Finite Group Databasebeta

← #410826

Subgroups — #410826

Order 512.

1558 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1558 conjugacy classes of subgroups — too many to draw. Only its 242 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34052C211yesmaximal, characteristic
2#13870C211yesmaximal, characteristic
3#13870C212yesmaximal
4#11287C214yesmaximal
5#11070C214yesmaximal
6#10726C211yesmaximal, characteristic
7#10726C212yesmaximal
8C22×#117C2211yescharacteristic
9C22×#117C2212yes—
10C23×C4⋊C4C2211yescharacteristic
11#2317—24no—
12#2316—22no—
13#2315C2214yes—
14#2296—24no—
15#2285—22no—
16C2×#381C2212yes—
17C2×#381C2214yes—
18C2×#381C2212yes—
19C2×#381C2211yescharacteristic
20C2×#342C2214yes—
21C2×#342C2218yes—
22#1441—22no—
23#1441—24no—
24#1436—22no—
25#1436—24no—
26C22⋊#95C2214yes—
27C22⋊#95C2212yes—
28C24×C4C2311yescharacteristic
29C24×C4C2311yescharacteristic
30C24×C4C2311yescharacteristic
31C2×#117—22no—
32C2×#117—22no—
33C2×#117—22no—
34C2×#117—22no—
35C2×#117—22no—
36C2×#117—22no—
37C2×#117—28no—
38C2×#117—48no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—24no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4D412yes—
46C22×C4⋊C4D414yes—
47C22×C4⋊C4Q814yes—
48C22×C22⋊C4C2314yes—
49C22×C22⋊C4C2312yes—
50C22×C22⋊C4C2314yes—
51C22×C22⋊C4C2312yes—
52C22×C42D412yes—
53C22×C42D414yes—
54#389—28no—
55#389—28no—
56#385—216no—
57#384—24no—
58#382—24no—
59#381—24no—
60#381—24no—
61#381—24no—
62#381—28no—
63#381—48no—
64#380—28no—
65C4×C4⋊C4—28no—
66C2×#95—28no—
67C2×#95—24no—
68C2×#95—21no—
69C2×#95—22no—
70C2×#95—28no—
71C2×#95—24no—
72C2×#95—21no—
73C2×#95—22no—
74#342—28no—
75#342—48no—
76C25C2411yescharacteristic
77C23×C4—24no—
78C23×C4—24no—
79C23×C4—21no—
80C23×C4—22no—
81C23×C4—21no—
82C23×C4—22no—
83C23×C4—22no—
84C23×C4—21no—
85C23×C4—22no—
86C23×C4—21no—
87C23×C4—24no—
88C23×C4—24no—
89C23×C4—22no—
90C23×C4—24no—
91C23×C4—44no—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D412yes—
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D412yes—
96C23×C4C2×D418yes—
97C23×C4C2×D414yes—
98C23×C4C2×Q812yes—
99C23×C4C2×Q812yes—
100C23×C4C2×Q812yes—
101C23×C4#4114yes—
102C23×C4#4112yes—
103#117—216no—
104#117—216no—
105#117—48no—
106#117—48no—
107#117—48no—
108#117—48no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—28no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—28no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—28no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—28no—
118C2×C4⋊C4—28no—
119C2×C4⋊C4—48no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—44no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—44no—
126C2×C4⋊C4—42no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—28no—
132C2×C22⋊C4—28no—
133C2×C22⋊C4—28no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—28no—
140C2×C22⋊C4#4118yes—
141C2×C42—22no—
142C2×C42—22no—
143C2×C42—28no—
144C2×C42—24no—
145C2×C42—24no—
146C2×C42—48no—
147#95—216no—
148#95—48no—
149#95—416no—
150#95—416no—
151#95—48no—
152#95—44no—
153#95—42no—
154#95—44no—
155#95—42no—
156C24—42no—
157C24—42no—
158C24C22×D411yescharacteristic
159C24C22×D412yes—
160C24C22×Q811yescharacteristic
161C24C2×#4112yes—
162C24C2×#4111yescharacteristic
163C24C2×#4114yes—
164C24H(5,𝔽2)14yes—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—216no—
