Finite Group Databasebeta

← #449609

Subgroups — #449609

Order 512.

1772 conjugacy classes of subgroups; only the 272 normal subgroups are shown.

normal subgroups only

This group has 1772 conjugacy classes of subgroups — too many to draw. Only its 272 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
1C2×#2487C212yesmaximal
2C2×#1852C214yesmaximal, complemented
3#11328C214yesmaximal, complemented
4#11200C214yesmaximal, complemented
5#10964C211yesmaximal, characteristic
6C22×C4×D4C2211yescharacteristic
7C23×C4⋊C4C2212yes—
8C22×#107C2214yes—
9C22×C22⋊C8C2214yes—
10#2487—216no—
11C2×#400—22no—
12C2×#399—22no—
13C2×#389C2214yes—
14C2×#386—22no—
15C2×#385—22no—
16C2×#380C2212yes—
17C2×#380C2214yes—
18#1852—216no—
19C2×#340C2218yes—
20C2×#340C2214yes—
21C2×#336C2212yes—
22C23×D4D411yescharacteristic
23C24×C4—22no—
24C24×C4C2312yes—
25C23×C8C2314yes—
26C2×C4×D4—28no—
27C2×C4×D4—42no—
28C22×C4⋊C4—22no—
29C22×C4⋊C4—24no—
30C22×C4⋊C4—28no—
31C22×C4⋊C4—28no—
32C22×C4⋊C4—44no—
33C22×C4⋊C4D411yescharacteristic
34C22×C4⋊C4C2314yes—
35C22×C4⋊C4C2314yes—
36C22×C22⋊C4—22no—
37C22×C22⋊C4—24no—
38C22×C22⋊C4D412yes—
39C22×C42C2311yescharacteristic
40C2×#107—216no—
41C2×#107—44no—
42C2×C22⋊C8—28no—
43C2×C22⋊C8—216no—
44#400—216no—
45#399—216no—
46#389—28no—
47#389—28no—
48#386—216no—
49#385—216no—
50#380—28no—
51#380—28no—
52#380D4116yes—
53C2×#95—22no—
54C2×#95—24no—
55C2×#95—24no—
56#340—28no—
57#340—216no—
58#336—42no—
59C25C2×D412yes—
60C22×D4—24no—
61C22×D4—24no—
62C22×D4—41no—
63C22×D4—41no—
64C22×D4Q1614yes—
65C23×C4—24no—
66C23×C4—28no—
67C23×C4—24no—
68C23×C4—28no—
69C23×C4—24no—
70C23×C4—24no—
71C23×C4—28no—
72C23×C4—44no—
73C23×C4—44no—
74C23×C4—88no—
75C23×C4C2×D411yescharacteristic
76C23×C4C2×D412yes—
77C23×C4C2×D418yes—
78C23×C4C2×D411yescharacteristic
79C23×C4C2411yescharacteristic
80C22×C8—28no—
81C22×C8—28no—
82C22×C8—24no—
83C22×C8—24no—
84C22×C8#4118yes—
85C4×D4—416no—
86C2×C4⋊C4—28no—
87C2×C4⋊C4—28no—
88C2×C4⋊C4—28no—
89C2×C4⋊C4—28no—
90C2×C4⋊C4—28no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—216no—
95C2×C4⋊C4—432no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—44no—
98C2×C4⋊C4—44no—
99C2×C4⋊C4—88no—
100C2×C4⋊C4C2×D418yes—
101C2×C4⋊C4C2×D418yes—
102C2×C22⋊C4—28no—
103C2×C22⋊C4—24no—
104C2×C22⋊C4—24no—
105C2×C22⋊C4—216no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—44no—
108C2×C22⋊C4—44no—
109C2×C22⋊C4—88no—
110C2×C42—24no—
111C2×C42—22no—
112#107—416no—
113C22⋊C8—232no—
114#95—28no—
115#95—216no—
116#95—216no—
117C24—22no—
118C24—22no—
119C24—28no—
120C24—22no—
121C24—22no—
122C24—28no—
123C24#12014yes—
124C24C2×Q1614yes—
125C24#13714yes—
126C24C22×D411yescharacteristic
127C2×D4—216no—
128C2×D4—44no—
129C2×D4—44no—
130C2×D4—44no—
