Finite Group Databasebeta

← #417004

Subgroups — #417004

Order 512.

1880 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1880 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#35020C211yesmaximal, characteristic
2#13922C212yesmaximal
3#13922C211yesmaximal, characteristic
4#11287C218yesmaximal
5#10854C212yesmaximal
6C4×#342C211yesmaximal, characteristic
7C24×Q8C2211yescharacteristic
8#2851—22no—
9#2646—22no—
10#2569—24no—
11#2442—24no—
12#2315C2214yes—
13C2×#391—22no—
14C2×#391—24no—
15C2×#380C2211yescharacteristic
16C2×#380C2212yes—
17C2×C4×C22⋊C4C2212yes—
18C2×C4×C22⋊C4C2211yescharacteristic
19C2×#342C2218yes—
20C2×#342C2214yes—
21#1446—24no—
22#1446—22no—
23C22⋊#95C2214yes—
24C22⋊#95C2218yes—
25C23×Q8—21no—
26C23×Q8—22no—
27C23×Q8—22no—
28C23×Q8—21no—
29C23×Q8—22no—
30C23×Q8—24no—
31C23×Q8—48no—
32C23×Q8Q814yes—
33C24×C4C2312yes—
34C24×C4C2311yescharacteristic
35C2×#125—22no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4D414yes—
38C22×C22⋊C4C2312yes—
39C22×C22⋊C4C2312yes—
40C22×C22⋊C4C2314yes—
41C22×C22⋊C4C2314yes—
42C22×C42—22no—
43C22×C42D412yes—
44#400—28no—
45#398—28no—
46#391—48no—
47#391—416no—
48#389—24no—
49#387—28no—
50#385—28no—
51#385—28no—
52#382—22no—
53#380—22no—
54#380—28no—
55#380—22no—
56#380—22no—
57#380—22no—
58#380—24no—
59C4×C22⋊C4—22no—
60C4×C22⋊C4—22no—
61C4×C22⋊C4—24no—
62C4×C22⋊C4—24no—
63C2×#95—216no—
64C2×#95—28no—
65C2×#95—22no—
66C2×#95—21no—
67C2×#95D414yes—
68C2×#95D412yes—
69#342—28no—
70#342—216no—
71C25C2411yescharacteristic
72C22×Q8—28no—
73C22×Q8—216no—
74C22×Q8—24no—
75C22×Q8—28no—
76C22×Q8—24no—
77C22×Q8—28no—
78C22×Q8—44no—
79C22×Q8—48no—
80C22×Q8—416no—
81C22×Q8—48no—
82C22×Q8—416no—
83C22×Q8—416no—
84C22×Q8—432no—
85C22×Q8—41no—
86C22×Q8—42no—
87C22×Q8—41no—
88C22×Q8—44no—
89C22×Q8—42no—
90C22×Q8—42no—
91C22×Q8—44no—
92C23×C4—22no—
93C23×C4—24no—
94C23×C4—24no—
95C23×C4—24no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—48no—
99C23×C4—44no—
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D412yes—
102C23×C4C2×D418yes—
103C23×C4C2×D414yes—
104C23×C4C2×D412yes—
105C23×C4C2×D411yescharacteristic
106C23×C4C2×Q814yes—
107C23×C4C2×Q812yes—
108C23×C4#4112yes—
109C23×C4#4114yes—
110#125—84no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—28no—
113C2×C4⋊C4—28no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—48no—
116C2×C4⋊C4—44no—
117C2×C4⋊C4—42no—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—24no—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—24no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—28no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—28no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—28no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4#4118yes—
134C2×C42—22no—
135C2×C42—22no—
136C2×C42—44no—
137C2×C42—48no—
138C2×C42—44no—
139#95—28no—
140#95—28no—
141#95—24no—
142#95—24no—
143#95—24no—
144#95—416no—
145#95—432no—
146#95—44no—
147#95—42no—
148C24—42no—
149C24—42no—
150C24C22×D411yescharacteristic
151C24C22×D412yes—
152C24C22×Q811yescharacteristic
153C24C2×#4112yes—
154C24C2×#4111yescharacteristic
155C24C2×#4114yes—
156C242−1+414yes—
157C2×Q8—216no—
158C2×Q8—232no—
159C2×Q8—48no—
160C2×Q8—416no—
161C2×Q8—432no—
162C2×Q8—416no—
