Finite Group Databasebeta

← #238539

Subgroups — #238539

Order 512.

1142 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35173C211yesmaximal, characteristic
2#21860C212yesmaximal
3C2×#1503C212yesmaximal
4#13910C211yesmaximal, characteristic
5#10406C218yesmaximal
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×#389—22no—
17C2×#384—21no—
18C2×#384C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21#1990—22no—
22#1956—22no—
23#1872C2218yes—
24#1803—24no—
25#1503—28no—
26C2×#328C2218yes—
27#1444C2214yes—
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⋊C4—24no—
47C22×C4⋊C4D411yescharacteristic
48C22×C4⋊C4D411yescharacteristic
49C22×C4⋊C4C2312yes—
50C22×C42—21no—
51C22×C42D412yes—
52C22×C42C2311yescharacteristic
53#431—24no—
54#431—24no—
55C2×C4⋊C8—22no—
56C2×#103—28no—
57C2×C22⋊C8C2314yes—
58C2×C4×C8—22no—
59#398—28no—
60#395—28no—
61#391—24no—
62#389—24no—
63#389—24no—
64#389—24no—
65#389—24no—
66#389—24no—
67#389—24no—
68#389—48no—
69#384—22no—
70#384—22no—
71#384—22no—
72#384—44no—
73#380—24no—
74#380—24no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4D414yes—
79C2×#95—24no—
80C2×#95C2312yes—
81C2×#95C2312yes—
82#340—28no—
83#328—216no—
84C22×Q8—22no—
85C22×Q8C2×D414yes—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4C2×D411yescharacteristic
89C23×C4C2×D412yes—
90C23×C4C2×D412yes—
91C23×C4C2×D411yescharacteristic
92C23×C4C2411yescharacteristic
93C22×C8—24no—
94C4⋊Q8—24no—
95C4⋊Q8—24no—
96C4⋊Q8—24no—
97C4⋊Q8—24no—
98C4⋊Q8—84no—
99#125—44no—
100#125—44no—
101C22⋊Q8—24no—
102C22⋊Q8—24no—
103C22⋊Q8—24no—
104C22⋊Q8—24no—
105C22⋊Q8—44no—
106C22⋊Q8—44no—
107C4×Q8—28no—
108C4×Q8—28no—
109C4×Q8—44no—
110C4×Q8—44no—
111#117—44no—
112#117—44no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—44no—
131C2×C4⋊C4—48no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—48no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—82no—
139C2×C4⋊C4—84no—
140C2×C4⋊C4SD1614yes—
141C2×C4⋊C4Q1614yes—
142C2×C4⋊C4C2×D414yes—
143C2×C22⋊C4—24no—
144C2×C42—24no—
145C2×C42—22no—
146C2×C42—24no—
147C2×C42—22no—
148C2×C42—42no—
149C2×C42—42no—
150C2×C42—44no—
151C2×C42C2×D412yes—
152C2×C42C2×D412yes—
153C2×C42C2×D412yes—
154C2×C42#4114yes—
155C2×C42#4112yes—
156C4⋊C8—44no—
157#103—88no—
158C22⋊C8—24no—
159C22⋊C8—24no—
160C4×C8—44no—
161#95—28no—
162#95—28no—
163#95—24no—
164#95—22no—
165#95—48no—
166#95C2×D414yes—
167C24C22×D411yescharacteristic
168C2×Q8—24no—
169C2×Q8—24no—
170C2×Q8—44no—
171C2×Q8—44no—
172C2×Q8—44no—
173C2×Q8—44no—
174C2×Q8—44no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—44no—
197C22×C4—44no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4#12011yescharacteristic
202C22×C4#12012yes—
203C22×C4#12011yescharacteristic
204C22×C4#12112yes—
205C22×C4#12112yes—
206C22×C4#12412yes—
207C22×C4#12412yes—
208C22×C4C2×SD1614yes—
209C22×C4C2×SD1612yes—
210C22×C4C2×Q1614yes—
211C22×C4C2×Q1612yes—
212C22×C4#13712yes—
213C22×C4#13712yes—
214C22×C4C22×D412yes—
215C22×C4C22×D411yescharacteristic
216C22×C4C2×#4112yes—
217C22×C4C2×#4111yescharacteristic
218C2×C8—44no—
219C2×C8—44no—
220C2×C8—44no—
221C4⋊C4—216no—
222C4⋊C4—28no—
223C4⋊C4—28no—
224C4⋊C4—24no—
225C4⋊C4—24no—
226C4⋊C4—48no—
227C4⋊C4—44no—
228C4⋊C4—44no—
229C4⋊C4—44no—
230C4⋊C4—44no—
231C4⋊C4—48no—
232C4⋊C4—44no—
233C4⋊C4—82no—
234C4⋊C4—84no—
235C4⋊C4—84no—
236C4⋊C4—84no—
237C4⋊C4—84no—
238C4⋊C4—816no—
239C4⋊C4—84no—
240C4⋊C4—82no—
241C22⋊C4—44no—
242C22⋊C4—44no—
243C42—48no—
244C42—44no—
245C42—48no—
246C42—82no—
247C42#12114yes—
248C42#12114yes—
249C42#12314yes—
250C42#12314yes—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23C2×#12011yescharacteristic
256C23C2×#12111yescharacteristic
257C23C2×#12411yescharacteristic
258C23C22×SD1611yescharacteristic
259C23C22×Q1611yescharacteristic
260C23C2×#13711yescharacteristic
261C23C2×#13711yescharacteristic
262Q8—48no—
263Q8—84no—
264Q8—88no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—28no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—84no—
298C2×C4—84no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4#39412yes—
303C2×C4#39412yes—
304C2×C4#39414yes—
305C2×C4#44814yes—
306C2×C4#45114yes—
307C2×C4#45312yes—
308C2×C4#45612yes—
309C2×C4#46114yes—
310C2×C4#46214yes—
311C2×C4#48714yes—
312C2×C4#49114yes—
313C2×C4C2×#12112yes—
314C2×C4C2×#12312yes—
315C8—84no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22C2×#39411yescharacteristic
329C22C2×#44811yescharacteristic
330C22C2×#45111yescharacteristic
331C22C2×#45311yescharacteristic
332C22C2×#45611yescharacteristic
333C22#302711yescharacteristic
334C22#302811yescharacteristic
335C22C2×#46111yescharacteristic
336C22C2×#46211yescharacteristic
337C22C2×#48711yescharacteristic
338C22C2×#49111yescharacteristic
339C4—22no—
340C4—22no—
341C4—22no—
342C4—48no—
343C4—44no—
344C4—44no—
345C4—84no—
346C4—84no—
347C4—162no—
348C4#203914yes—
349C4#204318yes—
350C2—21no—
351C2—21no—
352C2—21no—
353C2—21no—
354C2#1838911yescharacteristic
355C2#1839711yescharacteristic
356C2#1840411yescharacteristic
357C2#1841011yescharacteristic
358C2C2×#203911yescharacteristic
359C2C2×#204311yescharacteristic
360C2#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
#139102
#104062
#79082