Finite Group Databasebeta

← #127850

Subgroups — #127850

Order 512.

552 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 128 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#21853C211yesmaximal, characteristic
2#17366C212yesmaximal
3#7286C214yesmaximal, complemented
4C2×C4×#41C2211yescharacteristic
5#2920—22no—
6C2×#432C411yescharacteristic
7C2×#423C411yescharacteristic
8#1985C2212yes—
9#1955C412yes—
10#1870—22no—
11#1797—22no—
12#1465—22no—
13#1465C414yes—
14#1464—22no—
15#1464C414yes—
16#1461C2214yes—
17#1460—22no—
18C22×#41Q811yescharacteristic
19C22×M16—21no—
20C4×#41—42no—
21C4×#41—42no—
22C2×C4×Q8Q811yescharacteristic
23C2×C4×Q8C2311yescharacteristic
24C2×C4×D4C2×C412yes—
25C2×C4×D4C2×C411yescharacteristic
26C2×C4×D4D411yescharacteristic
27C2×C4×D4D412yes—
28C2×#117C2×C412yes—
29C2×#117D411yescharacteristic
30C22×C42C2×C411yescharacteristic
31C22×C42D412yes—
32#432—24no—
33#432—42no—
34#432—42no—
35#431—24no—
36#431—42no—
37#423—42no—
38#423—42no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8C2×C412yes—
42#407—24no—
43C2×C22⋊C8—21no—
44C2×C22⋊C8—24no—
45C4×M16—42no—
46C2×#97C2×C412yes—
47#336—24no—
48#324—24no—
49#324C2×C418yes—
50#323—24no—
51#323—24no—
52C2×#41—42no—
53C2×#41—44no—
54C22×Q8C2×Q811yescharacteristic
55C22×D4C4⋊C412yes—
56C22×D4C4⋊C411yescharacteristic
57C23×C4C4211yescharacteristic
58C23×C4C22⋊C412yes—
59C23×C4C22⋊C411yescharacteristic
60C23×C4C22⋊C412yes—
61C23×C4C4⋊C412yes—
62C23×C4C4⋊C411yescharacteristic
63C2×M16—42no—
64C2×M16—44no—
65C2×M16—82no—
66C22×C8—22no—
67C22×C8—22no—
68C4×Q8—42no—
69C4×Q8—41no—
70C4×Q8—41no—
71C4×D4—42no—
72C4×D4—42no—
73C4×D4—41no—
74C4×D4—41no—
75C4×D4—42no—
76C4×D4—42no—
77C4×D4—42no—
78#117—42no—
79#117—42no—
80#117—42no—
81C2×C4⋊C4C4⋊C412yes—
82C2×C4⋊C4C4⋊C411yescharacteristic
83C2×C4⋊C4C22×C412yes—
84C2×C4⋊C4C2×D411yescharacteristic
85C2×C22⋊C4C4212yes—
86C2×C22⋊C4C22⋊C412yes—
87C2×C22⋊C4C22⋊C412yes—
88C2×C42—22no—
89C2×C42—22no—
90C2×C42—22no—
91C2×C42—22no—
92C2×C42—22no—
93C2×C42—22no—
94C2×C42C4211yescharacteristic
95C2×C42C22⋊C412yes—
96C2×C42C22⋊C411yescharacteristic
97C2×C42C4⋊C412yes—
98C2×C42C4⋊C411yescharacteristic
99C2×C42C22×C411yescharacteristic
100C2×C42C2×D412yes—
101C4⋊C8—42no—
102C4⋊C8—44no—
103C4⋊C8—44no—
104C4⋊C8—42no—
105C4⋊C8—42no—
106C22⋊C8—28no—
107C22⋊C8—42no—
108C22⋊C8—44no—
109C22⋊C8—84no—
110#97—42no—
111#97—44no—
112C4×C8—44no—
113C24#9512yes—
114C24#9511yescharacteristic
115#41—416no—
116C2×Q8—24no—
117C2×Q8—41no—
118C2×D4—24no—
119C2×D4—42no—
120C2×D4—41no—
121C2×D4—42no—
122C2×D4—42no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—24no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—82no—
141C22×C4—81no—
142C22×C4#9511yescharacteristic
143C22×C4#9511yescharacteristic
144C22×C4#9512yes—
145C22×C4#9511yescharacteristic
146C22×C4C2×C4211yescharacteristic
147C22×C4C2×C22⋊C411yescharacteristic
148C22×C4C2×C22⋊C412yes—
149C22×C4C2×C4⋊C411yescharacteristic
150C22×C4C2×C4⋊C412yes—
151M16—88no—
152M16—84no—
153C2×C8—48no—
154C2×C8—42no—
155C2×C8—44no—
156C2×C8—42no—
157C2×C8—42no—
158C2×C8—42no—
159C4⋊C4—42no—
160C4⋊C4—41no—
161C4⋊C4—42no—
162C4⋊C4—41no—
163C22⋊C4—42no—
164C22⋊C4—42no—
165C22⋊C4—42no—
166C42—22no—
167C42—22no—
168C42—22no—
169C42—22no—
170C42—22no—
171C42—22no—
172C42—21no—
173C42—22no—
174C42—22no—
175C42—21no—
176C42#9912yes—
177C42#9912yes—
178C23—22no—
179C23—22no—
180C23—21no—
181C23—22no—
182C23—22no—
183C23—42no—
184C23—42no—
185C23—41no—
186C23—41no—
187C23C2×#9511yescharacteristic
188Q8—44no—
189D4—48no—
190D4—44no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—42no—
207C2×C4—44no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4#34212yes—
220C2×C4#34212yes—
221C2×C4C2×#9911yescharacteristic
222C2×C4C2×#9911yescharacteristic
223C2×C4#43014yes—
224C8—82no—
225C8—88no—
226C8—82no—
227C22—21no—
228C22—21no—
229C22—44no—
230C22—42no—
231C22—41no—
232C22—41no—
233C22—42no—
234C22—42no—
235C22—41no—
236C22—82no—
237C22—82no—
238C22#173812yes—
239C22C2×#34211yescharacteristic
240C4—22no—
241C4—21no—
242C4—22no—
243C4—24no—
244C4—42no—
245C4—41no—
246C4—41no—
247C4—42no—
248C4—41no—
249C4#213411yescharacteristic
250C4#213511yescharacteristic
251C2—22no—
252C2—82no—
253C2—81no—
254C2#805511yescharacteristic
255C2#1257711yescharacteristic
256C2#1258711yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×Q8C23
socleC22C2×#342
prime radicalC22×#41Q8
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7286C2semidirect

Maximal subgroups

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

subgroupindex
#218532
#173662
#72862