163C2×Q8—48no—
164C2×Q8—416no—
165C2×Q8—416no—
166C2×Q8—432no—
167C2×Q8—432no—
168C2×Q8—416no—
169C2×Q8—44no—
170C2×Q8—48no—
171C2×Q8—44no—
172C2×Q8—48no—
173C22×C4—24no—
174C22×C4—28no—
175C22×C4—28no—
176C22×C4—216no—
177C22×C4—24no—
178C22×C4—28no—
179C22×C4—24no—
180C22×C4—28no—
181C22×C4—24no—
182C22×C4—28no—
183C22×C4—24no—
184C22×C4—216no—
185C22×C4—44no—
186C22×C4—48no—
187C22×C4—44no—
188C22×C4—48no—
189C22×C4—44no—
190C22×C4—42no—
191C22×C4—416no—
192C22×C4—48no—
193C22×C4—44no—
194C22×C4—48no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—44no—
198C22×C4—44no—
199C22×C4—44no—
200C22×C4—48no—
201C22×C4—48no—
202C22×C4—44no—
203C22×C4—44no—
204C22×C4—48no—
205C22×C4—48no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—42no—
209C22×C4—44no—
210C22×C4#12011yescharacteristic
211C22×C4#12012yes—
212C22×C4#12011yescharacteristic
213C22×C4C22⋊Q812yes—
214C22×C4C22⋊Q814yes—
215C22×C4C22⋊Q812yes—
216C22×C4C22⋊Q814yes—
217C22×C4#12414yes—
218C22×C4#12412yes—
219C22×C4#12412yes—
220C22×C4#12414yes—
221C4⋊C4—48no—
222C4⋊C4—84no—
223C4⋊C4—84no—
224C4⋊C4—88no—
225C4⋊C4—88no—
226C22⋊C4—28no—
227C22⋊C4—28no—
228C22⋊C4—44no—
229C22⋊C4—44no—
230C22⋊C4—44no—
231C22⋊C4—44no—
232C22⋊C4—44no—
233C22⋊C4—48no—
234C22⋊C4—44no—
235C22⋊C4—48no—
236C42—44no—
237C42—82no—
238C42—88no—
239C23—22no—
240C23—24no—
241C23—22no—
242C23—24no—
243C23—41no—
244C23—42no—
245C23—41no—
246C23—41no—
247C23—48no—
248C23—42no—
249C23—41no—
250C23—48no—
251C23C2×#12011yescharacteristic
252C23C2×C22⋊Q811yescharacteristic
253C23C2×C22⋊Q812yes—
254C23C2×#12411yescharacteristic
255C23C2×#12412yes—
256C23#52914yes—
257C23#53212yes—
258C23#53612yes—
259C23#54114yes—
260C23#54714yes—
261C23#54814yes—
262C23#55214yes—
263C23#55714yes—
264Q8—416no—
265Q8—432no—
266Q8—416no—
267C2×C4—22no—
268C2×C4—24no—
269C2×C4—28no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—28no—
273C2×C4—28no—
274C2×C4—216no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—42no—
279C2×C4—44no—
280C2×C4—48no—
281C2×C4—416no—
282C2×C4—48no—
283C2×C4—44no—
284C2×C4—48no—
285C2×C4—416no—
286C2×C4—48no—
287C2×C4—44no—
288C2×C4—48no—
289C2×C4—48no—
290C2×C4—44no—
291C2×C4—48no—
292C2×C4—44no—
293C2×C4—48no—
294C2×C4—84no—
295C2×C4—88no—
296C2×C4—84no—
297C2×C4—84no—
298C2×C4—816no—
299C2×C4—82no—
300C2×C4—88no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4#39312yes—
304C2×C4#39314yes—
305C2×C4#39312yes—
306C2×C4B(4,𝔽2)14yes—
307C22—24no—
308C22—28no—
309C22—24no—
310C22—28no—
311C22—21no—
312C22—22no—
313C22—22no—
314C22—21no—
315C22—22no—
316C22—21no—
317C22—21no—
318C22—22no—
319C22—44no—
320C22—44no—
321C22—44no—
322C22—44no—
323C22C2×#39311yescharacteristic
324C22#243212yes—
325C22#245314yes—
326C22#248914yes—
327C22#256512yes—
328C22#264211yescharacteristic
329C22#286111yescharacteristic
330C22C2×B(4,𝔽2)12yes—
331C22#302812yes—
332C4—22no—
333C4—24no—
334C4—48no—
335C4—88no—
336C4—84no—
337C4—88no—
338C4—162no—
339C4#202914yes—
340C2—24no—
341C2—21no—
342C2—22no—
343C2—21no—
344C2—24no—
345C2#1523811yescharacteristic
346C2C2×#202911yescharacteristic
347C2#2347011yescharacteristic
348C2#2566314yes—
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
#350202
#139222
#112872
#108542
C4×#3422