Finite Group Databasebeta

← #150642

Subgroups — #150642

Order 512.

400 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 60 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×#2096C211yesmaximal, characteristic
2#12860C211yesmaximal, characteristic
3#11506C211yesmaximal, characteristic
4#7437C214yesmaximal, complemented
5C22×#106—21no—
6C22×#103—21no—
7C2×#391C2211yescharacteristic
8C2×#384C411yescharacteristic
9#2096—21no—
10#2096—21no—
11#2096C412yes—
12#2096C2212yes—
13#1784—21no—
14C2×#336C411yescharacteristic
15#1463—21no—
16#1394—21no—
17#1394C2212yes—
18#1319—21no—
19#1319C2212yes—
20C23×Q8D411yescharacteristic
21C23×C8—21no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—41no—
24C22×C4⋊C4D411yescharacteristic
25C22×C42C2×C411yescharacteristic
26C2×#106—22no—
27C2×#106—22no—
28C2×#106—42no—
29C2×#106—42no—
30C2×#103—22no—
31C2×#103—22no—
32C2×#103—42no—
33C2×#103—42no—
34C2×#101—22no—
35C2×#101—41no—
36#407—41no—
37#407—41no—
38#391—41no—
39#391C2×C411yescharacteristic
40#391D411yescharacteristic
41#391D411yescharacteristic
42#391C2311yescharacteristic
43#384—21no—
44#384—41no—
45#384C2×C412yes—
46C2×#95—41no—
47#336—21no—
48#336C2×C412yes—
49#324—22no—
50#324—41no—
51C22×Q8—21no—
52C22×Q8—22no—
53C22×Q8—41no—
54C22×Q8—41no—
55C22×Q8—41no—
56C22×Q8—42no—
57C22×Q8C22⋊C411yescharacteristic
58C22×Q8C2×D411yescharacteristic
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—41no—
63C23×C4C22⋊C411yescharacteristic
64C2×M16—42no—
65C2×M16—81no—
66C22×C8—22no—
67C22×C8—22no—
68C22×C8—22no—
69C22×C8—42no—
70C22×C8—42no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—84no—
83C2×C4⋊C4C22⋊C411yescharacteristic
84C2×C4⋊C4C2×D411yescharacteristic
85C2×C42—21no—
86C2×C42—42no—
87C2×C42C22⋊C411yescharacteristic
88C2×C42C22×C411yescharacteristic
89#106—44no—
90#106—44no—
91#103—44no—
92#103—44no—
93#101—42no—
94#101—42no—
95#101—82no—
96C22⋊C8—42no—
97C22⋊C8—81no—
98C22⋊C8—81no—
99C22⋊C8—81no—
100#95—41no—
101#95—41no—
102#95—41no—
103#95—41no—
104C24#9911yescharacteristic
105C2×Q8—21no—
106C2×Q8—21no—
107C2×Q8—21no—
108C2×Q8—42no—
109C2×Q8—41no—
110C2×Q8—41no—
111C2×Q8—41no—
112C2×Q8—41no—
113C2×Q8—44no—
114C2×Q8—44no—
115C2×Q8—42no—
116C2×Q8—81no—
117C2×Q8—82no—
118C2×Q8—82no—
119C2×Q8#13712yes—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—24no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—42no—
140C22×C4—81no—
141C22×C4—82no—
142C22×C4—84no—
143C22×C4—81no—
144C22×C4—81no—
145C22×C4#9911yescharacteristic
146C22×C4C2×C22⋊C411yescharacteristic
147M16—82no—
148M16—82no—
149M16—162no—
150C2×C8—24no—
151C2×C8—44no—
152C2×C8—44no—
153C2×C8—42no—
154C2×C8—42no—
155C2×C8—82no—
156C2×C8—82no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—81no—
162C4⋊C4—81no—
163C4⋊C4—81no—
164C4⋊C4—84no—
165C4⋊C4—84no—
166C42—41no—
167C42—41no—
168C42—42no—
169C23—21no—
170C23—22no—
171C23—22no—
172C23—21no—
173C23#35311yescharacteristic
174C23#35311yescharacteristic
175C23C2×#9911yescharacteristic
176Q8—42no—
177Q8—42no—
178Q8—81no—
179Q8—84no—
180Q8—81no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—48no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—84no—
210C2×C4—82no—
211C2×C4—161no—
212C2×C4—162no—
213C2×C4—162no—
214C2×C4#35212yes—
215C2×C4#41911yescharacteristic
216C2×C4#42111yescharacteristic
217C8—44no—
218C8—162no—
219C8—162no—
220C22—24no—
221C22—24no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22#140712yes—
229C22#141012yes—
230C22#151111yescharacteristic
231C22C2×#35211yescharacteristic
232C22C2×#35311yescharacteristic
233C4—21no—
234C4—22no—
235C4—21no—
236C4—42no—
237C4—42no—
238C4—42no—
239C4—81no—
240C4—81no—
241C4—81no—
242C4—162no—
243C2—21no—
244C2—22no—
245C2—22no—
246C2—21no—
247C2#1180911yescharacteristic
248C2C2×#140711yescharacteristic
249C2C2×#141011yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC2×C42C22×C4
Frattini, cosocle#391C23
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#391 ◁ G
LECSG ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7437C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20962
#128602
#115062
#74372