Finite Group Databasebeta

← #149433

Subgroups — #149433

Order 512.

884 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 884 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C2×#2565C211yesmaximal, characteristic
2C2×#1394C212yesmaximal
3C2×#1394C212yesmaximal
4#10763C212yesmaximal
5C22×#125C411yescharacteristic
6C22×C4×Q8C2211yescharacteristic
7C22×#101C2214yes—
8#2565—24no—
9#2565—24no—
10C2×#398—22no—
11C2×#391C2211yescharacteristic
12C2×#391C2211yescharacteristic
13C2×#387—22no—
14C2×#384C411yescharacteristic
15C2×#382C411yescharacteristic
16C2×C4×C4⋊C4C411yescharacteristic
17C2×#343—22no—
18#1728—22no—
19#1394—24no—
20#1394—24no—
21C23×Q8C2311yescharacteristic
22C22×M16—24no—
23C2×#125—22no—
24C2×#125—22no—
25C2×#125—42no—
26C2×#125—42no—
27C2×C4×Q8—22no—
28C2×C4×Q8—22no—
29C2×C4×Q8—42no—
30C2×C4×Q8—42no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4C2×C411yescharacteristic
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4D412yes—
39C22×C42C2×C411yescharacteristic
40C22×C42C2×C411yescharacteristic
41C22×C42C2×C411yescharacteristic
42C22×C42D412yes—
43C2×#101—24no—
44C2×#101—24no—
45C2×#101—24no—
46C2×#101—24no—
47C2×#101—24no—
48C2×#101Q818yes—
49#398—216no—
50#391—22no—
51#391—22no—
52#391D414yes—
53#391D414yes—
54#387—216no—
55#384—22no—
56#384—22no—
57#382—22no—
58#382—22no—
59C4×C4⋊C4—22no—
60C4×C4⋊C4—22no—
61C2×#95—22no—
62C2×#95—21no—
63C2×#95—22no—
64C2×#95—22no—
65#343—44no—
66#336—44no—
67C22×Q8—41no—
68C22×Q8—41no—
69C22×Q8—42no—
70C22×Q8C2×D412yes—
71C22×Q8C2×D412yes—
72C22×Q8C2×Q814yes—
73C22×Q8#4114yes—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4C22⋊C412yes—
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22×C411yescharacteristic
82C23×C4C2×D412yes—
83C2×M16—28no—
84C2×M16—44no—
85C2×M16—44no—
86C2×M16—44no—
87C2×M16—44no—
88C2×M16—44no—
89C22×C8—44no—
90#125—48no—
91#125—48no—
92C4×Q8—48no—
93C4×Q8—48no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—28no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—44no—
112C2×C4⋊C4—48no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4C22⋊C412yes—
118C2×C4⋊C4C22⋊C412yes—
119C2×C42—22no—
120C2×C42—22no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—24no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—24no—
127C2×C42—42no—
128C2×C42—42no—
129C2×C42—42no—
130C2×C42C22⋊C412yes—
131C2×C42C22⋊C412yes—
132#101—28no—
133#101—28no—
134#101—44no—
135#101—44no—
136#95—24no—
137#95—28no—
138#95—28no—
139#95—28no—
140C24C2×C22⋊C411yescharacteristic
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8—42no—
145C2×Q8—44no—
146C2×Q8—44no—
147C2×Q8—44no—
148C2×Q8—44no—
149C2×Q8C22⋊Q814yes—
150C2×Q8C22⋊Q814yes—
151C2×Q8#12412yes—
152C2×Q8#12412yes—
153C2×Q8C4⋊Q812yes—
154C2×Q8C4⋊Q812yes—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—24no—
163C22×C4—22no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—44no—
179C22×C4—42no—
180C22×C4—44no—
181C22×C4—42no—
182C22×C4—44no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—42no—
188C22×C4#9912yes—
189C22×C4#9912yes—
190C22×C4C2×C22⋊C411yescharacteristic
191C22×C4C2×C22⋊C411yescharacteristic
192C22×C4C4×Q814yes—
193C22×C4#12014yes—
194C22×C4C22⋊Q814yes—
195C22×C4#12412yes—
196C22×C4C4⋊Q812yes—
197M16—48no—
198M16—48no—
199M16—84no—
200M16—84no—
201C2×C8—48no—
202C2×C8—84no—
203C2×C8—84no—
204C4⋊C4—48no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—48no—
208C4⋊C4—416no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—48no—
214C4⋊C4—48no—
215C42—24no—
216C42—24no—
217C42—48no—
218C42—42no—
219C42—42no—
220C42—42no—
221C42—42no—
222C23—22no—
223C23—22no—
224C23#37911yescharacteristic
225C23#39112yes—
226C23C2×#9911yescharacteristic
227C23C2×#9911yescharacteristic
228C23#41012yes—
229Q8—44no—
230Q8—44no—
231Q8—42no—
232Q8—42no—
233Q8—42no—
234Q8—42no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—24no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—48no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—42no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—84no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4#39112yes—
278C2×C4#39112yes—
279C2×C4#39314yes—
280C2×C4#39514yes—
281C8—88no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—24no—
287C22—24no—
288C22#144611yescharacteristic
289C22#178111yescharacteristic
290C22#178211yescharacteristic
291C22#199612yes—
292C22#199612yes—
293C22#213212yes—
294C22#213312yes—
295C4—24no—
296C4—22no—
297C4—22no—
298C4—24no—
299C4—44no—
300C4—42no—
301C4—42no—
302C4—82no—
303C4—82no—
304C4—82no—
305C4—82no—
306C2—21no—
307C2—21no—
308C2—22no—
309C2#1138111yescharacteristic
310C2#1270111yescharacteristic
311C2#1270211yescharacteristic
312C2#1330314yes—
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC23×Q8C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#25652
C2×#13942
#107632