Finite Group Databasebeta

← #238542

Subgroups — #238542

Order 512.

1070 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35173C211yesmaximal, characteristic
2#21860C212yesmaximal
3C2×#1503C212yesmaximal
4#13911C211yesmaximal, characteristic
5#10407C218yesmaximal
6#7908C211yesmaximal, characteristic
7C2×#533C2212yes—
8C2×#431C2212yes—
9#2857—22no—
10#2659C2214yes—
11#2584—24no—
12#2567—22no—
13#2546—22no—
14#2316—21no—
15#2316—21no—
16C2×#384C2211yescharacteristic
17C2×#382—21no—
18C2×C4×C4⋊C4C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20#1990—22no—
21#1956—22no—
22#1872C2218yes—
23#1803—24no—
24#1503—28no—
25C2×#328C2218yes—
26#1442C2214yes—
27C4⋊#95—22no—
28C4⋊#95—21no—
29#1280C2214yes—
30#533—24no—
31#533—24no—
32#533—24no—
33#533—24no—
34C2×C4⋊Q8—22no—
35C2×C4⋊Q8D412yes—
36C2×C4⋊Q8D412yes—
37C2×#125—22no—
38C2×C22⋊Q8—22no—
39C2×C22⋊Q8C2314yes—
40C2×C4×Q8—22no—
41C2×C4×Q8D414yes—
42C2×#117—22no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4D411yescharacteristic
47C22×C4⋊C4D411yescharacteristic
48C22×C4⋊C4C2312yes—
49C22×C42—21no—
50C22×C42D412yes—
51C22×C42C2311yescharacteristic
52#431—24no—
53#431—24no—
54C2×C4⋊C8—22no—
55C2×#103—28no—
56C2×C22⋊C8C2314yes—
57C2×C4×C8—22no—
58#398—28no—
59#395—28no—
60#391—24no—
61#389—24no—
62#389—24no—
63#389—24no—
64#389—24no—
65#389—24no—
66#389—24no—
67#384—22no—
68#384—22no—
69#384—22no—
70#382—44no—
71#380—24no—
72#380—24no—
73C4×C4⋊C4—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4D414yes—
77C2×#95—24no—
78C2×#95—24no—
79C2×#95C2312yes—
80C2×#95C2312yes—
81#340—28no—
82#328—216no—
83C22×Q8—22no—
84C22×Q8C2×D414yes—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4C2×D411yescharacteristic
88C23×C4C2×D412yes—
89C23×C4C2×D412yes—
90C23×C4C2×D411yescharacteristic
91C23×C4C2411yescharacteristic
92C22×C8—24no—
93C4⋊Q8—24no—
94C4⋊Q8—24no—
95C4⋊Q8—24no—
96C4⋊Q8—24no—
97C4⋊Q8—84no—
98#125—44no—
99#125—44no—
100C22⋊Q8—24no—
101C22⋊Q8—24no—
102C22⋊Q8—24no—
103C22⋊Q8—24no—
104C22⋊Q8—44no—
105C22⋊Q8—44no—
106C4×Q8—28no—
107C4×Q8—28no—
108C4×Q8—44no—
109C4×Q8—44no—
110#117—44no—
111#117—44no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—44no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—82no—
136C2×C4⋊C4SD1614yes—
137C2×C4⋊C4Q1614yes—
138C2×C4⋊C4C2×D414yes—
139C2×C22⋊C4—24no—
140C2×C42—24no—
141C2×C42—22no—
142C2×C42—24no—
143C2×C42—22no—
144C2×C42—42no—
145C2×C42—42no—
146C2×C42—44no—
147C2×C42C2×D412yes—
148C2×C42C2×D412yes—
149C2×C42C2×D412yes—
150C2×C42#4114yes—
151C2×C42#4112yes—
152C4⋊C8—44no—
153#103—88no—
154C22⋊C8—24no—
155C22⋊C8—24no—
156C4×C8—44no—
157#95—28no—
158#95—28no—
159#95—24no—
160#95—22no—
161#95—48no—
162#95—84no—
163#95C2×D414yes—
