Finite Group Databasebeta

← #120221

Subgroups — #120221

Order 512.

603 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 603 conjugacy classes of subgroups — too many to draw. Only its 132 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#21940C211yesmaximal, characteristic
2#21933C211yesmaximal, characteristic
3#8107C212yesmaximal
4C2×#1293C211yesmaximal, characteristic
5#7983C212yesmaximal
6C2×#407C2212yes—
7C2×#407C2211yescharacteristic
8C22×C22⋊C8C2211yescharacteristic
9#2280C2211yescharacteristic
10C2×#380C2212yes—
11#1852—24no—
12#1851C414yescomplemented
13#1850—22no—
14#1808—22no—
15#1798—22no—
16#1305—24no—
17#1293—22no—
18#1289—24no—
19C24×C4C2311yescharacteristic
20C22×M16—22no—
21C22×M16C2×C412yes—
22C23×C8D412yes—
23C2×#117—21no—
24C22×C4⋊C4—22no—
25C22×C22⋊C4—22no—
26C2×#107—24no—
27C2×C4⋊C8—24no—
28#407—22no—
29#407—22no—
30#407—22no—
31#407—22no—
32#407—22no—
33#407—22no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—21no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—21no—
39C2×C22⋊C8C2×C412yes—
40C2×C22⋊C8D412yes—
41C2×C22⋊C8Q814yes—
42#381Q812yes—
43#380—21no—
44#380—22no—
45#380—22no—
46#380—22no—
47#380D414yes—
48#380D412yes—
49#379—21no—
50C4×C22⋊C4—21no—
51C4×C22⋊C4C2×C412yes—
52C2×#95—22no—
53#340—24no—
54#336—24no—
55C25#4111yescharacteristic
56C23×C4—42no—
57C23×C4—82no—
58C23×C4C22×C411yescharacteristic
59C23×C4C2×D411yescharacteristic
60C23×C4C2×D411yescharacteristic
61C23×C4C2×D412yes—
62C23×C4C2×D411yescharacteristic
63C23×C4C2×Q811yescharacteristic
64C23×C4C2×Q812yes—
65C23×C4#4112yes—
66C23×C4#4112yes—
67C23×C4#4111yescharacteristic
68C2×M16—24no—
69C2×M16—42no—
70C2×M16—42no—
71C2×M16—82no—
72C2×M16—84no—
73C22×C8—22no—
74C22×C8—22no—
75C22×C8—22no—
76C22×C8—24no—
77C22×C8—82no—
78C22×C8C22⋊C414yes—
79#117—82no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—84no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—24no—
88C2×C22⋊C4—42no—
89C2×C22⋊C4—84no—
90C2×C42—22no—
91#107—84no—
92C4⋊C8—84no—
93C22⋊C8—24no—
94C22⋊C8—24no—
95C22⋊C8—24no—
96C22⋊C8—24no—
97C22⋊C8—41no—
98C22⋊C8—42no—
99C22⋊C8—42no—
100C22⋊C8—42no—
101C22⋊C8—41no—
102#95—22no—
103#95—24no—
104#95—42no—
105#95C4⋊C414yes—
106C24—22no—
107C24—22no—
108C24—81no—
109C24—81no—
110C24#11711yescharacteristic
111C24#12111yescharacteristic
112C24C22⋊Q811yescharacteristic
113C24#12312yes—
114C24#12512yes—
115C22×C4—24no—
116C22×C4—44no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4—44no—
121C22×C4—44no—
122C22×C4—81no—
123C22×C4—82no—
124C22×C4—81no—
125C22×C4—81no—
126C22×C4—82no—
127C22×C4—84no—
128C22×C4—84no—
129C22×C4—84no—
130C22×C4C2×C22⋊C411yescharacteristic
131C22×C4C2×C4⋊C411yescharacteristic
132C22×C4C4×D412yes—
133C22×C4C4×Q812yes—
134C22×C4#12011yescharacteristic
135C22×C4#12112yes—
136C22×C4#12112yes—
137C22×C4#12111yescharacteristic
138C22×C4C22⋊Q812yes—
139C22×C4C22⋊Q812yes—
140C22×C4C22⋊Q812yes—
141C22×C4C22⋊Q811yescharacteristic
142C22×C4#12311yescharacteristic
143C22×C4#12411yescharacteristic
144C22×C4#12512yes—
145C22×C4#12612yes—
146C22×C4C4⋊Q812yes—
147C22×C4C4⋊Q811yescharacteristic
148M16—84no—
149M16—84no—
150M16—84no—
151C2×C8—24no—
152C2×C8—44no—
153C2×C8—44no—
154C2×C8—44no—
155C2×C8—42no—
156C2×C8—44no—
157C2×C8—82no—
158C2×C8—84no—
159C4⋊C4—48no—
160C4⋊C4—82no—
161C4⋊C4—84no—
162C22⋊C4—42no—
163C22⋊C4—42no—
164C22⋊C4—82no—
165C22⋊C4—82no—
166C22⋊C4—84no—
167C22⋊C4—84no—
168C42—44no—
169C23—22no—
170C23—22no—
171C23—22no—
172C23—22no—
173C23—24no—
174C23—42no—
175C23—41no—
176C23—42no—
177C23—41no—
178C23—41no—
179C23—41no—
180C23—41no—
181C23—42no—
182C23—41no—
183C23—42no—
184C23—84no—
185C23—84no—
186C23#38011yescharacteristic
187C23#38712yes—
188C23#39411yescharacteristic
189C23#39812yes—
190C23#39811yescharacteristic
191C23#47212yes—
192C23#47312yes—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—24no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—84no—
215C2×C4—84no—
216C2×C4—84no—
217C2×C4—168no—
218C2×C4#38611yescharacteristic
219C2×C4#38912yes—
220C2×C4#39111yescharacteristic
221C2×C4#39511yescharacteristic
222C2×C4#39612yes—
223C2×C4#40011yescharacteristic
224C8—44no—
225C8—84no—
226C8—82no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22—21no—
231C22—21no—
232C22—21no—
233C22—22no—
234C22—22no—
235C22—22no—
236C22—22no—
237C22—22no—
238C22—22no—
239C22—42no—
240C22—42no—
241C22—44no—
242C22—44no—
243C22—82no—
244C22—82no—
245C22—84no—
246C22#144111yescharacteristic
247C22#181612yes—
248C22#204511yescharacteristic
249C22#204711yescharacteristic
250C22#208912yes—
251C4—22no—
252C4—22no—
253C4—21no—
254C4—21no—
255C4—21no—
256C4—84no—
257C4—164no—
258C4#191312yes—
259C2—22no—
260C2—21no—
261C2—21no—
262C2—22no—
263C2—42no—
264C2—42no—
265C2#1120511yescharacteristic
266C2#1123911yescharacteristic
267C2#1124511yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #1851C4semidirect

Maximal subgroups

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

subgroupindex
#219402
#219332
#81072
C2×#12932
#79832