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—24no—
241C23—24no—
242C23—24no—
243C23—21no—
244C23—21no—
245C23—22no—
246C23—24no—
247C23—42no—
248C23—42no—
249C23—48no—
250C23—48no—
251C23#38512yes—
252C23#38512yes—
253C23#45412yes—
254C23#45612yes—
255C23C22×C4⋊C411yescharacteristic
256C23C2×#12011yescharacteristic
257C23C2×#12311yescharacteristic
258C23#52514yes—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—28no—
264C2×C4—28no—
265C2×C4—28no—
266C2×C4—28no—
267C2×C4—28no—
268C2×C4—28no—
269C2×C4—28no—
270C2×C4—416no—
271C2×C4—416no—
272C2×C4—416no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—48no—
278C2×C4—48no—
279C2×C4—44no—
280C2×C4—48no—
281C2×C4—84no—
282C2×C4—88no—
283C2×C4—82no—
284C2×C4—84no—
285C2×C4—82no—
286C2×C4—88no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4#38514yes—
291C2×C4C2×C4×D412yes—
292C2×C4C4×#4114yes—
293C2×C4C2×C22⋊Q812yes—
294C2×C4C2×#12314yes—
295C2×C4#52514yes—
296C2×C4#52918yes—
297C2×C4#53314yes—
298C8—48no—
299C22—22no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—22no—
304C22—21no—
305C22—24no—
306C22—24no—
307C22—24no—
308C22—24no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—48no—
313C22—48no—
314C22—42no—
315C22—42no—
316C22#185714yes—
317C22C2×#38511yescharacteristic
318C22#229612yes—
319C22#284912yes—
320C22#285312yes—
321C22C2×#45411yescharacteristic
322C22C2×#45611yescharacteristic
323C22#301912yes—
324C4—24no—
325C4—48no—
326C4—48no—
327C4—48no—
328C4—84no—
329C4—162no—
330C4C4×#12314yes—
331C4#285514yes—
332C2—22no—
333C2—24no—
334C2—22no—
335C2—42no—
336C2#1470511yescharacteristic
337C2C2×#185711yescharacteristic
338C2#2085611yescharacteristic
339C2#3299712yes—
340C2#3300012yes—
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
22. #1821C4semidirect
74. #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
#139242
#108992
#107332