131C2×D4—48no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—28no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—28no—
140C22×C4—28no—
141C22×C4—216no—
142C22×C4—216no—
143C22×C4—28no—
144C22×C4—28no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—24no—
148C22×C4—44no—
149C22×C4—44no—
150C22×C4—48no—
151C22×C4—48no—
152C22×C4—44no—
153C22×C4—44no—
154C22×C4—416no—
155C22×C4—44no—
156C22×C4—44no—
157C22×C4—48no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—48no—
161C22×C4—48no—
162C22×C4—48no—
163C22×C4—816no—
164C22×C4—816no—
165C22×C4—816no—
166C22×C4#12014yes—
167C22×C4C2×Q1612yes—
168C22×C4C22×D414yes—
169C22×C4C2×#4114yes—
170C22×C4H(5,𝔽2)12yes—
171C22×C42−1+414yes—
172C2×C8—216no—
173C2×C8—28no—
174C2×C8—44no—
175C2×C8—48no—
176C2×C8—44no—
177C4⋊C4—216no—
178C4⋊C4—216no—
179C4⋊C4—44no—
180C4⋊C4—416no—
181C4⋊C4—44no—
182C4⋊C4—44no—
183C4⋊C4—48no—
184C4⋊C4—816no—
185C22⋊C4—48no—
186C22⋊C4—416no—
187C22⋊C4—816no—
188C42—44no—
189C23—216no—
190C23—216no—
191C23—24no—
192C23—28no—
193C23—24no—
194C23—24no—
195C23—24no—
196C23—24no—
197C23—24no—
198C23—28no—
199C23—48no—
200C23—48no—
201C23—44no—
202C23—44no—
203C23—44no—
204C23—44no—
205C23#45118yes—
206C23C2×#12012yes—
207C23#53411yescharacteristic
208C23#54714yes—
209C23C22×Q1611yescharacteristic
210C23C2×#13712yes—
211C23#57511yescharacteristic
212C23#57614yes—
213D4—416no—
214C2×C4—24no—
215C2×C4—28no—
216C2×C4—22no—
217C2×C4—24no—
218C2×C4—41no—
219C2×C4—44no—
220C2×C4—41no—
221C2×C4—44no—
222C2×C4—42no—
223C2×C4—416no—
224C2×C4—416no—
225C2×C4—41no—
226C2×C4—48no—
227C2×C4—48no—
228C2×C4—42no—
229C2×C4—48no—
230C2×C4—48no—
231C2×C4—48no—
232C2×C4—44no—
233C2×C4—48no—
234C2×C4—832no—
235C2×C4—88no—
236C2×C4—88no—
237C2×C4—88no—
238C2×C4—88no—
239C2×C4#52514yes—
240C2×C4#53614yes—
241C2×C4#53914yes—
242C2×C4D4214yes—
243C2×C4#54718yes—
244C2×C4#55912yes—
245C2×C4#56312yes—
246C8—48no—
247C22—216no—
248C22—28no—
249C22—48no—
250C22—48no—
251C22—48no—
252C22—44no—
253C22—44no—
254C22—48no—
255C22—48no—
256C22#240311yescharacteristic
257C22#240614yes—
258C22#272412yes—
259C22C2×#45112yes—
260C22#300712yes—
261C22#300814yes—
262C22#318911yescharacteristic
263C22#319511yescharacteristic
264C22#319912yes—
265C22#333814yes—
266C22#334618yes—
267C22#335514yes—
268C4—44no—
269C4—42no—
270C4—88no—
271C4—81no—
272C4—88no—
273C4#245514yes—
274C4#268014yes—
275C2—48no—
276C2#1594911yescharacteristic
277C2#2392111yescharacteristic
278C2#2392511yescharacteristic
279C2#2409814yes—
280C2#2410214yes—
281C2#3246012yes—
282C2#3246712yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#1852C2semidirect
3. #11328C2semidirect
4. #11200C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
C2×#18522
#113282
#112002
#109642