Finite Group Databasebeta

← #127849

Subgroups — #127849

Order 512.

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

normal subgroups only

This group has 523 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#21822C211yesmaximal, characteristic
2#17365C212yesmaximal
3#7286C214yesmaximal, complemented
4C22×C4×Q8C2211yescharacteristic
5#2920—22no—
6C2×#432C411yescharacteristic
7C2×#432C411yescharacteristic
8#1985—22no—
9#1985C412yes—
10#1985C2212yes—
11#1817—22no—
12#1467—22no—
13#1467C414yes—
14#1463—22no—
15#1463C414yes—
16#1461—22no—
17#1461C2214yes—
18C23×Q8Q811yescharacteristic
19C2×C4×Q8—42no—
20C2×C4×Q8—42no—
21C2×C4×Q8C2×C412yes—
22C2×C4×Q8C2×C411yescharacteristic
23C2×C4×Q8D411yescharacteristic
24C2×C4×Q8D412yes—
25C2×C4×Q8Q811yescharacteristic
26C2×C4×Q8C2311yescharacteristic
27C22×C4⋊C4C2×C412yes—
28C22×C4⋊C4D411yescharacteristic
29C22×C42C2×C411yescharacteristic
30C22×C42D412yes—
31#432—24no—
32#432—42no—
33#432—42no—
34#432—42no—
35#431—24no—
36#431—42no—
37#423—42no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8C2×C412yes—
40#407—24no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43C4×M16—42no—
44C2×#97—21no—
45C2×#97C2×C412yes—
46#336—24no—
47#336—24no—
48#324—24no—
49#324—24no—
50#324—24no—
51#324C2×C418yes—
52C22×Q8—42no—
53C22×Q8—44no—
54C22×Q8C4⋊C412yes—
55C22×Q8C4⋊C411yescharacteristic
56C22×Q8C2×Q811yescharacteristic
57C23×C4C4211yescharacteristic
58C23×C4C22⋊C412yes—
59C23×C4C22⋊C411yescharacteristic
60C23×C4C4⋊C412yes—
61C23×C4C4⋊C411yescharacteristic
62C2×M16—44no—
63C22×C8—22no—
64C22×C8—22no—
65C4×Q8—42no—
66C4×Q8—42no—
67C4×Q8—41no—
68C4×Q8—41no—
69C4×Q8—42no—
70C4×Q8—42no—
71C4×Q8—42no—
72C4×Q8—42no—
73C4×Q8—41no—
74C4×Q8—41no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4C4212yes—
79C2×C4⋊C4C22⋊C412yes—
80C2×C4⋊C4C22⋊C412yes—
81C2×C4⋊C4C4⋊C412yes—
82C2×C4⋊C4C4⋊C411yescharacteristic
83C2×C4⋊C4C22×C412yes—
84C2×C4⋊C4C2×D411yescharacteristic
85C2×C42—22no—
86C2×C42—22no—
87C2×C42—22no—
88C2×C42—22no—
89C2×C42—22no—
90C2×C42—22no—
91C2×C42C4211yescharacteristic
92C2×C42C22⋊C412yes—
93C2×C42C22⋊C412yes—
94C2×C42C22⋊C411yescharacteristic
95C2×C42C4⋊C412yes—
96C2×C42C4⋊C411yescharacteristic
97C2×C42C22×C411yescharacteristic
98C2×C42C2×D412yes—
99C4⋊C8—42no—
100C4⋊C8—42no—
101C4⋊C8—44no—
102C4⋊C8—44no—
103C22⋊C8—28no—
104C22⋊C8—44no—
105C22⋊C8—42no—
106C22⋊C8—42no—
107#97—42no—
108#97—42no—
109#97—44no—
110C4×C8—44no—
111C24#9511yescharacteristic
112C2×Q8—24no—
113C2×Q8—24no—
114C2×Q8—42no—
115C2×Q8—41no—
116C2×Q8—41no—
117C2×Q8—42no—
118C2×Q8—42no—
119C2×Q8—416no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—24no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—82no—
131C22×C4—81no—
132C22×C4#9511yescharacteristic
133C22×C4#9512yes—
134C22×C4#9512yes—
135C22×C4#9511yescharacteristic
136C22×C4#9511yescharacteristic
137C22×C4#9912yes—
138C22×C4#9912yes—
139C22×C4C2×C4211yescharacteristic
140C22×C4C2×C22⋊C411yescharacteristic
141C22×C4C2×C22⋊C412yes—
142C22×C4C2×C4⋊C411yescharacteristic
143C22×C4C2×C4⋊C412yes—
144M16—88no—
145C2×C8—48no—
146C2×C8—44no—
147C2×C8—44no—
148C2×C8—42no—
149C2×C8—42no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—41no—
155C4⋊C4—42no—
156C4⋊C4—41no—
157C42—22no—
158C42—22no—
159C42—22no—
160C42—22no—
161C42—22no—
162C42—22no—
163C42—21no—
164C42—21no—
165C42—22no—
166C42—22no—
167C42—22no—
168C42—21no—
169C42—21no—
170C42—22no—
171C23—22no—
172C23—41no—
173C23—41no—
174C23#34212yes—
175C23C2×#9511yescharacteristic
176Q8—44no—
177Q8—48no—
178Q8—44no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—24no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—44no—
199C2×C4—44no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4#34212yes—
207C2×C4C2×#9911yescharacteristic
208C2×C4C2×#9911yescharacteristic
209C2×C4#43014yes—
210C8—82no—
211C8—88no—
212C8—82no—
213C22—21no—
214C22—21no—
215C22—22no—
216C22—22no—
217C22—22no—
218C22—44no—
219C22#173812yes—
220C22C2×#34211yescharacteristic
221C4—21no—
222C4—22no—
223C4—24no—
224C4—42no—
225C4—42no—
226C4—41no—
227C4—41no—
228C4—42no—
229C4—41no—
230C4—41no—
231C4#213312yes—
232C2—22no—
233C2—42no—
234C2#805511yescharacteristic
235C2#1257711yescharacteristic
236C2#1258611yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×Q8C23
socleC22C2×#342
prime radicalC24#95
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×Q8 ◁ 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
#218222
#173652
#72862