Finite Group Databasebeta

← #416719

Subgroups — #416719

Order 512.

1408 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1408 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#13904C211yesmaximal, characteristic
3#13904C212yesmaximal
4#11070C218yesmaximal
5#10854C212yesmaximal
6C4×#342C211yesmaximal, characteristic
7C23×C4⋊C4C2211yescharacteristic
8#2851—22no—
9#2646—22no—
10#2569—24no—
11#2442—24no—
12#2315C2214yes—
13C2×#381C2218yes—
14C2×#381C2214yes—
15C2×#380C2211yescharacteristic
16C2×#380C2212yes—
17C2×C4×C22⋊C4C2212yes—
18C2×C4×C22⋊C4C2211yescharacteristic
19C2×#342C2214yes—
20C2×#342C2218yes—
21#1445—24no—
22#1445—22no—
23C4⋊#95—22no—
24C4⋊#95—24no—
25C24×C4C2312yes—
26C24×C4C2311yescharacteristic
27C2×#125—22no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—22no—
30C22×C4⋊C4—22no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4—24no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4D414yes—
36C22×C4⋊C4Q814yes—
37C22×C22⋊C4C2312yes—
38C22×C22⋊C4C2312yes—
39C22×C22⋊C4C2314yes—
40C22×C22⋊C4C2314yes—
41C22×C42—22no—
42C22×C42D412yes—
43#400—28no—
44#398—28no—
45#389—24no—
46#387—28no—
47#385—28no—
48#385—28no—
49#382—22no—
50#381—216no—
51#381—48no—
52#381—48no—
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—48no—
71C25C2411yescharacteristic
72C23×C4—24no—
73C23×C4—22no—
74C23×C4—24no—
75C23×C4—24no—
76C23×C4—22no—
77C23×C4—28no—
78C23×C4—28no—
79C23×C4—22no—
80C23×C4—44no—
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D412yes—
83C23×C4C2×D418yes—
84C23×C4C2×D414yes—
85C23×C4C2×D412yes—
86C23×C4C2×D411yescharacteristic
87C23×C4C2×Q814yes—
88C23×C4C2×Q812yes—
89C23×C4#4112yes—
90C23×C4#4114yes—
91#125—84no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—28no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—28no—
97C2×C4⋊C4—28no—
98C2×C4⋊C4—28no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—44no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—44no—
107C2×C4⋊C4—48no—
108C2×C4⋊C4—44no—
109C2×C4⋊C4—42no—
110C2×C22⋊C4—24no—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—24no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—28no—
120C2×C22⋊C4—28no—
121C2×C22⋊C4—24no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—28no—
125C2×C22⋊C4#4118yes—
126C2×C42—22no—
127C2×C42—22no—
128C2×C42—44no—
129C2×C42—48no—
130C2×C42—44no—
131#95—28no—
132#95—28no—
133#95—24no—
134#95—24no—
135#95—24no—
136#95—432no—
137#95—416no—
138#95—44no—
139#95—42no—
140C24—42no—
141C24—42no—
142C24C22×D411yescharacteristic
143C24C22×D412yes—
144C24C22×Q811yescharacteristic
145C24C2×#4112yes—
146C24C2×#4111yescharacteristic
147C24C2×#4114yes—
148C24H(5,𝔽2)14yes—
149C22×C4—24no—
150C22×C4—28no—
151C22×C4—24no—
152C22×C4—28no—
153C22×C4—24no—
154C22×C4—28no—
155C22×C4—24no—
156C22×C4—28no—
157C22×C4—216no—
158C22×C4—24no—
159C22×C4—216no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4—48no—
