Finite Group Databasebeta

← #134129

Subgroups — #134129

Order 512.

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

normal subgroups only

This group has 760 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
1C2×#2660C211yesmaximal, characteristic
2#11512C212yesmaximal
3#11511C212yesmaximal
4#8551C212yesmaximal
5C22×C4⋊Q8C2211yescharacteristic
6C22×#125C411yescharacteristic
7#2660—24no—
8#2660—24no—
9C2×#398—22no—
10C2×#395—22no—
11C2×#384C411yescharacteristic
12C2×#384C2211yescharacteristic
13C2×#382C411yescharacteristic
14C2×#382C2211yescharacteristic
15C2×#376C411yescharacteristic
16#1784—22no—
17#1463—22no—
18#1463C2214yes—
19#1325—24no—
20#1324—24no—
21C23×Q8D412yes—
22C2×C4⋊Q8—22no—
23C2×C4⋊Q8—22no—
24C2×C4⋊Q8—42no—
25C2×C4⋊Q8—42no—
26C2×#125—22no—
27C2×#125—22no—
28C2×#125—42no—
29C2×#125—42no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4C2×C411yescharacteristic
37C22×C4⋊C4D412yes—
38C22×C4⋊C4C2311yescharacteristic
39C22×C42C2×C411yescharacteristic
40C22×C42C2×C411yescharacteristic
41C22×C42C2×C411yescharacteristic
42C2×#101—44no—
43#407—24no—
44#398—216no—
45#395—216no—
46#384—22no—
47#384—22no—
48#384—22no—
49#384D414yes—
50#382—22no—
51#382—22no—
52#382—22no—
53#382D414yes—
54#376—22no—
55#376—22no—
56C2×#95—21no—
57C2×#95—22no—
58C2×#95—21no—
59C2×#95—22no—
60#324—24no—
61#324—24no—
62#324—44no—
63C22×Q8—24no—
64C22×Q8—24no—
65C22×Q8—42no—
66C22×Q8—84no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4C22⋊C412yes—
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22×C411yescharacteristic
75C23×C4C2×D412yes—
76C2×M16—44no—
77C4⋊Q8—48no—
78C4⋊Q8—48no—
79#125—48no—
80#125—48no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—28no—
90C2×C4⋊C4—28no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—84no—
105C2×C4⋊C4—84no—
106C2×C4⋊C4C22⋊C412yes—
107C2×C4⋊C4C22⋊C412yes—
108C2×C4⋊C4C2×D412yes—
109C2×C4⋊C4C2×D412yes—
110C2×C42—22no—
111C2×C42—22no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—21no—
116C2×C42—24no—
117C2×C42—24no—
118C2×C42C22⋊C412yes—
119C2×C42C22⋊C412yes—
120#101—88no—
121C22⋊C8—44no—
122C22⋊C8—44no—
123#95—24no—
124#95—24no—
125#95—28no—
126#95—28no—
127C24C2×C22⋊C411yescharacteristic
128C2×Q8—48no—
129C2×Q8—48no—
130C2×Q8—84no—
131C2×Q8—88no—
132C2×Q8—88no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—24no—
138C22×C4—24no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4#9912yes—
167C22×C4#9912yes—
168C22×C4C2×C22⋊C411yescharacteristic
169C22×C4C2×C22⋊C411yescharacteristic
170C22×C4#12014yes—
171M16—88no—
172C2×C8—84no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—48no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—48no—
181C4⋊C4—48no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—88no—
185C4⋊C4—88no—
186C42—24no—
187C42—24no—
188C42—24no—
189C42—24no—
190C42—22no—
191C42—22no—
192C42#10114yescomplemented
193C42#10114yescomplemented
194C23—22no—
195C23—22no—
196C23—21no—
197C23—21no—
198C23#37911yescharacteristic
199C23C2×#9911yescharacteristic
200C23C2×#9911yescharacteristic
201Q8—816no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—24no—
209C2×C4—24no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—24no—
213C2×C4—24no—
214C2×C4—24no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—48no—
218C2×C4—48no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4C2×#10112yes—
241C2×C4C2×#10112yes—
242C8—164no—
243C22—24no—
244C22—24no—
245C22—21no—
246C22—21no—
247C22—21no—
248C22—21no—
249C22—21no—
250C22—21no—
251C22#178111yescharacteristic
252C22#178411yescharacteristic
253C22#178411yescharacteristic
254C22#212712yes—
255C22#213312yes—
256C4—22no—
257C4—22no—
258C4—44no—
259C4—44no—
260C4—44no—
261C4—42no—
262C4—42no—
263C4—82no—
264C4—82no—
265C4—82no—
266C2—21no—
267C2—21no—
268C2—22no—
269C2—22no—
270C2#855111yescharacteristic
271C2#1269611yescharacteristic
272C2#1270211yescharacteristic
subgroupG/H
center, socleC22#1781
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

Complements

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

NH ≅ G/Ntype
192. C42#101semidirect
193. C42#101semidirect

Maximal subgroups

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

subgroupindex
C2×#26602
#115122
#115112
#85512