Finite Group Databasebeta

← #135577

Subgroups — #135577

Order 512.

602 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 74 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#14155C211yesmaximal, characteristic
2#8551C212yesmaximal
3#7326C214yesmaximal
4C22×C4⋊Q8C2211yescharacteristic
5C2×#391—22no—
6C2×C4×C4⋊C4C411yescharacteristic
7C2×C4×C4⋊C4C411yescharacteristic
8#1784—22no—
9#1463—22no—
10#1463C2214yes—
11#1445C412yes—
12#1445C2212yes—
13C4⋊#95—22no—
14C23×Q8D412yes—
15C2×C4⋊Q8—24no—
16C2×C4⋊Q8—41no—
17C2×C4⋊Q8—41no—
18C2×C4⋊Q8—41no—
19C2×C4⋊Q8—41no—
20C22×C4⋊C4—21no—
21C22×C4⋊C4—21no—
22C22×C4⋊C4C2×C411yescharacteristic
23C22×C4⋊C4D412yes—
24C22×C4⋊C4C2311yescharacteristic
25C22×C42—21no—
26C22×C42—21no—
27C22×C42C2×C411yescharacteristic
28C2×#101—42no—
29C2×#101—42no—
30#407—24no—
31#391—48no—
32C4×C4⋊C4—41no—
33C4×C4⋊C4—41no—
34C4×C4⋊C4—41no—
35C4×C4⋊C4—41no—
36C2×#95—24no—
37C2×#95—24no—
38C2×#95C2×C412yes—
39C2×#95C2×C412yes—
40#324—28no—
41#324—42no—
42#324—42no—
43C22×Q8—22no—
44C22×Q8—22no—
45C22×Q8—24no—
46C22×Q8—42no—
47C22×Q8—44no—
48C22×Q8—44no—
49C23×C4—22no—
50C23×C4—22no—
51C23×C4C22⋊C411yescharacteristic
52C23×C4C22⋊C411yescharacteristic
53C23×C4C22⋊C412yes—
54C23×C4C22×C411yescharacteristic
55C23×C4C2×D412yes—
56C2×M16—44no—
57C4⋊Q8—44no—
58C4⋊Q8—44no—
59C4⋊Q8—44no—
60C4⋊Q8—44no—
61C2×C4⋊C4—24no—
62C2×C4⋊C4—24no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—24no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—42no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4#4114yes—
77C2×C42—21no—
78C2×C42—21no—
79C2×C42—21no—
80C2×C42—21no—
81C2×C42—22no—
82C2×C42—44no—
83C2×C42—44no—
84C2×C42—41no—
85C2×C42—41no—
86C2×C42—41no—
87C2×C42—41no—
88#101—84no—
89#101—84no—
90C22⋊C8—44no—
91C22⋊C8—44no—
92#95—42no—
93#95—42no—
94#95—48no—
95#95—48no—
96C24C2×C22⋊C411yescharacteristic
97C2×Q8—44no—
98C2×Q8—44no—
99C2×Q8—416no—
100C2×Q8—44no—
101C2×Q8—44no—
102C2×Q8—82no—
103C2×Q8—82no—
104C2×Q8—88no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—24no—
118C22×C4—24no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—44no—
128C22×C4—44no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—44no—
132C22×C4—44no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4#9911yescharacteristic
136C22×C4#9911yescharacteristic
137C22×C4#9911yescharacteristic
138C22×C4#9911yescharacteristic
139C22×C4#11712yes—
140C22×C4#12314yes—
141M16—84no—
142M16—84no—
143C2×C8—84no—
144C4⋊C4—44no—
145C4⋊C4—44no—
146C4⋊C4—44no—
147C4⋊C4—48no—
148C4⋊C4—48no—
149C4⋊C4—44no—
150C42—22no—
151C42—22no—
152C42—22no—
153C42—22no—
154C42—82no—
155C42—82no—
156C42—82no—
157C42—82no—
158C23—22no—
159C23—21no—
160C23—21no—
161C23—21no—
162C23—21no—
163C23#38111yescharacteristic
164C23C2×#9911yescharacteristic
165C23C2×#9911yescharacteristic
166Q8—88no—
167Q8—88no—
168C2×C4—24no—
169C2×C4—24no—
170C2×C4—24no—
171C2×C4—24no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—24no—
182C2×C4—24no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—44no—
188C2×C4—44no—
189C2×C4—44no—
190C2×C4—48no—
191C2×C4—48no—
192C2×C4—44no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—82no—
197C2×C4—84no—
198C2×C4—84no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4#35312yes—
205C2×C4#35412yes—
206C2×C4#35512yes—
207C2×C4#35612yes—
208C8—164no—
209C22—22no—
210C22—22no—
211C22—22no—
212C22—22no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22#181911yescharacteristic
220C22#183012yes—
221C22C2×#35311yescharacteristic
222C22C2×#35411yescharacteristic
223C22C2×#35511yescharacteristic
224C22C2×#35611yescharacteristic
225C4—22no—
226C4—22no—
227C4—22no—
228C4—22no—
229C4—42no—
230C4—44no—
231C4—84no—
232C4—84no—
233C4—82no—
234C2—21no—
235C2—21no—
236C2—22no—
237C2—21no—
238C2—21no—
239C2#875311yescharacteristic
240C2#1285611yescharacteristic
241C2#1286011yescharacteristic
subgroupG/H
center, socleC22#1819
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#141552
#85512
#73262