163C22×C4—44no—
164C22×C4—42no—
165C22×C4—44no—
166C22×C4—44no—
167C22×C4—48no—
168C22×C4—48no—
169C22×C4—42no—
170C22×C4—44no—
171C22×C4—42no—
172C22×C4—44no—
173C22×C4—48no—
174C22×C4—44no—
175C22×C4—44no—
176C22×C4—48no—
177C22×C4—48no—
178C22×C4—44no—
179C22×C4—48no—
180C22×C4—48no—
181C22×C4—48no—
182C22×C4—48no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—42no—
186C22×C4—44no—
187C22×C4#12011yescharacteristic
188C22×C4#12012yes—
189C22×C4#12011yescharacteristic
190C22×C4C22⋊Q812yes—
191C22×C4C22⋊Q814yes—
192C22×C4C22⋊Q812yes—
193C22×C4C22⋊Q814yes—
194C22×C4#12414yes—
195C22×C4#12412yes—
196C22×C4#12412yes—
197C22×C4#12414yes—
198C4⋊C4—44no—
199C4⋊C4—48no—
200C4⋊C4—44no—
201C4⋊C4—48no—
202C4⋊C4—84no—
203C4⋊C4—84no—
204C4⋊C4—88no—
205C4⋊C4—88no—
206C22⋊C4—28no—
207C22⋊C4—28no—
208C22⋊C4—44no—
209C22⋊C4—44no—
210C22⋊C4—44no—
211C22⋊C4—44no—
212C22⋊C4—44no—
213C22⋊C4—48no—
214C22⋊C4—44no—
215C22⋊C4—48no—
216C42—44no—
217C42—82no—
218C42—88no—
219C23—22no—
220C23—24no—
221C23—22no—
222C23—24no—
223C23—41no—
224C23—42no—
225C23—41no—
226C23—41no—
227C23—42no—
228C23—41no—
229C23—48no—
230C23—48no—
231C23C2×#12011yescharacteristic
232C23C2×C22⋊Q811yescharacteristic
233C23C2×C22⋊Q812yes—
234C23C2×#12411yescharacteristic
235C23C2×#12412yes—
236C23#52914yes—
237C23#53212yes—
238C23#53512yes—
239C23#54014yes—
240C23#54614yes—
241C23#55014yes—
242C23#55314yes—
243C23#55414yes—
244C2×C4—22no—
245C2×C4—24no—
246C2×C4—28no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—28no—
250C2×C4—44no—
251C2×C4—42no—
252C2×C4—44no—
253C2×C4—42no—
254C2×C4—416no—
255C2×C4—48no—
256C2×C4—48no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—48no—
260C2×C4—416no—
261C2×C4—48no—
262C2×C4—44no—
263C2×C4—48no—
264C2×C4—44no—
265C2×C4—48no—
266C2×C4—84no—
267C2×C4—84no—
268C2×C4—88no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—84no—
272C2×C4—88no—
273C2×C4—84no—
274C2×C4—816no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—88no—
278C2×C4—82no—
279C2×C4#39312yes—
280C2×C4#39314yes—
281C2×C4#39312yes—
282C2×C4#45814yes—
283C22—22no—
284C22—21no—
285C22—21no—
286C22—22no—
287C22—22no—
288C22—21no—
289C22—21no—
290C22—22no—
291C22—24no—
292C22—28no—
293C22—24no—
294C22—28no—
295C22—44no—
296C22—44no—
297C22—44no—
298C22—44no—
299C22C2×#39311yescharacteristic
300C22#243112yes—
301C22#245414yes—
302C22#249014yes—
303C22#256612yes—
304C22#264111yescharacteristic
305C22#286111yescharacteristic
306C22C2×#45812yes—
307C22#302812yes—
308C4—22no—
309C4—24no—
310C4—88no—
311C4—84no—
312C4—84no—
313C4—88no—
314C4—162no—
315C4#203014yes—
316C2—24no—
317C2—21no—
318C2—22no—
319C2—21no—
320C2—24no—
321C2#1522911yescharacteristic
322C2C2×#203011yescharacteristic
323C2#2347011yescharacteristic
324C2#2558614yes—
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
#139042
#110702
#108542
C4×#3422