Finite Group Databasebeta

← #173961

Subgroups — #173961

Order 512.

901 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 901 conjugacy classes of subgroups — too many to draw. Only its 176 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#59827C211yesmaximal, characteristic
2#29218C212yesmaximal, complemented
3#28509C211yesmaximal, characteristic
4#7824C218yesmaximal, complemented
5#7570C212yesmaximal, complemented
6#7569C211yesmaximal, characteristic
7C2×#559—21no—
8C2×#476—22no—
9#2868C2211yescharacteristic
10#2724C2214yes—
11#2582—21no—
12#2547—22no—
13#2486—22no—
14#2447—24no—
15D4×C4⋊C4—22no—
16C4×C4⋊Q8C2212yes—
17C4×#127C2211yescharacteristic
18#2100C2218yes—
19#2094C2214yes—
20#1990—22no—
21#1987—24no—
22#1957—21no—
23#1844—21no—
24#1834—22no—
25C4×C4⋊C8C2212yes—
26C4×C4⋊C8C2211yescharacteristic
27#1271C2218yes—
28#1270C2214yes—
29#559—44no—
30#559—44no—
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8D412yes—
34C2×#127D411yescharacteristic
35C2×#124—24no—
36C2×#123—24no—
37C2×C4×Q8—22no—
38C2×C4×D4—24no—
39C2×C4×D4—22no—
40C2×C4×D4D412yes—
41C22×C4⋊C4—22no—
42#476—48no—
43C2×#106—24no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—22no—
46C2×#102—24no—
47C2×C4×C8—22no—
48C2×C4×C8—21no—
49#398—24no—
50#397—24no—
51#397—28no—
52#391—24no—
53#391—22no—
54#390—22no—
55#390C2312yes—
56#389—24no—
57#389—22no—
58#389C2314yes—
59#385—28no—
60#384—22no—
61#384—21no—
62#384D411yescharacteristic
63#384D412yes—
64#384C2312yes—
65#381—24no—
66#380—24no—
67#380—22no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4D414yes—
70C43C2311yescharacteristic
71#340—28no—
72#340—28no—
73#336C2312yes—
74#336C2314yes—
75C22×Q8—22no—
76C22×Q8—41no—
77C22×D4—22no—
78C22×D4C2×D412yes—
79C23×C4—24no—
80C23×C4—22no—
81C22×C8—22no—
82C22×C8—24no—
83C4⋊Q8—24no—
84C4⋊Q8—24no—
85C4⋊Q8—48no—
86C4⋊Q8—44no—
87#127—22no—
88#127SD1614yes—
89#124—416no—
90#123—48no—
91#123—48no—
92C4×Q8—44no—
93C4×Q8—44no—
94C4×D4—44no—
95C4×D4—416no—
96C4×D4—44no—
97C4×D4—44no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—84no—
109C2×C4⋊C4C2×D414yes—
110C2×C4⋊C4C2×D412yes—
111C2×C4⋊C4C2×D414yes—
112C2×C22⋊C4—28no—
113C2×C22⋊C4—24no—
114C2×C22⋊C4—24no—
115C2×C22⋊C4—22no—
116C2×C42—24no—
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D412yes—
119C2×C42C2×D411yescharacteristic
120C2×C42C2×D412yes—
121C2×C42C2411yescharacteristic
122#106—48no—
123C4⋊C8—24no—
124C4⋊C8—44no—
125#102—48no—
126C4×C8—44no—
127C4×C8—42no—
128#95—28no—
129#95—24no—
130#95—24no—
131#95—24no—
132C24—22no—
133C2×Q8—44no—
134C2×Q8—44no—
135C2×Q8—44no—
136C2×Q8—82no—
137C2×Q8—84no—
138C2×D4—22no—
139C2×D4—28no—
140C2×D4—42no—
141C2×D4—42no—
142C2×D4—44no—
143C2×D4—44no—
144C2×D4C2×SD1614yes—
145C22×C4—24no—
146C22×C4—22no—
147C22×C4—24no—
148C22×C4—48no—
149C22×C4—48no—
150C22×C4—44no—
151C22×C4—42no—
152C22×C4—84no—
153C22×C4—84no—
154C22×C4#12011yescharacteristic
155C22×C4#12012yes—
156C22×C4#12011yescharacteristic
157C22×C4C22×D411yescharacteristic
158C22×C4C22×D412yes—
159C2×C8—24no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—44no—
163C2×C8—44no—
164C2×C8—44no—
165C4⋊C4—24no—
166C4⋊C4—24no—
167C4⋊C4—42no—
168C4⋊C4—44no—
169C4⋊C4—44no—
170C4⋊C4—48no—
171C4⋊C4—44no—
172C4⋊C4—44no—
173C4⋊C4—44no—
174C4⋊C4—44no—
175C4⋊C4—44no—
176C4⋊C4—44no—
177C4⋊C4—84no—
178C4⋊C4—82no—
179C4⋊C4—84no—
180C22⋊C4—416no—
181C22⋊C4—48no—
182C22⋊C4—44no—
183C22⋊C4—44no—
184C22⋊C4—82no—
185C42—22no—
186C42—24no—
187C42—41no—
188C42—48no—
189C42C2×SD1612yes—
190C42#13514yes—
191C42AGL(1,ℤ/8)12yes—
192C42#13714yes—
193C23—24no—
194C23—44no—
195C23—42no—
196C23—44no—
197C23—42no—
198C23C2×#12011yescharacteristic
199Q8—88no—
200Q8—164no—
201D4—44no—
202D4—416no—
203D4—82no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—42no—
215C2×C4—44no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—88no—
219C2×C4—82no—
220C2×C4—84no—
221C2×C4—168no—
222C2×C4#45014yes—
223C2×C4#45214yes—
224C2×C4#45214yes—
225C2×C4#45312yes—
226C2×C4#45414yes—
227C2×C4C22×SD1611yescharacteristic
228C2×C4C2×#13512yes—
229C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
230C2×C4C2×#13712yes—
231C8—44no—
232C8—84no—
233C22—48no—
234C22—44no—
235C22—82no—
236C22—82no—
237C22—84no—
238C22C2×#45011yescharacteristic
239C22C2×#45212yes—
240C22C2×#45311yescharacteristic
241C22C2×#45412yes—
242C22#302811yescharacteristic
243C4—21no—
244C4—21no—
245C4—22no—
246C4—22no—
247C4—41no—
248C4—82no—
249C4—84no—
250C4—84no—
251C4—162no—
252C4#163014yes—
253C4#163518yes—
254C2—84no—
255C2#1839511yescharacteristic
256C2#1839912yes—
257C2C2×#163011yescharacteristic
258C2C2×#163512yes—
259C2#1907711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #29218C2semidirect
4. #7824C2semidirect
5. #7570C2semidirect

Maximal subgroups

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

subgroupindex
#598272
#292182
#285092
#78242
#75702
#75692