Finite Group Databasebeta

← #349915

Subgroups — #349915

Order 512.

861 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 861 conjugacy classes of subgroups — too many to draw. Only its 156 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×#2382C211yesmaximal, characteristic
2#27956C212yesmaximal
3#25912C211yesmaximal, characteristic
4#9004C214yesmaximal, complemented
5#8913C214yesmaximal, complemented
6#8719C212yesmaximal
7#8335C211yesmaximal, characteristic
8C22×#125C2211yescharacteristic
9C2×#491C2212yes—
10#2857C2211yescharacteristic
11#2382—24no—
12#2382—22no—
13#2382C2214yes—
14C2×#389—22no—
15C2×#387—22no—
16C2×#387—22no—
17C2×#382C2212yes—
18#2085—24no—
19#2026—22no—
20#1993C2214yes—
21#1992C2212yes—
22#1872—22no—
23#1853—22no—
24#1852—22no—
25#1847C2214yes—
26#1749C2211yescharacteristic
27#1749C2212yes—
28C2×#330C2214yes—
29#1295C2218yes—
30C2×C4⋊Q8C2311yescharacteristic
31C2×#125—28no—
32C2×#125—22no—
33C2×#125D412yes—
34C2×#125C2312yes—
35C2×C22⋊Q8—21no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4D412yes—
39C22×C4⋊C4D412yes—
40C22×C4⋊C4D412yes—
41C22×C42—22no—
42C22×C42C2311yescharacteristic
43#491—28no—
44C2×#107—24no—
45C2×#106—24no—
46C2×#103—24no—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—21no—
49C2×#97C2312yes—
50C2×#97C2314yes—
51#391—21no—
52#389—28no—
53#389—28no—
54#389—22no—
55#387—28no—
56#387—28no—
57#387—28no—
58#387—28no—
59#384C2311yescharacteristic
60#382—22no—
61#382—22no—
62#382D414yes—
63#382C2314yes—
64#380—22no—
65C2×#95—22no—
66C2×#95—22no—
67C2×#95—22no—
68C2×#95—22no—
69#340—24no—
70#340—24no—
71#340—24no—
72#336—22no—
73#336—21no—
74#330—28no—
75C22×Q8—21no—
76C23×C4—22no—
77C23×C4—22no—
78C23×C4—22no—
79C23×C4C2×D412yes—
80C23×C4C2×D411yescharacteristic
81C22×C8—24no—
82C22×C8—22no—
83C4⋊Q8—24no—
84#125—216no—
85#125—28no—
86#125—24no—
87#125#4118yes—
88C22⋊Q8—44no—
89C2×C4⋊C4—216no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—28no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—28no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—82no—
102C2×C4⋊C4—82no—
103C2×C4⋊C4—84no—
104C2×C4⋊C4C2×D414yes—
105C2×C4⋊C4C2×D414yes—
106C2×C4⋊C4C2×D414yes—
107C2×C22⋊C4—41no—
108C2×C42—24no—
109C2×C42—28no—
110C2×C42—84no—
111C2×C42C2×D412yes—
112C2×C42C2×D411yescharacteristic
113C2×C42C2411yescharacteristic
114#107—84no—
115#106—84no—
116#103—48no—
117C22⋊C8—28no—
118C22⋊C8—42no—
119#97—28no—
120#97—24no—
121#95—28no—
122#95—28no—
123#95—24no—
124#95—24no—
125#95—24no—
126#95—24no—
127#95—42no—
128C24#12011yescharacteristic
129C2×Q8—42no—
130C2×Q8—42no—
131C2×Q8—42no—
132C22×C4—28no—
133C22×C4—24no—
134C22×C4—22no—
135C22×C4—24no—
136C22×C4—22no—
137C22×C4—24no—
138C22×C4—24no—
139C22×C4—24no—
140C22×C4—42no—
141C22×C4—41no—
142C22×C4—44no—
143C22×C4—44no—
144C22×C4—44no—
145C22×C4—44no—
146C22×C4—44no—
147C22×C4—44no—
148C22×C4—44no—
149C22×C4—44no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4#12011yescharacteristic
153C22×C4#12012yes—
154C22×C4C22×D411yescharacteristic
155C22×C4C22×D412yes—
156C2×C8—28no—
157C2×C8—44no—
158C2×C8—44no—
159C2×C8—42no—
160C2×C8—42no—
161C2×C8—42no—
162C4⋊C4—216no—
163C4⋊C4—28no—
164C4⋊C4—24no—
165C4⋊C4—216no—
166C4⋊C4—44no—
167C4⋊C4—44no—
168C4⋊C4—44no—
169C4⋊C4—42no—
170C4⋊C4—44no—
171C4⋊C4—82no—
172C4⋊C4—82no—
173C4⋊C4—82no—
174C4⋊C4—88no—
175C4⋊C4#12318yes—
176C22⋊C4—82no—
177C42—22no—
178C42—24no—
179C42—88no—
180C42C2×#4114yes—
181C23—21no—
182C23—22no—
183C23—21no—
184C23#45412yes—
185C23#45614yes—
186C23C2×#12011yescharacteristic
187Q8—84no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—28no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—44no—
200C2×C4—48no—
201C2×C4—416no—
202C2×C4—44no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4—88no—
211C2×C4—88no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—82no—
216C2×C4—81no—
217C2×C4C2×#12312yes—
218C2×C4#52514yes—
219C8—48no—
220C8—82no—
221C22—22no—
222C22—24no—
223C22—22no—
224C22—24no—
225C22#166314yes—
226C22#284911yescharacteristic
227C22C2×#45411yescharacteristic
228C22C2×#45612yes—
229C22#301912yes—
230C22#302011yescharacteristic
231C4—22no—
232C4—24no—
233C4—41no—
234C4—42no—
235C4—48no—
236C4—84no—
237C4—84no—
238C4—161no—
239C4—161no—
240C2—24no—
241C2C2×#166311yescharacteristic
242C2#1900111yescharacteristic
243C2#1900712yes—
244C2#3299111yescharacteristic
245C2#3300012yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9004C2semidirect
5. #8913C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23822
#279562
#259122
#90042
#89132
#87192
#83352