Finite Group Databasebeta

← #359916

Subgroups — #359916

Order 512.

1658 conjugacy classes of subgroups.

The subgroup lattice has 363 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2317C211yesmaximal, characteristic
2C2×#2313C211yesmaximal, characteristic
3C2×#1820C212yesmaximal
4#14342C211yesmaximal, characteristic
5#11203C214yesmaximal
6#10912C214yesmaximal
7#10726C212yesmaximal
8C22×#117C2211yescharacteristic
9C22×#117C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11#2317—24no—
12#2317—24no—
13#2313—28no—
14C2×#389C414yescomplemented
15C2×#387—24no—
16C2×#385C2212yes—
17C2×#385C2212yes—
18C2×#384C412yescomplemented
19C2×#382C412yescomplemented
20C2×#381C2212yes—
21C2×#381C2212yes—
22C2×#380C2212yes—
23C2×C4×C22⋊C4C2212yes—
24#1820—28no—
25C2×#342C2218yes—
26C2×#342C2218yes—
27#1445—24no—
28C22⋊#95C2212yes—
29C22⋊#95C2212yes—
30C24×C4C2311yescharacteristic
31C24×C4C2311yescharacteristic
32C24×C4C2311yescharacteristic
33C2×#117—22no—
34C2×#117—22no—
35C2×#117—22no—
36C2×#117—22no—
37C2×#117—24no—
38C2×#117—24no—
39C2×#117—28no—
40C22×C4⋊C4—24no—
41C22×C4⋊C4—24no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C412yes—
46C22×C4⋊C4C2×C412yes—
47C22×C4⋊C4C2×C412yes—
48C22×C4⋊C4C2×C414yes—
49C22×C4⋊C4C2×C414yes—
50C22×C4⋊C4D414yes—
51C22×C4⋊C4D414yes—
52C22×C22⋊C4C2314yes—
53C22×C22⋊C4C2312yes—
54C22×C22⋊C4C2314yes—
55C22×C22⋊C4C2312yes—
56C22×C42—22no—
57C22×C42C2×C412yes—
58C22×C42C2×C412yes—
59#389—216no—
60#387—232no—
61#385—24no—
62#385—24no—
63#385—24no—
64#385—24no—
65#384—28no—
66#382—28no—
67#381—24no—
68#381—24no—
69#381—24no—
70#381D418yes—
71#380—28no—
72C4×C22⋊C4—28no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—24no—
76C2×#95—24no—
77C2×#95—24no—
78C2×#95—24no—
79C2×#95C2×C414yes—
80C2×#95C2×C414yes—
81#342—216no—
82#342—216no—
83C25C2411yescharacteristic
84C23×C4—24no—
85C23×C4—24no—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—24no—
91C23×C4—24no—
92C23×C4—82no—
93C23×C4—82no—
94C23×C4C22×C411yescharacteristic
95C23×C4C22×C411yescharacteristic
96C23×C4C22×C412yes—
97C23×C4C22×C412yes—
98C23×C4C22×C412yes—
99C23×C4C22×C412yes—
100C23×C4C22×C414yes—
101C23×C4C2×D412yes—
102C23×C4C2×D412yes—
103C23×C4C2×D414yes—
104C23×C4C2×D412yes—
105C23×C4C2×D412yes—
106C23×C4C2×D412yes—
107C23×C4C2×D412yes—
108C23×C4#4114yes—
109#117—216no—
110#117—216no—
111#117—44no—
112#117—48no—
113#117—44no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—28no—
116C2×C4⋊C4—28no—
117C2×C4⋊C4—28no—
118C2×C4⋊C4—28no—
119C2×C4⋊C4—216no—
120C2×C4⋊C4—28no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—28no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—28no—
128C2×C4⋊C4—84no—
129C2×C4⋊C4—84no—
130C2×C4⋊C4—84no—
131C2×C4⋊C4C22⋊C418yes—
132C2×C4⋊C4C22⋊C418yes—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—28no—
137C2×C22⋊C4—28no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—28no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—28no—
147C2×C22⋊C4C2×D418yes—
148C2×C22⋊C4#4118yes—
149C2×C42—24no—
150C2×C42—28no—
151C2×C42—24no—
152C2×C42—28no—
153C2×C42—28no—
154C2×C42—84no—
155#95—28no—
156#95—28no—
157#95—28no—
158#95—28no—
159#95—216no—
160#95—48no—
161#95—84no—
162#95—84no—
