Finite Group Databasebeta

← #150520

Subgroups — #150520

Order 512.

280 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 54 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#29269C211yesmaximal, characteristic
2#12632C211yesmaximal, characteristic
3#11535C211yesmaximal, characteristic
4#7437C212yesmaximal, complemented
5#7429C212yesmaximal, complemented
6#2330C411yescharacteristic
7#2326C2211yescharacteristic
8C2×#355—21no—
9#2096C2211yescharacteristic
10#2094C411yescharacteristic
11#2026C2211yescharacteristic
12#2025C411yescharacteristic
13#1983—21no—
14#1948—21no—
15#1865—21no—
16#1864—21no—
17#1832C411yescharacteristic
18C4×#101—21no—
19#1465—21no—
20#1394C2212yes—
21#1325—21no—
22#1319C2212yes—
23C2×#125C2×C411yescharacteristic
24C2×#124D411yescharacteristic
25C2×C4×Q8D411yescharacteristic
26C2×C4×D4D411yescharacteristic
27C2×#106—21no—
28C2×#106—21no—
29C2×C4⋊C8—21no—
30C2×#103—21no—
31C2×#103—21no—
32C2×#102—21no—
33C2×#102—21no—
34C2×#101—22no—
35#407—41no—
36C4×M16—41no—
37#391C2311yescharacteristic
38#390C2×C411yescharacteristic
39#389—21no—
40#389—41no—
41#388—41no—
42#387—41no—
43#385—41no—
44#384C2×C411yescharacteristic
45#384D411yescharacteristic
46C4×C4⋊C4—21no—
47#376C2×C411yescharacteristic
48#355—24no—
49#343—41no—
50#340—21no—
51#340—21no—
52#336—21no—
53#336C2×C411yescharacteristic
54#336C2×C411yescharacteristic
55#324—22no—
56#323—41no—
57C22×Q8C2×D411yescharacteristic
58C22×D4C22⋊C411yescharacteristic
59C23×C4—21no—
60C2×M16—42no—
61C2×M16—81no—
62C22×C8—21no—
63C22×C8—21no—
64#125—42no—
65#124—22no—
66#124—22no—
67C4×Q8—81no—
68C4×D4—81no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4C22⋊C411yescharacteristic
77C2×C4⋊C4C2×D411yescharacteristic
78C2×C22⋊C4—21no—
79C2×C22⋊C4—41no—
80C2×C22⋊C4—41no—
81C2×C42—21no—
82C2×C42—41no—
83C2×C42C22⋊C411yescharacteristic
84C2×C42C22⋊C411yescharacteristic
85C2×C42C22×C411yescharacteristic
86#106—81no—
87#106—81no—
88C4⋊C8—42no—
89#103—42no—
90#103—81no—
91#102—42no—
92#102—81no—
93#101—24no—
94#101—42no—
95C22⋊C8—42no—
96C22⋊C8—81no—
97#97—81no—
98C4×C8—81no—
99#95—21no—
100#95—21no—
101#95—41no—
102#95—41no—
103#95—41no—
104C24—21no—
105C2×Q8—21no—
106C2×Q8—41no—
107C2×Q8—41no—
108C2×Q8#13712yes—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4—41no—
112C2×D4—41no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—41no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4—41no—
122C22×C4—161no—
123C22×C4#9911yescharacteristic
124C22×C4#9911yescharacteristic
125C22×C4C2×C22⋊C411yescharacteristic
126M16—44no—
127M16—82no—
128M16—162no—
129C2×C8—41no—
130C2×C8—41no—
131C2×C8—41no—
132C2×C8—41no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—82no—
136C2×C8—82no—
137C4⋊C4—22no—
138C4⋊C4—41no—
139C4⋊C4—41no—
140C4⋊C4—41no—
141C4⋊C4—41no—
142C4⋊C4—41no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—81no—
146C4⋊C4—161no—
147C4⋊C4—161no—
148C4⋊C4—161no—
149C22⋊C4—44no—
150C22⋊C4—81no—
151C22⋊C4—161no—
152C42—21no—
153C42—41no—
154C42—41no—
155C42—41no—
156C42—41no—
157C42—161no—
158C42#10412yes—
159C23—41no—
160C23—41no—
161C23—41no—
162C23—81no—
163C23—81no—
164C23C2×#9911yescharacteristic
165Q8—42no—
166Q8—81no—
167D4—42no—
168D4—81no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—82no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4—81no—
194C2×C4—81no—
195C2×C4—161no—
196C2×C4#41911yescharacteristic
197C2×C4C2×#10411yescharacteristic
198C8—81no—
199C8—81no—
200C8—84no—
201C8—162no—
202C22—21no—
203C22—21no—
204C22—82no—
205C22—81no—
206C22—81no—
207C22—81no—
208C22—161no—
209C22#150811yescharacteristic
210C22#212011yescharacteristic
211C22#212711yescharacteristic
212C4—21no—
213C4—21no—
214C4—42no—
215C4—42no—
216C4—41no—
217C4—41no—
218C4—81no—
219C4—82no—
220C4—81no—
221C4—81no—
222C4—161no—
223C2—21no—
224C2—21no—
225C2—161no—
226C2#1178411yescharacteristic
227C2#1360711yescharacteristic
228C2#1361111yescharacteristic
subgroupG/H
center, socleC22#2127
commutatorC2×C42C22×C4
Frattini, cosocle#391C23
prime radicalC22×D4C22⋊C4
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C42 ◁ #2326 ◁ G
LECSG ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7437C2semidirect
5. #7429C2semidirect

Maximal subgroups

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

subgroupindex
#292692
#126322
#115352
#74372
#74292