Finite Group Databasebeta

← #124532

Subgroups — #124532

Order 512.

458 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 94 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#14277C211yesmaximal, characteristic
2#8699C211yesmaximal, characteristic
3#8699C211yesmaximal, characteristic
4#7122C214yesmaximal
5C2×#389—21no—
6C2×#384C411yescharacteristic
7C2×#382C411yescharacteristic
8C2×#376C2211yescharacteristic
9#1817—21no—
10#1817—21no—
11#1444—21no—
12#1444C412yes—
13#1442—21no—
14#1442C2212yes—
15C4⋊#95—21no—
16#1280—21no—
17#1280—21no—
18#1280C2212yes—
19#1280C2212yes—
20C22×C4⋊C4—21no—
21C22×C4⋊C4—21no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—22no—
24C22×C4⋊C4C2×C412yes—
25C22×C42C2×C411yescharacteristic
26C22×C42D411yescharacteristic
27C22×C42D411yescharacteristic
28C2×C22⋊C8—22no—
29C2×C22⋊C8—22no—
30#389—44no—
31#384—42no—
32#382—22no—
33#382C814yescomplemented
34#376—42no—
35C2×#95—21no—
36C2×#95—22no—
37C2×#95—22no—
38C2×#95—22no—
39C2×#95C2×C412yes—
40C2×#95C2×C411yescharacteristic
41C2×#95D411yescharacteristic
42C2×#95D411yescharacteristic
43C2×#95C2311yescharacteristic
44#336—42no—
45#336—42no—
46C23×C4—22no—
47C23×C4—22no—
48C23×C4C22⋊C411yescharacteristic
49C23×C4C22⋊C411yescharacteristic
50C23×C4C22⋊C411yescharacteristic
51C23×C4C22⋊C411yescharacteristic
52C23×C4C22×C411yescharacteristic
53C23×C4C2×D411yescharacteristic
54C23×C4C2×D411yescharacteristic
55C22×C8—42no—
56C22×C8—42no—
57C2×C4⋊C4—24no—
58C2×C4⋊C4—42no—
59C2×C4⋊C4—42no—
60C2×C4⋊C4—42no—
61C2×C4⋊C4—42no—
62C2×C4⋊C4—42no—
63C2×C4⋊C4—42no—
64C2×C4⋊C4—44no—
65C2×C4⋊C4—42no—
66C2×C4⋊C4—82no—
67C2×C4⋊C4—82no—
68C2×C4⋊C4—82no—
69C2×C4⋊C4—82no—
70C2×C42—22no—
71C2×C42—22no—
72C2×C42—22no—
73C2×C42—22no—
74C2×C42—22no—
75C2×C42—22no—
76C2×C42—41no—
77C2×C42—41no—
78C2×C42C2×C812yes—
79C2×C42M1612yes—
80C22⋊C8—42no—
81C22⋊C8—42no—
82C22⋊C8—42no—
83C22⋊C8—42no—
84#95—24no—
85#95—22no—
86#95—41no—
87#95—41no—
88#95—41no—
89#95—41no—
90#95—44no—
91#95—82no—
92#95—82no—
93#95C2×C814yes—
94C24C2×C22⋊C411yescharacteristic
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—24no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—41no—
115C22×C4—41no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4—41no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4—41no—
122C22×C4—41no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4C22⋊C812yes—
140C22×C4C22⋊C812yes—
141C22×C4#9912yes—
142C22×C4#10112yes—
143C22×C4C22×C811yescharacteristic
144C22×C4C2×M1611yescharacteristic
145C22×C4AGL(1,ℤ/8)12yes—
146C22×C4#13712yes—
147C2×C8—82no—
148C2×C8—82no—
149C2×C8—82no—
150C2×C8—82no—
151C2×C8—82no—
152C2×C8—82no—
153C4⋊C4—48no—
154C4⋊C4—84no—
155C4⋊C4—84no—
156C4⋊C4—84no—
157C4⋊C4—88no—
158C42—24no—
159C42—24no—
160C42—44no—
161C42—44no—
162C23—21no—
163C23—21no—
164C23—21no—
165C23—21no—
166C23C2×C22⋊C811yescharacteristic
167C23C2×#9911yescharacteristic
168C23C2×#10111yescharacteristic
169C23#41811yescharacteristic
170C23#41911yescharacteristic
171C23#42111yescharacteristic
172C23#42111yescharacteristic
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—44no—
201C2×C4—44no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—44no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4#32414yes—
226C8—162no—
227C8—162no—
228C22—21no—
229C22—21no—
230C22—21no—
231C22—21no—
232C22—21no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—21no—
240C22C2×#32411yescharacteristic
241C22#147811yescharacteristic
242C22#147911yescharacteristic
243C22#151011yescharacteristic
244C22#151111yescharacteristic
245C22#153411yescharacteristic
246C22#155611yescharacteristic
247C22#212211yescharacteristic
248C22#212311yescharacteristic
249C22#212911yescharacteristic
250C22#213411yescharacteristic
251C4—24no—
252C4—42no—
253C4—42no—
254C4—42no—
255C4—41no—
256C4—41no—
257C4—84no—
258C4—81no—
259C4—81no—
260C4—162no—
261C2—21no—
262C2—21no—
263C2—21no—
264C2—21no—
265C2#759311yescharacteristic
266C2#1144911yescharacteristic
267C2#1145411yescharacteristic
268C2#1180111yescharacteristic
269C2#1180611yescharacteristic
270C2#1181211yescharacteristic
271C2#1181811yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#376 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
33. #382C8semidirect

Maximal subgroups

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

subgroupindex
#142772
#86992
#71222