163C24—42no—
164C24—42no—
165C24C23×C411yescharacteristic
166C24C22×D411yescharacteristic
167C24C22×D411yescharacteristic
168C24C22×D412yes—
169C24C2×#4114yes—
170C24C2×#4112yes—
171C242−1+414yes—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—216no—
181C22×C4—216no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—28no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—44no—
198C22×C4—44no—
199C22×C4—44no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—48no—
206C22×C4—48no—
207C22×C4—44no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—48no—
212C22×C4—48no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4—84no—
217C22×C4—84no—
218C22×C4—84no—
219C22×C4—84no—
220C22×C4C2×C22⋊C414yes—
221C22×C4C2×C22⋊C414yes—
222C22×C4C2×C22⋊C412yes—
223C22×C4C2×C22⋊C412yes—
224C22×C4#11718yes—
225C22×C4C4×D414yes—
226C22×C4C4×D414yes—
227C22×C4#12014yes—
228C22×C4#12118yes—
229C22×C4#12314yes—
230C4⋊C4—216no—
231C4⋊C4—216no—
232C4⋊C4—216no—
233C4⋊C4—44no—
234C4⋊C4—44no—
235C4⋊C4—88no—
236C4⋊C4—88no—
237C4⋊C4—88no—
238C22⋊C4—28no—
239C22⋊C4—28no—
240C22⋊C4—216no—
241C22⋊C4—42no—
242C22⋊C4—44no—
243C22⋊C4—42no—
244C22⋊C4—48no—
245C22⋊C4—48no—
246C22⋊C4—48no—
247C42—216no—
248C42—48no—
249C42—88no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—48no—
260C23—42no—
261C23—48no—
262C23C22×C22⋊C411yescharacteristic
263C23C2×#11712yes—
264C23C2×C4×D412yes—
265C23#51912yes—
266C23C2×#12011yescharacteristic
267C23C2×#12112yes—
268C23C2×#12311yescharacteristic
269C23#52512yes—
270C23#52512yes—
271C23#53612yes—
272C23#53612yes—
273C23#54718yes—
274C23#55218yes—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—28no—
278C2×C4—28no—
279C2×C4—28no—
280C2×C4—216no—
281C2×C4—28no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—28no—
285C2×C4—28no—
286C2×C4—28no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—48no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—48no—
297C2×C4—48no—
298C2×C4—48no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—48no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—84no—
305C2×C4—84no—
306C2×C4—84no—
307C2×C4—84no—
308C2×C4—88no—
309C2×C4—88no—
310C2×C4—84no—
311C2×C4—84no—
312C2×C4—88no—
313C2×C4—88no—
314C2×C4—84no—
315C2×C4—84no—
316C2×C4—84no—
317C2×C4#38614yes—
318C2×C4#38614yes—
319C22—24no—
320C22—24no—
321C22—24no—
322C22—24no—
323C22—24no—
324C22—24no—
325C22—21no—
326C22—21no—
327C22—22no—
328C22—22no—
329C22—21no—
330C22—22no—
331C22—22no—
332C22—21no—
333C22—48no—
334C22—48no—
335C22#228011yescharacteristic
336C22C2×#38611yescharacteristic
337C22#231011yescharacteristic
338C22#234414yes—
339C22#240612yes—
340C22#241314yes—
341C22#248412yes—
342C22#288312yes—
343C22#288412yes—
344C4—24no—
345C4—24no—
346C4—48no—
347C4—44no—
348C4—44no—
349C4—88no—
350C4—82no—
351C4—82no—
352C4—164no—
353C2—22no—
354C2—24no—
355C2—24no—
356C2—22no—
357C2#1387011yescharacteristic
358C2#2072512yes—
359C2#2098511yescharacteristic
360C2#2098611yescharacteristic
361C2#2182812yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
14. C2×#389C4semidirect
18. C2×#384C4semidirect
19. C2×#382C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
C2×#23132
C2×#18202
#143422
#112032
#109122
#107262