164C24C22×D411yescharacteristic
165C2×Q8—24no—
166C2×Q8—24no—
167C2×Q8—44no—
168C2×Q8—44no—
169C2×Q8—44no—
170C2×Q8—44no—
171C2×Q8—44no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4#12011yescharacteristic
199C22×C4#12012yes—
200C22×C4#12011yescharacteristic
201C22×C4#12112yes—
202C22×C4#12112yes—
203C22×C4#12412yes—
204C22×C4#12412yes—
205C22×C4C2×SD1614yes—
206C22×C4C2×SD1612yes—
207C22×C4C2×Q1614yes—
208C22×C4C2×Q1612yes—
209C22×C4#13712yes—
210C22×C4#13712yes—
211C22×C4C22×D412yes—
212C22×C4C22×D411yescharacteristic
213C22×C4C2×#4112yes—
214C22×C4C2×#4111yescharacteristic
215C2×C8—44no—
216C2×C8—44no—
217C2×C8—44no—
218C4⋊C4—28no—
219C4⋊C4—28no—
220C4⋊C4—24no—
221C4⋊C4—24no—
222C4⋊C4—48no—
223C4⋊C4—48no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C4⋊C4—44no—
227C4⋊C4—44no—
228C4⋊C4—48no—
229C4⋊C4—44no—
230C4⋊C4—82no—
231C4⋊C4—84no—
232C4⋊C4—84no—
233C4⋊C4—84no—
234C4⋊C4—84no—
235C4⋊C4—84no—
236C4⋊C4—82no—
237C22⋊C4—44no—
238C22⋊C4—44no—
239C42—48no—
240C42—44no—
241C42—48no—
242C42—82no—
243C42#12114yes—
244C42#12114yes—
245C42#12314yes—
246C42#12314yes—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23C2×#12011yescharacteristic
252C23C2×#12111yescharacteristic
253C23C2×#12411yescharacteristic
254C23C22×SD1611yescharacteristic
255C23C22×Q1611yescharacteristic
256C23C2×#13711yescharacteristic
257C23C2×#13711yescharacteristic
258Q8—48no—
259Q8—84no—
260Q8—88no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—84no—
294C2×C4—84no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4#39412yes—
299C2×C4#39412yes—
300C2×C4#39414yes—
301C2×C4#44814yes—
302C2×C4#45114yes—
303C2×C4#45412yes—
304C2×C4#45512yes—
305C2×C4#46114yes—
306C2×C4#46214yes—
307C2×C4#48714yes—
308C2×C4#49114yes—
309C2×C4C2×#12112yes—
310C2×C4C2×#12312yes—
311C8—84no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22C2×#39411yescharacteristic
325C22C2×#44811yescharacteristic
326C22C2×#45111yescharacteristic
327C22C2×#45511yescharacteristic
328C22C2×#45411yescharacteristic
329C22#302711yescharacteristic
330C22#302811yescharacteristic
331C22C2×#46111yescharacteristic
332C22C2×#46211yescharacteristic
333C22C2×#48711yescharacteristic
334C22C2×#49111yescharacteristic
335C4—22no—
336C4—22no—
337C4—22no—
338C4—44no—
339C4—44no—
340C4—84no—
341C4—84no—
342C4—84no—
343C4—162no—
344C4#204014yes—
345C4#204318yes—
346C2—21no—
347C2—21no—
348C2—21no—
349C2—21no—
350C2#1839011yescharacteristic
351C2#1839611yescharacteristic
352C2#1840311yescharacteristic
353C2#1841111yescharacteristic
354C2C2×#204011yescharacteristic
355C2C2×#204311yescharacteristic
356C2#2351611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#351732
#218602
C2×#15032
#139112
#104072
#79082