Finite Group Databasebeta

← #302891

Subgroups — #302891

Order 512.

877 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 877 conjugacy classes of subgroups — too many to draw. Only its 192 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#59864C211yesmaximal, characteristic
2#28687C212yesmaximal, complemented
3#28508C211yesmaximal, characteristic
4#10576C214yesmaximal, complemented
5#9864C212yesmaximal, complemented
6#8924C214yesmaximal
7C4×#331C211yesmaximal, characteristic
8C2×#498—22no—
9C2×#474—22no—
10#2868C2211yescharacteristic
11#2804—22no—
12#2580—24no—
13#2571C2212yes—
14#2563—22no—
15#2487—22no—
16#2366C2212yes—
17D4×C4⋊C4—22no—
18C4×C4⋊Q8C2212yes—
19C4×#127C2211yescharacteristic
20#2093—22no—
21#2084—24no—
22#1990—22no—
23#1957C2212yes—
24#1948C2218yes—
25#1845C2212yes—
26C4×C4⋊C8C2212yes—
27C4×C4⋊C8C2211yescharacteristic
28C2×#331C2212yes—
29#1296C2212yes—
30#1279C2218yes—
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8D412yes—
34C2×#127C2311yescharacteristic
35C2×C4×Q8—22no—
36C2×C4×D4—24no—
37C2×C4×D4—22no—
38C2×C4×D4D412yes—
39C22×C4⋊C4—22no—
40#498—48no—
41#474—48no—
42C2×#107—24no—
43C2×#106—28no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8C2314yes—
46C2×#102—24no—
47C2×C4×C8—22no—
48C2×C4×C8C2312yes—
49#400—24no—
50#399—24no—
51#393—28no—
52#391—22no—
53#390—24no—
54#390—22no—
55#390D412yes—
56#389—24no—
57#389—22no—
58#389D414yes—
59#386—24no—
60#385—24no—
61#385—28no—
62#384—22no—
63#384—21no—
64#384D412yes—
65#384C2312yes—
66#384C2311yescharacteristic
67#382—24no—
68#380—22no—
69#380—24no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4C2314yes—
72C43C2311yescharacteristic
73#340—28no—
74#336—22no—
75#336—21no—
76#331Q818yescomplemented
77C22×Q8—22no—
78C22×Q8—41no—
79C22×D4—22no—
80C22×D4C2×D412yes—
81C23×C4—24no—
82C23×C4—22no—
83C22×C8—24no—
84C22×C8—22no—
85C4⋊Q8—24no—
86C4⋊Q8—24no—
87C4⋊Q8—44no—
88C4⋊Q8—44no—
89C4⋊Q8—44no—
90#127—22no—
91#127C2×Q814yes—
92C4×Q8—84no—
93C4×D4—44no—
94C4×D4—44no—
95C4×D4—44no—
96C4×D4—88no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—84no—
108C2×C4⋊C4C2×D414yes—
109C2×C4⋊C4C2×D414yes—
110C2×C4⋊C4C2×D412yes—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—28no—
113C2×C22⋊C4—24no—
114C2×C22⋊C4—22no—
115C2×C42—24no—
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D412yes—
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D412yes—
120C2×C42C2411yescharacteristic
121#107—48no—
122#106—416no—
123C4⋊C8—24no—
124C4⋊C8—24no—
125C4⋊C8C2×Q818yes—
126#102—48no—
127C4×C8—24no—
128C4×C8—44no—
129#95—24no—
130#95—28no—
131#95—24no—
132#95—24no—
133C24—22no—
134C2×Q8—44no—
135C2×Q8—82no—
136C2×Q8—82no—
137C2×Q8—84no—
138C2×Q8—82no—
139C2×D4—22no—
140C2×D4—28no—
141C2×D4—42no—
142C2×D4—42no—
143C2×D4—42no—
144C2×D4—42no—
145C2×D4—44no—
146C2×D4C4⋊Q814yes—
147C22×C4—24no—
148C22×C4—22no—
149C22×C4—24no—
150C22×C4—48no—
151C22×C4—44no—
152C22×C4—42no—
153C22×C4—84no—
154C22×C4—88no—
155C22×C4#12011yescharacteristic
156C22×C4#12012yes—
157C22×C4#12011yescharacteristic
158C22×C4C22×D411yescharacteristic
159C22×C4C22×D412yes—
160C2×C8—24no—
161C2×C8—28no—
162C2×C8—42no—
163C2×C8—42no—
164C2×C8—44no—
165C2×C8—44no—
166C4⋊C4—24no—
167C4⋊C4—24no—
168C4⋊C4—28no—
169C4⋊C4—42no—
170C4⋊C4—44no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4—44no—
174C4⋊C4—44no—
175C4⋊C4—44no—
176C4⋊C4—44no—
177C4⋊C4—82no—
178C4⋊C4—84no—
179C4⋊C4—82no—
180C4⋊C4—84no—
181C4⋊C4—84no—
182C22⋊C4—48no—
183C22⋊C4—82no—
184C22⋊C4—84no—
185C22⋊C4—88no—
186C42—24no—
187C42—22no—
188C42—41no—
189C42—48no—
190C42#13514yes—
191C42AGL(1,ℤ/8)14yes—
192C42C22×Q812yes—
193C422−1+412yes—
194C23—24no—
195C23—44no—
196C23—44no—
197C23—42no—
198C23—42no—
199C23C2×#12011yescharacteristic
200Q8—88no—
201Q8—164no—
202D4—44no—
203D4—48no—
204D4—48no—
205D4—82no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4—44no—
216C2×C4—44no—
217C2×C4—42no—
218C2×C4—44no—
219C2×C4—44no—
220C2×C4—88no—
221C2×C4—84no—
222C2×C4—82no—
223C2×C4—168no—
224C2×C4#44914yes—
225C2×C4#44914yes—
226C2×C4#45312yes—
227C2×C4C2×C4⋊Q811yescharacteristic
228C2×C4#53611yescharacteristic
229C2×C4D4×Q814yes—
230C2×C4C2×#13512yes—
231C2×C4C2×AGL(1,ℤ/8)12yes—
232C2×C4#57714yes—
233C8—44no—
234C8—48no—
235C22—44no—
236C22—48no—
237C22—82no—
238C22—82no—
239C22—84no—
240C22#243411yescharacteristic
241C22C2×#44912yes—
242C22#300912yes—
243C22C2×#45311yescharacteristic
244C22#302211yescharacteristic
245C4—21no—
246C4—21no—
247C4—24no—
248C4—41no—
249C4—84no—
250C4—82no—
251C4—84no—
252C4—162no—
253C4#162014yes—
254C4#338114yes—
255C4#338414yes—
256C2—84no—
257C2C2×#162011yescharacteristic
258C2#1871511yescharacteristic
259C2#1892612yes—
260C2#2582411yescharacteristic
261C2#2583512yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #28687C2semidirect
4. #10576C2semidirect
5. #9864C2semidirect
76. #331Q8semidirect

Maximal subgroups

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

subgroupindex
#598642
#286872
#285082
#105762
#98642
#89242
C4×#3312