Finite Group Databasebeta

← #124121

Subgroups — #124121

Order 512.

634 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 146 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#13850C211yesmaximal, characteristic
2#7906C212yesmaximal
3#7132C212yesmaximal
4#7131C212yesmaximal
5C22×C4×Q8C411yescharacteristic
6C2×#431—22no—
7C2×C4×C4⋊C4C411yescharacteristic
8C2×#376C2211yescharacteristic
9#1446C411yescharacteristic
10#1446C2211yescharacteristic
11#1445C411yescharacteristic
12#1445C2211yescharacteristic
13#1441—22no—
14C4×#95—22no—
15#1280—22no—
16#1280C2214yes—
17C23×Q8C2×C411yescharacteristic
18C2×C4×Q8—22no—
19C2×C4×Q8—41no—
20C2×C4×Q8—41no—
21C2×C4×Q8—42no—
22C2×C4×Q8C814yescomplemented
23C22×C4⋊C4—21no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C42—21no—
27C22×C42—21no—
28C22×C42C2×C411yescharacteristic
29C22×C42D412yes—
30#431—28no—
31#431—44no—
32C2×C4⋊C8—42no—
33C2×C22⋊C8—24no—
34C2×C4×C8—42no—
35C4×C4⋊C4—42no—
36#376—41no—
37#376—41no—
38C2×#95—22no—
39C2×#95—22no—
40C2×#95—22no—
41C2×#95—22no—
42C2×#95C2×C411yescharacteristic
43C2×#95C2×C411yescharacteristic
44C2×#95C2×C411yescharacteristic
45C2×#95D412yes—
46C2×#95C2311yescharacteristic
47C22×Q8—21no—
48C22×Q8—44no—
49C22×Q8—41no—
50C22×Q8—41no—
51C22×Q8C2×C812yes—
52C23×C4—21no—
53C23×C4—21no—
54C23×C4—21no—
55C23×C4—21no—
56C23×C4C22⋊C411yescharacteristic
57C23×C4C22⋊C412yes—
58C23×C4C22⋊C411yescharacteristic
59C23×C4C22×C411yescharacteristic
60C23×C4C2×D412yes—
61C22×C8—44no—
62C4×Q8—44no—
63C4×Q8—44no—
64C4×Q8—44no—
65C4×Q8—44no—
66C2×C4⋊C4—24no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4C2×C812yes—
76C2×C42—21no—
77C2×C42—22no—
78C2×C42—21no—
79C2×C42—22no—
80C2×C42—22no—
81C2×C42—24no—
82C2×C42—24no—
83C2×C42—41no—
84C2×C42—41no—
85C2×C42—42no—
86C2×C42—42no—
87C2×C42—44no—
88C2×C42C2×C812yes—
89C2×C42M1612yes—
90C2×C42SD1614yes—
91C2×C42Q1614yes—
92C4⋊C8—48no—
93C22⋊C8—28no—
94C22⋊C8—44no—
95C4×C8—48no—
96#95—41no—
97#95—42no—
98#95—41no—
99#95—41no—
100#95—41no—
101#95—44no—
102#95—44no—
103#95—44no—
104#95—44no—
105C24C2×C22⋊C411yescharacteristic
106C2×Q8—44no—
107C2×Q8—42no—
108C2×Q8—44no—
109C2×Q8—42no—
110C2×Q8—48no—
111C2×Q8—44no—
112C2×Q8—44no—
113C22×C4—24no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—42no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—41no—
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×C4—42no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4C22⋊C812yes—
149C22×C4C22⋊C812yes—
150C22×C4#9911yescharacteristic
151C22×C4#9911yescharacteristic
152C22×C4#10012yes—
153C22×C4#10314yes—
154C22×C4#10314yes—
155C22×C4#10414yes—
156C22×C4C22×C811yescharacteristic
157C22×C4C2×M1611yescharacteristic
158C22×C4C2×SD1612yes—
159C22×C4C2×Q1612yes—
160C2×C8—48no—
161C2×C8—84no—
162C2×C8—84no—
163C4⋊C4—42no—
164C4⋊C4—48no—
165C4⋊C4—42no—
166C4⋊C4—48no—
167C42—22no—
168C42—22no—
169C42—22no—
170C42—22no—
171C42—24no—
172C42—24no—
173C42—24no—
174C42—24no—
175C42—44no—
176C42—44no—
177C42—84no—
178C23—22no—
179C23—21no—
180C23—21no—
181C23C2×C22⋊C811yescharacteristic
182C23C2×#9911yescharacteristic
183C23C2×#10011yescharacteristic
184C23C2×#10312yes—
185C23C2×#10412yes—
186Q8—44no—
187Q8—44no—
188Q8—48no—
189C2×C4—24no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—44no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—44no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—44no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—44no—
218C2×C4—44no—
219C2×C4—44no—
220C2×C4—42no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4#32312yes—
233C2×C4#32312yes—
234C2×C4Q8⋊C818yes—
235C2×C4#32814yes—
236C2×C4#32814yes—
237C2×C4#32914yes—
238C2×C4#33314yes—
239C8—88no—
240C22—21no—
241C22—21no—
242C22—21no—
243C22—21no—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22C2×#32311yescharacteristic
249C22C2×Q8⋊C812yes—
250C22C2×#32812yes—
251C22C2×#32911yescharacteristic
252C22C2×#33311yescharacteristic
253C22#212012yes—
254C22#212611yescharacteristic
255C22#213211yescharacteristic
256C4—44no—
257C4—41no—
258C4—41no—
259C4—42no—
260C4—42no—
261C4—42no—
262C4—44no—
263C4—82no—
264C4—82no—
265C4—82no—
266C4#127214yes—
267C4#127214yes—
268C2—21no—
269C2—21no—
270C2—22no—
271C2C2×#127211yescharacteristic
272C2#1142511yescharacteristic
273C2#1143111yescharacteristic
274C2#1171512yes—
275C2#1172112yes—
subgroupG/H
centerC22×C4#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
socleC23C2×#99
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ 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
22. C2×C4×Q8C8semidirect

Maximal subgroups

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

subgroupindex
#138502
#79062
#71322
#71312