Finite Group Databasebeta

← #174013

Subgroups — #174013

Order 512.

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

normal subgroups only

This group has 861 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#59864C211yesmaximal, characteristic
2#28717C212yesmaximal, complemented
3#28508C211yesmaximal, characteristic
4#7821C218yesmaximal, complemented
5#7571C212yesmaximal, complemented
6#7569C211yesmaximal, characteristic
7C2×#487—22no—
8C2×#474—22no—
9#2868C2211yescharacteristic
10#2804C2214yes—
11#2580—24no—
12#2571—21no—
13#2563—22no—
14#2487—22no—
15#2366—21no—
16D4×C4⋊C4—22no—
17C4×C4⋊Q8C2212yes—
18C4×#127C2211yescharacteristic
19#2093C2214yes—
20#2072C2218yes—
21#1990—22no—
22#1957—21no—
23#1845—21no—
24#1803—24no—
25C4×C4⋊C8C2212yes—
26C4×C4⋊C8C2211yescharacteristic
27#1272C2218yes—
28#1270C2214yes—
29C2×C4⋊Q8—22no—
30C2×C4⋊Q8—21no—
31C2×C4⋊Q8D412yes—
32C2×#127D411yescharacteristic
33C2×C4×Q8D414yes—
34C2×C4×D4—24no—
35C2×C4×D4—22no—
36C2×C4×D4D412yes—
37C22×C4⋊C4—22no—
38#487—48no—
39#474—48no—
40C2×#107—24no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—22no—
43C2×#103—28no—
44C2×#102—24no—
45C2×C4×C8—21no—
46C2×C4×C8—22no—
47#400—24no—
48#399—24no—
49#393—28no—
50#391C2314yes—
51#390—24no—
52#390—22no—
53#390C2312yes—
54#389—24no—
55#389—22no—
56#389—22no—
57#386—24no—
58#385—24no—
59#385—28no—
60#384—22no—
61#384—21no—
62#384D411yescharacteristic
63#384D412yes—
64#384C2312yes—
65#382—24no—
66#380—22no—
67#380—24no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—22no—
70C43C2311yescharacteristic
71#340—28no—
72#336C2314yes—
73#336C2312yes—
74C22×Q8—41no—
75C22×Q8C2×D414yes—
76C22×D4—22no—
77C22×D4C2×D412yes—
78C23×C4—24no—
79C23×C4—22no—
80C22×C8—22no—
81C22×C8—24no—
82C4⋊Q8—24no—
83C4⋊Q8—24no—
84C4⋊Q8—44no—
85C4⋊Q8—44no—
86C4⋊Q8—44no—
87#127—22no—
88#127Q1614yes—
89C4×Q8—48no—
90C4×D4—44no—
91C4×D4—44no—
92C4×D4—44no—
93C4×D4—88no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—84no—
106C2×C4⋊C4C2×D412yes—
107C2×C4⋊C4C2×D414yes—
108C2×C22⋊C4—24no—
109C2×C22⋊C4—28no—
110C2×C22⋊C4—24no—
111C2×C22⋊C4—22no—
112C2×C42—24no—
113C2×C42C2×D411yescharacteristic
114C2×C42C2×D412yes—
115C2×C42C2×D411yescharacteristic
116C2×C42C2×D412yes—
117C2×C42C2411yescharacteristic
118#107—48no—
119C4⋊C8—24no—
120C4⋊C8—44no—
121#103—416no—
122#102—48no—
123C4×C8—44no—
124C4×C8—42no—
125#95—24no—
126#95—28no—
127#95—24no—
128#95—24no—
129C24—22no—
130C2×Q8—24no—
131C2×Q8—24no—
132C2×Q8—44no—
133C2×Q8—44no—
134C2×Q8—84no—
135C2×Q8—82no—
136C2×D4—22no—
137C2×D4—28no—
138C2×D4—42no—
139C2×D4—42no—
140C2×D4—44no—
141C2×D4—44no—
142C2×D4C2×Q1614yes—
143C22×C4—24no—
144C22×C4—22no—
145C22×C4—24no—
146C22×C4—48no—
147C22×C4—44no—
148C22×C4—42no—
149C22×C4—84no—
150C22×C4—88no—
151C22×C4#12011yescharacteristic
152C22×C4#12011yescharacteristic
153C22×C4#12012yes—
154C22×C4C22×D411yescharacteristic
155C22×C4C22×D412yes—
156C2×C8—24no—
157C2×C8—42no—
158C2×C8—42no—
159C2×C8—44no—
160C2×C8—44no—
161C2×C8—44no—
162C4⋊C4—42no—
163C4⋊C4—44no—
164C4⋊C4—44no—
165C4⋊C4—44no—
166C4⋊C4—44no—
167C4⋊C4—44no—
168C4⋊C4—44no—
169C4⋊C4—44no—
170C4⋊C4—44no—
171C4⋊C4—44no—
172C4⋊C4—84no—
173C4⋊C4—82no—
174C4⋊C4—84no—
175C4⋊C4—84no—
176C22⋊C4—44no—
177C22⋊C4—44no—
178C22⋊C4—82no—
179C22⋊C4—84no—
180C22⋊C4—88no—
181C42—22no—
182C42—24no—
183C42—41no—
184C42—48no—
185C42C2×Q1612yes—
186C42#13514yes—
187C42AGL(1,ℤ/8)14yes—
188C42#13712yes—
189C23—24no—
190C23—44no—
191C23—42no—
192C23—44no—
193C23—42no—
194C23C2×#12011yescharacteristic
195Q8—48no—
196Q8—84no—
197Q8—164no—
198D4—44no—
199D4—416no—
200D4—82no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—44no—
208C2×C4—44no—
209C2×C4—42no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4—88no—
216C2×C4—84no—
217C2×C4—82no—
218C2×C4—168no—
219C2×C4#44914yes—
220C2×C4#44914yes—
221C2×C4#45114yes—
222C2×C4#45312yes—
223C2×C4#45514yes—
224C2×C4C22×Q1611yescharacteristic
225C2×C4C2×#13512yes—
226C2×C4C2×AGL(1,ℤ/8)12yes—
227C2×C4C2×#13711yescharacteristic
228C8—44no—
229C8—84no—
230C22—48no—
231C22—44no—
232C22—82no—
233C22—82no—
234C22—84no—
235C22C2×#45111yescharacteristic
236C22C2×#44912yes—
237C22C2×#45311yescharacteristic
238C22C2×#45512yes—
239C22#302811yescharacteristic
240C4—21no—
241C4—21no—
242C4—22no—
243C4—22no—
244C4—41no—
245C4—82no—
246C4—84no—
247C4—84no—
248C4—162no—
249C4#162014yes—
250C4#164418yes—
251C2—84no—
252C2#1839711yescharacteristic
253C2#1841212yes—
254C2C2×#162011yescharacteristic
255C2C2×#164412yes—
256C2#1907411yescharacteristic
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. #28717C2semidirect
4. #7821C2semidirect
5. #7571C2semidirect

Maximal subgroups

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

subgroupindex
#598642
#287172
#285082
#78212
#75712
#75692