Finite Group Databasebeta

← #174041

Subgroups — #174041

Order 512.

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

normal subgroups only

This group has 721 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#29218C212yesmaximal
3#28511C211yesmaximal, characteristic
4#7831C218yesmaximal
5#7571C211yesmaximal, characteristic
6#7570C212yesmaximal
7C2×#477—22no—
8#2868C2211yescharacteristic
9#2809C2214yes—
10#2571—21no—
11#2568—22no—
12#2546—22no—
13#2508—24no—
14#2369—21no—
15#2355—22no—
16C4×C4⋊Q8C2212yes—
17C4×C4⋊Q8C2211yescharacteristic
18#2100C2218yes—
19#2095C2214yes—
20#1990—22no—
21#1987—24no—
22#1956—21no—
23#1845—21no—
24#1834—22no—
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#477—48no—
37C2×#107—24no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—22no—
40C2×#103—24no—
41C2×C4×C8—22no—
42C2×C4×C8—21no—
43#400—24no—
44#400—24no—
45#398—28no—
46#391—24no—
47#391—24no—
48#391—22no—
49#391C2312yes—
50#389—24no—
51#389—24no—
52#389—22no—
53#389—22no—
54#389C2314yes—
55#387—28no—
56#384—22no—
57#384—21no—
58#384D411yescharacteristic
59#384D412yes—
60#384C2312yes—
61#382—24no—
62#382—24no—
63C4×C4⋊C4—22no—
64C4×C4⋊C4—24no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4D414yes—
68C43C2311yescharacteristic
69#340—28no—
70#340—28no—
71#336C2312yes—
72#336C2314yes—
73C22×Q8—22no—
74C22×Q8—41no—
75C22×Q8C2×D412yes—
76C22×C8—22no—
77C22×C8—24no—
78C4⋊Q8—24no—
79C4⋊Q8—22no—
80C4⋊Q8—24no—
81C4⋊Q8—48no—
82C4⋊Q8—44no—
83C4⋊Q8—84no—
84C4⋊Q8Q1614yes—
85C4×Q8—44no—
86C4×Q8—44no—
87C4×Q8—44no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4C2×D414yes—
102C2×C4⋊C4C2×D412yes—
103C2×C4⋊C4C2×D414yes—
104C2×C42—24no—
105C2×C42—22no—
106C2×C42—24no—
107C2×C42C2×D411yescharacteristic
108C2×C42C2×D412yes—
109C2×C42C2×D411yescharacteristic
110C2×C42C2×D412yes—
111C2×C42C2411yescharacteristic
112#107—48no—
113C4⋊C8—24no—
114C4⋊C8—44no—
115#103—48no—
116C4×C8—44no—
117C4×C8—42no—
118#95—24no—
119#95—28no—
120#95—28no—
121#95—24no—
122#95—24no—
123#95—24no—
124C2×Q8—22no—
125C2×Q8—42no—
126C2×Q8—42no—
127C2×Q8—44no—
128C2×Q8—44no—
129C2×Q8—44no—
130C2×Q8—82no—
131C2×Q8—84no—
132C2×Q8C2×Q1614yes—
133C22×C4—22no—
134C22×C4—24no—
135C22×C4—22no—
136C22×C4—24no—
137C22×C4—42no—
138C22×C4#12011yescharacteristic
139C22×C4#12012yes—
140C22×C4#12011yescharacteristic
141C22×C4C22×D411yescharacteristic
142C22×C4C22×D412yes—
143C2×C8—24no—
144C2×C8—42no—
145C2×C8—42no—
146C2×C8—44no—
147C2×C8—44no—
148C2×C8—44no—
149C4⋊C4—28no—
150C4⋊C4—24no—
151C4⋊C4—24no—
152C4⋊C4—42no—
153C4⋊C4—44no—
154C4⋊C4—44no—
155C4⋊C4—48no—
156C4⋊C4—44no—
157C4⋊C4—44no—
158C4⋊C4—44no—
159C4⋊C4—44no—
160C4⋊C4—44no—
161C4⋊C4—44no—
162C4⋊C4—82no—
163C4⋊C4—84no—
164C4⋊C4—84no—
165C4⋊C4—82no—
166C4⋊C4—84no—
167C42—22no—
168C42—24no—
169C42—41no—
170C42—48no—
171C42—44no—
172C42—84no—
173C42C2×Q1612yes—
174C42#13514yes—
175C42#13712yes—
176C42#13714yes—
177C23C2×#12011yescharacteristic
178Q8—44no—
179Q8—82no—
180Q8—88no—
181Q8—164no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—24no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—44no—
192C2×C4—44no—
193C2×C4—44no—
194C2×C4—42no—
195C2×C4—44no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—44no—
199C2×C4—82no—
200C2×C4—84no—
201C2×C4#45114yes—
202C2×C4#45214yes—
203C2×C4#45214yes—
204C2×C4#45312yes—
205C2×C4#45414yes—
206C2×C4C22×Q1611yescharacteristic
207C2×C4C2×#13512yes—
208C2×C4C2×#13711yescharacteristic
209C2×C4C2×#13712yes—
210C8—44no—
211C8—84no—
212C22C2×#45111yescharacteristic
213C22C2×#45212yes—
214C22C2×#45311yescharacteristic
215C22C2×#45412yes—
216C22#302811yescharacteristic
217C4—21no—
218C4—21no—
219C4—22no—
220C4—22no—
221C4—41no—
222C4—44no—
223C4—84no—
224C4—82no—
225C4—84no—
226C4—162no—
227C4#163414yes—
228C4#163618yes—
229C2#1839711yescharacteristic
230C2#1839912yes—
231C2C2×#163612yes—
232C2C2×#163411yescharacteristic
233C2#1907711yescharacteristic
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
#292182
#285112
#78312
#75712
#75702