174C22×C4—28no—
175C22×C4—28no—
176C22×C4—28no—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—28no—
183C22×C4—24no—
184C22×C4—22no—
185C22×C4—28no—
186C22×C4—28no—
187C22×C4—28no—
188C22×C4—44no—
189C22×C4—48no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—42no—
193C22×C4—41no—
194C22×C4—42no—
195C22×C4—41no—
196C22×C4—44no—
197C22×C4—48no—
198C22×C4—44no—
199C22×C4—48no—
200C22×C4—41no—
201C22×C4—42no—
202C22×C4—44no—
203C22×C4—42no—
204C22×C4—41no—
205C22×C4—44no—
206C22×C4—42no—
207C22×C4—48no—
208C22×C4—44no—
209C22×C4—48no—
210C22×C4—44no—
211C22×C4—48no—
212C22×C4—44no—
213C22×C4—42no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4—42no—
217C22×C4—44no—
218C22×C4—42no—
219C22×C4#12011yescharacteristic
220C22×C4#12011yescharacteristic
221C22×C4#12012yes—
222C22×C4C22⋊Q814yes—
223C22×C4C22⋊Q814yes—
224C22×C4C22⋊Q812yes—
225C22×C4C22⋊Q812yes—
226C22×C4#12318yes—
227C22×C4#12414yes—
228C22×C4#12414yes—
229C22×C4#12412yes—
230C22×C4#12412yes—
231C4⋊C4—216no—
232C4⋊C4—48no—
233C4⋊C4—48no—
234C4⋊C4—44no—
235C4⋊C4—44no—
236C4⋊C4—48no—
237C4⋊C4—88no—
238C4⋊C4—88no—
239C22⋊C4—216no—
240C22⋊C4—48no—
241C22⋊C4—48no—
242C22⋊C4—48no—
243C22⋊C4—48no—
244C22⋊C4—48no—
245C42—28no—
246C42—28no—
247C42—48no—
248C42—48no—
249C23—24no—
250C23—22no—
251C23—24no—
252C23—22no—
253C23—41no—
254C23—42no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—42no—
259C23—48no—
260C23—48no—
261C23C2×#12011yescharacteristic
262C23C2×C22⋊Q812yes—
263C23C2×C22⋊Q811yescharacteristic
264C23C2×#12312yes—
265C23#52514yes—
266C23C2×#12412yes—
267C23C2×#12411yescharacteristic
268C23#53414yes—
269C23#53512yes—
270C23#55318yes—
271C23#55314yes—
272C23#55414yes—
273C2×C4—28no—
274C2×C4—28no—
275C2×C4—28no—
276C2×C4—28no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—28no—
288C2×C4—24no—
289C2×C4—28no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—48no—
293C2×C4—44no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—48no—
299C2×C4—416no—
300C2×C4—48no—
301C2×C4—48no—
302C2×C4—42no—
303C2×C4—44no—
304C2×C4—48no—
305C2×C4—48no—
306C2×C4—84no—
307C2×C4—84no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—82no—
311C2×C4—88no—
312C2×C4—84no—
313C2×C4—82no—
314C2×C4—84no—
315C2×C4—82no—
316C2×C4—88no—
317C2×C4—84no—
318C2×C4—84no—
319C2×C4—88no—
320C2×C4—84no—
321C2×C4—82no—
322C2×C4#39312yes—
323C2×C4#39312yes—
324C2×C4#39314yes—
325C22—22no—
326C22—21no—
327C22—22no—
328C22—21no—
329C22—22no—
330C22—21no—
331C22—22no—
332C22—21no—
333C22—28no—
334C22—24no—
335C22—24no—
336C22—28no—
337C22—44no—
338C22—48no—
339C22—44no—
340C22C2×#39311yescharacteristic
341C22#248712yes—
342C22#250412yes—
343C22#257914yes—
344C22#259114yes—
345C22#261311yescharacteristic
346C22#284911yescharacteristic
347C22#302512yes—
348C22#302812yes—
349C4—22no—
350C4—22no—
351C4—24no—
352C4—44no—
353C4—48no—
354C4—88no—
355C4—82no—
356C4—164no—
357C4—162no—
358C2—24no—
359C2—22no—
360C2—24no—
361C2—22no—
362C2#1509111yescharacteristic
363C2#2346711yescharacteristic
364C2#2347011yescharacteristic
365C2#2963712yes—
366C2#3302312yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, prime radical, cosocleC25C24
LCSG ▷ C25 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#340522
#138702
#112872
#110702
#107262