Finite Group Databasebeta

← #135241

Subgroups — #135241

Order 512.

572 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 90 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#13942C211yesmaximal, characteristic
2#8303C211yesmaximal, characteristic
3#8303C211yesmaximal, characteristic
4#7189C214yesmaximal
5C22×C4×Q8C2211yescharacteristic
6C2×#391C412yes—
7C2×#382C411yescharacteristic
8C2×C4×C4⋊C4C411yescharacteristic
9#1817—21no—
10#1817—21no—
11C2×#324—21no—
12C2×#324—21no—
13C2×#324C2212yes—
14C2×#324C2212yes—
15#1441—21no—
16#1441—21no—
17C4⋊#95C2212yes—
18#1434—21no—
19#1434—21no—
20C23×Q8C2×C411yescharacteristic
21C2×C4×Q8—24no—
22C2×C4×Q8—42no—
23C2×C4×Q8—42no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4C2×C412yes—
26C22×C4⋊C4D411yescharacteristic
27C22×C4⋊C4D411yescharacteristic
28C22×C4⋊C4C2311yescharacteristic
29C22×C42C2×C411yescharacteristic
30C22×C42C2×C412yes—
31C22×C42D411yescharacteristic
32C22×C42D411yescharacteristic
33C2×C22⋊C8—22no—
34C2×C22⋊C8—22no—
35#391—44no—
36#382—42no—
37C4×C4⋊C4—42no—
38C2×#95—22no—
39C2×#95—22no—
40C2×#95—21no—
41C2×#95—21no—
42C2×#95—21no—
43C2×#95—22no—
44C2×#95—22no—
45#336—42no—
46#336—42no—
47#324—24no—
48#324—24no—
49#324—42no—
50#324—42no—
51C22×Q8—22no—
52C22×Q8—22no—
53C22×Q8—22no—
54C22×Q8—22no—
55C22×Q8—22no—
56C22×Q8—41no—
57C22×Q8—41no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4C22⋊C411yescharacteristic
61C23×C4C22⋊C411yescharacteristic
62C23×C4C22⋊C411yescharacteristic
63C23×C4C22⋊C411yescharacteristic
64C23×C4C22×C411yescharacteristic
65C23×C4C2×D411yescharacteristic
66C23×C4C2×D411yescharacteristic
67C22×C8—42no—
68C22×C8—42no—
69C4×Q8—416no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—44no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4#4114yes—
84C2×C42—22no—
85C2×C42—22no—
86C2×C42—22no—
87C2×C42—22no—
88C2×C42—22no—
89C2×C42—22no—
90C2×C42—22no—
91C2×C42—24no—
92C2×C42—41no—
93C2×C42—41no—
94C2×C42—42no—
95C2×C42—42no—
96C22⋊C8—42no—
97C22⋊C8—42no—
98C22⋊C8—42no—
99C22⋊C8—42no—
100#95—44no—
101#95—44no—
102#95—42no—
103#95—42no—
104#95—44no—
105#95—42no—
106#95—44no—
107C24C2×C22⋊C411yescharacteristic
108C2×Q8—28no—
109C2×Q8—44no—
110C2×Q8—44no—
111C2×Q8—44no—
112C2×Q8—44no—
113C2×Q8—44no—
114C2×Q8—44no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—24no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4#9912yes—
158C22×C4#9912yes—
159C22×C4#10012yes—
160C22×C4#10112yes—
161C22×C4#11712yes—
162C22×C4#12312yes—
163C22×C4#12312yes—
164C22×C4#13112yes—
165C22×C4#13112yes—
166C2×C8—82no—
167C2×C8—82no—
168C2×C8—84no—
169C2×C8—84no—
170C4⋊C4—44no—
171C4⋊C4—44no—
172C4⋊C4—44no—
173C4⋊C4—48no—
174C4⋊C4—84no—
175C42—28no—
176C42—44no—
177C42—44no—
178C42—84no—
179C23—22no—
180C23—22no—
181C23#38111yescharacteristic
182C23#40811yescharacteristic
183C23#40811yescharacteristic
184C23C2×#9911yescharacteristic
185C23C2×#9911yescharacteristic
186C23C2×#10011yescharacteristic
187C23C2×#10111yescharacteristic
188Q8—416no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4—82no—
239C2×C4#43314yes—
240C8—162no—
241C8—162no—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—22no—
246C22—21no—
247C22—21no—
248C22—21no—
249C22—21no—
250C22#147111yescharacteristic
251C22#147111yescharacteristic
252C22#147211yescharacteristic
253C22#147211yescharacteristic
254C22#181811yescharacteristic
255C22#181911yescharacteristic
256C22#182111yescharacteristic
257C22#212612yes—
258C22#213312yes—
259C4—24no—
260C4—42no—
261C4—42no—
262C4—42no—
263C4—42no—
264C4—42no—
265C4—42no—
266C4—84no—
267C4—162no—
268C2—21no—
269C2—21no—
270C2—21no—
271C2—21no—
272C2#870411yescharacteristic
273C2#1156512yes—
274C2#1157112yes—
275C2#1285711yescharacteristic
276C2#1286211yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ 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
#139422
#83032
#71892