Finite Group Databasebeta

← #150913

Subgroups — #150913

Order 512.

379 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 379 conjugacy classes of subgroups — too many to draw. Only its 64 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#28509C211yesmaximal, characteristic
2#12797C211yesmaximal, characteristic
3#11475C211yesmaximal, characteristic
4#7435C214yesmaximal, complemented
5C2×#476—21no—
6C2×#476—21no—
7#2868C411yescharacteristic, complemented
8C4×#127C2211yescharacteristic
9C2×#353—21no—
10#2094C412yescomplemented
11#2094C2212yes—
12#1957—21no—
13#1844—21no—
14#1799—21no—
15C4×C4⋊C8C411yescharacteristic, complemented
16#1459—21no—
17#1393C2212yes—
18#1325—21no—
19#1318C2212yes—
20C2×C4⋊Q8—21no—
21C2×C4⋊Q8—41no—
22C2×#127D411yescharacteristic
23C2×C4×D4—41no—
24C2×C4×D4D411yescharacteristic
25C2×C4×D4D411yescharacteristic
26#476—42no—
27#476—42no—
28#476—42no—
29#476—42no—
30C2×#106—22no—
31C2×#106—22no—
32C2×C4⋊C8—21no—
33C2×#102—22no—
34C2×#102—22no—
35C2×#99—22no—
36C2×C22⋊C8—41no—
37C2×C4×C8—21no—
38#390—41no—
39#390C2×C411yescharacteristic
40#390C2311yescharacteristic
41#384—21no—
42#384—41no—
43#384C2×C412yes—
44#384D411yescharacteristic
45#380—41no—
46C43C2×C411yescharacteristic
47#353—42no—
48#342—41no—
49#336C2×C412yes—
50#324—22no—
51#323—41no—
52C22×Q8—41no—
53C22×D4—41no—
54C22×D4C22⋊C411yescharacteristic
55C22×D4C2×D411yescharacteristic
56C23×C4—21no—
57C23×C4—21no—
58C22×C8—22no—
59C22×C8—81no—
60C4⋊Q8—41no—
61C4⋊Q8—41no—
62C4⋊Q8—41no—
63C4⋊Q8—41no—
64C4⋊Q8—84no—
65#127—41no—
66#127SD1612yes—
67#127Q1612yes—
68C4×D4—41no—
69C4×D4—41no—
70C4×D4—41no—
71C4×D4—41no—
72C4×D4—84no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—81no—
79C2×C4⋊C4C22⋊C411yescharacteristic
80C2×C4⋊C4C2×D411yescharacteristic
81C2×C22⋊C4—21no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—41no—
84C2×C22⋊C4—41no—
85C2×C22⋊C4—42no—
86C2×C42—21no—
87C2×C42—21no—
88C2×C42C22⋊C411yescharacteristic
89C2×C42C22⋊C411yescharacteristic
90C2×C42C22×C411yescharacteristic
91#106—44no—
92#106—44no—
93C4⋊C8—22no—
94C4⋊C8—22no—
95#102—44no—
96#102—44no—
97#99—42no—
98#99—42no—
99C22⋊C8—42no—
100C22⋊C8—81no—
101C4×C8—42no—
102#95—81no—
103C24—21no—
104C24—21no—
105C2×Q8—81no—
106C2×Q8—81no—
107C2×Q8—84no—
108C2×D4—21no—
109C2×D4—21no—
110C2×D4—41no—
111C2×D4—41no—
112C2×D4—41no—
113C2×D4—41no—
114C2×D4—42no—
115C2×D4—42no—
116C2×D4—82no—
117C2×D4—82no—
118C2×D4#10312yes—
119C2×D4C2×SD1611yescharacteristic
120C2×D4C2×Q1611yescharacteristic
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—42no—
130C22×C4—82no—
131C22×C4—82no—
132C22×C4—82no—
133C22×C4#9911yescharacteristic
134C22×C4#9911yescharacteristic
135C22×C4C2×C22⋊C411yescharacteristic
136C2×C8—24no—
137C2×C8—42no—
138C2×C8—42no—
139C2×C8—82no—
140C2×C8—162no—
141C4⋊C4—41no—
142C4⋊C4—41no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—82no—
146C4⋊C4—84no—
147C4⋊C4—84no—
148C4⋊C4—81no—
149C4⋊C4—81no—
150C4⋊C4—162no—
151C22⋊C4—81no—
152C22⋊C4—81no—
153C22⋊C4—82no—
154C22⋊C4—82no—
155C22⋊C4—82no—
156C22⋊C4—82no—
157C42—24no—
158C42—21no—
159C42—21no—
160C42—21no—
161C42—41no—
162C42—42no—
163C42#10312yes—
164C23—21no—
165C23—22no—
166C23—21no—
167C23—41no—
168C23—41no—
169C23—41no—
170C23—41no—
171C23—81no—
172C23—81no—
173C23—81no—
174C23—81no—
175C23C2×#9911yescharacteristic
176Q8—162no—
177Q8—162no—
178D4—42no—
179D4—41no—
180D4—41no—
181D4—81no—
182D4—81no—
183D4—84no—
184D4—84no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4—84no—
203C2×C4—82no—
204C2×C4—161no—
205C2×C4—161no—
206C2×C4—162no—
207C2×C4—162no—
208C2×C4—162no—
209C2×C4#35311yescharacteristic
210C2×C4#35311yescharacteristic
211C2×C4C2×#10311yescharacteristic
212C2×C4#42111yescharacteristic
213C8—44no—
214C8—162no—
215C22—21no—
216C22—21no—
217C22—41no—
218C22—42no—
219C22—41no—
220C22—81no—
221C22—82no—
222C22—81no—
223C22—81no—
224C22—81no—
225C22—81no—
226C22—81no—
227C22—161no—
228C22—161no—
229C22#152011yescharacteristic
230C22#212311yescharacteristic
231C22C2×#35311yescharacteristic
232C4—24no—
233C4—21no—
234C4—21no—
235C4—21no—
236C4—42no—
237C4—41no—
238C4—81no—
239C4—81no—
240C4—162no—
241C4—322no—
242C4#140912yes—
243C2—21no—
244C2—21no—
245C2—81no—
246C2—81no—
247C2—82no—
248C2#1190111yescharacteristic
249C2#1357911yescharacteristic
250C2C2×#140911yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC2×C42C22×C4
Frattini, cosocle#390C23
prime radicalC2×#127D4
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#127 ◁ G
LECSG ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7435C2semidirect
7. #2868C4semidirect
10. #2094C4semidirect
15. C4×C4⋊C8C4semidirect

Maximal subgroups

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

subgroupindex
#285092
#127972
#114752
#74352