Finite Group Databasebeta

← #174001

Subgroups — #174001

Order 512.

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

normal subgroups only

This group has 757 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#59869C211yesmaximal, characteristic
2#28689C212yesmaximal
3#28512C211yesmaximal, characteristic
4#7802C218yesmaximal
5#7571C211yesmaximal, characteristic
6#7570C212yesmaximal
7C2×#500—22no—
8C2×#475—22no—
9#2868C2211yescharacteristic
10#2809C2214yes—
11#2571—21no—
12#2568—22no—
13#2546—22no—
14#2508—24no—
15#2369—21no—
16#2355—22no—
17C4×C4⋊Q8C2212yes—
18C4×C4⋊Q8C2211yescharacteristic
19#2096C2214yes—
20#2083C2218yes—
21#1990—22no—
22#1956—21no—
23#1947—24no—
24#1844—21no—
25C4×C4⋊C8C2212yes—
26C4×C4⋊C8C2211yescharacteristic
27#1272C2214yes—
28#1271C2218yes—
29C2×C4⋊Q8—22no—
30C2×C4⋊Q8—21no—
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8D411yescharacteristic
33C2×C4⋊Q8D412yes—
34C2×C4×Q8—22no—
35C2×C4×Q8D412yes—
36#500—48no—
37#475—48no—
38C2×#107—28no—
39C2×#106—24no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—22no—
42C2×#103—24no—
43C2×C4×C8—21no—
44C2×C4×C8—22no—
45#400—24no—
46#400—24no—
47#398—28no—
48#391—24no—
49#391—24no—
50#391—22no—
51#391C2312yes—
52#389—24no—
53#389—24no—
54#389—22no—
55#389—22no—
56#389C2314yes—
57#387—28no—
58#384—21no—
59#384—22no—
60#384D411yescharacteristic
61#384D412yes—
62#384C2312yes—
63#382—24no—
64#382—24no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—24no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4D414yes—
70C43C2311yescharacteristic
71#340—28no—
72#336C2314yes—
73#336C2312yes—
74C22×Q8—22no—
75C22×Q8—41no—
76C22×Q8C2×D412yes—
77C22×C8—24no—
78C22×C8—22no—
79C4⋊Q8—24no—
80C4⋊Q8—24no—
81C4⋊Q8—22no—
82C4⋊Q8—44no—
83C4⋊Q8—44no—
84C4⋊Q8—44no—
85C4⋊Q8—84no—
86C4⋊Q8Q1614yes—
87C4×Q8—44no—
88C4×Q8—44no—
89C4×Q8—44no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4C2×D414yes—
104C2×C4⋊C4C2×D412yes—
105C2×C4⋊C4C2×D414yes—
106C2×C42—24no—
107C2×C42—22no—
108C2×C42—24no—
109C2×C42C2×D411yescharacteristic
110C2×C42C2×D412yes—
111C2×C42C2×D411yescharacteristic
112C2×C42C2×D412yes—
113C2×C42C2411yescharacteristic
114#107—416no—
115#106—48no—
116C4⋊C8—28no—
117C4⋊C8—24no—
118#103—48no—
119C4×C8—44no—
120C4×C8—42no—
121#95—24no—
122#95—28no—
123#95—28no—
124#95—24no—
125#95—24no—
126#95—24no—
127C2×Q8—22no—
128C2×Q8—42no—
129C2×Q8—42no—
130C2×Q8—44no—
131C2×Q8—44no—
132C2×Q8—44no—
133C2×Q8—84no—
134C2×Q8—82no—
135C2×Q8C2×Q1614yes—
136C22×C4—22no—
137C22×C4—24no—
138C22×C4—22no—
139C22×C4—24no—
140C22×C4—42no—
141C22×C4#12011yescharacteristic
142C22×C4#12011yescharacteristic
143C22×C4#12012yes—
144C22×C4C22×D411yescharacteristic
145C22×C4C22×D412yes—
146C2×C8—24no—
147C2×C8—28no—
148C2×C8—44no—
149C2×C8—42no—
150C2×C8—44no—
151C2×C8—42no—
152C4⋊C4—28no—
153C4⋊C4—24no—
154C4⋊C4—24no—
155C4⋊C4—42no—
156C4⋊C4—44no—
157C4⋊C4—44no—
158C4⋊C4—44no—
159C4⋊C4—44no—
160C4⋊C4—44no—
161C4⋊C4—44no—
162C4⋊C4—44no—
163C4⋊C4—44no—
164C4⋊C4—44no—
165C4⋊C4—44no—
166C4⋊C4—82no—
167C4⋊C4—82no—
168C4⋊C4—84no—
169C4⋊C4—84no—
170C4⋊C4—84no—
171C42—22no—
172C42—24no—
173C42—41no—
174C42—48no—
175C42—44no—
176C42—84no—
177C42C2×Q1612yes—
178C42#13514yes—
179C42AGL(1,ℤ/8)14yes—
180C42#13712yes—
181C23C2×#12011yescharacteristic
182Q8—44no—
183Q8—82no—
184Q8—88no—
185Q8—164no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—24no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—44no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—42no—
199C2×C4—44no—
200C2×C4—44no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4—84no—
204C2×C4—82no—
205C2×C4#44914yes—
206C2×C4#44914yes—
207C2×C4#45114yes—
208C2×C4#45312yes—
209C2×C4#45614yes—
210C2×C4C22×Q1611yescharacteristic
211C2×C4C2×#13512yes—
212C2×C4C2×AGL(1,ℤ/8)12yes—
213C2×C4C2×#13711yescharacteristic
214C8—44no—
215C8—48no—
216C22C2×#45111yescharacteristic
217C22C2×#44912yes—
218C22C2×#45311yescharacteristic
219C22C2×#45612yes—
220C22#302811yescharacteristic
221C4—21no—
222C4—21no—
223C4—22no—
224C4—22no—
225C4—41no—
226C4—44no—
227C4—84no—
228C4—82no—
229C4—84no—
230C4—162no—
231C4#162414yes—
232C4#164818yes—
233C2#1839711yescharacteristic
234C2#1839812yes—
235C2C2×#164812yes—
236C2C2×#162411yescharacteristic
237C2#1907811yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#120
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#598692
#286892
#285122
#78022
#75712
#75702