Finite Group Databasebeta

← #120285

Subgroups — #120285

Order 512.

449 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 449 conjugacy classes of subgroups — too many to draw. Only its 124 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#20815C211yesmaximal, characteristic
2#17815C211yesmaximal, characteristic
3#7980C211yesmaximal, characteristic
4#7954C212yesmaximal
5#7940C212yesmaximal
6#2920C2212yes—
7C2×#432C2211yescharacteristic
8C2×#431C2211yescharacteristic
9#2283C2212yes—
10C4×#117C2211yescharacteristic
11#1842C412yescomplemented
12#1831C412yescomplemented
13#1798—22no—
14#1768—21no—
15#1768—21no—
16#1572—24no—
17#1567—21no—
18#1566—21no—
19#1289—22no—
20#1285—24no—
21#1284—24no—
22C2×#117—21no—
23C22×C42C2311yescharacteristic
24#432—22no—
25#432—22no—
26#432—22no—
27#432D414yes—
28#431—22no—
29#431—22no—
30#431—22no—
31#431—22no—
32#423—22no—
33C2×C4⋊C8—22no—
34C2×C4⋊C8—22no—
35C2×C4⋊C8C2×C412yes—
36C2×C4⋊C8C2×C412yes—
37#407—22no—
38C2×C22⋊C8—21no—
39C2×C22⋊C8D412yes—
40C4×M16—22no—
41C2×#97—21no—
42C2×C4×C8D411yescharacteristic
43C2×C4×C8D411yescharacteristic
44#383—22no—
45#382—22no—
46#381—22no—
47#380—22no—
48C4×C4⋊C4—21no—
49C4×C4⋊C4—22no—
50C4×C4⋊C4Q812yes—
51C4×C22⋊C4—21no—
52C4×C22⋊C4—22no—
53C4×C22⋊C4D412yes—
54#376—21no—
55#376Q814yes—
56C43C2×C411yescharacteristic
57C43C2×C411yescharacteristic
58#336—22no—
59#336—22no—
60#335—24no—
61#334—24no—
62C23×C4C2×D411yescharacteristic
63C23×C4C2×D411yescharacteristic
64C23×C4#4111yescharacteristic
65C2×M16—42no—
66C22×C8—22no—
67C22×C8C22⋊C412yes—
68C22×C8C22⋊C412yes—
69#117—44no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—42no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—42no—
74C2×C42—21no—
75C2×C42—21no—
76C2×C42—21no—
77C2×C42—21no—
78C2×C42—24no—
79C2×C42C4⋊C412yes—
80C2×C42C4⋊C412yes—
81C2×C42C22×C411yescharacteristic
82C2×C42C2×D412yes—
83C2×C42C2×D411yescharacteristic
84C2×C42C2×Q812yes—
85C2×C42C2×Q811yescharacteristic
86C2×C42#4112yes—
87C2×C42#4112yes—
88C2×C42#4111yescharacteristic
89C4⋊C8—24no—
90C4⋊C8—24no—
91C4⋊C8—24no—
92C4⋊C8—44no—
93C4⋊C8—44no—
94C4⋊C8—42no—
95C22⋊C8—24no—
96C22⋊C8—24no—
97C22⋊C8—42no—
98C22⋊C8—42no—
99#97—24no—
100#97—41no—
101#97—41no—
102C4×C8—21no—
103C4×C8—21no—
104C4×C8—21no—
105C4×C8—21no—
106C4×C8—42no—
107#95—22no—
108#95—22no—
109#95—24no—
110#95—24no—
111#95—42no—
112#95—42no—
113C24#12111yescharacteristic
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—44no—
119C22×C4—81no—
120C22×C4C2×C22⋊C411yescharacteristic
121C22×C4C2×C4⋊C411yescharacteristic
122C22×C4#11711yescharacteristic
123C22×C4#12112yes—
124C22×C4#12111yescharacteristic
125C22×C4#12112yes—
126C22×C4C22⋊Q812yes—
127C22×C4C22⋊Q812yes—
128C22×C4C22⋊Q812yes—
129C22×C4C22⋊Q811yescharacteristic
130C22×C4C22⋊Q811yescharacteristic
131C22×C4#12612yes—
132M16—84no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—42no—
136C2×C8—42no—
137C2×C8—42no—
138C2×C8—42no—
139C2×C8—42no—
140C2×C8—44no—
141C4⋊C4—44no—
142C4⋊C4—84no—
143C22⋊C4—44no—
144C22⋊C4—84no—
145C42—24no—
146C42—42no—
147C42—42no—
148C42—42no—
149C42—81no—
150C42—81no—
151C42—84no—
152C42C4×D412yes—
153C42C4×Q812yes—
154C42#12011yescharacteristic
155C42#12311yescharacteristic
156C42#12312yes—
157C42#12411yescharacteristic
158C42#12512yes—
159C42#12512yes—
160C42C4⋊Q812yes—
161C42C4⋊Q811yescharacteristic
162C23—22no—
163C23—41no—
164C23—41no—
165C23#38011yescharacteristic
166C23#39612yes—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4—24no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—84no—
186C2×C4—84no—
187C2×C4—84no—
188C2×C4#38611yescharacteristic
189C2×C4#38712yes—
190C2×C4#38912yes—
191C2×C4#39111yescharacteristic
192C2×C4#39411yescharacteristic
193C2×C4#39511yescharacteristic
194C2×C4#39812yes—
195C2×C4#39811yescharacteristic
196C2×C4#40011yescharacteristic
197C8—42no—
198C8—42no—
199C8—84no—
200C8—82no—
201C22—21no—
202C22—21no—
203C22—22no—
204C22—22no—
205C22—22no—
206C22—82no—
207C22#144111yescharacteristic
208C22#181212yes—
209C4—21no—
210C4—22no—
211C4—21no—
212C4—22no—
213C4—21no—
214C4—22no—
215C4—21no—
216C4—21no—
217C4—44no—
218C4—82no—
219C4—164no—
220C4#188412yes—
221C2—22no—
222C2—42no—
223C2#1121211yescharacteristic
224C2#1122911yescharacteristic
225C2#1124611yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #1842C4semidirect
12. #1831C4semidirect

Maximal subgroups

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

subgroupindex
#208152
#178152
#79802
#79542
#79402