Finite Group Databasebeta

← #120222

Subgroups — #120222

Order 512.

567 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 567 conjugacy classes of subgroups — too many to draw. Only its 124 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#21948C211yesmaximal, characteristic
2#21933C211yesmaximal, characteristic
3#8108C212yesmaximal
4#7982C211yesmaximal, characteristic
5#7946C212yesmaximal
6C2×#407C2212yes—
7C2×#407C2211yescharacteristic
8C22×C22⋊C8C2211yescharacteristic
9#2280C2211yescharacteristic
10#2280C2212yes—
11#1855—24no—
12#1851C414yescomplemented
13#1850—22no—
14#1810—22no—
15#1798—22no—
16#1305—24no—
17#1289—22no—
18#1284—24no—
19C24×C4C2311yescharacteristic
20C22×M16—22no—
21C22×M16C2×C412yes—
22C23×C8D412yes—
23C2×#117—21no—
24C2×#117—22no—
25C2×#108—24no—
26C2×C4⋊C8—24no—
27#407—22no—
28#407—22no—
29#407—22no—
30#407—22no—
31#407—22no—
32#407D414yes—
33C2×C22⋊C8—22no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—21no—
38C2×C22⋊C8—21no—
39C2×C22⋊C8C2×C412yes—
40C2×C22⋊C8D412yes—
41#381—22no—
42#381D412yes—
43#380—21no—
44#380—22no—
45#380—22no—
46#380Q812yes—
47#379—21no—
48#379—22no—
49C4×C22⋊C4—21no—
50C4×C22⋊C4—22no—
51C4×C22⋊C4C2×C412yes—
52C4×C22⋊C4Q814yes—
53#341—24no—
54#336—24no—
55C25#4111yescharacteristic
56C23×C4—82no—
57C23×C4C22×C411yescharacteristic
58C23×C4C2×D411yescharacteristic
59C23×C4C2×D411yescharacteristic
60C23×C4C2×D412yes—
61C23×C4C2×D411yescharacteristic
62C23×C4C2×Q811yescharacteristic
63C23×C4C2×Q812yes—
64C23×C4#4112yes—
65C23×C4#4112yes—
66C23×C4#4111yescharacteristic
67C2×M16—24no—
68C2×M16—24no—
69C2×M16—42no—
70C2×M16—82no—
71C2×M16—84no—
72C22×C8—22no—
73C22×C8—22no—
74C22×C8—22no—
75C22×C8—42no—
76C22×C8—82no—
77C22×C8C22⋊C414yes—
78#117—82no—
79#117—84no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—42no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4—42no—
87C2×C22⋊C4—42no—
88C2×C42—22no—
89C2×C42—24no—
90#108—84no—
91C4⋊C8—84no—
92C22⋊C8—24no—
93C22⋊C8—24no—
94C22⋊C8—24no—
95C22⋊C8—24no—
96C22⋊C8—41no—
97C22⋊C8—42no—
98C22⋊C8—42no—
99C22⋊C8—41no—
100C22⋊C8—42no—
101#95—22no—
102#95—24no—
103#95—42no—
104#95C4⋊C414yes—
105C24—42no—
106C24—81no—
107C24—81no—
108C24#11711yescharacteristic
109C24#12111yescharacteristic
110C24C22⋊Q811yescharacteristic
111C24C22⋊Q812yes—
112C24#12612yes—
113C22×C4—24no—
114C22×C4—42no—
115C22×C4—44no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—44no—
119C22×C4—44no—
120C22×C4—81no—
121C22×C4—82no—
122C22×C4—81no—
123C22×C4—81no—
124C22×C4—82no—
125C22×C4C2×C22⋊C411yescharacteristic
126C22×C4C2×C4⋊C411yescharacteristic
127C22×C4C4×D412yes—
128C22×C4C4×Q812yes—
129C22×C4#12011yescharacteristic
130C22×C4#12112yes—
131C22×C4#12112yes—
132C22×C4#12111yescharacteristic
133C22×C4C22⋊Q812yes—
134C22×C4C22⋊Q812yes—
135C22×C4C22⋊Q811yescharacteristic
136C22×C4#12311yescharacteristic
137C22×C4#12312yes—
138C22×C4#12411yescharacteristic
139C22×C4#12512yes—
140C22×C4#12512yes—
141C22×C4C4⋊Q812yes—
142C22×C4C4⋊Q811yescharacteristic
143M16—48no—
144M16—84no—
145M16—84no—
146C2×C8—24no—
147C2×C8—44no—
148C2×C8—44no—
149C2×C8—44no—
150C2×C8—42no—
151C2×C8—44no—
152C2×C8—82no—
153C2×C8—84no—
154C4⋊C4—84no—
155C4⋊C4—82no—
156C22⋊C4—44no—
157C22⋊C4—82no—
158C22⋊C4—82no—
159C22⋊C4—84no—
160C22⋊C4—84no—
161C22⋊C4—84no—
162C42—44no—
163C42—84no—
164C23—22no—
165C23—22no—
166C23—22no—
167C23—41no—
168C23—41no—
169C23—41no—
170C23—42no—
171C23—41no—
172C23—42no—
173C23—41no—
174C23—42no—
175C23—42no—
176C23—42no—
177C23—41no—
178C23—42no—
179C23—88no—
180C23#38011yescharacteristic
181C23#38712yes—
182C23#39511yescharacteristic
183C23#39612yes—
184C23#40011yescharacteristic
185C2×C4—22no—
186C2×C4—24no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—24no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—84no—
207C2×C4—84no—
208C2×C4—84no—
209C2×C4#38611yescharacteristic
210C2×C4#38912yes—
211C2×C4#39111yescharacteristic
212C2×C4#39411yescharacteristic
213C2×C4#39812yes—
214C2×C4#39811yescharacteristic
215C8—44no—
216C8—84no—
217C8—82no—
218C22—21no—
219C22—21no—
220C22—22no—
221C22—22no—
222C22—21no—
223C22—21no—
224C22—22no—
225C22—22no—
226C22—21no—
227C22—21no—
228C22—22no—
229C22—22no—
230C22—21no—
231C22—21no—
232C22—44no—
233C22—82no—
234C22—82no—
235C22—84no—
236C22—84no—
237C22#144111yescharacteristic
238C22#181412yes—
239C4—21no—
240C4—22no—
241C4—21no—
242C4—22no—
243C4—21no—
244C4—84no—
245C4—164no—
246C4#191312yes—
247C2—22no—
248C2—42no—
249C2—41no—
250C2—41no—
251C2—42no—
252C2#1120511yescharacteristic
253C2#1124011yescharacteristic
254C2#1124611yescharacteristic
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
#219482
#219332
#81082
#79822
#79462