Finite Group Databasebeta

← #149818

Subgroups — #149818

Order 512.

372 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 62 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#20800C211yesmaximal, characteristic
2#11513C211yesmaximal, characteristic
3#11513C211yesmaximal, characteristic
4#7372C214yesmaximal, complemented
5C2×#428—21no—
6C2×#428—21no—
7C2×C4×M16C411yescharacteristic
8#2285C2211yescharacteristic
9C2×#384C411yescharacteristic
10#1847C412yes—
11#1847C2212yes—
12#1845—21no—
13#1844—21no—
14#1464—21no—
15#1464—21no—
16#1325—21no—
17#1325—21no—
18#1324C2212yes—
19#1324C2212yes—
20C22×M16—21no—
21C2×#117D411yescharacteristic
22C2×#117D411yescharacteristic
23C22×C4⋊C4—21no—
24C22×C4⋊C4—41no—
25C22×C42C2×C411yescharacteristic
26#432—41no—
27#432—41no—
28#428—42no—
29#428—42no—
30#428—42no—
31#428—42no—
32C2×#107—22no—
33C2×#107—22no—
34C2×#106—22no—
35C2×#106—22no—
36C4×M16—24no—
37C2×#97C2×C412yes—
38C2×C4×C8—21no—
39#384—21no—
40#384—41no—
41#384C2×C412yes—
42#384D411yescharacteristic
43#384D411yescharacteristic
44#382C2×C411yescharacteristic
45#382C2311yescharacteristic
46#380—41no—
47C4×C4⋊C4—41no—
48#324—22no—
49#324—22no—
50#323—41no—
51#323—41no—
52C23×C4—21no—
53C23×C4—41no—
54C23×C4C22⋊C411yescharacteristic
55C2×M16—24no—
56C2×M16—24no—
57C2×M16—22no—
58C2×M16—22no—
59C22×C8—22no—
60#117—41no—
61#117—41no—
62#117—41no—
63#117—41no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—84no—
75C2×C4⋊C4C22⋊C411yescharacteristic
76C2×C4⋊C4C22⋊C411yescharacteristic
77C2×C4⋊C4C2×D411yescharacteristic
78C2×C4⋊C4C2×D411yescharacteristic
79C2×C22⋊C4—21no—
80C2×C22⋊C4—21no—
81C2×C42—21no—
82C2×C42—22no—
83C2×C42—22no—
84C2×C42—21no—
85C2×C42—21no—
86C2×C42C22⋊C411yescharacteristic
87C2×C42C22×C411yescharacteristic
88#107—44no—
89#107—44no—
90#106—44no—
91#106—44no—
92C4⋊C8—81no—
93C4⋊C8—81no—
94C22⋊C8—42no—
95C22⋊C8—42no—
96#97—22no—
97#97—81no—
98#97—81no—
99#97M1614yes—
100C4×C8—24no—
101#95—41no—
102#95—41no—
103C24#9911yescharacteristic
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—24no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—81no—
120C22×C4—81no—
121C22×C4—84no—
122C22×C4#9911yescharacteristic
123C22×C4C2×C22⋊C411yescharacteristic
124M16—28no—
125M16—28no—
126C2×C8—24no—
127C2×C8—24no—
128C2×C8—24no—
129C2×C8—82no—
130C2×C8—82no—
131C4⋊C4—41no—
132C4⋊C4—41no—
133C4⋊C4—41no—
134C4⋊C4—41no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4—82no—
138C4⋊C4—82no—
139C4⋊C4—84no—
140C4⋊C4—84no—
141C4⋊C4—81no—
142C4⋊C4—81no—
143C22⋊C4—81no—
144C22⋊C4—81no—
145C42—24no—
146C42—21no—
147C42—21no—
148C42—41no—
149C42—41no—
150C42—41no—
151C42—41no—
152C42#10112yes—
153C42C2×M1612yes—
154C23—21no—
155C23—21no—
156C23—41no—
157C23—41no—
158C23#35612yes—
159C23C2×#9911yescharacteristic
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—48no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—81no—
180C2×C4—81no—
181C2×C4—81no—
182C2×C4—81no—
183C2×C4—84no—
184C2×C4—82no—
185C2×C4—162no—
186C2×C4—162no—
187C2×C4#35311yescharacteristic
188C2×C4#35311yescharacteristic
189C2×C4#40711yescharacteristic
190C2×C4C2×#10111yescharacteristic
191C8—28no—
192C8—162no—
193C8—162no—
194C22—21no—
195C22—21no—
196C22—22no—
197C22—42no—
198C22—42no—
199C22—41no—
200C22—41no—
201C22#146311yescharacteristic
202C22C2×#35311yescharacteristic
203C22C2×#35611yescharacteristic
204C4—24no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—42no—
210C4—81no—
211C4—81no—
212C4—81no—
213C4—81no—
214C4—162no—
215C2—21no—
216C2—21no—
217C2—42no—
218C2#1151211yescharacteristic
219C2#1357911yescharacteristic
220C2#1358011yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC2×C42C22×C4
Frattini, cosocle#382C23
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2285 ◁ G
LECSG ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7372C2semidirect

Maximal subgroups

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

subgroupindex
#208002
#115132
#73722