Finite Group Databasebeta

← #173960

Subgroups — #173960

Order 512.

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

normal subgroups only

This group has 937 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#28687C212yesmaximal, complemented
3#28508C211yesmaximal, characteristic
4#7819C218yesmaximal, complemented
5#7570C212yesmaximal, complemented
6#7569C211yesmaximal, characteristic
7C2×#559—21no—
8C2×#498—22no—
9C2×#474—22no—
10#2868C2211yescharacteristic
11#2724C2214yes—
12#2582—21no—
13#2547—22no—
14#2486—22no—
15#2447—24no—
16D4×C4⋊C4—22no—
17C4×C4⋊Q8C2212yes—
18C4×#127C2211yescharacteristic
19#2093C2214yes—
20#2084C2218yes—
21#1990—22no—
22#1957—21no—
23#1948—24no—
24#1845—21no—
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#498—48no—
43#474—48no—
44C2×#107—24no—
45C2×#106—28no—
46C2×C4⋊C8—22no—
47C2×C4⋊C8—21no—
48C2×#102—24no—
49C2×C4×C8—21no—
50C2×C4×C8—22no—
51#398—24no—
52#397—24no—
53#397—28no—
54#391—24no—
55#391—22no—
56#390—22no—
57#390C2312yes—
58#389—24no—
59#389—22no—
60#389C2314yes—
61#385—28no—
62#384—21no—
63#384—22no—
64#384D411yescharacteristic
65#384D412yes—
66#384C2312yes—
67#381—24no—
68#380—24no—
69#380—22no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4D414yes—
72C43C2311yescharacteristic
73#340—28no—
74#336C2314yes—
75#336C2312yes—
76C22×Q8—22no—
77C22×Q8—41no—
78C22×D4—22no—
79C22×D4C2×D412yes—
80C23×C4—24no—
81C23×C4—22no—
82C22×C8—24no—
83C22×C8—22no—
84C4⋊Q8—24no—
85C4⋊Q8—24no—
86C4⋊Q8—44no—
87C4⋊Q8—44no—
88C4⋊Q8—44no—
89#127—22no—
90#127SD1614yes—
91#124—416no—
92#123—48no—
93#123—48no—
94C4×Q8—44no—
95C4×Q8—44no—
96C4×D4—44no—
97C4×D4—416no—
98C4×D4—44no—
99C4×D4—44no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—84no—
111C2×C4⋊C4C2×D414yes—
112C2×C4⋊C4C2×D412yes—
113C2×C4⋊C4C2×D414yes—
114C2×C22⋊C4—28no—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—22no—
118C2×C42—24no—
119C2×C42C2×D411yescharacteristic
120C2×C42C2×D412yes—
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D412yes—
123C2×C42C2411yescharacteristic
124#107—48no—
125#106—416no—
126C4⋊C8—28no—
127C4⋊C8—24no—
128#102—48no—
129C4×C8—42no—
130C4×C8—44no—
131#95—28no—
132#95—24no—
133#95—24no—
134#95—24no—
135C24—22no—
136C2×Q8—44no—
137C2×Q8—44no—
138C2×Q8—44no—
139C2×Q8—84no—
140C2×Q8—82no—
141C2×D4—22no—
142C2×D4—28no—
143C2×D4—42no—
144C2×D4—42no—
145C2×D4—44no—
146C2×D4—44no—
147C2×D4C2×SD1614yes—
148C22×C4—24no—
149C22×C4—22no—
150C22×C4—24no—
151C22×C4—48no—
152C22×C4—48no—
153C22×C4—44no—
154C22×C4—42no—
155C22×C4—84no—
156C22×C4—84no—
157C22×C4#12011yescharacteristic
158C22×C4#12011yescharacteristic
159C22×C4#12012yes—
160C22×C4C22×D411yescharacteristic
161C22×C4C22×D412yes—
162C2×C8—24no—
163C2×C8—28no—
164C2×C8—44no—
165C2×C8—42no—
166C2×C8—44no—
167C2×C8—42no—
168C4⋊C4—24no—
169C4⋊C4—24no—
170C4⋊C4—42no—
171C4⋊C4—44no—
172C4⋊C4—44no—
173C4⋊C4—44no—
174C4⋊C4—44no—
175C4⋊C4—44no—
176C4⋊C4—44no—
177C4⋊C4—44no—
178C4⋊C4—44no—
179C4⋊C4—44no—
180C4⋊C4—44no—
181C4⋊C4—82no—
182C4⋊C4—84no—
183C4⋊C4—84no—
184C22⋊C4—416no—
185C22⋊C4—48no—
186C22⋊C4—44no—
187C22⋊C4—44no—
188C22⋊C4—82no—
189C42—22no—
190C42—24no—
191C42—41no—
192C42—48no—
193C42C2×SD1612yes—
194C42#13514yes—
195C42AGL(1,ℤ/8)14yes—
196C42AGL(1,ℤ/8)12yes—
197C23—24no—
198C23—44no—
199C23—42no—
200C23—44no—
201C23—42no—
202C23C2×#12011yescharacteristic
203Q8—88no—
204Q8—164no—
205D4—44no—
206D4—416no—
207D4—82no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—44no—
215C2×C4—44no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—42no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—88no—
223C2×C4—84no—
224C2×C4—82no—
225C2×C4—168no—
226C2×C4#44914yes—
227C2×C4#44914yes—
228C2×C4#45014yes—
229C2×C4#45312yes—
230C2×C4#45614yes—
231C2×C4C22×SD1611yescharacteristic
232C2×C4C2×#13512yes—
233C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
234C2×C4C2×AGL(1,ℤ/8)12yes—
235C8—44no—
236C8—48no—
237C22—48no—
238C22—44no—
239C22—82no—
240C22—82no—
241C22—84no—
242C22C2×#45011yescharacteristic
243C22C2×#44912yes—
244C22C2×#45311yescharacteristic
245C22C2×#45612yes—
246C22#302811yescharacteristic
247C4—21no—
248C4—21no—
249C4—22no—
250C4—22no—
251C4—41no—
252C4—82no—
253C4—84no—
254C4—84no—
255C4—162no—
256C4#162014yes—
257C4#165418yes—
258C2—84no—
259C2#1839511yescharacteristic
260C2#1839812yes—
261C2C2×#162011yescharacteristic
262C2C2×#165412yes—
263C2#1907811yescharacteristic
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. #28687C2semidirect
4. #7819C2semidirect
5. #7570C2semidirect

Maximal subgroups

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

subgroupindex
#598272
#286872
#285082
#78192
#75702
#75692