Finite Group Databasebeta

← #127384

Subgroups — #127384

Order 512.

563 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 563 conjugacy classes of subgroups — too many to draw. Only its 136 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#21818C211yesmaximal, characteristic
2#20727C212yesmaximal
3#7238C214yesmaximal, complemented
4C22×C4×Q8C2211yescharacteristic
5#2920—22no—
6C2×#431C411yescharacteristic
7C2×#431C411yescharacteristic
8#1984—22no—
9#1984C412yes—
10#1984C2212yes—
11#1821C414yes—
12#1817—22no—
13#1801—22no—
14#1801C2214yes—
15#1784—22no—
16C4×#101—22no—
17C4×#101C414yes—
18C23×Q8D411yescharacteristic
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⋊C4Q811yescharacteristic
29C22×C42C2×C411yescharacteristic
30C22×C42D412yes—
31#432—42no—
32#432—42no—
33#431—42no—
34#431—42no—
35#431—42no—
36#431—42no—
37#423—24no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8C2×C412yes—
40C2×#101—24no—
41C2×#101—24no—
42#407—24no—
43C2×C22⋊C8—21no—
44C2×C22⋊C8—21no—
45C4×M16—24no—
46C2×C4×C8—21no—
47C2×C4×C8C2×C412yes—
48#343—24no—
49#343C2×C418yes—
50#336—24no—
51#336—24no—
52C22×Q8—24no—
53C22×Q8—42no—
54C22×Q8—42no—
55C22×Q8C22⋊C412yes—
56C22×Q8C4⋊C411yescharacteristic
57C22×Q8C2×D411yescharacteristic
58C23×C4C4211yescharacteristic
59C23×C4C22⋊C412yes—
60C23×C4C22⋊C411yescharacteristic
61C23×C4C4⋊C411yescharacteristic
62C23×C4C4⋊C412yes—
63C2×M16—28no—
64C22×C8—22no—
65C22×C8—22no—
66C4×Q8—42no—
67C4×Q8—42no—
68C4×Q8—41no—
69C4×Q8—41no—
70C4×Q8—42no—
71C4×Q8—42no—
72C4×Q8—42no—
73C4×Q8—42no—
74C4×Q8—41no—
75C4×Q8—41no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4C4212yes—
80C2×C4⋊C4C22⋊C412yes—
81C2×C4⋊C4C4⋊C412yes—
82C2×C4⋊C4C4⋊C412yes—
83C2×C4⋊C4C4⋊C411yescharacteristic
84C2×C4⋊C4C22×C412yes—
85C2×C4⋊C4C2×Q811yescharacteristic
86C2×C42—22no—
87C2×C42—22no—
88C2×C42—22no—
89C2×C42—22no—
90C2×C42—22no—
91C2×C42—22no—
92C2×C42C4211yescharacteristic
93C2×C42C22⋊C412yes—
94C2×C42C22⋊C412yes—
95C2×C42C22⋊C411yescharacteristic
96C2×C42C4⋊C412yes—
97C2×C42C4⋊C411yescharacteristic
98C2×C42C22×C411yescharacteristic
99C2×C42C2×D412yes—
100C4⋊C8—42no—
101C4⋊C8—42no—
102C4⋊C8—44no—
103C4⋊C8—44no—
104#101—48no—
105#101—48no—
106C22⋊C8—44no—
107C22⋊C8—44no—
108C22⋊C8—42no—
109C22⋊C8—42no—
110#97—44no—
111C4×C8—42no—
112C4×C8—42no—
113C4×C8—44no—
114C24#9511yescharacteristic
115C2×Q8—24no—
116C2×Q8—24no—
117C2×Q8—22no—
118C2×Q8—42no—
119C2×Q8—41no—
120C2×Q8—41no—
121C2×Q8—41no—
122C2×Q8—41no—
123C2×Q8—416no—
124C2×Q8#13514yes—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—24no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—82no—
137C22×C4—81no—
138C22×C4#9512yes—
139C22×C4#9511yescharacteristic
140C22×C4#9512yes—
141C22×C4#9511yescharacteristic
142C22×C4#9511yescharacteristic
143C22×C4#9912yes—
144C22×C4#9912yes—
145C22×C4C2×C4211yescharacteristic
146C22×C4C2×C22⋊C411yescharacteristic
147C22×C4C2×C22⋊C412yes—
148C22×C4C2×C4⋊C411yescharacteristic
149C22×C4C2×C4⋊C412yes—
150M16—48no—
151M16—48no—
152C2×C8—48no—
153C2×C8—44no—
154C2×C8—44no—
155C2×C8—42no—
156C2×C8—42no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—41no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—41no—
164C42—22no—
165C42—22no—
166C42—22no—
167C42—22no—
168C42—22no—
169C42—22no—
170C42—21no—
171C42—21no—
172C42—21no—
173C42—21no—
174C42—22no—
175C42—22no—
176C42—22no—
177C42—22no—
178C23—21no—
179C23—21no—
180C23—41no—
181C23—41no—
182C23#34212yes—
183C23C2×#9511yescharacteristic
184Q8—44no—
185Q8—48no—
186Q8—44no—
187C2×C4—28no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4H(3,ℤ/4)12yes—
215C2×C4#34212yes—
216C2×C4C2×#9911yescharacteristic
217C2×C4C2×#9911yescharacteristic
218C2×C4#41614yes—
219C2×C4#42712yes—
220C8—82no—
221C8—88no—
222C8—82no—
223C22—21no—
224C22—21no—
225C22—22no—
226C22—22no—
227C22—22no—
228C22—42no—
229C22—42no—
230C22C2×H(3,ℤ/4)11yescharacteristic
231C22#173511yescharacteristic
232C22C2×#34211yescharacteristic
233C4—21no—
234C4—22no—
235C4—24no—
236C4—22no—
237C4—22no—
238C4—42no—
239C4—42no—
240C4—41no—
241C4—41no—
242C4—42no—
243C4#213312yes—
244C2—21no—
245C2—21no—
246C2—42no—
247C2#798211yescharacteristic
248C2#1257911yescharacteristic
249C2#1258611yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×Q8C23
prime radicalC24#95
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ 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. #7238C2semidirect

Maximal subgroups

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

subgroupindex
#218182
#207272
#72382